Proceedings of the 11th IFToMM International Conference on Rotordynamics: Volume 1 (Mechanisms and Machine Science, 139) [1st ed. 2024] 3031404548, 9783031404542

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Mechanisms and Machine Science

139

Series Editor Marco Ceccarelli , Department of Industrial Engineering, University of Rome Tor Vergata, Roma, Italy

Advisory Editors Sunil K. Agrawal, Department of Mechanical Engineering, Columbia University, New York, NY, USA Burkhard Corves, RWTH Aachen University, Aachen, Germany Victor Glazunov, Mechanical Engineering Research Institute, Moscow, Russia Alfonso Hernández, University of the Basque Country, Bilbao, Spain Tian Huang, Tianjin University, Tianjin, China Juan Carlos Jauregui Correa , Universidad Autonoma de Queretaro, Queretaro, Mexico Yukio Takeda, Tokyo Institute of Technology, Tokyo, Japan

This book series establishes a well-defined forum for monographs, edited Books, and proceedings on mechanical engineering with particular emphasis on MMS (Mechanism and Machine Science). The final goal is the publication of research that shows the development of mechanical engineering and particularly MMS in all technical aspects, even in very recent assessments. Published works share an approach by which technical details and formulation are discussed, and discuss modern formalisms with the aim to circulate research and technical achievements for use in professional, research, academic, and teaching activities. This technical approach is an essential characteristic of the series. By discussing technical details and formulations in terms of modern formalisms, the possibility is created not only to show technical developments but also to explain achievements for technical teaching and research activity today and for the future. The book series is intended to collect technical views on developments of the broad field of MMS in a unique frame that can be seen in its totality as an Encyclopaedia of MMS but with the additional purpose of archiving and teaching MMS achievements. Therefore, the book series will be of use not only for researchers and teachers in Mechanical Engineering but also for professionals and students for their formation and future work. The series is promoted under the auspices of International Federation for the Promotion of Mechanism and Machine Science (IFToMM). Prospective authors and editors can contact Mr. Pierpaolo Riva (publishing editor, Springer) at: [email protected] Indexed by SCOPUS and Google Scholar.

Fulei Chu · Zhaoye Qin Editors

Proceedings of the 11th IFToMM International Conference on Rotordynamics Volume 1

Editors Fulei Chu Department of Mechanical Engineering Tsinghua University Beijing, China

Zhaoye Qin Department of Mechanical Engineering Tsinghua University Beijing, China

ISSN 2211-0984 ISSN 2211-0992 (electronic) Mechanisms and Machine Science ISBN 978-3-031-40454-2 ISBN 978-3-031-40455-9 (eBook) https://doi.org/10.1007/978-3-031-40455-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The two volumes printed by Springer Nature constitute the proceedings of the eleventh in a series of IFToMM International Conference on Rotordynamics. The primary aim of the conference is to bring together the expertise of scientists and engineers in academia and industry in the field of rotordynamics and related areas and to exchange information with a particular emphasis on scientific and technical development. The themes of the conference reflect current interests in a wider field of rotordynamics. The series of IFToMM International Conference on Rotordynamics has been established as a major forum for discussion and dissemination of recent advances in rotordynamics. This quadrennial conference continues a tradition that started with the inaugural meeting in Rome in 1982. Over the years, the conference has traveled to diverse locations, including Tokyo (1986), Lyon (1990), Chicago (1994), Darmstadt (1998), Sydney (2002), Vienna (2006), Seoul (2010), Milano (2014), and Rio De Janeiro (2018), creating a rich history of collaborations and knowledge exchanges. Due to the impact of the COVID-19 pandemic, the 11th IFToMM International Conference on Rotordynamics held in Beijing, China, was postponed for one year. This has affected the number of submissions to the conference to a certain extent. However, delightingly, researchers from across the globe have exhibited tremendous enthusiasm for the conference, and the themes of submissions span a vast array of subjects. This reflects the flourishing and popularity of the field of rotordynamics. After undergoing rigorous peer review, a total of 75 papers have been carefully selected and organized into two volumes for inclusion in this conference proceeding. Volume 1 focuses on the themes including: • • • • • •

Active Components and Vibration Control Balancing Bearings: Fluid Film Bearings, Magnetic Bearings, Rolling Bearings, and Seals Blades, Bladed Systems, and Impellers Condition Monitoring, Fault Diagnostics, and Prognostics Dynamic Analysis and Stability Volume 2 delves into the following themes:

• • • • • • • •

Electromechanical Interactions in Rotordynamics Fluid Structure Interactions in Rotordynamics Nonlinear Phenomena in Rotordynamics Numerical and Analytical Methods in Nonlinear Rotordynamics Parametric and Self-excitation in Rotordynamics Uncertainties, Reliability, and Life Predictions of Rotating Machinery Torsional Vibrations and Geared Systems Dynamics Aero-Engines

vi

Preface

• Automotive Rotating Systems • Optimization of Rotor Systems • Smart Rotor Systems The conference organizing committee is very grateful to the authors and keynote speakers for their efforts in producing the papers and to the IFToMM TC of Rotordynamics for their valuable time in reviewing papers in line with the journal standard. We would also like to thank Professors Hongguang Li, Xuejun Li, Zhong Luo, Hui Ma, Weimin Wang, Chaofeng Li, Dayi Zhang, Yongfeng Yang, Tian He, Zhongliang Xie, Yeyin Xu, Xingrong Huang, and Xueping Xu for handling the review tasks of late papers. The organizers are grateful to the Natural Science Foundation of China (NSFC) for the financial support to this conference and to Chinese Society for Vibration Engineering and Tsinghua University, who kindly supported the promotion of this event. Finally, we would like to mention Dr. Jun Wang and Dr. Wenliang Gao, who have been working hard for the organization of all the papers. Fulei Chu Zhaoye Qin

Contents

Dynamic Analysis and Safety Design for Aero-Engine Rotor-Support System Under the Blade-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jian Zhang, Shiping Song, Chao Li, Yanhong Ma, and Jie Hong

1

Application of Metamaterial in Vibration Suppression of Rotor System . . . . . . . . Zhenping Li, Hongliang Yao, and Hui Li

15

A Novel Approach to Model the Shrink-Fit Connection . . . . . . . . . . . . . . . . . . . . . R. D. Firouz-Abadi, Fahimeh Mehralian, Hadi Amirabadizadeh, and Masoud Yousefi

25

Numerical Analysis of the Effect of Different Surface Textures on the Steady State Characteristics of Tilting Pad Journal Bearings . . . . . . . . . . . Bin Wei, Xiuli Hu, Renwei Che, and Yinghou Jiao

36

Effect of Support Parameters on the Vibrations of a Cracked Rotor Passing Through Critical Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fahimeh Mehralian, R. D. Firouz-Abadi, and Masoud Yousefi

46

Active Suppression of Multi-frequency Vibration of Rotor Based on Adaptive Immersion & Invariant Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xinyu Gao, Yifan Bao, Zeliang Zhang, and Jianfei Yao

54

Simulation Study for Hole Diaphragm Labyrinth Seal at Synchronous Whirl Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiang Zhang, Renwei Che, Yinghou Jiao, and Huzhi Du

71

Effect of Flow Rate on the Performance of an Evacuated Tilting Pad Journal Bearing: Load on Pad vs. Load-Between-Pad Configurations . . . . . . . . . . Luis San Andrés and Andy Alcantar

78

Dynamic Performance of an O-Ring Sealed Squeeze Film Damper and a Simple Way to Estimate the (Ingested) Gas Content in a Squeeze Film . . . Luis San Andrés and Bryan Rodríguez

93

Bearing Fault Diagnosis Using Transfer Learning with ICCEMDAN . . . . . . . . . . 111 Chenghui Pan, Song Xue, Yinwei Zhang, Lihui Chen, Peiyuan Lian, Congsi Wang, Qian Xu, and Wulin Zhao

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Contents

Dynamic Modeling and Stability Analysis of the Rod-Fastening Rotor System with the Different Preload Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Chongyang Wang, Xilong Hu, Fan Qin, and Lihua Yang Modeling Rotordynamic Effects of Angular Contact Ball Bearing in Xand O-arrangements with Full Bearing Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Giota Goswami, Iikka Martikainen, Eerik Sikanen, Charles Nutakor, Janne Heikkinen, and Jussi Sopanen Research on the Nonlinear Response of the Rotor System with Bolted Joint with Spigot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Yongbo Ma, Jie Hong, Shaobao Feng, and Yanhong Ma On the Influence of the Lubricant Feed Orifice Size and End Plate Seals’ Clearance on the Static and Dynamic Performance of Integral Squeeze Film Dampers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Xueliang Lu, Luis San Andrés, and Bonjin Koo Dropdown Analysis of High-Speed Thin-Shaft Coupled Rotor System Integrated with Three Active Magnetic Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Gyan Ranjan, Juuso Narsakka, Tuhin Choudhury, and Jussi Sopanen Parametric Analysis of Journal Bearings with Chevron Textures on the Shaft Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Luis F. dos Anjos, Alfredo Jaramillo, Gustavo C. Buscaglia, and Rodrigo Nicoletti Prediction of Remaining Useful Life of Passive and Adjustable Fluid Film Bearings Using Physics-Based Models of Their Degradation . . . . . . . . . . . . . . . . . 211 Denis Shutin, Maxim Bondarenko, Roman Polyakov, Ivan Stebakov, and Leonid Savin Application of Machine Learning in Simulation Models and Optimal Controllers for Fluid Film Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Yuri Kazakov, Ivan Stebakov, Denis Shutin, and Leonid Savin Study on Solution Algorithm of Reynolds Equation of Self-acting Gas Journal Bearings Based on Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . 236 Haijun Zhang, Qin Yang, Wei Zhao, and Feilong Jiang The Dynamic Equilibrium Surfaces Calculation in the Liquid Friction Conical Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 A. Yu. Korneev, Shengbo Li, A. Yu. Koltsov, E. V. Mishchenko, and L. A. Savin

Contents

ix

Early Warning Signal Based on Global Dynamics for Instability Responses in Rotor/Stator Rubbing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Xinxin Dong, Zigang Li, Ling Hong, and Jun Jiang Research on Rub-Impact Fault Quantification of Rotor System Based on Effective Singular Value Noise Reduction and Minimum Mutual Entropy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Jintao Li, Zhaobo Chen, and Dong Yu Influence of Slight Gravitational Effect on the Characteristics of Onset Speeds of Instability and Stability in the Vertical Rotating Shaft Supported by Journal Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Li Fan, Tsuyoshi Inoue, and Akira Heya Investigation of Journal Gas Foil Bearing Characteristics with Foils Prestress from Assembling Taken into Account . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 M. Yu. Temis and A. B. Meshcheryakov Investigation on Feature Attribution for Remaining Useful Life Prediction Model of Cryogenic Ball Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Byul An, Yunseok Ha, Yeongdo Lee, Wonil Kwak, and Yongbok Lee Numerical Investigation on Leakage Characteristics of a Novel Honeycomb Seal with Wall Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 Huzhi Du, Yinghou Jiao, Xiang Zhang, and Renwei Che Dynamic Performance of Spacecraft Flywheel Ball Bearing with Different Type and Distribution of Cage Pocket Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Shuai Gao, Lanyu Liu, and Qinkai Han High Speed Rotor Drop-Downs in Different Planetary Touch-Down Bearings Differing in the Number of Bearing Units . . . . . . . . . . . . . . . . . . . . . . . . . 320 Benedikt Schüßler and Stephan Rinderknecht Model Based Identification of the Measured Vibration Multi-fault Diagnostic Signals Generated by a Large Rotating Machine . . . . . . . . . . . . . . . . . 338 Tomasz Szolc, Robert Konowrocki, and Dominik Pisarski Vibration Reduction in an Unbalanced Rotor System Using Nonlinear Energy Sinks with Varying Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 Harikrishnan Venugopal, Kevin Dekemele, and Mia Loccufier Modelling and Validation of Rotor-Active Magnetic Bearing System Considering Interface Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Yang Zhou, Jin Zhou, Jarir Mahfoud, Yue Zhang, and Yuanping Xu

x

Contents

Experiment and CFD Analysis of Plain Seal, Labyrinth Seal and Floating Ring Seal on Leakage Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Yunseok Ha, Yeongdo Lee, Byul An, and Yongbok Lee Rolling Element Bearing Fault Diagnosis Using Hybrid Machine Learning Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 Mario Antunovi´c, Sanjin Braut, Roberto Žiguli´c, Goranka Štimac Ronˇcevi´c, and Mario Lovri´c Fluid-Structure Analysis of a Hybrid Brush Seal . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 Joury Temis and Alexey Selivanov Effects of Relief Hole on the Static Characteristics of Externally Pressurized Steam-Lubricated Hybrid Journal-Thrust Bearing . . . . . . . . . . . . . . . . 430 Zhansheng Liu, Jinlei Qi, Xiangyu Yu, Peng He, Bing Han, Yishun Yang, and Jiaqi Wang Simulation Analysis of Main Bearing Vibration Characteristics of Wind Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 Liang Xuan, Ao Shen, Xiaochi He, Shuai Dong, and Jiaxin Dong Study of Multiplicative Load on the Misaligned Rotor-AMB System . . . . . . . . . . 462 Atul Kumar Gautam and Rajiv Tiwari Intelligent Fault Classification of a Misaligned Geared-Rotor Machine Equipped with Active Magnetic Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 Pantha Pradip Das, Rajiv Tiwari, and Dhruba Jyoti Bordoloi Eigenvalues of the Free Rotation Mode of the Multi-bladed Rotor . . . . . . . . . . . . 495 Chao Peng and Alessandro Tasora The Passive Vibration Control in Bridge Configured Winding Cage Rotor Induction Motor: An Experimental Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 Rakesh Deore, Bipul Brahma, Shahrukh, and Karuna Kalita Identification Method for Cage Rubbing Faults of Flywheel Bearings Based on Characteristic Frequency Ratio and Convolutional Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Jianwen Wang, Hong Wang, Tian He, and Tao Qing Unbalance Measurement Traceability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 Weijian Guo Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

Dynamic Analysis and Safety Design for Aero-Engine Rotor-Support System Under the Blade-off Jian Zhang1 , Shiping Song1 , Chao Li1(B) , Yanhong Ma1,2 , and Jie Hong2,3 1 Research Institute of Aero-Engine, Beihang University, Beijing 100191, China

[email protected]

2 Collaborative Innovation Center of Advanced Aero-Engine, Beijing 100191, China 3 School of Energy and Power Engineering, Beihang University, Beijing 100191, China

Abstract. Considering turbine blade-off event in the typical turboshaft engine, dynamic model of the rotor-support system is established, and the modal characteristic, as well as vibration response under the turbine blade-off are investigated with centered finite difference method. Impact load is detected at the moment of blade loss due to the sudden unbalance, and the vibration response goes to extreme when the rotor decelerates though the critical speeds. The bearing loads on each support are investigated and the response on the rear supports is much high than the front supports, possible leads to bearing failure and structure damage. Structural safety designs are proposed, including the sealing brake structure and the metal-rubber ring, to ensure the integrity of the bearings and the support structure. The validity of these applications is quantitatively analyzed. It is demonstrated that the vibration amplitude and the bearing loads are both significantly reduced during the rotor deceleration when applied with the safety designs. Some other conclusions instructive for further engineering applications are also obtained. Keywords: blade-off · rotor-support system · dynamic analysis · bearing load · safety design

1 Introduction Blade-off is rare but possible event where part of blade or even the whole blade breaks away from the disk, leading to sudden increase of the rotor unbalance and applying large impact load to the support structure, which would seriously threaten the safety of aero-engine [1]. Additionally, a rapid declaration from the operating speed down to the wind-milling speed is required after blade loss, and thus violent vibration arises near the critical speed due to the rotor unbalance [2]. This is rather harmful to flexible rotor system, such as power turbine rotor in turboshaft engine, which operates above multi-order critical speed and would further induce severe damage during the declaration process. For the safety of flight, it is important to keep the integrity of the support structure and ensure the engine continuous operating at the wind-milling speed after blade-off event [3]. Therefore, analytical modeling and numerical simulation to predict the vibration response under the blade-off are practically employed, which helps structural safety design for the rotor-support system. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 1–14, 2024. https://doi.org/10.1007/978-3-031-40455-9_1

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In the last decades, a large number of researches have been conducted concerned with the vibration response analysis of blade-off event, mainly focused on the rotor dynamics problems including the sudden unbalance load [4], the influence of the deceleration process [5], the asymmetry of blade-disk inertia [6] and the rub-impact effect between blade and casing [7], etc. Genta established the dynamic model if a Jeffcott rotor with sudden unbalance and the effects of rotation speed, excitation amplitude, as well as other parameters on the vibration displacements, and the rotor orbits were also investigated [8]. Ma simulated the vibration response of an overhung rotor to study the effects of gyroscopic moment under the sudden unbalance [9]. The impact effects due to the sudden unbalance were demonstrated in the above works, and low order natural frequencies can be observed in the frequency domain. Gunter [10], Miao [11] considered the declaration after blade loss based on the Jeffcott rotor model, while Yamamoto [12] focused on the influence of asymmetric characteristics in use of the single-disk rigid rotor model. The results showed that, considering the varying velocity, the beating vibration appears near the critical speed due to the coupling between natural frequency and speed frequency, and lags in peak response. Muszynaka [13], Sinha [14, 15] respectively studied the nonlinear dynamic response of the rotor system. The rub-impact effect was considered in their study, and the backward whirl, quasi-periodic and even chaotic motions have been revealed. Thiery [16] simulated the global dynamic of a misaligned turbine with blade-casing contacts, and the detailed parametric influence was obtained by analyzing rotor response in terms of Poincare sections, bifurcation diagrams and maximum displacements at steady state. In consideration of structure integrity and engine safety, investigations on rotor system safety design method should be conducted to ensure the rotor-support system survives from sudden shock and operates with large unbalance excitation. Some technologies were proposed, including bearing structure fusing design, variable supporting stiffness structure design, bearing load optimization, damping structure and deformation control, etc. [3]. Kastl [17] put forward a hollow cone diverging between the bearing seat and mounting flange as the bearing structure fuse, which would fail in shear under abnormal unbalance load, decreasing the stiffness and reaction of the bearing. Keven [18] proposed a radially-inward extending portion as bearing support, whose radial stiffness and angular stiffness could be optimized by modifying the frustoconical parameters, decreasing the bearing damage from the shaft bending. Jadczak [19] interposed a plurality of flexible bags between the bearing and shaft as a damping buffer, which would attenuate the vibration under extreme condition. It can be noted that a great deal of damage mechanism researches and simulation analysis have been conducted about the blade-off problem. However, the dynamic models used in the past researches usually are very simple, with little freedom and can hardly consider the vibration response of the support structure. In fact, the rotor systems in aeroengines usually operate above several critical speeds and the coupling vibration between the rotor and its support structure should not be neglected [20]. Besides, safety strategies in the above works still stayed on the level of theoretical analysis and qualitative design. Therefore, it is of great value to investigate the theory of the safety design strategies for the rotor-support system in areo-engine, and the dynamic analysis, as well as the parameter influence are still key questions needed to be solved.

Dynamic Analysis and Safety Design for Aero-Engine Rotor-Support System

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Therefore, this paper proposes dynamic analysis method for the rotor-support system in typical turboshaft engine, and the vibration response for the system under turbine blade-off condition is predicted. Based on which, two strategies of safety design are put forward for vibration reduction and structure integrity. Finally, theoretical investigation and simulation analysis are carried out to demonstrate the validity of the strategies.

2 Dynamic Model of Rotor-Support System with Blade-Off 2.1 Structural Characteristics of Power Turbine Rotor and Support Structure The power turbine rotor (PT-rotor) in typical turboshaft engine usually consists of slender rotor shaft and overhung blade-disk segment, as shown in Fig. 1. The front and rear ends of the shaft are respectively supported by two bearings, and the PT-rotor is designed to be flexible, operating above several critical speeds. The mass and the moment of inertia of the rotor are concentrated on the power turbine at the rear end. The details of the PTrotor’s structural characteristics are quantificationally listed in Table 1. Different from fan blade in high bypass ratio turbofan engine, the turbine blade considered here has much smaller mass and inertia compared with the disk, and thus the loss of single blade can hardly induce the asymmetry of the blade-disk. The rear bearings (3#, 4#) are supported by the shared squirrel-cage spring, and the radial loads on the bearings are propagated through the turbine support structure including the frustoconical casing and the rear frame, and finally transmitted to the auxiliary mounting on the outer casing. Auxiliary mounting Rear frame

Outer casing

1st bearing

Overhung blade-disk segment

2nd bearing

Frustoconical casing

Rotor shaft 3th bearing

4th bearing

Squirrel-cage spring

Fig. 1. Diagram of rotor-support system in typical turboshaft engine

Table 1. Structural characteristic parameters of the PT-rotor Symbol

Content

Values

ms , mt , mt

Mass of shaft, blade-disk, single blade

3/12/0.04 (kg)

Jps , Jpt , Jpb

Polar inertia of shaft, blade-disk, single blade

1/100/0.008 (×10–3 kg·m2 )

Jdt , Jdb

Diameter inertia of blade-disk, single blade

64/0.01 (×10–3 kg·m2 )

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2.2 Physical Process of Turbine Blade-Off Event The physical process of rotor after blade-off event in aero-engine can be divided into three stages in time sequences, including the instantaneous stage of the blade loss, deceleration of the damaged rotor and the steady-state operation at the wind-milling speed [2]. The time scale of the instantaneous stage is about 0.1 s, during which the mass centroid the blade-disk is altered suddenly and extreme large impact is applied on the rotor, possibly causing fracture of the slender shaft and failure of the bearings. After the blade loss, the fuel supply is cut off, and the rotor will decelerate rapidly down to the wind-milling speed, in the time scale of several seconds. In the deceleration status, the vibration response reaches the peak in the vicinity of the critical speed, and the support structure must maintain integrity. Finally, the wind-milling process lasts about 10 –103 till landing, and large unbalance is still applied on the rotor, which might result in severe rub-impact and other damages. The primary loads and influence factors at each stage are different, and it is critical to catch the key loads and corresponding response during the dynamic analysis and structural safety design. 2.3 Dynamic Model and Governing Equations The key principle of dynamic analysis is to accurately model the distribution of the system’s mass and stiffness. Thus, finite element method is used with solid, shell, and spring elements to establish the dynamic model for the rotor-support system [21], as shown in Fig. 2. Most components of the PT-rotor and turbine support structure are modeled by solid elements, and the turbine blades and the frame struts are modeled by shell element. Spring elements are used to model the front supports (1#, 2#) and the bearing connection at the rear supports (3#, 4#). The stiffness coefficients ki and damping coefficients ci of the spring elements at each support are listed in Table 2.

Fig. 2. Finite element model of the rotor-support structure system (sectional view)

The dynamic model of the PT-rotor consists of the solid part and shell blade segment, and the dynamic equation of the PT-rotor is written in Eq. (1):            J r + C r [0] K r [0] f rs M r [0] q¨ r q˙ r qr (1) + + = {0} [0] Mb (t) q¨ b [0] J b (t) q˙ b [0] K b (t) qb

Dynamic Analysis and Safety Design for Aero-Engine Rotor-Support System

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Table 2. Proprieties of the spring elements used in the finite element model Symbol

Content

Values

ki

Stiffness coefficient

8/4/50/50 (×107 N/m)

ci

Viscous damping coefficient

0/1200/1500/0 (N·s/m)

where, qr , qb are respectively freedom degree vectors of the PT-rotor without blades and the turbine blades segment, and f rs is the spring force vector at the rear supports (3#, 4#), depending on the system displacement. M r , J r , C r , K r are respectively time independent mass, damping, gyroscopic and stiffness matrixes of the PT-rotor without blades. While Mb , J b , K b are time dependent mass, gyroscopic and stiffness matrixes of the turbine blades segment, which would change at the moment of blade loss. The dynamic equation of the support structure is written in Eq. (2): M s q¨ s + K s qs = f sr

(2)

where, qs is freedom vector of the support structure, and M s , K s are respectively mass and stiffness matrixes. f sr is the spring force vector at the rear supports (3#, 4#) applied on the support structure and has different dimensions from f sr . Thus, the two force vectors can be expressed respectively as Eq. (3) and Eq. (4):      (3) f rs = Ars K sp Asb qs − Arb qr + C sp Asb q˙ s − Arb q˙ r      f sr = Asr K sp Arb qr − Asb qs + C sp Arb q˙ r − Asb q˙ s

(4)

where, K sp , C sp are the stiffness and damping matrixes of the spring elements, and Arb , Asb are transformation matrixes transferring the freedom of rotor and support structure to the rear bearings (3#, 4#), and Ars , Asr are matrixes transferring the dynamic loads on the rear bearings to the rotor and support structure. The dynamic model of the whole rotor-support system consists of the rotor model part (Eq. (1)) and the support structure part model (Eq. (2)). These two parts are assembled with the equilibriums of the rear bearings (Eqs. (3), (4)). Therefore, the dynamic equation of the rotor-support system is written in Eq. (5): ⎤⎧ ⎫ ⎡ ⎤⎧ ⎫ ⎡ J r + C r + Ars C sp Arb [0] −Ars C sp Asb ⎨ q˙ r ⎬ Mr ⎨ q¨ r ⎬ ⎦ q¨ b + ⎣ ⎦ q˙ b ⎣ Mb (t) J b (t) [0] [0] ⎩ ⎭ ⎩ ⎭ Ms −Asr C sp Arb q¨ s q˙ s [0] Asr C sp Asb ⎡ ⎤⎧ ⎫ K r + Ars K sp Arb [0] −Ars K sp Asb ⎨ qr ⎬ ⎦ qb = 0 +⎣ (5) K b (t) [0] [0] ⎩ ⎭ −Asr K sp Arb qs [0] K s + Asr K sp Asb Considering the influence of turbine blade-off event, the mechanical parameters, like mass and stiffness distribution, are all time dependent. It should be noted that the freedom degrees of the lost blade are deleted from the blade segment at the moment of blade loss, and the freedom vector qb , as well as the matrixes Mb , J b , K b are all time dependent in the dimension. Sudden impact effect is induced at the blade-off instant, and large unbalance stays on in the status of following operation.

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2.4 Solution Flow of the Governing Equation Based on Explicit Dynamic The dynamic model of the rotor-support system in the typical turboshaft engine is established with finite element method, and the governing equation Eq. (5) can be written in short form shown as Eq. (6): M(t)¨q + C(t)˙q + K(t)q = 0

(6)

where, q is the degree of freedom vector of the whole system, and M(t), C(t), K(t) are respectively time dependent mass, damping and stiffness matrix. Second order accuracy in time centered finite difference method is used to solve the governing equation [22]. Assuming that the time domain [0, t] is equally divided, and the timestep is dt. Then, the velocity and acceleration at the moment of t n can expressed as the function of displacement, shown as Eq. (7): ⎧  ⎪ ⎨ q˙ (tn ) = 1 q(tn+1 ) − q(tn−1 ) 2t (7)     ⎪ 1 ⎩ q¨ (t ) = 1 q˙ t ˙ tn−1/2 = t2 q(tn+1 ) + q(tn−1 ) − 2q(tn ) n n+1/2 − q t Then, the response state recursive scheme is established as shown in Eq. (8):       M(tn ) C(tn ) 2M(tn ) M(tn ) C(tn ) q(tn+1 ) = q(tn−1 ) + − K(tn ) q(tn ) − − 2t 2t t 2 t 2 t 2 (8) The basic steps of the calculation procedure for dynamic equations of the rotorsupport system are listed as follow: (1) Discretize the rotor-support system in space with finite element method, and establish the dynamic differential equations. (2) Start the quasi-static analysis to acquire the initial displacement/velocity state. (3) Discretize the time domain into equal step, and calculate the dynamic response with centered finite difference method.

3 Dynamic Analysis of the Rotor-Support System Based on the above dynamic model and calculation method, modal characteristics and dynamic response of the rotor-support system considering the turbine blade loss event are quantitatively analyzed. 3.1 Modal Characteristics of the System The eigenvalues of Eq. (5) are obtained, and the Campbell diagram is shown in Fig. 3.

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The first four natural frequencies are investigated, and the forward mode diagrams are presented in Fig. 4 respectively. Bending deformation of the flexible shaft occurs at the first three modes, while the turbine frame vibrates at the fourth mode, as shown in Fig. 4d, revealing the coupling vibration between the PT-rotor and the support structure. Additionally, two critical speeds are found below the operation speed (21000 rpm), and the speed value are respectively 7401 rpm and 12989 rpm.

Fig. 3. Campbell diagram of the rotor-support system in the typical turboshaft engine

a)

1st-order forward mode

b)

2nd-order forward mode

c)

3th-order forward mode

d)

4th-order forward mode

Fig. 4. Forward mode diagrams of the rotor-support system under critical speeds

3.2 Dynamic Response of the System Under the Turbine Blade-Off The PT-rotor of the typical turboshaft engine works in supercritical state, and needs to decelerate to wind-milling status rapidly after the blade-off event. Assumed that the turbine blade-off occurs at the moment t 0 = 0.2 s, and the deceleration starts at t 1 = 1 s and end at t 2 = 4 s, during which the PT-rotor decelerates from the operation speed 21000 rpm to the wind-milling speed 3000 rpm, as shown in Fig. 5a. The dynamic response of the rotor-support system is analyzed with centered finite difference method,

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and the displacement of the turbine disk is obtained, as shown in Fig. 5b. The amplitude of response suddenly increases at the moment of blade-off, and reaches the peaks as the rotor passing through the critical speeds during the deceleration. The peak value at 7400 rpm (1st-oder mode) is about twice the value at 13000 rpm (2nd-order mode). The reason for that lies in the difference of the rotor displacement distribution at the two states. As shown in the Fig. 4, turbine lateral displacement and shaft bending deformation both take place at the 1st-order mode, while only shaft bending deformation takes place at the 2nd-order mode. As a result, the response amplitude of turbine disk at 7400 rpm would be higher than the amplitude at 13000 rpm. Besides, the bearing loads at each support are also investigated for bearing damage assessment, as shown in Fig. 6. Similarly, the bearing loads suddenly increase at the moment of blade-off, and reach the peaks the rotor decelerating through the critical speeds. The bearing loads at the rear supports is much higher in comparison with the front supports because of the turbine inertial load, and the 4th bearing is applied the maximum load, which reaches 22.997 kN at speed of 13000 rpm and 14.61 kN at speed of 7400 rpm.

Fig. 5. (a) Rotation speed curve and (b) the displacement of the turbine disk

4 Safety Design for the Rotor-Support System 4.1 Structure Design In order to reduce the vibration of the PT-rotor, maintaining the integrity of the bearing and turbine support, two typical structures for the safety design are introduced, including the sealing brake structure and the metal-rubber ring, as shown in Fig. 7. The sealing brake structure consists of the drum shaft protruding from the disk edge and the conical shell extended from the inner ring of the rear frame, which can act as sealing arrangement in the normal operation. When the turbine blade is lost and the rotor vibration reaches a certain extent, the drum shaft and the conical shell come in contact, and thus the friction damping is generated to reduce the vibration. The conical shell also provides additional supporting stiffness for the rotor when contact to the drum shaft, which increases the critical speeds and benefits rotor deceleration.

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Fig. 6. Bearings load at (a) 1# support, (b) 2# support, (c) 3# support and (d) 4# support

The metal-rubber ring is made of a kind of widely used metal wire porous material [23], as shown in Fig. 8, mounted inside the frustoconical casing. Clearance between the ring and the squirrel-cage spring exists in the normal operation, and the spring compresses the ring when the displacement of the cage exceeds the clearance. Buffering effect will be induced as the ring deforming due to the damping capacity of metal-rubber material, and then help vibration suppression.

Metal-rubber damping ring

Sealing brake structure

Fig. 7. Diagram of the safety designs applied in the typical turbine segment

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Fig. 8. Schematic diagram of (a) metal-rubber material and (b) metal-rubber ring

4.2 Update of the Dynamic Model Additional stiffness and damping characteristics are introduced to the rotor-support system when the sealing brake structure and the metal-rubber ring take effect. Thus, the ∼

∼

updated stiffness matrix K (t) and the damping matrix C (t) are expressed in Eqs. (9) and (10) respectively: ˜ K(t) = K(t) + T Tsbs K sbs (t)T sbs + T Tmrr K mrr (t)T mrr

(9)

˜ C(t) = C(t) + T Tsbs C sbs (t)T sbs + T Tmrr C mrr (t)T mrr

(10)

where, K sbs , C sbs are the local stiffness and damping matrixes at the sealing brake structure location, and K mrr , C mrr are the local stiffness and damping matrixes at the metal-rubber ring location, while T sbs and T mrr are transformation matrixes transferring the local freedoms to the whole system. The safety designs only take effect when the relative displacement of structures is large enough, which can be expressed as Eqs. (11) and (12).   ⎤ ⎤ ⎡ ⎡ kxsbs [0] cxsbs [0] ssbs,w ≥ ssbs ⎥ ssbs,w ≥ ssbs ⎥ ⎢ ⎢ K sbs = ⎣ [0] kysbs ⎦, C sbs = ⎣ [0] cysbs ⎦, (11) ssbs,w ≤ ssbs ssbs,w ≤ ssbs [0] [0]   ⎤ ⎤ ⎡ ⎡ kxmrr [0] cxmrr [0] ≥ ssbs ⎥ s smrr,w ≥ ssbs ⎥ ⎢ ⎢ K mrr = ⎣ [0] kymrr mrr,w ⎦, C mrr = ⎣ [0] cymrr ⎦ (12) [0]

smrr,w ≤ ssbs

[0]

smrr,w ≤ ssbs

where, kxsbs , kysbs , cxsbs , cysbs are the local stiffness and damping coefficients of the sealing brake structure in different directions, and ssbs and ssbs,w are respectively the clearance between the two components at initial state and working state. kxmrr , kymrr , cxmrr , cymrr are the local stiffness and damping coefficients of the metal-rubber ring, and smrr and smrr,w are respectively the clearance between the two components at initial state and working state.

Dynamic Analysis and Safety Design for Aero-Engine Rotor-Support System

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Above all, update the governing equations of the rotor-support system applied with the safety design structures, as shown in Eq. (13). 

˜ q + C(t)˙q + K(t)q ˜ M(t)¨ =0

(13)

4.3 Dynamic Analysis The local stiffness coefficients and damping coefficients are determined by structure configuration, for example, the thickness and width of the metal-rubber ring, and the number of the grate teeth. Liu [24] and Hong [3] respectively proposed the safety design strategies with the buffer-damp support structure, and put forward the optimized values of the local stiffness and damping coefficients. Referring to the existed literature researches mentioned above and engineering experience, the local stiffness coefficients and damping coefficients, as well as the initial clearance of the safety design structures are listed in Table 3. The vibration response of the rotor-support system is analyzed, considering the application of safety designs respectively. The disk displacement and the bearing load on the rear supports are investigated, as shown in Fig. 9. Table 3. Basic parameters of the safety design structures Safety design

Stiffness

Damping

Clearance

Sealing brake structure

3 × 106 N/m 1 × 106 N/m

8000 N·s/m

2 mm

15000 N·s/m

0.1 mm

Metal-rubber ring

Under the extreme condition of turbine blade-off, the vibration response of the rotorsupport system applied with safety design are lower than the original system. The validity of vibration reduction is proved effective for both design structures, while the damping performance appears different. The metal-rubber ring is mounted inside the stator components, and the clearance of the structure can be set to a small value, even to zero, which allows the damping ring to take effect in a large operation range. While the initial clearance of the sealing brake structure is set above the threshold that clearance maintains during the normal operation, and the braking effect only occurs when the drum shaft is under large displacement and in contact with the conical shell. As a consequence, the metal-rubber ring application shows stronger damping capacity while also introduces the shift on the critical speeds of the rotor-support system.

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Fig. 9. Comparison between different design in the response of (a) disk displacement and bearing load at (b) 3# support and (c) 4# support.

5 Conclusions Considering the turbine blade-off event of turboshaft engine, the dynamic model for the typical flexible PT-rotor and turbine support structure is established, and the vibration response of the rotor-support system under the blade-off is solved in the use of centered finite difference method. Besides, two strategies for the structural safety design are proposed and the validity of the applications is quantitatively analyzed. The conclusions can be drawn as follow: (1) The coupling mode between the flexible rotor and the support structure is detected, and the coupling vibration can be excited due to the impact load of the sudden unbalance under the blade-off event. Therefore, it is necessary to establish the dynamic model of the whole rotor-support system.

Dynamic Analysis and Safety Design for Aero-Engine Rotor-Support System

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(2) When turbine blade-off happens, large unbalance load occurs, and the vibration response goes to extreme when the rotor decelerates though the critical speeds. The bearing loads at the rear supports is much high than the front supports, and structural safety design should be applied to maintain the integrity of the bearings and the support structure. (3) The application of the sealing brake structure and the metal-rubber ring can effectively increase the damping capacity of the turbine segment. Thus, the vibration amplitude and the bearing loads are both significantly reduced during the rotor deceleration. Acknowledgements. The authors gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (Nos. 52205082) and National Science and Technology Major Project (Nos. J2019-VIII-0008-0169).

References 1. Kalinowski, P., Bargen, O., Liebich, R.: Vibrations of rotating machinery due to sudden mass loss. In: The 8th IFToMM International Conference on Rotor Dynamics, pp 584–590. Korea (2010) 2. Ma, Y., Wang, Y., Hong, J.: Dynamic model and theoretical investigation for the fan-blade out event in the flexible rotor system of aero-engine. In: Cavalca, K., Weber, H. (eds) Proceedings of the 10th International Conference on Rotor Dynamics – IFToMM. IFToMM 2018. Mechanisms and Machine Science, vol. 63, pp. 18–33. Springer, Cham (2019). https://doi. org/10.1007/978-3-319-99272-3_2 3. Hong, J., Li, T., Liang, Z., Zhang, D., Ma, Y., Wang, Y.: Safety design methods for rotorbearing system and dynamic analysis in aero-Engines. In: Proceedings of the ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition, vol. 7A: Structures and Dynamics. Oslo, Norway (2018) 4. Roberts, G., Revilock, D., Binienda, W.: Impact testing and analysis of composites for aircraft engine fan cases. J. Aerosp. Eng. 15(3), 104–110 (2002) 5. Lawrence, C., Carney, K., Gallardo, V.: Simulation of aircraft engine blade-out structural dynamics. In: Worldwide Aerospace Conference and Technology Showcase. NASA, Glenn Research Center, Toulouse, France (2001) 6. Ganesan, R.: Effects of bearing and shaft asymmetries on the instability of rotors operating at near-critical speeds. Mech. Mach. Theory 35(5), 737–752 (2000) 7. Chu, F., Zhang, Z.: Periodic, quasi-periodic and chaotic vibrations of a rub-impact rotor system supported on oil film bearings. Int. J. Eng. Sci. 35(10/11), 963–973 (1997) 8. Genta, G.: Dynamics of rotating systems. Spring Science and Business Media, New York (2007). https://doi.org/10.1007/0-387-28687-X 9. Ma Y., Liang Z., Zhang D.: Experimental investigation on dynamical response of an overhung rotor due to sudden unbalance. In: Proceedings of the ASME Turbo Expo 2015: Turbine Technical Conference and Exposition, vol. 7B: Structures and Dynamics. Montreal, Quebec, Canada (2015) 10. Hassenpflug, H., Flack, R., Gunter, E.: Influence of acceleration on the critical speed of a Jeffcott rotor. Gas Turbines Power 103(1), 108–113 (1981) 11. Miao, H., Gao, J., Xu, H.: Transient response of unbalanced rotor system through its critical speed. J. Vibr. Shock 23(3), 3–6 (2004). (in Chinese)

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12. Yamamoto, T., Ishida, Y., Ikeda, T.: Summed-and-differential harmonic oscillations of an unsymmetrical shaft. Bullet. JSME 24(187), 192–199 (2008) 13. Muszynska, A.: Rotordynamics. Taylor and Francis, New York (2005) 14. Sinha, S., Dorbala, S.: Dynamic loads in the fan containment structure of a turbofan engine. J. Aerosp. Eng. 22(3), 260–269 (2009) 15. Sinha, S.: Dynamic characteristics of a flexible bladed-rotor with Coulomb damping due to tip-rub. J. Sound Vib. 273(4–5), 875–919 (2004) 16. Thiery, F., Gustavsson, R., Aidanpaa, J.: Dynamics of a misaligned Kaplan turbine with blade-to-stator contacts. Int. J. Mech. Sci. 99, 251–261 (2015) 17. Kastl, J.: Bearing support fuse. United States Patent and Trademark Office, Virginia (2002) 18. Keven, G.: Gas turbine engine bearing support. European Patent Office, Bulletin (2013) 19. Jadczak, E.: Uncoupling system for an aircraft turbojet engine rotary shaft. United States Patent and Trademark Office, Virginia (2014) 20. Zhang, J., Zhang, D., Wang, Y.: Coupling vibration characteristics analysis of shared supportrotors system. J. Beijing Univ. Aeronaut. Astronaut. 45(9), 1902–1910 (2019). (in Chinese) 21. Ortiz, R., Herran, M., Chalons, H.: Blade loss studies in low-pressure turbines-from blade containment to controlled blade-shedding. WIT Trans. Model. Simul. 48, 559–567 (2009) 22. Liu, S., Hong, J., Chen, M.: Numerical simulation of the dynamic process of aero-engine blade-to-case rub-impact. J. Aerospace Power 26(6), 1282–1288 (2009). (in Chinese) 23. Ma, Y., Zhang, Q., Zhang, D.: The mechanics of shape memory alloy metal rubber. Acta Mater. 96, 89–100 (2015) 24. Li, C., Liu, D., Ma, Y.: Dynamic characteristics of the rotor and safety design of support structure with fan blade off. J. Aerospace Power 35(11), 2263–2274 (2020). (in Chinese)

Application of Metamaterial in Vibration Suppression of Rotor System Zhenping Li1 , Hongliang Yao2(B) , and Hui Li2 1 China North Vehicle Research Institute, Beijing 100000, People’s Republic of China 2 Northeastern University, Shenyang 110819, People’s Republic of China

[email protected]

Abstract. Focusing on the problem of narrow frequency band and large size of vibration suppression devices for rotor systems, metamaterials are applied in vibration suppression of rotor system. Firstly, the fundamental design methods of vibration isolators and absorbers are introduced. Then, the inertial method to increase the vibration suppress ability is introduced, which can overcome the overlarge size and weight problem of traditional isolators and absorbers. An experimental platform is built and carried out to verify the torsional vibration isolation performance of the proposed meta-shaft. According to the research, the designed meta-shaft is capable of achieving torsional vibration attenuation in the low frequency range of 25 Hz–200 Hz by adjusting the mass of the inerter, with a maximum vibration isolation efficiency of 88%. Keywords: rotor system · acoustic metamaterials · vibration suppression · isolator

1 Introduction Torsional vibrations have significant negative impact on normal operation of the rotor system thus must be suppressed. Currently, the methods for torsional vibration suppression include flywheel vibration isolators [1, 2], vibration absorbers [3], nonlinear energy sinks [4, 5] and active control [6]. Compare to conventional vibration control methods, locally resonant metamaterial has been extensively studied in recent years due to advantages in vibration suppression, such as bending vibration suppression [7–11], tube [12–14] and shaft [15, 16] torsional vibration suppression issues. Nevertheless, the band gap boundary frequency of the local resonant metamaterial is determined by the stiffness and mass of the elastic layer and resonator, and the geometric structure is significantly restricted in handling ultra-low frequency vibrations and noise. To address the existing problem of oversized metamaterial for ultra-low frequency torsional vibration suppression, this paper proposes a method of combining inerters and metamaterial together. By making the inerter with the metamaterial, most part of the mass of the metamaterial resonator is transferred to the inerter, so that the difficult problem of achieving low frequency torsional vibration suppression without increasing the mass of the resonator is solved. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 15–24, 2024. https://doi.org/10.1007/978-3-031-40455-9_2

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2 Fundamental Design Methods of Vibration Isolators and Absorbers for Rotor System Meta-shaft or meta-absorbers have long been used in vibration suppression of rotor system. The fundamental mechanism of vibration suppressing is: the vibration of the rotor will be absorbed by the meta-shaft unit cell, which is a local resonator. A band gap of low vibration will occur in the meta-shaft, and the shaft can remain in low vibration in this vibration range. The fundamental structure of the traditional meta-shaft is shown in Fig. 1(a), which is composed of base shaft, rubber disc and metal disc, and one cell made up of a rubber disc, a metal disc and a small segment of base shaft is called a local resonator, as shown in Fig. 1(b).

Fig. 1. Structure of the traditional meta-shaft.

When only torsional vibration is considered, the dynamic model of a unit cell is shown in Fig. 2(a), in which J1 , J2 are the moments of inertia of the base shaft and the resonator, respectively. k1 , k2 and c1 , c2 are the corresponding stiffness coefficients and damping coefficients, respectively. θ1 , θ2 are the twist angles, respectively. TL and TR are the force of the j-1th and j + 1th unit cells on the jth unit cell, respectively. ⎧     ⎪ ⎪ ⎪ J1 θ¨1 = TL − k1 (θ1 − θR ) − c1 θ˙1 − θ˙R − k2 (θ1 − θ2 ) − c2 θ˙1 − θ˙2 ⎪ ⎪ ⎨     (1) J2 θ¨2 = k2 (θ1 − θ2 ) + c2 θ˙1 − θ˙2 − k3 (θ2 − θ3 ) − c3 θ˙2 − θ˙3 ⎪ ⎪ ⎪ ⎪   ⎪ ⎩ 0 = TR + k1 (θ1 − θR ) + c1 θ˙1 − θ˙R The band gap of the meta-shaft is studied by carrying out the band gap calculation and the results are shown in Fig. 2(b). It can be seen from Fig. 2(b) that the rotor system will have a low torsional vibration frequency band gap of 140–380 Hz. But this band gap is still rather high, as most rotor systems operate in speed lower than 6000 rpm, which is 100 Hz. To lower the band gap, so as to have a lower frequency vibration suppression range, one has to enlarge the resonator, which will result in the large size and large deformation of the shaft.

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17

Fig. 2. Band structure of the resonator

3 Inerter Enhanced Meta-shaft 3.1 Structure of the Inerter Enhanced Meta-shaft To solve the large deformation and large size of the traditional resonator, inerter enhanced meta-shaft is proposed. Figure 3 shows the assembly drawing and exploded diagram of the designed inerter enhanced resonator of the meta-shaft. The resonator is mainly composed of 3 parts: the drive part, the lever part and the driven part. The first part is the drive part consisting of a rubber disc, a metal disc and a driving inertial disc. The second part is the lever part, which composed of an intermediate wheel with two flexible beams, and the intermediate wheel is fixed on the base shaft. The third part is the driven part composed of a bearing, a metal disc, and a driven inertial disc. The resonator is installed on the shaft, and the multi resonators arranged regularly on the shaft and the meta-shaft is formed. Figure 4 is a meta-shaft with 5 local resonators. 3.2 Intermediate Wheel Drive Amplification Mechanism Figure 5 shows the intermediate wheel drive amplification mechanism. Stop 1 and stop 2 represent the point at which the drive part and the driven part come into contact with the lever, as shown in Fig. 5(a). θ1 and θ2 represent the torsion angles of the drive part and the lever part, respectively. r1 and r2 are the distance stop 1 and stop 2 to the axial line of the shaft. l1 , l2 represent distance from the link point of the lever to stop 1, stop 2 to the link point of the lever, respectively. In Fig. 5 (b), the lever has two relative arc lengths under the action of stop 1 and stop 2. They are ⎧ ⎨ L = r (θ − θ ) 1 1 r b (2) ⎩ l2 L2 = l1 r1 (θr − θb ) The angle of twist of the driven disc can be approximated as θi ≈ where, b =

l2 r1 l1 r2

L2 + θb = b(θr − θb ) + θb r2

is the inertance of the lever part.

(3)

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Fig. 3. The structure of the inerter enhanced resonator

Fig. 4. The meta-shaft with 5 local resonators.

(a) 2D sketch

(b) Geometric relationship

(c) Motion relationship

Fig. 5. Intermediate wheel drive amplification mechanism

3.3 Dynamic Equations of the Proposed Meta-shaft For the meta-shaft proposed in this paper, it can be simplified as a mass-spring system for analytical analysis. Figure 6 shows the one-dimensional periodic local resonance mass-damping-spring system of the meta-shaft. The system in the dotted box in the

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19

figure represents a unit cell of the periodic system, and a is the lattice constant, which represents the distance between the centers of two adjacent unit cells.

Fig. 6. Mathematical model diagram of the proposed meta-shaft.

In Fig. 6, J1 , J2 and J3 , are the moments of inertia of the resonator, the elastic layer and the base shaft, respectively. k1 , k2 , k3 and c1 , c2 , c3 are the corresponding stiffness j j j coefficients and damping coefficients, respectively. θ1 , θ2 and θ3 are the twist angles, respectively. FL and FR are the force of the j-1th and j + 1th unit cells on the jth unit cell, respectively. Considering the transfer of harmonics in the simplified model, the equations of motion of the jth unit cell can be written as J θ¨ + C θ˙ + Kθ = f

(4)

where, ⎡ ⎢ ⎢ J=⎢ ⎣

J1

⎤

⎡

k1 + k2 ⎥ ⎢ J2 ⎥ ⎢ −k2 ⎥K=⎢ ⎣0 J3 ⎦ −k1 0

−k2 k2 + k3 −k3 0

0 −k3 k3 0

⎤ ⎡ −k1 c 1 + c2 ⎥ ⎢ 0 ⎥ ⎢ −c2 ⎥C=⎢ ⎣0 0 ⎦ −c1 k1

−c2 c2 + c3 −c3 0

0 −c3 c3 0

⎤ −c1 ⎥ 0 ⎥ ⎥ 0 ⎦ c1

(5)

3.4 Vibration Transmission Loss and Band Gap Analysis The vibration transmission loss curves of a rotor system consisting of 5 meta-shaft unit cells are analyzed to verify the effectiveness of the designed meta-shaft. The unit cell material parameters and dimensions are shown in Table 1. The vibration transmission loss curves of the rotor system with and without the meta-shaft are shown in Fig. 7(a), and the effect of different distances of the meta-shaft unit cell in the rotor system on the vibration transmission loss curves is shown in Fig. 7(b). As shown in Fig. 7, the torsional vibration of the rotor system is suppressed significantly when the meta-shaft is installed: when the meta-shaft unit cell spacing is 80mm, the maximum vibration suppression effect of the rotor system occurs at 165 Hz, which is 94 dB. As the spacing of the meta-shaft unit cell increases, the vibration suppression effect decreases slightly, but the frequency range where the maximum vibration isolation effect occurs decreases significantly, contributing to the low frequency vibration suppression of the rotor system.

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Z. Li et al. Table 1. Material parameters and dimensions of the meta-shaft Material types

Young’s modulus(Pa)

Density (kg/m3 )

Poisson’s ratio

Steel

1.2 × 107

1150

0.28

10

100

Rubber disc Rubber

1.2 × 107

1300

0.47

60

10

Metal disc of the drive part

2.1 × 1011

7850

0.3

100

10

Drive inertia PLA disc

3 × 109

1250

0.3

100

2

Metal disc Steel of the driven part

2.1 × 1011

7850

0.3

100

10

Driven inertia dis

3 × 109

1250

0.3

100

2

Base shaft

Steel

PLA

80

40 20 0 -20 -40

Thickness (mm)

Spacing between meta-shaft unit cell 100mm Spacing between meta-shaft unit cell 120mm

20

Transmission loss(dB)

Transmission loss(dB)

40

Spacing between meta-shaft unit cell 80mm Bare shaft

60

Diameter (mm)

0 -20 -40 -60

-60

-80

-80 -100

0

50

100

150

200

250

Frequency(Hz)

(a) With or without meta-shaft

300

-100

0

50

100

150

200

250

300

Frequency(Hz)

(b) Different meta-shaft unit cell spacing

Fig. 7. Vibration suppression effect verifications

Then the band gap of the meta-shaft is studied, the diameters of the driven metal discs were selected as 40 mm and 160 mm respectively, thus carrying out the band gap calculation, the calculated results are shown in Fig. 8. As shown in Fig. 8, the meta-shaft band gap width is relatively large when the diameter of the driven part metal disc is varied. In particular, when the diameter of the metal disc is 40 mm, the band gap width is the narrowest at 274.49 Hz and the energy band diagram is shown in Fig. 8(a). When the diameter of the metal disc is 160 mm, the band gap width is the widest at 348 Hz and the band structure diagram is shown in Fig. 8(b).

Application of Metamaterial in Vibration Suppression of Rotor System 1000

700 600

Frequency(Hz)

800 Frequency(Hz)

21

600 400

500 400 300 200

200

100 0

Γ

Χ Wave vector k

Γ

(a) The diameter of the metal disc is 40mm

0

Γ

Χ Wave vector k

Γ

(b) The diameter of the metal disc is 160mm

Fig. 8. Band structure of different metal disc diameter.

4 Experimental Study 4.1 Experiment Setup In order to verify the vibration suppression effect of the inerter enhanced meta-shaft designed in this paper, an experimental bench was built to test the vibration suppression capability. Figure 9 shows the experimental setup. Five unit cells are used to form a periodic shaft, the intermediate wheel is fixed on the shaft, and the driven part is driven by the intermediate wheel. In order to reduce the influence of bending vibration on the experimental results, bearing supports are used to counteract the bending moment caused by the elastic layer and the resonator weight force on the shaft. In this paper, a vibration exciter is used to apply a sinusoidal torque excitation at one end of the periodic structure, and the frequency is swept in the range of 0–200 Hz. Laser displacement sensors are arranged on the left and right sides of the periodic structure to measure the torsional deformation of the excitation point and the response point. The data output by the sensor can be processed by the LMS system to obtain the transmission curve. 4.2 Experiment Results A rotor system with five meta-shaft unit cells was selected for analysis, the transmission loss curve is compared with the simulation results, as shown in Fig. 10. It can be seen from Fig. 10 that the transmission loss curve drawn from the experimental data is basically consistent with the corresponding simulation results. There are two main reasons for the deviation. One reason is that the material parameters of the experimental equipment deviate slightly from the theoretical parameters, and another reason is that there are inevitable errors in the process of building the experiment bench. The experimental results show that after 20 Hz, the meta-shaft with inerter enhanced resonators begins to exhibit the vibration suppression effect, and around 50 Hz, the vibration suppression effect reaches the maximum, which is about 50 dB low frequency, broadband vibration suppression. Changing the base shaft material to steel, repeating the above experiments and comparing them with the simulation results, the comparison graph is shown in Fig. 10(b). As

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Z. Li et al. Laser displacement sensor

Coupling

Shaker

Base axis

Bearing housing

Inertial device

Fig. 9. Experimental setup of inerter enhanced meta-shaft with 5 unit cells.

can be seen from Fig. 10(b), the transmission loss curves plotted from the experimental data are also in general agreement with the corresponding simulation results, and the reasons for the deviations are the same as those described above. The experimental results show that from 10 Hz onwards, the meta-shaft with inerter enhanced resonators starts to show a vibration suppression effect, which reaches a maximum of around 35 dB around 15 Hz and stabilizes at over 12 dB after 30 Hz. It can be seen that this structure can also achieve low frequency and wide frequency vibration suppression of the rotor system. 40

30

Simulation Experiment

Simulation Experiment

20

Transmission loss(dB)

Transmission loss(dB)

20

0

-20

-40

10 0 -10 -20 -30

-60

-40 -80

0

50

100

150

Frequency(Hz)

(a) Base shaft material is nylon

200

-50 0

50

100

150

Frequency(Hz)

(b) Base shaft material is steel

Fig. 10. Experimental simulation comparison of vibration transmission curve.

200

Application of Metamaterial in Vibration Suppression of Rotor System

23

5 Conclusions Through the simulation analysis and experimental verification of the inerter enhanced meta-shaft in suppressing the torsional vibration of the rotor system, this paper mainly draws the following conclusions: 1) An inerter enhanced meta-shaft structure scheme that can be implemented is designed, which has a certain degree of vibration suppression effect on the vibration of the rotor system in the torsional vibration direction; 2) The designed meta-shaft unit cell band gap has a low initial frequency and a wide band gap range, which provides theoretical support for its torsional vibration in the rotor system; 3) The influence of the meta-shaft parameters on its vibration suppression ability is explored, and these laws provide a reference for the experiments of the selected meta-shaft; 4) Compared with the traditional metamaterial vibration absorber, the inerter enhanced meta-shaft designed in this paper greatly reduces the mass of the metamaterial and greatly reduces the impact of the vibration isolator on the rotor system. Experiment and simulation all show that the inerter enhanced meta-shaft has a good vibration reduction effect. Acknowledgments. The authors would like to gratefully acknowledge the Foundation of Equipment Pre-research Area (Grant No. 50910050302) and the National Natural Science Foundation of China (Grant No. 52075084) for the financial support for this study.

Conflict of Interests. The authors declare that there is no conflict of interests regarding the publication of this paper.

References 1. Wang, Y., Qin, X., Huang, S., Deng, S.: Design and analysis of a multi-stage torsional stiffness dual mass flywheel based on vibration control. Appl. Acoust. 104, 172–181 (2016). https:// doi.org/10.1016/j.apacoust.2015.11.004 2. Tang, X., Hu, X., Yang, W., Yu, H.: Novel torsional vibration modeling and assessment of a power-split hybrid electric vehicle equipped with a dual-mass flywheel. In: IEEE Transactions on Vehicular Technology, p. 1 (2017). https://doi.org/10.1109/TVT.2017.2769084 3. Gao, P., Walker, P.D., Liu, H., Zhou, S., Xiang, C.: Application of an adaptive tuned vibration absorber on a dual lay-shaft dual clutch transmission powertrain for vibration reduction. Mech. Syst. Signal Process. 121, 725–744 (2019). https://doi.org/10.1016/j.ymssp.2018.12.003 4. Geng, X.F., Ding, H., Mao, X.Y., Chen, L.Q.: Nonlinear energy sink with limited vibration amplitude. Mech. Syst. Signal Process. 156, 107625 (2021). https://doi.org/10.1016/j.ymssp. 2021.107625 5. Haris, A., Alevras, P., Mohammadpour, M., Theodossiades, S., O’ Mahony, M.: Design and validation of a nonlinear vibration absorber to attenuate torsional oscillations of propulsion systems. Nonlinear Dyn. 100(1), 33–49 (2020). https://doi.org/10.1007/s11071-020-05502-z

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6. Wenzhi, G., Zhiyong, H.: Active control and simulation test study on torsional vibration of large turbo-generator rotor shaft. Mech. Mach. Theory 45(9), 1326–1336 (2010). https://doi. org/10.1016/j.mechmachtheory.2010.04.005 7. Hyun, J., Choi, W., Kim, M.: Gradient-index phononic crystals for highly dense flexural energy harvesting. Appl. Phys. Lett. 115, 173901 (2019). https://doi.org/10.1063/1.5111566 8. Fan, L., He, Y., Zhao, X., Chen, X.: Subwavelength and broadband tunable topological interface state for flexural wave in one-dimensional locally resonant phononic crystal. J. Appl. Phys. 127, 235106 (2020). https://doi.org/10.1063/5.0001548 9. Yuan, L., Cai, Z., Zhao, P., Ding, Y., Ma, T., Wang, J.: Flexural wave propagation in periodic tunnels with elastic foundations. Mech. Adv. Mater. Struct. 29, 1–8 (2020). https://doi.org/ 10.1080/15376494.2020.1769233 10. Zhang, S., Gao, Y.: Flexural wave band structure of magneto-elastic phononic crystal nanobeams based on the nonlocal theory. Phys. Lett. A 390, 127090 (2020). https://doi. org/10.1016/j.physleta.2020.127090 11. Wang, Z., Li, T.: All-angle negative refraction of flexural wave propagation on phononic thin plates with multilayer inclusions. Waves Random Complex Media 31, 1–15 (2019). https:// doi.org/10.1080/17455030.2019.1598601 12. Liu, J., Yu, D., Zhang, Z., Shen, H., Wen, J.: Flexural wave bandgap property of a periodic pipe with axial load and hydro-pressure. Acta Mech. Solida Sin. 32(2), 173–185 (2018). https:// doi.org/10.1007/s10338-018-0070-2 13. Plisson, J., Pelat, A., Gautier, F., Garcia, V., Bourdon, T.: Experimental evidence of absolute bandgaps in phononic crystal pipes. Appl. Phys. Lett. 116, 201902 (2020). https://doi.org/10. 1063/5.0007532 14. Wu, J., Zhu, H., Sun, Y., Yin, Z., Su, M.: Reduction of flexural vibration of a fluid-filled pipe with attached vibration absorbers. Int. J. Pressure Vessels Piping 194, 104525 (2021). https:// doi.org/10.1016/j.ijpvp.2021.104525 15. Song, Y., Wen, J., Yu, D., Wen, X.: Analysis and enhancement of torsional vibration stopbands in a periodic shaft system. J. Phys. D Appl. Phys. 46, 145306 (2013). https://doi.org/10.1088/ 0022-3727/46/14/145306 16. Wang, K., Zhou, J., Xu, D., Ouyang, H.: Tunable low-frequency torsional-wave band gaps in a meta-shaft. J. Phys. D: Appl. Phys. 52, 055104 (2018). https://doi.org/10.1088/1361-6463/ aaf039

A Novel Approach to Model the Shrink-Fit Connection R. D. Firouz-Abadi1(B) , Fahimeh Mehralian1,2 , Hadi Amirabadizadeh1,2 , and Masoud Yousefi1 1 Department of Aerospace Engineering, Sharif University of Technology,

P.O. Box 11155-8639, Tehran, Iran [email protected] 2 Boshra Institute of Science and Technology, Tehran, Iran

Abstract. This study develops an analytical approach for modeling the behavior of shrink-fit connection, widely used in industries. A novel method to determine the equivalent contact stiffness of shrink-fit connection is proposed and its accuracy is shown. The identified relationships indicate the nonlinear correlation between the angular displacement and bending moment capacity for shrink-fit connection. The results indicate the importance of considering the occurrence of the relative slip on the fitting surface of the shrink-fit connection. Keywords: Rotordynamics · Shrink-fit connection · Bending moment · Angular displacement

1 Introduction Given the increasing demand for joint systems, pressure couplings are widely used in industries and industrial connections. Among the pressure couplings, the assembling through shrink-fit connection is widespread because of low costs, withstand high loads and simple implementation process. This interference connection is usually used in telescopic booms like in cranes, antennas and satellites according to Fig. 1.

Fig. 1. The shrink-fit connection in telescopic boom. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 25–35, 2024. https://doi.org/10.1007/978-3-031-40455-9_3

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This interference connection due to differences between the internal diameter of the hub and external diameter of the shaft, called radial interference, causes a contact pressure on the fitting surfaces. Moreover, the relative slip between the fitting surfaces leads to friction forces, which results in connection tolerances. It is of great importance for technically reliable application, model the shrink-fit connection accurately. However, many researchers have investigated this connection through finite element based commercial software, in which their results are only specified for specific working conditions. For example, Schmitz et al. presented a finite element modeling to calculate the stiffness and equivalent viscous damping of thermal shrink-fit connection between tool and holder [1]. The continuous contact profile was approximated in this study. The axisymmetric finite element analysis was performed by Pedersen [2]. The super element technique was utilized to evaluate the distribution of contact pressure due to shrink-fit interference. In this study, friction is neglected. Arslan et al. investigated the shrink-fit performance at rotation and elevated temperature with solid homogeneous inclusion and functionally graded hub [3]. The improper operation of the shrink-fit interference of the 34 MW turbine rotor was studied by Rusin et al. [4]. The malfunction of the shrink-fit connection leads to arising vibration at some phases of the turbine start-up. Increasing the interference stress and using extra protection in form of a circumferential weld between the linked components solved this problem. The shrink-fit connection was investigated by Golbakhshi et al. by considering heating and mounting process and backing to ambient condition through a 3D coupled thermal and structural simulation based on finite element simulation [5]. Nagae et al. enhanced the accuracy of the rotor vibration analysis by calculating the bending stiffness of a rotor with interference connection using finite element method [6]. Since the shrink-fit connections transfer loads between linked components, they can be modeled as torsional spring and damping by evaluating their equivalent stiffness and damping. The force applied to the system transmits by the shaft and shrink-fit connection. Also, the accuracy of bending stiffness of the corresponding shrink-fit connection could affect the accuracy of the predicted rotor vibration. In the present study, the shrink-fit interference connection is modeled analytically to improve the designing procedures of connection parts in industries. Therefore, the behavior of this interference connection is modeled by bending stiffness, and its magnitude is obtained by developing a novel formulation. The accuracy of this model is then verified using commercial software.

2 Analytical Modeling The shrink-fit interference connection is indicated for a telescopic boom with two elements in Fig. 2. Its related parameters including the length of shrink-fit interference connection (2c), diameter of a shaft (2d) and radial interference (δ) are shown in Fig. 3. The force (F) applied to the shaft end transmits by the shaft and shrink-fit connection. This shrink-fit connection due to differences between the internal diameter of the hub and external diameter of the shaft, called radial interference, causes a contact pressure on the fitting surfaces. If the contact pressure between fitting surfaces is not enough, the relative circumferential slip, depending on the magnitude of the slip can cause instability

A Novel Approach to Model the Shrink-Fit Connection

27

Fig. 2. Representation of the shrink-fit interference connection.

Fig. 3. The shrink-fit connection parameters.

and even failure in system. Additionally, the pressure producing in shrink-fit connection must not exceed the maximum admissible value to avoid displacements in members. For example, a loose of capacity to transmit load on members of shrink-fit connection due to decreasing the connection pressure between fitting members causes a malfunction of the interference connection in 34 MW turbine rotor [4]. This turbine was found to operate in an unstable manner and indicated itself by increase in vibration. Consequently, it is important to accurately design the shrink-fit connection and the highest priority would be the best estimation of its dimension. However, there is a little theoretical study in the literature and many practical engineering problems are calculated by FE based commercial software. Due to small angular displacement range, a connection is fixed and indicates the linear behavior, according to Fig. 4. The interference connection can indicate nonlinear behavior for large angular displacement range. In this condition, besides continuous pressure, two other source of pressure at the first and the end of connection surfaces appear according to Fig. 5a. The distribution of this pressure is assumed according to Eq. (1) which can be analytically obtained. Moreover, in large angular displacements, the relative slip between the fitting surfaces leads to friction forces as is shown in Fig. 5b, which results in connection

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Fig. 4. The continuous pressure distribution of the shrink-fit interference connection with small angular displacement.

tolerances. Note that the rigid rotor system is assumed in this study. P = k θ sin2 (φ), φ = 0, ..., 2π

(a) Pressure distribution

(1)

(b) Friction force distribution

Fig. 5. The force distribution of shrink-fit interference connection with large angular displacement.

The magnitude of force supported by shaft and connection is dependent on the relative rotation of that member. The preliminary stage also would be calculating the rotation which can be obtained by computing the equivalent damping and bending stiffness. According to the beam theory, the bending stiffness of the shaft member can be calculated. To obtain the damping and stiffness of the interference connection, the free body diagram of the shaft is indicated according to Fig. 6. As is visible, the continuous pressure distribution and a normal force at the end of a cycle of shaft rotation at the maximum angular displacement (θ max = θ s ) besides

A Novel Approach to Model the Shrink-Fit Connection

29

Fig. 6. The free body diagram of a shaft with the shrink-fit interference connection.

friction force are appeared. Thus, a formulation to determine the capacity of shrink-fit connection to transfer bending moment is developed as:  M =

(Ps + P)μk 2c2d sgn(θ˙ )d φ +

 P(2c − 2d sin(θ) − 2c(1 − cos(θ))sgn(θ)d φ

(2)

where P is obtained through Eq. (1), Ps is shrink-fit pressure distribution, μk is friction coefficient, θ is relative rotation of shaft, φ is angle of pressure. It should be noted that the equivalent damping is achieved by determining the internal area of bending moment versus angular displacement diagram. Moreover, the capability of shrink-fit to transfer moment is vitally dependent on the connection pressure between fitting members. Note that the damping and bending stiffness of interference connection has nonlinear nature. Another important issue comes from calculating the maximum capability of the interference connection to transmit the bending moment. To this end, the maximum bending moment can be determined by determining the maximum relative rotation of a shrink-fit connection, according to Eq. (3). ⎧ ⎪ ⎪ ⎪ M μ=μk θ˙ = 0 ⎪ ⎪ ⎨ (3) M = M μ=μ θ˙ = 0, θ¨ = 0, M μ=μ ,θ=θ > M μ=μ ,θ=θ s s s k k ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ M μ=μ θ˙ = 0, θ¨ = 0, M μ=μ ,θ=θ < M μ=μ ,θ=θ s s s k k

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3 Discussion 3.1 Comparison with FEM In order to evaluate the proposed formulation, a system according to Fig. 7 is considered. This system consists of two similar rotors mounted on linear spring supports with the respective properties proposed in Tables 1 and 2. The excitation force is considered as F (t) = 2.2e3 sin(7π t) and the properties of shaft 2 is similar to shaft 1. The connection between two shafts is modeled according to the formulations presented in Sect. 2.

Fig. 7. The system considered for comparison purpose.

Table 1. Definition of properties of the system according to Fig. 7. Parameters

a1 (m)

c1 (m)

D1 (m)

ρ 1 (kg/m3 )

k x1 (N/m)

Magnitude

1

1

0.1

7800

1e7

Table 2. Definition of properties of the shrink-fit connection according to Fig. 7. Parameters

μs

μk

c (m)

Ps (Pa)

k (N/m)

Magnitude

0.7

0.6

0.04

25e4

3e7

A Novel Approach to Model the Shrink-Fit Connection

31

This system is simulated by means of ADAMS commercial software and the shrinkfit connection is simulated by torsional spring and friction force. According to Fig. 8, the similar behavior is obtained between analytical procedure and ADAMS simulation. According to Fig. 9, the trend of response of shrink-fit connection is completely similar to the response of friction force to displacement in a simple spring mass system.

Fig. 8. A comparison between the results obtained in this paper with ADAMS results.

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3.2 Parameter Study Stiffness To evaluate the influences of equivalent stiffness of shrink-fit connection on dynamic and stability behavior of a rotor system, Fig. 10 is presented. This analysis provides the design insights into select the rotor parameters for stable operation and failure prevention. As is shown in Fig. 10, the vibration amplitude and stability are affected by the stiffness parameters. Thus, it is suitable to tune stiffness parameters for specific purpose. Such that by increasing stiffness, the vibration amplitude is diminished and the system becomes more stable.

Fig. 9. The response of friction force versus displacement.

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33

Prepressure To effectively capture the influence of prepressure in the shrink-fit interference connection, Fig. 11 is plotted for a rotor system. According to Fig. 11, the higher values of prepressure reduces the amplitude of vibration and causes a rotor system more stable.

Fig. 10. The behavior of a rotor system affected by stiffness parameter

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Fig. 11. The behavior of a rotor system affected by prepressure parameter.

4 Conclusion This paper established a novel approach for modeling the shrink-fit interference connection between the rotor members and an analytical model is developed. The bending stiffness of the interference connection was obtained and the accuracy of results was shown by means of ADAMS modeling software. The proposed formulation assisted researchers to model the shrink-fit connection without doing experimental tests and this formulation is applicable in a rigid rotor system with nonlinear behavior for large angular displacement range. In this regard, the continuous pressure distribution and a normal force at the end of a cycle of shaft rotation at the maximum angular displacement besides friction force are appeared.

A Novel Approach to Model the Shrink-Fit Connection

35

References Schmitz, T.L., Powell, K., Won, D., Duncan, G.S., Sawyer, W.G., Ziegert, J.C.: Shrink fit tool holder connection stiffness/damping modeling for frequency response prediction in milling. Int. J. Mach. Tools Manuf 47(9), 1368–1380 (2007) Pedersen, P.: On shrink fit analysis and design. Comput. Mech. 37(2), 121–130 (2006) Arslan, E., Mack, W.: Shrink fit with solid inclusion and functionally graded hub. Compos. Struct. 121, 217–224 (2015) Rusin, A., Nowak, G., Piecha, W.: Shrink connection modelling of the steam turbine rotor. Eng. Fail. Anal. 34, 217–227 (2013) Golbakhshi, H., Namjoo, M., Mohammadi, M.: A 3d comprehensive finite element based simulation for best shrink fit design process. Mech. Ind. 14(1), 23–30 (2013) Nagae, N., Murayama, T., Yamaji, S., Goto, T.: Modeling of the fitting part to improve the calculation accuracy of the rotor shaft vibration analysis model. JSME Proc. 15–00 174 (2015)

Numerical Analysis of the Effect of Different Surface Textures on the Steady State Characteristics of Tilting Pad Journal Bearings Bin Wei(B) , Xiuli Hu, Renwei Che, and Yinghou Jiao School of Mechanical Electronic Engineering, Harbin Institute of Technology, Heilongjiang, Harbin 150001, China [email protected]

Abstract. The purpose of this paper is to investigate the effect of three different types of textures on the steady-state performance of tilting pad journal bearings. It has been shown that surface texture can improve the steady-state performance of bearings, but there are few studies on tilting pad journal bearings. This paper solves the thermal elastohydrodynamic (TEHD) numerical model of tilting pad journal bearings using the central difference method and the bi-conjugate gradient stabilization algorithm. The results agree well with the simulation results proving the model’s effectiveness. The effects of ellipsoidal, cylindrical, and square textures on the steady-state performance of tilting pad journal bearings are studied and compared with non-textured bearings. The influence of texture on oil film pressure, oil film maximum temperature, and bearing capacity was analysed. The results show that the square texture can significantly enhance the steady-state performance of tilting pad journal bearings, and the effects of ellipsoidal texture and cylindrical texture are basically the same, which provides a reference for surface texture research on tilting pad journal bearings. Keywords: Tilting Pad Journal Bearing · Texture · Thermal Elasto-hydrodynamic · Steady-state

1 Introduction The rapid development of modern heavy rotary machinery brings forwards higher tribological properties of bearings, thus surface texture technology is introduced. Surface texture technology is designed to process regular texture features on the contact surface of bearings and journals by physical or chemical methods. Compared with nontextured bearings, it has the advantages of providing higher bearing capacity, reducing the maximum temperature, and prolonging the service life. The texture acts as a micro-trap to collect worn particles and reduce the damage of debris to the working surface. Texture can also be viewed as a micro-oil groove, providing lubricant in the case of mixing or boundary lubrication. The increasing effect of texture on bearing capacity is called micro-bearing effect [1, 2]. The enhancement of tribological properties of dynamic pressure lubricated bearings is usually related to the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 36–45, 2024. https://doi.org/10.1007/978-3-031-40455-9_4

Numerical Analysis of the Effect of Different Surface Textures

37

local cavitation, inertia effect, inlet suction, balance wedge action, and collective texture function played by texture [3]. Researchers have conducted much work on the influence of surface texture on the performance of journal bearings. Niranjan Singh [4] et al. studied the influence of spherical, cylindrical, triangular and diamond texture parameters on the performance of journal bearings. The results show that the spherical texture has better improvement than the others. Xiang Guo [5] et al. studied the influence of triangular texture on the tribological dynamic performance of water-lubricated bearings during start-up. The results show that the acceleration time and surface roughness significantly affect the friction dynamics, while the roughness curvature has little effect on the friction dynamics. Sanjay Sharma [6] et al. studied the influence of triangular texture on the dynamic and steady-state performance of bearings. It is found that shallow texture depth can obtain the maximum direct stiffness coefficient and threshold velocity, while deep texture depth can obtain the maximum direct damping coefficient. Zhang Xiangyuan [7] et al. studied the influence of rectangular grooves on journal bearing performance and optimized the groove parameters by a particle swarm optimization algorithm. The results show that under different eccentricities and rotational speeds, the groove distribution is best in trapezoidal shape in the circumferential direction, and the trapezoid becomes thinner with increasing eccentricity. At lower eccentricity, the optimum groove texture has more significant effect on the reduction of the friction coefficient. Chen Dongjun [8] et al. studied the effects of square, spherical, and composite textures on the performance of journal bearings, and the results showed that the existence of textures can improve the bearing capacity and reduce the friction coefficient of bearings. The optimal unit area ratio of microtexture maximizes the bearing capacity and minimizes the friction coefficient. The relative depth of the texture is the most important factor, followed by the number of textures, and finally the unit area ratio. In conclusion, there have been many studies on the influence of surface texture on journal bearing performance, but relatively few studies on the influence of texture on the performance of tilting pad bearings. Tilting pad journal bearings are widely used in high-speed rotating machinery because of their excellent stability. In this work, the finite difference method is used to solve the Reynolds equation of tilting pad journal bearings. The effects of ellipsoidal, cylindrical, and square textures on the steady-state performance of tilting pad journal bearings are studied and compared with those of non-textured bearings.

2 Governing Equation 2.1 The Reynolds Equation The hydrodynamic pressure distribution in the tilting pad journal bearings is described by the dimensionless Reynolds equation under the isoviscosity and incompressibility assumptions.  3     3  ∂ h ∂P D 2 ∂ h ∂P ∂h (1) + = −3 ∂ϕ η ∂ϕ L ∂λ η ∂λ ∂ϕ

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where, ϕ is circumferential coordinates, λ is axial coordinates, h is oil film thickness, η is viscosity, P is pressure, D is width of bearing, L is length of bearing. 2.2 Moment on Pad At steady state, the resultant force of the fluid dynamic pressure colinear with the line passing through the pivot and the center of pad, and the net bending moment acting is zero. The bending moment acting on the pad is expressed as follows.    1 ϕi,trailing  2 M = − ηT0 ωR LR R + tp P sin μd μd λ (2) −1 ϕi,leading

where, M is bending moment, R is dimensionless radius of pad, tp is dimensionless thickness of pad, ϕi,leading is leading edge of i-th pad, ϕi,trailing is trailing edge of the i-th pad, μ is angle from the line which passing through the pivot and the center of pad. 2.3 Thermal Effects The heat generated by friction causes the bearing temperature to rise, while the viscosity of the lubricant also changes with increasing temperature. Assuming that all the heat generated by friction is taken away by the flowing lubricant, the equivalent temperature can be expressed as follows [9]. Te = Tin +

1.6FU Qρσ

(3)

where, Te is the equivalent temperature, Tin is the oil supply temperature, F is the friction force, U is the linear velocity of journal surface, Q is the side leakage, ρ is the density of oil, σ is the specific heat of oil. 2.4 Elastic Deformation The pad is deformed under the action of oil film pressure. It is assumed that the inner surface of the pad remains circular when deformation occurs. Then the bending stiffness of the pad is [10] 2

2  Ep Lt γ 2 − 1 − 2γ ln(γ )

 kp = (4) γ − 1 4 1 − γ 2 1 − 2 ln(γ ) where, Ep is Young’s module of the pad, and γ = (R + t)/R is the ratio of the outer and inner surface radii of the pad. The average bending moment acting on the pad is  1 ϕtrailing Maverage = Md μ (5) α ϕleading Then, the average elastic deformation of the pad is as follows. δp,e =

Maverage kp

(6)

Numerical Analysis of the Effect of Different Surface Textures

39

2.5 Lubricant Film Thickness In non-textured tilting pad journal bearings, the oil film thickness is given by   hs = c + ex cos ϕ + ey sin ϕ − c − c cos(ϕ − βi ) + δi R sin(ϕ − βi )

(7)

where, ex = e cos θ ey = e sin θ , e is eccentricity, θ is attitude angle, c is bearing clearance, and δi is tilt angle of the i-th pad. When there is a texture on the pad surface, the clearance equation consists of two parts. One part is expressed by Eq. (7). The other part is the sum of the texture depth of the texture region and Eq. (7). The different textures shown in Fig. 1 are considered, and the equation of depth of different textures is shown as follows. The depth at any point in the ellipsoidal texture region is given by 

  2 2 2   h 2  2  rd r hd d − + d (8) − xc − xoi + zc − zoi hd = + 2 2hd 2 2hd When considering the elastic deformation of the pad, the corresponding equation of thickness is as follows.  hs + δp,e / e (xc , zc ) ∈ h= (9) hs + hd + δp,e (xc , zc ) ∈ e where, hd is depth of texture, rd is radius of texture cross circle on the pad surface, xc ,zc is the projection of one point on the inner surface of the texture onto the x-z plane, xoi , zoi is coordinates of the texture cross circle in the x-z plane. In the cylindrical texture area c and square texture area sq , the depth of any point inside the texture is the same, so the clearance equation of thickness is as follows.  / c hs + δp,e (xc , zc ) ∈ (10) h= hs + hd + δp,e (xc , zc ) ∈ c  / sq hs + δp,e (xc , zc ) ∈ (11) h= hs + hd + δp,e (xc , zc ) ∈ sq

3 Numerical Method The transient tilting pad journal bearing model considering the thermal effect and elastic deformation requires simultaneous solution of the Reynolds equation, pad tilting equation, thermal balance equation and elastic deformation equation. Although a more precise model can improve the accuracy of the solution results, it also increases the cost. Compromise considerations are made to improve the numerical calculation performance. The central difference scheme is used to discretize the Reynolds equation, and the bi-conjugate gradient stable algorithm is used to solve the difference equation until the solution accuracy is met or the maximum number of iterations is exceeded. The inverse quadratic interpolation method is used to solve the value of the tilting angle of the pad under moment balance. When the pad tilting angle meets the solution accuracy or exceeds the maximum number of iterations, the process is stopped.

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a) ellipsoidal texture

b) cylindrical texture

c) square texture

Fig. 1. Different types of surface texture

4 Results and Discussion 4.1 Validation of the Model Before further analysis, the accuracy of the model result is verified with the rotor dynamics analysis software MADYN 2000 which is widely used in research [11–15]. Although the MADYN 2000 software is not able to model and analyze tilt pad journal bearing with textural features. When the depth of texture is zero, the model degrades to an untextured tilt pad journal bearing, which can be processed by MADYN 2000. In this way, the results of the C + + program can be verified using the MADYN 2000 software. The parameters are shown in Table 1. Finally, the oil film force is 29775.7 N, and the comparison of the results is shown in Fig. 2. Table 1. Parameters of the bearing Item

Value

Item

Value

Diameter of bearing/m

0.479

Preload

0

Bearing length-diameter ratio

0.6263

Thickness of pad/m

0.121

Bearing clearance/m

0.000612

Young’s module of pad/Pa

2.06e11

Number of pad

4

Lubricant

ISO VG32

Pad arc/°

80

Rotation speed/rpm

3000

Pivot offset

0.5

Bearing load/N

30000

Numerical Analysis of the Effect of Different Surface Textures

41

Fig. 2. Comparison of calculation results

Figure 2 shows that the oil film pressure in tilting pad journal bearings is mainly generated at loaded pad 1 and pad 2, and the pressure center passes through the pivot, which is determined by the characteristics of the geometry. The comparison shows that the pressure distribution and pressure center are basically the same as those of MADYN 2000. The reason for the difference (red dotted box in the figure) is that the MADYN 2000 software sets the default inlet pressure, while the model in this paper uses the Reynolds boundary condition (the boundary pressure is zero). However, it is also observed that the pressure shows good consistency on pad 2 in the circumferential direction. In conclusion, it can be considered that the calculation result of the model is reasonable and reliable. It can be inferred that the results obtained from the model calculations in this paper are also reasonable when the depth of texture is not zero. 4.2 Effect on Pressure The eccentricity is 0.2, the dimensionless texture depth is 1, the texture radius is 0.02 m, the number of circumferential and axial textures of tilting pad journal bearings is 6x6, and the textures are distributed on the surface of pad 1 and pad 2. Other parameters and numerical analysis parameters remain unchanged. The influence of three different types of textures on the maximum oil film pressure was calculated, and the corresponding calculation results were compared with those of non-textured bearings, as shown in Fig. 3. Figure 3 shows that the maximum hydrodynamic pressure on pad 1 and pad 2 increases gradually as the journal center moves downwards, and different textures have different influences. Square texture caused the largest change, and cylindrical texture and ellipsoidal texture caused almost the same change. This is because under the same parameter conditions, the square texture has the largest area proportion on the inner surface of the pad, while the cylindrical texture and ellipsoidal texture have the same area. At the same time, the increase in the maximum pressure leads to an increase in the leakage amount of the end face of the pad, as shown in Fig. 4, and the law is the same as that of the former.

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a) maximum pressure on pad 1

b) maximum pressure on pad 2

Fig. 3. The effect of different types of textures on maximum pressure

a) side leakage on pad 1

b) side leakage on pad 2

Fig. 4. The effect of different types of textures on side leakage

4.3 Effect on Maximum Temperature Keeping the bearing parameters and numerical analysis parameters unchanged, the influences of three different types of textures on the maximum oil film temperature were calculated. The calculated results were compared with those of non-textured bearings, as shown in Fig. 5. Figure 5 shows that the maximum oil film temperature on pad 1 and pad 2 decreases with the gradual downwards movement of the journal center. This is because under the same parameter conditions, the variation in pressure caused by the square texture is the largest, which leads to the largest variation in leakage, and thus the maximum oil film temperature decreases. Cylindrical texture and ellipsoidal texture cause the same variation in pressure and leakage, so the variations in maximum oil film temperature are basically the same.

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a) maximum oil film temperature on pad 1 b) maximum oil film temperature on pad 2 Fig. 5. The effect of different types of textures on maximum oil film temperature

4.4 Effect on Bearing Capacity Keep the bearing parameters and numerical analysis parameters unchanged, calculate the influence of three different types of texture on the bearing capacity, and compare the calculation results with those corresponding to the non-textured bearing. The calculation results are shown in Fig. 6.

Fig. 6. Effect of different types of textures on the resultant hydrodynamic force

Figure 6 shows that, as the journal center gradually moves downwards, the bearing capacity of tilting pad bearings increases significantly compared with non-textured bearings due to different types of textures. The square texture causes the largest variation, because under the same parameter conditions, the pressure variation and leakage variation caused by the square texture are the largest, resulting in the maximum oil film temperature reduction, resulting in the increase in lubricating oil viscosity, which is

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conducive to the further formation of fluid dynamic pressure, and the bearing capacity increases significantly under the combined action of the above reasons. The cylindrical texture and ellipsoidal texture cause the same pressure and leakage variation, so the bearing capacity changes are basically the same.

5 Conclusion The thermoelastohydrodynamic steady-state model of tilting pad journal bearings is established and solved by the central difference method and bi-conjugate gradient stable algorithm. The effects of three different textures on the performance of tilting pad journal bearings are analysed, and the conclusions are as follows. 1) Three different surface textures significantly affect the steady-state performance of tilting pad journal bearings. The influence of square texture on oil film pressure, maximum oil film temperature and bearing capacity is better than that of cylindrical and ellipsoidal surfaces texture. The effects of cylindrical and ellipsoidal surface textures on the performance are basically the same. 2) The main reason for the influence of different types of surface texture is the area ratio of surface texture on the inner surface of pad, while the effect of the depth curve of the texture is not obvious. Acknowledgements. This work was supported by the National Natural Science Foundation of China (Grant No.11972131) and the National Natural Science Foundation of China (Grant No.12072089).

References 1. de Matos Reis, J.O., Rodrigues, G.W., Bittencourt, M.L.: Virtual texturing of lightweight engine crankshaft bearings. J. Braz. Soc. Mech. Sci. Eng. 41(6), 1–15 (2019). https://doi.org/ 10.1007/s40430-019-1740-9 2. Tala-Ighil, N., Fillon, M.: A numerical investigation of both thermal and texturing surface effects on the journal bearings static characteristics. Tribol. Int. 90, 228–239 (2015) 3. Gropper, D., Wang, L., Harvey, T.J.: Hydrodynamic lubrication of textured surfaces: a review of modeling techniques and key findings. Tribol. Int. 94, 509–529 (2016) 4. Singh, N., Awasthi, R.K.: Influence of texture geometries on the performance parameters of hydrodynamic journal bearing. P I Mech Eng J-J Eng. 235, 2056–2072 (2021) 5. Xiang, G., Han, Y.F.: Study on the tribo-dynamic performances of water-lubricated microgroove bearings during start-up. Tribol Int. 151, 106395 (2020) 6. Sharma, S., Jamwal, G., Awasthi, R.K.: Dynamic and stability performance improvement of the hydrodynamic bearing by using triangular-shaped textures. P I Mech. Eng. J-J Eng. 234, 1436–1451 (2020) 7. Zhang, X.Y., Liu, C.P., Zhao, B.: An optimization research on groove textures of a journal bearing using particle swarm optimization algorithm. Mech Ind. 22, 1 (2021) 8. Chen, D.J., Zhao, Y., Zha, C.Q., Pan, R., Fan, J.W.: Performance evaluation of different types of micro-textured hydrostatic spindles under the main influencing factors. P I Mech. Eng. J-J Eng. 235, 2169–2183 (2021)

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9. Stachowiak, G.W., Batchelor, A.W.: Engineering Tribology, 4th edn (2013) 10. Wilkes, J.C., Childs, D.W.: Improving tilting pad journal bearing predictions-part I: model development and impact of rotor excited versus bearing excited impedance coefficients. J. Eng. Gas Turb. Power 135 (2013) 11. Gaulard, F., Schmied, J., Fuchs, A.: State-of-the-art rotordynamic analyses of pumps. Bull. Polish Acad. Sci.-Tech. Sci. (2022) 12. Ghalea, A., Varshitha, H., Kumar, M.V., et al.: Design validation of high inertia generator rotor. In: Advanced Trends in Mechanical And Aerospace Engineering: ATMA-2019 (2021) 13. Kicinski, J., Zywica, G., Baginski, P.: The dynamic performance analysis of the microturbine’s rotor supported on slide and rolling element bearings. In: International Conference on Vibration Problems (2011) 14. Schmied, J., Fedorov, A., Grigoriev, B.S.: Non-synchronous tilting pad bearing characteristics. In: IFToMM International Conference on Rotor Dynamics (2012) 15. Schmied, J., Perucchi, M., Pradetto, J.C.: Application of MADYN 2000 to rotordynamic problems of industrial machinery. Asme Turbo Expo: Power for Land, Sea, & Air (2007)

Effect of Support Parameters on the Vibrations of a Cracked Rotor Passing Through Critical Speed Fahimeh Mehralian1,2 , R. D. Firouz-Abadi2(B) , and Masoud Yousefi2 1 Boshra Institute of Science & Technology, Tehran, Iran 2 Department of Aerospace Engineering, Sharif University of Technology,

P.O. Box 11155-8639, Tehran, Iran [email protected] Abstract. The aim of the present study is to improve the efficiency of a rotor bearing system for passing through critical speed. The current study assesses the influences of support parameters on the cracked rotor response for passing through critical speeds. The actual breathing mechanism of the transverse crack is addressed herein. The Finite element model is developed for more realistic analysis. The Newmark-β method to solve the nonlinear lateral vibration of rotorbearing systems is approached and the effects of various parameters are shown. Keywords: Rotordynamics · Support parameters · Shaft crack · Dynamic response

1 Introduction To attain a specified speed in high speed machinery, they require to pass through critical speed. The maximum response of unbalanced rotating machinery can be affected by a modification in operating speed or rate of acceleration. The fatigue cracks among the various faults in rotor bearing systems is known as a grave threat to an uninterrupted operation of rotating machinery due to their potential to cause catastrophic failure. The dynamic response of a slant-cracked rotor passing through critical speed have been calculated by Prabhakar et al. using finite element method and the transient response of the cracked rotor for passing through critical speed has been analyzed for various crack depths, angular acceleration and torsional excitation for finding crack detection and monitoring techniques [1]. Since the vibration response during start up and shut down is important to detect cracks, a signal processing technique has been applied by Babu et al. to transient response of a cracked rotor [2]. The Hilbert Huang transform found to be better compared to fast Fourier transform and continuous wavelet transform. Chandra et al. utilized the rotor start up vibrations to perform fault identification using time frequency techniques using three signal processing tools namely short time Fourier transform, continuous wavelet transform and Hilbert Huang transform [3]. The computational time of Hilbert Huang transform is obtained very less in comparison to continuous wavelet transform however for noisy data continuous wavelet transform is preferred over Hilbert Huang transform. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 46–53, 2024. https://doi.org/10.1007/978-3-031-40455-9_5

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Since the maximum response of rotating machinery can be affected by a modification in operating speed or rate of acceleration, to traverse a machine safely through the critical speed, it can be quickly accelerated to decrease the destructive influence of the presence of critical speeds [4–9]. Since during run down the deceleration rates are slower than acceleration rates in startup, many resonant problems occur on run down, resulting in machine damage which necessitates the study of this phenomenon and its effective parameters. Broad review of the state of the art in rotating machinery with particular regard to passing through critical speed is presented herein. The overcritical induction motor under electromagnetic forces accelerated passing through the critical speed was studied by Pennacchi et al. [10]. A finite element model was developed and dependency of magnetic stiffness and damping on the rotational speed of the sleeve bearings is considered. In this study the influences of support parameters on the cracked rotor response is investigated. A model consists of crack fault for passing through critical speed is analyzed. An in depth study on support parameters that are effective in the transient response of rotating machinery is presented.

2 Rotor System Model To perform a preliminary analysis of the rotor bearing system, a rotor mounted symmetrically on two bearings is indicated in Fig. 1. The generalized coordinates of the rotor system are considered as X, Y, Z, φ, θ and ψ.

Fig. 1. Representation of a rotor system.

The equations of motion of the rotor system can be achieved using the Lagrangian approach. The Lagrange ‘s equations can be expressed as: d ∂L ∂R ∂L + − =0 dt ∂ q˙ ∂ q˙ ∂q

(1)

In order to apply Lagrange’s equations, the kinetic energy T, including rotational and translational kinetic energy, and potential energy U ∗ , should be calculated: L = T − U∗

(2)

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In order to derive the kinetic and potential energy see Refs. [11] and [14]. The crack is modeled according to Fig. 2. The dashed segment present the crack segment. The centroidal area moment of inertia of the cracked element about X - and Y -axis are IX and IY , respectively. According to [12], the time varying area moments of inertia IX and IY and IXY about X - and Y -axis during the shaft rotation are given as. IX (t) =

Ix − Iy I x + Iy + cos(2ψ) − Ixy sin(2ψ) 2 2

IY (t) =

Ix − Iy I x + Iy − cos(2ψ) + Ixy sin(2ψ) 2 2

IX Y (t) =

(3)

I x − Iy sin(2ψ) + Ixy cos(2ψ) 2

where Ix = I x − Ac e2 , Iy =I y and I x and I y are the area moments of inertia of the cracked element about the rotating x − and y − axis, Ac is the area of the cracked element and e is its centroid location on the y − axis. Since y is the axis of symmetry of the cracked element cross section area during rotation, Ixy = 0. Ac and e have been calculated in [13] as A = R2 (π − cos−1 (1 − μ) + (1 − μ)γ ) e=

2R3 3 γ 3A1

(4) (5)

where μ = h/R is √ the non-dimensional crack depth, h is the crack depth in the radial direction and γ = μ(2 − μ). The area moments of inertia I x and I y of the cracked element about x- and y-axis, respectively, have also been determined in Ref. [13] for 0 ≤ μ ≤ 1 as Ix =

R4 π R4 + ((1 − μ)(2μ2 − 4μ + 1)γ + sin−1 (1 − μ)) 4 4

(6)

R4 π R4 − ((1 − μ)(2μ2 − 4μ − 3)γ + 3 sin−1 (γ )) 4 12

(7)

Iy =

Accordingly, the area moment of inertia and the transversal and polar moment of inertia, i.e. J t and J P , of the cracked element should be modified. Since the transversal and polar moment of inertia per length are J t = 41 ρπ R4 and J P = 2J t of a rotor element without crack, the relations (Eq. 3) can be utilized to modify the transversal and polar moment of inertia of a cracked one. Note that the moment of inertia is calculated for a rotating system, as a result it is not accurate to use Fourier series to model the breathing mechanism of the cracked element. Therefore, using Eq. (1) the governing equations of motion in matrix form for a rotor system in a rotor system can be obtained as: ˙ q˙ + C q˙ + K q = F M q¨ + ψG

(8)

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Fig. 2. Representation of the cracked element cross section a) before rotation b) after rotation.

where M, G, C, K and F are mass matrix, gyroscopic matrix, damping matrix, stiffness matrix and force vector, respectively and q is displacement vector. To set up the dynamic model of the rotor system, the finite element method is utilized. In this method, the solution is subdivided into finite elements on which shape functions are defined to approximate the solution. The rotor shaft is modeled by the Euler-Bernoulli beam element and the cross-section of the shaft is assumed to be uniform in each element. The shaft is discretized into N finite elements with 6 degrees of freedom (DOFs), including three rotational and three translational DOFs at each node.

3 Results and Discussion In this section, the results based on the proposed formulation are also presented and discussed. The six-degree-of-freedom rotor bearing models are used to demonstrate the influences of various parameters based on the FEM method using the Euler-Bernoulli beam element. 3.1 Validation of Results The accuracy of the proposed formulation is evaluated herein. The dynamic response of a simple rotor system, according to Fig. 3, is investigated for comparison study. The rotor parameters are tabulated in Table 1. The time response in x and y directions of node 1 and node 6 are achieved according to Fig. 4. It can be found that the results are coincident with Dyrobes ones. Dyrobes is a rotordynamics software which offers rotordynamic analysis, vibration analysis, bearing performance and balancing calculations based in Finite Element Analysis. The software combines an intuitive Windows-based interface with sophisticated modeling and analysis capabilities that can satisfy the most demanding industry requirements.

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Fig. 3. Representation of a rotor system for comparison study.

3.2 Support Parameter Study The influences of support parameters on the dynamic behavior of the cracked rotor bearing system to pass through critical speed is conducted herein to choose the right ones. For this purpose, a rotor system with one shaft and two supports is considered. The distribution of residual shaft bow is assumed as the first order bending mode of shaft.

(a) Displacement in x direction of node 1

(b)Displacement in y direction of node 1

(c) Displacement in x direction of node 6

(d) Displacement in y direction of node 6

Fig. 4. Comparison study between Dyrobes and present study - Time response.

Additionally, the rotor parameters are considered according to Table 1 unless otherwise stated. In this section, δ parameter means the radial displacement of a node, i.e.  2 δ6 = U6 + V62 , and node 6 is in the middle of the rotor. The influences of support parameters including stiffness and damping are evaluated based on the aforementioned formulation and the results are described in Figs. 5 and

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Table 1. Definition of geometric and physical properties of a rotor bearing system. Parameter

Value

Shaft Length

L = 1200 mm

Shaft Radius

r = 25 mm

Eccentricity

ε = 25 mm

Mass unbalance

mε = 3 gr

Density

ρ = 2700 Kgm−3

Young modulus

E = 70e9 Nm−2

Support Stiffness

k = 100000 Nm−1

Support Damping

c = 20 Nsm−1

6. This analysis provides the design insights into select the rotor parameters and failure prevention for passing through critical speed.

Fig. 5. The influences of stiffness parameters on angular velocity and transient response.

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As is shown in Figs. 5 and 6, the vibration amplitude and critical frequency of a rotor system are affected by the support parameters. Thus, it is suitable to tune support parameters by acting on the amount of damping and stiffness for specific purpose. As a result, by increasing damping, the vibration amplitude is diminished but stiffness affects in different way. According to Figs. 5a, b and c, for a rotor passing through critical speed, the increase in stiffness reduces the vibration amplitude but according to Figs. 5g, h and i, the rise of stiffness increases the vibration amplitude for a rotor which cannot pass through critical speed.

Fig. 6. The influences of damping parameters on angular velocity and transient response.

4 Conclusion The main purpose of the present study was to examine the dynamic response of a cracked rotor system. The equations of motion for 6 degrees of freedom system were obtained using Lagrange’s equations and a FE model of the rotor system was developed. The Newmark-β integration scheme was utilized to calculate the model’s equations of motion and find the transient start up response. The influences of support parameters were studied

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and it was shown that the support parameters have significant effect in changing dynamic characteristics of the rotor system to assist the system to pass through critical speeds.

References 1. Prabhakar, S., Sekhar, A., Mohanty, A.: Transient lateral analysis of a slant- cracked rotor passing through its flexural critical speed. Mech. Mach. Theory 37(9), 1007–1020 (2002) 2. Babu, T.R., Srikanth, S., Sekhar, A.: Hilbert–huang transform for detection and monitoring of crack in a transient rotor. Mech. Syst. Signal Process. 22(4), 905–914 (2008) 3. Chandra, N.H., Sekhar, A.: Fault detection in rotor bearing systems using time frequency techniques. Mech. Syst. Signal Process. 72, 105–133 (2016) 4. K. Millsaps and G. Reed, “Reducing lateral vibrations of a rotor passing through critical speeds by acceleration scheduling,” 1998 5. Wang, S.-M., Lu, Q.-S., Twizell, E.: Reducing lateral vibration of a rotor passing through critical speeds by phase modulating. J. Eng. Gas Turbines Power 125(3), 766–771 (2003) 6. Zapomˇel, J., Ferfecki, P.: A computational investigation on the reducing lateral vibration of rotors with rolling-element bearings passing through critical speeds by means of tuning the stiffness of the system supports. Mech. Mach. Theory 46(5), 707–724 (2011) 7. S. Yanabe, S. Kaneko, Y. Kanemitsu, N. Tomi, and K. Sugiyama, “Rotor vibration due to collision with annular guard during passage through critical speed,” 1998 8. Prabhakar, S., Sekhar, A., Mohanty, A.: Vibration analysis of a misaligned ro- torcouplingbearing system passing through the critical speed. Proceedings of the institution of mechanical engineers, Part C: Journal of mechanical engineering sci- ence 215(12), 1417–1428 (2001) 9. Hu, X., Gao, F., Cui, C., Liu, J., Wang, H., Wang, H., Dai, Y., Ni, Z., Li, Y., Yu, G. et al.: Active control method for passing through critical speeds of rotating super- conducting rotor by changing stiffness of the supports with use of electromagnetic force. IEEE transactions on applied superconductivity, vol. 23, no. 3, pp. 5 201 304–5 201 304, 2012 10. C. Bauer and U. Werner, “Rotordyamic analysis of a 2-pole induction motor consid- ering magnetic excitation due to dynamic rotor eccentricity during startup,” in Pro- ceedings of the 9th IFToMM International Conference on Rotor Dynamics. Springer, 2015, pp. 661–676 11. F. Mehralian, S. M. R. Mousavi, R. D. Firouz-Abadi, M. Farajolahi, and A. Chasalevris, “Stability assessment of bowed asymmetric rotors on nonlinear sup- ports,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 44, no. 586, 2022 12. W. D. Pilkey, Analysis and design of elastic beams: Computational methods. John Wiley & Sons, 2002 13. Al-Shudeifat, M.A., Butcher, E.A.: New breathing functions for the transverse breathing crack of the cracked rotor system: approach for critical and subcritical harmonic analysis. J. Sound Vib. 330(3), 526–544 (2011) 14. Amirzadegan, S., Rokn-Abadi, M., Firouz-Abadi, R.D., Mehralian, F.: Nonlinear responses of unbalanced flexible rotating shaft passing through critical speeds. Meccanica 57(1), 193–212 (2021). https://doi.org/10.1007/s11012-021-01447-8

Active Suppression of Multi-frequency Vibration of Rotor Based on Adaptive Immersion & Invariant Theory Xinyu Gao1,2 , Yifan Bao1,2 , Zeliang Zhang1,2 , and Jianfei Yao1,2,3(B) 1 School of Mechanical and Electrical Engineering, Beijing University of Chemical Technology,

Beijing 100029, China [email protected] 2 Beijing Key Laboratory of Health Monitoring and Self-Recovering for High-End Mechanical Equipment, Beijing University of Chemical Technology, Beijing 100029, China 3 Key Lab of Engine Health Monitoring-Control and Networking of Ministry of Education, Beijing University of Chemical Technology, Beijing 100029, China

Abstract. This paper describes an active suppression method based on Immersion and Invariance Theory (I&I) for the multi-frequency vibration of rotor in rotating machinery. The approach decomposes the multi-frequency excitation vectorially in Cartesian coordinate system. And the high-order non-linear system is immersed into the low-order system. AMA is used as the actuator in control system. The model of rotor-bearing-AMA system is established, integrating the model of unbalance and misalignment. A zero bias current strategy combined PD-I&I control algorithm is proposed by analyzing the influence of electromagnetic force nonlinearity. An adaptive notch filter is designed to meet the need of multi-speed control and cope with the influence of speed fluctuation. Finally, simulation and experimental research are carried out to verify and analyze the effectiveness of the proposed method to suppress multi-frequency vibration. Keywords: Vibration suppression · Multiple frequencies · Adaptive immersion and invariance theory · Active magnetic actuator

1 Introduction Rotating machinery operated in the conditions of high temperature, high pressure, high strength, high density energy and shock, such as aero-engine, gas turbine, steam turbine and large compressor unit, are core equipment in the industrial field. Rotor of rotating machinery has become a fault-prone part in the equipment system, because it is subjected to the action of various excitation, such as flow, solid, heat and other excitation sources. Rotor vibration failure seriously affects the normal operation of the unit, and is one of the key factors affecting the safe, efficient and long-term operation of large rotating machinery. If the excessive rotor vibration is not effectively suppressed in time, the vibration state will further deteriorate and finally lead to shutdown accidents and cause huge losses. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 54–70, 2024. https://doi.org/10.1007/978-3-031-40455-9_6

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Rotor of rotating machinery is frequently excited by periodic interference force in operation, and the excitation frequency is mostly related to rotor speed, presenting the characteristics of the same frequency, integer multiples and fractional multiples with the rotor speed. By analyzing the time/frequency domain characteristics of rotor vibration, the frequency component of rotor vibration can be taken as the vibration suppression target according to the analysis of the time/frequency domain characteristics of the rotor vibration. Then the appropriate control laws and actuators can be designed to realize active suppression of rotor vibration. More and more attention has been paid to the effective suppression of rotor vibration by active control devices during the operation of rotating machinery. The devices such as active suspension [1], bearing with variable geometry size [2], shape memory alloy spring [3], metal rubber [4], active tilting bearing [5], and integral squeeze film damper [6] have been used in active control of rotor vibration. The device mentioned above, however, produces a control force that is in direct contact with the rotor, which has a great influence on the characteristics of the rotor system. The magnetic bearing or electromagnetic actuator generates a non-contact electromagnetic force, which has no mechanical friction and no lubrication, and is more suitable for active vibration control of the rotor. Active magnetic bearings have been extensively studied in the field of rotor vibration control [7, 8] in recent years. R.Siva Srinivas et at. [8] has paid attention to the research status of magnetic bearings in the field of vibration suppression and condition monitoring, and summarized the research progress of magnetic bearings in the field of the suppression of rotors vibration from the views of function and vibration components of magnetic bearings. The magnetic bearing is used to support the rotor while compensating for the rotor imbalance and suppressing the synchronous vibration of the rotor [9–13]. Jin Zhou et al. [14] have proposed an identification method of closed-loop stiffness and damping coefficient of electromagnetic bearing based on unbalanced response and carried out experimental verification. Shaolin Ran et al. [15] have designed a flexible rotor test bench with active magnetic bearings, and developed a model-based robust controller for the first-order critical speed of bending through the rotor system. In terms of vibration suppression at different frequencies, Bernd Riemann [16], Alexander H. Pesch [17] and Wataru Tsunoda [18] have carried out in-depth research on oil film eddy and stability control of oil film oscillation of the rotor using active magnetic bearings and achieved a series of research results. Matthew O. T. Cole et al. [19] have realized parallel dynamic feedback control of multiple vibration frequencies of flexible rotor in magnetic bearing system through frequency matching control signal and carried out experimental verification. Kejian Jiang et al. [20] have proposed a multi-frequency periodic vibration compensation method for active magnetic bearing rotor system based on adaptive timedomain finite-length pulse response filter. Jie Yu et al. [21] also have proposed a method to suppress multi-frequency interference of a modulated self-sensing magnetic bearing rotor system. Nasser Abdul-Fadeel Abdul-Hameed Saeed et al. [22] have simulated the non-linear vibration suppression of Jeffcott rotor system using electromagnetic actuator and non-linear PD controller. Alireza Ebrahimi et al. [23] have studied rotor vibration control with cracks using electromagnetic actuators. Yao et al. have used active magnetic actuator (AMA) to suppress the multi-frequency vibration of rotor study [24, 25]. In

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a word, researchers have carried out in-depth theoretical and Experimental Research on active vibration suppression of rotor. There are many studies on active vibration suppression of rotor caused by single excitation sources such as rotor imbalance and oil film instability have been performed. There are however, a few studies on active control of rotor vibration with multi-frequency components caused by multiple excitation sources. “Immersion and invariant theory (I&I theory)” are adaptive control laws proposed by Astolfi [26, 27] for asymmetrical convergence of nonlinear systems. Lyapunov function does not need to be constructed for the I&I theory, or the control law based on Lyapunov method is a high-dimensional special case of the I&I theory. Immersion refers to a onedimensional target system in which the high-dimensional primitive system is immersed and converges asymmetrically at the origin. Invariant means that the designed control law can make the image of the target system possess invariable manifold attraction to satisfy the stability of the control system [28]. The I&I theory has good control robustness for nonlinear systems with unknown parameters or unmodeled dynamics and unknown quality. The parameter estimation based on the I&I theory and the design of control algorithm can be considered as two modules, which can be designed independently [29]. Since The I&I theory has been studied extensively since it was proposed in 2003. Yao et al. have designed a feedback controller based on the I&I theory to solve the problems of track tracking and vibration suppression of elastic drive system [30]. I&I theory is particularly used in the field of aircraft control. Liu et al. have proposed an output feedback control strategy based on I&I theory for speed and height tracking of suction super-high speed aircraft [31]. Lee et al. have established a state variable feedback control law based on I&I theory for spacecraft hovering control near asteroids [32]. Li et al. have designed a parameter estimator based on I&I theory for the uncertainty of aircraft model [32]. Liu Jiaqi et al. have combined the I&I theory with disturbance observer and proposed an adaptive robust I&I dynamic surface controller for magnetic suspension ball system from the theoretical and simulation perspectives [33, 34]. Chen et al. applied I&I theory to the field of active vibration suppression of rotor firstly. Unbalance parameters of rotor system were estimated in real time, and three-pole AMB structure was adopted to realize compensation suppression of rotor unbalance by combining I&I theory and sliding mode algorithm [35, 36]. This paper has proposed an active control method for multi-frequency vibration of rotor based on adaptive immersion and invariant theory and the convergence of the algorithm has been proved. The multi-frequency excitation force vector is decomposed by using transformation coordinate system method. AMA is used as the actuator in control system. The model of rotor-bearing-AMA system is established, integrating the model of unbalance and misalignment. An adaptive notch filter is designed to meet the need of multi-speed control and cope with the influence of speed fluctuation. A test rig of the rotor-bearing-AMA system is built. Finally, simulation and experimental research are carried out to verify and analyze the effectiveness of the proposed method to suppress multi-frequency vibration.

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2 Method 2.1 Model of Rotor-Bearing-AMA System The motion equation of rotor-bearing-AMA system is established by the finite element method: ¨ + CZ˙ + KZ = F1 + F2 MZ

(1)

where, M, C, K and Z are the mass matrix, the damping matrix, the stiffness matrix and the displacement column vector respectively. Equation 1 does not consider gyroscope effect. F1 is a generalized interference force vector, such as unbalanced excitation Fd and misalignment excitation Fc . F2 is the control force. Electromagnetic force is used as the control force here. A 12-pole E-type AMA with a y-direction structure is shown in Fig. 1, where g0 is a static air gap. β = 33° is the angle between the magnetic poles, then the resultant magnetic force in the y direction is expressed as [35]: ⎡ 2  − 2 ⎤ + iy i y y ⎦ − (2) Fmag = ka ⎣ g0 + y g0 − y where, ka = 49μ0 N 2 A20 (1 + cos β)/128g02 , μ0 is the vacuum permeability, y is the displacement of rotor in Y direction, N is the number of turns of the large coil, iy± is the coil current (iy± = i0 ± icy , where icy is the y-direction control current and i0 is the bias current), and A0 is the area of the large magnetic pole.

Fig. 1. Diagram of AMA structure (y direction)

The electromagnetic force is a binary quadratic nonlinear function of displacement and current in Eq. (2). To reduce the influence of electromagnetic force nonlinearity, the first-order Taylor expansion of Eq. (2) is linearized near the working point (0, 0) [36] : y

Fmag = −ks y + ki icy

(3)

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where, k s is displacement stiffness, k s = 4k a i0 2 /g0 3 , and k i is current stiffness, k i = 4k a i0 /g0 2 . It is assumed that the air gap g0 in the x and y directions is uniform, and the magnetic pole area A0 is the same. Moreover, the influence of magnetic leakage and magnetic field coupling is not considered. The current stiffness and displacement stiffness in the x and y directions can be equal. The Eq. (4) can be obtained. 



 x F x i e mag (4) = −ks + ki cx = −ks za + ki ic Fmag = y icy y Fmag Fmag in the local coordinate system is used as the control force, which is transferred to the global coordinate system. The expression of the control force F2 can be obtained. F2 = TTc [−ks Tc Z + ki ic ]

(5)

where, T c is the conversion matrix, ic is the control current, k s is the displacement stiffness, and k i is the current stiffness, respectively. Equation (5) is substituted into Eq. (1). Equation (6) can be obtained.

¨ + CZ˙ + K + ks TTc Tc Z = F1 + ki TTc ic MZ (6) The matrix K1 is defined, shown in Eq. (7). K1 = K + ks TTc Tc So, Eq. (6) can be expressed as Eq. (8).   −1 ¨ Z=M F1 + M−1 ki TTc ic − M−1 CZ˙ − M−1 K1 Z Equation (8) is transformed into a state-space equation, shown in Eq. (9).    



Z Z˙ 0 0 0 I T ¨ = −M−1 K1 −M−1 C Z˙ + M−1 ki Tc ic + M−1 F1 Z The state space vector is defined by Eq. (10).



 Z Z˙ q = ˙ , q˙ = ¨ Z Z

(7)

(8)

(9)

(10)

So, Eq. (9) can be expressed as Eq. (11). q˙ = As q + Bsa ic + Bsu F1

(11)

where, As is the system matrix, Bsa and Bsu are the input matrices of AMA current and interference force respectively. Equation (11) is the governing equation of the rotor-bearing-AMA system. The differential equation can be solved by ode series solver (e.g. variable step ode45) in simulation.

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2.2 Adaptive Immersion and Invariance Theory The local simplified non-linear system based on Eq. (1) is considered to design an adaptive I&I control law, as shown in Eq. (12). ⎤ ⎡ ⎤ ⎡ ⎡ ⎤ 0 x˙ x˙ ⎢ 0 ⎥ ⎢ Fxmag ⎥ ⎢ x¨ ⎥ e ⎥ ⎢ ⎥ ⎢ ⎥ M⎢ ⎣ y˙ ⎦ = F1 + M ⎣ y˙ ⎦ + ⎣ 0 ⎦ Fymag 0 y¨

(12)

where, M is the generalized mass and F1 e is the coupling interference force vector. Equation (12) is simplified to: ς˙ = g(ς )F(ς ) + h(ς )ξ + g(ς )u

(13)

where, h(ς ) and g(ς ) are the transformation matrix; ς, and ξ are the system state vector,  T ς = x, x˙ , y, y˙ , and ξ =[˙x, y˙ ]T ; u is the system input vector, u = [ux , uy ] T ; and F(ς ) is the generalized external interference force vector of the system. Equation (13) can be expressed as Eq. (14). ς˙ = f (x, t, θ ) + g(ς )u

(14)

where, θ is assumed unknown parameters, f (x, t, θ ) is an interference and unmodeled dynamic function. The real-time estimation of f (x, t, θ ) is one of the keys to control system design. A non-iterative estimation method based on S-G filter is proposed in the reference [35], as shown in Eq. (15): fest (t) = f (x, t − 1, θ ) + g(ς )u + g(ς )u(t)

(15)

where, the f est (t) is t for estimates of f (x, t, θ ). The control law is designed based on the adaptive I&I principle to realize active suppression of multi-frequency vibration of rotor. First, let θ * be the calibration parameter of the f (x, t, θ ) with respect to θ. If the calibration parameters θ * are known, then there is a control law (Eq. (16)) to make the closed loop system (Eq. (17)) converge asymptotically at the original point [36]. u = (ς, t, θ ∗ )

(16)

ς˙ = f (ς, t, θ ∗ ) + g(ς )(x, t, θ ∗ )

(17)

The functions f (x, t, θ ) in Eq. (14) is expressed as a linearly independent form with respect to θ as shown in Eq. (18). f (ς, t, θ )=f0 (ς ,t)+f1 (ς ,t)θ where, f0 (ς, t) = [˙x, 0, y˙ , 0]T .

(18)

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An adaptive compensation algorithm based on I&I principle is designed to estimate unknown parameters in real time. The essence of immersion principle is that the highorder system of AMA based on I&I adaptive compensation PD algorithm with unknown parameter θ is immersed into the low-order system of AMA based on PD algorithm with known calibration parameter θ * . The manifold attractiveness and trajectory boundedness [36] are also required in the control law design. There exist systems (Eq. (19–20)) whose states are bounded and satisfies the conditions of Eq. (21).   ς˙ = f (x, t, θ ∗ ) + g(ς ) (x, t, z + θ ∗ ) − (x, t, θ ∗ ) (19)

 ∂β1 (ς, t) Z˙ = − f1 (ς, t) z ∂ς

(20)

  lim g(ς (t)) (ς (t), t, z(t) + θ ∗ ) − (ς (t), t, θ ∗ ) =0

(21)

t→∞

The function β 1 (ς, t) can be constructed by Eq. (20). The function β2 (ς, t, θˆ ) can be designed, as shown in Eq. (22) [37]: β2 (ς, t, θˆ ) = −

   ∂β1 (ς, t)  f0 (ς ) + f1 (ς ) θˆ + β1 (ς, t) + g(ς )u ∂ς

(22)

Finally, a control law based on adaptive I&I principle can be obtained: u = (x, t, θˆ + β1 (ς, t))

(23)

θ˙ˆ = β2 (ς, t, θˆ )

(24)

where, θˆ is an adaptive variable, θˆ + β1 (ς, t) is an online estimate of the unknown parameter θ. 2.3 Design of Controller The multi-frequency component vibration of rotor is taken as the suppression target. The parameter θ and the function f (x, t, θ ) are evaluated in real time using the adaptive I&I principle to design the controller for suppressing the multi-frequency vibration of rotor. Firstly, the vector force F(ς ) with multi-frequency component is decomposed according to the order of the multi-frequency vibration of rotor, as shown in Eq. (25). ⎤ ⎡ n      F cos(i × t + ϕ ) n i ⎥  ⎢  i ⎥  i (ς ) = ⎢ i=0 (25) F F(ς ) =   n ⎦ ⎣    Fi  sin(i × t + ϕi ) i=0 i=0

 i represents the vector force with i times the where, i ∈ N is a non-negative integer,F      i . ϕ i is the frequency component, Fi  represents the magnitude of the vector force F initial phase angle of the vector force and Ω is the rotating speed of the rotor, respectively.

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 i with multi-frequency components Fig. 2. The decomposition diagram of the vector F

Figure 2 shows the decomposition diagram of the vector forces for each frequency component. A Cartesian transformation coordinate system x i Oyi with rotation speed and time is constructed. The vector forces in fixed coordinate system x o Oyo are transformed to the rotating coordinate system x i Oyi , as shown in Eq. (26).

   T cos(i × t) − sin(i × t) ai  = R(i × , t) ai bi Fi (ς ) = (26) bi sin(i × t) cos(i × t)         where, ai = F i  cos ϕi , bi = F i  sin ϕi , and ϕ i = arctan(bi /ai ). Then, the term of f1 (ς ,t) θ in Eq. (18) is split by the frequency component of the interference force, as shown in Eq. (27).  f (i) (ς, t, θ )=f0(i) (ς ,t)+f1(i) (ς ,t)θi× , (i ∈ N) (27) (i)  i (ς ) = g(ς )R(i × , t)θi× f1 (ς, t)θi× = g(ς )F where, the expression of θ i×Ω is shown in Eq. (28). T  θi× = ai bi

(28)

Subsequently, the design of the controller is performed and the convergence of  i of each frequency the controller is proved after decomposing the vector forces F multiplication with the transformation coordinate system. The proposed method can therefore be applied in the control of multi-frequency vibration of rotor. The block diagram of the proposed method is shown in Fig. 3.

3 Experiments 3.1 Description of Test Rig Test rig of the rotor-bearing-AMA system is mainly composed of rotor, tilting-pad sliding bearing, AMA, coupling, sensors and motor (Fig. 4). The temperature and operating pressure of the lubricating oil ISO VG32 for bearing are 20°C and 0.2 MPa respectively.

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Fig. 3. The block diagram of active control of multi-frequency vibration based on adaptive immersion and invariant theory

There are 6 displacement probes installed in the test rig to measure real-time rotational speed and rotor vibration. The sampling rate of control system is f s = 10.24 kHz. The parameters of the test rig are shown in Table 1.

Fig. 4. Rotor-Bearing-AMA System Test rig

3.2 Results at Multi Speeds The original vibration of the rotor in test rig is mainly composed of 1X frequency and 2X frequency components (as shown in Fig. 17). The 1X and 2X frequency components of vibration of rotor are therefore taken as the suppression targets in the experiment of active suppression of multi-frequency vibration. That is, the parameter n is 2. The parameters of the proposed algorithm are shown in Table 2. The PD-I&I algorithm and the PD algorithm were respectively performed for rotor vibration suppression at the speeds of 1050 rpm, 1200 rpm, 1350 rpm, 1500 rpm, 1650 rpm, 1800 rpm and 1950 rpm. In this

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Table 1. Parameters of test rig

AMA

Rotor

Bearings

Coupling

Parameters

Values

Magnetic pole area (A)

10–4 m2

Turns of large coils(N)

100

Turns of small coils

75

Rotor diameter

125 mm

Stator diameter

126.2 mm

Air gap(g0 )

0.6 mm

Length of shaft

1m

Outer diameter

50 mm

Thickness of disc

25 mm

Diameter of disc

270 mm

Number of pads

5

Preloading

0.3

Clearance

0.05 mm

Half coupling length

65 mm

case, a vibration suppression rate Q is proposed to describe the effect of rotor vibration suppression with and without control in the experiment. The parameters Amp (V b ) and Amp (V a ) are the amplitudes of vibration without and with control in the period of T n , receptively. The vibration suppression rate Q is defined as shown in Eq. (29). Q=

Amp(Vb ) − Amp(Va ) Amp(Vb )

(29)

Table 2. The parameters of the algorithms Parameters

K px

PD

104

PD-I&I

104

K dx

K py

K dy

h

10

1.3 × 104

13

——

10

1.3 × 104

13

3 × 10–5

For the amplitude comparison experiment of point #1 in the y-direction with and without control. The PD algorithm based on the I&I principle (PD-I&I) has a suppression rate Q of 45.5%~59%, while the suppression rate Q of the PD algorithm is only 18.7% ~ 35.0%. So, it is obviously that the suppression effect of the PD-I&I algorithm is significantly better than the PD algorithm. Moreover, the robustness of PD-I&I algorithm is also significantly stronger than that of traditional PD algorithm with the increase of the rotation speed.

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(a) Waveform of vibration in Y-direction

(c) Shaft orbits of rotor vibration with and without control

(b) Coil current in Y-direction

Fig. 5. Experimental results of the measurement #1 at 1350 rpm

The red and blue lines in (c) of Fig. 5 represent the motion trajectories of the control front and rear axes, respectively. The following is the same as above. The waveform, output coil current iy ± and shaft orbits of rotor vibration with and without control at 1350 rpm, 1650 rpm and 1950 rpm are shown in Fig. 5, Fig. 6 and Fig. 7, respectively. The PD-I&I controller is activated at about 5 s for 1350 rpm and at about 10 s for 1650 rpm and 1950 rpm. In Fig. 5 (a), 6(a) and 8(a), the time-domain amplitude of rotor vibration is effectively suppressed with PD-I&I and PD controller. The shaft orbits of rotor vibration are closer to the origin from Fig. 5 (c), 6(c) and 7(c). Distinctly, the suppression effect of PD-I&I control algorithm is better than that of PD control algorithm. The experimental results shown in Fig. 5, Fig. 6 and Fig. 7 also show that the effectiveness of the PD-I&I algorithm proposed in this work for the suppression of the multi-frequency vibration of rotor in time-domain. Figures 8, 9 and 10 show the waterfall diagrams of rotor vibration of the measuring point #1 at the speeds of 1350 rpm, 1650 rpm and 1950 rpm with control and without control, respectively. Obviously, the 1X frequency and 2X frequency components of vibration are suppressed with PD-I&I and PD controller in Figs. 8–10. The PD-I&I control algorithm is better than the PD control algorithm in suppressing multi-frequency vibration components. The effectiveness of the PD-I&I control algorithm designed in this paper is proved in frequency domain. Figures 11 (a) and (b) are frequency domain waterfall diagrams of vibration of the measuring point #2 in y-direction (near the bearing #1) with and without control at the speeds of 1350 rpm and 1950 rpm, respectively. It can be found that the rotor vibration at the measuring point#2 decreases significantly when the active control force is applied by AMA. The effectiveness of the proposed method for the suppression of multi-frequency vibration of rotor is clearly at different measuring points.

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(a) Y-direction time-domain waveform

(c) Shaft orbits of rotor vibration with and without control (b)Y-direction output current

Fig. 6. Experimental results of the measurement #1 at 1650 rpm

(a) Y-direction time-domain waveform

(c) Shaft orbits of rotor vibration with and without control (b)Y-direction output current

Fig. 7. Experimental results of the measurement #1 at 1950 rpm

3.3 Comparison of PD-I&I and PD Algorithms with Different Parameters The effects of PD-I&I and PD algorithms on rotor vibration suppression are compared with different parameters. The control algorithm parameters are shown in Table 3. Figure 12 shows the comparison of PD-I&I and PD algorithms on the amplitude of rotor vibration in y-direction at the measuring point #1 at multiple speeds. The vibration suppression rate of the PD-I&I algorithm is significantly higher than that of the PD

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(a) Before filtering

(b) After filtering

Fig. 8. Waterfall diagram of the measurement point #1 in y direction at 1350 rpm

(a) Before filtering

(b) After filtering

Fig. 9. Waterfall diagram of the measurement point #1 in y direction at 1650 rpm

(a) Before filtering

(b) After real-time filtering

Fig. 10. Waterfall diagram of the measurement point #1 in y direction at 1950 rpm

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(a) 1350 rpm

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(b) 1950 rpm

Fig. 11. Waterfall diagrams of the measurement point #2 at 1350 rpm and 1950 rpm

Table 3. Parameters of algorithms Parameters

K px

K dx

K py

K dy

h

PD

104

10

2 × 104

20

——

PD-I&I

104

10

1.3 × 104

13

3 × 10–5

algorithm, and the maximum difference between them is 17.55%, and the minimum difference is 3.14% for various speeds. It can be seen from Fig. 12 that the vibration suppression of PD-I&I control algorithm is still better than that of PD control algorithm, under various speeds with different parameters.

Fig. 12. Comparison of PD-I&I and PD algorithms at different speeds

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It can however be seen that the suppression rate Q of the proposed algorithm also decreases when the speed is greater than 1650 rpm. Therefore, the PD-I&I control algorithm proposed in this work still needs further improvement.

4 Conclusion An active suppression method of multi frequency vibration rotor is proposed based on the adaptive immersion and invariance theory in this work. The multi frequency vector force is decomposed and synthesized by the transformation coordinate system. The model of rotor-bearing-AMA system is established, integrating the model of unbalance and misalignment. The convergence of the proposed control algorithm is proved. A zero bias current PD-I&I control algorithm is designed by analyzing the influence of electromagnetic force nonlinearity. Finally, simulation and experimental research are carried out to verify and analyze the effectiveness of the proposed method to suppress multi-frequency vibration. The following conclusions are drawn. (1) The PD control algorithm based on the immersion and invariance principle is designed for the multi frequency excitation of the rotor system at multiple speeds. It has an excellent effect on the suppression of the multiple frequencies vibration of the rotor. The proposed method can theoretically suppress the higher frequency components of the vibration by increasing the frequency order n of the controller. It will however increase the real time of the computation. (2) The PD-I&I algorithm proposed in this work has better robustness and suppression effect than the traditional PD algorithm on the suppression of the multi-frequency vibration of rotor. (3) The zero bias current PD-I&I control algorithm designed in this work can also overcome the shortcomings of electromagnetic nonlinearity. (4) The proposed method can provide methodological support for active vibration suppression of rotor in rotating machinery.

References 1. Arias-Montiel, M., Silva-Navarro, G., Antonio-Garcia, A.: Active vibration control in a rotor system by an active suspension with linear actuators. J. Appl. Res. Technol. 12, 898–907 (2014) 2. Chasalevris, A., Dohnal, F.: A journal bearing with variable geometry for the suppression of vibrations in rotating shafts: Simulation, design, construction and experiment. Mech. Syst. Signal Process. 52–53, 506–528 (2015) 3. Enemark, S., Santos, I.F.: Rotor-bearing system integrated with shape memory alloy springs for ensuring adaptable dynamics and damping enhancement-Theory and experiment. J. Sound Vibration, 369, 29–49 (2016) 4. Yan-hong, M.A., Zhang, Q.-C., Zhang, D.-Y., et al.: Tuning the vibration of a rotor with shape memory alloy metal rubber supports. J. Sound Vibration, 351, 1–16 (2015) 5. Salazar, J.G., Santos, I.F.: Active tilting-pad journal bearings supporting flexible rotors: part ll-the model-based feedback-controlled lubrication. Tribol. Int. 107, 106–115 (2017)

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6. Ferraro, R., Catanzaro, M., Kim, J., et al.: Suppression of subsynchronous vibrations in a 11 MW steam turbine using integral squeeze film damper technology at the exhaust side bearing. Proceedings of ASME Turbo Expo 2016, GT2016–57410 7. Siva Srinivas, R., Tiwari, R., Kannabalu, C.: Application of active magnetic bearings in flexible rotordynamic system-a state-of-the-art review. Mech. Syst. Signal Process. 106, 537–572 (2018) 8. Schweitzer, G., Maslen, E.H.: Magnetic Bearings Theory, Design, and Application to Rotating Machinery. Springer-Verlag Berlin Heidelberg (2009) 9. Hutterer, M., Kalteis, G., Schrodl, M.: Redundant unbalance compensation of an active magnetic bearing system. Mech. Syst. Signal Process. 94, 267–278 (2017) 10. Chen, Q., Liu, G., Han, B.: Unbalance vibration suppression for AMBs system using adaptive notch filter. Mech. Syst. Signal Process. 93, 136–150 (2017) 11. Chuan, M., Changsheng, Z.: Unbalance compensation for active magnetic bearing rotor system using a variable step size real-time iterative seeking algorithm. IEEE Trans. Industrial Electron. 65(5), 4177-4186 (2018) 12. Xiang, M., Wei, T.: Autobalancing of high-speed rotors suspended by magnetic bearings using LMS adaptive feedforward compensation. J. Vib. Control 20, 1428–1436 (2014) 13. Fang, J., Xu, X., Xie, J.: Active vibration control of rotor imbalance in active magnetic bearing systems. J. Vib. Control 21, 684–700 (2015) 14. Zhou, J., Di, L., Cheng, C., et. al.: A rotor unbalance response based approach to the identification of the closed-loop stiffness and damping coefficients of active magnetic bearings. Mech. Syst. Signal Process. 66–67, 665–678 (2016) 15. Ran, S., Yefa, H., Huachun, W.: Design, modeling, and robust control of the flexible rotor to pass the first bending critical speed with active magnetic bearing. Adv. Mech. Eng. 10(2), 1–13 (2018) 16. Riemann, B., Perini, E.A., Cavalca, K.L., et. al.: Oil whip instability control using μ-synthesis technique on a magnetic actuator. J. Sound Vib. 332, 654–673 (2013) 17. Pesch, A.H., Sawicki, J.T.: Stabilizing hydrodynamic bearing oil whip with μ-synthesis control of an active magnetic beaing. Proceedings of ASME Turbo Expo 2015 (2015), GT2015–44059 18. Tsunoda, W., Hijikata, W., Shinshi, T., et. al.: Utilizing a radial magnetic bearing to stabilize self-excited vibrations in a rotor-oil film bearing system. In: The proceedings of 15th International Symposium on Magnetic Bearings, pp. 239–244 (2015) 19. Cole, M.O.T., Keogh, P.S., Burrows, C.R.: Robust control of multiple discrete frequency vibration components in rotor-magnetic bearing systems. JSME Int. J. 46(3), 891–899 (2003) 20. Jiang, K., Changsheng, Z.: Multi-frequency periodic vibration suppressing in active magnetic bearing-rotor systems via response matching in frequency domain. Mech. Syst. Signal Process. 25(4), 1417–1429 (2011) 21. Yu, J., Changsheng, Z.: A multifrequency disturbances identification and suppression method for the self-sensing AMB rotor system. IEEE Trans. Industrial Electron. 65(8) (2018) 22. Saeed, N.A.F.A.H., Kamel, M.: Nonlinear PD-controller to suppress the nonlinear oscillations of horizontally supported Jeffcoott-rotor system. Int. J. Non-Linear Mech. 87, 109–124 (2016) 23. Ebrahimi, A., Heydari, M., Behzad, M.: Optimal vibration control of rotors with an open edge crack using an electromagnetic actuator. J. Vib. Control 24(1), 37–59 (2018) 24. Jianfei, Y., Jinji, G., Weimin, W.: Multi-frequency rotor vibration suppressing through selfoptimizing control of electromagnetic force. J. Vib. Control 23(5), 701–715 (2017) 25. Yao, J.-F., Dai, J.-B., Liu, L., Yang, F., Gao, S.-C.: Active vibration suppression of rotor unbalance through an adaptive control method based on self-adjusting of PD gain. J. Vib Test. Syst. Dyn. 2(1), 69–81 (2018) 26. Astolfi, A., Ortefa, R.: Immersion and Invariance: a new tool for stabilization and adaptive control of nonlinear systems. IEEE Trans. Autom. Control 48(4), 590–606 (2003)

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Simulation Study for Hole Diaphragm Labyrinth Seal at Synchronous Whirl Frequency Xiang Zhang, Renwei Che, Yinghou Jiao(B) , and Huzhi Du School of Mechanical Electronic Engineering, Harbin Institute of Technology, Heilongjiang Province, Harbin 150001, China [email protected]

Abstract. Labyrinth seals are widely used in fluid mechanics due to their simple structure. However, better leakage performance and good rotordynamic stability cannot be obtained at the same time in traditional labyrinth seals (TLS). A new type of hole diaphragm labyrinth seal (HDLS) was proposed by our previous research that combined the advantages of the TLS and the damping seal. Further flow analyses of working gas in HDLS at synchronous whirl frequency under different preswirl rates were studied in this paper. The computational fluid dynamics method was used to obtain the flow details of the HDLS. The local pressure and circumferential velocity at the central panel of the first cavity were compared. The results show that the effective stiffness decreases with increasing preswirl rate, while the opposite is true for effective damping. The maximum pressure gradient in the circumferential cavity is observed at a positive 1 preswirl rate. A stronger low-pressure vortex core appears under negative preswirl conditions, which causes at most a 0.8% reduction in leakage. Reduction of circumferential velocity in the holes also decreases the circumferential flow of working gas and enhances the damping effect of HDLS. Current studies on hole diaphragm labyrinth seals show better performance under slightly negative preswirl conditions. Keywords: Labyrinth Seal · Damping Seal · Circumferential Velocity · Preswirl Rate

1 Introduction Inlet preswirl technologies were first proposed in annular seal studies and extended into labyrinth seal applications in the first decade of the twenty-first century [1, 2]. Different performances of positive and negative preswirls were found in the studies [3, 4]. In most studies, a negative preswirl presents better stability performance [5], while there is research showing an advantage of a positive preswirl used in short labyrinth seals by Zhang [3]. Because the hole diaphragm labyrinth seal (HDLS) is one kind of combination seal designed by combining the labyrinth seal (LS) and damper seal (DS), a similar performance of LS can be obtained by HDLS [6]. Previous studies of HDLS showed a promising future in applications in gas turbine and steam turbine areas [6, 7].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 71–77, 2024. https://doi.org/10.1007/978-3-031-40455-9_7

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Fig. 1. Diagram of hole diaphragm labyrinth seal [5]

In this paper, a series of preswirl conditions were used to study the performance of a hole diaphragm labyrinth seal (HDLS) at a synchronous whirl frequency. The computational fluid dynamics (CFD) method was used to perform the calculations. Different preswirl rates conditions were considered. Detailed physical variables in the circumferential direction were discussed. Changes in the inner flow under different preswirl rates further explained the mechanism of using the preswirl conditions.

2 Model Establishment As a relatively mature method, the CFD method has been widely used in fluid mechanics research [8–10]. A quasi-steady state model of the rotor-seal system was also used in the calculation [3, 11]. Because the mesh of the HDLS in this paper is same as that in our previous work [5], grid verification can be omitted here. An 8.93 million element grid was used to perform the calculations under synchronous whirl frequency conditions. Detailed structural parameters and other calculation conditions are listed in Table 1 and Table 2. Table 1. Design parameters of the HDLS Parameters name

Values

Rotor radius/mm

30

Radial clearance cr /mm

0.2

Cavity depth h/mm

3.5

Cavity width l2 /mm

3.8

Blade thickness t/mm

0.25

Blade space width l1 /mm

2.3

Diaphragm width θ/°

2

Diagram of holes φ/mm

1

3 Results and Discussion According to Zahorulko’s research [12], effective stiffness and effective damping coefficients can be easily obtained by reading the CFD data. Rotordynamic coefficients and leakage of HDLS at different preswirl rates can be drawn as shown in Fig. 2.

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Table 2. Calculation conditions Parameters’ name

Values

Turbulence model

Standard k-ε

Inlet pressure/MPa

0.69

Outlet pressure/MPa

0.1

Preswirl rate

-1, -0.75, -0.5, -0.25, 0, 0.25, 0.5, 0.75, 1

Whirl frequency / Hz

50

Rotation speed ω/ rpm

3000

Maximum flow difference /%

0.01

Residual control

≤ 10−5

It can be observed easily in Fig. 2(a) that the effective stiffness, which is positively related to rotor stability, decreases when the preswirl rate increases. The relationship between the effective stiffness and preswirl rate is almost linear. The effective damping coefficient in Fig. 2(b) shows a different stability trend compared with the effective stiffness, and rotor stability increases with increasing preswirl rate if only considering the effective damping coefficient. Leakage of the HDLS increases with increasing preswirl rate. The effective stiffness in the HDLS shows the same trend in most studies that the negative preswirl rates mostly play a positive role in rotor stability, which is different from Zhang’s research [3]. However, the calculated structure in Zhang’s research is an original labyrinth seal with 5 stages fins, which is different from the current structure in this paper. To further understand the different preswirl influences on the stiffness and damping coefficients in the HDLS working at a synchronous whirl frequency, a local pressure contour map at the center of the first cavity can be drawn, as shown in Fig. 3. The local pressure in the circumferential direction in the HDLS is different from that in the traditional labyrinth seal. Complete pressure gradients can be observed at most preswirl rates in Fig. 3(a), 3(b), 3(c) and 3(d). The distance of the nearby contour lines in circumferential cavities presents little change at negative preswirl rate conditions, as shown in Fig. 3, while it presents more influence at positive preswirl rate conditions, especially at a positive 1 preswirl rate. The pressure difference at the different sides of the holes is clearly enough to produce a jet flow in the hole area. However, the increase in the pressure gradient in circumferential cavities seems to reduce the pressure differences between the two sides of the holes. The maximum pressures in the two down circumferential cavities in Fig. 3(d) and 3(e) are much smaller than those in Fig. 3(a), 3(b) and 3(c). Besides, the seal force, which is caused by the circumferential unbalanced pressure, appears a rotation with preswirl rate increasing. It can be observed in Fig. 3(b) and 3(e) that tangential component Ft , which is positively related to cross-coupled stiffness, increases with preswirl rate increasing. However, the radial component Fr , which is

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Fig. 2. Rotordynamic coefficients and leakage performance of HDLS at different preswirl rates

positively related to effective stiffness, decreases at the same time. Force changes in the circumferential direction explain the different changing tendencies for effective stiffness and effective damping in Fig. 2(a) and 2(b). Corresponding streamlines and circumferential velocity contour maps of HDLS in the center panel of the first cavity, as shown in Fig. 4, verified this inference. The low-pressure vortex core tends to disappear as the preswirl rate increases. The whole dissipation effect of the vortexes decreases even when the high-pressure reflux becomes much stronger at a 1 preswirl rate than at a -1 preswirl rate. Such a phenomenon causes the increase in leakage in Fig. 2(c). Meanwhile, the circumferential velocity obtained its minimum values at the positive 1 preswirl rate condition, which actually reduces the circumferential flow of the working gas and enhances the damping effect of the HDLS, as shown in Fig. 2(b).

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Fig. 3. Local pressure contour map at the center of the first cavity of the HDLS

Fig. 4. Streamlines and circumferential velocity contour map of HDLS in the hole areas

4 Conclusions In this paper, the rotordynamic characteristics and leakage performance of hole diaphragm labyrinth seal (HDLS) at synchronous whirl frequencies under different preswirl rates were studied. The CFD method was used to perform the calculations. The local pressure and circumferential velocity at the center panel of the first cavity were compared. Several results were obtained.

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Leakage of HDLS increases with preswirl rates increasing. The effective stiffness of the HDLS decreases with increasing preswirl rate; however, the effective damping increases at this time. Analyses on flow details explain the increase of leakage and effective damping because of the reducing of vortex dissipation and the rotation of seal force. Such different changing trends cannot usually be observed in one type of seal. The maximum pressure gradient in the circumferential cavity is observed at a positive 1 preswirl rate. A phenomenon similar to pressure gradient conservation, which shows a much smaller gradient, was observed in the hole areas. A larger circumferential velocity was obtained at the negative preswirl rate condition, which causes the corresponding stronger low-pressure vortex behind the outlet of the hole region. This causes at most a 0.8% reduction in leakage. Meanwhile, reduction of circumferential velocity in the hole region actually decreases the mass transfer of working gas and enhances the damping effect of HDLS. This eventually causes an abnormal increase in effective damping when the preswirl rate increases. Acknowledgements. This work was supported by the National Natural Science Foundation of China (No.12072089, No.11972131).

References 1. Hendricks, R.C., Tam, L.T., Muszynska, A.: Turbomachine Sealing and Secondary Flows: Part 2: Review of Rotordynamics Issues in Inherently Unsteady Flow Systems With Small Clearances. NASA (2004) TM—2004–211991 2. Moore, J.J.: Three-dimensional CFD rotordynamic analysis of gas labyrinth seals. J. Vib. Acoustics-Trans. ASME 125, 427–433 (2003) 3. Zhang, W., Gu, Q., Cao, H., Wang, Y., Yin, L.: Improving the rotordynamic stability of short labyrinth seals using positive preswirl. J. Vibroeng. 22, 1295–1308 (2020) 4. Li, Z., Li, J., Feng, Z.: Numerical comparisons of rotordynamic characteristics for three types of labyrinth gas seals with inlet preswirl. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 230, 721–738 (2016) 5. Zhang, X., Jiao, Y.H., Qu, X.Q., Zhou, J.H., Zhao, Z.Q.: Rotordynamic analysis and leakage performance study of a hole diaphragm labyrinth seal using the CFD method. Alex. Eng. J. 61, 9921–9928 (2022) 6. Zhang, X., Jiao, Y., Qu, X., Zhao, Z., Huo, G., Huang, K.: Inlet preswirl dependence research on three different labyrinth seals. Tribol. Int. 176, 107929 (2022) 7. Zhang, X., Jiao, Y.H., Qu, X.Q., Huo, G.H., Zhao, Z.Q.: Simulation and flow analysis of the hole diaphragm labyrinth seal at several whirl frequencies. Energies 15, 379 (2022) 8. Zhang, W.F., Gu, C.J., Yang, X.C., Wu, K.X., Li, C.: Effect of hole arrangement patterns on the leakage and rotordynamic characteristics of the honeycomb seal. Propulsion Power Res. 11, 181–195 (2022) 9. Savvakis, S., Mertzis, D., Nassiopoulos, E., Samaras, Z.: A design of the compression chamber and optimization of the sealing of a novel rotary internal combustion engine using CFD, Energies 13 (2020) 10. Li, Z.G., Li, J., Feng, Z.P.: Numerical Investigations on the Leakage and Rotordynamic Characteristics of Pocket Damper Seals-Part II: Effects of Partition Wall Type. Partition Wall Number, and Cavity Depth, Journal of Engineering for Gas Turbines and Power-Transactions of the ASME 137, 032504 (2015)

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11. Yan, X., He, K., Li, J., Feng, Z.: Numerical techniques for computing nonlinear dynamic characteristic of rotor-seal system. J. Mech. Sci. Technol. 28(5), 1727–1740 (2014). https:// doi.org/10.1007/s12206-014-0318-9 12. Zahorulko, A.V., Lee, Y.-B.: Computational analysis for scallop seals with sickle grooves, part II: Rotordynamic characteristics. Mech. Syst. Signal Process. 147, 107154 (2021)

Effect of Flow Rate on the Performance of an Evacuated Tilting Pad Journal Bearing: Load on Pad vs. Load-Between-Pad Configurations Luis San Andrés1(B) and Andy Alcantar2 1 J. Mike Walker ’66 Department of Mechanical Engineering, Texas A&M University College

Station, Houston, TX 77843, USA [email protected] 2 Bearing Group Miba Industrial Bearings U.S. (Houston) LLC, Houston, TX 77536, USA [email protected]

Abstract. Reducing the oil flow supplied to a fluid film bearing decreases the drag power loss and increases efficiency, though with increased pad temperatures and a drop in damping. Savings in pumping and lubricant storage make the case for low flow. The paper reports further measurements of performance in a tilting pad journal bearing with evacuated ends and configured as load on pad (LOP) and load between pads (LBP). The supplied flow (Q) ranges from 150% to 25% of a nominal flow rate (Qs ). Operation is at a constant shaft speed of 6 krpm and with unit loads as large as 2.07 MPa. Both bearings operate slightly more eccentrically as Q < Qs , the LBP bearing displacing more. Pad temperatures are similar for both bearings, although the LOP bearing runs cooler and its oil exit temperature is also lower as Q decreases. The bearings main stiffness (K yy ) increases with load, the LOP bearing producing 25% more stiffness that increases as Q decreases. The LOP bearing produces small K xx that quickly decreases for Q = 25% Qs . The LBP bearing does produce significant K xx ~ K yy , both slightly affected by flow reduction. The main damping C yy for the LBP bearing drops quickly for the lowest Q, whereas C yy for the LOP bearing remains unchanged. Similarly, C xx for the LBP bearing varies little with Q decreasing, while for the LBP bearing, C xx → 0 for the lowest Q. The measurements show the LBP bearing performs better at low flows whereas the LOP bearing losses both stiffness and damping for the lowest Q. At this low flow, the bearing produced low frequency shaft motions (SSV Hash). Keywords: Tilting Pad Bearing · Rotordynamic Force Coefficients · Stability

1 Introduction Modern operation of tilting pad journal bearings (TPJBs) demands of reduced flow rates while keeping the pad temperatures within an acceptable limit for the Babbitt material ( K xx ) and damping (C yy ~ C xx ) coefficients increase with both unit load (along Y ) and shaft speed. Reduced flow rates Q → ½ Qs do not significantly affect the bearings force coefficients. However, for very low flow rates, Q ~ ¼ Qs and lesser, there is a precipitous loss in both stiffness and damping coefficients. The LOP bearing performs the worst since both K xx and C xx → 0 as Q < < Qs . SSV Hash emerged for extreme starved flow conditions and low loads. Please refer to the References cited [4–8]1 and students theses [9, 10] for more details on the experimental campaigns, the major results and a thorough discussion of the findings. This paper reports further measurements of performance in a four-pads TPJB with evacuated ends, and configured as load on pad (LOP) and load between pads (LBP). The measurements complement those in Ref. [8] and compare the performance of a LOP bearing vs. that of a LBP bearing as the supplied flow rate varies from 1.5 Qs to ¼ Qs.

2 Description of Test Rig and Bearings Figure 1 portrays the test rig used for the force evaluation, static and dynamic, of hydrodynamic fluid film bearings. Earlier references [4–8] fully describe the test rig, its major components and ancillary systems. A 40kW air turbine drives a solid rigid rotor supported on high-precision rolling element bearings housed in massive steel pedestals. A 1 References [5] and [6] received best paper awards by the Structures and Dynamics Committee

of the ASME International Gas Turbine Institute in 2022 and 2023, respectively.

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split-parts bearing stator carries the test bearing, both installed in the middle section between the supporting rolling element bearings. An external system supplies ISO VG 46 oil to the test bearing at a constant inlet temperature T in = 60 °C, and a turbine type flow meter measures the lubricant volumetric supplied Q. Long rods (pitch stabilizers) with very low stiffness, hold the bearing stator assembly which includes ports for oil exit, as well as instrumentation for measurement of accelerations and displacements relative to the rotor along two orthogonal directions (X, Y ).

Fig. 1. Side view of test rig and list of major components.

Figure 2 depicts the evacuated ends TPJB with oil supplied through spray bars. Individual separate arcuate plates retain each of the Babbitted pads constructed from AISI 1018 steel. The shaft diameter D = 101.77 mm and the bearing has four pads, each with axial length L = 67 mm and thickness t = 19 mm, and weighing 635 g. A pad is 72° in arc length and rests on a spherical pivot located at 50% pad offset. Table 1 lists other important bearing geometry and materials.

Fig. 2. Photographs of front and back sides of test bearing with evacuated ends.

Effect of Flow Rate on the Performance of an Evacuated Tilting Pad Journal Bearing Table 1. Four pad TPJB geometry and materials, Ref. [8] C p = 134 μm = C r (1 + r)

Pad Clearance Pad Preload, r

r = 0.30

Bearing Room T clearance

C rc = 115 μm measured

— Hot (After Test at 6 krpm)

C rh = 106 μm measured

Spray bar (5 mm from pad edge)

Five orifices, diameter = 5/64 inch

Lubricant

ISO VG 46

Supply Temperature, T s

60 °C

Viscosity (μ) at T s

16.43 cPoise

Density (ρ) at T s

838 kg/m3

Specific heat (cP )

2.08 kJ/(kg °C)

Viscosity-temperature Coefficient

0.0369 1/°C

(a) LOP bearing

(b) LBP bearing

Fig. 3. Schematic views of load on pad (LOP) and load between pads (LBP) bearings.

Fig. 4. Schematic depiction of thermocouples location for the LBP bearing.

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3 Experimental Procedure The experiments aim to quantify the performance of the test bearings, LOP and LBP, to significant variations in the supplied lubricant (Q), from 150% of a nominal condition to 25% (or less). In the tests, the rotor angular speed  = 6 krpm and the applied load ranged from 2.13 kN to 12.8 kN. The specific load (W/ (LD)) ranged from 0.345 MPa to 2.07 MPa for the LBP bearing, and 0.345 MPa to 1.38 MPa for the LOP bearing. At 6 krpm, the rotor surface speed U = ½ D  = 32 m/s, and the nominal flow rate (Qs ) is proportional to the number of pads, ½ U, the pad axial length L, the bearing clearance C r , and an empirical factor (λ < 1) that denotes the fraction of the flow leaving a pad trailing edge that enters the downstream pad at its leading edge. That is, Qs is the flow that must fully wet (fill in) the bearing clearance at a no-load condition (W = 0). Presently, as determined from prior tests in Ref. [10], Qs = 14.4 L per minute (LPM). Do realize the nominal supplied flow is only a function of shaft speed and operating clearance, and has no direct relation to the lubricant viscosity. Prior art [11] details the dynamic load measurement procedure along with the identification process carried in the frequency domain. The bearing and its stator are represented as a lumped mass (M BC ) that displaces along two degrees of freedom (X,Y ), as shown in Fig. 3. The stator elastic supports and the test bearing produce reactions to external forces applied by the shakers. The dynamic forces encompass an ensemble of loads with pseudo-random amplitude and frequencies, typically 10 Hz. First,  spaced f k eiωkt , f y = 0]T that one shaker excites the test bearing with load FX = [ f x = produces dynamic motion zX = [x x , yx ]T and acceleration aX = [axx , ayx ]T . Note that zX is the relative displacement between the rotor and the bearing, and aX is the absolute accelerationof the bearing stator. Next, the other shaker exerts the excitation FY = [ f x = 0, f y = f k eiωkt ]T to produce the displacement zY = [x y , yy ]T and acceleration aY = [axy , ayy ]T . Sensors and a data acquisition system record the applied forces and ensuing motions, displacement and acceleration, of the bearing stator along the X and Y directions. Next, a Discrete Fourier Transformation (DFT) transfers the recorded time domain data into the frequency domain, i.e., FX(ω) = DFT(FX(t) ). The procedure calls for forming the (2x2) matrices of external forces F(ω) = [FX(ω) | FY(ω) ], bearing stator acceleration a(ω) = [aX(ω) | aY(ω) ] and displacements z(ω) = [zX(ω) | zY(ω) ]. The test system has a complex dynamic stiffness matrix (H) determined from.2 H(ω) = [F − MBC a](ω) z−1

(1)

where H = [ H xx H xy | H yx H yy ] is the matrix of complex dynamic stiffnesses, and whose real and imaginary parts are curve fitted with stiffness, damping and virtual mass coefficients (K, C, M) α,β =x,y .     Re H(ω) → K − ω2 M, Im H(ω) → iωC

(2)

2 For simplicity in the discussion, Eq. (1) omits the (small) forces from the elastic rods holding

the bearing stator.

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In general for a TPJB, the cross-coupled force coefficients are much smaller than the direct force coefficients, i.e., |H xy | and |H yx | < < |H xx | or |H yy |, except for Q < < Qs . For more details see Ref. [10].

4 Experimental Results and Discussion 4.1 Bearing Eccentricity Figure 5 depicts the bearing eccentricity (e) for increasing specific loads and with flow rates ranging from 1.5 Qs to 0.2 Qs (=3.4 LPM) for the LBP bearing, and from 1.5 Qs to 0.35 Qs (=5 LPM) for the LOP bearing. In Fig. 5b, the graphs include dashed vertical lines denoting the nominal flow (Qs = 14.4 LPM) at the operating speed of 6 krpm. Compared to the LBP bearing, the LOP bearing operates with a smaller eccentricity (e) that is not affected as the flow rate drops to 50% nominal. On the other hand, the LBP bearing shows a sharp increase in e for the smallest load (345 kPa) as Q drops to 20% of nominal.

(a) ey vs. ex

(b) e vs. Q. Load increases Fig. 5. (a) Locus of bearing center (ey vs. ex ) for increasing loads and a range of flow rates Q, (b) Bearing eccentricity (e) vs. flow rate (Q) for a range of specific loads (W/ (LD)).

Note that the bearing eccentricity shown is relative to the hot center identified without any imposed load. Recall, the applied static load on the bearing is along the Y-direction

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for both LOP or LBP configurations. Thus, as shown in Fig. 5b, the LOP bearing center (e) displaces nearly parallel to the load direction, i.e., ex < < ey . The LBP bearing, on the other hand, shows a sizeable displacement (ex ) orthogonal to the applied load W along Y. For the highest specific load of 2.07 MPa, the LBP bearing displaces to an eccentricity (e) that exceeds the cold clearance (C rc = 0.115 mm). This behavior is not unusual as found in past measurements conducted with the LBP bearing configured with end seals (flooded condition); see Refs. [5, 6]. 4.2 Pad Temperatures and Oil Exit Temperature Figure 6 depicts the pad sub-surface temperature rise  T = (T-T s ) vs. flow rate for increasing specific loads. Figure 6a shows the largest temperature rise in the loaded pads. Note the specific loads and smallest flow rates (Q) are distinct for the LOP and LBP bearing cases; hence, a direct comparison of results is not immediately apparent, except for the smallest unit loads of 345 kPa and 1,034 kPa. For the 100% nominal flow (Qs ), both bearings produce the same temperature rise that increases as the unit load increases. In general, as Q reduces to 20% Qs for the LBP bearing and 35% Qs for the LOP bearing,  T rises by at most 10 °C and 5 °C respectively. Figure 6b shows the largest temperature rise in the unloaded pads. For both bearings, LOP and LBP,  T is not a function of the applied load and slowly increases as Q decreases (< Qs ). At the lowest flow,  T is at most 10 °C higher than the temperature rise recorded for the nominal (100%) flow. Note that the largest  T at the loaded pads does not reach the Babbitt recommended safe limit of 130 °C (70 o C above T s = 60 °C). More temperature measurements reported in Refs. [5–7] do show temperature rises reaching the operational limit for very low flow rates and under high loads. Figure 7 presents the measured oil exit temperature rise  T E = (T E -T s ) vs. flow and various unit loads. For the LOP bearing, independent of the applied load, the oil exit temperature rises by at most 5 °C as the supplied flow (Q) drops to 35% Qs . On the other hand, for the LBP bearing,  T E increases by at most 15 °C as Q = 0.25 Qs and the largest applied load of 2.07 MPa. The photograph in the inset depicts a thermographic image of the bearing housing. Note the red color denotes the highest temperatures that occur on the bearing caps located on the sides of the test bearing. The caps collect the exit oil and a drain holes route the hot lubricant to the return sump and heat exchanger. Incidentally, the recorded oil exit temperatures are not representative of the temperatures in the bearing pads (shown in Figs. 6). Typically, the shear induced drag power loss (Ploss ) in a fluid film bearing is estimated from the product of the flow rate (Q) and the oil exit temperature rise  T E . That is, Ploss ~ (cP ρ Q  T E ), not shown here for brevity. See Refs. [4, 5] for complete details including the measurement of the drag torque that directly leads to Ploss . However, one can readily infer from the supplied Q and the measured  T E that Ploss decreases by at most 25% for the lowest Q and a unit load of 1.034 MPa. The savings in Ploss are not too significant at the operating speed of 6 krpm. Not so for operation at 12 krpm, as shown in Ref. [8]. In the same reference, Ploss for the evacuated bearing is significantly lower than that for the flooded bearing.

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(a) Peak temperature rise in loaded pads of evacuated ends TPJB

(b) Peak temperature rise in unloaded pads of test bearing Fig. 6. (a) Loaded pads largest temperature rise and (b) unloaded pads temperature rise vs. flow rate (Q) over a range of specific loads (W/ (LD)). Temperature rise relative to oil inlet temperature T s = 60 °C.

4.3 Force Coefficients for LOP and LBP Bearings with Evacuated Ends Figure 8 presents samples of the experimentally obtained LOP bearing complex stiffnesses, real and imaginary parts, vs. frequency under two applied loads and two flow rates, 35% and 150% of Qs . For the bearing type, Re(H yy ) > Re(H xx ) and Im(H yy ) > Im(H xx ), in particular for the moderate unit load of 1,379 kPa. Note the cross coupled coefficients H yx and H xy have the same order of magnitude as H xx . In general, the direct complex stiffnesses significantly increase as the applied load increases and as the flow rate (Q) decreases. The Im(H)’s are proportional to the excitation frequency, hence the bearing provides constant damping coefficients since Im(H xx ) → (ω C xx ),for example. It is important to realize that Im(H xx ) for a 35% Qs is much smaller than the one for the 150% nominal flow.

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Fig. 7. Oil exit temperature rise vs. flow rate (Q) over a range of specific loads (W/ (LD)). Temperature rise relative to oil inlet temperature T s = 60 °C.

Figures 9 and 10 depict the direct stiffnesses (K yy , K xx ) and damping coefficients (C yy , C xx ) estimated from the complex stiffnesses (H) vs. the supplied flow. Recall that for both the LBP and LOP bearings the Y direction coincides with the applied load (W ). From the results in Fig. 9, for the LOP bearing notice that the (load) stiffness K yy > > K xx , whereas K yy ~ K xx for the LPB bearing. For both bearings, K yy substantially increases as the applied load increases. Importantly enough, K yy slightly grows as the flow rate decreases below the nominal condition (Q < Qs ), an expected outcome since the film thickness decreases as the flow reduces. The most surprising outcome relates to the LOP bearing stiffness K xx (orthogonal to W direction) that is not a function of the applied load and quickly decreases as the flow reduces. In fact, K xx → 0 for the smallest Q, which means the lateral pads could become statically unstable. The curve fits Re(H) → (K-ω2 M) deliver virtual mass coefficients (M) with significant magnitudes. The said coefficients are not shown for brevity; see Refs. [7, 8] for example. Presently, the LOP bearing gives M yy ~ 35 kg and slightly increasing with load; hence Re(H yy ) softens with frequency, as seen in Fig. 8. Similarly, M xx ~ 40 kg and quickly decreases as Q drops. On the other hand, the LBP bearing has M yy ~ - 20 kg and which decreases further as Q falls below Qs . Hence, Re(H yy ) hardens with frequency. In the same fashion, M xx ~ -10 kg at Qs and begins to increase for the lowest Q = 0.25 Qs .

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(a) Real part of LOP bearing complex stiffnesses (H)

(b) Imaginary part of LOP bearing complex stiffnesses (H)

Fig. 8. Real and imaginary parts of complex stiffness (H) vs. excitation frequency. Results for LOP bearing: unit loads (W /(LD)) = 345 kPa (left graphs) and 1,379 kPa (right graphs) and two flow rates Q = 35% Qs (top graphs) and 150% Qs (bottom graphs). Operation at 6 krpm (surface speed = 32 m/s).

Similarly, shown in Fig. 10, the direct damping coefficients (C yy , C xx ) for the LBP bearing are similar in magnitude and C yy decreases as Q drops. The applied load does increase C yy but not as strongly as the direct stiffness K yy does. The LBP bearing, on the other hand, shows C yy > C xx , and with the coefficient C yy not affected by a reduction on the supplied flow below the nominal condition. As with the coefficient K xx , the damping

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(a) Stiffness Kyy vs. Q

(b) Stiffness Kxx vs. Q Fig. 9. Direct stiffnesses K yy and K xx (top and bottom graphs) vs. flow rate (Q) and various specific loads (W/ (LD)). Results for LBP bearing (left graphs) and LOP bearing (right graphs). Operation at 6 krpm (surface speed = 32 m/s). Y is the direction of applied static load (W ).

coefficient C xx shows a precipitous drop as soon as the flow rate falls below nominal, Q < Qs . The sudden reduction in damping is independent of the applied load and evidences the unloaded pads (along X direction) severely starve of lubricant. Clearly, since K xx and C xx → 0, then the bearing would be both statically and dynamically unstable, and which would certainly induce SSV Hash at very low frequencies. Ref. [8] provides details on the amplitude and frequency content of the bearing SSV motions.

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(a) Damping Cyy vs. Q

(b) Damping Cxx vs. Q Fig. 10. Direct damping C yy and C xx (top and bottom graphs) vs. flow rate (Q) and various specific loads (W/ (LD)). Results for LBP bearing (left graphs) and LOP bearing (right graphs). Operation at 6 krpm (surface speed = 32 m/s).

5 Conclusion The paper presented measurements of performance for a four-pad tilting pad journal bearing with evacuated ends. The bearing is configured as load on pad (LOP) and load between pads (LBP). Operation is at a constant shaft speed of 6 krpm (surface speed = 32 m/s) and with unit loads as large as 2.07 MPa. In the tests, the supplied flow (Q) of an ISO VG oil delivered at 60 °C ranges from 150% to 25% of a nominal 100% rate Qs = 14.4 L per minute. The major findings of the experimental campaign are: • A reduction in Q (< Qs ) makes both bearings operate slightly more off-center, the LBP bearing displaces more. • Pad subsurface temperatures are similar for both bearings, although the LOP bearing is colder by a few Celsius degrees, and its oil exit temperature is also lower as the flow decreases well below nominal. • The bearings direct stiffness K yy along the load direction (Y ) increases with load, the LOP bearing producing 25% more stiffness and which increases as Q decreases. The LOP bearing produces a smaller stiffness K xx , that quickly drops to reach nearly zero as Q drops to 25% Qs . The LBP bearing does produce significant stiffness K xx (~ K yy ), and both stiffnesses increase as Q reduces.

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• The LBP bearing shows a negative virtual mass coefficient M yy , hence the bearing dynamic stiffness hardens with excitation frequency. In opposition, the LOP bearing produces M yy > 0, and the bearing softens as the excitation frequency increases. • The direct damping coefficient C yy for the LBP bearing drops quickly for the lowest Q, whereas C yy for the LOP bearing remains impervious to flow changes. Similarly, C xx varies little with a flow reduction. • On the other hand, C xx for the LOP bearing dramatically drops toward null for the lowest flow rate, Q =0.25 Qs . • Hence, the LBP bearing is more tolerant to significant flow reductions than the LOP bearing is. For the lowest Q, this last bearing type entirely lost its stiffness and damping coefficients in a direction orthogonal to the applied load. Under this extreme operating condition, the LOP bearing produced low frequency subsynchronous shaft motions (SSV Hash). Lastly, under no circumstances the test bearings operated for extended periods of time with a too low flow rate that could have degraded the lubricant and bearing pads’ Babbitted surfaces. Acknowledgment. The authors acknowledge the financial support of the Turbomachinery Research Consortium and the personal research funds of Dr. L. San Andrés. Thanks to undergraduate student Mr. Zihan Ouyang for assisting to operate and maintain the test rig.

Abbreviations NOMENCLATURE Cr C αβ cP D e H αβ K αβ L M αβ M BC Ploss Q T T s, T E X,Y W T ρ, μ  ω

Bearing radial clearance [m] Bearing damping coefficients, α, β = x,y [Ns/m] Lubricant specific heat [J/(kg o C)] Shaft diameter [m] Bearing eccentricity [m] Bearing complex dynamic stiffnesses, α, β = x,y [N/m] Bearing static stiffness coefficients, α, β = x,y [N/m] Bearing length [m] Bearing virtual mass coefficients, α, β = x,y [N/m] Bearing housing and assembly mass [kg] ~ cP ρ Q  T E . Estimated drag power loss [W] Supplied flow rate [m3 /s], Qs nominal flow rate [m3 /s] Temperature [o C] Oil inlet and exit temperatures [o C] Coordinate system Applied static load (along Y ) [N] (T-T s ). Temperature difference [o C] Oil density [kg/m3 ] and viscosity [Pa.s] Rotor angular speed [rad/s] Excitation frequency [rad/s

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Vectors and matrices a F H z

[ax , ay ]T . Vector of bearing acceleration [m/s2 ] [F x , F y ]T . Vector of applied forces [N] [H xx , H xy | H yx , H yy ]. Matrix of complex dynamic stiffnesses [N/m] [X , y]T . Vector of bearing displacements relative to shaft [m]

Abbreviations LBP LOP LPM SSV TBJB

Load between pads Load on pad Liters per minute Subsynchronous vibration Tilting pad journal bearing

References 1. Leopard, A.J.: Tilting pad bearings-limits of operation. ASLE Lubrication Eng. 32(2), 637– 644 (1976) 2. Dmochowski, W.M., Blair, B.: Effect of oil evacuation on the static and dynamic properties of tilting pad journal bearings. Trib. Trans. 49 (2006) 3. DeCamillo, S., He, M., Cloud, C.H., Byrne, J.: Journal bearing vibration and SSV hash. In: Proceedings of the 37th Turbomachinery Symposium, The Turbomachinery Laboratory, Texas A&M University, September 7–11, Houston, TX, USA, pp. 179–194 (2008). https:// doi.org/10.21423/R1DH1J 4. San Andrés, L., Jani, H., Kaizar, H., Thorat, M.: On the effect of supplied flow rate to the performance of a tilting-pad journal bearing. - static load and dynamic force measurements. ASME J. Gas Turbines Power 142(2), 121006 (2020). https://doi.org/10.1115/1.4048798 5. San Andrés, L., Toner, J., Alcantar, A.: Measurements to quantify the effect of a reduced flow rate on the performance of a tilting pad Journal Bearing (LBP) with flooded ends. ASME J. Eng. Gas Turbines Power 143(11), 111012 (2021). ASME Paper GT2021–58771. https://doi. org/10.1115/1.4052268 6. San Andrés, L., Alcantar, A.J.: Effect of reduced oil flow rate on the static and dynamic performance of a tilting pad journal bearing running in both flooded and evacuated conditions. ASME J. Eng. Gas Turbines Power 145(6), 061012 (2023). ASME Paper GT2022-81839. https://doi.org/10.1115/1.4056535 7. San Andrés, L., Alcantar, A.: Effect of reduced oil flow rate on the static and dynamic performance of a tilting pad journal bearing running in both the flooded and evacuated conditions. In: Proceedings of the 50th Turbomachinery & Pump Symposia, Houston, TX, December 2021 8. San Andrés, L., Ouyang, Z., Qin, Y.: Effect of reduced oil flow on the performance of a load on pad, tilting pad journal bearing: flooded vs. evacuated conditions. In: Proceedings of ASME Turbo Expo 2023, Turbomachinery Technical Conference and Exposition, Boston. ASME Paper GT2023-103242 (2023). https://hdl.handle.net/1969.1/196726 9. Jani, H.: Measurements of the Static and Dynamic Force Performance on a Five-Pad, Spherical-Pivot Tilting-Pad Journal Bearing: Influence of Oil Flow Rate. M.S. thesis, Texas A&M University, College Station, TX, USA (2018)

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10. Toner, J.: Measurements of the Static and Dynamic Force Performance on a Five-Pad, Spherical-Pivot Tilting-Pad Journal Bearing: Influence of Oil Flow Rate. M.S. thesis, Texas A&M University, College Station, TX, USA (2020) 11. San Andrés, L.: Experimental identification of bearing force coefficients. Modern Lubrication Theory, Notes 14, Libraries Texas A&M University Repository (2009). http://oaktrust.library. tamu.edu/handle/1969.1/93197. Accessed Jan 2021

Dynamic Performance of an O-Ring Sealed Squeeze Film Damper and a Simple Way to Estimate the (Ingested) Gas Content in a Squeeze Film Luis San Andrés1(B) and Bryan Rodríguez2 1 J. Mike Walker ’66 Department of Mechanical Engineering, Texas A&M University College

Station, Texas, TX 77843, USA [email protected] 2 L.A. Turbine, 91355 Valencia, CA, USA [email protected]

Abstract. Completing a long-term project characterizing squeeze film dampers (SFDs) for air breathing engines, the paper presents complete measurements of the forced performance of an elastic structure hosting an OR-sealed damper (OR-SFD) and the identification of physical parameters, namely stiffness and damping, for the dry structure, the ORs, and the lubricated OR-SFD. The damper with slenderness ratio L/D = 0.2 and radial clearance c ~ 0.002 D is lubricated a light oil supplied at 0.69 bar(g). Shakers produce circular whirl motions of the test system with amplitude r = 0.05c to 0.45c and frequency ω = 10 Hz to 70 Hz (max. Squeeze velocity vs = r ω=55 mm/s). The ORs force coefficients are a function of the orbit amplitude and nearly invariant with frequency. The ORs centering stiffness quickly decreases with amplitude, likely due to the extensive deformation and slow elastic recovery in the ORs’ material bonds. As the whirl amplitude grows, the ORs viscous damping (C OR ) reduces from 10% to 3% of the lubricated system damping. For small orbit radii, r ≤ 0.25 c, the SFD added mass (M SFD ) and viscous damping (C SFD ) coefficients approach predictions for a fully sealed damper. For moderate size orbits (r = 0.45 c), M SFD drops by 75% and C OR decreases by ~ 40% due to persistent air ingestion for motions with vs > 24.5 mm/s. Videos show a bubbly mixture in the return line and also through the damper top end exposed to ambient. Over a certain elapsed time, a simple procedure draws into a deflated balloon the material contents in the film. The novel approach estimates the gas volume fraction (GVF) that rapidly increases as vs grows. The findings quantify the deterioration of ORs force coefficients, their inability to keep the lubricant sealed, and their limited ability to prevent persistent air ingestion into the film land. Keywords: Squeeze film damper · Force coefficients · Rotordynamics

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 93–110, 2024. https://doi.org/10.1007/978-3-031-40455-9_9

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1 Introduction Rotor-bearing systems employing squeeze film dampers (SFDs) have better control of rotor displacements while crossing a critical speed and can effectively suppress rotordynamic instabilities. Incidentally, gas turbines in aircraft engines rely on SFDs mounted in series with rolling element bearings, see Fig. 1, to withstand maneuver load conditions and isolate the engine frame from harmful vibrations [1, 2]. Often, a centering support (squirrel cage) [1] serves as a soft mount for the damper; and in other instances, a dowel pin restricts rotation of the ball bearing outer race. Rotor motions squeezing the lubricant within the small clearance (c) generate a pressure field that produces reaction forces characterized by both damping (C) and inertia (M) force coefficients [1]. Used as sealing devices, O-rings (ORs) reduce the lubricant side leakage, whilst amplifying the damping amidst a limited axial length. O-rings are commodity products that also add viscoelastic damping [3] and a mean to statically support the damper. Limitations to the applicability of ORs include temperature (high or low), ageing and creeping, compatibility with the lubricant, and sensitivity to the amplitude and frequency of whirl motion [3]. Nonetheless, ORs are low cost, simple to install, and can perfectly contain lubricant in a confined space.

Fig. 1. Schematic views (not to scale) of a hole fed SFD with O-rings as end seals. Graph adapted from an original rendition in Ref. [1].

2 Appraisal of Prior Work O-rings are molded from an elastomer, such as Nitrile Butadiene Rubber (NBR or BunaN) or Fluorocarbon (Viton) shaped with a round cross-section and have a broad range of industrial applications and solutions. Advantages in the use of ORs as static seals include their low cost, small weight and volume, durability, and easily identifiable failure modes [3]. Besides sealing a SFD land, O-rings are a centering stiffness to support lightweight rotor-bearing systems, as detailed in Refs. [4–8]. Moreover, ORs are ubiquitous in micro

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turbomachinery (MTM), power 250 kW or less, offering low-cost elastic supports with plenty of damping, as reported in Refs. [9–11]. In practice, O-ring manufacturers [3] spell out a range of operating and installation conditions, that if not followed, diminish an OR useful life, impairs its sealing ability and increase the probability of a sudden failure. For example, OR material extrusion is present under excessive pressures, large static eccentricity, improper gland machining, etc. Worse yet, inappropriate installation or lubrication could provoke a spiral type failure distorting the O-ring geometry. In a seminal paper appearing in 1978, Smalley et al. [12] first report OR stiffness (K OR ) and visco-elastic damping, characterized as a loss factor η, as a function of frequency and seven parameters that include ring thickness, groove width, and amplitude of motion with frequencies ranging from 70 Hz to 1,000 Hz. Stemmed from their experimental results, the authors determine the amplitude of motion provides the most significant effect on the ORs’ physical parameters. For instance, increasing the amplitude of motion 3.5 times decreases K OR by 30% and η by 20%. However, other parameters, such as OR stretch and thickness render a lesser effect on the tested ORs dynamic forced performance. Later in 1986, Green and Etsion [13], further identify the effect of static pressure and squeeze level on ORs force coefficients. To advance aircraft gas turbine engine design, an engine manufacturer funded a decade long SFD research program at the authors’ laboratory. The concerted effort characterized the dynamic response of multiple SFD types for ready implementation in jet engines. In 2016, San Andrés et al. [14] summarize the major outcomes and provide fundamental knowledge regarding damper operation and design to the practicing engineer. Relevant to the current research, San Andrés and students [15–18] conduct dynamic load tests with piston-rings (PRs) and ORs sealing the ends of ultra-short SFDs and report the dynamic force coefficients of the end seals alone and the squeeze film section when lubricated. Both PR-SFD and OR-SFD produce similar magnitude viscous damping and virtual mass coefficients. However, the OR-SFD adds a measurable viscoelastic damping coefficient, 20% or so of the lubricated one, and produce a quadrature stiffness (K ⊥ ) indicative of a significant material loss factor. Of interest, San Andrés and Rodríguez [18] when testing an OR-SFD supplied with a low lubricant pressure (PS = 0.7 barg) also report the ORs render a significant direct stiffness (K OR ). This paper details the outcome of measurements conducted in the same test rig and with larger amplitude whirl motions producing sizeable squeeze velocities (vs ). The procedure quantifies the force coefficients of both the ORs and the squeeze film land. At sufficiently large vs , air ingestion and entrapment degrades the SFD forced performance. This work also introduces a simple method to determine the gas volume fraction ingested while the damper operates. Please refer to the MS thesis of Rodríguez [19] for a full account of the work that includes a thorough review of the past literature.

3 Description of Test Rig, SFD and Seals Figure 2 depicts the SFD test rig comprising a rigid pedestal, four support rods, a journal, and a bearing cartridge (BC). For a full description of the experimental apparatus, see Refs. [15, 16]. A pair of orthogonally placed electromagnetic shakers connected to

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stingers directly apply dynamic loads to the BC and produce circular centered orbits (CCO) with amplitude r and frequency ω. The squeeze velocity vs = r x ω. Installed in the BC, along the X and Y axes, pairs of accelerometers, eddy current sensors, and load cells measure the BC accelerations, the displacements relative to the fixed journal, and the applied shaker loads.

Fig. 2. (a) Photograph and (b) schematic top view of SFD test rig with electromagnetic shakers and static loader.

Figure 3 shows a top view and a cross-section view of the journal and BC. The drawings include labels for all components. The SFD section is short in axial length, L = 25.4 mm, and the journal diameter D = 127 mm (L/D = 0.2). The film radial clearance c = 0.279 mm. The journal comprises two parts: a base affixed to a rigid pedestal, and a top sleeve with a hollow center or plenum that connects to the lubricant supply line. A pump supplies ISO VG 2 lubricant at a low pressure PS = 0.69 bar(g) to fill in the plenum upstream of the squeeze film. Then, oil flows through the mechanical check valve, and into the middle plane of the film land (z = 0). The lubricant leaves the damper through one discharge orifice that connects to a return line. A return pump pushes the lubricant through a bubble eliminator and into a storage tank. At a supply temperature of 25 °C, the ISO VG2 viscosity is μ = 2.81 ± 0.01 mPa-s and its density ρ= 800 ± 0.02 kg/m3 . Figure 4 displays (a) a photograph of the test journal with its dimensions and (b) a cross-section schematic view of the journal with its grooves. The Buna-N AS568 #244 ORs have a thickness of 3.53 mm with inner and outer diameters equaling 107.54 mm and 114.56 mm. Once installed, the ORs are effective to fully prevent oil leakage under static conditions (no BC displacement).

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Fig. 3. SFD schematic views: (a) top view featuring one lubricant feedhole (θ in = 45°, z = 0) and a discharge hole (θ out = 240°, z = + ¼ L); and (b) section A-A showing ORs installed. (Drawings not to scale and with exaggerated features).

Fig. 4. (a) Photograph of assembled journal and (b) schematic cross-section view with dimensions for O-ring grooves. Drawing in (b) not to scale and with exaggerated features.

4 Test Procedure and Identification of Force Coefficients The experiments comprise single-frequency dynamic loads over a frequency range ω = 10 to 130 Hz, in increments of 10 Hz. The test rig is idealized as a two degree of freedom mechanical system with the cartridge mass (M BC ) attached to the elastic support structure, the O-rings, and the squeeze film section. Under lubricated conditions, the equation of motion in the frequency domain is HL(ω) Z(ω) = F(ω) − MBC a(ω)

(1)

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where F( ω) , Z( ω) and a( ω) are complex vectors having as rows the X and Y components of applied external force, BC displacement relative to the journal, and BC absolute acceleration, respectively. Above, HL is a 2x2 matrix representing the lubricated system complex dynamic stiffness, and which includes the physical parameters from the structure, O-rings and squeeze film. Following San Andrés and Koo [15], let HL = Hst+OR + HSFD where Hst+OR = Hst + HOR

⎧   ⎨ Hst = Kst − ω2 Mst + i ω Cst = ⎩ H = K + i [ω C + K ] OR OR OR ⊥

(2)

(3)

Above, the structure is modeled with a set of stiffness, viscous-damping (C), and virtual mass coefficients, (K, C, M)st . The O-rings dynamic forced response is modeled with a viscous damping coefficient (COR ), a restoring stiffness (KOR ), and a quadrature stiffness (K⊥ ). Without the O-rings installed and no lubricant in the system, dynamic load measurements produce the support complex stiffness (Hst ), and curve fits over a frequency range deliver the physical parameters. The matrices of coefficients are isotropic; that is, Kst = K st I, Mst = M st I, and Cst = C st I, where I is the (2x2) identity matrix. The elastic rods have a structural stiffness K st = 6.3 MN/m, and a small damping coefficient C st = 0.2 kNs/m. The BC effective mass is M BC = 15.6 kg.√The natural frequency of the dry system (without lubricant or ORs installed) is f n = Kst /MBC = 100.5 Hz and the system damping ratio ζ st = √ Cst /(2 Kst MBC ) = 0.01. Next, the ORs are installed in the journal and still with a dry structure, a new set of dynamic loads produce the parameters of the ORs plus structure, i.e., Hst+OR in Eq. (3). The ORs force coefficients are obtained from HOR = [ Hst+OR - Hst ]. Lastly, lubricant is supplied into the damper at a set pressure (Ps ), and a new round of dynamic load tests performed to obtain the whole complex dynamic stiffness HL . The squeeze film parameters follow from HSFD = HL - Hst+OR . Curve fits over a certain frequency range deliver the SFD stiffness, damping and added mass coefficients from     (4) HSFD = HL − Hst+OR → KSFD − ω2 MSFD + i ω CSFD Note that the magnitudes of the lubricated system cross-coupled complex stiffnesses (H XY , H YX ) are at least an order smaller than the direct coefficients (H XX , H YY ). For brevity, the said cross-coupled coefficients are not reported. See Ref. [19] for details.

5 Force Coefficients for O-rings With the ORs installed in the journal and pressing against the BC, static (push) load tests produce BC displacements from which the static stiffness of the O-rings is determined. The experiments produced an isotropic K OR,static = 11.6 MN/m.

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Next, with the system unlubricated or dry and no external pressurization, dynamic load tests generated circular centered orbits (CCO) with frequency ω = 10 Hz to 130 Hz and with amplitude r = 0.05c to 0.45c. In the tests, the squeeze velocity vs = (r ω) ranged from 0.9 mm/s to 102.5 mm/s. Figures 5 and 6 depict the real and imaginary components of Hst+OR vs. the orbit radius (r) and the squeeze velocity (vs ). In Fig. 5, the two horizontal planes denote the static stiffnesses K st and K st + K OR,static . The identification reveals Re(H XX ) ~ Re(H YY ), i.e. an isotropic system. Re(H XX )st+OR decreases with amplitude of motion (r) although varying little with vs for orbit radii r > 0.15c. Importantly, Re(H XX )st+OR > (K st + K OR,static ) for small amplitude orbital motions, r = 0.05c to 0.20c. On the other hand, for the largest orbit sizes, r > 0.25c, Re(H st+OR ) < (K st+OR,static ) and approaches K st . This means the O-rings lose their centering stiffness for whirl motions of significant amplitude.

Fig. 5. Dry system structure + ORs: Re(H XX , H YY )st+OR vs. squeeze velocity (vs ) and (r/c). Frequency range ω = 10 Hz to 130 Hz. PS = 0.0 bar(g).

Fig. 6. Dry system structure + ORs: Im(H XX , H YY )st+OR vs. squeeze velocity (vs ) and (r/c). Frequency range ω = 10 Hz to 130 Hz. PS = 0.0 bar(g).

The imaginary parts of [H XX and H YY ]st+OR show similar trends as their real counterparts. That is, Im(Hst+OR ) decreases as r increases and increases with frequency.

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Hence Im(Hst+OR ) is not a strong function of the squeeze velocity. Note that as ω → 0, Im(Hst+OR ) produces a sizeable quadrature stiffness (K ⊥ ). From the results in Figs. 5 and 6, the O-rings complex stiffness HOR = [ Hst+OR Hst ]. Im(HOR ) is fitted with the model (K⊥ + ω COR ) while Re(HOR ) → KOR . Figure 7 depicts the identified ORs coefficients (K OR , K ⊥ and C OR ) vs. orbit radius (r/c) with error bars denoting the uncertainty of a parameter. The ORs’ force coefficients decrease as the orbit amplitude increases. Note Fig. 7 also depicts the derived material loss factor η = (K ⊥ /K OR ). η = 0.50 for moderately large amplitude orbits (r > 0.15 c), whereas η = 0.35 for small size orbits as in Ref. [18]. Incidentally, the correlation coefficients (R2 ) quantifying the goodness of fit are larger than 0.90. That is, the assumed model fits the experimental data.

Fig. 7. ORs’ stiffness (K OR ), quadrature stiffness (K ⊥ ), viscous damping (C OR ), and loss factor (η) vs. orbit radius (r/c). Coefficients estimated from circular centered orbits over frequency range ω = 10 – 130 Hz.

Prior art in Refs. [20–22] report experiments consisting of unidirectional dynamic loads of different amplitudes and over a range of frequencies exerted onto elastomeric specimens, similar in composition to the ORs herein used. The results in Refs. [20– 22] also document a characteristic drop in stiffness and damping coefficients as the amplitude of motion grows. The authors in Refs. [20–22] attribute the reduction to the continuous breakdown and recovery of the polymer inner structure under dynamic loads. Under large amplitudes of motion, the breakdown is so extensive that its reformation is much slower than the strain cycle time, hence leading to both low stiffness and viscous damping coefficients. That is, for large displacements, the ORs lose their elastic support and energy dissipation characteristics.

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6 Force Coefficients for Sealed SFD This section presents the lubricated system complex stiffness (HL ) and the identified SFD force coefficients (C SFD , M SFD ). The experiments comprise circular centered orbits with r/c = 0.05 to 0.45 and frequencies ω = 10 Hz to 70 Hz (vs = 0.9 mm/s to 55.2 mm/s). The ISO VG2 lubricant enters the film land at PS = 0.69 bar(g) (10 psig) and the supplied flow rate QS = 0.8 LPM. Installed in the journal end grooves, O-rings seal the film land and prevent lubricant through flow. Figures 8 and 9 present the real and imaginary parts of HL vs. orbit amplitude (r/c) and squeeze velocity (vs = r ω). In Fig. 8, Re(H XX, H YY )L at vs = 0 renders the direct stiffnesses K XX ~ K YY , both similar in magnitude to the stiffness of the dry system (K st+ORs ). That is, the squeeze film does not generate a static stiffness. For small r < 0.20c, Re(HL ) decays quadratically with an increase in vs , hence revealing a strong fluid inertia effect. However, for r ≥ 0.25c, Re(HL ) increases as ω → 70 Hz (vs = 31 mm/s); and more noticeably, for r = 0.30c, Re(HL ) grows for ω > 55 Hz (vs > 37 mm/s). This effect continues until Re(HL ) shows an upward trend as frequency increases, see Fig. 8 (b), and thus resulting in a dramatic reduction in the film inertia coefficient as r → 0.45c.

Fig. 8. Lubricated system: (a) Re(HXX ) L and (b) Re(HYY )L vs. orbit radius (r/c) and squeeze velocity (vs ). Frequency range ω = 10 – 70 Hz. PS = 0.69 bar(g).

Conversely, in Fig. 9, the sizeable magnitude of Im(H XX, H YY )L as ω → 0 Hz indicates the ORs quadrature stiffness (K ⊥ ). Im(H)L shows a linear increase with frequency ω → 70 Hz, typical of a viscous damping coefficient (C SFD ). However, for amplitudes r > 0.35c and ω>55 Hz, corresponding to vs = 34 mm/s and 43 mm/s, Im(HL ) reduces in magnitude. This means C SFD decreases likely due to the effect of air ingestion into the squeeze film, as discussed later. Figure 10 depicts the squeeze film damper HSFD = (HL - Hst+OR ), real and imaginary parts, vs. frequency for three orbit amplitudes r/c = 0.05, 0.20 and 0.45. The symbols (squares and circles) denote test data, whereas the dotted lines represent the curve built with the estimated parameters using the physical model Re(HSFD ) = (KSFD – ω 2 MSFD ) and Im(HSFD ) = (ω CSFD ). The estimation of the SFD stiffnesses spans a frequency range from ω = 10 to 40 Hz. Employing a larger frequency range (or vs > 24.5 mm/s) would demand of a higher order fit model, likely having no physical significance.

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Fig. 9. Lubricated system: (a) Im(HXX ) L and (b) Im(HYY )L vs. orbit radius (r/c) and squeeze velocity (vs ). Frequency range ω = 10 – 70 Hz. PS = 0.69 bar(g).

For r = 0.05c and 0.20c, Re(HSFD ) decreases as frequency increases to show a significant added mass coefficient (M SFD ). However, Re(HSFD ) rather grows at vs = 24.5mm/s, hence revealing a lesser virtual mass. For vs ≥ 24 mm/s, Re(HSFD ) ~ 0; hence M SFD ~ 0 or slightly negative. Conversely, Fig. 10 shows Im(HSFD ) increases as ω → 70 Hz and r < 0.45c, hence evidencing a dominant viscous damping coefficient (C SFD ). Alas, Im(HSFD ) drops in magnitude for vs > 35 mm/s and which denoting a rapid reduction in the film viscous damping effect. The reduction forced the authors to select a shorter frequency span to identify the SFD force coefficients. Note the experimental cross-coupled coefficients are (at least) one order lower than the direct force coefficients and not shown for brevity. Figure 11 showcases the normalized squeeze film damping (CSFD = C SFD /C* ) and added mass coefficients (MSFD = M SFD /M * ) as obtained from curve fits of the physical model (KSFD – ω 2 MSFD ) → Re(HSFD ), and (ω CSFD ) → Im(HSFD ). The shown coefficients are strictly valid over ω = 10 Hz to 40 Hz. For the normalization, the reference force coefficients are [1]

3 D πL D 3 ∗ = 31.4kNs/m; M = ρ = 58kg C = 12π μL 2c c 2 ∗

(5)

applicable to a full film, tightly sealed SFD undergoing small amplitude (r → 0) motions. M* and C* above apply for a squeeze film Reynolds number ReS = (ρωc2 )/μ < 10. In the tests, the maximum ReS = 9.8. Regarding the squeeze film damping coefficients shown in Fig. 11, their largest magnitudes occur at r = 0.05c, and C SFD = ½(C XX + C YY )SFD , ~ (0.9 C * ) as the orbit amplitude grows to r = 0.25c. C SFD is roughly constant for r < 30c but quickly reduces to ~ (0.6 C * ) as r → 0.45c. Note that the reduction in C SFD is due to the presence and pervasiveness of air ingestion for operation with a large enough vs .

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Similarly, (M XX, M YY )SFD ~ M * for r < 0.10c, and slightly decrease as r → 0.25c. At r = 0.30c, M SFD has reduced to ~ (0.8M * ) and continues to decrease with r, a wellknown effect. Please see Ref. [6] for a physical explanation of the outcome. At the largest r = 0.45c, M SFD ~ (0.3 M*).

Fig. 10. Lubricated SFD: Real and imaginary parts of (H XX , H YY )SFD vs. whirl frequency (ω) for three orbit radii. PS = 0.69 bar(g). Identification of parameters (K, C, M) in range ω = 10 – 40 Hz.

Fig. 11. Experimental (normalized) squeeze film damping and inertia force coefficients vs. orbit radius (r). Physical parameters estimated from circular orbits and over frequency range ω = 10 – 40 Hz. Oil supply pressure PS = 0.69 bar(g).

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The experimental SFD coefficients are slightly orthotropic with C XX ~ 1.05 C YY and M XX ~ 1.1 M YY . The orthotropy is due to the location of the oil supply and discharge lines, a source and sink of flow, that likely distort the film dynamic pressure field and hence, the SFD reaction forces. Incidentally, the identified parameters have a correlation coefficient R2 > 0.90 to the physical model Re(HSFD ) → (KSFD –ω 2 MSFD ), Im(HSFD ) → (ω CSFD ) within the frequency range ω = 10 to 40 Hz. As a comparison between the damping coefficients from the pair of ORs and those for the lubricated system (C L = C OR + C SFD ), Fig. 12 shows C OR /C L vs. orbit radius. The contribution of the ORs’ damping to the lubricated system damping is nearly 10% for r < 0.10c. However, as r grows, C OR /C L rapidly decreases to reach only 3% of the whole lubricated system damping for r = 0.45c.

Fig. 12. Ratio of ORs’ damping to lubricated system damping vs. orbit radius (r/c). Lubricated system supplied with PS = 0.69 bar(g). Experimentally estimated parameters from CCOs and frequency range ω = 10 – 40 Hz.

For completeness, Fig. 13 shows the SFD normalized coefficients CSFD and MSFD and compares to predictions using the orbit model in Ref [23]. The agreement of the results derived from a physically sound model against the experimenal record is rather

Fig. 13. Normalized damping (CSFD ) and inertia (MSFD ) coefficients vs. orbit radius (r/c). Comparison with experimental data in Refs. [16, 18] and prediction model from Ref. [23]. PS = 0.69 bar(g). Experiments from CCOs and frequency range ω = 10 – 40 Hz.

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good. Simiularly, the current experimental force coefficients agree with prior test data obtained from a smaller range of whirl orbit sizes in Refs. [16, 18].

7 Estimation of Gas Volume Fraction in O-ring Sealed Damper In a separate set of experiments, a series of videos and photographs record the operating test rig to identify the onset of air entrainment in the film land. The hyperlink https:// youtu.be/R5WPD6t1fWs depicts the SFD test rig operating at r = 0.25c, 0.35c, 0.40c and 0.45c, at ω = 70 Hz. Originally yellow in color, the lubricant is dyed blue to visualize bubbles more easily. While a direct view of the damper film land is not possible, the video graphic records depict the damper discharge line and the top sealed end at multiple operating conditions. Figure 14 portrays the lubricated system operating at r = 0.20c and ω = 70 Hz, corresponding to vs = 24.5 mm/s and ReS = 9.8. During the operation of the damper for vs ≤ 24.5 mm/s, the sealed ends do not allow oil leakage (through top and bottom ends), and the lubricant discharge line shows the returning oil has a bright blue color.

Fig. 14. ORs sealed ends damper. Photographs depicting SFD operating at r = 0.20c and PS = 0.69 bar(g). Whirl frequency ω = 70 Hz, vs = 24.5 mm/s.

Figure 15 shows pictures of the damper top end and lubricant discharge line while operating at vs = 31, 43 and 55 mm/s. At vs = 31 mm/s, when the damper operates at ω= 70 Hz and r/c = 0.25, small bubbles and a minute amount of oil leave through the top side of the journal. In addition, the discharge line shows oil colored light blue mixed with small air bubbles of air when returning to the reservoir; hence denoting the presence of air leaving the film land, see Fig. 15 (a). With the system operating at vs = 43 mm/s, shown in Fig. 15(b), the bubbles leaving the top side of the journal are more numerous, and the bubbly oil exiting the damper appears lighter in color. When the damper operates at vs = 55 mm/s, oil droplets and bubbles leave the top surface of the journal, and the return line presents a turquoise lubricant gaseous mixture, see Fig. 15 (c). Hence, at this condition the gas content in the squeeze film land is larger.

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Fig. 15. ORs sealed ends damper. Photographs depicting SFD operating at whirl frequency ω = 70 Hz. (a) r = 0.25c and vs = 31mm/s, (b) r = 0.35c and vs = 43 mm/s, and (c) r = 0.45c and vs = 55 mm/s. Oil supply pressure PS = 0.69 bar(g).

Other experiments aimed to quantify the gas volume fraction (GVF) in the film land while the damper is operating. The tests extract small volume of lubricant from the film land into an initially deflated balloon. Then, a scale determines the weight of the balloon holding the lubricant sample (mballoon = wballoon /g), which may contain air dissolved within. The volume of the sample assuming it solely contains lubricant is V oil = wballoon /(ρ g). Next, the enclosed sample is fully submerged in a cylindrical container filled with lubricant. The lubricant in the container rises a certain height ( h), and the estimated volume of the sample is V balloon = ( h π d 2 /4). Where d is the inner diameter of the cylindrical container. Afterwards, the ratio between the volume of the sample assuming its mass is full of lubricant (V oil ) and the volume estimated by submerging the balloon in lubricant (V balloon ) is equivalent to the sample liquid volume fraction (LVF). Hence, the gas volume fraction (GVF = 1-LVF). The measurements estimating the damper GVF span orbit amplitudes r = 0.15c to 0.45c and whirl frequencies ω = 55 and 70 Hz. These are the operating conditions in which gas entrainment is more likely to happen. The hyperlink https://youtu.be/AtV Bw5XMdcw shows a video of the procedure and measurements. For vs > 31 mm/s, the contents filling the balloons show an increased amount of gas. In some cases, air coming out of lubricant sample is visible. The test was repeated three times. As the orbit amplitude increases, Fig. 16 shows the estimated GVF vs. (a) whirl frequency and vs. (b) squeeze velocity. GVF ~ 7% for vs < 30 mm/s (r ≤ 0.20c and ω = 70 Hz). The GVF increases to ~ 25% from 34 mm/s ≤ vs ≤ 43 mm/s, and then jumps to GVF ~ 45% at vs ~ 44 mm/s. At the largest squeeze velocity of vs = 55.2 mm/s, r = 0.45c x ω = 70 Hz, the gas content surpasses oil, with GVF ~ 58%.

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The figures include the coefficient of variability CV = standard deviation/mean value. Note CV < 10% for r ≥ 0.40c and ω ≥ 55 Hz (vs ≥ 39 mm/s), whereas CV > 20% for r ≤ 0.35c and ω ≤ 50 Hz. In sum, CV is small for the largest GVFs found.

Fig. 16. (a) Estimated GVF vs. orbit radius (r/c) and whirl frequency (ω) for OR sealed SFD. (b) Estimated GVF vs. (r/c) and squeeze velocity (vs ).

8 Conclusion The paper presents measurements of the dynamic forced performance of an O-ring sealed SFD supplied with ISO VG 2 lubricant at PS = 0.69 barg. The lubricant fills an oil delivery plenum, flows through a mechanical check valve and leaves into the squeeze film land through a single feedhole located halfway between the top and bottom ORs. Experiments with single-frequency circular orbit motions of increasing amplitude (r = 0.05c to 0.45c) serve to identify the ORs and SFD force coefficients at squeeze velocities up to vs = r ω = 55 mm/s. The ORs direct stiffness and damping coefficients quickly decrease as the amplitude of motion increases. The ORs’ quadrature stiffness (K ⊥ ) produces a significant material loss factor η = K ⊥ /K OR → 0.60 for the largest motions with r = 0.45c. The ORs viscous damping (C OR ) represents ~ 10% of the damping in the lubricated system for r < 0.10c. However, the damping contribution of the ORs to the lubricated system reduces to just 3% for motions with orbits r > 0.25c. The measurements with a lubricated system produce complex dynamic stiffnesses (HL ) that have a peculiar behavior for operation with large vs . The identification of parameters is restricted to a narrow frequency range. In short, the squeeze film damping and the added mass coefficients, C SFD and M SFD , agree with full film predictions while also quickly decreasing as the amplitude of motion increases. The drop in damping is due to the ingestion of air into the film as seen in a comprehensive video recording (see hyperlink).

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Lastly, a simple yet unique method allowed the estimation of the gas volume fraction (GVF) while the damper operates. At the largest squeeze velocity vs = 55 mm/s = r = 0.45c x ω = 70 Hz, the GVF ~ 58%, hence the air content surpasses that of the oil in the squeeze film region.

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References 1. San Andrés, L.: Modern lubrication theory. Squeeze Film Dampers, Notes 13, Texas A&M University Digital Libraries, [Jun 16, 2021] (2010). http://oaktrust.library.tamu.edu/handle/ 1969.1/93253 2. Spakovszky, Z.: Instabilities Everywhere! Hard Problems in Aero-Engines. ASME Paper No. GT2021–60864 (2021). https://doi.org/10.1115/GT2021-60864 3. Parker O-ring Handbook ORD 5700, Parker Hannifin Corporation, Lexington, KY, USA, [Sep 30th, 2020]. https://www.parker.com/Literature/O-Ring%20Division%20Literature/ORD% 205700.pdf 4. Edney, S.L., Nicholas, J.C.: Retrofitting a large steam turbine with a mechanically centered squeeze film damper. In: Proceedings of the 27th Turbomachinery Symposium, Houston, TX, pp. 29–40 (1999). https://doi.org/10.21423/R1X660 5. Bansal, P.N., Hibner, D.H.: Experimental and analytical investigation of squeeze film bearing damper forces induced by offset circular whirl orbits. ASME J. Mech. Des. 100(3), 549–557 (1978). https://doi.org/10.1115/1.3453967 6. San Andrés, L.: Effect of fluid inertia effect on squeeze film damper force response. Ph.D. Dissertation, Texas A&M University, College Station, TX (1985) 7. Marmol, R.A., Vance, J.M.: Squeeze film damper characteristics for gas turbine engines. ASME J. Mech. Design 100(1), 139–146 (1978). https://doi.org/10.1115/1.3453878 8. Leader, M.E., Whalen, J.K., Grey, G.G., Hess, T.D.: The design and application of a squeeze film damper bearing to a flexible steam turbine rotor. In: Proceedings of the 24th Turbomachinery Symposium, Houston, TX, pp. 49–58 (1995). https://doi.org/10.21423/ R1R36D 9. Belforte, G., Colombo, F., Raparelli, T., Viktorov, V.: High-speed rotor with air bearings mounted on flexible supports: test bench and experimental results. ASME J. Tribol. 130(2), 021103 (2008). https://doi.org/10.1115/1.2908905 10. Waumans, T., Peirs, J., Al-Bender, F., Reynaerts, D.: Aerodynamic journal bearing with a flexible, damped support operating at 7.2 million DN. J. Micromech. Microeng. 21, 104014 (2011). https://iopscience.iop.org/article/10.1088/0960-1317/21/10/104014 11. Cunningham, C., Ransom, D., Wilkes, J., Bishop, J., White, B.: Mechanical design features of a small gas turbine for power generation in unmanned aerial vehicles. ASME Paper No. GT2015–43491 (2018). https://doi.org/10.1115/GT2015-43491 12. Smalley, A., Darlow, M., Mehta, R.: The dynamic characteristics of O-Rings. ASME J. Mech. Design 100(1), 132–138 (1978). https://doi.org/10.1115/1.3453877 13. Green, I., Etsion, I.: Pressure and squeeze effects on the dynamic characteristics of elastomer o-rings under small reciprocating motion. ASME J. Tribol. 108(3), 439–444 (1986). https:// doi.org/10.1115/1.3261231 14. San Andrés, L., Jeung, S.-H., Den, S., Savela, G.: Squeeze film dampers: an experimental appraisal of their dynamic performance. In: Proceedings of the 2016 Asia Turbomachinery & Pump Symposium, Marina Bay Sands, Singapore, 22–25 February, pp. 1–23 (2016). https:// doi.org/10.21423/R12Q4N 15. San Andrés, L., Koo, B.: Effect of lubricant supply pressure on SFD performance: ends sealed with o-rings and piston rings. In: Cavalca, K.L., Weber, H.I. (eds.) IFToMM 2018. MMS, vol. 60, pp. 359–371. Springer, Cham (2019). https://doi.org/10.1007/978-3-319-99262-4_26 16. San Andrés, L., Koo, B., Jeung, S.-H.: Experimental force coefficients for two sealed ends squeeze film dampers (piston rings and o-rings), an assessment of their similarities and differences. ASME J. Eng. Gas Turb. Power 141(2), 021024 (2019). https://doi.org/10.1115/1. 4040902

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17. San Andrés, L., Koo, B.: Model and experimental verification of the dynamic forced performance of a tightly sealed squeeze film damper supplied with a bubbly mixture. ASME J. Eng. Gas Turb. Power 142(1), 011023 (2020). https://doi.org/10.1115/1.4044994 18. San Andrés, L., Rodríguez, B.: On the experimental dynamic force performance of a squeeze film damper supplied through a check valve and sealed with o-rings. ASME J. Eng. Gas Turb. Power (2021). https://doi.org/10.1115/1.4051964 19. Rodríguez, B.: Measurements of the dynamic forced response of an o-rings sealed squeeze film damper supplied with low feed supply pressure. M.S. thesis, Texas A&M University, College Station, TX (2022) 20. Payne, A.R.: The dynamic properties of carbon black-loaded natural rubber vulcanizates. Part I. J. App. Pol. Sci. 6(19), 57–63 (1962). https://doi.org/10.1002/app.1962.070061906 21. García, M.J., Kari, L., Vinolas, J., Negrete, N.G.: Frequency and amplitude dependence of the axial and radial stiffness of carbon-black filled rubber bushings. Polym. Testing 26(5), 629–638 (2007). https://doi.org/10.1016/j.polymertesting.2007.03.011 22. Rendek, M., Lion, A.: Amplitude dependence of filler-reinforced rubber: experiments, constitutive modeling and FEM – implementation. Int. J. Solid Struc. 47(21), 2918–2936. https:// doi.org/10.1016/j.ijsolstr.2010.06.021 23. San Andrés, L., Jeung, S.-H.: Orbit-model force coefficients for fluid film bearings: a step beyond linearization. ASME J. Eng. Gas Turbines Power 138(2), 022502 (2016). https://doi. org/10.1115/1.4031237

Bearing Fault Diagnosis Using Transfer Learning with ICCEMDAN Chenghui Pan1 , Song Xue1(B) , Yinwei Zhang1 , Lihui Chen1 , Peiyuan Lian1 , Congsi Wang2 , Qian Xu3 , and Wulin Zhao4 1 School of Mechano-Electronic Engineering, Xidian University, Xi’an, Shaanxi, China

[email protected]

2 Guangzhou Institute of Technology, Xidian University, Guangzhou, Guangdong, China 3 XinJiang Astronomical Observatory, China Academy of Sciences, Urumqi, China 4 The 39Th Research Institute of China Electronics Technology Group Corporation, Xi’an,

Shaanxi, China

Abstract. Bearing is one of the fundamental components in the mechanical system and sometimes, it works for a long time under extreme conditions of high speed, which inevitably induces various faults. In recent years, many machine learning methods have been widely used in bearing fault diagnosis under varying working conditions. Currently, a major assumption in machine learning is that the training and test data must have the same distribution. However, it is very difficult to effectively obtain data that meets the conditions of independent and identical distribution of training data and test data, which may result in unsatisfactory fault diagnosis results. To solve the problem, a transfer learning fault diagnosis method with Improved Complete Ensemble EMD (ICEEMDAN) has been proposed. Firstly, the fault features in the original signal are extracted by ICEEMDAN, and a vibration signal can be decomposed into multiple IMF components. Next, the correlation value between the IMF component and the original signal is calculated to obtain the related IMF. At the end, the pretrained 1-D VGG16 model is trained by the target dataset, and the related parameters are fine-tuned. The iterated 1-D VGG16 model is applied for fault identification and diagnosis on the different distribution. Based on two online bearing datasets, transfer fault diagnosis comparison experiments under different working conditions have been carried out. The diagnosis accuracy of this method can reach up to 99%. Compared with 1-D VGG16 method and 1-D CNN method, the results show that the proposed method have better performance in fault diagnosis. Keywords: Bearing · Fault Diagnosis · Data Distribution shift · Transfer Learning · ICEEMDAN

1 Introduction Bearings are one of the basic components in mechanical systems and their performance directly affects the reliability of the equipment. Sometimes working under extreme conditions at high speeds for a long time inevitably causes various failures. When the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 111–129, 2024. https://doi.org/10.1007/978-3-031-40455-9_10

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equipment is damaged, it will inevitably result in huge economic losses. Therefore, it is of great significance to quickly discover and monitor faults for safety and production efficiency [1]. Rolling bearings, with their advantages of simple manufacturing and low price, play an important role. However, bearings sometimes work under extreme conditions at high speeds for a long time and inevitably cause various failures. Therefore, monitoring the operating status of rolling bearings plays a crucial role in ensuring safe operation. In recent years, with the rapid development of artificial intelligence and machine learning, mechanical intelligent fault diagnosis has attracted the attention of many scholars. Intelligent fault diagnosis analyzes signals, automatically learns failure features, and then judges the health of the equipment, with the aim of establishing an end-to-end diagnosis model. Intelligent fault diagnosis can be roughly divided into three stages: traditional machine learning methods, deep learning methods, and transfer learning methods [2]. Traditional machine learning methods mainly include support vector machines (SVM), artificial neural networks (ANN) [3], and k-nearest neighbor classification algorithm (kNN) [4]. TUERXUN et al. [3] constructed an SSA-SVM wind turbine fault diagnosis model, which diagnoses wind turbine faults through the SVM model. SSA optimizes the model parameters and a fault set is formed, showing that the SSA-SVM diagnostic model effectively improves the accuracy and optimization capability. Rohit et al. [4] researched bearing fault classification using advanced signal processing methods and artificial intelligence technologies such as ANN and KNN. An adaptive algorithm based on wavelet transform is used to extract bearing fault features from the time-domain signal, which are then used as inputs to the ANN model, and the same features are also used in KNN. Traditional machine learning methods have a shallow network structure and are not suitable for handling large volumes of data. In recent years, some scholars have focused on researching deep learning algorithms and have achieved good results. Chen et al. [5] used convolutional neural networks (CNN) and extreme learning machines to complete the fault classification problems of gears and bearings. Hyun et al. [6] established a convolution neural network, which converted the acoustic emission signal acquired from bearings into a spectrum image in time-frequency domain, and then diagnosed the state of bearings through deep learning of various CNN models. Xie et al. [7] proposed an end-to-end fault diagnosis model based on NM-optimized adaptive DBN, which extracts deep representative features from rotating machinery and identifies bearing fault types and degrees. Deep learning method has attracted much attention and research from many scholars in feature extraction. Currently, a major assumption in deep learning methods is that the training data and test data must have the same distribution and often require a large amount of labeled data from different healthy states. However, it is very difficult to effectively obtain data that meets the independent and identically distributed conditions of training data and test data, which may result in unsatisfactory fault diagnosis results [8]. With the development of intelligent diagnostic algorithms, transfer learning methods have attracted some scholars’ research to meet the practical needs of engineering. Transfer learning algorithms can extract knowledge from one or more application scenarios to help improve learning performance in the target scenario and can solve challenging learning problems in the target domain with few or even no labeled samples [9]. With the continuous research on transfer learning algorithm, some

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research results have been achieved on transfer learning fault diagnosis method. Lei et al. [10] proposed a deep convolutional transfer learning network (DCTLN) fault diagnosis method. It helps the 1D CNN to learn domain-invariant features through conditional recognition and identify the health status and domain of the machine by adapting the minimum probability distribution distance. Reference [11] proposes a new architecture for deep adaptation network (DAN) that extends deep convolutional neural networks to domain adaptation scenarios, reducing differences in different domain distributions. For the fault diagnosis of bearings, Su et al. [12] proposed a fault diagnosis method based on a two-dimensional VGG16 convolutional neural network (CNN) and transfer learning by applying a fine-tuned VGG16 model to fault diagnosis. Han et al. [13] proposed a deep adversarial CNN (DACNN) to address the fault diagnosis problem of bearings by introducing an additional discriminative classifier as an adversarial learning in CNN. The diagnostic accuracy of transfer learning models continues to improve, but the training time of these models also becomes longer. However, in practical engineering applications, there is a pressing need for transfer learning models that are both highprecision and efficient in terms of computation time [14]. Therefore, this paper proposes an improved transfer learning fault diagnosis method called the Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN). ICEEMDAN extracts fault features from the raw signal, calculates the correlation values between the Intrinsic Mode Function (IMF) components and the original signal, and selects the IMFs with high correlation. The pre-trained one-dimensional VGG16 model is trained on the target dataset, and the relevant parameters are fine-tuned for use in the recognition and diagnosis of faults with different distributions. In the fault diagnosis using deep learning, time-domain signals, frequency-domain signals, and time-frequency domain signals can all be utilized as the inputs for the deep learning net. In our study, the goal is to provide an end-to-end diagnosis process with using one-dimensional time-domain signals. We primarily focus on using time-domain data for fault diagnosis. This approach also speeds up model training and improves fault diagnosis efficiency.

2 Signal Processing Methods ICEEMDAN is an improved version of the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) signal processing method, proposed by Colominas et al. [15]. ICEEMDAN is developed based on CEEMDAN, which uses an adaptive noise method to decompose the training dataset into a limited number of Intrinsic Mode Function (IMF) sub-sequences and a residue. Compared with other decomposition methods such as ICEEMDAN and EEMD, results show that ICEEMDAN greatly reduces the residual noise problem in the IMF while solving the mean problem caused by EEMD generating different numbers of IMF. The specific steps are as follows: 1) First, empirical mode decomposition is performed on each group of white noise signals, and the frequency of each column of noise IMF is arranged from high to low,

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resulting in a noise IMF matrix:  ⎤ ⎡  (1)   (2)  E1 w [n] E1 w [n] ... E1 w(l) [n]   ⎢ E w(1) [n] E w(2) [n] ... E w(l) [n] ⎥ 2 2 2 ⎥ E w(i) [n] = ⎢ ⎦ ⎣ ... ...  (1)   (2)   ...  (l) Ek w [n] Ek w [n] ... Ek w [n]

(1)

where, w(i) [n] denotes of noise, E(∗) denotes the  the ithgroup  of noise,  thereare l groups  IMF component, E1 w(i) [n] , E2 w(i) [n] , ..., Ek w(i) [n] denotes the 1st, 2nd , ..., kth order IMF component for the ith group of noise, respectively.   2) First-order IMF with each set of noise E1 w(i) [n] added to the initial signal separately,   (2) x(i) [n] = x[n] + ε0 E1 w(i) [n] i = 1, 2, ... where, x[n] denotes the initial sequence, the noise weighting factor ε0 = (x(i) [n]) β0 std std , and Scale factor β0 represents the inverse of the signal-to-noise ratio (E1 (w(i) [n])) of the input signal to the first added noise 3) The first-order residuals are obtained by averaging each set of signals after adding noise    (3) r1 [n] = M (x(i) [n] + ε0 E1 w(i) [n] i = 1, 2, ... where, M (∗) denotes the local mean, and x(i) [n] denotes the sequence after adding noise. 4) First-order IMF is obtained by differentiating the initial signal x[n] from the first-order residuals r1 [n], (4) IMF1 = x[n] − r1 [n] i = 1, 2, ...  (i)  5) Second-order IMF with each set of noise E2 w [n] added to the first-order residuals, and the second-order residuals r2 [n] are obtained by averaging,    (i) (5) r2 [n] = M r1 [n] + ε1 E2 w(i) [n] i = 1, 2, ... 6) Second-order IMF is obtained by differentiating the first-order residuals from the second-order residuals,    (i) (6) r2 [n] = M r1 [n] + ε1 E2 w(i) [n] i = 1, 2, ... 7) For k-order residuals and k-order IMFs, the calculations are as follows,    (i) rk [n] = M (rk−1 [n] + εk−1 Ek w(i) [n] i = 1, 2, ... k = 3, 4, ... IMFk = rk−1 [n] − rk [n] k = 3, 4, ... (i) where, the noise weight factor εk−1 = β0 std (rk−1 [n])

(7) (8)

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3 Convolutional Neural Network Model and Transfer Learning Model 3.1 1-D VGG16 Convolution Neural Network Model The VGG16 convolutional neural network model was proposed by the University of Oxford in 2014 [16]. Among many variants of VGG, VGG16 performs well in image classification and object detection tasks. The one-dimensional VGG16 model is a special form of the two-dimensional VGG16 model and consists of an input layer, convolutional layers, fully connected layers, and a softmax output layer. Compared to the two-dimensional model, the one-dimensional VGG16 model has higher computational efficiency, so this paper chooses to use the 1-D VGG16 convolutional neural network. The structure of the 1-D VGG16 convolutional neural network is shown in Fig. 1. This network uses a stack of continuous 3 × 3 convolutional layers to increase depth, which can reduce parameters on the one hand, and on the other hand, perform more nonlinear mappings to increase the fitting and expression ability of the network. The reduction in convolutional size is processed by the max pooling layer, which performs the max pooling operation in a 2 × 2 window with a stride of 2. Convolutional and pooling layers act as a feature extraction process for the input data and make the originally single linear transformation more diversified through the ReLU activation function. Then there are two fully connected layers, each containing 4096 nodes, followed by a dense layer containing 1000 units, and the last layer is a softmax layer that can classify samples.

Fig. 1. 1-D VGG16 Model

3.2 1-D VGG16 Transfer Learning Model During the process of 1-D VGG16 transfer learning, we modified the network by fixing the parameters of several initial layers and fine-tuning several subsequent layers for our own task. This greatly accelerated the training speed of the network and had a significant impact on improving the performance of our task. As shown in Fig. 2, we applied finetuning to the 1-D VGG16 model by unfreezing the last two modules (module 4 and module 5) so that their weights could be updated in each epoch of the training process.

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Fig. 2. 1-D VGG16 fine tuning model

3.3 This Paper Presents a Method In this section, ICEEMDAN and 1-D VGG16 model transfer learning are used to diagnose bearing faults, as shown in Fig. 3. ICEEMDAN is used to denoise the data, calculate the correlation, and select important IMF components. The 1-D VGG16 model, as an excellent lightweight CNN model, is chosen as the baseline model. By transferring the parameters trained in the source domain to the target model, and then fine-tuning the target model using the target domain, the training process of the model in the new domain problem is greatly simplified. As shown in Fig. 4, the specific process is as follows: 1) Vibration data under different working conditions were obtained through an accelerometer, and a source domain and a target domain were constructed. The source domain data were labeled, while the target domain data were unlabeled. 2) The data were decomposed into multiple IMF components using ICEEMDAN, and the correlation of each IMF component was calculated to select the highly correlated components, which filtered out the noise in the original signal and further extracted fault features. 3) A transfer learning model based on 1-D VGG16 convolution neural network was constructed.

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Fig. 3. ICEEMDAN and 1-D VGG16 transfer learning model

4) The model was trained using the source domain data, and the model parameters and features were trained and extracted to optimize the features for better transfer learning performance. 5) The training was continued until the given number of iterations was reached, and the trained model was obtained. 6) The model was fine-tuned with the target domain data to achieve better performance, and the training process and results were visualized and analyzed.

4 Experimental Verification 4.1 Description of Experimental Data Set This paper presents an experimental study on fault diagnosis of bearings based on vibration analysis. The experimental dataset is obtained from the bearing database of Case Western Reserve University [17]. The experimental setup, as shown in Fig. 5, consists of a 2HP (1HP = 735W) motor, a transmission system, vibration acceleration sensors, and control electronics. Single point faults are arranged on the bearings using electric spark machining, including inner race, outer race, and rolling element faults. Fault images of the bearing is shown in Fig. 6. Vibration data of the drive-end bearings are collected using accelerometers. The sampling frequency is 12 kHz, and the loads are 0, 1, and 2HP. This paper presents the IMS bearing dataset for fault diagnosis. The dataset was generated by the Intelligent Maintenance Systems (IMS) Center and selected from the bearing test platform [18]. Each bearing was equipped with an accelerometer to record its vibration data. Multiple runs to failure tests were conducted, with four bearings in each run, and the vibration signals were recorded from the beginning of the test until the bearing failed. The bearing faults were classified as inner race, outer race, and rolling element faults. Therefore, we have signals of three kinds of health condition, inner race fault condition, outer race fault and ball fault, to conduct fault diagnosis. However, we are aware of its fault type and approximate fault size, which is similar to the fault size of 0.07 inches in datasets A, B, and C. Therefore, for the following experiments, we only used fault sizes of 0.07 inches in datasets A, B, and C for tasks A-D, B-D, and C-D.

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Fig. 4. ICEEMDAN and 1-D VGG16 transfer learning diagnosis process

Fig. 5. The bearing test bench in Case Western Reserve University

To verify the generalization and accuracy of the proposed method, this study selected the bearing vibration signals from the drive end in Dataset 1, as shown in Table 1. The

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Fig. 6. Fault images of the bearing

bearing faults were of sizes 0.07, 0.014, and 0.021 inches, and the fault states were normal, inner race fault, outer race fault, and rolling element fault. The bearing fault states in Dataset 2 were normal, inner race fault, outer race fault, and rolling element fault. The sample size for each dataset is also shown in Table 1. Table 1. Experimental dataset of different Condition ID

Load

Condition

Fault size

Sample number

Dataset

A

0HP

N/I/O/B

0.07/0.014/0.021

60000*10

CWRU

B

1HP

N/I/O/B

0.07/0.014/0.021

60000*10

C

2HP

N/I/O/B

0.07/0.014/0.021

60000*10

D

/

N/I/O/B

60000*10

IMS

4.2 Data Preprocessing In general, noise is usually a high-frequency signal, while useful vibration signals are low-frequency signals. Wavelet threshold denoising has excellent ability and calculation speed in dealing with such signals. Experimental results show that the signal-to-noise ratio (SNR) of the signal decomposed by the db10 wavelet packet and the hard threshold function is better than other methods. In order to ensure that the signal features are

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distinct after decomposition, a decomposition level of 5 is selected. Figure 7 shows the original signals and the signals after wavelet filtering for four health conditions.

Fig. 7. Signal after wavelet denoising. (a) normal. (b) rolling element fault. (c) bearing inner ring fault. (d) bearing outer ring fault.

In order to further extract the features of the vibration signals, the ICEEMDAN method proposed in this paper is used for feature extraction. In order to avoid the adverse effects of unseparated noise on the diagnostic performance of the neural network model, ICEEMDAN extracts the fault features of the denoised signal. The correlation between each IMF component and the original signal is calculated, as shown in Fig. 8. Useful and distinct IMF components are obtained, and then the signal is reconstructed. The correlation calculation is given by the following equation: N i=1 Cki (n)Si (n)

(9) ρK = N C 2 (n) N S 2 (n) i=1 ki i=1 i

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Fig. 8. IMF relevance. (a) normal. (b) rolling element fault. (c) bearing inner ring fault. (d) bearing outer ring fault.

4.3 Experiments and Results 4.3.1 Bearing Experiment In order to shorten the training period, reduce the number of training samples, and improve the performance of the model, the fine-tuning method is commonly used. The most common fine-tuning method is to freeze the parameters of the model’s upper layers while continuously updating the parameters of the lower layers. Although this method has been applied to a large number of networks, it is still worth exploring in the VGG16 model. In this paper, four datasets, A, B, C, and D, are set to nine transfer tasks, A-B, A-C, B-A, B-C, C-A, C-B, as well as A-D, B-D, and C-D, as shown in Table 2. In tasks A-D, B-D, and C-D, due to the similarity between the fault size in data D and the fault size of 0.07 inches in data A, B, and C, only the data with a fault size of 0.07 inches from A, B, and C are used in the following transfer experiments. Before transfer training, the signals are first subjected to feature extraction using ICEEMDAN, and the correlation between the IMFs and the original signal is calculated using Eq. (9). The sensitivity of classification accuracy from different IMFs selections is

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Training dataset

Testing dataset

A-B

Dataset A:100%

Dataset B:60%

A-C

Dataset A:100%

Dataset C:60%

B-A

Dataset B:100%

Dataset A:60%

B-C

Dataset B:100%

Dataset C:60%

C-A

Dataset C:100%

Dataset A:60%

C-B

Dataset C:100%

Dataset B:60%

A-D

Dataset A:100%

Dataset D:60%

B-D

Dataset B:100%

Dataset D:60%

C-D

Dataset C:100%

Dataset D:60%

analyzed to select the optimal number of IMFs for reconstruction. The sensitivity analysis of classification accuracy for different tasks to this parameter is shown in Table 3. Table 3. The sensitivity analysis of classification accuracy to correlation Correlation

A-B

A-C

B-A

B-C

C-A

C-B

A-D

B-D

C-D

> 0.2

0.96

0.96

0.97

0.91

0.92

0.94

0.75

0.74

0.71

> 0.3

0.97

0.96

0.95

0.96

0.95

0.97

0.74

0.76

0.73

> 0.4

0.95

0.92

0.95

0.94

0.9

0.92

0.7

0.75

0.69

From Table 3, it is evident that the ICEEMDAN + 1-D VGG16 model achieves the highest diagnostic accuracy when using the IMFs with a correlation greater than 0.3. Therefore, in the study, the IMFs with a correlation greater than 0.3 were selected for reconstruction. The loss value between the actual label of the sample and the model prediction value is calculated through the cross-entropy function, and the weight of the proposed model is updated through Adma optimization. The proposed method is tested on various transfer tasks, and in this particular task, the diagnostic results of different models are compared, as shown in Fig. 9. The results in Fig. 9 demonstrate that, compared to the VGG16 model, ICEEMDAN and 1-D VGG16 exhibit superior diagnostic accuracy for bearing fault transfer diagnosis across different datasets and loads. This further confirms that the proposed model not only reduces noise in the bearing vibration signal, but also captures more useful features during the feature extraction stage. To intuitively understand the impact of transfer learning on the difference of feature distributions between the source and target domains, we use the confusion matrix method to map high-dimensional features to a two-dimensional space. Taking transfer task A-B

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Fig. 9. The classification accuracy at different transfer task

as an example, we present the confusion matrix of the proposed model and 1-D VGG16 model in the target domain, as shown in Fig. 10. Table 4 presents the diagnostic accuracy, training time and testing time of these two methods for transfer tasks A - B. Table 4. Diagnostic accuracy, training time and testing time of these two methods Method

Accuracy

Training time

Testing time

1-D VGG16

0.93

61s

3s

ICEEMDAN + 1-VGG16

0.97

86s

17s

From Table 4, it can be clear that the proposed method ICEEMDAN + 1-D VGG16 achieves higher diagnostic accuracy compared to 1-D VGG16, with a negligible difference in runtime. This effectively demonstrates the effectiveness of the proposed method for cross-domain diagnosis. Figure 10 (a) shows that samples with category labels 5 and 8, 6 and 8, 2 and 6, 3 and 9, 2 and 8, 3 and 7, as well as 4 and 5 are incorrect due to noise interference in the vibration signals. Additionally, as shown in Fig. 10 (b), the aforementioned issue has been largely resolved through a combined approach of ICEEMDAN and 1-DVGG16. The effective fault features obtained by ICEEMDAN make it easier for 1-DVGG16 to distinguish between different types of faults under various operating conditions. In conclusion, the proposed method has achieved significant results in bearing fault detection and localization under different operating conditions.

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Fig. 10. Confusion matrices of 1-D VGG16 and ICEEMDAN + 1-D VGG16. (a) Confusion matrix of 1-D VGG16. (b) Confusion matrix of ICEEMDAN + 1-D VGG16

To further validate the effectiveness of the proposed method, we present the confusion matrix for the proposed model and 1-D VGG16 model in diagnosing transfer task A-D, as shown in Fig. 11. The diagnostic accuracy, training time and testing time of both methods are provided in Table 5. Table 5. Diagnostic accuracy, training time and testing time of both methods Method

Accuracy

Training time

Testing time

1-D VGG16

0.56

36s

3s

ICEEMDAN + 1-VGG16

0.74

47s

10s

Similarly, from Table 5, it can be observed that the proposed method ICEEMDAN + 1-D VGG16 achieves higher diagnostic accuracy compared to 1-D VGG16, with comparable runtime. This effectively demonstrates the effectiveness of the proposed method for cross-domain diagnosis. From Fig. 11, it can be observed that the ICEEMDAN + 1-D VGG16 method achieves superior diagnostic accuracy. In conclusion, the proposed method has demonstrated significant effectiveness in bearing fault detection and localization under different operating conditions.

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Fig. 11. Confusion matrices of 1-D VGG16 and ICEEMDAN + 1-D VGG16. (a) Confusion matrix of 1-D VGG16. (b) Confusion matrix of ICEEMDAN + 1-D VGG16

4.3.2 Gearbox Experiment To further validate the proposed method, we established a parallel gearbox experimental platform in the laboratory. The structure of the gearbox experimental platform fault diagram of gearbox is shown in Fig. 12. The experimental platform mainly consists of servo stepper motor, planetary gearbox, parallel gearbox, magnetic powder brake, etc. The sampling frequency is set to 10240 Hz and the sampling time is 10s. The forms of gearbox failures include health, cracks, missing, and surface wear. Divide the collected data into training data and testing data according to 8:2. Train the ICEEMDAN + 1-D VGG16 model with 80% training data, and then test with 20% test data. Test the confusion matrix as shown in the Fig. 13. From Fig. 13, it can be seen that the diagnostic accuracy for different faults in the gearbox can reach 85%. This demonstrates that the proposed method is also effective in diagnosing faults in other machines defects.

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Fig. 12. The structure of the gearbox experimental platform

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Fig. 13. The confusion matrix of parallel gearbox fault

5 Conclusion This paper proposes an improved transfer learning fault diagnosis method based on the complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) for bearing fault diagnosis under different loads and operating conditions. The main innovations of this method are as follows: 1) This paper proposes an efficient signal decomposition and feature extraction method using ICEEMDAN to decompose the original vibration sensor signal into multiple intrinsic mode functions (IMFs), calculate the correlation between the IMFs, and then reconstruct the signal through filtering, greatly reducing the complexity of feature extraction. 2) This paper designs a one-dimensional VGG16 transfer learning model that preserves the shallow structure and weights of the pre-trained VGG16 convolutional neural network model and fine-tunes the weights in the higher layers of the 1-D VGG16 model with the target dataset. The fine-tuned VGG16 model can be applied to bearing fault diagnosis under different operating conditions. In this study, this paper proposed a method based on ICEEMDAN and 1-DVGG16 for bearing fault detection and localization under different operating conditions. Experimental results show that the proposed method can effectively extract fault features and achieve excellent classification and localization performance. Compared to using ICEEMDAN or 1-DVGG16 alone, the proposed method has higher accuracy and robustness in fault detection and localization under different operating conditions. Therefore, this method has practical application potential and important significance in bearing fault monitoring and maintenance. Acknowledgement. This research work was supported the National Natural Science Foundation of China under No. 52005377, National Key Research and Development Program of China under

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No. 2021YFC2203600 and National Natural Science Foundation of China under No. 52275269, which are greatly appreciated.

Acronyms Acronyms ICEEMDAN CNN SVM ANN kNN DCTLN DAN DACNN IMF CEEMDAN IMS SNR

Full Form Improved Complete Ensemble EMD Convolutional Neural Network The support vector machines Artificial neural networks k-nearest neighbor Deep convolutional transfer learning network Deep adaptation network Deep adversarial CNN The Intrinsic Mode Function The Complete Ensemble Empirical Mode De-composition with Adaptive Noise The Intelligent Maintenance Systems The signal-to-noise ratio

References 1. Zhong, H., Lv, Y., Yuan, R., Yang, D.: Bearing fault diagnosis using transfer learning and selfattention ensemble lightweight convolutional neural network. Neurocomputing 501, 765–777 (2022) 2. Lei, Y., Yang, B., Jiang, X., Jia, F., Li, N., Nandi, A.K.: Applications of machine learning to machine fault diagnosis: a review and roadmap. Mech. Syst. Signal Process. 138, 106587 (2020) 3. Tuerxun, W., Chang, X., Hongyu, G., Zhijie, J., Huajian, Z.: Fault diagnosis of wind turbines based on a support vector machine optimized by the sparrow search algorithm. IEEE Access. 9, 69307–69315 (2021) 4. Gunerkar, R.S., Jalan, A.K., Belgamwar, S.U.: Fault diagnosis of rolling element bearing based on artificial neural network. J. Mech. Sci. Technol. 33(2), 505–511 (2019). https://doi. org/10.1007/s12206-019-0103-x 5. Chen, Z., Gryllias, K., Li, W.: Mechanical fault diagnosis using convolutional neural networks and extreme learning machine. Mech. Syst. Signal Process. 133, 106272 (2019) 6. Hyun, P.J., Kim, C.H.: Analysis of accuracy and computation complexity of bearing fault diagnosis methods using CNN-based deep learning. J. Korea Next Generation Comput. Soc. 18, 7–18 (2022) 7. Xie, J., Du, G., Shen, C., Chen, N., Chen, L., Zhu, Z.: An end-to-end model based on improved adaptive deep belief network and its application to bearing fault diagnosis. IEEE Access. 6, 63584–63596 (2018) 8. He, J., Li, X., Chen, Y., Chen, D., Guo, J., Zhou, Y.: Deep transfer learning method based on 1D-CNN for bearing fault diagnosis. Shock. Vib. 2021, e6687331 (2021) 9. Cao, H., Shao, H., Zhong, X., Deng, Q., Yang, X., Xuan, J.: Unsupervised domain-share CNN for machine fault transfer diagnosis from steady speeds to time-varying speeds. J. Manuf. Syst. 62, 186–198 (2022)

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10. Guo, L., Lei, Y., Xing, S., Yan, T., Li, N.: Deep convolutional transfer learning network: a new method for intelligent fault diagnosis of machines with unlabeled data. IEEE Trans. Ind. Electron. 66, 7316–7325 (2019) 11. Long, M., Cao, Y., Wang, J., Jordan, M.I.: Learning Transferable Features with Deep Adaptation Networks (2015) 12. Su, J., Wang, H.: Fine-tuning and efficient VGG16 transfer learning fault diagnosis method for rolling bearing. In: Zhang, H., Feng, G., Wang, H., Gu, F., and Sinha, J.K. (eds.) Proceedings of IncoME-VI and TEPEN 2021. pp. 453–461. Springer International Publishing, Cham (2023) 13. Han, T., Liu, C., Yang, W., Jiang, D.: A novel adversarial learning framework in deep convolutional neural network for intelligent diagnosis of mechanical faults. Knowl.-Based Syst. 165, 474–487 (2019) 14. Qian, C., Zhu, J., Shen, Y., Jiang, Q., Zhang, Q.: Deep transfer learning in mechanical intelligent fault diagnosis: application and challenge. Neural Process. Lett. 54, 2509–2531 (2022)https://doi.org/10.1007/s11063-021-10719-z 15. Huang, D., Li, S., Qin, N., Zhang, Y.: Fault diagnosis of high-speed train bogie based on the improved-CEEMDAN and 1-D CNN algorithms (vol 70, 3508811, 2021). IEEE Trans. Instrum. Meas. 70, 9900601 (2021) 16. Simonyan, K., Zisserman, A.: Very Deep Convolutional Networks for Large-Scale Image Recognition. CoRR. (2014) 17. Case Western Reserve University, https://engineering.case.edu/bearingdatacenter. Accessed 26 Jan 2023 18. IMS Bearing Dataset, https://data.nasa.gov/download/7wwx-fk77/application%2Fzip. Accessed 26 Jan 2023

Dynamic Modeling and Stability Analysis of the Rod-Fastening Rotor System with the Different Preload Status Chongyang Wang1(B) , Xilong Hu1 , Fan Qin1 , and Lihua Yang1,2,3 1 State Key Laboratory for Strength and Vibration of Mechanical Structures,

Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China [email protected] 2 Shaanxi Key Laboratory of Environment and Control for Flight Vehicle, Xi’an Jiaotong University, No.28 Xianning West Road, Xi’an 710049, Shaanxi, People’s Republic of China 3 School of Aerospace Engineering, Xi’an Jiaotong University, No.28 Xianning West Road, Xi’an 710049, Shaanxi, People’s Republic of China

Abstract. With the development of gas turbine technology, disk-distributed rodfastening combined rotor systems are widely used in power machines such as heavy-duty gas turbines and aero engines. The connection structure in rodfastening combined rotor is subjected to high temperature, unbalanced forces, and moments during operation, which can affect the vibration characteristics and dynamic stability of the combined rotor. However, few studies have been conducted to investigate the rotor vibration characteristics considering the dynamic parameters of the rod-fastening connection structure. Therefore, this paper establishes the dynamic equations of the rod-fastening combined rotor, which considers the contact effect between disks and the equivalent simplification of the rod preload force. The New-mark-β method is applied to numerically solve the kinetic equations to investigate the kinetic and bifurcation characteristics of rotor systems with different preload states under nonlinear oil film forces. The timedomain vibration waveform, three-dimensional spectrograms, the bifurcation diagram, and the poincaré section are applied to investigate the dynamic behavior of the rod-fastening rotor under different preload conditions. The results show that different preload states cause various frequency components of the system, and insufficient preload raises the system’s instability threshold, which even leads to oil whip. The research results help to detect the state of the rod-fastening rotorbearing system and prevent failures due to changes in the preload state of the fastened rod . Keywords: rod-fastening rotor · preload · dynamics · nonlinear dynamic response

1 Introduction The rod-fastening combined rotor (RFCR) system represents a prevalent mode of mechanical design, extensively utilized across various industrial sectors. The system consists of a rotor, supported by bearings and fastened through a series of tie-rods, also © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 130–148, 2024. https://doi.org/10.1007/978-3-031-40455-9_11

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referred to as rod-fasteners. These tie-rods provide a secure and stable connection to the system while reducing the overall mass of the rotor. Owing to its remarkable robustness and longevity, the RFCR is utilized in diverse industries such as power generation, oil and gas, and heavy machinery. In the context of the RFCR, the tie-rod constitutes a crucial component. Nevertheless, within the current engineering practices, the dynamic analysis of RFCRs is predominantly simplified by applying a continuous model. This approach fails to account for the complex structural reality of the system, thereby hindering the acquisition of precise rotor dynamic characteristics during fault analysis and structural design. This lack of accuracy has posed significant challenges in the industry’s creation of gas turbines and related diagnostic work. To understand the behavior of tie-rod assemblies comprehensively, it is imperative to examine the impact of preload on the rotor dynamic properties. The rotor dynamic behavior of RFCR differs significantly from standard rotors due to the presence of multiple contact interfaces. The traditional methods of rotor dynamics calculation, which often overlook the effects of these contact interfaces and treat the rotor as a homogeneous entity, have been shown to have a substantial error when applied to RFCRs. This necessitates the conduct of numerous experiments to correct the predictions. To reduce the dependence on experimental data and account for the contact interfaces in the analysis of tie-rod combined rotors, Rao [1] proposed a model in which the inter-disk contact was modeled as a distributed spring. In this model, the rotor and disk ends were treated as beams, and the elastic-plastic connection between the disks was considered to calculate the equivalent bending stiffness. This method was used to establish the motion equation of RFCR and to determine the critical speed. The work of He et al. [2–4] focuses on investigating the elastic-plastic contact between the disks, determining the equivalent bending stiffness between the disks, and formulating the motion equation of the RFCR. Subsequently, the critical speed of the system is calculated through analysis of the motion equation. Yuan et al. [5] considered the effect of preload and calculated the normal contact stiffness at the contact interface. At the same time, Yuan et al. [6] studied the equivalent stiffness of a heavy-duty gas turbine incorporating coupling and main shaft bolts. Gao et al. [7] calculated the bending stiffness of the contact surface of the tie-rod rotor under both integral contact and separation contact states. For a fixed preload, the bending stiffness of the contact layer was found to decrease with increasing bending moment during contact and increase with growing bending moment during separation. Zhang et al. [8] conducted a comparative study of the modal test results of a series of RFCRs under various preload and roughness conditions. Through numerical simulation, he obtained the natural frequencies for different contact stiffnesses. He established a relationship between contact stiffness and tension and between contact stiffness and surface roughness. The gas turbine operates in a high-temperature environment for extended periods. For a rotor with a tie-rod structure, the preload force of the tie-rod has a significant impact on the characteristics of the gas turbine rotor. Inadequate uniformity in the fastening force of different tie-rods can result in relaxation, material creep, and other detrimental effects that impact the nonlinear dynamic behavior of the combined rotor system, leading to potential failures. Wu et al. [9] employed the nonlinear contact stiffness matrix to characterize the interaction between the discs and developed a mechanical model of

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the circumferentially distributed connecting rods. He analyzed the influence of preload, rod diameter, contact surface roughness, deformation phase, and uneven preload on the stiffness coefficient. Li et al. [10, 11] investigated the impact of the bending stiffness of the bolted disc on the system response and validated the results of the simulation study through rotor dynamic response experiments. Wang et al. [12] studied the rotor dynamic behavior of a rod-fastening rotor (RFR) system incorporating internal damping. He analyzed the response amplitude and instability threshold at varying speeds, considering the impact of internal damping. The study’s results revealed that internal damping has a dual effect on the RFR response, with vibration amplitude decreasing at low speeds and increasing significantly at high speeds. Xu et al. [13] developed the dynamic model of the rod-fastening Jeffcott rotor (RFJR). They analyzed the effect of preload and rod detuning rate on the nonlinear dynamic characteristics of the rotor system. In the study by Zhao et al. [14], the effect of contact stiffness on the bifurcation characteristics of displacement and spectral components of the RFCR was investigated. A comparison was made between the preload’s uniform and non-uniform relaxation on the rotor system’s dynamic response. Hu et al. [15–17] conducted a comprehensive investigation of the nonlinear dynamic properties of the RFCR, incorporating several important factors such as the nonlinear rubbing force, nonlinear oil film force, and unbalanced mass. In his analysis, the nonlinear contact characteristics between the discs were modeled using a bending spring with nonlinear stiffness. Hei et al. [18–20] analyzed the nonlinear dynamic behavior of RFCR by a combined bearing system consisting of fixed and tilting pads. The study investigated the impact of tilting pad block moment of inertia, fulcrum ratio, and preload on the nonlinear dynamic response of the rotor. Zhang et al. [21, 22] comprehensively analyzed RFCR, considering various influential factors such as rotor cracks and rubbing. This study revealed the interplay between the tie-rod combined rotor system’s parameters and fault characteristics. Despite the extensive examination of the mechanical properties of RFCR in previous literature, the impact of tie-rod preload on the nonlinear dynamics of the rotor system often relies on the lumped mass method and overlooks the gyroscopic effect. Hence, there is a need for more comprehensive research that incorporates these essential features into a suitable rotor model. In prior work ([13]), a mathematical model of the tie-rod connection was established to investigate the influence of tie-rod connection parameters on the rotor dynamics, particularly on the critical speed. This paper aims to provide a comprehensive overview of the current research literature on the effect of tie-rod preload on rotodynamic characteristics and to emphasize the significance of this topic in the design and analysis of rotor systems.This study utilizes the lumped mass method (as cited in references [11, 12]) to analyze the dynamic performance of RFCR, incorporating the tie-rod connection into the overall dynamics model of the rotor. A mathematical model incorporating the nonlinear restoring force of the tie-rod is employed to analyze the system’s nonlinear dynamic response and stability under different preload statuses through bifurcation diagrams, waterfall diagrams, time-domain waveform diagrams, axis center trajectory diagrams, and frequency response function colormaps. The findings of this paper can aid in preventing failures caused by loose tie-rod preload.

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2 Modeling of RFCR System To advance the study of the dynamics on the RFCR system, a comprehensive mathematical model is established to analyze and understand the behavior of such systems. The dynamic model incorporates various factors that impact the system’s behavior, including joint stiffness, the nonlinear restoring force of the tie-rods, and the nonlinear oil film force. By integrating these factors into the rotor model, the resulting motion equation of the 4-node rotor provides a comprehensive understanding of the system’s behavior. 2.1 Structure of Tie-rod Combined Rotor The significant distinction between the RFCR and the traditional continuous rotor lies in the presence of multiple contact interfaces and structural discontinuity in the former. To analyze the nonlinear dynamic behavior of combined rotors under varying tie-rod preloads, a classical single-interface RFCR model (depicted in Fig. 1) is utilized. The two discs of the model are connected by eight tie rods. It not only significant structural properties of heavy-duty gas turbine rotors are preserved, and the calculation process is also simplified. The relevant parameters of the tie-rod combined rotor-bearing system are presented in Table 1. At node i, the concentrated mass, polar moment of inertia, and diameter moment of inertia are denoted as mi , J pi , and J di , respectively. Table 1. Parameters of the RFR-bearing system. Parameter

Value

Density ρ (kg/m3 )

7800

Elastic modulus of shaft E (Gpa)

200

Length l 1 , l 2 , l 3 (mm)

200,90,200

Outer diameter D1 , D2 , D3 (mm) Inner diameter d 1 , d 2 , d 3 (mm)

50,200,160

m1 , m2 , m3 , m4 (kg)

0.747,4.451,4.451,0.747

(kg m2 )

25,25,140

J p1 , J p2 , J p3 , J p4 J d1 , J d2 , J d3 , J d4 (kg m2 )

2.07 × 10–4 , 8.79 × 10–3 , 8.79 × 10–3 , 2.07 × 10–4

Stiffness coefficient k1 , k1

1e7,1.5e7

Bearing clearance c (mm)

0.11

Bearing diameter D (mm)

50

1.08 × 10–3 , 9.03 × 10–3 , 9.03 × 10–3 , 1.08 × 10–3

Bearing length L (mm)

12

Lubricating viscosity η (Pa s)

0.018

Coefficient ξ 1 , ξ 2

0.02, 0.02

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Fig. 1. Schematic diagram of the rotor-bearing system

2.2 Motion Equation of Tie-rod Combined Rotor To study the nonlinear dynamic behavior of the RFCR system depicted in Fig. 1, a finite element approach utilizing the Timoshenko beam theory is employed to discretize the rotor. In the discretization process, the corresponding mass and moment of inertia are concentrated in each node to form a mass matrix. A mechanical model of the Timoshenko beam-shaft element is combined with the Lagrange equation to derive the stiffness matrix and gyro force matrix of the bending vibration. The contact effect between the discs can be modeled as the interaction of a shear spring with a tangential stiffness and a bending spring with a bending stiffness, contributing to the overall stiffness matrix of the RFCR system. A schematic representation of the stiffness matrix assembly is presented in Fig. 2, where the expressions of K x and K y are shown in the Appendix A.

Fig. 2. Schematic diagram of rotor matrix assembly

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A 16-DOF bending vibration differential equation describing the dynamic behavior of the tie-rod combined rotor-bearing system is established as follows: M q¨ − (C − ωG)˙q + Kq = Q

(1)

where q is the displacement vector of the rotor system, which can be expressed as:  T q = x1 , θy1 , x2 , θy2 , x3 , θy3 , x4 , θy4 , y1 , θx1 , y2 , θx2 , y3 , θx3 , y4 , θx4

(2)

M, G, C, Kand Q are the mass matrix, gyro matrix, damping matrix, connection stiffness matrix, shaft section stiffness matrix, and load matrix of the tie-rod combined rotor-bearing system, which is expressed as:   Mx 0 , Mx = My = diag([m1 , Id 1 , m2 , Id 2 , m3 , Id 3 , m4 , Id 4 ]), M = (3) 0 My     0 J1 G=J = , J1 = diag 0, Jp1 , 0, Jp2 , 0, Jp3 , 0, Jp4 (4) T −J1 0 K = KR + KL

(5)

K R is the connection stiffness matrix, and K L is the element stiffness matrix of the shaft section. Based on the first two flexural vibration modes ωn1 , ωn2 and damping coefficient ξ1 , ξ2 measured in experiments, the shaft’s flexural vibration damping matrix is obtained using the Rayleigh damping model. C = αM + βK 4π ωn1 ωn2 (ξ1 ωn2 − ξ2 ωn1 ) ξ2 ωn2 − ξ1 ωn1 ,β = 2 2 2 − ω2 ) (ωn2 − ωn1 ) π(ωn2 n1 ⎤ ⎤ ⎡ e ⎡ e Ky1 Kx1   Kx 0 ⎥ ⎥ ⎢ ⎢ .. .. , Kx = ⎣ KL = ⎦, Ky = ⎣ ⎦ . . 0 Ky e e Kxi Kyi ⎡ ⎤ ⎡ ⎤ 12 6li −12 6li 12 −6li −12 −6li 2 2 ⎥ ⎢ EIi ⎢ 6li 4li2 −6li 2li2 ⎥ ⎥, K e = EIi ⎢ −6li 4li 6li 2li ⎥ Kxei = 3 ⎢ y i li ⎣ −12 −6li 12 −6li ⎦ li3 ⎣ −12 6li 12 6li ⎦ 2 2 6li 2li −6li 4li −6li 2li2 6li 4li2 α=

(6) (7)

(8)

(9)

where E is the modulus of elasticity, I i is the area moment of inertia, and li is the length of the beam element, ⎤ ⎡ .. . ⎤ ⎡ ⎥ ⎢ y ks 0 −ks 0 ⎥ ⎢ K R ⎥ ⎢ ⎢ 0 kθ 0 −kθ ⎥ ⎥ x ⎢ y .. ⎥ (10) KR = ⎢ ⎥, KR = KR = ⎢ . ⎣ −ks 0 ks 0 ⎦ ⎥ ⎢ y ⎥ ⎢ KR ⎦ ⎣ 0 −kθ 0 kθ .. .

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It should be noted that the stiffness of the contact interface is a function of the pre-tension force. Please note that the stiffness matrix K R of the contact surfaces is a function of the preload force. Upon the application of the preload force by the tie rod, the actual contact between the two surfaces occurs on the surface of the discontinuous micro-convex body, owing to the influence of the surface morphology, as depicted in Fig. 3.

Fig. 3. Schematic diagram of planes contacts

When the two rough surfaces initially make contact, a few microconvexes are deformed first, and recoverable elastic deformation occurs under the external load. As the external load increases, the surfaces come closer, and the number of microconvexes in contact increases to balance the increasing external load. When the external load reaches a certain value, the microconvexes experience elastic or even plastic deformation. To investigate the contact stiffness characteristics between the discs, the contact between the two is modeled as a rough surface in contact with a smooth rigid plane, according to the GW model and Hertz contact theory. This allows for the determination of the stress on the unit nominal area as ∞ 2 3 − z 4 ∗ ∗ 12 (z − h − yθ ) 2 e 2σ 2 dz (11) py = √ nE R 3σ 2π h+yθ The equivalent contact stiffness of the contact interface is ∞ 2 dpy 1 1 1 − z = 2ηE ∗ R∗ 2 (z − h − yθ ) 2 √ e 2σ 2 dz kc = − d (h) σ 2π h+yθ The equivalent bending stiffness of the contact interface is [13] ∞ 2 1 3 − z 2 ∂M0 = − √ ηE ∗ R∗ 2 (z − h − yθ ) 2 e 2σ 2 · y2 dzdAnom Gc = ∂θ σ 2π Anom h

(12)

(13)

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Table 2. Parameters of the RFR-bearing system. σ /μm 2.01

R/μm

η/(1/m2 )

θ

95

5.62 × 107

1 × 10−7 rad

The detailed derivation of Eqs. (11–13) are in Appendix B. The surface contact model parameters [2] are as shown in the Table 2. The relationship between the preload (P) and the connection stiffness of the contact shaft, as described by the GW model, can be represented mathematically in Fig. 4 [2, 13, 14].

Fig. 4. The relationship between preload and bending stiffness.

Given the nonlinear relationship present in the bending vibration of the tie-rod combined rotor, this paper employs dimensionless treatment for each dynamic parameter in the relevant equation(τ = ωt; u = q/c; Xi = xi /c; Yi = yi /c (i = 1,2,3,4)), effectively reducing the parameters of the shaft system to a small value and transforming the motion into a weak linear problem. Furthermore, the dimensionless treatment of the vibration dynamics differential equation minimizes the differences in response among the tie-rod rotor with multiple degrees of freedom, thereby avoiding computational issues associated with ill-conditioned problems and reducing calculation time. This dimensionless processing leads to the differential equation of the bending dynamics of RFCR as: ω2 M u¨ + ω(C + ωG)˙u + Ku =

Qu + Qb − Qg − Qcx c

(14)

where Qu , Qg , and Qcx are the vector representation of the unbalanced force, gravity force, and the nonlinear restoring force generated by the tie-rod and the contact region,

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respectively, as specified in reference [16]. Qu =

[0, 0, m2 e1 ω2 cos(τ ), 0, m3 e2 ω2 cos(τ + φ), 0, 0, 0, 0, 0, m2 e1 ω2 sin(τ ), 0, m3 e2 ω2 sin(τ + φ), 0, 0, 0]T

(15)

Qg = [0, 0, 0, 0, 0, 0, 0, 0, 0, m1 g, 0, m2 g, 0, m3 g, 0, m4 g, 0]T

(16)

Qcx = [0, 0, Qcx1 , 0, Qcx2 , 0, 0, 0, 0, 0, Qcy1 , 0, Qcy2 , 0, 0, 0]T

(17)



Qcx1 Qcy1



 =

−Qcx2 −Qcy2



 =

k1 (X2 − X3 ) + k1 (X2 − X3 )3 k1 (Y2 − Y3 ) + k1 (Y2 − Y3 )3

 (18)

In the formula, e1 and e2 are the unbalanced eccentric distances of discs 1 and 2; ϕ is the initial phase angle of the two discs; g is the gravitational constant; k 1 , k 1 ’ are the linear and nonlinear stiffness coefficients between the tie rod and the disc. Qb is the nonlinear oil film force vector based on short bearing theory. According to the Capone model [18–20], the nonlinear oil film force acting on nodes 1 and 4 is determined as follows: [Qbx (X1 , Y1 , X1 , Y1 ), 0, 0, 0, 0, 0, 0, Qbx (X4 , Y4 , Y4 ), 0, (19) Qby (X1 , Y1 , X1 , Y1 ), 0, 0, 0, 0, 0, 0, Qby (X4 , Y4 , Y4 ), 0]T 



 (X − 2Y˙ )2 + (Y + 2X˙ )2 Qbx X × V − sin ϕ × G − 2 cos ϕ × S = −sW × Qby 3Y × V + cos ϕ × G − 2 sin ϕ × S 1 − X2 − Y2 (20) Qb =

V (X , Y , ϕ) =

2 + (Y cos ϕ − X sin ϕ) × G(X , Y , ϕ) 1 − X2 − Y2

(21)

X cos ϕ + Y sin ϕ 1 − (X cos ϕ + Y sin ϕ)2

(22)

S(X , Y , ϕ) =

Y cos ϕ − X sin ϕ 2 π G(X , Y , ϕ) = √ × ( + arctan √ ) 2 2 2 1−X −Y 1 − X2 − Y2 ϕ = arctan

Y + 2X˙ Y + 2X˙ π π − sign − sign(Y + 2X˙ ) 2 2 X − 2Y˙ X − 2Y˙

(23) (24)

R 2 L 2 where s = ηωRL W ( c ) ( 2R ) is the Sommerfeld coefficient, η is the oil film viscosity, and W is the weight of the rotor.

3 Numerical Results and Discussion In this study, a numerical investigation is conducted to examine the nonlinear dynamic behavior of the RFCR bearing system while considering the impact of different preload states of the tie-rods. The system is equipped with sliding bearings on both sides and is

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characterized by consistent structural parameters and lubrication conditions. The nonlinear differential equation outlined in Eq. (10) is solved using the efficient and suitable Newmark-β method, with a numerical integration step size of 2π/512. The response of the rotor is calculated for 500 cycles, with initial values for the displacement, velocity, and acceleration vectors being randomized and subject to correction via fixed-point iterations. After disregarding the first 300 cycles, the steady-state response of the rotor in the x/y directions is obtained. The nonlinear dynamic characteristics are analyzed using bifurcation diagrams, three-dimensional spectrums (waterfall diagrams), Poincaré diagrams, axis trajectories, vibration time-domain waveforms, and response displacements, with rotational speed and connection stiffness serving as control parameters. The results provide a comprehensive understanding of the nonlinear dynamic behavior of the RFCR system, taking into account the effects of preload state, and demonstrate unique and intriguing phenomena. 3.1 Validation To validate this paper’s established mathematical model and solution process, the calculation parameters were replaced with those of the bolted thin-shaft rotor in Reference [11].

(a)

(b) Fig. 5. Comparison of the time-domain diagrams of the rotor system in this paper and the literature [10] at different speeds: (a) the results of the literature [10], (b) the results of this paper.

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The responses were then calculated and compared. Figure 5 displays the time domain waveforms of the rotor system at rotational speeds of 2800, 4400, and 6000 rpm. The results from this study are consistent with those reported in the literature, thus verifying the validity of the mathematical formulas and calculation codes proposed in this paper. 3.2 Nonlinear Dynamics Comparison of RFCR Under Different Preload Conditions To examine the dynamic behavior of the RFCR system, the Newmark-β method and MATLAB simulation software were employed to solve the dynamic equations. The analysis of the disk-to-disk contact effect indicated that the pre-tensioning force impacts the rotor’s radial displacement and displacement angle by altering the shear stiffness and the bending stiffness.

Fig. 6. Bifurcation diagram and three-dimensional spectrum at lumped mass point 2

Figure 6 illustrates a bifurcation diagram and a three-dimensional frequency spectrum of the dimensionless displacement X of disk 1, in the x-direction, as a function of rotational speed ω, at a preload force of 400 kN. The rotational speed is varied in the range of 600–12000 rpm. The results reveal that only the fundamental frequency is observed when the rotational speed is below 5800 rpm, and the system response is synchronous, with stable motion. Upon reaching 5800 rpm, period-doubling bifurcation is observed. In the local three-dimensional spectrum, the 1/2 harmonic frequency 0.5fr indicates that the rotor system has transitioned to a 2-period motion. With further increased rotational speed, the rotor system enters a quasi-periodic movement at 11000 rpm (Figs. 7, 8, 9 and 10). The dynamic characteristics of the RFCR at different rotational speeds, under a preload force of 400 kN, are depicted in Fig. 7 to Fig. 11 through time domain plots, spectral plots, axial trajectories, and Poincaré plots. The representation enables more visual comparison of the nonlinear dynamic behavior of the system at varying speeds. The figure shows that when the rotational speed is 800 rpm, the axis trajectory is not a stable circle. Some scattered points are present on the Poincaré diagram, indicating a chaotic motion. As the speed increases to 1200 rpm, the axis center trajectory becomes a closed

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(b)

(c)

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(d)

Fig. 7. Vibration response at ω = 800 rpm: (a) time-domain, (b) frequency domain, (c) axis center trajectory, and (d) Poincaré map

(a)

(b)

(c)

(d)

Fig. 8. Vibration response at ω = 1200 rpm: (a) time-domain, (b) frequency domain, (c) axis center trajectory, and (d) Poincaré map

(a)

(b)

(c)

(d)

Fig. 9. Vibration response at ω = 2800 rpm: (a) time-domain, (b) frequency domain, (c) axis center trajectory, and (d) Poincaré map

(a)

(b)

(c)

(d)

Fig. 10. Vibration response at ω = 6000 rpm: (a) time-domain, (b) frequency domain, (c) axis center trajectory, and (d) Poincaré map

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(a)

(b)

(c)

(d)

Fig. 11. Vibration response at ω = 12000 rpm: (a) time-domain, (b) frequency domain, (c) axis center trajectory, and (d) Poincaré map

circle, the Poincaré diagram converges to a single point, and the system enters periodone(P1) motion. At 5800 rpm, the rotor enters a period-two(P2) motion, as evident from the two isolated points in the Poincaré diagram and the bifurcation point in the bifurcation diagram. The system response is dominated by the vibration amplitude generated by 0.5f r . With a further increase in speed, the bifurcation diagram demonstrates a gradual increase in response. The isolated points in the Poincaré diagram form a closed loop, indicating a quasi-periodic motion. Thus, as the speed increases, the dynamic behavior of the system can be summarized as chaos-(P1)-(P2)-(quasi-periodic motion). The preload applied to the tie-rods significantly impacts the rotor-bearing system’s dynamic behavior. As depicted in Figs. 12, 13, 14 and 15, the bifurcation diagrams, waterfall diagrams, and spectrum colormap diagrams demonstrate the tie-rod preload’s effect on the system’s nonlinear dynamic characteristics. The results indicate that the preload is a crucial parameter that influences the stability of the rotor. As the preload decreases, the critical speed for half-frequency whirl instability decreases, and the vibration amplitude of RFCR gradually increases. When the preload is reduced to 5 kN, the main components of the frequency components change. The colormap shows that the oil whirl gradually changes into oil whip with the increase of speed. From the perspective of response amplitude, the system exhibits more severe vibration, and the main frequency components are locked. Hence, the results suggest that insufficient preload causes increased instability at high speeds, reducing the instability threshold.

(a)

(b)

Fig. 12. Bifurcation diagram and spectrum (colormap) at lumped mass point 2 when P = 400 kN

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Fig. 13. Bifurcation diagram and spectrum (colormap) at lumped mass point 2 when P = 200 kN

(a)

(b)

Fig. 14. Bifurcation diagram and spectrum (colormap) at lumped mass point 2 when P = 100 kN

(a)

(b)

Fig. 15. Bifurcation diagram and spectrum (colormap) at lumped mass point 2 when P = 50 kN

4 Conclusions This study used lumped mass modeling to establish a dynamic model of a rotor-bearing system with tension rod restoring force and nonlinear oil film force. This model considered the Timoshenko beam theory and the gyroscopic effect. The study investigated the impact of tie-rod preload on the nonlinear dynamic response of the tie-rod composite

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rotor by using rotational speed and tie-rod preload as control parameters for parametric research. The results were presented through various visual representations, including bifurcation diagrams, three-dimensional waterfall diagrams, Poincaré diagrams, axis locus diagrams, time-domain waveform diagrams, frequency spectra, response amplitudes, and colormaps. The findings demonstrated that the tie-rod preload directly influenced the contact characteristics between the rotor discs, leading to alterations in the nonlinear dynamic characteristics of the RFCR system. Specifically, the preload of the tie-rod modified the instability threshold of the system and reduced the instability threshold of the oil film oscillation by affecting the contact interface between the rotor disks. Additionally, the vibration amplitude of the RFCR system decreased with increasing preload and became more stable. The study highlights the importance of considering the preload state of the tie-rod for rotors with sliding bearings and tie-rods, provides valuable insights for fault diagnosis and vibration reduction, and a deeper understanding of the nonlinear dynamic characteristics of RFCRs. In this paper, it is assumed that the preload force of the tie rod is uniformly distributed and that the material is in complete elastic contact. However, in an actual combined rotor, the preload force in the circumferential direction of the tie rod may differ due to uneven preload force during assembly, stress relaxation of the tie rod, high temperature creep of the material, and other factors, which may result in some plastic deformation. Therefore, further research is required to optimize the above mentioned influencing factors. Acknowledgement. The authors are very grateful for the support of the National Major Science and Technology Major Project [grant numbers J2019-IV-0021-0089]; the National Natural Science Foundation of China [grant numbers 11872288].

Appendix A ⎡

12EI L3

6EI L2

⎢ ⎢ ⎢ 6EI 4EI ⎢ L2 L ⎢ ⎢ ⎢ 12EI 6EI ⎢− 3 − 2 L ⎢ L ⎢ ⎢ 2E1 ⎢ 6EI ⎢ L2 L ⎢ Kx = ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ ⎢ ⎢ ⎢ 0 0 ⎢ ⎣ 0 0

− 12EI L3

6EI L2

0

0

0

− 6EI L2

2EI L

0

0

0

− 6EI L2

−ks

0

0

0

kθ

0

6EI L2

− 12EI L3

12EI L3

+ ks

− 6EI L2

4EI L

+ kθ

−ks

0

0

kθ

6EI L2

0

0

− 12EI L3

− 6EI L2

12EI L3

0

0

6EI L2

2EI L

− 6EI L2

12EI L3

+ ks

4EI L

+ kθ − 6EI L2

0

⎤

⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 6EI ⎥ ⎥ L2 ⎥ ⎥ ⎥ 2EI ⎥ L ⎥ ⎥ ⎥ 6EI ⎥ − L2 ⎥ ⎥ ⎦ 4EI L

Dynamic Modeling and Stability Analysis of the Rod-Fastening Rotor System

⎡

12EI L3

⎢ ⎢ ⎢ − 6EI ⎢ L2 ⎢ ⎢ ⎢ 12EI ⎢− 3 ⎢ L ⎢ ⎢ ⎢ − 6EI ⎢ L2 ⎢ Ky = ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎣ 0

− 6EI L2

− 12EI L3

− 6EI L2

0

0

0

4EI L

6EI L2

2EI L

0

0

0

6EI L2

−ks

0

0

0

kθ

0

− 6EI L2

− 12EI L3

6EI L2

12EI L3

+ ks

2E1 L

6EI L2

0

−ks

0

0

0

kθ

− 6EI L2

0

0

0

− 12EI L3

− 6EI L2

12EI L3

0

0

0

− 6EI L2

2EI L

6EI L2

4EI L

+ kθ 12EI L3

+ ks

4EI L

+ kθ − 6EI L2

0

145

⎤

⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ − 6EI L2 ⎥ ⎥ ⎥ 2EI ⎥ L ⎥ ⎥ ⎥ 6EI ⎥ − L2 ⎥ ⎥ ⎦ 4EI L

Appendix B According to the Hertz contact theory, the contact between two elastic bodies can be modeled as the contact between a sphere with an equivalent radius of curvature R* and an equivalent elastic modulus E* with a rigid plane under the load W. As depicted in the figure, W denotes the load, δ represents the deformation, a stands for the radius of the actual contact area, and e denotes the actual contact area (Fig. 16).

Fig. 16. Schematic diagram of contact between elastic sphere plane and rigid plane

The equivalent radius R* and equivalent elastic modulus E* can be calculated using the following equation: 1 − ν12 1 − ν22 1 1 1 1 = + , ∗ = + ∗ R R1 R2 E E1 E2

(B1)

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The radius “a”, deformation “δ”, and contact load “W ” of a micro-convex body can be determined through calculations based on the principles of elastic mechanics, as follows: 3WR∗ 1 )3 4E ∗

(B2)

9W 2 1 a2 = ( )3 R∗ 16R∗ E ∗2

(B3)

4 ∗ ∗1 3 E R 2δ2 3

(B4)

a=( δ=

W =

In a real rough surface, multiple micro-convex bodies are in contact with each other. According to the G-W model, assuming that the micro-convex body height z follows a normal distribution with a distribution function ρ(z)(N(μ, σ 2 )), the probability that the height of a micro-convex body falls within the range of z to dz is ρ(z)dz. The tips of the micro-convex bodies on the rough surface are all spherical with radii of curvature being equal. The probability of contact occurring for a given microconvex body on a rough surface can be expressed as the equation ∞ 2 1 − z ρ(z)dz, ρ(z) = √ e 2σ 2 (B5) P(z > h) = σ 2π h If there are n microconvex bodies on the surface, then the number of microconvex bodies in contact can be calculated as ∞ ρ(z)dz (B6) N =n h

The actual contact area of the two planes can be determined using the above formula: ∞ ∗ Anom = π NR (z − h)ρ(z)dz (B7) h

The aggregate load produced by the microconvex body in contact is: 3 4 ∗ ∗1 ∞ W = NE R 2 (z − h) 2 ρ(z)dz 3 h

(B8)

The load per unit area that is applied on the two contact planes is known as the contact pressure. ∞ 2 1 3 − z W 4 P= = (z − h) 2 e 2σ 2 dz (B9) √ ηAnom E ∗ R∗ 2 Anom 3σ 2π h The contact stiffness is k=−

1 dpunit = 2ηE ∗ R∗ 2 d (d )



∞ h

2 1 1 − z (z − h) 2 √ e 2σ 2 dz σ 2π

(B10)

Dynamic Modeling and Stability Analysis of the Rod-Fastening Rotor System

The bending stiffness of the contact layer is: Gc = kc y2 dA = kc Ia

147

(B11)

A

During rotor vibration, the two rough contact surfaces do not remain perfectly parallel, which results in a relative rotation angle denoted by θ. The distance between the two contact surfaces in the radial direction varies and is denoted by d. d = h + yθ

(B12)

The stress per unit area corresponds to different positions, depending on the aforementioned parameters. ∞ 2 1 3 − z 4 (z − h − yθ ) 2 e 2σ 2 dz (B13) py = √ nE ∗ R∗ 2 3σ 2π h+yθ The stress integral is performed over the entire contact surface, resulting in the bending moment generated at the center of the contact surface when the rotation angle between the discs is θ. M0 = −

py · ydAnom = −

Anom

4 √

3σ 2π

1

ηE ∗ R∗ 2

Anom

∞

3

(z − h − yθ) 2 e

−

z2 2σ 2

dz · ydAnom

(B14)

h+yθ

Since θ is a very small value, Eq. (7) can be simplified to ∞ 2 3 − z 4 ∗ ∗ 12 (z − h − yθ ) 2 e 2σ 2 dz · ydAnom M0 = − √ ηE R 3σ 2π An0m h

(B15)

The equivalent bending stiffness of the contact interface is expressed as follows: ∞ 2 1 3 − z 2 ∂M0 = − √ ηE ∗ R∗ 2 Gc = (z − d0 − yθ ) 2 e 2σ 2 · y2 dzdAnom (B16) ∂θ σ 2π Anom d0

References 1. Rao, Z.S.: A study of dynamic characteristic and contact stiffness of the rod fastening composite special rotor. Ph.D. Dissertation. Harbin Institute of Technology, Harbin, China (1992) 2. He, P., Liu, Z., Wang, G., Min, Z.: Rotor dynamic analysis of tie-bolt fastened rotor based on elastic-plastic contact. In: ASME 2011 Turbo Expo: Turbine Technical Conference and Exposition (2011) 3. He, P., Liu, Z., Zhang, G.: A study of overall contact behavior of an elastic perfectly plastic hemisphere and a rigid plane. In: ARCHIVE Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 1994–1996, vol. 208–210, 227, no. 3, pp. 259–274 (2013) 4. He, P., Liu, Z., Huang, F., Ma, R.: A study of elastic–plastic contact of statistical rough surfaces. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol. 227(10), 1076–1089 (2013)

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5. Yuan, Q., Gao, R., Feng, Z., Wang, J.: Analysis of dynamic characteristics of gas turbine rotor considering contact effects and pre-tightening force. In: Proceedings of the ASME Turbo Expo, pp. 983–988 (2008) 6. Yuan, S., Zhang, Y., Zhu, Y., Jiang, X.: Study on the equivalent stiffness of heavy-duty gas turbines composite rotor with curvic couplings & spindle tie-bolts. In: ASME 2011 Power Conference (2011) 7. Gao, J., Yuan, Q., Li, P., Zhang, H., Lv, Z.: Effects of bending moments and pre-tightening forces on the flexural stiffness of contact interfaces in rod-fastened rotors. J. Eng. Gas Turbines Power 134, 1–8 (2012) 8. Zhang, Y., Du, Z.: Determination of contact stiffness of rod-fastened rotors based on modal test and finite element analysis. J. Eng. Gas Turbines Power 132(9), 275–282 (2010) 9. Wu, X., Jiao, Y., Chen, Z., Ma, W.: Establishment of a contact stiffness matrix and its effect on the dynamic behavior of rod-fastening rotor bearing system. Arch. Appl. Mech. 91(7), 3247–3271 (2021). https://doi.org/10.1007/s00419-021-01963-9 10. Li, Y., Luo, Z., Liu, Z., Hou, X.: Nonlinear dynamic behaviors of a bolted joint rotor system supported by ball bearings. Arch. Appl. Mech. 89(11), 2381–2395 (2019). https://doi.org/10. 1007/s00419-019-01585-2 11. Li, Y., Luo, Z., Liu, J., Ma, H., Yang, D.: Dynamic modeling and stability analysis of a rotor-bearing system with bolted-disk joint. Mech. Syst. Signal Process. 158(24), 107778 (2021) 12. Wang, L., Wang, A., Jin, M., Huang, Q., Yin, Y.: Nonlinear effects of induced unbalance in the rod fastening rotor-bearing system considering nonlinear contact. Arch. Appl. Mech. 90(5), 917–943 (2019). https://doi.org/10.1007/s00419-019-01645-7 13. Xu, H., Yang, L., Xu, T.: The influence of the preload on the nonlinear dynamic performance of the rod-fastened jeffcott rotor system. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 236(15), 8245–8260 (2022) 14. Zhao, L., Zhang, H., Shen, P., Liu, Y.: Nonlinear dynamic characteristics of rod fastening rotor with preload relaxation. Energies 15, 1052 (2022) 15. Hu, L., Liu, Y.B., Teng, W., Zhou, C.: Nonlinear coupled dynamics of a rod fastening rotor under rub-impact and initial permanent deflection. Energies 9(11), 883 (2016) 16. Hu, L., Liu, Y., Zhao, L., Zhou, C.: Nonlinear dynamic response of a rub-impact rod fastening rotor considering nonlinear contact characteristic. Arch. Appl. Mech. 86(11), 1869–1886 (2016). https://doi.org/10.1007/s00419-016-1152-6 17. Hu, L., Liu, Y., Zhao, L., Zhou, C.: Nonlinear dynamic behaviors of circumferential rod fastening rotor under unbalanced pre-tightening force. Arch. Appl. Mech. 86(9), 1621–1631 (2016). https://doi.org/10.1007/s00419-016-1139-3 18. Hei, D., Lu, Y., Zhang, Y., Lu, Z., Gupta, P., Müller, N.: Nonlinear dynamic behaviors of a rod fastening rotor supported by fixed–tilting pad journal bearings. Chaos Soliton Fract. 69, 129–150 (2014) 19. Hei, D., Lu, Y., Zhang, Y., Liu, F., Zhou, C., Müller, N.: Nonlinear dynamic behaviors of rod fastening rotor–hydrodynamic journal bearing system. Arch. Appl. Mech. 85, 855–875 (2015) 20. Hei, D., Zheng, M.: Investigation on the dynamic behaviors of a rod fastening rotor based on an analytical solution of the oil film force of the supporting bearing. J. Low Freq. Noise V. A. (2020) 21. Zhang, Y., Xiang, L., Su, H., Hu, A., Yang, X.: Dynamic analysis of composite rod fastening rotor system considering multiple parameter influence. Appl. Math. Model. 105, 615–630 (2022) 22. Zhang, Y., Xiang, L., Su, H., Hu, A., Chen, K.: Nonlinear dynamic response on multi-fault rod fastening rotor with variable parameters. Appl. Math. Model. 114, 147–161 (2022)

Modeling Rotordynamic Effects of Angular Contact Ball Bearing in X- and O-arrangements with Full Bearing Matrix Giota Goswami(B)

, Iikka Martikainen, Eerik Sikanen, Charles Nutakor, Janne Heikkinen, and Jussi Sopanen

Department of Mechanical Engineering, Lappeenranta University of Technology, Skinnarilankatu 34, 53850 Lappeenranta, Finland {Giota.Goswami,Iikka.Martikainen,Eerik.Sikanen,Charles.Nutakor, Janne.Heikkinen,Jussi.Sopanen}@lut.fi

Abstract. Since ball bearings are one of the most commonly used bearing type in rotating machinery, predicting their dynamic performance is of utmost importance. The use of one-dimensional (1D) beam elements and bearing stiffness matrices with diagonal terms only is prevalent in the modeling and simulation of rotor-bearing systems. This study analyses the rotordynamic effects of angular contact ball bearings in X- and O-arrangements. A modeling method employing three-dimensional (3D) solid finite elements (FE) to model complex rotor geometries combined with a ball bearing stiffness formulation that accounts for the off-diagonal terms is presented. The case of an electric machine with an outer rotor permanent magnet design is studied with the O- and X-arrangements of the ball bearings. These arrangements are compared with the bearing stiffness matrix considering only the diagonal terms using both 3D solid FE and 1D beam element to model the rotor. The results show large variations between the natural frequencies and mode shapes of the compared models. It is observed that accounting for the off-diagonal terms in the bearing stiffness matrix in different arrangements, and using the 3D solid FE rotor model, can have a significant impact on the results. Therefore, the proposed method can improve the accuracy in the prediction of critical speeds for safe operation of the rotor-bearing system. Keywords: Bearing modelling · 3D solid elements · Angular contact ball bearing

1 Introduction Ball bearings are machine elements commonly used for supporting rotating structures. Due to the widespread nature of these structures, vibration-free operation of the rotorbearing system is of utmost importance. Dynamic modeling and simulation of these systems can help in predicting their performance. By modeling the bearing more accurately the rotor-bearing systems can be designed much closer to the design limit with improved performance. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 149–158, 2024. https://doi.org/10.1007/978-3-031-40455-9_12

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Research in ball bearing modeling has come a long way since the empirical formulabased models [1] to multiple-degree-of-freedom (DOF) models [2]. A fiveDOF dynamical bearing model including effects of non-linear Hertzian contact deformation and thickness of the lubrication film was introduced in [3]. This five-DOF bearing model was compared with model including centrifugal and gyroscopic effect in [4] to better evaluate the bearing stiffness at higher speeds. The bearing stiffness remains constant up to a certain speed and then starts to decrease [4]. While the modeling of bearings can include the full bearing stiffness matrix for higher accuracy, these models only included diagonal stiffness terms. The off-diagonal terms of the bearing stiffness matrix were introduced to the bearing model in [5]. This study categorizes the relevant off-diagonal terms for different load cases and bearing contact angles [5]. The full bearing stiffness matrix was further studied and improved upon in [6] by applying centrifugal and gyroscopic effects. The effects of the bearing arrangement on the rotor dynamics were studied and it was found that the Oarrangement has notably higher natural frequencies compared to the X-arrangement. The cross-coupling terms of the full bearing stiffness matrix was also numerically calculated and validated in [7]. The coupling of axial, radial and tilting deflections are captured in these terms. Although the above studies have focused on modeling ball bearings, the rotor must also be modeled accurately for complete rotor-dynamic analyses. In the above studies, one-dimensional (1D) beam elements have been used. Timoshenko beam elements have been used in [8] to model rotors for finite element rotor dynamic analysis. Even in the study of cross-coupling terms of the bearing stiffness matrix in a geared-rotor-bearing system in [9], 1D finite beam elements have been used in the rotor-dynamic model. However, 1D beam elements do not take into account cross-sectional deformations of the rotor structure. Cross-sectional deformations can be included by modeling the rotor using the threedimensional (3D) solid finite elements (FE). This kind of modeling can include many features such as stress-stiffening and non-linear contacts [10]. The use of 3D solid elements in rotor dynamics has been covered in [11], where the construction of the equations of motion for co-rotating or fixed reference frame was shown. The 3D solid FE formulation proposed in [12] achieved improved accuracy compared to other approaches while accounting for centrifugal stiffening effect and pre-stress deformations. In the reviewed literature, full-ball bearing stiffness formulations have been investigated with beam elements, but not with 3D solid FE models. The objective of this study is to combine the 3D solid FE rotor model with the full bearing stiffness matrix for Oand X-arrangements of angular contact ball bearings. It is hypothesized that the use of full bearing stiffness matrices and 3D solid FE rotor model will improve the accuracy of the rotor dynamic analysis. The modelling method is studied in a case example of a complicated outer rotor high speed electrical machine. The cases of O- and X-arrangements of the bearing with 3D solid FE rotor are compared to the cases of the bearing stiffness matrix containing only diagonal terms with both 3D solid FE and 1D beam element rotor.

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2 Theoretical Background In this section, the method for modeling the rotor-bearing system using the 3D solid finite element method and spiderweb beam elements is explained. The formulation of the full bearing stiffness matrix including the cross-coupling terms in O- and X-arrangements is also detailed. 2.1 System Equations The rotor-bearing system is modeled by obtaining the global mass, stiffness, and damping matrices of the finite element rotor model and the bearings. These matrices are used in the equation of motion that represents the entire system: M¨x + (Ce + Cs + Cb + ωG)˙x + (Ke + Ks + Kb )x = F

(1)

where M is the mass matrix, Ce the damping matrix of the 3D solid elements, Cs the damping matrix of the spiderweb beam elements, Cb the damping matrix of the bearings, ω is the rotational speed, G is the gyroscopic matrix, Ke is the stiffness matrix of the 3D solid elements, Ks is the stiffness matrix of the spiderweb beam elements, Kb is the stiffness matrix of the bearings, F is the vector of external forces, and x, x˙ and x¨ are the generalized displacements, velocities, and accelerations, respectively. 2.2 3D Solid FE and Spiderweb Beam Elements The 3D solid FE mesh for the rotor bodies are created in Ansys using quadratic tetrahedron elements. The mesh has 3 translational DOFs and no rotational DOFs. The element matrices are computed according to [10], and assembled for numerical integration according to [13] and [14]. To connect the bearing model to the 3D mesh, a spiderweb consisting of six-DOF beam elements was used. These beam elements connect the bearing surface nodes to a virtual center point. The spiderweb beam elements were modeled according to Ansys BEAM4 [15]. The material properties considered for the beam elements are artificial and represent those of a rigid and low-weight element. Therefore, the mass matrix of the spiderweb beam elements has negligible effect on the system. The length of each beam element is calculated as:  (2) L = (xb + xc )2 + (yb + yc )2 + (zb + zc )2 where xb , yb and zb are the X , Y and Z coordinates of the bearing surface node, and xc , yc and zc are the X , Y and Z coordinates of the bearing center node. The length of the projection of each beam on the XY -plane is given by:  Lxy = L2 − (zc − zb )2 (3) To rotate each beam element in the spiderweb in the correct orientation of the bearing nodes, the following transformation matrix is used: ⎡ ⎤ Tm 0 0 T = ⎣ 0 Tm 0 ⎦ (4) 0 0 Tm

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⎤ C1 C2 S1 C2 S2 Tm = ⎣ −S1 C1 0 ⎦ −C1 S2 −S1 S2 C2

(5)

where S1 = (yc −yb )/Lxy , S2 = (zc −zb )/Lxy , C1 = (xc −xb )/Lxy , and C2 = Lxy /L. The local spiderweb stiffness matrix, Klocal is, thus, transformed into the global coordinate system as: Ks = TT Klocal T

(6)

2.3 Ball Bearing Stiffness Formulation For an angular contact ball bearing, as shown in Fig. 1, the bearing forces acting on the shaft can be calculated using the relative displacements between the rings. Referring to Fig. 1, the eccentricities of the shaft in X , Y and Z directions are eX , eY and eZ , respectively, the misalignments of the inner race about the X and Y axes are γX and γY , respectively, the contact angle and the azimuth angle of ball j are φj and βj , respectively, and the radius of the ball is r. Rin and Rout are the inner and outer raceway radii, respectively, and rin and rout are the inner and outer groove radii, respectively.

Fig. 1. Axial cross-section of the ball bearing and transverse cross-section of ball j in the A-A plane [4].

In the presented ball bearing model, centrifugal and gyroscopic forces not included. For bearings with perfect geometry, when the thicknesses of the lubricant film between out the inner and outer contact surfaces are hin 0 and h0 , respectively, and the distance between the surfaces of the inner and outer races along the line of contact is dj the expression for total elastic deformation is given as [3]: in out δtot j = 2r + h0 + h0 − dj

(7)

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the contact force acting on ball j can be expressed as [3]:  1.5 Fj = Kctot δtot j

(8)

where Kctot is the total stiffness coefficient involving both inner and outer race contacts. The resultant forces and moments that the bearing exerts on the shaft in all directions, when elastic compression occurs, i.e. δjtot > 0, can be given by [3]: FX = − FY = −

Z j=1

Z j=1

FZ = − TX = − TY = −

Fj cos φj cos βj

(10)

Z

(11)

Fj (Rin + rin )Sinφj Sinβj

(12)

Fj (Rin + rin )Sinφj (− cos βj )

(13)

j=1

j=1

(9)

Fj Sinφj

Z

Z

Fj cos φj cos βj

j=1

The bearing stiffness matrix can be symbolically calculated by partial differentiation of the resultant forces (F = [FX, FY , FZ , TX , TY ]) with respect to the relative displacements (d = eX , eY , eZ , γX , γY ) in all directions [6]: ⎡

kxx ⎢ k ⎢ yx ⎢ K = ⎢ kzx ⎢ ⎣ kγX x kγY x

kxy kyy kzy kγX y kγY y

kxz kyz kzz kγX z kγY z

kxγX kyγX kzγX kγX γX kγY γX

kxγY kyγY kzγY kxγY kγY γY

⎤

⎡ ∂F

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎦ ⎢ ⎣

X

∂eX ∂FY ∂eX ∂FZ ∂eX ∂TX ∂eX ∂TY ∂eX

∂FX ∂eY ∂FY ∂eY ∂FZ ∂eY ∂TX ∂eY ∂TY ∂eY

∂FX ∂eZ ∂FY ∂eZ ∂FZ ∂eZ ∂TX ∂eZ ∂TY ∂eZ

∂FX ∂γX ∂FY ∂γX ∂FZ ∂γX ∂TX ∂γX ∂γX ∂γX

∂FX ∂γY ∂FY ∂γY ∂FZ ∂γY ∂TX ∂γY ∂TY ∂γY

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(14)

3 Results and Discussions The case of an electric machine with an outer rotor permanent magnet design has been studied in this work. This unconventional rotor design consists of three main components – a hollow shaft, an outer rotor and a disk like flange. The outer rotor houses all the electromagnetic components. The bearings are connected on the shaft. The flange connects the shaft and the outer rotor. The outer rotor has significant mass properties and its connection to supports via flange is relatively flexible. This makes the structure dynamically sub-optimal. Since the studied rotor system is of overhang type, the system is sensitive to bearing stiffness, which in turn requires comprehensive bearing modeling. A cross section of the structure with the bearings is shown in Fig. 2a where the diameters of the rotor structure can be seen. The length of the outer rotor is 116 mm, the length of the shaft is 212.5 mm, and the thickness of the flange is 15 mm. The shaft

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(a)

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Fig. 2. (a) Cross section of the studied rotor. (b) Cross section of the 3D FE mesh of the three bodies assembled with the spiderweb beam elements connecting the bearing surface nodes to the bearing center points.

and outer rotor are made of steel while the flange is made of aluminium. Two bearings, SKF 71919 CD/P4A [16] and SKF 71916 CE/P4A [17], are added at opposite ends of the shaft as shown in Fig. 2a. The bearing rings are made of bearing steel and the balls are made of silicon nitride. The rotor and bearing parameters are given in Table. 1. This rotor-bearing system has been selected for the case study because its special geometry and high speed application makes it a dynamically demanding case. A cross-sectional view of the 3D solid FE mesh created in Ansys and the spiderweb beam elements connecting the bearing surface nodes to a virtual bearing center point for each bearing are shown in Fig. 2b. The Campbell diagrams for the X- and O-arrangements of the bearing are shown in Fig. 3a and Fig. 3b, respectively. The case with bearing stiffness matrix with diagonal terms only using 3D solid FE rotor model is shown in Fig. 3c. The Campbell diagram obtained from modeling the rotor as a 1D beam element is also presented in Fig. 3d. It is observed that the off-diagonal elements have a significant effect on the natural frequencies of the systems when compared to the diagonal stiffness matrix. In comparison to the diagonal stiffness matrix, the full bearing matrix can either reduce, or increase the natural frequency depending on the bearing arrangement. The Campbell diagrams show that the O-arrangement can be stiffer than the Xarrangement of the bearings and also the diagonal stiffness case in the studied system. The Campbell diagram of the diagonal stiffness case is observed to be closely following the O-arrangement case in the first two modes. The axial mode at 489 Hz and the two

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Table 1. Parameters of the rotor-bearing system Rotor cross-section area Outer rotor (mm2 )

18155

Flange (mm2 )

56145

Shaft (mm2 )

1800

Rotor moments of inertia Outer rotor (mm4 )

211.85 · 106

Flange (mm4 )

295.76 · 106

Shaft (mm4 )

1.19 · 106

Bearing loads Axial preload (N)

900

Gravitational load (N)

400

Lubricant properties Pressure viscosity coefficient (mm2 /N)

0.023

Absolute viscosity at zero pressure, (Ns/mm2 )

0.04 · 10−6

whirling modes at 720 Hz appear similarly in all three models using the 3D solid FE rotor. However, in the last two modes large deviations can be see between all the four compared cases. It can be seen in the diagrams that there are differences in the critical speeds of the different cases, indicating higher resonance frequencies of the O-arrangement when compared to the X-arrangement. These findings are consistent with [6] where the X- and O-arrangements of the bearing have been previously compared. The diagonal stiffness matrix has also been used in the case of the 1D beam element. In the Campbell diagram of this model (Fig. 3d) fewer modes are seen. The modes that do appear in this diagram have different frequencies compared to the other cases. When the 1D beam element are compared to the diagonal stiffness case results for the 3D solid FE model, which uses the same bearing stiffness matrix, large deviations are observed. This indicates that the use of 3D solid FE method has a significant impact on the results. The variations in the results of each studied case show that with more accurate models that consider the off-diagonal bearing stiffness matrix terms, it is possible to safely get closer to the critical speed during operation. The first six mode shapes of the three cases are shown in Fig. 4. The first four modes appear similar for the three bearing stiffness matrices. However, modes five and six vary largely depending on the bearing arrangement and considered stiffness elements. This shows the significant effect of the bearing arrangement and off-diagonal stiffness terms on the resulting mode shapes.

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Fig. 3. Campbell diagrams of the (a) X-arrangement, (b) O-arrangement and (c) diagonal stiffness matrix cases of the 3D solid FE model and the (d) diagonal stiffness matrix case of the 1D beam element model.

4 Conclusions This study details and investigates the 3D solid FE modeling of a rotor-bearing system with full bearing stiffness matrices for bearings in different arrangements. The presented method is applied on an electric machine with outer rotor permanent magnet design. The X- and O-arrangements of the bearings modeled with 3D solid FE rotor were compared with the case of diagonal bearing stiffness matrix modeled with both 3D solid FE and 1D beam element rotor. The results show that the natural frequencies and mode shapes differ significantly between the compared cases. In the Campbell diagrams, the O-arrangement is observed to be stiffer than the Xarrangement, and the O-arrangement and the diagonal stiffness matrix cases closely follow each other in most modes. The comparison of the 3D solid FE rotor model and

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Fig. 4. Mode shapes of the three cases for modes 1 to 6.

1D beam element rotor model using diagonal stiffness matrix of the bearing produces large differences in the results. This indicates that 3D solid FE modeling of the rotor can have a significant impact on the accuracy of the model. The developed modeling method aims to improve the accuracy of rotor-bearing simulations by accounting for the bearing arrangement and the off-diagonal terms in the bearing stiffness matrix. The results show that the effects of the studied variations in the bearing matrices are significant and can provide valuable insight on the critical speeds of the system. The use of 3D solid FE methods is expected to increase the accuracy of the results. Such simulation models can improve the design of ball bearings in the industry by considering the effects of bearing arrangement. Validation of the achieved results could be done during future studies. Other cases that include contacts, stress-stiffening of thermal effects can also be studied in the future to further utilize the 3D solid FE model. The bearing stiffness matrix can be further developed to include and study the effects of centrifugal and gyroscopic forces, angular misalignments and load variations.

References 1. Stribeck, R., et al.: Ball bearings for various loads. Trans. ASME 29(4), 420–463 (1907) 2. Jones, A.: A general theory for elastically constrained ball and radial roller bearings under arbitrary load and speed conditions. J. Basic Eng. 82(2), 309–320 (1960) 3. Sopanen, J., Mikkola, A.: Dynamic model of a deep-groove ball bearing including localized and distributed defects. Part 1: theory. Proc. Inst. Mech. Eng. Part K: J. Multi-body Dyna. 217(3), 201–211 (2003) 4. Kurvinen, E., Sopanen, J., Mikkola, A.: Ball bearing model performance on various sized rotors with and without centrifugal and gyroscopic forces. Mech. Mach. Theory 90, 240–260 (2015) 5. Lim, T.C., Singh, R.: Vibration transmission through rolling element bearings, part i: bearing stiffness formulation. J. Sound Vib. 139(2), 179–199 (1990)

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6. Fang, B., Yan, K., Hong, J., Zhang, J.: A comprehensive study on the off-diagonal coupling elements in the stiffness matrix of the angular contact ball bearing and their influence on the dynamic characteristics of the rotor system. Mech. Mach. Theory 158(104), 251 (2021) 7. Guo, Y., Parker, R.G.: Stiffness matrix calculation of rolling element bearings using a finite element/contact mechanics model. Mech. Mach. Theory 51, 32–45 (2012) 8. Nelson, H.: A finite rotating shaft element using Timoshenko beam theory (1980) 9. Xu, H., et al.: Effects of supporting stiffness of deep groove ball bearings with raceway misalignment on vibration behaviors of a gear-rotor system. Mech. Mach. Theory 177(105), 041 (2022) 10. Sikanen, E., Heikkinen, J.E., Sopanen, J.: Shrink-fitted joint behavior using three-dimensional solid finite elements in rotor dynamics with inclusion of stress-stiffening effect. Adv. Mech. Eng. 10(6), 1687814018780054 (2018) 11. Kirchgäßner, B.: Finite elements in rotordynamics. Procedia Eng. 144, 736–750 (2016) 12. Shen, Z., Chouvion, B., Thouverez, F., Beley, A.: Enhanced 3D solid finite element formulation for rotor dynamics simulation. Finite Elem. Anal. Des. 195(103), 584 (2021) 13. Liu, G.R., Quek, S.S.: The Finite Element Method: A Practical Course. ButterworthHeinemann, Oxford (2013) 14. Logan, D.L.: A First Course in the Finite Element Method. Cengage Learning, Boston (2016) 15. Kohnke, P.: Theory Reference for the Mechanical APDL and Mechanical Applications. Ansys Inc, release 12 (2009) 16. SKF: 71919 cd/p4a super-precision angular contact ball bearing (2023). https://www.skf. com/in/products/super-precision-bearings/angular-contact-ball-bearings/productid-71919% 20CD%2FP4A 17. SKF: 71916 ce/p4a super-precision angular contact ball bearing (2023). https://www.skf. com/in/products/super-precision-bearings/angular-contact-ball-bearings/productid-71916% 20CE%2FP4A

Research on the Nonlinear Response of the Rotor System with Bolted Joint with Spigot Yongbo Ma1(B) , Jie Hong1,2 , Shaobao Feng3 , and Yanhong Ma2 1 School of Energy and Power Engineering, Beihang University, Beijing 100191,

People’s Republic of China [email protected] 2 Research Institute of Aero-Engine, Beihang University, Beijing 100191, People’s Republic of China 3 AECC Shenyang Engine Design and Research Institute, Aero Engine (Group) Corporation of China, Shenyang 110015, People’s Republic of China

Abstract. The bolted joint with spigot is the most common form of joint in the aero-engine rotor system. There will be slip damage at the spigot interface under the complex load, which will influence the dynamic characteristic of the rotor system. However, there are few researches on the spigot interface, so the mechanical characteristic of the non-continuous rotor system cannot be described accurately enough. In this paper, a new mechanical model of bolted joint with spigot has been proposed, which can accurately describe the force-displacement characteristic of the bolted joint with spigot. In addition, the dynamic characteristic of the non-continuous rotor system with the bolted joint with spigot has been calculated, which has been used to analyze the influence of the nonlinear characteristic on the rotor whirl. The results show that the slip process of the spigot interface can been divided into three stages: sticking, microslip and macroslip, with obvious nonlinear characteristic of stiffness, and there may be bifurcation phenomenon in the dynamic response of the non-continuous rotor system. In addition, the influence of the slip damage will vary according to the structure parameters and the excitation, which is necessary to be considered in the robust design of aero-engine rotor system. Keywords: rotor dynamics · Nonlinear response · spigot interface · Bolted joint

1 Introduction The aero-engine rotor system usually consists of multiple components, and assembled by various joints [1, 2]. As shown in the Fig. 1, the bolted joint with spigot is one of the most common joints, which is beneficial to the high-speed rotors because of the higher centering accuracy and lower unbalance. However, affected by complex load, there may be slip damage at the interface of joints during operating, which will affect the rotor mechanical characteristic and seriously threaten to safety and stability of rotor system. Therefore, it’s necessary to study the slip damage at the interface during operating, and analyze the impact on the dynamic characteristic of the rotor system. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 159–172, 2024. https://doi.org/10.1007/978-3-031-40455-9_13

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Fig. 1. The bolted joints with spigot in aero-engine rotor system

Different from the continuous structure system, the contact state at the interface of the joints will change under complex load, and affect the mechanical properties [3]. The development of contact mechanics, tribology and other disciplines, provided a solution for the interface contact. Matthew [4], from Sandia National Laboratory, divided the whole slip process into four stages: sticking, microslip, macroslip and pinning, by studying the bolted joints under the tangential force. Gaul [5] applied a periodic tangential load to the bolted joint, and found the contact state of the interface gradually changed from microslip to macroslip with the load increasing. Marshall [6] used a nonintrusive ultrasonic technique to quantify the contact pressure distribution in a bolted joint. The famous Iwan model [7, 8], which explained the nonlinear mechanical behavior of the contact interface, had become the most widely used phenomenological model. Based on the research of contact state, extensive researches have been done on the joint mechanical characteristic. Argatov [9] derived the analytical nonlinear constitutive equation of bolted joint. Kim [10] established a fine finite element model of bolted joint by three-dimensional solid elements according to the actual size, obtained the interface stress distribution and structural deformation, and then studied the stiffness of the joint structures. Beaudoin [11] presented a new lump model that accounts for the nonlinear phenomena of partial clearance and friction, and had analytical parameters related to the flange geometry. Fan [12] established the model of bolted joint in aero-engine, revealed the mechanism of stiffness nonlinearity, and calculated the steady-state response of dual-rotor system when the stiffness of the joint structure is interval distributed. Yue [13] established the stiffness damage model of the rabbet joint structure and proposed the robust design method of the rabbet joint structure. Wang [14] established a detailed dynamic model of non-continuous rotor, and introduced interval analysis method. Based on the research on the joint mechanical characteristic, the research on the rotor system with joints has also been carried out. Caddemi [15, 16] derived the closed-form solution method for the rotor system with singularity change in mechanical characteristic by generalized functions, and analyzed the dynamic characteristic of discontinuous rotor, and gives the analytical solution. Yang [17] analyzed the asymmetric and specific residual pre-tightening force of the bolts after the bolted joint operated, established the parametric excitation system of the time-varying stiffness of the aero-engine rotor system, and found the combined frequency in the rotor response. Chen [1] proposed a discontinuous rotor dynamics optimization method according to the joint stiffness characteristic. Dai [18] studied the rotor dynamic characteristic with the loose supports, and

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proposed a dynamic optimization method for the multi-support flexible rotor system. Lei [19] established quantitative evaluation parameters based on interface deformation coordination, and optimized geometric characteristic parameters of rotor. However, there have been few studies about the spigot. Hong [20] established a slip damage mechanical model to analyze the stiffness loss in the aero-engine rotor system. Yu [21] established a nonlinear analysis model to describe the mechanical behavior of the aero-engine rotor under bending moment considering the spigot interface, and calculated the nonlinear stiffness and damping of the joint structure. It can be found that there are sufficient studies on the joint mechanical characteristic, but the research on the bolted joint with spigot is relatively lacking. Aiming at the bolted joint with spigot in the aero-engine rotor system, the mechanical model has been proposed to analyze the slip damage during operating and the dynamic response of the non-continuous rotor system considering slip damage has been calculated in this paper.

2 Mechanical Model of Slip Damage at the Spigot Interface In this section, the slippage of the joint interface under axial external load is modeled based on the classic Iwan model to describe the Force-Displacement characteristic of the bolted joint with the spigot. 2.1 The Distribution Function of Slippage at the Spigot Interface There is usually a certain amount of interference at the spigot interface, to achieve the centering of the different components. The contact stress distribution at the spigot interface is generally complex, as shown in the Fig. 2, and the normal stress distribution has been roughly simplified as a quadratic distribution along the axial direction in this paper.

Pmax Pmin D

interference

Fig. 2. The normal contact stress distribution at the spigot interface

Therefore, it is assumed that the normal contact stress distribution at the spigot interface is P(d ) = Pmin + (Pmax − Pmin )

d2 D2

(1)

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where, D is the axial length of spigot interface, Pmax and Pmin are respectively the maximum and minimum of the normal contact stress at the spigot interface and d represents the axial position at the interface. Thus, the maximum friction force for different positions can be provided is   d2 (2) τ (d ) = μ Pmin + (Pmax − Pmin ) 2 D where, μ is the coefficient of friction of the spigot interface, which is regarded as a constant in this model. The spigot interface will slip under the axial tension load. The relationship between slip area and critical slip position is s(d ) = d · t

(3)

where, t is the width of spigot interface, d is the critical position at which slips. By substituting Eq. (3) into Eq. (2), the maximum friction expressed in slip area can be obtained:   s 2  (4) τ (s) = μ Pmin + (Pmax − Pmin ) Dt Conversely, the slip area is expressed by the maximum friction: ⎧ ⎪ ⎪0 0 ≤ τ < μPmin ⎪ ⎪ ⎨

τ −μPmin s(τ ) = Dt μ(Pmax −Pmin ) μPmin ≤ τ < μPmax ⎪ ⎪ ⎪ ⎪ ⎩ Dt τ > μPmax

(5)

Normalize the slip area: ⎧ ⎪ ⎪ ⎪0 ⎪ ⎨

s(τ ) s˜ (τ ) = = ⎪ smax ⎪ ⎪ ⎪ ⎩1

0 ≤ τ < μPmin τ −μPmin μ(Pmax −Pmin )

μPmin ≤ τ < μPmax

(6)

τ > μPmax

where, smax = Dt. Taking the derivative of Eq. (6) with respect to τ : (τ ) =

1 1 d s˜ (τ ) = [(τ − μPmin )(μPmax − μPmin )]− 2 dτ 2

μPmin ≤ τ ≤ μPmax

(7)

2.2 Force-Displacement Relationship of Spigot Interface W. D. Iwan [7, 8] had proposed a phenomenological model for the nonlinear mechanical behavior at the interface in 1966. Because of the simple structure and clear physical

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k N

k N

f1* N

f 2* N

k N

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k N

fi* N

f N* N

f

x

Fig. 3. Schematic diagram of the Iwan model

meaning, this model has been widely used and sufficiently developed in recent years. In this part, the Iwan model has been used to derive the force-displacement relationship of the contact interface. The Iwan model includes multiple Jenkins elements, as shown in the Fig. 3, the friction force F and the displacement x satisfies 

kx

F(x) = 0

f ·  f ∗ df ∗ + ∗



∞

kx ·  f ∗ df ∗

(8)

kx

where, (f ∗ ) is the distribution function of the maximum friction provided by different Jenkins elements. In this model, it is considered that the relevant material parameters of the interface are the uniform, so it is assumed that the stiffness of each Jenkins element in the sticking stage is the same, ki = k, i = 1, 2, 3... Let τ = λf ∗ , substitute it into Eq. (7): 1  ∗

− 1 λf − μPmin (μPmax − μPmin ) 2  f∗ = 2

μPmin ≤ τ ≤ μPmax

(9)

Substitute Eq. (9) into Eq. (8), the force-displacement relationship of this model can be obtained. where, λkϕ2 = τmax = μPmax . ϕ1 and ϕ2 respectively indicate the critical displacement of mircroslip and macroslip at the spigot interface, and can be obtained: (ϕ) =

1 1 (ϕ − ϕ1 )− 2 ϕ1 ≤ ϕ ≤ ϕ2 √ 2λk ϕ2 − ϕ1

(10)

Segalman [22, 23] proposed a coefficient to replace k in Eq. (8): ρ(ϕ) = k 2 · (ϕ) ϕ=f /k

(11)

Substitute Eq. (11) into Eq. (9), the new expressions of Iwan model and interface contact normal stress distribution can be obtained respectively: (12)

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(13) and then:

⎧ ϕ1 ⎪ ρ(ϕ)d ϕ = K0 ⎪ ⎪ ⎨ 0  ϕ2 ϕ1 ρ(ϕ)d ϕ = K0 − K∞ ⎪ ⎪ ⎪  ⎩ ∞ ϕ2 ρ(ϕ)d ϕ = K∞

(14)

where, K0 represents the initial stiffness, K∞ represents the residual stiffness. Solve Eq. (14) and obtain the density function ρ(ϕ): (15)

where, H ( ) and δ( ) are respectively the Heaviside function and the Dirac delta function. Substitute Eq. (15) into Eq. (13) and obtain the force-displacement relationship is: ⎧ ⎪ ⎪ K0 x 0 ≤ x < ϕ1 ⎪ ⎪ ⎨ 3 F(x) = K0 x − 2(K (16) √0 −K∞ ) (x − ϕ1 ) 2 ϕ1 ≤ x < ϕ2 3 ϕ2 −ϕ1 ⎪ ⎪ ⎪ ⎪ ⎩ 1 (K0 − K∞ )(2ϕ1 + ϕ2 ) + K∞ x ϕ2 ≤ x 3 and the stiffness-displacement relationship is: ⎧ 0 ≤ x < ϕ1 ⎪ K0 1 dF(x) ⎨ √0 −K∞ ) (x − ϕ1 ) 2 ϕ1 ≤ x < ϕ2 = K0 − (K K(x) = ϕ2 −ϕ1 ⎪ x ⎩ K∞ ϕ2 ≤ x

(17)

where, 0 ≤ x < ϕ1 represents the sticking state, ϕ1 ≤ x < ϕ2 represents the microslip state and ϕ2 ≤ x represents the macroslip state. The force-displacement relationship and stiffness-displacement relationship of the bolted joint with spigot are shown in the Fig. 4. According to the above results, the slip process at the spigot can be divided into three stages: sticking, microslip and macroslip. There will stiffness loss at the microslip state with significant nonlinear characteristic. It’s also known that the bolted joint with spigot contains three structural parameters, including the microslip displacement ϕ1 , the macroslip displacement ϕ2 and the stiffness loss coefficient η = K∞ /K0 , which all have decisive impacts on the mechanical characteristic of the bolted joint with spigot.

3 Dynamic Characteristic of the Non-continuous Rotor System In this section, the dynamic response of a rotor system with a bolted joint with spigot has been calculated, which has been used to analyze the dynamic characteristic of the non-continuous rotor system.

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K0 x

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3 2

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K0

K x C

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F

K0

K0

K 2

x

1

1 2

1

K

F

K0 x Displacement

(a) Force-Displacement curve

Displacement

(b) Stiffness-Displacement curve

Fig. 4. Theoretical Derivation Results

3.1 The Rotor Example A high-speed rotor system with two supporting has been selected as an example in this paper, as shown in the Fig. 5. There is a bolted joint with spigot at the drum shaft between the compressor and the turbine disk, which is the weakest structure of the rotor system. Therefore, the stiffness of the whole rotor system will be strongly influenced by the slip damage of this spigot interface.

The Front Supporting

The bolted joint with spigot

k1 Compressor

The Back Supporting

k2 Turbine disk

Fig. 5. Schematic diagram of The Rotor Example

The beam element model of the rotor system has been established, and the mass, stiffness, damping and gyroscope matrix have been respectively extracted. Based on the previous analysis, the force-displacement relationship during slippage has been substituted into the stiffness matrix, and a non-continuous rotor system with a bolted joint with spigot can be obtained. The modal characteristic and unbalanced response of the continuous rotor system (the rotor example without considering the bolted joint with spigot) have been calculated, as shown in the Fig. 6. According to the database, the rotor example operates at the range of about 11000~14000 rpm. As can be seen from the Fig. 6, the first and the second critical rotation speed are respectively 2790 rpm and 6630 rpm from the calculation results. Therefore, the operating rotation speed of the rotor system has been set between the second and the third critical rotation speed, and there will be low level dynamic response during the operating, which well satisfies the vibration requirement.

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The First-order Mode The Second-order Mode The Third-order Mode

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Fig. 6. The Dynamic Characteristic of the Continuous Rotor System

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Vibration amplitude (mm)

Next, the bolted joint with spigot has been considered in the rotor example system, and the dynamic response has been calculated in the full rotation speed range. It is known from the literature [20] that the magnitude of displacement of slip damage is within 10 μm. Therefore, the structural parameters have been taken separately the microslip displacement ϕ1 = 4 μm, the macroslip displacement ϕ2 = 6 μm and the stiffness loss coefficient η = 90% in this calculation. The calculated non-continuous rotor dynamic response under the 100 g · mm unbalance has been calculated through the Newmark-β method, as shown in the Fig. 7.

The front support The back support

2540rpm

10.0

5750rpm

7.5

Operating Rotation Speed Range

5.0

2.5

0.0

0

5000

10000

Rotation speed (rpm)

15000

Fig. 7. The Dynamic Characteristic of the Non-Continuous Rotor System

Different from the calculation results of continuous rotor system, it can be found that the first and the second critical rotation speeds have descended respectively by 8.96% and 13.3% due to the slip damage of the bolted joint with spigot. In addition, the dynamic response at almost each speed has increased compared with the continuous rotor system. It’s necessary to pay attention to the change of the rotor whirl within the operating rotation speed range due to the slip damage of the bolted joint with spigot. It can be seen from the Fig. 8 that the bifurcation phenomenon of the dynamic response is relatively

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obvious in the follow two situations: (1) in the low speed range and close to the first and the second critical rotation speed; (2) in the operating rotation speed range. For the former, which is caused by the large dynamic response; and for the latter, the cause is that the nonlinear characteristic of stiffness plays a leading role at the deformation range of the bolted joint. 3.0 2.5

Displacement (mm)

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The time, frequency and orbit responses at different rotation speeds have been extracted, as shown in the Fig. 9. It can be seen from the subgraph (a) that the response is large but the slip damage at the bolted joint with spigot is slight at the speed of 2500 rpm, which is close to the first critical rotation speed. The fundamental frequency dominated the frequency spectrum with the weak nonlinear characteristic of the rotor system. As shown in the subgraph (b), the response has been significantly reduced as the rotation speed exceeds the second critical rotation speed. However, there has been the slip damage at the spigot interface because of the large deformation of the bolted joint with spigot, which made the non-circle orbit response. In addition, there are fundamental frequency and octave frequencies of rotational speed in the spectrum, which shows apparent nonlinear characteristic. The nonlinear characteristic has become obvious due to the serious slip damage at the bolted joint with spigot in the operating rotation speed range as shown in the subgraph

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(c). The orbit response has become a ring with chaotic and irregular frequencies in the frequency response, which indicates the complex and unhealthy rotor whirl. The slip damage at the bolted joint with spigot has depressed when the rotation speed is up to around 15000 rpm, as shown in the subgraph (d). Also, the nonlinear characteristic has weakened, which is similar to the response at the 8000 rpm shown in the subgraph (b). Based on the previous analysis, it can be found that the impact of slip damage at the bolted joint with spigot is mainly concentrated in the specified rotation speed, rather than the whole rotation speed range. It’s necessary to pay attention to the slip damage at the bolted joint with spigot while its effect concentrated on the operating rotation speed range. 3.3 Discussion It can be known that structural parameters, such as microslip displacement ϕ1 , the macroslip displacement ϕ2 and the stiffness loss coefficient η, have significant influence

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on the rotor dynamic characteristic from the previous theoretical analysis. In addition, the value of excitation also has impact on the dynamic response on the non-continuous rotor system with the slip damage. 3.0 2.5

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The structural parameters have been changed (microslip displacement ϕ1 = 6 μm, the macroslip displacement ϕ2 = 8 μm and the stiffness loss coefficient η = 90%), and the dynamic response has shown in the Fig. 10. It can be known that the dynamic response at the most rotation speeds is similar to the previous results and the difference is concentrated at the operating rotation speed range, by compared with the Fig. 8 and Fig. 10. The most obvious is there is serious bifurcation phenomenon in the range of 12000 to 15000 rpm shown in the subgraph (b), which is little in the Fig. 8. Because of the similar response under different structural parameters, this paper will not repeat. Similarly, the dynamic response can be changed by different value of excitation. It can be found from the Fig. 11 that the response amplitude is almost half of that shown in the Fig. 8 after the unbalance changed to 50 g·mm. However, the bifurcation phenomenon has changed during the operation rotation speed, which has strong performance at the back supporting. It can be seen from the subgraph (b) that there is slight bifurcation phenomenon at about 13000 rpm, but which has sharply strong after 13500 rpm and until 17000 rpm.

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Fig. 11. Bifurcation diagram of the Non-Continuous Rotor System under the 50 g·mm Unbalance

It can be found that when the structure parameters of the bolted joint with spigot and the value of excitation have changed, the rotation speed range and severity of bifurcation phenomenon will be greatly different through the above comparison. If thinking from another perspective, the adaptability of the rotor system to the slip damage can be adjusted by the robust design of the structure parameters of the bolted joint with spigot. Therefore, the author expresses a view that for the high-speed non-continuous rotor system, the slip damage is necessary to be considered in the design phase, and be placed in the appropriate rotation speed range through robust design, so as to ensure the robustness of the dynamic characteristic of the rotor system within the working speed range.

4 Conclusion The bolted joint with spigot, as a widely used joint in aero-engine rotor system, has been lack of sufficient research of dynamic characteristic. The mechanical model has been proposed to analyze the slip damage during operating and the dynamic response of the non-continuous rotor system considering slip damage has been calculated in this paper. The results of this research have led to the following conclusions: 1) Aiming at the bolted joint with spigot, the slip damage mechanical model has been proposed to describe the process of the interface slippage and get the force-displacement relationship based on the Iwan model;

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2) Based on the proposed mechanical model, the dynamic characteristic of the aeroengine rotor system considering the slip damage has been calculated. It has been found the prominent nonlinear characteristic during the slip damage, which is manifested as obvious bifurcation phenomenon. 3) The simulation of the non-continuous rotor system has indicated that the influence of the slip damage will vary according to the structure parameters and the value of excitation. It’s necessary to fully consider the effect of slip damage in the rotor design stage to improve the robustness of rotor dynamic characteristic. In this paper, a mechanical model of slip damage at bolted joint with spigot in the aero-engine rotor system has been established. However, due to the complexity of the load and structure of aero-engine, the depth analysis of the dynamic characteristic has been lack in this paper. In addition, the quantitative evaluation method of the influence of slip damage on the dynamic characteristic of aero-engine rotor system is also the future research direction.

References 1. Hong, J., Chen, X., Wang, Y., et al.: Optimization of dynamics of non-continuous rotor based on model of rotor stiffness. Mech. Syst. Signal Process. 131, 166–182 (2019) 2. Hong, J., Ma, Y.: Structure and Design of Aircraft Gas Turbine Engine. Science Press, Beijing (2021) 3. Farrar, C.R., Worden, K.: An introduction to structural health monitoring. Philos. Trans. R. Soc. Lond. A: Math. Phys. Eng. Sci. 2007(365), 303–315 (1851) 4. Brake, M.R.W. (ed.): The Mechanics of Jointed Structures. Springer, Cham (2018). https:// doi.org/10.1007/978-3-319-56818-8 5. Gaul, L., Lenz, J.: Nonlinear dynamics of structures assembled by bolted joints. Acta Mech. 125(1–4), 169–181 (1997) 6. Marshall, M.B., Lewis, R., Dwyerjoyce, R.S., et al.: Characterisation of contact pressure distribution in bolted joints (2006) 7. Iwan, W.D.: A Distributed-Element Model for Hysteresis and Its Steady-State Dynamic Response (1966) 8. Iwan, W.D.: On a Class of Models for the Yielding Behavior of Continuous and Composite Systems (1967) 9. Argatov, I.I., Butcher, E.A.: On the Iwan models for lap-type bolted joints. Int. J. Non Linear Mech. 46(2), 347–356 (2011) 10. Kim, J., Yoon, J.C., Kang, B.S.: Finite element analysis and modeling of structure with bolted joints. Appl. Math. Model. 31(8), 895–911 (2007) 11. Beaudoin, M.A., Behdinan, K.: Analytical lump model for the nonlinear dynamic response of bolted flanges in aero-engine casings. Mech. Syst. Signal Process. 115, 14–28 (2019) 12. Ning, F.: Studies on Dynamic Characteristics of the Joint in the Aero-Engine Rotor System. BUAA, Beijing (2018) 13. Yue, W., Mei, Q., Zhang, D., et al.: Robust design method of rabbet joint structure in high speed assemble rotor. J. Aero-Space Power 32(7), 1754–1761 (2017) 14. Cun, W.: Dynamic Model and Interval Analysis Method of Noncontinuous Rotor. BUAA, Beijing (2018) 15. Caddemi, S., Caliò, I.: Exact closed-form solution for the vibration modes of the EulerBernoulli beam with multiple open cracks. J. Sound Vib. 327, 473–489 (2009)

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16. Caddemi, S., Caliò, I., Cannizzaro, F.: Closed-form solutions for stepped Timoshenko beams with internal singularities and along-axis external supports. Arch. Appl. Mech. 83, 559–577 (2013) 17. Hong, J., Yang, Z., Wang, Y., et al.: Combination resonances of rotor systems with asymmetric residual preloads in bolted joints. Mech. Syst. Signal Process. 183, 109626 (2023) 18. Hong, J., Dai, Q., Wu, F., et al.: Dynamic characteristics analysis of flexible rotor system with pedestal looseness. In: Turbo Expo: Power for Land, Sea, and Air, vol. 85031, p. V09BT28A008. American Society of Mechanical Engineers (2021) 19. Lei, B., Li, C., Hong, J., Ma, Y.: Robust design of mechanical characteristics of rotor connection structure. Aeroengine 47(02), 3844 (2021) 20. Hong, J., Ma, Y., Feng, S., et al.: The mechanism and quantitative evaluation of slip damage of bolted joint with spigot. In: ASME, Turbo Expo: Power for Land, Sea, and Air. American Society of Mechanical Engineers (2022) 21. Yu, P., Li, L., Chen, G., et al.: Dynamic modelling and vibration characteristics analysis for the bolted joint with spigot in the rotor system. Appl. Math. Model. 94, 306–331 (2021) 22. Segalman, D.: A four parameter Iwan model for lap-type joints, Sandia report, SAND20023828. Sandia Laboratories, Albuquerque, NM (2002) 23. Segalman, D.J., Gregory, D.L., Starr, M.J., et al.: Handbook on dynamics of jointed structures. Sandia National Laboratories, Albuquerque (2009)

On the Influence of the Lubricant Feed Orifice Size and End Plate Seals’ Clearance on the Static and Dynamic Performance of Integral Squeeze Film Dampers Xueliang Lu1(B) , Luis San Andrés2 , and Bonjin Koo3 1 Technology Center, Hunan SUND Technological Corporation, Hunan, China

[email protected]

2 J. Mike Walker ‘66 Department of Mechanical Engineering, Texas A&M University, College

Station, USA [email protected] 3 Applied Development Center, Daikin Applied, Minneapolis, USA [email protected]

Abstract. In rotating machinery squeeze film dampers (SFDs) reduce rotor synchronous response amplitude motions, provide structural isolation and enhance rotordynamic stability. Compared to conventional squirrel cage supported SFDs, integral squeeze film dampers (ISFDs) are more compact and require a shorter axial span. This paper presents predictions of pressure profile, lubricant flow rate and dynamic force coefficients of a four pads ISFD configured with distinct inlet orifices (d 0 – 3 d 0 ) and ends’ seal clearances (b1 - 3b1 , b1 = 1/3 c). The dimensions of the inlet orifice (d 0 ) and end seal clearance (b1 ) are based on an ISFD used in an industrial supercritical CO2 expander. Thin S-shape beams acting as springs divide the annular clearance into several individual fluid domains. The analysis quantifies the effect of the lubricant feed holes’ size and the end seals’ clearance on the static and dynamic forced performance of an ISFD for a typical compressor application. An increase in diameter for the inlet orifice, from d 0 to 2d 0 , increases significantly the fluid film pressure in the all pads, delaying the whirl speed when vapor cavitation occurs. However, with a nominal clearance b1 = 0.191 mm, the damping and added inertia coefficients reduces by almost 1/3 as the inlet orifice diameter increases from d 0 to 2d 0 . Although the damping and inertia coefficients of the ISFD are more sensitive to the end seal clearance than to the diameter of the inlet orifice, an increase in the size of the inlet orifice does weaken the relationship “damping ~ 1/clearance3 ”. In addition, predictions for the ISFD operating with an air in oil mixture shows that the damping and added inertia coefficients drop almost linearly as the inlet gas volume fraction (GVF) increases from 0.0 (all liquid) to 0.2.

1 Introduction The demand for high pressure and high aerodynamic efficiency continuously pushes a compressor/turbine to operate at ever fast rotor speeds and with extremely tight clearances in the seals controlling secondary leakage paths. To keep the shaft surface speed at © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 173–189, 2024. https://doi.org/10.1007/978-3-031-40455-9_14

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the support bearings under a reasonable limit, the journal diameter must reduce inversely to the increase in angular speed. The reduction in diameter makes the rotor more flexible when compared to the resilience of the journal bearings. Further, conventional gas seals such as labyrinth seals are prone to produce large cross coupled stiffnesses for operation at high pressure (large gas density) while having small clearances. A flexible rotor plus large cross coupled stiffnesses make conventional style fluid film bearings (both fixed pads or with tilting pads) inadequate to overcome stability issues. In such a case, SFDs can be installed in series with the journal bearings to render a soft support that drops the system critical speeds while introducing additional damping to stabilize the rotor [1, 2]. Zeidan [3] summarizes the characteristics of four types of squeeze film dampers (SFDs), the ones without a centering spring being the simplest. Due to the lack of a centering spring, these SFDs are bottomed out at start-up, and as the shaft speed increases, the damper develops a centering like-stiffness due to persistent oil cavitation. Elastomeric O-rings are simple means to provide a centering stiffness and added damping in light weight rotors [3, 4]. However, the stiffness of O-rings is hard to predict as it varies with the elastomer material properties and the operating temperature and frequency. Aging is also a problem for O-ring sealed SFDs. Gooding et al. [5] report a vibration issue in a test facility to study aerodynamics of an overhung centrifugal compressor for aero-engine applications. The compressor installed with O-rings sealed SFDs operate with adequate stability for several years, but goes unstable at some speed range as time goes by. The authors suspect the degradation of the sealing O-rings reduces the effective length of the damper and leading to a decrease in the damper’s stiffness and damping, which cause rotordynamic instability to the rotor. Squirrel cage supported SFDs are the most commonly used design in aircraft engines [3, 6]. The squirrel cage not only provide means to center the rotor, but also can be used to tune the system natural frequency as its stiffness is adjustable over a wide range. However, squirrel cages require large axial space for installation and are difficult to center during the assembly process. To overcome the difficulties of the above mentioned three types of SFDs, Zeidan [3] introduced a type of integral damper that features S-shaped flexible webs as centering springs. The S-shaped springs are manufactured from the same piece of raw material as the SFD using electrical discharge method (EDM). Thus, the centering spring and the SFD are integrated as a single piece, and does not require extra axial space. As such, this mechanical element is referred as integral centering spring squeeze film damper (ISFD) [1]. ISFDs can be fitted on the outer housing of ball bearings or fluid film bearings to guarantee smooth operation of a turbomachinery [1]. Early versions of the ISFDs mainly served to tune the natural frequency of the rotor-bearing system to improve separation margin between running speed and critical speeds [7, 8]. Agnew and Childs [8] identify the force coefficients of a four pad flexure pivot tilting pad bearing in series with an ISFD. The ISFD has four pads, each pad has arc length of 73°, diameter of 158 mm, and the axial length is 76 mm. The experimental results demonstrate that the ISFD bearing has a lower direct stiffness and added mass compared to those of the tilting-pad bearing itself. The experimentally estimated damping coefficient remains practically unchanged. ISFDs also serve to provide additional damping thus increasing the stability of an otherwise unstable system in steam turbines [9, 10] and high-speed high-pressure process

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gas compressors, i.e., integrally geared centrifugal compressors [11]. Ertas et al. [10] discuss stabilizing a 46 MW multistage utility steam turbine (nominal speed 5500 rpm, rotor mass 6,320 kg) using ISFD supported bearings. The original design of the turbine has two 50% offset rocket pivot 5 pad load on pad tilting pad journal bearings (TPJBs). During the initial commissioning, the turbine trips with high subsynchronous vibration (SSV) at 30 Hz (0.33X) when the power is above 35 MW. A detailed rotordynamic analysis shows that the flexible rotor plus the seal and turbine stage destabilizing forces [12, 13] yield a system with negative damping ratio at the first forward mode (~30 Hz). To solve the SSV issue, the authors introduce an ISFD and replace the 5 pad TIJBs with 4 pads load between pads TPJBs. The combined 4 pad TPJB and ISFD has direct stiffness 20% lower than the original 5 pads TPJBs. After the retrofit, the turbine operates without SSV. Since the invention in the early 1990s, ISFDs have been frequently mentioned to suppress different types of system vibration issues. However, there are very limited number of component level studies on the static and dynamic performance of ISFDs. Ertas et al. [14] measure and predict damping and inertia coefficients for a four pads ISFD with a diameter D = 141 mm and axial length L = 56 mm, and several end seals’ clearances. The authors find that the ISFD damping and inertia coefficients dramatically decrease as the end seal clearance increases. A companion test program by Lu et al. [15] for a four pads ISFD shows similar trends in damping and inertia coefficients as the end seal clearance increases. The test ISFD has diameter D = 157 mm (L/D = 0.48), pad arc extend = 73°, film clearance c = 0.353 mm, and axial seal gap = 1.5c, 1.21c, and 0.8c. The damper with the tightest seals (gap = 0.8c) produces a unique stiffness hardening as the excitation frequency increases. The large increase in dynamic stiffness is due to fluid compressibility. As is well known, in conventional 360° SFDs the oil feed condition such as supply pressure and the number and diameter of feed hole affect the SFD damping and inertia force coefficients. Zhang and Roberts [16] analyze the effect of lubricant supply mechanism (LSM) on the force coefficients of a centrally grooved short squeeze film damper, and find a nonzero fluid static force which is linearly related to the supply pressure. San Andrés et al. [17] present an experimental effort to quantify the similarities and differences between two piston rings and O-ring sealed short length SFDs. The SFD has diameter D = 127 mm (L/D = 0.2) and clearance c = 0.373 mm (c/D = 0.003). One or three evenly distributed feed holes with diameter 2.5 mm feed an ISO VG 2 oil into the film land at 0.6 barg. The results show that the SFD having three oil feed holes produces 20%–40% less damping compared the SFD having only one feed hole. The feed holes disturb the film pressure and cause reduction in damping. Iacobellis et al. [18] conduct experiments to study the effect of feed hole on the unbalance response of a rotor supported on SFDS with either sealed ends or with open ends SFD. The rotor supported by two ball bearings in series with two SFDs on the two ends weights 20.84 kg and has a maximum speed of 20,000 rpm. Four feed holes evenly distributed along the SFD circumference at a supply pressure of 5.5 bar. The feed holes can be plugged so that the SFD can operate with any combination of the 1 to 4 feed holes conditions. The research find out that with the same amount of unbalance the rotor supported by the SFD with four feed holes (both the sealed and open ends) has smaller

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amplitude of vibration when crossing the critical speed, indicating that the SFD with four feed holes generates more damping than the SFD with only one feed hole. This finding is in contradictory with the results in Ref. [17]. The authors argue that the SFD with four feed holes are less likely to develop air ingestion because the four oil feed hole configuration produce a surplus of oil leading to higher and more stable pressure in the SFD. The extensive research shows that both the inlet feed holes and end seal clearances greatly affect the dynamic force coefficients of conventional SFDs. However, there is very scant literature on the influence of feed hole on the static and dynamic performance of ISFDs. Therefore, this paper provides insight on how the supply oil flow rate, pressure profile and dynamic force coefficients change with the diameter of inlet feed orifices and end seal clearances.

2 Analysis Figure 1 shows a schematic view of an integral squeeze film damper [15]. Four pairs of S-springs connect the damper outer ring to the inner ring that supports the housing of a roller element bearing or a fluid film bearing. The damper pads have arcuate extent θ p and are filled with lubricant. The graph includes a coordinate system (x, y) with the angle  starting from the –x axis. According to an experimental study in Ref. [15], the damping coefficients of an ISFD is not a strong function of eccentricity and is isotropic. Circular orbit of the damper inner ring is adequate to study the performance of an ISFD. Undergoing a circular centered orbit (CCO) with amplitude r o and frequency ω, the displacement vector of the center of the inner ring is [19]     cos(ωt) esX (t) (1) = ro z= sin(ωt) esY (t) Accordingly, the film thickness h(t) in the segments of the integral squeeze film damper is h(t) = c + esX (t) cos() + esY (t) sin()

(2)

where c is the nominal damper radial clearance that is determined by the electrical discharge manufacturing (EDM) process. The first and second time derivatives of the film thickness are

 where

˙ = e˙ sX (t) cos() + e˙ sY (t) sin() h(t)

(3)

¨ = e¨ sX (t) cos() + e¨ sY (t) sin() h(t)

(4)

       e˙ sX (t) cos(ωt) − sin(ωt) e¨ (t) . = ro ω = −ro ω2 , and sX sin(ωt) cos(ωt) e˙ sY (t) e¨ sY (t)

On the Influence of the Lubricant Feed Orifice Size

Fig. 1. A schematic view of an integral squeeze film damper and coordinates [15].

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Fig. 2. A cross section view of an ISFD showing the flow path [15].

Figure 2 shows a cross section view of the ISFD whose two ends are sealed with end plates with radial a length l1 and an axial gap b1 . Each damper segment features an orifice with diameter d 0 to deliver lubricant into the middle plane fluid film land. As shown in the graph, the lubricant feeds the orifice at a supply pressure1 Ps , and enters the damper film land at pressure Pi . After exiting the film land with pressure Pe , the lubricant flows through the gap between the end plates and the outer ring and discharges to ambient at pressure Pa . Delgado and San Andrés [20] introduce an extended Reynolds Equation for squeeze film dampers and oil seal rings.  3    ρh ∂P ∂ ρh3 ∂P ∂ ∂ ρh2 ∂ 2 + = (ρh) + (5) (ρh) R ∂ 12μ R ∂ ∂z 12μ ∂z ∂t 12μ ∂t 2 where μ and ρ are the lubricant viscosity and density, respectively. For operation with a two phase flow, the effective viscosity and density are: μ = μg α + μl (1 − α)

(6)

ρ = ρg αg + ρl (1 − α)

(7)

where μg and μl and ρ g and ρ l are the viscosity and density of the gas and liquid components, respectively, and α is the gas volume fraction (GVF). For a homogeneous mixture, the local gas volume fraction is a direct function of the fluid pressure [21]: α(P) =

1 1+

P(,z,t) −Pv Ps −Pv



1−αs αs



(8)

where Pv ~ 1 kPa [21] is the liquid cavitation pressure, and α s is gas volume fraction at a supply condition. 1 In a centrifugal compressor, a typical range for the oil supply pressure is between 1.7 to 3.5

barg.

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At the two sides along the circumferential direction, the S-springs block the circumferential flow, i.e., qθ ∼

∂P =0 ∂

(9)

The film axial ends (z = ± ½ L) can be either sealed or open to ambient. For an open ends condition, the lubricant exit pressure is ambient (Pa ). For a sealed ends damper, the mass flow (q1z ) per unit circumferential length through the end seal is proportional to the local pressure drop and a local flow resistance [15], ρh3 q1z |

1 ,z=± 2 L

=−



z=±

1 2L

12μ

 ρb31 ∂P  = − (Pe − Pa ) ∂z z=± 1 L 12μl1

(10)

2

Note that if Pe > Pa , the flow is from the damper toward ambient; whereas if Pe < Pa , the flow reverses direction. The mass of the lubricant flowing through an inlet orifice is [19] 

(11) Qin = sgn(Ps − Pi ) Cf Aeq 2ρ|Ps − Pi | where C f is an orifice flow coefficient, and Aeq is an equivalent flow area2 . The computational program implements a finite volume method (FVM) to interactively solve the extended Reynolds Eq. (5). A difference of 1‰ or less between consecutive iteration pressure fields determines convergence in the fluid film land. A Newton-Raphson Method balances the lubricant flowing into/out of the damper pads from the orifice. Once the pressure field is obtained, the fluid force (F) acting on the inner ring of the ISFD is [19]: 

fx(t) fy(t)

 =

N K=1

L θ2K

R 0

θ1K

 K P(,z)

 cos() d dz sin()

(12)

where θ1K and θ2K denote the angles for the leading edge and trailing edge of the K th pad. The steps to obtain numerically the force coefficients (C and M) are similar to those in the experimental procedure detailed in Ref. [22]. The inner ring of the ISFD whirls forward (|z|eiωt ) and backward (|z|e−iωt ) over a specified frequency range. Correspondingly, the forward and backward whirl motions produce reaction forces (|F|eiωt and |F|e−iωt ). The whirl motions and the calculated reaction forces then render the ISFD complex dynamic stiffness (H), from which linearized dynamic force coefficient are extracted, i.e., −1 H(ω) = F(ω) z(ω)

(13)

A0 ·Ac 2 Ertas et al. [14] utilizes A for the equivalent orifice area to predict the ISFD eq =  A20 +A2c πd2

orifice flow, where A0 = 4 0 ,Ac = π d0 c, d 0 is the orifice diameter (Aeq ~ Ac for large orifice diameter = 3 × 1.98 mm). Ref. [14] uses C f = 0.98 to match the predicted oil flow to the measured flow.

On the Influence of the Lubricant Feed Orifice Size

Re(H) → K − ω2 M, Ima(H) → i ω C

179

(14)

where F(ω) = [F(+ω) | F(-ω) ], z(ω) = [z(ω) | z(-ω) ]T are matrices with force and displacements, forward and backward. Re(H) and Ima(H) are the real and imaginary parts of the complex dynamic stiffness, and K, C and M are the matrices of stiffnesses, viscous damping and added mass coefficients.

3 Fluid Film Pressure Field and Lubricant Flow Rate Table 1 details the basic dimensions for the ISFD and the lubricant properties, as in Ertas et al. [14]. The film lands diameter D = 141 mm, length L = 56 mm, and radial clearance c = 0.56 mm. Each pad with arc length θ p = 54° features an inlet orifice with a diameter d 0 = 1.98 mm. End plate seals with radial length l1 = 5.1 mm and axial clearance b1 = 0.191 mm restrict the axial flow. The lubricant is ISO VG32 oil with a viscosity of 19.1 cP at a supply temperature of 49 °C. To produce a complete parametric study on how the end seals’ gap and the size of the inlet orifices affect the static and dynamic forced performance of the ISFD, this work also includes predictions for the same ISFD with end seals’ clearance varying from b1 to 3b1 and the diameter of the inlet orifice changing from d 0 to 3d 0 . The range of the dimensions chosen cover the tolerance range of the inlet orifice and end seal clearance. For the prediction of force coefficients, the whirl frequency (ω) of the ISFD inner ring ranges from 20 Hz to 120 Hz, and the amplitude of the circular centered orbits (CCOs) varies from r 0 = 23 μm (e = r 0 /c = 4%) to r 0 = 112 μm (e = r 0 /c = 20%). The maximum squeeze film speed vs = ωr 0 = 84.5 mm/s, and the maximum squeeze film Reynolds number Res = (ρ/μ) ωc2 = 10.9. Table 1. Dimension of the sample ISFD and fluid properties [14]. Diameter at film land, D

141 mm

Length, L

56 mm

Film land clearance, c

0.56 mm

Number of pads, n

4

Arc radius θ p

54°

Inlet orifice diameter, d o

1.98 mm, 3.96 mm, 5.94 mm

End seals clearance, b1

0.191 mm, 0.382 mm, 0.573 mm

radial length, l 1

5.1 mm

ISO VG32 Viscosity, μ

19.1 cP at 49 °C

Density, ρ oil

871 kg/m3

Supply pressure

2.4 bar(a)

Ambient pressure

1.0 bar(a)

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Figure 3 shows the pressure profile in a single pad of a sealed ends ISFD (b1 = 0.191 mm = 0.34 c) and supplied with lubricant through orifices with diameter d 0 = 0.198 mm and 0.396 mm. The damper is centered and the whirl frequency of the inner ring is null, i.e. a stationary condition. The pressure at the two ends is above ambient pressure due to the large flow resistance induced by the tightness of the end seals. The sealed ends ISFD with d 0 = 0.198 mm has an orifice discharge pressure of 1.61 bar(a), whereas the ISFD with inlet orifice diameter = 2d 0 has a higher orifice discharge pressure of 2.03 bar(a)3 .

Fig. 3. Pressure profiles in the film lands of a centered ISFD with sealed ends and two size oil inlet orifices. Supply pressure Ps = 2.4 bar, ambient pressure Pa = 1.0 bar, whirl frequency ω = 0 Hz.

Figure 4 depicts the ISFD oil flow rate (Q) versus supply pressure (Ps ). The end seals’ clearance is b1 = 0.191 mm, whereas the diameter of the inlet orifice equals d 0 = 1.98 mm, 2 d 0 and 3d 0 . The lines denote predictions while the symbols correspond to test data in Ref. [14]. As the diameter of the feed orifices increases from d 0 to 2d 0 , the flow rate almost doubles. Note that API 614 [23] requires two to fifteen minutes of retention time in an oil reservoir4 . The larger orifice diameter will inevitably increase capital expenditures as the larger supply flow calls for a larger volume reservoir. Figure 5 shows pressure profiles in a pad of a sealed ends ISFD (b1 = 0.191 mm, d 0 = 1.98 mm) as the inner ring describes circular centered orbits (CCO) with amplitude r 0 = 56 μm (r 0 /c = 0.10. The supply pressure Ps = 2.4 bar(a) and the ambient pressure Pa = 1.0 bar(a). The top graph shows the pressure profile in Pad 1 (see Fig. 1 for reference) at the instant t = 0 when the ring eccentricity center aligns with the + x axis, and the whirl frequency is 70 Hz (vs = 24.6 mm/s, Res = 6.4). The bottom graph shows the 3 bar(a) represents absolute pressure. 4 The retention time is the amount of time that the lubricant stays in the oil reservoir before its

return into the piping system. A typical retention time is between three to five minutes.

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Fig. 4. ISFD oil flow rate (Q) vs supply absolute pressure (Ps ). Lubricant orifice diameter varies, end seal b1 = 0.191 mm. Comparisons to measured flow in Ref. [14]

pressure and film thickness versus normalized time (t/T ) at z = 1 /4 L and θ = ½ θ p over one period of whirl motion (T = 2π/ω). The curves denote operation with frequency ω = 70 Hz, 80 Hz (vs = 28.1 mm/s, Res = 7.3), and 100 Hz (vs = 35.2 mm/s, Res = 9.1). For operation at ω = 80 Hz, the pressure wave is close to the one for 0 bar(a) at t/T ~ 0.3. As the whirl frequency of the inner ring reaches 100 Hz, the pressure at (z = 1 /4 L and θ = ½ θ p ) is below cavitation pressure during the time period t/T = 0.2 ~ 0.4. Thus, the pressure is set as P = Pcav , where Pcav is the vapor pressure of the liquid lubricant.

4 Dynamic Force Coefficients Figure 6 shows the predicted complex dynamic stiffness, Re(H XX ) and Ima(H XX ), versus frequency (ω). The axial clearance of the end seals is constant (b1 = 0.191 mm) whereas the diameter of the oil feed orifices equals d 0 , 2 d 0 and 3d 0 . The amplitude of whirl motion is r 0 = 56 μm (r 0 /c = 10%). Note the predictions produce H XX = H YY . Re(H XX ) is a parabolic function of frequency (ω), except for the configuration having orifice diameter d 0 = 1.98 mm. Thus, Re(H XX ) can be characterized with a static stiffness (K) and a virtual mass (M), i.e., Re(H XX ) → (K-ω2 M). The fluid films in the ISFD do not produce a significant static stiffness as Ps = 2.4 bar(a) is low. The increase in orifice diameter from d 0 to 3d 0 causes Re(H XX ) to reduce less with frequency. The imaginary part of the complex dynamic stiffness, Ima(H XX ), is a linear function of frequency and reducing in magnitude as the orifice diameter increases. Recall that for the ISFD with inlet orifice diameter d 0 , the cavitation occurs when ω > 80 Hz (ωr 0 = 28.1 mm/s). Thus, the curve representing Ima(H XX ) reaches a turning point at ω = 80 Hz. Note that the increase of orifice diameter from d 0 to 2 d 0 removes the turning point of Ima(H XX ), indicating that the onset of oil cavitation is delayed to a higher whirl frequency.

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Fig. 5. Top graph: Snapshot of the pressure profile in a single pad at t = 0 s (whirl frequency 70 Hz). Bottom graph: pressure and film thickness versus normalized time (t/T ) at z = 1 /4 L and θ = ½ θ p (whirl frequency 70 Hz, 80 Hz, and 100 Hz). Inlet orifice diameter = d 0 . End seals’ clearance b1 = 0.191 mm. Supply pressure Ps = 2.4 bar(a), ambient pressure Pa = 1.0 bar(a), amplitude of whirl motion r 0 = 56 μm (r 0 /c = 10%).

Figure 7 shows the real and imaginary parts of the cross coupled complex dynamic stiffness H XY versus frequency. The predicted results show H XY = −H YX , and for simplicity, the graphs only show H XY . Compared to the direct coefficient H XX , H XY is very small in magnitude. For the ISFD with orifice diameter d 0 , Re(H XY ) switches from positive to negative at high frequencies, and Ima(H XY ) becomes nonlinear at high frequencies where cavitation occurs in the fluid film.

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(a) Re(HXX)

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(b) Ima(H HXX)

Fig. 6. Real and imaginary parts of direct complex dynamic stiffness H XX = H YY vs. frequency. Inlet orifice diameter = d 0 , 2d 0 , and 3d 0 . End seals’ clearance b1 = 0.191 mm. Supply pressure Ps = 2.4 bar(a), ambient pressure Pa = 1.0 bar(a), amplitude of whirl motion r 0 = 56 μm (r 0 /c = 10%).

(a) Re((HXY)

(b) Ima(H HXY)

Fig. 7. Real and imaginary parts of cross coupled complex dynamic stiffness H XY = - H YX. vs. whirl frequency. Inlet orifice diameter = d 0 , 2d 0 , and 3d 0 . End seals’ clearance b1 = 0.191 mm. Supply pressure Ps = 2.4 bar(a), ambient pressure Pa = 1.0 bar(a), amplitude of whirl motion r 0 = 56 μm (r 0 /c = 10%).

Figure 8 depicts the direct damping coefficients C, derived from a linear curve fit of Ima(H XX ), versus end seal clearance b1 and three orifice diameters. The lines and open symbols represent predictions for journal amplitudes of motion r = 0.04 c and 0.1c respectively. The solid symbols stand for test data in Ref. [14], an ISFD with d0 = 1.98 mm and b1 = 0.191 mm. Ref. [14] states that the outer film land contributes to about 70% of the total damping (See Fig. 1 for reference). Therefore, the predicted damping coefficients in this paper are scaled by a factor equaling (1.0/0.7) = 1.43 to obtain the damping of the whole ISFD. The change of orifice diameter is very effective to alter C when the end seals’ clearance is small. For example, when b1 = 0.191 mm, the increase of orifice diameter from d 0 = 1.98 mm to 3d 0 = 5.94 mm reduces C by ~51%. However, for the damper with a 3b1 = 0.573 mm end gap, the damping coefficient reduces by only ~15%.

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On the other hand, when the inlet orifice diameter is fixed at d 0 = 1.98 mm, an increase in the end seals’ clearance, from b1 = 0.191mm to 3b1 = 0.573 mm, reduces the damping greatly by ~88%. For a damper with 3d 0 orifice diameter, the damping coefficient reduces by ~78%. The results shows that the increase in the size of the inlet orifice weakens the classic lubrication theory relationship “damping coefficient ~ 1/c3 ” [14]. Figure 9 shows the predicted added inertia coefficient (M) versus the end seals’ clearance (b1 ). An increase in b1 and in the inlet orifice diameter greatly reduces M. The model in Eq. (5) overestimates as much as four times the virtual inertia coefficient. However, in a rotordynamic analysis, the added mass term is always neglected as most of the time the inclusion of added mass term does not affect the “logdec” too much.

Fig. 8. Predicted direct damping coefficient C vs. end seals’ clearance b1 . Inlet orifice diameter = d 0, 2 d 0 , 3d 0 . Supply pressure Ps = 2.4 bar(a), ambient pressure Pa = 1.0 bar(a). Lines: r 0 = 23 μm (r 0 /c = 4%), Open symbols: prediction for r 0 = 56 μm (r 0 /c = 10%), Solid symbols: test data from Ref. [14].

Figure 10 shows the direct damping coefficient C = Ima(H XX )/ω versus squeeze velocity (vs = rω) and operation with orbit radii r/c = 0.04, 0.10 and 0.20. C is independent of the whirl motion for vs < 0.22 mm/s. A further increase in vs , due to either a larger r or a higher frequency ω or both, produces lubricant vapor cavitation in the film and thus a (linear) drop in the damping coefficient. San Andrés and Koo [19] analyze the dynamic force coefficients of a sealed ends SFD supplied with an air in oil bubbly mixture. The goal of the research in Ref. [19] was to study the performance of the SFD with a bubbly mixture of known gas volume fraction at the damper inlet orifice. The experimentally derived and predicted damping coefficients decrease by ~20% as the air volume fraction in the mixture increases to 50% while the inertia rapidly decreases.

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Fig. 9. Predicted direct inertia coefficient M vs. end seals’ clearance b1 . Inlet orifice diameter = d 0, 2 d 0 , 3d 0 . Supply pressure Ps = 2.4 bar(a), ambient pressure Pa = 1.0 bar(a). Lines: r 0 = 23 μm (r 0 /c = 4%), and Open symbols: prediction for r 0 = 56 μm (r 0 /c = 10%), Solid symbols: test data from Ref. [14].

Fig. 10. Predicted direct damping coefficients (C) vs squeeze film velocity (vs ). Inlet orifice d 0 varies and end seals’ clearance = b1 . Supply pressure Ps = 2.4 bar(a), ambient pressure Pa = 1.0 bar(a), amplitude of whirl motion varies.

Figure 11 and Fig. 12 show the predicted direct damping and added inertia versus inlet GVF. The amplitude of whirl motion r 0 = 23 μm (r 0 /c = 10%), and the maximum squeeze velocity (vs ) is 42.2 mm/s). The end seals’ clearance b1 = 0.191 mm and the

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feed hold diameter equals d 0 , 2 d 0 and 3 d 0 . As the predictions show, both the direct damping and added inertia reduce continuously as the inlet GVF increases from 0 to 0.2. Note that the test by San Andrés and Lu [24] for a bubbly seal also reveals that the damping and inertia reduces almost linearly with the increase in inlet GVF.

Fig. 11. Predicted direct damping coefficient C versus inlet GVF. Inlet orifice d 0 = 1.98 mm, end seal clearance b1 = 0.191 mm. Supply pressure Ps = 2.4 bar(a), ambient pressure Pa = 1.0 barg, amplitude of whirl motion r 0 = 23 μm (r 0 /c = 10%).

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Fig. 12. Predicted added inertia versus inlet GVF. Inlet orifice d 0 = 1.98 mm, end seal clearance b1 = 0.191 mm. Supply pressure Ps = 2.4 bar(a), ambient pressure Pa = 1.0 barg, amplitude of whirl motion r 0 = 23 μm (r 0 /c = 10%).

5 Conclusion Squeeze film dampers (SFDs) help to reduce the amplitude of rotor whirl motions, as well as to provide structural isolation and enhance stability where conventional type fluid film bearings are inadequate to meet certain requirements. Compared to conventional squirrel cage supported SFDs, ISFDs provide a compact solution along with a short axial length. This paper presents a physical model and predictions oil flow rate and dynamic force coefficients for a four pads ISFD supplied through inlet orifices whose diameter d 0 = 1.98 mm, to 2d 0 , to 3d 0 . The damper has end seals with axial clearance b1 = 0.191 mm, and increased to 3 b1 . As the diameter of the oil inlet orifice increases from d 0 to 2d 0 the flow rate into the ISFD almost doubles, thus requiring of a larger oil reservoir. Compared to the ISFD with inlet orifice diameter d 0 , the ISFD having a seal clearance b1 = 0.191 mm but supplied through larger orifices (2d 0 , 3d 0 ) produces a lower damping and added inertia coefficients. The magnitude of the damping and added inertia coefficients generated by the ISFD with end seals having a tight clearance (b1 ) is more sensitive to the feed hole diameter than to an increase in the ISFD end seals’ clearance. An increase in the size of the inlet orifice weakens relationship “damping ~ 1/clearance3 ” [14]. Predictions for the ISFD operating with an air in oil mixture shows that the damping and added inertia reduce almost linearly as the inlet GVF reduces from 0 to 0.2, suggesting that if a bubbly mixture exists in the ISFD, both the damping and added inertia will reduce linearly with the increase in inlet GVF. The paper calls for further experiments

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of the ISFD lubricated with a bubbly mixture with controlled inlet GVF to study the influence of free gas on its performance. Acknowledgement. The first author would like to thank Hunan SUND Technological Corporation for allowing the publication of this paper.

References 1. Zeidan, F.Y., San Andrés, L., Vance, J.M.: Design and application of squeeze film dampers in rotating machinery. In: Proceedings of the 25th Turbomachinery Symposium, Texas A&M University, Turbomachinery Laboratory, Houston, September, pp. 169–188 (1996). https:// doi.org/10.21423/R1694R 2. Gunter, E.: Ekofisk Revisited-Bearing Optimization for Improved Rotor Stability. Dyrobes Rotordynamic Software (2020). https://dyrobes.com/paper/ekofisk-revisited-bearing-optimi zation-for-improved-rotor-stability 3. Zeidan, F.Y.: Application of Squeeze Film Dampers. Turbomachinery International, pp. 50–53 (1995) 4. Leader, M.E., Whalen, J.K., Grey, G.G.: The design and application of a squeeze film damper bearing to a flexible steam turbine rotor. In: 24th Turbomachinery Symposium, Dallas, TX, pp. 49–58 (1995). https://doi.org/10.21423/R1R36D 5. Gooding, W.J., Meier, M.A., Gunter, E.J., Key, N.L.: Nonlinear Response and Stability of an Experimental Overhung Compressor Mounted with a Squeeze Film Damper. ASME Paper No. GT2020–15212 (2020) 6. Jeung, S.-H., San Andrés, L., Den, S., Koo, B.: Effect of oil supply pressure on the force coefficients of a squeeze film damper sealed with piston rings. J. Tribol. 141, 061701 (2019) 7. De Santiago, O., San Andrés, L.: Imbalance Response and Damping Force Coefficients of a Rotor Supported on End Sealed Integral Squeeze Film Dampers. ASME Paper 99-GT-203 (1999). https://doi.org/10.1115/99-GT-203 8. Agnew, J., Childs, D.: Rotordynamic Characteristics of a Flexure Pivot Pad Bearing with an Active and Locked Integral Squeeze Film Damper. ASME Paper No. GT2012-68564 (2012). https://doi.org/10.1115/GT2012-68564 9. Ferraro, R., Catanzaro, M., Kim, J., Massini, M., Betti, D., Hardy, R.L.: Supression of Subsynchronous Vibration in a 11 MW Steam Turbine Using Integral Squeeze Film Damper Technology at the Exhaust Side Bearing. ASME Paper GT2016-57410 (2016). https://doi. org/10.1115/GT2016-57410 10. Ertas, B., Cerny, V., Kim, J., Polreich, V.: Stabilizing a 46 MW multistage utility system turbine using integral squeeze film bearing support dampers. J. Eng. Gas Turbines Power 137, 052506 (2015) 11. Saeki, K., Sanari, H., Baba, Y., Ito, M., Shibata, S., Kurohashi, M.: Integrally geared centrifugal compressors for high pressure process gas services. Kobelco Technol. Rev. (29), 042–046 (2010). https://www.kobelco.co.jp/english/ktr/pdf/ktr_29/042-046.pdf 12. Alford, J.: Protecting turbomachinery from self-excited rotor whirl. J. Eng. Power 87(4), 333–343 (1965) 13. Lerche, A.H., Musgrove, G.O., Moore, J.J., Kulhanek, C.D., Nordwall, G.: Rotordynamic Force Prediction of an Unshrouded Radial Inflow Turbine Using Computational Fluid Dynamics. ASME paper No. GT2013-95137 (2013) 14. Ertas, B., Delgado, A., Moore, J.: Dynamic characterization of an integral squeeze film bearing support damper for a supercritical CO2 expander. J. Eng. Gas Turbines Power 140(5), 052501 (2018)

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15. Lu, X., San Andrés, L., Koo, B., Scott, T.: On the effect of the gap of end seals on force coefficients of a test integral squeeze film damper: experiments and predictions. J. Eng. Gas Turbines Power 143(1), 011014 (2021) 16. Zhang, J.X., Roberts, J.B.: Force coefficients for a centrally grooved short squeeze film damper. J. Tribol. 118(3), 608–616 (1996) 17. San Andrés, L., Koo, B., Jeung, S.-H.: Experimental force coefficients for two sealed ends squeeze film dampers (piston rings and O-Rings): an assessment of their similarities and differences. J. Eng. Gas Turbines Power 141(2), 021024 (2019) 18. Iacobellis, V., Behdinan, K., Chan, D., Beamish, D.: Effect of hole feed system on the response of a squeeze film damper supported rotor. Tribol. Int. 151, 106450 (2020) 19. San Andrés, L., Koo, B.: Model and experimental verification of the dynamic forced performance of a tightly sealed squeeze film damper supplied with a bubbly mixture. J. Eng. Gas Turbines Power 142, 011023 (2020) 20. Delgado, A., San Andrés, L.: A model for improved prediction of force coefficients in grooved squeeze film dampers and oil seal rings. J. Tribol. 132(3), 032202 (2007) 21. Diaz, S., San Andrés, L.: A model for squeeze film dampers operating with air entrainment and validation with experiments. J. Tribol. 123(1), 125–133 (2000) 22. San Andres, L., Jeung, S.-H.: Orbit-model force coefficients for fluid film bearings: a step beyond linearization. J. Eng. Gas Turbines Power 138(2), 233502 (2016) 23. Patel, V.P., Coppins, D.G.: Selection of API 614, Fourth Edition, Chapter 3-general purpose lube oil system components for rotating process equipment. In: Proceedings of the 18th International Pump Users Symposium, Texas A&M University, Turbomachinery Laboratory, Houston (September 2001). https://doi.org/10.21423/R1WT2C 24. San Andrés, L., Lu, X.: Leakage, drag power, and rotordynamic force coefficients of an air in oil (wet) annular seal. J. Eng. Gas Turbines Power 140(1), 012505 (2018)

Dropdown Analysis of High-Speed Thin-Shaft Coupled Rotor System Integrated with Three Active Magnetic Bearings Gyan Ranjan(B) , Juuso Narsakka, Tuhin Choudhury, and Jussi Sopanen Department of Mechanical Engineering, Lappeenranta-Lahti, University of Technology, LUT Skinnarilankatu 34, 53850 Lappeenranta, Finland {Gyan.Ranjan,Juuso.Narsakka,Tuhin.Choudhury, Jussi.Sopanen}@lut.fi

Abstract. Rotating machines supported by Active Magnetic Bearings (AMBs) are also equipped with touchdown bearings (TDBs) to support the rotor in cases of power failure or overload of AMBs system. In these dropdown events, the TDBs absorb the impact of the dynamic loads. The simulation and analysis of the orbit responses from such dropdown events can be utilized to prevent critical damage to the machinery. To that end, this study presents the dropdown simulation of a megawatt class induction machine driveline, levitated on three AMBs. The driveline has two AMBs for the motor and one AMB supporting the impeller of a compressor on an extension shaft. The shafts are connected via a thin shaft coupling. The system is first levitated with the designed LQR controller. To simulate a dropdown event, the destabilized system is allowed to fall under the influence of gravity load at the three TDBs. The time domain displacement amplitudes of the FE model are generated for different contact conditions based on the tangential velocity of the rotor. The vibrational amplitudes with different values of unbalance forces are analyzed to identify the most critical bearing and determine the operational limits of rotor system. Keywords: Active Magnetic Bearings · Dropdown Analysis · High-Speed Rotor · LQR control · Finite Element Method

1 Introduction The first direct drive high-speed machines were designed for the applications where high power and low mass and volume were required with less emphasis on cost of the machine, such as racing and aeroengines [1, 2]. The first high-speed machines were equipped with conventional roller or sliding bearings. The development of active magnetic bearings (AMBs) has opened the field of machines where frictionless rotational systems are possible. The advantages of a direct high-speed AMB driveline, such as oil-free and low-maintenance operation and a small footprint, have increased interest in more general applications such as compressors, turbines, and pumps. Downside of the direct highspeed machines has been a long development time due to demanding multidisciplinary © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 190–200, 2024. https://doi.org/10.1007/978-3-031-40455-9_15

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design process. Also, the nature of case specific design is limiting the usability of same design for various application, which results in small manufacturing batches and high unit costs [3]. To overcome the usability challenge, a novel three radial AMB rotor configuration is proposed to be used in part of the modular high-speed driveline [4]. With the new technical solution is always necessary to study its capabilities from various aspects. The main target of this study is to investigate the performance of touchdown bearings in new rotor configuration. To be able to study the performance of TDBs under different usage situations, a simulation model including the rotor, AMBs and TDBs must be built. From previous studies of TDBs and rotor AMB systems [5–7] can be derived the methods for constructing a simulation in which operational levitation and dropdown event transient analysis can be studied. Also, the analysis of orbit responses at TDBs locations and the design of different AMB controller are important areas of research for rotor-AMB configuration [8–10]. The scope of this work is to analyze the behavior of the three AMB configuration in a dropdown situation, and the influencing parameters for maximum load at TDBs. Also, the critical bearing is identified in different dropdown conditions to avoid failure to the machinery. The structure of the work is as follows: In Sect. 2 is presented the theory of the simulation model. In Sect. 3, the configuration and case study of the proposed new solution are described. In Sect. 4 are presented the results and analysis of simulation. Sect. 5 includes the conclusion of the work.

2 Mathematical Modelling of Thin-Shaft Coupled Rotor-AMB System with TDBs The mathematical model of the rotor system is developed using Timoshenko beam elements. The system is analyzed for the lateral vibrations in the axis perpendicular to the shaft axis (z). Two translations and two rotational degrees of freedoms (DoFs) are considered at each node locations. The compressor drive and extension shaft with an impeller are supported on AMBs. A thin-shaft coupling is used to connect the compressor drive with an extension shaft. The rotor system also comprises of TDBs to support the system in case of a dropdown event or when the rotor is not levitated. 2.1 Equations of Motion of Rotor System The flexible shaft is discretized into several elements and the impeller is added as a rigid disc to the respective location on the rotor. The assembled equation of motion of the rotor system is given as (Ms + Md )¨r(t) + (Cs + ω(Gs + Gd ))˙r(t) + Ks r(t) = fe (t) + fg

(1)

Cs = α (Ms + Md ) + βKs ; fe = meω2 ej(ωt+θ ) x

(2)

with,

where Ms , Cs , Gs and Ks are the mass, damping, gyroscopic and stiffness matrices of the shaft. Md and Gd are the mass and gyroscopic matrices of the rigid disc. Proportional

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damping matrix (Rayleigh damping) is obtained using coefficients α and β. f g is the gravity load vector of the rotor system. f e is the unbalance force at the impeller location. m, e, θ are the mass, eccentricity, and phase of the unbalance in the rotor system. ω is the operating speed of the rotor system. x is the position vector of the unbalance force and is given as x = [0 0 0 ... 1 ...]T

(3)

where, ‘1’ is the location of the unbalance. 2.2 AMB Support System Model The rotor system is supported on AMBs at different locations. A linearized AMB force model is utilized in the present work with differential driving mode control and is given as fa (t) = −Ka ra + Ki ic

(4)

where, Ka and Ki are the displacement and the current stiffness matrices of AMBs. ra and ic are the displacements at AMB locations and control currents respectively. The controller of the AMB system is designed based on the LQR control strategy. The state space representation of the rotor system is given as   0 1 (5) A= −1 (K − K −1 −Mm a,m ) −Mm (Cm + ωGm ) m   0 B= ; Q; R (6) −1 K −Mm i,m where, Mm , Cm , Gm , Km , Ka,m and Ki,m are the reduced order matrices of the rotorAMB system with required DoFs to control. The Q and R are weights given to states and control vectors [11]. The controller matrix, H is obtained solving the Algebraic Ricatti Equation. The state space matrix with LQR control is given as Ae = A − BH

(7)

The AMB force with the LQR control model combined with integral action is given as

  fa (t) = −Ka ra + Ki H{rs r˙ s }T + KI rs

(8)

where, KI is the integral gain of the controller and rs is the responses estimated through Kalman observer. rs is the integration of the estimated responses.

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2.3 Touchdown Bearing Force Model The touchdown bearing forces are modelled based on displacements (rt ) exceeding the air gap (c) between rotor and TDB [6, 12]. The relative responses at TDBs are given as re = (|rt | − c)ejφ ; φ = arg (rt )

(9)

The resulting normal force, f n,s , due to support stiffness can be given as follows while avoiding discontinuity at r = 0 using arctan functions fn,s = (kt f(re )re + ct f(re )˙re )ejφ , for re > 0 for re > 0 fn,s = 0

(10)

  1 arctan(π γk re )+0.5 f(re )= π

(11)

with,

where, k t and ct are the support stiffness and damping coefficients of TDBs; the constant γ k ensures the continuity of the function f(re ) around r = 0. The resulting normal force f n,c due to contact stiffness is given as   p q p fn,c = kc f(re )re + bf re r˙ e ejφ , for re > 0 (12) fn,c = 0 for re > 0 with, b = 1.5λk c where, k c is the contact stiffness of the TDB; λ is the contact parameter. The tangential contact force can be written as ft,c = jμ(vr )fn,c ejφ , for re > 0 for re > 0 ft,c = 0

(13)

with sliding velocity, (x˙y - y˙x) vr =

+ωR x2 + y2

(14)

and the friction coefficient given as vr μ(vr ) = 2vg



1−γ 1+ζ + 1 + ζ |vr | 1 + ζ |vr |2

 (15)

with, γ = (1 − μd /μs )1/2 ; ζ = (1−γ )/ 2vg μd . vg is the regularized parameter. R is the radius of the rotor at TDBs location.

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2.4 Overall Equation of Motion of Rotor-AMB System Integrated with TDBs The assembled equation of motion of rotor-AMB system is given as  M¨r(t) + (Cs + ωG)˙r(t) + Ks r(t) = fe (t) + fg − La fa − Lt fn,s + fn,c + ft,c (16) with, M = Ms +Md ; G = Gs +Gd . Where, La and Lt are the location matrices of AMBs and TDBs respectively. The equation of motion is utilized to simulate the dropdown analysis of rotor-AMB system and analyze the bearing forces developed in the system under different dynamic conditions.

3 Dropdown Analysis of the Developed System Model In the events of dropdown, the TDBs absorb the impact of the dynamic loads. The orbit responses from such dropdown events are analyzed to identify the level of impact loads and displacements arising in the system. Also, the analysis is carried out at different operating speed in presence of unbalances. The operational limits and critical TDB location can be identified with the investigation to prevent damage to the machinery. 3.1 Test Rig Description and Simulation Model The developed finite element model of the rotor system consisting of compressor drive and extension shaft is divided into 75 elements as shown in Fig. 1.

Fig. 1. Wireframe model of the rotor-AMB system with discretized elements. Key nodes such as the AMB actuators, sensors, TDBs and impeller are shown.

The compressor drive is supported on two AMBs at node locations 14 and 32, while the extension shaft is supported by one AMB at node location 70. The touchdown bearings are located at node positions 7, 38 and 73. The displacement at the nodes 11, 29 and 68 are considered for AMB control and at TDBs nodes for the dropdown analysis. An impeller of mass 10 kg is added to the node 75. The equation of motion of the rotor system is solved using SIMULINK model designed on the MATLAB platform to obtain the translational displacements at required nodal positions. The physical parameters of the system are mentioned in Table 1. The Campbell diagram of the rotor-AMB system is shown in Fig. 2. The system is first levitated by the AMBs and then after system is stabilized, the controller is stopped, and the system is allowed to fall on the TDBs under the action of gravity. The displacement responses at the TDBs are analyzed to identify the impact forces generated in the rotor system.

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Fig. 2. Campbell diagram of the thin-shaft coupled rotor-AMB system.

Table 1. Physical parameters of the rotor-AMB system integrated with TDBs. Parameters

Values

Support stiffness (k t )

1.305e08 N/m

Support damping (ct )

3.27e03 N-s/m

Contact Stiffness (k c )

2.02e09 N/m

Power exponents (p and q)

10/9 and 1

Contact parameter (λ)

0.08

Friction coefficients (μd ); (μs )

0.1; 0.2

Continuity parameter (γ k )

1e11

AMB displacement stiffness factor

7.5e06 N/m

AMB current stiffness factor

735 N/A

Impeller mass and diameter

10 kg and 0.3 m

4 Results and Discussions Initially, the rotor system is placed on the TDBs at 2e-04 m below the bearing axis as shown in Fig. 3. The system is then levitated by AMBs using the designed LQR controller. The static load of the rotor system is balanced by the AMBs forces generated due to static control currents as shown in Fig. 3. The system gets stabilized under less than 0.1 s and then after 0.5 s the system is allowed to fall freely on TDBs under the effect of gravity. The dropdown analysis is carried with two different dynamic conditions as mentioned in subsequent sections.

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Fig. 3. (a) Displacement and (b) control currents along y axis upon levitation.

4.1 Displacement Response for Dropdown at Spin Condition (ω = 50 Hz, Anticlockwise), Without Unbalance The system is dropped on the TDBs at pure spinning condition, without considering the effect of unbalance in the system. The displacement responses generated in the rotor at the TDBs location are shown in Fig. 4 below. The displacement of the system is scattered along the x-axis due to the frictional contact force between the rotor and TDBs. The reaction forces generated at the location of TDBs are in different proportionate due to non-uniform distribution of the gravity load along the shaft axis and the distance between the rotor and TDBs node locations as shown in Fig. 5. The maximum bearing force is generated at the TDB-CD2 is equivalent to 25.6 kN and the selection of TDBs based on load rating must be considered higher than the maximum force value.

Fig. 4. Displacement response orbits at the TDBs location (speed = 50 Hz).

Fig. 5. Plot of the reaction forces at the TDBs location (speed = 50 Hz).

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In case the rotor falls while vibrating with different amplitude in presence of unbalances, it would be necessary to analyze the magnitude of bearing force. The effect of unbalance in the system in discussed in the subsequent section. 4.2 Displacement Response for Dropdown at Spin Condition (ω = 50 Hz, 100 Hz and 150 Hz Anticlockwise), with Unbalance With the inclusion of unbalance in the system, the amplitude of vibration effects the intensity of the reaction forces. Also, the number of bounce off of rotor increases, which in turn increases the stress cycles the TDBs are required to withstand. The acceleration of rotor at the time of release and the direction of motion affects the intensity of the reaction forces. For the rotor with centre of mass above the bearing axis at the time of dropdown, the reaction force is higher at TDBs as the distance between the rotor and the TDBs increases. The analysis is carried out for an unbalanced system in which an additional unbalance is added with G 2.5 grade to show the effect of unbalance on the system. Therefore, the unbalance magnitude of 500 g-mm is assumed at the impeller corresponding to node 75. The bearing forces at the TDBs of the system released with different phase of unbalances from 0 to 360° (with step of 22.5°) is shown in Fig. 6. The maximum force of 28.9 kN is observed at TDB-CD2 along vertical (y) axis and 7.8 kN at TDB-ES for horizontal (x) axis. The TDBs are required to withstand several numbers of bounce off during dropdown. The number of bounce off of rotor with bearing load above 10 kN with variation in phase of unbalance is also estimated as mentioned in Table 2. The maximum number bounce off i.e., 20 is observed for TDB-CD1 at the compressor drive side.

Fig. 6. Maximum force at the TDBs with unbalance phase (0–360°) along (a) y and (b) x axes. (c) Number of bounce off at TDBs for bearing load above 10 kN with unbalance phase (0–360°) along y axis (spin speed = 50 Hz).

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Table 2. Estimated Bearing forces and the number of bounce off at the TDBs location with different operating speed (50, 100, 150 Hz) of the rotor system. Frequency (Hz)

TDB Bearing forces (kN)

Bearing forces (kN) y-axis

50

100

150

Bounce offs

x-axis

TDB-CD1

23.9

6.6

20

TDB-CD2

28.9

5.8

14

TDB-ES

16.3

7.8

6

TDB-CD1

34.2

10.6

18

TDB-CD2

33.5

9.5

14

TDB-ES

25.0

14.2

15

TDB-CD1

26.1

8.3

23

TDB-CD2

27.6

8.9

22

TDB-ES

22.7

16.6

53

The same procedure is performed for spin speeds of 100 and 150 Hz to check the severity in the reaction forces and bounce off with change in phase of unbalance during dropdown. The orbit plot of displacement response for dropdown event at operating speed of 150 Hz is shown in Fig. 7. The amplitude of the displacement due to unbalance is higher at the TDB-ES. As a result, the number of bounce off with higher bearing load is greater at TDB-ES location as shown in Fig. 8. It can be seen from the Table 2 that in case of dropdown in different dynamic condition, the maximum forces are around 34.2 kN at TDB-CD1.

Fig. 7. Displacement response orbits at the TDBs location (speed = 150 Hz).

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Fig. 8. Plot of reaction forces at the TDBs location (speed = 150 Hz).

There is an increment in the magnitude of bearing forces along x-axis due to increase in amplitude of vibration with spin speed. However, the number of bounce off is increased up to 53 at the TDB-ES location due to increase in the amplitude of vibration. The rotor remains bouncing off in TDB until the operating speed of the system is reduced to zero or the system is switched off. The excessive bounce off of rotor may result in the failure of the TDB-ES at high-speed condition.

5 Conclusions The dropdown analysis of the 3 AMB-rotor system in event of rotating condition with different operating speed and unbalance force is carried out. The maximum force generated at the TDBs is critical at the compressor drive side at different operating speed with unbalance. However, at higher speed due to increase in vibration amplitude at the extension shaft, the number of bounce off increases at TDB-ES. As a result of this, the TDB at extension shaft is subjected to large magnitude of repetitive load and increase its chances of failure in dropdown event. Therefore, the presence of unbalance has significant effect in identifying the critical TDB in the dropdown event for high speed coupled rotor-AMB systems. Also, when heavier loads are in extension shaft side possible failures of TDBs occurs more on extension shaft side. That can be considered in the design of extension shaft on maintenance point of view.

References 1. Gerada, D., Mebarki, A., Brown, N.L., Gerada, C., Cavagnino, A., Boglietti, A.: High-speed electrical machines: Technologies, trends, and developments. IEEE Trans. Industr. Electron. 61(6), 2946–2959 (2013) 2. Uzhegov, N., Smirnov, A., Park, C.H., Ahn, J.H., Heikkinen, J., Pyrhönen, J.: Design aspects of high-speed electrical machines with active magnetic bearings for compressor applications. IEEE Trans. Industr. Electron. 64(11), 8427–8436 (2017) 3. Kurvinen, E., et al.: Design and manufacturing of a modular low-voltage multimegawatt high-speed solid-rotor induction motor. IEEE Trans Ind App 57(6), 6903–6912 (2021) 4. Pyrhönen J, et al.: An electric machine system. Worldwide Appl. WO2022258880A1 (2022) 5. Saket, F., Sahinkaya, M., Keogh, P.: Measurement and calibration of rotor/touchdown bearing contact in active magnetic bearing systems. Mech. Syst. Signal Process. 122, 1–18 (2019)

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6. Neisi, N., Heikkinen, J., Sillanpää, T., Hartikainen, T., Sopanen, J.: Performance evaluation of touchdown bearing using model-based approach. Nonlinear Dyn. 101(1), 211–232 (2020). https://doi.org/10.1007/s11071-020-05754-9 7. Jarroux, C., Mahfoud, J., Defoy, B., Alban, T.: Stability of rotating machinery supported on active magnetic bearings subjected to base excitation. J. Vibr. Acoust. 142(3), 031,004 (2020) 8. Pesch, A.H., Sawicki, J.T.: Active magnetic bearing online levitation recovery through μsynthesis robust control. Actuators, MDPI 6, 2 (2017) 9. Lyu, M., Liu, T., Wang, Z., Yan, S., Jia, X., Wang, Y.: Orbit response recognition during touchdowns by instantaneous frequency in active magnetic bearings. J. Vibr. Acoust. 140(2) (2018) 10. Liu, T., Lyu, M., Wang, Z., Yan, S.: An identification method of orbit responses rooting in vibration analysis of rotor during touchdowns of active magnetic bearings. J. Sound Vib. 414, 174–191 (2018) 11. Jastrzebski, R.P., Putkonen, A., Kurvinen, E., Pyrhönen, O.: Design and modeling of 2 MW AMB rotor with three radial bearing-sensor planes. IEEE Trans. Ind. Appl. 57(6), 6892–6902 (2021) 12. Jarroux, C., Mahfoud, J., Dufour, R., Legrand, F., Defoy, B., Alban, T.: Investigations on the dynamic behaviour of an on-board rotor-AMB system with touchdown bearing contacts: modelling and experimentation. Mech. Syst. Signal Process. 159(107), 787 (2021)

Parametric Analysis of Journal Bearings with Chevron Textures on the Shaft Surface Luis F. dos Anjos1

, Alfredo Jaramillo2 , Gustavo C. Buscaglia2 and Rodrigo Nicoletti1(B)

,

1 Sao Carlos School of Engineering, University of Sao Paulo, Sao Carlos, Brazil

[email protected], [email protected]

2 Institute of Mathematical Sciences and Computation, University of Sao Paulo, Sao Carlos,

Brazil [email protected]

Abstract. Surface texturing has proven to be a good technique to improve the characteristics of lubricated contacts. In journal bearings, texturing can increase the load-carrying capacity and reduce friction between the surfaces depending on some texture parameters such as geometry, size, and position. Although many works are studying the influence of textures on journal bearings, most of them apply the texture on the bearing surface (static part of the contact). In this work, chevron-shaped textures are applied on the shaft surface (moving part of the bearing) and a parametric analysis is performed considering some geometric parameters of the chevron. The fluid flow is modeled using the Reynolds equation and a mass-conserving boundary condition is used to deal with the cavitation zones. Due to the moving nature of the texture (imposed on the rotating surface of the shaft), a moving grid technique is employed in the Finite Volumes scheme. The results show the existence of optimal texture shapes depending on the Sommerfeld number of the bearing (operating condition). The textures with optimum shapes can reduce the shaft eccentricity, which indicates an increase in load-carrying capacity. The reason for this increase in load-carrying capacity is explained by an analysis of the lubricant’s flow dynamics in the bearing during operation, where a pumping effect is observed which reduces the leakage flow (the bearing with textured shaft keeps more lubricant in the bearing gap than the plain journal bearing). Keywords: Moving Grid · Finite Volumes Method · Hydrodynamic Lubrication

1 Introduction The increasing demand for more energetically efficient machines led Engineering to seek ways of reducing losses during operation. One of these ways that have been investigated is the texturization of surfaces in sliding and/or lubricated contacts. In this case, texturization can be considered as any imposed alteration of the surface topology, be it of a structured or random nature. The first studies on the subject focused on the unstructured textures imposed on the surface by conventional machining manufacturing processes [1, 2]. It was observed that © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 201–210, 2024. https://doi.org/10.1007/978-3-031-40455-9_16

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the presence of irregularities on the surface of sliding contacts did improve some hydrodynamic properties due to the pressure build-up and cavitation effects in the region of these irregularities. Hence, there came the idea of purposely imposing such irregularities on the surface of sliding contacts in a more organized (or structured) way, trying to take advantage of the resultant hydrodynamic effects to improve the behavior of the lubricated contact. Such an idea was fostered in the last decades with the advent of modern machining techniques that allowed the controlled imposition of micrometric textures on surfaces within acceptable tolerances: ion etching [3], abrasive jet machining [4, 5], laser surface texturing [5, 6], mechanical micro drawing [7], grinding texturing [8]. In the last decades, much has been investigated on the application of structured textures in lubricated contacts, especially in journal bearings and sliding bearings. Texturized surfaces were studied under different lubricating regimes [9–11], and they resulted in similar reductions in the friction coefficient. It has been assumed that the textures work as lubricant reservoirs and they provide additional lubrication under starved or boundary lubricating regimes. Such an explanation is not consensual among researchers, and the most accepted hypothesis is that textures locally change the hydrodynamic pressure distribution in the contact (pressure build-up) due to the presence of local cavitation regions. As a consequence, one achieves an overall increase in the load-carrying capacity of the contact [12]. In journal bearings, surface texturing can modify important static characteristics, such as the rotor eccentricity, the attitude angle, the driving torque, and the oil flow. In this case, the geometry, size, and position of the textures play an important role to achieve positive results. However, despite the high number of works investigating the subject, there is no consensus in the literature about the best texturing parameters in general terms. For example, the adoption of chevron-type textures in the loaded area of the bearing bush presented good results [13], whereas the adoption of square-type textures in the same region was detrimental in comparison to a full texturized bearing bush [14]. Many different texture geometries have been studied in the literature, including circular [15], rectangular [16], triangular [17], and chevron [18], with all presenting promising results under different operating conditions of the bearing. However, in the majority of the works, the texture is applied to the static surface of the contact. Few works focused on the texturization of the moving parts of the lubricated contact. For example, one observed friction reduction in the sliding contact of a cylinder piston when the surface of the piston had dimple patterns [19]. By adopting grooves on the surface of the cylinder, one observed a bigger retention of lubricant on the sliding surface (an increase of lubricant wetness) [20]. In the case of journal bearings, four different texture geometries were tested experimentally on the surface of the rotating shaft: chevron, sawtooth, oblong dimple, and aligned dimple [21]. In this case, the chevron-type texture significantly decreased the rotor eccentricity in comparison to the non-textured case, as evidenced by the increase in the load-carrying capacity of the bearing. In another work, the sawtooth pattern showed to be a better solution in comparison to the trapezoidal pattern, which is still better than the bearing with a non-textured shaft [22].

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The present work focuses on the analysis of a journal bearing supporting a shaft with chevron textures on its surface. The results bring some insight into the physics behind the increase of the load-carrying capacity of the bearing due to the presence of the textures. It is observed that chevron-type textures have a positive pumping effect on the lubricant flow in the bearing gap, thus keeping more fluid in the centerline of the bearing. As a result, the load capacity increases. In addition, the orientation of the chevron pattern plays an important role in the results: if the shaft rotates in the wrong direction, detrimental effects are observed (reduction of load capacity due to pumping of lubricant out of the bearing).

2 Mathematical Model By adopting the classic assumptions for the fluid flow in the lubricated bearing gap, one achieves the Reynolds equation in adimensional form, considering the film fraction in the cavitation zones:   ∂(θ h) ∂(θ h) (1) ∇ · h3 ∇p = α +2 ∂x1 ∂t where h(x, y, t) is the adimensional oil film thickness, p(x, y, t) is the adimensional oil hydrodynamic pressure, α is the adimensional shaft velocity, t is the adimensional time, θ (x, y, t) is the oil film fraction, and (x, y) is a local coordinate system in the planified surface of the bearing (x in tangential direction and y in axial direction). When θ = 1, there is full film lubrication and p ≥ 0, whereas when 0 ≤ θ < 1, there is cavitation and p = 0. The oil film thickness is composed of two terms:

Fig. 1. Global and local coordinate system in the journal bearing.

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h(x, y, t) = hf (x, y, t) + ht (x, y, t)

(2)

where hf is the oil film thickness in the regions without texture, and ht is the oil film thickness in the regions with texture. The oil film thickness in the regions without texture (hf ) can be written as a function of the position of the shaft in the global coordinate system: hf (x, y, t) = 1 + X (t) cos(2π x) + Y (t) sin(2π x)

(3)

where X (t) and Y (t) are the coordinated of the center of the shaft in the global coordinate system of the bearing (Fig. 1). To find the position of the shaft, one solves the force equilibrium problem by the Newmark method:  2 M ddtX2 = WX (t) + Wa (4) 2 M ddt 2Y = WY (t) where M is the shaft mass, WX and WY are the components of the hydrodynamic forces in the (X , Y ) global coordinate system, and Wa is the external loading applied on the shaft. The second term of Eq. (2) depends on the adopted texture geometry and on the rotating speed of the shaft. For a given instant of time, one has:  d if (x, y) ∈  ht (x, y, t) = (5) 0 if (x, y) ∈ / where d is the texture depth, and  is the texture domain. The main challenge in modelling the lubricated contact with textures in the moving surface is the fact that the  region moves (ht varies with time). To tackle this problem numerically, one moves the domain of the numerical solution  (in which Eq. (1) is solved) instead of , thus artificially keeping the texture domain in place. Hence, for every instant of time, the numerical domain moves by d  = Rdt, where  is the shaft rotating speed and R the shaft radius. Figure 2 shows the motion of the numerical domain  in two different instants of time. In a fixed reference frame, the domain  moves from left to right. However, in the local reference coordinate system (fixed at the numerical domain ), the texture domain  moves from right to the left. The use of multigrid techniques [23] also proved advantageous to reduce computational costs during the numerical simulations of the system.

3 Parametric Analysis The parameters of the journal bearing in analysis and the operating conditions are listed in Table 1. One considers ambient pressure as the boundary condition for solving the Reynolds equation in the numerical domain :  p(0, y, t) = p(L, y, t) = pamb (supply groove) (6) p(x, 0, t) = p(x, B, t) = pamb (axial boundary)

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Fig. 2. The domain of numerical solution  at two different instants of time.

where L is the total circumferential length of the bearing, and B is the bearing width. The geometric parameters of the chevron texture are the chevron angle (α), the chevron thickness (e), and the number of chevrons in the texture (Nt ) – Fig. 3. The depth of the texture is 25 µm in all studied cases. Table 1. Parameter values of the bearing in analysis. Parameter

Value

Unit

Parameter

Value

Unit

Bearing width (B)

25

mm

Rotating speed

4,000

rpm

Bearing nominal clearance (hN )

50

µm

Oil viscosity

0.032

Pa.s

Shaft radius (R)

25

mm

External load (Wa )

320

N

Shaft mass (M)

10

kg

The parametric analysis is performed by keeping the chevron thickness (e) constant and varying the values of the chevron angle (α) between 30 and 70°, and the number of chevrons (Nt ) between 0 and 15. One observes the effect of the texture in the resultant eccentricity of the shaft for the operating condition in the study: √ X2 + Y2 (7) ε= hN The obtained results are shown in Fig. 3 for different values of chevron thickness (3 ≤ e ≤ 7 mm). As one can see, for every chevron thickness, there is an optimum combination of angle α and number of chevrons Nt that decreases the shaft eccentricity (an evidence of an increase of the bearing load-carrying capacity). In this case, the best results were obtained for Nt = 7, e = 7 mm, and α = 60°. The resultant eccentricity was ε = 0.184, which is 9.4% smaller than that of the shaft without textures (ε = 0.203) (Fig. 4).

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Fig. 3. Geometric parameters of the chevron texture on the shaft surface: chevron angle (α), chevron thickness (e), and the number of chevron textures (Nt ).

Fig. 4. Shaft eccentricity as a function of the texture angle α and the number of textures Nt : (a) e = 3 mm, (b) e = 4 mm, (c) e = 5 mm, (d) e = 7 mm.

To better understand why the textured shaft leads to a smaller eccentricity, let’s analyze the case of the shaft with the best texture obtained in the parametric analysis (Nt = 7, e = 7 mm, α = 60°). Figure 5 presents the hydrodynamic pressure distribution in the bearing gap, for a given instant of time, for the cases of non-textured and textured shafts. In the case of the non-textured shaft (Fig. 5a), one observes a conventional hillshape distribution of the pressure around the position of minimum film thickness, with a maximum hydrodynamic pressure of 5.9 × 105 Pa. This pressure distribution has a

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convex shape in the direction of the shaft rotation, which tends to push the lubricant towards the axial boundaries of the bearing (leakage flow – Fig. 6a). In the case of the textured shaft (Fig. 5b), the pressure distribution is completely different: the pressure distribution is concentrated in the regions between the chevrons because of the cavitation and pressure build-up zones created by them. For this reason, these pressure “hills” tend to follow the shape of the chevrons, thus presenting a concave shape in the direction of the rotating speed. As a consequence, the lubricant is not totally pushed toward the axial boundaries of the bearing. It is rather pushed towards the bearing centerline by the texture (Fig. 6b), in a clear pumping effect that keeps more lubricant in the bearing gap. As a result, more lubricant remains in the region of minimum film thickness, thus reducing the leakage flow, increasing the hydrodynamic pressure (maximum value of 9.1 × 105 Pa) and, consequently, increasing the load-carrying capacity of the bearing.

Fig. 5. Hydrodynamic pressure distribution at a given instant of time: (a) non-textured shaft, (b) chevron textured shaft (Nt = 7, e = 7 mm, α = 60°).

This pumping effect is reversed if the shaft runs in the opposite direction. Figure 7 presents the hydrodynamic pressure and lubricant flow in the bearing gap for this situation. As one can see, the pressure distribution still concentrates in the regions between the chevron textures. However, due to the direction of the shaft surface motion, the pressure “hill” now has a convex shape which reverses the pumping effect. The lubricant is pushed towards the axial boundaries of the bearing, thus decreasing the volume of lubricant in the bearing gap and, consequently, increasing the leakage flow and reducing the hydrodynamic pressure. This reversed pumping effect increases the leakage flow in comparison to the non-textured case, thus resulting in a much higher shaft eccentricity (ε = 0.420 – an increase of 107% in comparison to the non-textured case). Hence, the direction of shaft rotation and the orientation of the chevron textures on the shaft are important parameters that define the success or failure to increase the load-carrying capacity of the bearing.

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Fig. 6. Lubricant flow in the bearing gap: (a) non-textured shaft, (b) chevron textured shaft.

Fig. 7. Effect of running the shaft with chevron textures in the opposite direction: (a) hydrodynamic pressure distribution, (b) lubricant flow.

The above results clearly show that there are optimum values for the texture parameters (Nt , e, α) according to the bearing operating condition. Although the optimization of the chevron texture is beyond the scope of the present work, further investigations in this direction showed that: • Nt decreases for increasing Sommerfeld numbers; • e increases for increasing Sommerfeld numbers; • α increases for increasing Sommerfeld numbers. In all these cases, one achieved optimum load-carrying capacity with the optimum chevron geometry.

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4 Conclusion The parametric analysis of the chevron texture applied to the surface of the rotating shaft in journal bearings showed that the rotor eccentricity can be reduced in the textured cases, as evidence of the increase of bearing load-carrying capacity. Such an increase in the bearing load-carrying capacity is a consequence of the pumping effect that occurs in the lubricant flow caused by the cavitation and pressure build-up zones created by the textures as they run over the lubricant. For this reason, the direction of shaft rotation and the orientation of the chevron textures on the shaft play important roles in the supporting mechanism of the journal bearing: the convex side of the chevron texture must be in the direction of motion of the shaft surface. Optimum texture geometries can be found for the given operating conditions of the bearing. Acknowledgements. This project was supported by the Brazilian research foundations Conselho Nacional de Desenvolvimento Científico e Tecnológico (Grant no.: 304212/2021-0) and Fundaçãao de Amparo à Pesquisa do Estado de São Paulo (Grant no.: 2021/00327-6).

References 1. Anno, J.N., Walowit, J., Allen, C.: Microasperity lubrication. J. Lubr. Technol. 90(2), 351–355 (1968) 2. Hamilton, D., Walowit, J., Allen, C.: A theory of lubrication by microirregularities. J. Basic Eng. 88(1), 177–185 (1966) 3. Wang, X., Kato, K., Adashi, K., Aizawa, K.: Loads carrying capacity map for the surface texture design of SiC thrust bearing sliding in water. Tribol. Int. 36(3), 189–197 (2003) 4. Wakuda, M., Yamauchi, Y., Kanzaki, S., Yasuda, Y.: Effect of surface texturing on friction reduction between ceramic and steel materials under lubricated sliding contact. Wear 254(3– 4), 356–363 (2003) 5. Etsion, I., Halperin, G., Brizmer, V., Kligerman, Y.: Experimental investigation of laser surface textured parallel thrust bearings. Tribol. Lett. 17(2), 295–300 (2004) 6. Hao, X., Sun, H., Wang, L., Ali, Q., Li, L., He, N.: Fabrication of micro-texture on cylindrical inner surface and its effect on the stability of hybrid bearing. Int. J. Adv. Manuf. Technol. 109(5–6), 1671–1680 (2020). https://doi.org/10.1007/s00170-020-05750-8 7. Costa, H., Hutchings, I.: Effects of die surface patterning on lubrication in strip drawing. J. Mater. Process. Technol. 209(3), 1175–1180 (2009) 8. Silva, E.J., Kirsch, B., Bottene, A.C., Simon, A., Aurich, J.C., Oliveira, J.F.G.: Manufacturing of structured surfaces via grinding. J. Mater. Process. Technol. 243, 170–183 (2017) 9. Shen, C., Khonsari, M.: Texture shape optimization for seal-like parallel surfaces: theory and experiment. Tribol. Trans. 59(4), 698–706 (2016) 10. Galda, L., Sep, J., Olszewski, A., Zochowski, T.: Experimental investigation into surface texture effect on journal bearings performance. Tribol. Int. 136, 372–384 (2019) 11. Yue, H., Deng, J., Ge, D., Li, X., Zhang, Y.: Effect of surface texturing on tribological performance of sliding guideway under boundary lubrication. J. Manuf. Process. 47, 172–182 (2019) 12. Gropper, D., Wang, L., Harvey, T.J.: Hydrodynamic lubrication of textured surfaces: a review of modeling techniques and key findings. Tribol. Int. 94, 509–529 (2016)

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13. Sharma, S., Jamwal, G., Awasthi, R.K.: Enhancement of steady state performance of hydrodynamic journal bearing using chevron-shaped surface texture. J. Eng. Tribol. 233(12), 1833–1843 (2019) 14. Liang, X., Liu, Z., Wang, H., Zhou, X., Zhou, X.: Hydrodynamic lubrication of partial textured sliding journal bearing based on three-dimensional CFD. Ind. Lubr. Tribol. 68(1), 106–115 (2016) 15. Shen, C., Khonsari, M.: Effect of dimple’s internal structure on hydrodynamic lubrication. Tribol. Lett. 52(3), 415–430 (2013) 16. Papadopoulos, C., Kaiktsis, L., Fillon, M.: Computational fluid dynamics thermohydrodynamic analysis of three-dimensional sector-pad thrust bearings with rectangular dimples. J. Tribol. 136(1), 011702 (2014) 17. Manser, B., Belaidi, I., Hamrani, A., Khelladi, S., Bakir, F.: Texture shape effects on hydrodynamic journal bearing performances using mass-conserving numerical approach. Tribol. Mater. Surf. Interfaces 14(1), 33–50 (2020) 18. Jamwal, G., Sharma, S., Awasthi, R.: The dynamic performance analysis of chevron shape textured hydrodynamic bearings. Ind. Lubr. Tribol. 72, 1–8 (2019) 19. Yin, B., Li, X., Fu, Y., Yun, W.: Effect of laser textured dimples on the lubrication performance of cylinder liner in diesel engine. Lubr. Sci. 24(7), 293–312 (2012) 20. Tanaka, H., Ichimaru, K.: Numerical simulation of oil-film behavior on cylinder liners in sliding contacts with piston rings. Tribol. Ser. 43, 283–290 (2003) 21. Filgueira Filho, I.C.M., Bottene, A.C., Silva, E.J., Nicoletti, R.: Static behavior of plain journal bearings with textured journal experimental analysis. Tribol. Int. 159, 106970 (2021). https:// doi.org/10.1016/j.triboint.2021.106970 22. Sinanoglu, C.: The analysis of the effects of surface texture on the capability of load carriage of journal bearings using neural network. Ind. Lubr. Tribol. 57(28–40), 293–312 (2005) 23. Gerya, T.: Introduction to Numerical Geodynamic Modelling. Cambridge University Press (2019). https://doi.org/10.1017/9781316534243

Prediction of Remaining Useful Life of Passive and Adjustable Fluid Film Bearings Using Physics-Based Models of Their Degradation Denis Shutin(B) , Maxim Bondarenko, Roman Polyakov, Ivan Stebakov, and Leonid Savin Orel State University, Orel 302026, Russian Federation [email protected]

Abstract. Systems for online prediction of remaining useful life (RUL) of technological equipment, and, in particular, fluid film bearings, are usually based on the analysis of a large amount of data received from the operated objects. In practice, formation of a data set meeting the size and quality requirements often encounters a number of difficulties. The work presents the approach to the possible overcoming of such difficulties through the use of physics-based models of degradation of fluid film bearings. The most common reasons for replacing them in rotating machines are considered as the criteria for the end of the service life. They include achievement of the wear limit in accordance with the current standard, and disruption of the bearing surfaces after reaching the material’s fatigue strength limit. The work focuses mainly on the last factor and demonstrates mathematical and numerical simulation models of rotor-bearing systems considering this phenomenon and allowing generating data on the bearing degradation process. The generated data is used to train a predictive model that estimates online the current state and RUL of the bearing. In addition, the proposed physics-based models also allow to evaluate the impact of the adjustable design of the fluid film bearings on their expected service life. The variable parameters of the adjustable bearings are also taken into account by the proposed predictive model. The work shows the results of numerical studies demonstrating the change in the service life taking in account the adjustable bearing parameters. Keywords: RUL Prediction · Fluid Film Bearings · Adjustable Bearings · Physics-Based Models · Machine Learning

1 Introduction Bearings are one of the most loaded parts of rotary machines. Their reliability largely determines the reliability of the entire system. Bearing failure almost inevitably leads to the loss of the machine’s workability. Therefore, both the timely maintenance and condition monitoring are of great importance, especially when the direct access to the bearings is difficult. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 211–223, 2024. https://doi.org/10.1007/978-3-031-40455-9_17

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Condition monitoring systems are aimed to detect defects in bearings during their operation. For example, works [1–3] propose several methods for estimating the amount of bearing wear from the dynamic rotor behavior. In works [4, 5] other measured parameters are used for the same purposes, allowing an indirect assessment of the bearing condition. Some solutions make it possible to explicitly determine the amount of wear, if it is possible to integrate sensor elements into the bearing pads [6]. Unlike wear, the accumulation of fatigue stresses in the material of the bearing shells does not appear until the very beginning of the destruction. An objective analysis of the amount of material fatigue on the basis of direct or indirect signs is extremely difficult, or even impossible. At the same time, fatigue failures are the common causes of failure of both rolling [7, 8] and sliding bearings in various applications [9, 10]. Adequate prediction of their service life from fatigue strength is an urgent task. In most cases modern condition monitoring systems make the predictions of the residual life (RUL) after training on large amounts of historical data on the operation and malfunctions of machines of the same type [11–14]. Adjustable design of bearings makes such an analysis even more difficult to implement, since the operation of control systems can affect the operation of the whole system significantly [11, 15, 16], and little predictably. In this regard, the importance of physics-based models increases, since they are able to assess the current physical processes in the machine more objectively. The approach associated with the joint work of physics-based and data-driven predictive models is increasingly being used in technology [17–19]. Previously, in [20], the application of such a principle was demonstrated in predicting the wear of a journal bearing and the resulting RUL. This work supplements it with the elements of a methodology for assessing the risk of fatigue failure of shell material. The presented approach takes into account a range of factors influencing these processes, including the operation of the control system of adjustable bearings. As in the case of wear prediction in [20], the present work offers theoretical and methodological foundations for assessing the current state and RUL of the bearing, and also requires additional refinement data for individual applications.

2 Materials and Methods 2.1 Subject Fatigue damage is a common cause of bearings failure in highly loaded rotary machines. The reason for the exhaustion of the resource in terms of fatigue strength is the cyclic loading of the shell material during oscillations of an unbalanced rotor [21]. Adjustable bearings are usually designed to reduce the amplitude of rotor oscillations in certain modes, for example, when overcoming resonant zones, or to compensate the influence of loads and distortions. A journal bearing for a heavy turbine rotor [22] with a displaceable upper half based on the principle similar to the demonstrated in [23, 24] is considered in this work as an illustrative example of the adjustable rotor-bearing system. Its schematic and principle of operation are shown in Fig. 1. This design allows the rotor motion to be stabilized compared to a simple circular bearing due to an additional hydrodynamic wedge at the upper shell, although this also leads to an increase in loads on the bearing surface.

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Fig. 1. Schematic of the journal bearing with a displaceable upper half.

Pressure distribution is determined using the Reynolds equation for the twodimensional flow of incompressible lubricant:     ∂ 3 ∂p ∂ ∂ 3 ∂p h + h = 6μ (V1 h) − 12μV2 , (1) ∂x ∂x ∂z ∂z ∂x where V 1 and V 2 are the lubricant velocities in circumferential and radial directions correspondingly, μ is the lubricant dynamic viscosity, h is the radial gap, p is the lubricant pressure. Equations of the radial gap considering the adjustable displacement of the upper half δ:  h0 − X sin(α) − Y cos(α), 0 < α < π ; (2) h(α) = h0 − (X + δ) sin(α) − Y cos(α), π ≤ α ≤ 2π. Equations (1) and (2) are solved together numerically using the finite differences method. The bearing force is calculated by the numerical integration of the pressure distribution in the bearing. This force is taken into account during solving Lagrange equations for calculation of the rotor motion. A simple rigid rotor model without distortions was considered. The same mathematical model of a rotor-bearing system is described in more detail in [25], only Eq. (2) was introduces specifically for this work. A typical 320 MW turbine generator with the set of parameters as in [22] was considered as the basic object for simulation in this study. The parameters of the rotorbearing system have been initially calculated and presented in [26], their values are shown in Table 1. 2.2 Approach to Considering the Bearing Degradation Wear. Wear in fluid film bearings occurs at extremely high rotor eccentricities under dry, boundary or mixed lubrication, often during startup and shutdown of the machine. In heavy machines, like the one considered in this paper, a hydraulic lifting system of the rotor is used for such modes for minimizing wear. In this case, in the normal operation of all subsystems, the fatigue destruction may become the main cause of the bearing failure, which is discussed in the following chapters.

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Unit

Value

Rotor diameter

mm

561

Bearing length L

mm

670

Radial clearance h0

µm

560

Rotor mass m

kg

106490

Oil dynamic viscosity μ

Pa · s

0.03

Oil density ρ

kg/m3

886

In the absence of hydraulic lift systems, and under extremely high rotor loads, wear is more likely to be a key contributor to bearing life. Previous work [20] details the wear on-line prediction approach based on a combination of physics-based and data-driven models. The approach is based on the approximation of the dependence of the wear rate on the force factors affecting it, including the operation of the control system of the adjustable bearing. Its physics-based model generates a dataset of the form [ω d 0 U RULw I], where ω is the rotation speed, d 0 is the current wear value, U is the control actions vector, RULw is the RUL estimate by the wear factor, and I is the wear rate under current operating conditions. The wear rate is calculated based on Archard’s law combined with a bearing model similar to that shown in Sect. 2.1. The dependencies embedded in the dataset are approximated by an artificial neural network (ANN) for fast calculation of the current wear rate and RUL depending on the current measured values of other parameters. In this paper, a similar principle of interaction between physics-based and data-driven models is transferred to the online calculation of fatigue degradation of the bearing material. Depending on the operating modes of a particular rotor-bearing system, the models for predicting degradation by wear and fatigue strength can be used either jointly or the most relevant one can be selected. Fatigue Destruction. Cyclic pressure variations in the fluid film during rotor oscillations in journal bearings are a key factor leading to fatigue failure of the shell material. The resistance of a material to fatigue failure is described by SN diagram [9]. The SN characteristic differs not only for different materials, but even for different manufacturing ways [24]. In this work the data from [9] are used for the babbitt SnSb11Cu6 used for bearing shells. Figure 2 shows the SN diagram with inverted axes for the convenience of determining the residual number of cycles from the stress amplitude. In the regression equation in Fig. 2, N is the number of cycles to failure, S is the stress amplitude. The stress amplitudes for the considered bearing were calculated using the model of the rotor-bearing system presented above, which is also described in more detail in [20]. A number of parameters connected with the fatigue degradation were calculated in a series of computational experiments at various values of the rotation speed, rotor unbalance, and controlled shift of the upper half of the bearing.

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Fig. 2. SN diagram with inverted axes for the babbitt SnSb11Cu6.

The rotation speed varied from 300 to 3300 rpm in 300 rpm steps, which covers the operating range of many heavy rotors. The unbalance values varied in the range from 1 · 10−4 to 5 · 105 m, which gives a range of vibration amplitudes from insignificant to unacceptable during the operation of such machines. The controlling displacement values δ varied from 0 to 60% of the initial gap h0 in 10% steps. After reaching the steady rotor orbit, the simulation model determines the steady oscillation amplitude, as well as the highest pmax and the lowest pmin pressures in the most loaded area of the bearing. The stress amplitude for the SN diagram is calculated then as 0.5(pmax − pmin ). The dependences of the vibration and stress amplitudes on the rotation speed and controlling displacement at a fixed rotor imbalance are shown in Fig. 3. Thus, a dataset [A n S δ] is obtained after the calculations. The generated dataset in this work includes 540 combinations. The ranges of parameter values in the dataset should be chosen so that they would overlap the possible operating ranges for the machine in question. The dataset is used for on-line calculation of S values according to the data of the monitoring and control system of the machine. The amplitude and the rotation speed values are measured by the corresponding sensors. The bearing displacement value δ is known from the control system at each moment of time. At the next step a regression function S = f(n A δ) was calculated in order to interpolate the dataset data and obtain actual stress amplitude value during the machine operation. The linear regression was used in this work, however, any other regression method can be used for better accuracy, speed and/or dataset size. Also, the generation of polynomial and interactive features was used to increase the accuracy of the regression model. Training was performed using a modification of the Adam gradient descent method to minimize the mean square loss function (MSE loss). Optimization of three hyperparameters was carried in order to tune the algorithm. The optimized parameters were the learning step, the polynomial degree, and the regularization coefficient. The dataset was divided into training and validation parts. PyTorch library for Python3

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Fig. 3. Dependencies of vibration and stress amplitudes on rotation speed and controlling displacement

programming language was used for building the regression model. The input and the output values were normalized by adjusting to zero mean and unit variance. The values of the average and maximum relative error were calculated to assess the quality of the models. The accuracy was obtained for the polynomial degree of 5, the average relative validation error was 8.4%. The results of training with the analysis of the accuracy of predicting sigma values are shown in Fig. 4. The fatigue damage due to its cumulative nature can be mathematically expressed by the Palmgren-Miner damage summation rule [27, 28]: D=

 ni , Ni i

(3)

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where ni is the number of cycles at a certain load level, N i is the number of cycles to failure under a certain loading mode.

Fig. 4. Linear regression accuracy scores during training (a) and at prediction of S (b).

According to the linear summation rule, damage under non-stationary loading is equal to the sum of partial damages in each mode. In this case, it is considered that the destruction occurs when the damage parameter D reaches the value of 1. Thus, the value of the number of cycles to failure will proportionally decrease with the accumulation of the damage parameter D. The decrease in the bearing life is reflected in the change in the SN diagram. Figure 5 shows the examples of the SN dependencies for babbitt SnSb11Cu6 at different RUL values. On-line calculation of the accumulated fatigue stress is implemented as follows. A basic time interval Δt is selected based on the operating conditions of the machine so that the changes in the vector [A n S δ] can be considered insignificant. For the i-th time interval, the number of loading cycles is calculated as ni = Δt·n/60. The current S value is determined using the obtained regression. The increase in fatigue stress is taken into account according to Eq. (3). Thus, a modified SN diagram is obtained at the end of each step. The RUL value calculated as RUL = (1 − D)/100% reflects the probability of occurrence of fatigue failure. For a rough estimate of RUL in the remaining load cycles

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Fig. 5. Babbitt SnSb11Cu6 SN dependencies at different RUL values.

or in time units, it is necessary to have data on the values of [A n S δ] in the upcoming periods, for example, from the statistics of the operation of the machine. Otherwise, it can be difficult to make such RUL estimations. The following section shows an example of RUL calculation using a simulation model of a turbine generator rotor-bearing system.

3 Results and Discussion The simulation tests below were performed based on the typical data on the operation of a turbine generator, obtained from a real thermal power plant. The data showing the average rotor speed for each hour of operation are shown in Fig. 6. The presented sample was applied cyclically during the simulation for the entire turbo generator bearing’s life. The rotor imbalance value was 3.3·10−4 m during all the simulation period. The control signal δ was considered constant during one calculation. The simulation results are shown in Fig. 7. As can be seen, the RUL value is gradually decreasing in accordance with the applied loads specified by the rotor speed diagram. It should be noted that the expected life significantly depends on the magnitude of the control displacement δ. It leads to an increase in the bearing stiffness and the characteristic pressures in the fluid film. Although Fig. 3 shows a decrease in vibration amplitudes with increasing displacement δ, this is accompanied by a multiple increase in pressure in the bearing and, accordingly, in the intensity of fatigue stress accumulation. It can be concluded that only a short-term increase in the displacement δ seems appropriate for a situational vibration reduction, for example, when overcoming the critical frequencies, or in the case of emergency increase in rotor imbalance. In other cases it is advisable to set the bearing clearance to be circular, i.e. δ = 0, in order to maximize the bearing life.

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Fig. 6. Typical turbine generator rotor speed diagram.

Fig. 7. Simulation of reducing the RUL at various values of bearing displacement.

Also, an additional analysis was carried out in order to compare the obtained results with the requirements of the standards and operating experience of turbine machines. Figure 8 shows the results of calculating the expected bearing life for the fixed stress amplitudes S at different [A n δ] combinations from the original dataset. The analysis was carried out by correlating the results with the data of the GOST 27165-97 standard [29]. At a speed of 3000 rpm, the vibration amplitude of 165 µm is the upper limit of the machine’s operation without time limits. Inadmissible vibration amplitude values start from 260 µm. At A = 165 µm, the stress amplitude is S = 5.5 MPa, which gives about 2·106 h of operation. At A = 260 µm, the stress amplitude S = 11 MPa, which gives about 1.1 · 106 h of operation. From this we can conclude that the margin for fatigue strength of the bearing shells significantly exceeds the typical service life

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Fig. 8. Simulation of the fatigue degradation and the corresponding RUL changes.

of machines in the case of operation without rotor misalignment. Many works show that the misalignments significantly affect the pressure distribution in journal bearings [30, 31]. Even without the direct contact between the rotor and the bearing resulting in wear, the increased pressure amplitudes in the lubricant film under misalignments can lead to intense local accumulation of fatigue stresses with the risk of accelerated material fatigue destruction. Since misalignments are present in most turbine machines, it is necessary to improve the presented approach and the developed model by taking this factor into account. Most likely, this will not have a fundamental impact on the shown methodology for estimating fatigue stresses. The additional data on the movements of the rotor at several different points, as well as modal analysis data will be required to calculate the current misalignment parameters, as well as the data on the settings of the bearings during their installation. The dataset obtained from the physics-based model should also include the values of the misalignment angles for a given bearing. In addition, separate consideration of fatigue stresses in different areas of the bearing surface may be required. In this case, the total bearing RUL will be determined by its minimum value among the considered areas. The rest of the approach can be implemented as described in this paper.

4 Conclusion The work proposes and demonstrates the basics of an approach to online prediction of residual useful life (RUL) of adjustable journal bearings using data from their physicsbased models. This work is focused mainly on predicting the probability of their fatigue failure due to cyclic pressure changes in the lubricant film. The controlled bearing parameters are considered as an additional factor influencing the intensity of degradation. The following main conclusions can be drawn.

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1. The operation of the machine in normal and even boundary conditions in the absence of a significant misalignment of the rotor in the bearing does not lead to a rapid exhaustion of the fatigue strength resource. A sharp drop in the resource begins only at critically high vibration amplitudes. The effect of misalignments should be taken into account as an additional measurable parameter when implementing the presented approach, since they can increase the risk of local fatigue failures. 2. Increasing the rigidity of the bearing, in particular, by shifting its upper half, leads to some reduction in vibrations, but also to a rapid reduction in fatigue life. It is recommended to apply such impacts only for a short time, for example, when overcoming critical frequencies, in order to extend the bearings life. 3. The application of regression methods to the data from physics-based models makes it possible to avoid computationally expensive calculations in the system of online monitoring of degradation and RUL prediction. At the same time, it remains possible to take into account the full range of factors affecting the degradation, including the influence of the control system of adjustable bearings. 4. Considering bearing degradation, both due to wear and due to loss of fatigue strength, the characteristics of specific materials and even methods of manufacturing the bearing elements should be taken into account. The most accurate data for increasing the prediction accuracy can be obtained by experimental studies. Acknowledgements. The study was supported by the Russian Science Foundation grant No. 22-19-00789, https://rscf.ru/en/project/22-19-00789/.

Nomenclature x, y, z h0 , h α O, O1 , O2 e V1 V2 μ ρ p, pmax , pmin n, ω r R, D L d mg δ d0 I N

Cartesian coordinates initial and local radial gap angular coordinate center of shaft and bearing eccentricity circumferential velocity radial velocity dynamic viscosity of lubricant density of lubricant pressure rotation speed and angle speed shaft radius bearing radius and diameter bearing length rotor imbalance rotor weight adjustable displacement of the upper half of the bearing current wear value wear rate under current operating conditions number of cycles to failure

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S A D ni Ni U RULw

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stress amplitude amplitude of the rotor oscillations fatigue damage parameter number of cycles at a certain load level number of cycles to failure under a certain loading mode control actions vector RUL estimate by the wear factor

References 1. Machado, T., Cavalca, K.: Modeling of hydrodynamic bearing wear in rotor-bearing systems. Mech. Res. Commun. 69, 15–23 (2015) 2. Machado, T., Alves, D., Cavalca, K.: Investigation about journal bearing wear effect on rotating system dynamic response in time domain. Tribol. Int. 129, 124–136 (2019) 3. Alves, D., Fieux, G., Machado, T., Keogh, P., Cavalca, K.: A parametric model to identify hydrodynamic bearing wear at a single rotating speed. Tribol. Int. 153(6), 106640 (2021) 4. Li, N., et al.: Multi-sensor data-driven remaining useful life prediction of semi-observable systems. IEEE Trans. Ind. Electron. 68, 11482–11491 (2021) 5. Wen, P., Li, Y., Chen, S., Zhao, S.: Remaining useful life prediction of IoT-enabled complex industrial systems with hybrid fusion of multiple information sources. IEEE Internet Things J. 8, 9045–9058 (2021) 6. RU2750542C1 – Mechatronic Journal Bearing –Google Patents. https://patents.google.com/ patent/RU2750542C1/ru?oq=2750542. Accessed 25 Feb 2023 7. Hase, A.: Early detection and identification of fatigue damage in thrust ball bearings by an acoustic emission technique. Lubricants 8(3), 37 (2020) 8. Shen, G., Xiang, D., Zhu, K., Jiang, L., Shen, Y., Li, Y.: Fatigue failure mechanism of planetary gear train for wind turbine gearbox. Eng. Fail. Anal. 87, 96–110 (2018) 9. Dong, Q., Yin, Z., Li, H., Gao, G., Zhong, N., Chen, Y.: Simulation and experimental verification of fatigue strength evaluation of journal bearing bush. Eng. Fail. Anal. 109, 104275 (2020) 10. El-Daher, C., Kebir, H., Bouvier, S., Pont, M., Hay, M.: Prediction of fatigue damage and spalling in a multilayered journal bearing shell. Tribol. Int. 175, 107850 (2022) 11. Ding, H., Yang, L., Cheng, Z., Yang, Z.: A remaining useful life prediction method for bearing based on deep neural networks. Measurement 172(3), 108878 (2021) 12. Chen, X., van Hillegersberg, J., Topan, E., Smith, S., Roberts, M.: Application of data-driven models to predictive maintenance: bearing wear prediction at TATA steel. Expert Syst. Appl. 186(2), 115699 (2021) 13. Ding, N., Li, H., Yin, Z., Zhong, N., Zhang, L.: Journal bearing seizure degradation assessment and remaining useful life prediction based on long short-term memory neural network. Measurement 166, 108215 (2020) 14. Suh, S., Jang, J., Won, S., Jha, M., Lee, Y.: Supervised health stage prediction using convolutional neural networks for bearing wear. Sensors 20, 1–19 (2020) 15. Xinyu, P., Xuanyi, X., Xiaowu, J.: Experimental study on wear life of journal bearings in the rotor system subjected to torque. Trans. Can. Soc. Mech. Eng. 44(2), 272–278 (2019) 16. Yan, J., He, Z., He, S.: A deep learning framework for sensor-equipped machine health indicator construction and remaining useful life prediction. Comput. Ind. Eng. 172(1), 108559 (2022)

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17. Huang, W., Zhang, X., Wu, C., Cao, S., Zhou, Q.: Tool wear prediction in ultrasonic vibrationassisted drilling of CFRP: a hybrid data-driven physics model-based framework. Tribol. Int. 174(1), 107755 (2022) 18. Yang, K., Duan, W., Huang, L., Zhang, P., Ma, S.: A prediction method for ship added resistance based on symbiosis of data-driven and physics-based models. Ocean Eng. 260(3), 112012 (2022) 19. Ye, S., Wang, C., Wang, Y., Lei, X., Wang, X., Yang, G.: Real-time model predictive control study of run-of-river hydropower plants with data-driven and physics-based coupled model. J. Hydrol. 617, 128942 (2023) 20. Shutin, D., Bondarenko, M., Polyakov, R., Stebakov, I., Savin, L.: Method for on-line remaining useful life and wear prediction for adjustable journal bearings utilizing a combination of physics-based and data-driven models: a numerical investigation. Lubricants 11(1), 33 (2023) 21. Zhu, H., Huang, Z., Lu, B., Zhou, C.: Bearing remaining useful life prediction of fatigue degradation process based on dynamic feature construction. Int. J. Fatigue 164, 107169 (2022) 22. Shutin, D., Polyakov, R.: Active hybrid bearings as mean for improving stability and diagnostics of heavy rotors of power generating machinery. IOP Conf. Ser. Mater. Sci. Eng. 862(3), 032098 (2020) 23. Zhang, S., Xu, H., Zhang, L., Xing, Y., Guo, Y.: Vibration suppression mechanism research of adjustable elliptical journal bearing under synchronous unbalance load. Tribol. Int. 132, 185–198 (2019) 24. Chasalevris, A., Dohnal, F.: Improving stability and operation of turbine rotors using adjustable journal bearings. Tribol. Int. 104, 369–382 (2016) 25. Polyakov, R., Shutin, D., Savin, L., Babin, A.: Peculiarities of reactions control for rotor positioning in an active journal hybrid bearing. Int. J. Mech. 10, 62–67 (2016) 26. Khutoretskiy, G., Tokov, M., Tolvinskaya, E.: Design of Turbogenerators. Energoatomizdat, Leningrad (1987) 27. Miner, M.: Cumulative damage in fatigue. J. Appl. Mech. 12, A159-A164 (1945). https://www. scirp.org/reference/ReferencesPapers.aspx?ReferenceID=1751417. Accessed 25 Feb 2023 28. Die Lebensdauer von Kugellagern, P.A.: Life length of roller bearings or durability of ball bearings. Zeitschrift des Vereines Deutscher Ingenieure (ZVDI) 14, 339–341 (1924). https://www. scirp.org/reference/referencespapers.aspx?referenceid=1754153. Accessed 15 Feb 2023 29. GOST 27165-97 Stationary Steam Turbine Units. Shafting Vibration Standards and General Requirements for Measurements – docs.cntd.ru. https://docs.cntd.ru/document/1200011635. Accessed 25 Feb 2023 30. Jang, J., Khonsari, M., Glovnea, R., Fillon, M.: On the characteristics of misaligned journal bearings. Lubricants 3, 27–53 (2015) 31. Feng, H., Jiang, S., Ji, A.: Investigations of the static and dynamic characteristics of waterlubricated hydrodynamic journal bearing considering turbulent, thermohydrodynamic and misaligned effects. Tribol. Int. 130, 245–260 (2019)

Application of Machine Learning in Simulation Models and Optimal Controllers for Fluid Film Bearings Yuri Kazakov(B) , Ivan Stebakov, Denis Shutin, and Leonid Savin Orel State University, Orel 302026, Russian Federation [email protected]

Abstract. Machine learning methods offer some alternatives to the conventional approaches to the development of passive and adjustable fluid film bearings. Data-based bearing models typically show an advantage over conventional numerical models in terms of computational speed, and can either replace or supplement them in certain applications. The most promising application of machine learning is to create high-performance models and optimal controllers for fluid film bearings. It covers a range of tasks connected with the rotor trajectory planning, like active vibration and friction reduction, that is the main scope of this work. On-line rotor position assessment considering the measured or estimated loads can also be implemented using fast data-driven models in diagnostics and predictive analytics systems. The work presents an analysis of this approach in terms of the accuracy of solutions, the time required for preparing data, and training the models. The results show that the calculation speed using data-driven models can be increased at least 10 times compared to the numerical models. Two ANN-based models with different structure were analyzed in accuracy and performance. A model consisting from three separate ANNs was introduced in addition to a single-ANN model based on the analysis of the bearing forces nonlinearities and demonstrated better accuracy and the training time reduced by 26%. The calculation speed increased 12 time compared to the reference numerical model. The use of approximation models is demonstrated for the case of active conical bearing with rotor motion control with intellectual DQN controller. Also the applicability of the approach is analyzed regarding the implementation of intellectual and predictive controllers of active bearings. Keywords: Fluid film bearings · Active bearings Data-based models · Optimal control

1

· Machine learning ·

Introduction

The Industry 4.0 paradigm implies integration of information technologies and control means into machines and equipment. Based on the data received from The study was supported by the Russian Science Foundation grant No. 22-19-00789, https://rscf.ru/en/project/22-19-00789/.. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2024  F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 224–235, 2024. https://doi.org/10.1007/978-3-031-40455-9_18

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them, analytical, predictive models and digital twins are formed for more effective production management. Bearings are the critical components of rotary machines. Fluid film bearings (FFB) are indispensable in many cases, although the non-linearity of their properties can make it difficult to make their proper design, especially with modern highly loaded machines. Nonlinear models of FFB are used in analysis [1,2], design [3,4] and control [5,6] tasks related to rotor-bearing systems. The calculation of main design parameters of both passive and active FFB is usually performed by solving the Reynolds equation using numerical methods [7–9]. Such calculations require significant computational resources to achieve good accuracy. However, the on-board computing facilities of rotary machine control systems almost always have limited performance. The linearization of the dynamic parameters of FFB [7,10] allows to reduce the amount of calculations while solving the rotor dynamics problems. However, this approach does not consider the nonlinear properties of the lubricant film. This makes it hardly applicable in a number of conditions, such as a significant eccentricity of the rotor position, the presence of cavitation phenomena, and especially in the use of adjustable bearings with rotor motion control systems [11,12]. Therefore, there is a need for more efficient models providing sufficient accuracy. A possible approach is the use of models based on machine learning methods. These models are created using experimental data and/or physics-based mathematical models of nonlinear objects [5,10,13–15]. Currently, machine learning methods are widely used in solving a number of different tasks, for example, defect detection [16–18] and condition diagnostics [15,19], as well as control of mechanical systems: rotors, hydraulic systems and manipulators [5,20,21]. It is even more important to take into account the nonlinear properties of the lubricant film in the problems of controlling rotor motion parameters in active FFB. Models of mechanical systems are often used for tuning and evaluating the controllers, as well as for training more advanced intelligent controllers [5,22]. It is also possible to use models directly to solve control problems, for example, using model predictive control (MPC) [13]. This paper analyses the possibilities and advantages of using machine learning for solving modeling and control problems in FFB. The proposed approach to modeling bearing forces makes it possible to obtain solutions to rotor dynamics problems comparable in accuracy, but many times faster than conventional numerical methods.

2

Modeling and Methods

Journal FFB is considered in this work as a well-known and understandable system. Its schematic is shown in Fig. 1. As a bearing model, we consider the flow of a viscous incompressible fluid in a channel of length l formed by two cylinders, a shaft and a bearing. The bearing with radius R = r + h0 , where r is the shaft radius, h0 is the radial clearance, is stationary. X1 , X2 , V1 and V2 are the rotor displacements and velocities on

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Fig. 1. Schematic of a shaft-bearing system with a fluid film bearing.

the abscissa and ordinate axes. X3 and V3 are the rotor axial displacement and velocity. In turn, the shaft rotates in the bearing with a constant frequency and angular velocity ω. Lubricant is supplied from one end of the bearing under pressure p0 . The velocity field is three-dimensional and is characterized by the vector V ≡ [(V1 , V2 , V3 )]. It is convenient to represent the equations describing the motions of the medium in cylindrical coordinates βi , where β1 , β2 , β3 are the radial, angular, and axial coordinates, respectively. Cylindrical coordinates are characterized by Lame coefficients: H1 = H3 = 1, H2 = β1 . The model of a fluid film bearing is based on the Reynolds equation in the form [5]:   ∂h 1 ∂ 3 ∂p − μ12u3 β1 (1) + h = μu1 6h + μu2 6 β1 ∂β2 ∂β2 ∂β2    −   −  β −β − β β β β where u1 = V3 , u2 = ω r0 + 1 l 1 (r1 − r0 ) + V1 cos 1r0 2 − V2 sin 1r0 2 ,  −   −  β β β β u3 = V1 sin 1r0 2 + V2 cos 1r0 2 are the components of the fluid velocity vector at the shaft surface, μ is viscosity. The radial clearance function is as follows:  −   −  β1 β2 β1 β2 h = h (β2 ) = h0 − X1 sin − X2 cos , (2) r0 r0 ∂ h ∂β1 3



∂p β1 ∂β1



where Xi are the coordinates of the center of mass of the rotor. Cavitation is based on the Gumbel hypothesis [5,23]. The Reynolds equation is solved by the finite differences method [5,7]. The result of the solution is the pressure field of the lubricant film. The bearing reactions are calculated by integrating the pressure field using the Simpson numerical integration method:

Application of Machine Learning in Simulation Models

R1 = R2 =

 β2+  β1+ β−

β−

β2−

β1−

 β22+  β11+

p cos φβ1 dβ1 H2 dβ2

227

(3)

p sin φβ1 dβ1 H2 dβ2

The dynamic model of the rotor system is based on a single-mass oscillatory model. The rotor is represented as a point mass oscillating in the bearing under the influence of gravity, the reaction forces and the imbalance forces. The imbalance forces are modeled as the centrifugal forces. The motion equation of the center mass of rotor in matrix form [24]:         R1 cos(ωt) 0 dV1 /dt = + mu dω 2 +m (4) m R2 dV2 /dt sin(ωt) g where V1 ,V2 is the velocity of the center of the shaft, t is time, mu d is imbalance, g is free fall acceleration. The journal bearing reactions are non-linear and increase significantly at high shaft eccentricities. Also, the calculation results using numerical methods, such as the finite difference method, are sensitive to the size of the computational grid. Figure 2 shows that the lower density grid gives a big calculation error. This phenomenon is most pronounced for the range of eccentricities from 0.9 to 1. Figure 2 demonstrates the stabilization of the calculated results in this area with an increase in the grid dimension from 30 × 30 to 60 × 60. However, it leads to an increase in the calculation time.

Fig. 2. The value of the reactions of the lubricating layer depending on the position of the rotor at a lower (30 × 30) and higher (60 × 60) density of the calculated grid.

Numerical solving of Lagrange equations in rotor dynamics problems requires recalculation of the bearing forces at each step so the modeling time can be enormous. Therefore, an alternative approach is the approximation of solutions obtained with numerical methods. Thus, artificial neural networks (ANN) can be used for approximation. A verified numerical model was considered as the reference solution. The Reynolds equation was solved by the finite difference method with a computational grid of 30 × 30 for low eccentricities and 55 × 55

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for higher eccentricities. A denser grid was used to obtain the sufficient accuracy at higher rotor eccentricities. The obtained bearing forces were used to test the accuracy of the developed approximation models. The choice of approximation methods is significantly affected by the nature of the initial dependencies. The bearing forces exponentially increase with increasing rotor eccentricity. Also, the compression rate of the fluid film significantly affects their values. The dependence of the fluid film reactions on the compression rate also is strongly nonlinear, see Fig. 3.

Fig. 3. Dependence of the reaction of the fluid film on the velocities of the rotor.

As Fig. 3 shows, the dependencies have both relatively linear and substantially non-linear regions. The latter require more values to obtain an approximation with acceptable accuracy. Therefore, as in the case of a finite difference grid when solving the Reynolds equation, it is advisable to use a grid of different density for different regions in the initial dataset for approximation.

3 3.1

Results and Discussion Approximation of FFB Models

A dataset was generated using the model described in Sect. 2. The model was implemented in Matlab software. The dataset describes the bearing forces R1 and R2 for various rotor positions and velocities. A bearing with the following parameters was taken as a test system: the bearing length is 64 mm, the bearing diameter is 40 mm, the clearance is 120 µm, the rotation speed is 3000 rpm, the lubricant is water at a temperature of 40 ◦ C. A polar coordinate system was used for the grid during building the dataset on the bearing forces. Their values were calculated for a uniform distribution of 12 points along the radial (i.e., along the eccentricity of the rotor position) and 40 points along the circumferential (i.e., along the angle of the rotor position in the bearing) coordinates. This distribution was used to collect data at 0-0.9 and 0.9-1 eccentricities to deal with the large nonlinearity at high eccentricity values.

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The entire eccentricities region was divided into 2 main sections, 0.1-0.5 and 0.51, for which different velocity distributions were used. The distributions differed in the boundary values of the velocities, as well as in the number of points in the critical region. The same number of points equal to 80 was used for all sections according to the velocity values, 40 points each for the positive and negative regions. For low eccentricities, we collected the data for the range of velocities from −0.2 m/s to 0.2 m/s, and 70% of all points were located within the range from −0.05 m/s to 0.05 m/s. At high eccentricities, the range was increased to −0.5 m/s to 0.5 m/s, and 50% of all points were located within the range from −0.2 m/s to 0.2 m/s. Thus, the final dataset included 4,700,000 training samples. The dataset input data were the coordinates of the rotor position (X1 , X2 ) and its velocities (V1 , V2 ). Bearing forces values (F1 , F2 ) were the output parameters. A fully connected ANN was used for training. ANN was with one hidden layer and contained 50 neurons The main evaluation parameters were the network training time, the trajectory calculation time, and the calculation accuracy. The accuracy and the calculation time were compared with those of the reference model based on the numerical solution of the Reynolds equation. A computer of the following configuration was used for all calculations: Intel Core i5-11600K 3.90 GHz processor, NVIDIA T1000 graphics card, 16 GB of RAM. Two approaches to the construction of approximation based on ANN were tested in this study. The first is to use a single ANN trained on the entire dataset. The network training time was 5 h 42 min. The second approach was to divide the samples into several parts, based on the nature of the approximate dependence in the corresponding region. In this work, the dataset was divided into 3 parts depending on the eccentricity: 0–0.5, 0.5–0.7, >0.7. Each range was used to train a separate ANN. The training of each ANN was about 1.5 h, which is about 4.5 h in total. After training ANNs, a number of computational experiments were carried out to check the accuracy of the obtained models. A number of rotor trajectories in the bearing were calculated with two trained and the reference models. The rotor imbalance was a variable value for evaluating the operation of models in different ranges of eccentricities. The calculation results are shown in Fig. 4. Firstly, the accuracy of the ANNs was tested at small imbalance values which is characterized by small values of velocities and eccentricities, the results are shown in Fig. 4 a. Figure 4 b and c show the rotor trajectories for the imbalance value of 5e−5 and 1e−4, respectively, while the trajectory span is approximately 20% of the radial clearance. After that, the imbalance value was increased to 5e−4 and 10e−4, which gives a span of about 65% of the radial clearance (see Fig. 4 d, e). Also, the operation of the models was tested for the cases when a constant force is applied to the rotor. The force was applied along the X1 coordinate and was equal to 30N and −40N. The results are shown in Fig. 4 f,g. Based on the data obtained, when using a single ANN, the rotor trajectories differ significantly from the reference ones in most tests, except for cases with the largest imbalance value. This suggests that the best approximation quality

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Fig. 4. Comparison of rotor trajectories: a) without imbalance, b) mu d equal to 5e−5, c) mu d equal to 1e−4, d) mu d equal to 5e−4, e) mu d equal to 10e−4, f) radial force equal to 30N, g) radial force equal to −40N.

was achieved for data from the region with low non-linearity. In other ranges, the approximation accuracy cannot be considered satisfactory. When using three ANNs, tests with all imbalance values show a satisfactory accuracy of approximation of the initial data. The model provides correct information about both the steady-state amplitudes of rotor oscillations and the shape of trajectories and transients. At the same time, as in the case of a single ANN, the discrepancy between the trajectories and the reference ones was large during the test with an external load. Although, in the case with three ANNs the

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discrepancy is less than for a single ANN. The form of the resulting trajectories is similar to the reference model, but their location and amplitude differ. Thus, the results cannot be assessed as completely satisfactory for any of the tested cases. Only the model based on three ANNs can be considered as limitedly applicable for calculations in systems without complex schemes for loading rotors with external forces. However, the nature of the results obtained allows us to conclude that the combination of several ANNs for data approximation is a direction that can lead to satisfactory results when it is developed. It is advisable to conduct a more in-depth analysis of the nonlinear properties of the initial data. Probably, dividing them into a larger number of sub-ranges will make it possible to achieve an acceptable modeling accuracy. In addition, the improvement of the results can be achieved by optimizing the approximation methods themselves, e.g., using more complex ANNs of a different architecture and/or increasing the size/number of hidden layers, applying different activation functions, etc. It should be noted that despite the more complex structure of the model consisting of three ANNs, the training time was reduced by 26% compared to the one ANN model. As for another key point, namely the calculation time, the use of a single ANN reduced the estimated time by 12 times in comparison with the reference numerical model. 3.2

Data Driven FFB Models in Control Tasks

Object models are often used in control tasks to tune controllers. Synthesis of intelligent controllers based on reinforcement learning should be notes particularly since the process of their training implies continuous interaction between the agent and the model of the system. In other cases, for example, in modelpredictive control (MPC), an object model is used to predict its response to a planned sequence of control signals. In all the cases noted, sufficient accuracy and high model speed are the essential factors, especially relating to MPC. As mentioned above, the use of machine learning methods can significantly reduce the calculation time in comparison with numerical methods. This also makes it possible to significantly reduce the controller training time. However, it should be noted that achieving high accuracy of such models is a laborious task. The least resource-intensive approach to obtaining data-driven bearing models is to train them using a limited amount of data about the typical ways of the rotor motion in a limited bearing area. Such an approach, for example, was used in one of our previous studies of actively lubricated journal bearing in [13]. In this work such a model was used to synthesize and test several optimal controllers. A similar approach was also used in [22], where an intelligent controller based on a DQN agent was trained to control the position of a rotor in adjustable conical bearings. The bearings were equipped with the axially displaceable sleeves for influencing the type of the rotor motion by changing the average fluid film thickness. ANN was also used to calculate the bearing forces. However, in contrast to this work, the original dataset included a limited number of precalculated rotor trajectories. The trajectories were obtained for different values of the axial

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dis placement of the rotor in bearings. An example of a calculation with such a model is shown in Fig. 5 [22].

Fig. 5. Rotor movement patterns: a) DQN-agent control, b) no control.

The tests shown in Fig. 5 presented a simulation of occurrence of a malfunction in a rotary machine associated with an increase in rotor imbalance. The imbalance increased uniformly over 5 s, the timing of the presented trajectories is shown in the legend of Fig. 5. As can be seen in the passive system (Fig. 5 b), this leads to an increase in the oscillations amplitude, while in an adjustable bearing (Fig. 5 a) it tends to restrain the growth of this parameter. In one of the presented results, the rotor trajectories at the final moment of time cross the bearing boundaries. In a physical system it would mean the contact of the rotor with the bearing and would lead to the occurrence of shock processes and, probably, chaotic oscillations. The continuation of the calculation indicates a significant approximation error near the boundaries of the training dataset. This phenomenon is strongly manifested in the region of high eccentricities and at the boundary of the control action. It can be concluded that the use of this modeling method imposes significant restrictions on the range of control actions, as well as on the possibility of numerical evaluation of controllers. However, they can be used for preliminary training and qualitative assessment in a strictly defined operating range. The proposed approach to modeling in this work allows developing models that can calculate the bearing forces for any rotor position and at its different velocities. This makes such a model more flexible with respect to the range of control actions, and also increases the accuracy of the qualitative and quantitative

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assessment. In addition, such models are more versatile in their application. For training intelligent controllers with reinforcement learning methods, the speed of computing on approximation models allows reducing the training time by at least 10 times while maintaining the accuracy of the results. Implementation of MPC may additionally require reducing the order of approximating models to further improve their performance. In this case, it will be necessary to find a balance between the accuracy and speed of the model. The disadvantage of the presented approximation approach is the need for preliminary generation of a significant amount of data describing all the nonlinear bearing forces properties with sufficient quality. In addition, in adjustable bearings the fluid film forces are also affected by controlled parameters (gap shape, lubricant supply pressure, its viscosity in the case of magnetorheological fluids, etc.) in addition to the rotor position and velocities. In some cases, like in [22], the controller impact is described by a single parameter. However, in other cases more of them may be required. Adding each parameter to a dataset multiplies its dimension and, accordingly, the generation time. The generation time can be reduced by accelerating the initial calculation methods, but in this case, the accuracy of the data decreases. Additionally, time is also spent on training approximation models. Reducing these time costs while meeting the requirements for the accuracy of the models is possible by choosing the most effective training methods and optimizing their parameters. It also should be noted that if the parameters of the bearing change during its designing, the above actions for approximating the models will have to be repeated, which further reduces the process flexibility.

4

Conclusion

An approach to modeling the fluid film bearings forces by approximation of their conventional numerical models with artificial neural networks (ANNs) was studied in this work. As the results show, such approximation models are able to exceed the numerical models in calculation speed at least an order of magnitude. However, obtaining such models is associated with a number of limiting factors. The strong non-linear dependence of bearing forces in on the rotor eccentricity and its radial velocity require generating a significant amount of data by the initial numerical model to provide the appropriate accuracy of approximation. A denser computational grid also should be used in numerical model for the same purpose. These factors make the generation of the initial dataset a timeconsuming task. It should be considered when making decisions on applying of ANNs to such approximation tasks. The speed and accuracy of the ANN-based approximation models also depends on their structure. Two different model structures have been tested in this work using the trajectories method. The model based on the combination of three separate ANNs showed better accuracy compared to the single-ANN model, as well as the reduction in the training time by 26%. The three-ANN bearing model also exceeds the reference numerical model by 12 times in calculation speed. However, its accuracy was still insufficient in some scenarios, like

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adding the radial force to the rotor, to obtain adequate solutions to rotor dynamics problems. So, the further improvement in the model and/or data structure is required. The described approach is mostly applicable to the tasks where a lot of calculations of rotor dynamic behavior is required. The ANN-based models were successfully utilized for training intelligent controllers of fluid film bearings. Using them instead of conventional numerical models reduces the controller training time. Fast approximation models are also suitable for model predictive controllers for active bearings, where fast on-line response analysis is required, as well as for predictive systems for monitoring and diagnostics of fluid film bearings.

References 1. Tala-Ighil, N., Fillon, M.: A numerical investigation of both thermal and texturing surface effects on the journal bearings static characteristics. Tribol. Int. 90, 228– 239 (2015) 2. Gropper, D., Harvey, T.J., Wang, L.: Numerical analysis and optimization of surface textures for a tilting pad thrust bearing. Tribol. Int. 124, 134–144 (2018) 3. Kumar, V., Sharma, S.C., Jain, S.C.: On the restrictor design parameter of hybrid journal bearing for optimum rotordynamic coefficients. Tribol. Int. 22, 356–368 (2006) 4. Cui, S., Zhang, C., Fillon, M., Gu, L.: Optimization performance of plain journal bearings with partial wall slip. Tribol. Int., 106–137 (2020) 5. Kazakov, Y.N., Kornaev, A.V., Shutin, D.V., Li, S., Savin, L.A.: Active fluidfilm bearing with deep q-network agent-based control system. J. Tribol. 144, 1–12 (2022) 6. Bre´ nkacz, L  ., Witanowski, L  ., Drosi´ nska-Komor, M., Szewczuk-Krypa, N.: Research and applications of active bearings: a state-of-the-art review. Mech. Syst. Signal Process. 151, 107423 (2021) 7. Kornaev, A.V., Kornaeva, E.P., Savin, L.A., Kazakov, Y.N., Fetisov, A., Rodichev, A.Y., Mayorov, S.V.: Enhanced hydrodynamic lubrication of lightly loaded fluidfilm bearings due to the viscosity wedge effect. Tribology International, vol. 160. 107027 (2021) 8. Peixoto, T.F., Cavalca, K.L.: Thrust bearing coupling effects on the lateral dynamics of turbochargers. Tribol. Int. 145, 106166 (2020) 9. Momoniat, E.: A reynolds equation modelling coriolis force effects on chemical mechanical polishing. Int. J. Non-Linear Mech. 92, 111–117 (2017) 10. Iseli, E., Schiffmann, J.: Prediction of the reaction forces of spiral-groove gas journal bearings by artificial neural network regression models. J. Comput. Sci. 48, 101256 (2021) 11. Chasalevris, A., Dohnal, F.: Vibration quenching in a large scale rotor-bearing system using journal bearings with variable geometry. J. Sound Vib. 333, 2087– 2099 (2014) 12. Santos, I.F.: Controllable sliding bearings and controllable lubrication principles-an overview. Lubricants 6, 1–16 (2018) 13. Li, S., et al.: Active hybrid journal bearings with lubrication control: towards machine learning. Tribol. Int. 175. 107805 (2022) 14. lmqvist, A.: Fundamentals of physics-informed neural networks applied to solve the reynolds boundary value problem. Lubricants 9, 1–9 (2021)

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15. Kornaev, A.V., Kornaev, N.V., Kornaeva, E.P., Savin, L.A.: Application of artificial neural networks to calculation of oil film reaction forces and dynamics of rotors on journal bearings. Int. J. Rotating Mach. 2017, 1–11 (2017) 16. Kumar, A., Gandhi, C.P., Zhou, Y., Kumar, R., Xiang, J.: Improved deep convolution neural network (CNN) for the identification of defects in the centrifugal pump using acoustic images. Applied Acoustics, vol. 167. 107399 (2020) 17. Misra, S., et al.: Fault detection in induction motor using time domain and spectral imaging-based transfer learning approach on vibration data. Sensors 22, 1–22 (2022) 18. Chen, H.Y., Lee, C.H.: Vibration signals analysis by explainable artificial intelligence (XAI) approach: application on bearing faults diagnosis. IEEE Access 8, 134246–134256 (2020) 19. Kornaeva, E.P., Kornaev, A.V., Kazakov, Y.N., Polyakov, R.N.: Application of artificial neural networks to diagnostics of fluid-film bearing lubrication. In: IOP Conference Series: Materials Science and Engineering 734, pp. 1–9 (2020) 20. Yeo, S., Naing, Y., Kim, T., Oh, S.: Achieving balanced load distribution with reinforcement learning-based switch migration in distributed SDN controllers. Electronics 10, 1–10 (2021) 21. Kim, J.B., Lim, H.K., Kim, C.M., Kim, M.S., Hong, Y.G., Han, Y.H.: Imitation reinforcement learning-based remote rotary inverted pendulum control in openflow network. IEEE Access 7, 36682–36690 (2019) 22. Kazakov, Y.N., Kornaev, A.V., Shutin, D.V., Kornaeva, E.P., Savin, L.A.: Reducing rotor vibrations in active conical fluid film bearings with controllable gap. Nonlinear Dyn. 18, 863–873 (2022) 23. Hori, Y.: Hydrodynamic lubrication. Hydrodynamic Lubrication, 1–231 (2006) 24. Friswell, M.I., Garvey, S.D., Lees, A.W.: Rotor dynamics: modeling and analysis modeling and analysis of rotating machines, 1–544 (2010)

Study on Solution Algorithm of Reynolds Equation of Self-acting Gas Journal Bearings Based on Finite Difference Method Haijun Zhang1(B) , Qin Yang2 , Wei Zhao1 , and Feilong Jiang1 1 College of Information Science and Engineering, Jiaxing University, Jiaxing 314001, China

[email protected] 2 College of Data Science, Jiaxing University, Jiaxing 314001, China

Abstract. It is very crucial to solve Reynolds equation quickly and accurately using the numerical methods in the research field of fluid lubrication. For the static Reynolds equation of self-acting gas journal bearings, the typical solution algorithm of the finite difference method is put forward and a new solution algorithm of the finite difference method is proposed. The typical solution algorithm discretizes the transformed Reynolds equation directly and the quadratic equation in one variable of the pressure can be obtained. Different from the typical solution algorithm, the both sides of the transformed Reynolds equation are firstly divided by the term of PH3 and then discretized in the new solution algorithm, thus the linear equation about the pressure can be obtained. Secondly, the Reynolds equation is solved numerically with the same parameters and the pressure distribution of gas bearings is obtained. Finally, the numerical solution from the new solution algorithm can be obtained with less number of iterations and the less computing time under the different computational grids and bearing numbers. Therefore, the new solution algorithm of the finite difference method is superior to the typical solution algorithm of the finite difference method. Keywords: self-acting journal bearings · static Reynolds equation · finite difference method · bearing number · computational grid

1 Introduction As a high-speed and high-precision supporting technology, gas journal bearings have the characteristics of no pollution, low power consumption and high temperature resistance, etc., and are widely used in gyroscope instruments, high-speed machine tools, centrifuges, compressors, fans and other rotating machinery [1–4]. To analyze the dynamic characteristics of the journal bearing and its mechanical system, the key is to solve the Reynolds equation and obtain the pressure distribution of the gas film accurately and quickly. As a nonlinear partial differential equation [5], the Reynolds equation for the selfacting gas journal bearing is usually solved by numerical methods [6]. Numerical methods include finite difference method and other numerical methods. In this paper, we © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 236–244, 2024. https://doi.org/10.1007/978-3-031-40455-9_19

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only focus on finite difference method. In 1961, Raimondi [7] used the finite difference method to solve the Reynolds equation of finite length self-acting gas journal bearing and obtained a series of detailed bearing design data charts, which are still used as a typical reference for the design data of self-acting gas journal bearing. Compared with other numerical methods, the finite difference method is not complex and easily obtain high-order accuracy of the spatial discretization. Therefore many researchers [2, 8–16] still used the finite difference method in the field of fluid film lubrication although its application limitation of simple geometry and structured grids. The content of this paper is arranged as follows: Firstly, the static Reynolds equation for the self-acting gas journal bearings is put forward; Secondly, the Reynolds equation is discretized, and the typical solution algorithm and a new solution algorithm based on the finite difference method are given respectively. Thirdly, the new solution algorithm is validated according to the comparison of the pressure profiles between the new solution algorithm and the typical solution algorithm. Fourthly, the number of iterations and computing time of the typical solution algorithm and the new solution algorithm are compared and analyzed for the different calculation grids and bearing numbers. Finally, the conclusion of this paper is given.

2 Static Reynolds Equation for Self-acting Gas Bearing

Fig. 1. Sketch of self-acting gas journal bearings

The sketch of the self-acting gas journal bearing is shown in Fig. 1. The outer shadowed ring represents the bearing, and the inner solid line circle represents the journal. The bearing is stationary and the journal rotates counterclockwise with angular velocity . The space between the bearing and the journal is filled with gas. When the self-acting gas bearing is working, the center of the journal deviates from the center of the bearing, and the distance between the both centers represents the eccentricity e. The radius of the journal is r and the width of the bearing is l. According to the fluid lubrication theory, the following assumptions are made: (1) The curvature of the lubricating air film is not considered; (2) In the direction of gas film thickness, the change of pressure is not considered; (3) The flow of the air film is laminar flow without turbulence; (4) Inertial force of air flow is not considered; (5)

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The lubricating air film is not affected by external forces; (6) There is no slip on the bearing surface; (7) Only the velocity gradient across the gas film thickness direction is considered, and the higher-order derivative term is ignored. The Navier-Stokes equations are simplified as ∂p ∂ 2u =μ 2 ∂x ∂ z

(1)

∂ 2v ∂p =μ 2 ∂y ∂ z

(2)

It is assumed that the lubrication film thickness direction is z direction. Without considering the flow velocity slip on the wall, the boundary conditions at the upper and lower walls are z = 0, u = rω, v = 0

(3)

z = h, u = 0, v = 0

(4)

According to the boundary conditions (3) and (4), the integral of Eqs. (1) and (2) is obtained, u=

 1 ∂p z z(z − h) + rω 1 − 2μ ∂x h

(5)

1 ∂p ∂p z(z − h) 2μ ∂y ∂x

(6)

v=

Let qx and qy be the circumferential and axial volume flow through the unit width gas film respectively, then  qx =

h

udz = −

0 h

h3 ∂p rωh + 12μ ∂x 2

qy = ∫ vdz = − 0

h3 ∂p 12μ ∂y

According to the continuum equation of the gas flow in the gas film,   ρqy ∂(ρqx ) ∂(ρh) + + =0 ∂x ∂y ∂t

(7) (8)

(9)

Substituting Eqs. (7) and (8) into Eq. (9), and simplifying, the following equation can be obtained,     ∂ ρh3 ∂p ∂ ρh3 ∂p ∂(ρh) ∂(ρh) + = 6rω + 12 (10) ∂x μ ∂x ∂y μ ∂y ∂x ∂t

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Without considering the temperature change of the gas film, the relation between the pressure and density of the gas film is, pρ−1 = constant

(11)

Substituting Eq. (11) into Eq. (10), without considering the change of gas viscosity, Eq. (10) can be written,     ∂ ∂ ∂(ph) ∂(ph) 3 ∂p 3 ∂p ph + ph = 6μrω + 12μ (12) ∂x ∂x ∂y ∂y ∂x ∂t The dimensionless variables are introduced, P = p/pa , θ = x/r, ζ = y/r, H = h/c =  r 2 12μω  r 2 1 + εcosθ, τ = ωt, = 6μω pa c , σ = pa c , Eq. (12) is changed into     ∂ ∂(PH ) ∂ ∂(PH ) 3 ∂P 3 ∂P +σ PH + PH = ∂θ ∂θ ∂ζ ∂ζ ∂θ ∂τ

(13)

This is the dynamical Reynolds equation for the self-acting gas journal bearings and denotes bearing number and σ denotes squeezing number. When the change of P or H with time is not taken into account, Eq. (13) becomes     ∂ ∂ ∂(PH ) ∂P ∂P PH 3 + PH 3 = (14) ∂θ ∂θ ∂ζ ∂ζ ∂θ This is the static Reynolds equation for the self-acting gas journal bearings. 2.1 Algorithm of Numerical Solution The static Reynolds equation of self-acting gas journal bearing is an elliptic partial differential equation, which is difficult to obtain an analytical solution. It is usually solved by numerical methods, such as the finite difference method and the finite element method. This paper deals only with finite difference methods. Before applying the finite difference method, Eq. (2) needs to be expanded as    2   2

∂P 2 ∂H ∂P ∂P ∂H ∂ 2P ∂P 3 ∂ P 3 = P + H PH + 2 +H + + 3PH 2 2 ∂θ ∂ζ ∂θ ∂ζ ∂θ ∂θ ∂θ ∂θ (15)

2.2 Typical Algorithm of Numerical Solution Firstly, the computing region is discretized and divides into n × m grid elements, including n elements in the circumferential direction of the bearing and m elements in the width direction of the bearing, as shown in Fig. 2.

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Fig. 2. Grid using finite difference method

According to the finite difference method, the partial derivative of the pressure P at any grid point (i,j) along the θ and ζ directions can be approximated by the pressure of the surrounding grid points. Here, the central difference scheme is used for discretization. ⎧ ⎪ Pi+1,j −Pi−1,j ⎪ ∂P ⎪ ⎪ ∂θ ≈ 2θ ⎪ ⎪ ⎪ ⎪ P −Pi,j−1 ⎪ i,j+1 ∂P ⎨ ∂ζ ≈ 2ζ (16) ⎪ Pi+1,j −2Pi,j +Pi−1,j ∂2P ⎪ ⎪ ≈ ⎪ ∂θ 2 (θ )2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ 2 P2 ≈ Pi,j+1 −2Pi,j2+Pi,j−1 ∂ζ (ζ )

After substituting Eq. (16) into Eq. (15) and arranging, the following quadratic equation in one variable can be obtained, 2 APi,j − BPi,j + C = 0

where =

2 (θ)2

+

2 , (ζ )2

B=

Pi+1,j +Pi−1,j (θ )2

Pi+1,j − Pi−1,j − C= 2 H 2θ



+

Pi,j+1 +Pi,j−1 (ζ )2

Pi+1,j − Pi−1,j 2θ

(17) +

2 −

3 ∂H Pi+1,j −Pi−1,j H ∂θ 2θ

−



2

Pi,j+1 − Pi,j−1 2ζ

∂H H 3 ∂θ

In B and C, the grid point pressure is the result after the last iteration, namely, is the known value. Then, Eq. (17) is a quadratic equation with constant coefficients in one variable, and its solution is   2 B B C + (18) Pi,j = − 2A 2A A

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2.3 New Algorithm of Numerical Solution For self-acting gas journal bearings, the value of the non-dimensional term PH 3 is finite. It is feasible that both sides of Eq. (15) are divided by the term PH 3 , that is,    2   2

∂ P 1 ∂P 2 1 ∂H 1 ∂P ∂ 2P ∂P 1 ∂P ∂H = 3 + + 2 + + +3 ∂θ 2 ∂ζ P ∂θ ∂ζ H ∂θ ∂θ H ∂θ PH 2 ∂θ (19) The central difference scheme is used and the following expression can be obtained, Pi,j = D/A

(20)

where D= 1 + Pi,j



+

Pi+1,j + Pi−1,j (θ)

2

Pi+1,j − Pi−1,j 2θ

+

2

Pi,j+1 + Pi,j−1 (ζ )2 

+

Pi,j+1 − Pi,j−1 2ζ

2

3 ∂H Pi+1,j − Pi−1,j 1 ∂H − 3 H ∂θ 2θ H ∂θ −

1 Pi+1,j − Pi−1,j Pi,j H 2 2θ

Comparing Eq. (20) with Eq. (17), the calculation of Eq. (20) is simpler and the computing time will be less. Once the pressure value of arbitrary nodes is obtained, the formula of relaxation iteration is adopted, that is,   (k) (k−1) (k−1) (k) (21) P i,j = P i,j + ω Pi,j − P i,j where ω is the iteration factor.

3 Program Verification According to the above finite-difference algorithms, the Reynolds equation for the selfacting gas journal bearing is solved and the pressure distribution is obtained, as shown in Fig. 3. In the calculating process, the bearing number Λ = 5.0 and eccentricity ratio ε = 0.4 are chosen. For the same bearing number, the pressure profiles of self-acting gas journal bearing under the different eccentricity ratios are shown in Fig. 4. It can be seen that the pressure distributions calculated from Reynolds equation are the same according to the present solution algorithm and typical solution algorithm respectively.

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Fig. 3. Pressure distribution of self-acting gas journal bearings 2 =0.4, present algorithm =0.4, typical algorithm =0.2, present algorithm =0.2, typical algorithm

1.8

nondimensional pressure

1.6

1.4

1.2

1

0.8

0

10

20

30

40

50

60

circumferential position

Fig. 4. Pressure profiles of self-acting gas journal bearings under the same bearing number

4 Algorithm Comparison In order to evaluate the computing efficiency of the present solution algorithm, the same computer configuration was selected: Windows 10 operating system, processor Intel(R) Core(TM)i7-4790, memory (RAM) 8G. The calculation software MATLAB R2015A (8.5.0.197613) was used. The comparison of the number of iterations and computing time in solving the Reynolds equation for self-acting gas journal bearings under different grid numbers and bearing numbers was made. Firstly, the same bearing number Λ = 3.26 is selected and the calculation grids are different. The number of iterations and computing time used in solving the Reynolds equation are shown in Table 1. It can be seen that with the increase of the computing grid number, both the number of iterations and the computing time of the two difference algorithms increase. When the grid number is the same, the number of iterations and computing time of the new difference algorithm are less than that of the typical difference algorithm. Secondly, the same calculation grid (61 × 121) is selected and the bearing numbers are different. The number of iterations and computing time are shown in Table 2. It can be seen that with the increase of the bearing number, the number of iterations and computing time used by the new difference algorithm first decreases slightly and then gradually increases, while the number of iterations and computing time used by the typical difference algorithm gradually increases. When the bearing number is the same,

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Table 1. Number of iterations and computing time Grids

Present algorithm

Typical algorithm

Number of iterations

Computing time (s)

Number of iterations

Computing time (s)

41 × 121

315

21.07

428

30.04

61 × 121

252

23.05

523

54.22

81 × 121

249

30.98

593

81.34

101 × 121

280

45.40

649

113.53

121 × 121

316

57.80

693

141.82

the number of iterations and computing time of the new difference algorithm are less than that of the typical difference algorithm, and when the bearing number is 3.0, the number of iterations and computing time of both difference algorithms are significantly different. Table 2. Number of iterations and computing time Bearing number

Present algorithm

Typical algorithm

Number of iterations

Computing time (s) Number of iterations

Computing time (s)

1.0

282

26.81

273

28.95

2.0

273

25.38

303

31.60

3.0

245

23.11

483

50.16

4.0

483

45.25

607

63.95

5.0

586

54.05

681

71.15

5 Conclusion A new solution algorithm of Reynolds equation for self-acting gas journal bearings has been proposed. The performance of new solution algorithm is compared with the one of typical solution algorithm, which proves its effectiveness. The conclusions are as follows: (1) The number of iterations and computing time of the new solution algorithm are less than those of the typical solution algorithm when the calculation grid and journal velocity are the same, which proves the superiority of the new solution algorithm. (2) The new solution algorithm of Reynolds equation can also be applicable to Reynolds equation of the self-acting gas slider bearings.

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Acknowledgement. The work was supported financially by the National Natural Science Foundation of China (No. 11102071 and No.11805085).

References 1. Szeri, A.Z.: Fluid Film Lubrication. Cambridge University Press, New York (2011) 2. Gu, L., Guenat, E., Schiffmann, J.: A review of grooved dynamic gas bearings. Appl. Mech. Rev. 72(1), 010802 (2020) 3. Kumar, J., et al.: A review of thermohydrodynamic aspects of gas foil bearings. Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 236(7), 1466–1490 (2022) 4. Pattnayak, M.R., et al.: An overview and assessment on aerodynamic journal bearings with important findings and scope for explorations. Tribol. Int. 174, 60 (2022) 5. Gustafsson, T., et al.: Nonlinear Reynolds equation for hydrodynamic lubrication. Appl. Math. Model. 39(17), 5299–5309 (2015) 6. Castelli, V., Pirvics, J.: Review of numerical method in gas bearing film analysis. J. Lubr. Technol. 90(4), 777–792 (1968) 7. Raimondi, A.A.: A numerical solution for the gas lubricated full journal bearing of finite length. ASLE Trans. 4, 131–155 (1961) 8. Zhang, H.-J., Zhu, C.-S., Yang, Q.: New numerical solution for self-acting gas journal bearings. J. Zhejiang Univ. Sci. A 10(5), 685–690 (2009). https://doi.org/10.1631/jzus.A08 20532 9. Nicoletti, R.: Comparison between a meshless method and the finite difference method for solving the Reynolds equation in finite bearings. J. Tribol. Trans. ASME 135(4), 044501 (2013) 10. Li, W., Liu, W., Feng, K.: Effect of microfabrication defects on the performance of rarefaction gas-lubricated micro flexure pivot tilting pad gas bearing in power MEMS. Microsyst. Technol. 23(8), 3401–3419 (2016). https://doi.org/10.1007/s00542-016-3155-1 11. Guenat, E., Schiffmann, J.: Real-gas effects on aerodynamic bearings. Tribol. Int. 120, 358– 368 (2018). https://doi.org/10.1016/j.triboint.2018.01.008 12. Yao, W., et al.: Analysis on dynamic characteristics of the ultra thin gas film lubrication in cylindrical gas journal bearing. J. Vibr. Eng. 32(5), 908–917 (2019) 13. Peterson, W., et al.: A strongly coupled finite difference method-finite element method model for two-dimensional elastohydrodynamically lubricated contact. J. Tribol. Trans. ASME 142(5), 051601 (2020) 14. Shahdhaar, M.A., Yadawad, S.S., Khamari, D.S., Behera, S.K.: Numerical investigation of slip flow phenomenon on performance characteristics of gas foil journal bearing. SN Appl. Sci. 2(10), 1–18 (2020). https://doi.org/10.1007/s42452-020-03494-4 15. Hu, H.Y., Feng, M., Ren, T.M.: Study on the performance of gas foil journal bearings with bump-type shim foil. Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 235(3), 509–523 (2021) 16. Kumar, J., Khamari, D.S., Behera, S.K., Sahoo, R.K.: Investigation of thermohydrodynamic behaviour of gas foil journal bearing accounting slip-flow phenomenon. J. Braz. Soc. Mech. Sci. Eng. 44(1), 1–19 (2021). https://doi.org/10.1007/s40430-021-03330-9

The Dynamic Equilibrium Surfaces Calculation in the Liquid Friction Conical Bearings A. Yu. Korneev1(B) , Shengbo Li2 , A. Yu. Koltsov1 , E. V. Mishchenko3 , and L. A. Savin1 1 Orel State University Named After I.S. Turgenev, Orel 302030, Russia

[email protected]

2 Xiamen University of Technology, 600 Ligong Road, Jimei District, Xiamen 361024, China

[email protected] 3 Orel State Agrarian University Named After N.V. Parakhin, Orel 302019, Russia

Abstract. The dynamic equilibrium surfaces calculation in the liquid friction conical bearings is considered on the basis of pressure fields and reactions in fluid film. The results of the numerical simulation of stable balance position with different parameters of bearing are presented. The comparative analysis of theoretical and experimental data is carrying out. Keywords: Liquid Friction Conical Bearing · Dynamic Equilibrium Surfaces · Stability of Motion · Dynamic Coefficients · Bearing Parameter · Load-Carrying Capacity · Mathematical Model · Experimental Results

1 Introduction When designing and calculating any rotor system an important point should be taken into account: to ensure the stable rotor operation in journal bearings. The number of published works on this subject in modern literature is relatively small and the majority of them considers the problem of determining the stability of the rotor on radial bearings. It is closely related to the definition of the dynamic equilibrium curve, which is one of the basic elements of the force and dynamic analysis of liquid friction bearings. The position of the shaft in various operating modes is schematically shown in Fig. 1. As noted in works on the hydrodynamic theory of lubrication [1–4] during the start-up period at low rotation speed and the predominance of boundary lubrication, the shaft moves to the side opposite the rotation to an angle ϕ, the tangent of which is equal to the friction coefficient of the boundary lubrication (Fig. 1, a). As the rotating speed increases, semi-liquid lubrication occurs and it leads to the decrease of friction coefficient. So the shaft journal moves under the action of the circumferential component of the lubricating layer reaction to another half-plane in the direction of rotation, occupying a certain position on the dynamic equilibrium curve. In this case, the microroughnesses of the shaft and the bearing are separated [1–4]. In the regions of liquid lubrication (Fig. 1, b) the position of the center of a balanced rotor on journal bearings with a static radial load is determined by the dynamic © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 245–252, 2024. https://doi.org/10.1007/978-3-031-40455-9_20

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equilibrium curve [4]. Dynamic equilibrium curves represent the trajectories of the shaft center movement in the lubricating layer, starting from the position corresponding to zero rotation speed to a point close to the center of the bearing (Fig. 1).

Fig. 1. The position of the shaft in the bearing: a – start-up period, b – liquid lubrication mode, c – dynamic equilibrium curves

These curves represent the locus of points relative to the stable equilibrium of the rotating rotor under the influence of external forces and reactions of the bearing lubricating layer. In other words, they describe the position of the center of the shaft at various values of the load parameter λ = μω/k, where μ is the dynamic viscosity (Pa·s), ω – angular velocity (rad/s), k – specific load (Pa), defined as: k = P/dl, where P – external load, d and l are the diameter and length of the bearing, respectively. With the increase of the parameter λ, the center of the rotor moves towards the center of the bearing along the trajectory close to a semicircle with a diameter approximately equal to the average radial clearance h0 [1–4]. With an infinitely large value of the parameter λ and ideal operating conditions, the center of the rotor coincides with the center of the bearing. But, in this case, the thickness of the lubricating layer becomes equal to the average radial clearance (hmin = h0 ), which accordingly leads to the disappearance of the wedge-shaped gap, and, as a result, the absence of pressure in the lubricating layer. Such an idealized state can occur in the absence of external load F [1–4]. Relative thickness of the lubricating layer h is called the minimum thickness hmin of the lubricating layer at the point of closest approach between the shaft and the bearing, referred to the average radial clearance h0 = h = hmin /h0 . The trajectories of the shaft movement in the bearing are shown in Fig. 1, c. For l/d = ∞, the trajectory is a regular semicircle (Humbel semicircle). At finite values of l/d, the trajectory is modified remaining generally close to a semicircle [4].

2 Modelling the Rotor-Bearing System In accordance with the accepted approach to the linearization of hydrodynamic reactions [6–11], a dynamic model of a rotor-support unit on a conical bearing is presented in the form of a rotor supported by a system of springs and dampers (Fig. 2). The angular movements of the rotor are not shown for sake of simplicity, but can be taken into account by introducing additional angular stiffnesses.

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As mentioned above, the equilibrium position of a balanced rotor on liquid friction bearings with a stationary load is determined by the dynamic equilibrium curve [6–11], i.e. the locus of points describing the position of the rotor center at various combinations of load and speed (Figs. 3 and 4).

Fig. 2. Dynamic model rotor-support unit on a conical bearing

In this case the rotor has a conical surface, i.e. each section has its own dynamic equilibrium curve. By calculating a given number of dynamic equilibrium curves for each specific section with a certain step along the length of the bearing, one can get a family of dynamic equilibrium curves in three-dimensional place, so called threedimensional dynamic equilibrium surfaces (Fig. 4). The dynamic equilibrium curve is found by varying the load parameter G = μωD * L/mg (D* – the current diameter of the bearing), achieved by changing the rotor speed ω at a constant value of gravity force (mg = const), the specific form of which depends on the geometric and operating parameters of the bearing, lubricant, etc. Each point (the point of static equilibrium) of this surface will correspond to a certain combination of axial and radial loads and rotational speed. In this case, the external load is balanced by the reactions of the lubricating layer. To find the dynamic characteristics of the lubricating layer, it is necessary to know the stationary position of the rotor, i.e. construct a dynamic equilibrium curve. Curve calculation results, including three-dimensional ones, with the help of special programs for a conical bearing lubricated with water, liquid hydrogen and turbine oils TP-22 and TP-30 at various angular speeds of rotation (rad/s) are shown in Figs. 3 and 4 [12]. Analyzing the above plots, it can be noted that with the increase of the lubricant viscosity the point of dynamic equilibrium moves to the bearing center and it fully confirms the classical theory. When lubricated with water, an increase in the angular velocity leads to a significant displacement of the dynamic equilibrium point towards the center of the bearing, while when lubricated with liquid hydrogen, the rotor practically does not float, i.e. “lies” on the bearing (Fig. 3, a, b). When lubricated with liquid hydrogen with a positive axial displacement of the rotor, the necessary bearing capacity is not provided, as a result of which only a limited number of dynamic equilibrium surfaces can be plotted. In the case of lubrication with turbine oils, the rotor is located

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Fig. 3. Dynamic equilibrium curves in conical bearing lubricated with: a – water; b – liquid hydrogen; c – turbine oil TP-22; d – turbine oil TP-30

near the geometric center of the bearing even at low angular speeds, thus providing the required load-carrying capacity (Fig. 3, c, d, Fig. 4, b).

Fig. 4. Dynamic equilibrium surfaces in conical bearing lubricated with: a – water; b – turbine oil TP-22

What makes the calculation of conical bearings different is, in particular, the fact that both radial and axial stability of the rotor movement is required to consider. In this case, the task of calculating the lubricating layer reactions RZ is connected by means of pressure fields with the determination of the lubricating layer reactions RX and RY . Radial stability can be considered by analogy with the classical type definition of stability for the case of a radial (cylindrical) bearing. To do this, a section is selected,

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for example, the middle one (n-n), for which a calculation scheme is drawn up (Fig. 5). In addition to radial stability, an important point is the preservation of axial stability, i.e. the bearing capacity in the axial direction must compensate the external load F a . The specified clearance in the conical bearing in the axial direction must provide the necessary load-carrying capacity, it is selected based on technological considerations.

Fig. 5. Calculation scheme and pressure profile in conical bearing

Each point (static equilibrium point) of the dynamic equilibrium curve will correspond to a certain combination of external load and angular velocity. In this case, the external load in the radial bearings is balanced by the lubricating layer reactions (RX and RY ) and, therefore, the equilibrium condition can be written as:  WR = R2X + R2Y = FR + mg, WA = RZ = FA . where W R and W A – the radial and axial load-carrying capacities, respectively. In this case, the lubricating layer reactions are determined by the following ratios: ¨ ¨ ¨ RX0 = p sin β cos(α/2)dF; RY0 = p cos β cos(α/2)dF; RZ = p sin(α/2)dF, F

F

F

where p – pressure; β – the angular coordinate in the circumferential direction; α – the cone angle; F – the area of the supporting surface; index “0” indicates a stationary state. On the basis of the approach based on the theory of small oscillations, the perturbed rotor states are considered as some deviations from the dynamic equilibrium curve. For this purpose, characteristic equations are composed, the solution of which makes it possible to define the stability of the system’s motion. Using the dynamic equilibrium curve, the stiffness and damping coefficients can also be calculated, however, unlike

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radial bearings, the dynamic coefficients of conical bearings are represented by threedimensional matrices [5]: ⎡ ⎤ ⎡ ⎤ KXX KXY KXZ BXX BXY BXZ K = ⎣ KYX KYY KYZ ⎦; B = ⎣ BYX BYY BYZ ⎦, KZX KZY KZZ BZX BZY BZZ which considerably complicates the modal and nonlinear analysis of the rotor systems dynamics.

3 Results and discussion To carry out a complex of theoretical calculations, special software was developed for calculating the static characteristics of conical externally pressurized bearings [13]. By varying the operating (pressure in the lubricating layer, angular velocity) and geometric (radial eccentricity) parameters, the radial load-carrying capacity is calculated and this value is compared with the load per bearing (value mg/2). If these values are equal, it can be stated that the initial value of the eccentricity and position angle of the rotor’s center indicates the location of dynamic equilibrium point. Analyzing the results of theoretical calculations, we obtain a family of dynamic equilibrium points and their connection by a smooth curve gives the dynamic equilibrium curve (Fig. 7).

Fig. 6. Calculation scheme (a) and 3-D model (b) of the rotor–conical bearing system

The simplifications used in this problem consisted in the fact that the dynamic equilibrium curves were constructed for one bearing in the rotor zero movement in the axial direction, i.e. in equilibrium in the axial direction. To confirm the results of theoretical calculations, a complex of experimental studies was carried out. For this purpose, an experimental rig was designed to study the dynamics of externally pressurized conical bearings [14–17]. The experimental rig includes a rotor-support unit based on an externally pressurized conical bearing, the design scheme (a) and the three-dimensional model (b) which is shown in Fig. 6 [14–17]. To measure and control the position of the rotor, two displacement sensors are mounted in the experimental rig, which are non-contact sensors ST-2-U-0500. To register the movement of the rotor in the conical bearing, the sensors are installed in the horizontal (along the X axis) and vertical (along the Y axis) directions at an angle of 90° to each other.

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Fig. 7. Dynamic equilibrium curves of the rotor at supply pressure: a – 0.16 MPa; b – 0.26 MPa; c – 0.36 MPa

Processing the results of experimental studies carried out with the same operating and geometric parameters as in the computational experiment, made it possible to obtain the position of the rotor in the bearing. The comparison of the theoretical and experimental calculation results is presented in the series of plots devoted to the study of the dynamic equilibrium curves of the rotor in the bearing gap (Fig. 7). The results obtained by changing the lubricant supply pressure (p0 = 0.16…0.36 MPa) and the rotation speed (n = 0…7500 rpm). The analysis results of the plots show that the supply pressure of 0.16 MPa is not enough to lift the shaft, while a further increase in the lubricant supply pressure leads to the floating of the rotor in the absence of rotation. In this case, the dynamic equilibrium curves take on a shape close to a semicircle. At the same time, it can be noted that an increase in pressure leads, on the one hand, to compression of the semicircle, and, on the other hand, to a state where the semicircle approaches to take on a correct shape.

4 Conclusions The analysis of the given trajectories shows that an increase in the lubricant supply pressure and the rotor speed correctly affects the rotor position stability – it takes a more stable position, i.e. it is located closer to the center of the bearing. Comparative analysis of theoretical (solid line) and experimental (dash-line) results shows a qualitative agreement between them. The calculating principle of dynamic equilibrium surfaces used in this article is quite general and allows, in addition to conical bearings, to calculate dynamic equilibrium surfaces for bearings of various geometries, which can be used in further calculations of rotor bearing units [18]. Acknowledgement. This project is supported by Fujian Provincial Natural Science Foundation, China (№ 2022J011249).

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References 1. Korovchinsky, M.V.: Theoretical Principles of the Operation of Journal Bearings. Mashgiz, Moscow (1959) 2. Voskresensky, V.A., Dyakov, V.I.: Calculation and Design of Sliding Bearings (Liquid Lubrication). Mashinostroenie, Moscow (1980) 3. Rippel, G.: Design of Hydrostatic Bearings. Mashinostroenie, Moscow (1967) 4. Orlov, P.I.: Principles of Design, vol. 2. Mashinostroenie, Moscow (1988) 5. Korneev, A.Yu., Savin, L.A., Solomin O.V.: Conical Liquid Friction Bearings. Mashinostroenie-1, Moscow (2008) 6. Poznyak, E.L.: Oscillations of rotors. Vibrations in engineering. Oscillations of Machines, Structures and Their Elements, vol. 3, pp. 130–189. Mashinostroenie, Moscow (1980) 7. Rao, J.S.: Rotor dynamics comes of age. In: Sixth International Conference on Rotor Dynamics: Proceedings, vol. 1, pp. 15–26. The University of New South Wales, Sydney, Australia (2002) 8. Yamamoto, T., Ishida, Y.: Linear and Nonlinear Rotordynamics: A Modern Treatment With Applications. John Wiley & Sons, New York (2001) 9. Korneev, A.Yu.: On the issue of the rotors stability on the conical bearings. In: International Scientific and Technical Symposium. 120 Years of the Hydrodynamic Theory of Lubrication, pp. 588–596. Orel (2006) 10. Lund, J.W.: Development of the concept of dynamic coefficients of liquid friction radial bearings. Probl. Frict. Lubr. 1, 40–45 (1987) 11. Korneev, A.Yu., Solomin, O.V., Alekhin, A.V.: Generalization of the concept of the dynamic coefficients of the lubricating layer to the liquid friction conical bearings. In: Proceedings of the VI International Scientific and Technical Conference. Vibration Machines and Technologies, pp. 132–135. Kursk (2003) 12. Koltsov, A., Korneev, A., Savin, L.A., Li, S.: Dynamic equilibrium surfaces for conical fluidfilm bearings. IOP Conf. Ser. Mater. Sci. Eng. 233(1), 012041 (2017) 13. Savin, L.A., Solomin, O.V., Korneev, A.Yu., et al.: The software for calculating the characteristics of journal bearings with cryogenic lubrication “Podshipnik-Cryogen”. Certificate of official registration of the software No. 2000610593. Registered in the Register of software (2000) 14. Korneev, A.Yu., Li, S.: Experimental rig for the study of conical bearings with an MR-damper. In: Proceedings of the IV International Scientific Symposium “Shock-Vibration Systems, Machines and Technologies”, pp. 155–164. Orel (2010) 15. Li, S.B., Ao, H.R., Jiang, H.Y., Korneev, A., Savin, L.A.: Steady characteristics of the waterlubricated conical bearings. J. Donghua Univ. Engl. Ed. 2(29), 115–122 (2012) 16. Li, S.B., Ao, H.R., Jiang, H.Y., Chen, L., Korneev, A.: Lubrication characteristics of deep cavity hybrid conical bearing. J. Harbin Inst. Technol. 1(45), 60–66 (2013) 17. Li, S.B.: Study of Dynamic Characteristics of Rotor System with Metal Rubber Ring and Conical Bearing Combined Support. Dissertation for the Doctoral Degree in Engineering, Harbin Institute of Technology, China (2012) 18. Korneev, A., Savin, L.A.: Features of the design calculation of the liquid friction conical bearings. Fundam. Appl. Probl. Eng. Technol. 3(299), 3–8 (2013)

Early Warning Signal Based on Global Dynamics for Instability Responses in Rotor/Stator Rubbing System Xinxin Dong1 , Zigang Li1(B) , Ling Hong2 , and Jun Jiang2 1 College of Sciences, Xi’an University of Science and Technology, Xi’an 710054, China

[email protected] 2 State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong

University, Xi’an 710049, China

Abstract. In a rotor system with a tiny clearance between blade and stator, the undesired dry friction backward whirl may be suddenly aroused by the global bifurcation to cause a destructive response up to clearance tens of times. To detect in advance the resulting sharp increase in amplitude, the evolution of Poincaré return points of response time-series is proposed as an index of stability in this paper to identify if the dangerous bifurcation happen. The uptrend of the proposed index in the Poincaré space indicates the change of response stability. Focusing on a disturbed rotor-rubbing system, the global dynamics before dry friction backward whirl occurs is investigated to reveal the mechanism of the proposed early warning signal. By setting multi-level warning thresholds, the signal can work nicely prior to rubbing happen when the dry friction backward whirl is coming on to avoid irreparable accidents as much as possible. Keywords: Early warning · Global dynamics · Rotor rubbing · Critical slowing down · Multi-level warning threshold

1 Introduction In a rotor system with a tiny clearance between blade and stator, the dry friction backward whirl of rotor may be coexisting in state space even if the system is working at a low rotating speed [1], which buries a potential danger leading to a sudden aggravation on vibration. The undesired responses could be aroused for the case that the essential properties of some parameters, such as support stiffness and structure damping coefficients in the rotor system, change slowly to deviate from the original design indexes as the increase of operating time, causing a destructive response up to clearance tens of times [2]. From the global dynamics point of view, the catastrophe could be induced by the “blue-sky” disappearance in the intrinsic dynamic structure when the system parameter is closing to and passing through a critical value, in other words, the dangerous global bifurcations like crisis happen. Thus, how to detect the amplitude of rotor increases significantly triggered by the global bifurcation is of great concern in engineering. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 253–260, 2024. https://doi.org/10.1007/978-3-031-40455-9_21

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In recent years, the characteristics of system resilience weakening when approaching the bifurcation point have been widely used to predict the critical state [3–5]. The phenomenon can be exemplified by the global structure of dynamical system where an approach between attractor and saddle on the boundary will expect an upcoming catastrophe, indicating that the dominant real eigenvalue on the manifold near the attractor is moving towards zero [3], tending to make response recovering from a small perturbation more sluggish. Meanwhile, a variety of early warning signals (EWSs) detecting the critical transition of different systems have also been derived from the principle of critical slowing down [6], which played a certain application value in predicting climate change [7], financial crashes [8], psychological depression [9], and epileptic seizure [10], etc. Previous researches suggest that increasing lag-1 autocorrelation (AR(1)) in time series is one of the well-known early warning signals [3, 11] because its “historical memory” can reflect that the state of the system at any given moment becomes more and more similar to its past state, characterizing the slow-down before a tipping point is reached. Nevertheless, the accuracy and reliability of the indicator are susceptible to insufficient historical data on response, filtering bandwidth, size of rolling windows, and disturbance intensity [9, 12, 13]. In global dynamics, the resilience loss approaching the bifurcation point can be characterized by the changes of underlying global structures. For example, the manifolds surrounding the attractor where a global bifurcation is about to occur will be severely distorted to slow down the movement of the attractor, which can be observed in the stroboscopic patterns. Thus, we propose a novel early warning indicator from the perspective of global dynamics. This paper focuses on a disturbed rotor-rubbing system whose responses will transition from a non-rubbing motion to another dry friction backward whirl as the system parameters change slowly. The global dynamics of the system are first investigated to identify the response’s resilience loss when its dynamics closes to the bifurcation. The evolution of a defined Poincaré return point of time series is proposed to measure the dynamical state of system. The uptrend of the proposed index in the Poincaré space is treated as an early warning signal to detect changes in response stability. Meanwhile, by setting multi-level warning thresholds, the signal can work nicely prior to rubbing occurring when the dry friction backward whirl is approaching, which can gain more time and opportunity for the early diagnosis and protection of such faults.

2 Global Dynamics in Rotor/Stator Rubbing System 2.1 Rotor/Stator Rubbing Model The considered rotor/stator rubbing system [1, 2] is characterized with a piecewise contact stiffness and a piecewise Coulomb dry friction, as illustrated in Fig. 1. The non-dimensional governing equations are written in the form: ⎧ R ⎨ X¨ + 2ζ X˙ + βX + (1 − R0 )(X − μsign(Vrel )Y ) = 2 cos(t) + εW1 (t) R (1) Y¨ + 2ζ Y˙ + βY + (1 − R0 )(μsign(Vrel )X + Y ) = 2 sin(t) + εW2 (t) ⎩ Vrel = Rdist  + Rωb

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Fig. 1. Schematic diagrams of a rotor/stator rubbing system: (a) Jeffcott rotor with a stator clearance; (b) section view of rotor/stator rubbing system with rubbing forces.

where X = √ x/e, Y = y/e are the dimensionless deflections of the rotor, β = ks /kb and ζ = c/2 kb m denote, respectively, the stiffness ratio and the damping ratio, μ is the friction coefficient, c is the damping coefficient of the rotor, m is the unbalanced mass of the rotor, kb is the stator stiffness, ks is the stiffness of the rotating shaft; vrel represents the relative velocity between the rotor and the stator at the contact point, ωb is the dimensionless frequency ratio of the whirling frequency and natural frequency, Rdisk is the disk radius, e is the mass eccentricity,  is the rotating speed of the rotor; is Gaussian white noise. In the ε presents the noise intensities and Wi (t)(i = 1, 2) √ cartesian coordinate system, the rotor amplitude R = X 2 + Y 2 and R0 = r0 /e is the initial clearance between the rotor and the stator.  is the Heaviside function to represent the judgment criterion of rubbing. When R < R0 ,  = 0, no rubbing behavior occurs. When R ≥ R0 ,  = 1, rubbing motion occurs. 2.2 Global Dynamic Structures with Slow-Varying Parameters For the deterministic case, namely ε = 0, global responses of the rotor system when ζ = 0.05,  = 0.75, μ = 0.15 is drawn in Fig. 2. The bifurcation diagram of Fig. 2a with the variation of stiffness β shows the coexistence of the no-rub motion and the backward whirl appearing in the range 0.01 < β < 0.0318. When the β is passing through the critical value 0.0318, the global bifurcation occurs, indicating the disappearance of the no-rub attractor (red dots in Fig. 2a) together with its basin of attraction, which means that even if the initial state of rotor is in the no-rub attractor, the catastrophic tipping may be triggered by the global bifurcation to force the transition of response from the no-rub motion with low amplitude (Fig. 2b) to the backward whirl with high amplitude oscillation (Fig. 2c). The resilience of response is closely associated with the shape and size of basins of attraction, which can be measured by the integrity factor (IF) [14]. Here, the resilience loss caused by slowly varying parameters is detected by the varying IFs. Figure 3a exhibits changes in the shape of basin of attraction for no-rub with the variation of β and a clear downward trend of the corresponding IF factor is presented in Fig. 3b. The IF tends to zero when the rotor system is in β = 0.03 with low resilience, indicating the impending occurrence of instability response. We intervene in the rotor system with β = 0.01 and β = 0.03 at a time point t = 4 × 104 and the response recovering to equilibrium from a small external perturbation is depicted in Fig. 4. As can be observed, the system

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with lower resilience tends to make response recovering from a small disturbance more sluggish, whereas the more resilient system recovers swiftly (at about t = 4.05 × 104 system response returns to steady state again).

Fig. 2. Global responses of the rotor system when ζ = 0.05,  = 0.75, μ = 0.15, R0 = 1.05, Rdisk = 20R0 : (a) global bifurcation diagram against the parameter β; (b) and (c) indicate the two-coexisting steady-state responses when 0.01 < β < 0.0318, that is, no-rub motion and dry whip backward whirl, respectively.

Fig. 3. The shape and size of basin of attraction for no-rub attractor: (a) 3D projection of the basin of attraction; (b) the integrity factor (IF) with the variation of β.

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Fig. 4. Response in the rotor system recovering to equilibrium from a small perturbation at a time point t = 4 × 104 with different parameters: (a) β = 0.01; (b) β = 0.03.

3 Multi-level Warning Indicator In this section, the Poincaré mapping is adopted in transforming the response of rotor collision with the periodicity into equilibrium problems in order to capture the movement of the response trajectory in the Poincaré space as the parameter slowly changes more efficiently. Therefore, the indicator sequence in continuous periods is defined as follow:   ξi = X(ti+1 ) − X(ti )2 = (P(X 0 ; ti+1 ) − P(X 0 ; ti ))2 , i ∈ (1, 2, 3 · · · ) (2) where X(t) denotes the state variable at time t and P(·) is an operator representing the Poincaré mapping, which can be implemented by stroboscopic or phase-space reconstruction. This expression represents the spatial Euclidean distance of the response trajectory in the Poincaré space in continuous time.

Fig. 5. Response trajectory in the Poincaré space under slowly varying parameter β: (a) Variations of parameter β with stepwise changes in the interval β ∈ [0.01, 0.05] and β = 0.0001 (b) The response trajectory initially starts from the attractor for no-rub (red star) and going through the bifurcation (green star).

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In this paper, the evolution of the Poincaré return points of a time-series with the variation of β in Fig. 5 is proposed to reveal the onset of transition from the no-rub motion with low amplitude to the dry friction backward whirl with large amplitude. From the enlarged view in Fig. 5b, the movement of the response trajectory in the Poincaré space initially surrounding the no-rub attractor (red star) demonstrates that the fluctuation of indicator ξ is minimal as the parameter gradually towards critical point (green star), stemming from its loss of the resilience of response (as in Fig. 4b where the IF factor tends to zero). When the bifurcation (green star) occurs, the Poincaré return points deviate from no-rub attractor with a large step towards the attractor of dry friction backward whirl dramatically, in this case the system will no longer back to no-rub motion eventually and correspondingly the fluctuation of indicator ξ will increase. Therefore, the rising trend of indicator ξ can be treated as an early warning signal for the dry friction backward whirl with high-amplitude oscillation. Furthermore, the changing trend of indicator ξ is extracted to the maximum possible extent by means of logarithmic transformation and the Savitzky-Golay filter. In order to improve the warning sensitivity, the multi-level warning threshold is set up in accordance with the Pauta criterion(3σ ), that is, monitoring whether the indicator ξ falls within the following inequality ξi − η > N · σ

(3)

where η and σ stand for the mean and variance of the indicator sequence ξ under normal conditions, as well as N stands for the warning threshold, N = 1, 2, 3, 4 indicate that the confidence probability of system instability is 68.26%, 95.45%, 99.73% and 99.99%, respectively. The 3rd-level and 2nd-level warning signals should be sent to reduce speed as quickly as feasible when N = 2, 3. On the other hand, the system instability probability converges to 1 when N = 4, suggesting that it is almost inevitable to issue a 1st-level warning signal, and then emergency shutdown or other countermeasures must be taken immediately. The suitable warning level in the project can be adjusted in accordance with practical engineering requirements.

4 Early Warning of Instability Responses in the Rotor System In order to evaluate the results of early warning indicator ξ (2) on the rotor system (1), the numerical simulations are carried out based on the classical fourth-order Runge–Kutta integral scheme with the forward scanning of parameter β changing period by period according to Fig. 5a. The good performance of early warning indicator ξ on warning of instability responses for the backward whirl is verified through the proposed multi-level warning method (3), as shown in Fig. 6 and Table 1. For the deterministic case, namely ε = 0, the obvious upward trend of the characteristic indicator ξ is identified prior to the onset of significant amplitude oscillation (t2 = 2624.6), as illustrated in Fig. 6a (Bottom). Moreover, the three-level warning is accurately executed before the occurrence of large amplitude oscillation and even the amplitude of rotor is still in a normal state when the highest-level warning signal (t1 = 1842.9) is issued (Table 1 displays the transitional interval up to 93 cycles). As shown in Fig. 6b and Fig. 6c, in the case of simulated noise disturbances, despite the

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fact that the high-amplitude oscillations occur earlier, the transition intervals are shortened, and the warning indicators ξ fluctuate, the multi-level warning signal still achieve warning effectiveness before the large oscillation emerge. Furthermore, as presented in Table 1, the 1st-level transition interval for warning instability responses of the backward whirl is up to 52 cycles at a disturbance capacity of 0.1% of the normal amplitude, preserving sufficient time and opportunity for the early diagnosis and protection of such faults.

Fig. 6. Rotor response and the warning signal under different ε: (a)ε = 0; (b)ε = 0.0005; (c)ε = 0.001. Top: Time history and Poincaré mapping as β increases with stepwise changes in the interval β ∈ [0.01, 0.05] and β = 0.0001. Bottom: The uptrend of early warning indicator ξ , where the red line t1 indicates the time when a 1st-level warning is issued, t2 indicates the time when the large oscillation caused by backward whirl occurs and yellow, purple, green horizontal dashed line stand for 3rd-level(η + 2σ ), 2nd-level(η + 3σ ), 1st-level(η + 4σ ) warning threshold, respectively.

Table 1. The early warning performance of multi-level indicator. noise intensity

the time when warning signal is issued (the transition interval) 3rd-level

2nd-level

large amplitude oscillation (t2 )

1st-level (t1 )

0

1837.5(93)

1841.1(93)

1842.9(93)

2624.6

0.0005

1622.2(69)

1629.3(68)

1634.8(68)

2204.5

0.001

1267.6(54)

1276.1(53)

1285.9(52)

1726.2

5 Conclusion The early warning indicator proposed only relies on the real-time vibration response series and can achieve the warning before the amplitude of the rotor motions increase dramatically via mathematical transformation and some other algorithms. Moreover, the

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suggested early warning method is based on the global dynamic structure for instability responses in the rotor/stator rubbing system, and in this framework, the signal and strategy are derived from the generalized dynamic laws. Therefore, even with varying scale and components in specific rotors, the adaptability and reliability of the indicator has not been lost. As compared to the existing conventional detections such as amplitude, frequency, and phase of the rotor, it offers the advantages of being more sensitive, more reliable, and earlier. However, Detection of the patterns in real data is challenging and may lead to false positive results as well as false negatives. More research is needed to find out how robust these signals in settings where spatial complexity, chaos and stochastic perturbations govern the dynamics. Nonetheless, Our work may provide some new ideas for the early warning of catastrophic high-amplitude oscillations in rotor systems and rotor engineers in preventing the occurrence of catastrophic critical jump phenomena in engineer applications.

References 1. Jiang, J.: Determination of the global responses characteristics of a piecewise smooth dynamical system with contact. Nonlinear Dyn. 57(3), 351–361 (2009) 2. Li, Z., Jiang, J., Hong, L.: Noise-induced transition in a piecewise smooth system by generalized cell mapping method with evolving probabilistic vector. Nonlinear Dyn. 88(2), 1473–1485 (2017) 3. Scheffer, M., Bascompte, J., Brock, W.A., et al.: Early-warning signals for critical transitions. Nature 461(7260), 53–59 (2009) 4. van Nes, E.H., Scheffer, M.: Slow recovery from perturbations as a generic indicator of a nearby catastrophic shift. Am. Nat. 169(6), 738–747 (2007) 5. Arnoldi, J.-F., Bideault, A., Loreau, M., et al.: How ecosystems recover from pulse perturbations: a theory of short- to long-term responses. J. Theor. Biol. 436, 79–92 (2018) 6. Scheffer, M., Carpenter, S.R., Lenton, T.M., et al.: Anticipating critical transitions. Science 338(6105), 344–348 (2012) 7. Dakos, V., Scheffer, M., van Nes, E.H., et al.: Slowing down as an early warning signal for abrupt climate change. PNAS 105(38), 14308–14312 (2008) 8. Ismail, M.S., Hussain, S.I., Noorani, M.S.: Detecting early warning signals of major financial crashes in bitcoin using persistent homology. IEEE Access 8, 42–57 (2020) 9. Dablander, F., Pichler, A., Cika, A., et al.: Anticipating critical transitions in psychological systems using early warning signals: theoretical and practical considerations. Psycholog. Meth. (2022) 10. McSharry, P.E., Smith, L.A., Tarassenko, L.: Prediction of epileptic seizures: are nonlinear methods relevant? Nat. Med. 9(3), 241–242 (2003) 11. Ghanavati, G., Hines, P.D.H., Lakoba, T.I., et al.: Understanding early indicators of critical transitions in power systems from autocorrelation functions. IEEE Trans. Circuits Syst. I Regul. Pap. 61(9), 2747–2760 (2014) 12. Lenton, T.M., Livina, V.N., Dakos, V., et al.: Early warning of climate tipping points from critical slowing down: comparing methods to improve robustness. Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 370(1962), 1185–1204 (2012) 13. Nazarimehr, F., Jafari, S., Hashemi Golpayegani, S.M.R., et al.: Predicting tipping points of dynamical systems during a period-doubling route to chaos. Chaos Interdisc. J. Nonlinear Sci. 28(7), 073102 (2018) 14. Azami, H., Mohammadi, K., Bozorgtabar, B.: An improved signal segmentation using moving average and Savitzky-Golay filter (2012)

Research on Rub-Impact Fault Quantification of Rotor System Based on Effective Singular Value Noise Reduction and Minimum Mutual Entropy Principle Jintao Li1,2 , Zhaobo Chen1,2(B) , and Dong Yu1,2 1 School Mechatronics Engineering, Harbin Institute Technology, Harbin 150000, Heilongjiang,

China [email protected] 2 Lab Vibration and Noise Control, Harbin Institute Technology, Harbin 150000, Heilongjiang, China

Abstract. The improved effective singular value noise reduction algorithm is able to reduce the noise of the signal while retaining as many effective frequencies of the signal as possible. Meanwhile, the difference between two signals is calculated by the minimum mutual entropy principle. Therefore, the improved effective singular value noise reduction algorithm and the minimum mutual entropy principle are combined to quantify the rotor system rub-impact faults. Firstly, the obtained signals are performed noise reduction using the improved algorithm. Secondly, the difference between signals in the healthy state and the signal in the fault state of the rotor system is calculated by the minimum mutual entropy principle. Finally, the faults of the rotor system are quantified based on the difference between signals. Keywords: Effective singular value noise reduction · Minimum mutual entropy principle · Rotor system · Fault quantification

1 Introduction When the signal is analyzed, if there is interference from other signals, it will affect the final analysis results and cause unnecessary costs, so we expect that there is no other interfering information in the signal. Therefore, it is extremely important to perform noise reduction on the signal. Noise reduction algorithms are used to obtain a clearer signal waveform by removing the interference information from the original signal. Wavelet packet, empirical mode decomposition (EMD) and singular value decomposition are commonly used. Klionskiy [1] et al. illustrated denoising using soft and hard thresholding with EMD. The results show that EMD is an effective noise reduction tool for both homoscedastic and heteroscedastic noise examples. Yuan [2] et al. proposed Ensemble noise-reconstructed empirical mode decomposition (ENEMD) to overcome the shortcomings of the original ENEMD and its derivative methods in terms of high accuracy with critical noise © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 261–272, 2024. https://doi.org/10.1007/978-3-031-40455-9_22

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estimation, and more accurately extracted multiple and weak fault features of rotating machinery. He [3] et al. adopted EMD to decompose the weld acoustic emission signal adaptively into several Intrinsic Mode Functions (IMFs), then selected effective IMFs based on the distribution frequency of the weld crack acoustic emission signal, and finally constructed wavelet packets based on the effective IMFs for noise reduction of the original weld acoustic emission signal. Deng [4] et al. proposed a high-order threshold function wavelet packet noise reduction algorithm for chaotic signals. Based on the continuity and derivability of the threshold function, this method is more consistent with the continuous signal and retains useful high-frequency information. Yang [5] et al. proposed the concept of singular entropy (SE) based on singular value decomposition (SVD). Based on SE, the distribution characteristics of noise and pure signal have been discovered, so that the noise reduction method is proposed. The results show that the method can handle both smooth and non-smooth signals. Golafshan [6] et al. performed noise reduction of rolling bearing vibration signals based on SVD and Hankel matrix. After eliminating the background noise, the reliability of fault detection was improved. Jiang [7] et al. introduced the ratio of adjacent singular values for feature extraction of rolling bearing vibration signals, and the feature was automatically classified by continuous hidden Markov model (CHMM). However, SVD is superior to EMD and wavelet packet transform in the purification of rotating mechanical axial trajectories [8]. The core of SVD noise reduction is to compute the effective singular values among all singular values of the original signal. Yang [5] et al. used the SE method for calculating the number of effective singular values, Lv [9] used the signal-to-noise ratio empirical method for calculating it, and Zhao [10] et al. used the difference spectrum method for automatically calculating it. However, the SE method requires analysis of the entropy change, and the signal-to-noise ratio empirical method relies on experience. Neither of them can automatically calculate the number of effective singular values. The differential spectrum method is able to calculate it automatically based on the peak of the spectrum, but the method fails for signals with similar amplitudes. To address the above problems, the parameter  is proposed to optimize the differential spectrum method. The Hankel matrix is constructed from the original signal, then all singular values are calculated. During iteration, the valid singular values are automatically identified among these singular values using the improved difference spectrum method. Furthermore, the discrepancies between the noise-reduced signals are performed by adopting the minimum mutual entropy principle (MMEP) [11]. Finally, the rotor faults are quantified based on these discrepancies.

2 Related Theories 2.1 SVD The Hankel matrix is used for SVD with low distortion and can handle high signal-tonoise ratio [12], so the Hankel matrix is to be constructed from the original signal x(t), which is shown in Eq. (1). The method of taking the Hankel matrix dimension (m, n)

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will be introduced in Sect. 2.3. ⎡ x(1) ⎢ x(2) ⎢ H=⎢ . ⎣ ..

x(2) x(3) .. .

x(3) x(4) .. .

⎤ · · · x(n) · · · x(n + 1) ⎥ ⎥ ⎥ .. .. ⎦ . .

x(m) x(m + 1) x(m + 2) · · ·

263

(1)

x(L)

In Eq. (1), L is the length of the signal x(t), H ∈ Rm×n . The SVD computational equation is shown in Eq. (2). H = UDV T

(2)

In Eq. (2), both U and V are orthogonal matrixes, and U ∈ Rm×m , V ∈ Rn×n ; D ∈ Rm×n is a diagonal matrix, and its form is shown in Eq. (3). ⎧ m n In Eq. (3), r = min(m, n), 0 is the zero matrix. According to the singular value property [8], there are the following relation between σ 1 , σ 2 , …, σ r . σ1 ≥ σ2 ≥ · · · ≥ σr ≥ 0 Rewrite the matrixes U and V in Eq. (2), as follows,  U = [u1 , u2 , · · · , um ] V = [v1 , v2 , · · · , vn ]

(4)

(5)

In Eq. (5), ui ∈ Rm×1 , vi ∈ Rn×1 . Bringing Eq. (5) into Eq. (2) H = σ1 u1 vT1 + σ2 u2 vT2 + · · · + σr ur vTr

(6)

From Eqs. (1) and (2), there are r singular values in the matrix H. Among these values, the non-zero singular values of a pure signal are effective, but most of the nonzero singular values of a signal with noise are not effective, which result from the noise [13]. The main content of noise reduction using singular values is to perform SVD on the matrix H and identify the frequencies of the pure signal s(t) from the signal with noise, and then extract these frequency components to reconstruct the signal. Each frequency of the signal corresponds to two singular values and they are adjacent in the ordering of all singular values [15]. Therefore, how to precisely identify the number of effective singular values, i.e., the number of frequencies of the signal s(t), becomes the main research of effective singular value noise reduction. The process of reconstructing the signal is shown in Eq. (7). Ai = U i Di V Ti ⎧ ⎨ U i = [u1i , u2i ] ∈ R m×2 D = diag σi1 , σi2 ⎩ i V i = [v1i , v2i ] ∈ Rn×2

(7)

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In Eq. (7), Ai is the reconstructed matrix at the ith frequency of reconstructed signal. The superscripts of σ, u and v denote the corresponding singular values and matrixes at the ith frequency, and the signal s(t) is reconstructed through the matrix Ai using the averaging method [14]. 2.2 MMEP Suppose there are two distributions: a known distribution P and an unknown distribution Q. The distance between these two distributions is denoted as, k 

D(P : Q) =

qi log2 (

i=1

qi ) pi

(8)

In Eq. (8), D is called the pseudo-distance or Kullback_Liebler distance. The optimal qi needs to be found under the given conditions such that D is minimized. Assume that there are k + 1 constraints, the first of which is the full probability. k 

qi = 1

(9)

qi ϕhi = θh , h = 1, 2, ..., l

(10)

i=1

The rest of them are k  i=1

In Eq. (10), θ h is the center distance of this distribution. Construct Lagrange function, L=

k  i=1

qi log2 (

   qi ) + (λ0 − 1)( qi − 1) + λh ( qi ϕhi − θh ) pi k

l

k

i=1

i=1

i=1

(11)

In Eq. (11), find the partial derivative of qi and make it equal to zero,  qi ) + λ0 + λh ϕhi = 0 pi l

log2 (

(12)

i=1

In Eq. (12), qi = pi exp (-λ0 -λ1 ϕ 1i -…-λl ϕ li ), i = 1, 2, …, k. The individual qi can be solved by Eq. (12). The pseudo-distance D has the following properties. (1) D is a function of p and q continuous, and the pseudo-distance D = 0 when p = q. (2) D(g : q) + D(q : p) ≥ D(g : p) (3) When the numerator and denominator are interchanged, the two pseudo-distances are not equal, i.e., D(q : p) = D(p : q). To solve the problem of property (3), define  as =

D(q : p) + D(p : q) 2

(13)

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2.3 Calculation the Number of Effective Singular Value In Eq. (7), the energy of the matrix Ai can be denoted  2   |Ai |2 = U i Di V Ti  2  2  2    2  2         2   2   2,T 2 σ v u = u1i  σi1  v1,T +       i i i i   2  2     = σi1  + σi2 

(14)

 2 In Eq. (14), σi1  denotes the square of the absolute value of the first singular value at the ith frequency, and the other symbolic meanings follow in the same way. The matrices U i and V i are orthogonal matrices, so they satisfy the following relationship ⎧  m   ⎪  j 2  j 2 ⎪ ⎪ ui,f = 1 ui  = ⎪ ⎪ ⎪ ⎪ f =1 ⎪ ⎪ ⎪ ⎪ n    2 ⎪  2 ⎪  j j ⎪ ⎪ vi  = vi,f = 1  ⎪ ⎪ ⎨ f =1 (15) m ⎪  ⎪ ⎪ 1 2 1 2 ⎪ ui · ui = ui,k ui,f = 0 ⎪ ⎪ ⎪ ⎪ f =1 ⎪ ⎪ ⎪ ⎪ n ⎪  ⎪ ⎪ 1 2 1 2 ⎪ v · v = vi,f vi,f = 0 ⎪ i i ⎩ f =1

In Eq. (15), j = 1, 2, it represents the jth u or v vector at the ith frequency. From Eq. (14), the greater the energy of the matrix Ai , the greater the corresponding two singular values. What’s more, the energy is proportional to the square of the signal amplitude [8]. Therefore, the ranking of all singular values is based on the magnitude of the corresponding amplitude at that frequency. As mentioned above, the number of effective singular values can be calculated automatically by the differential spectrum method from the amplitude of the signal (Taking the signal x 1 (t) = 0.46sin(4πt + 3) + 0.58cos(9πt) + 0.64sin(12πt + 2) + n(t) as an example, n(t) is Gaussian white noise). However, the method fails when the difference in the amplitude of the interested frequencies is large or when the energy of the interested frequencies is similar to the energy of the noise signal (Taking the signal x 2 (t) = 0.02sin(4πt + 3) + 0.26cos(9πt) + 0.35sin(12πt + 2) + n(t) as an example). Since one frequency corresponds to two singular values, the pair of singular values is called the one-order singular value here. From Table 1, x 1 (t) shows the step phenomenon of singular values (The difference spectrum method is based on this phenomenon to automatically calculate the number of effective singular values) at the 3rd order to the 4th order, while x 2 (t) shows this phenomenon at from the 2nd order to the 3rd order. In this case, the number of effective singular values of these two signals is 6 and 4 respectively, which corresponds to the number of effective frequencies of 3 and 2. Therefore, the differential spectrum method is invalid for such signal just like x 2 (t).

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2nd

3rd

4th

5th

6th

7th

8th 1.1 × 10−12 5.8 × 10−13

x 1 (t)

481.1

479.4

435.4

433.3

344.4

342.7

1.1 × 10−12

x 2 (t)

262.7

261.9

194.5

194.3

15.0

15.0

5.8 × 10−13

The above problem is solved by optimizing the differential spectrum method. The introduction of the parameter  allows the optimized algorithm to more accurately identify the number of effective singular values of the signal, as shown in Fig. 1.

Fig. 1. The flowchart of improved differential spectrum.

The details of the improved method are as follows, (1) Construct the H matrix using the original signal. First, calculate the length L of the signal; second, based on the parity of L compute the dimension (m, n) of the matrix. (2) Perform SVD on the H. (3) Preprocess these values obtained in Step (2) (The data are calculated differentialby-differential and normalized in this paper) to increase the difference between them. (4) Determine whether there is a frequency number N. If it is Yes, output N; else, assign the data to the parameters ϕ 1 and ϕ 2 . (5) Determine if the condition ϕ 1 >  && ϕ 2 <  is satisfied. If it is Yes, output N = n / 2; else, skip to Step (4). (6) Reconstruct the signal to complete the noise reduction process.

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3 Performance of the Improved Method 3.1 Simulation Results Process the signals in Sect. 2.3 adopting improved method, and the top 8 singular values are listed in Table 2. Table 2. Top 8 singular values of x 1 (t) and x 2 (t) after processing by improved method.

x 1 (t) x 2 (t)

1st

2nd

3rd

4th

5th

6th

7th

8th

0.005

0.128

0.007

0.259

0.005

1.000

≈0

≈0

1.000

6.2 × 10–7

0.083

≈0

≈0

0.004

0.376

0.007

Fig. 2. Time waveform of signal with noisy and s2 (t)

According to Table 2, the parameter  = 0.05 was set ( is a parameter to be determined due to the different methods of data preprocessing and the random nature of the noise). The parameters ϕ 1 and ϕ 2 mentioned in Fig. 1 are explained. The first iteration: ϕ 1 = 0.128, ϕ 2 = 0.259; The second iteration: ϕ 1 = 0.259, ϕ 2 = 1.000; The third iteration: ϕ 1 = 1.000, ϕ 2 = the 8th value; …… Figure 2 shows the time waveform of the signal before and after noise reduction (s2 (t) = x 2 (t) − n(t)). Figure 3 shows the results of noise reduction using the differential spectrum method and the improved method. From Fig. 3 the reconstructed signal obtained by the improved method coincides exactly with s2 (t), while the differential spectrum method does not precisely identify the effective frequency components of the signal, so the reconstructed signal using this method has a small difference (That is 0.02 sin(4πt + 3)) with s2 (t).

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Reconstructed signal

Improved differential spectrum

s2(t)

0.6

Amplitude

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0 0.8

0 .5

1

Differential spectrum

Reconstructed signal

1.5 s2(t)

0.6

Amplitude

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0

0 .5

1

1.5

Time

Fig. 3. Time waveform of reconstructed signal and s2 (t) by improved differential spectrum and differential spectrum

3.2 Experiment Results The rotor vibration experiments were conducted using the rotor test bench demonstrated in Fig. 4 to verify the improved noise reduction method proposed in this paper. The rotor speed is set to 1200 rmp. The parameters of experimental platform are listed in Table 3. The parameter  is set as 0.05 in the improved differential spectrum method. After modal analysis, the first four orders of nature frequencies of the test rig were obtained as 10 Hz, 40 Hz, 97 Hz, and 318 Hz, respectively. Eddy current sensor and BK acquisition system were used to collect the vibration signals of the rotor during the experiment. The data in Table 4 shows that the number of effective frequencies of the experimental signal is 3. The results of noise reduction of this signal using the improved method are shown in Fig. 5. The curves in Fig. 5b can illustrate the conclusion of Table 4. Comparing the results before and after noise reduction, the time waveform after noise reduction shows strong regularity. Figure 5 illustrate the interference signal generated in the experiment has been

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1 2

3

4 7 5

1. Motor

2. Coupling

3. Bearing

6

4. Shaft

5. Rub-impact rod

6. Sensor

7. Bearing

Fig. 4. Experimental platform Table 3. The parameters of experimental platform.

Values

Rotor mass

Polar moment of inertia

Transverse moment of inertia

153 kg

1.2471 kg·m2

35.132 kg·m2

Table 4. Singular values of experiment after processing by improved method. 1st Values

2nd

0.008

1.000

3rd

4th

0.002

5th

0.134

0.002

6th

7th

8th

0.099

2.3 × 10−4

0.007

completely filtered, while the effective frequency components of the experimental signal have been retained as much as possible to lay the foundation for subsequent quantitative fault analysis. Before noise reduction 0.06

a

Amplitude / mm

Amplitude / mm

0.05

0

-0.05 0

0.2

0.4

0.6

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b

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3fr

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fr

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fr

0.04 0.02 0

2fr 0

3fr

Fig. 5. Comparison of experimental signal before and after noise reduction

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4 Fault Quantification Based on Eq. (13), the amplitude of the signal is introduced into it to proposed a new index J measuring pseudo-distance D, its calculation is written in Eq. (16).  L   1   S1l S2l ( J (S1 , S2 ) = + − 2)  2M S2l S1l

(16)

l=1

where, L is the length of the signal, S jl is the lth amplitude of the jth curve ( j = 1, 2). Meanwhile, J has the following properties. 1. The two curves overlap, J = 0. 2. The greater the variability between the curves, the greater the value of J. 3. J(S 1 , S 2 ) = J(S 2 , S 1 ). A quantitative study is performed for the rotor system with rub-impact fault. Three working conditions are set up in the experiment: no fault and two different levels of faults. The time waveforms of these signals are shown in Fig. 6. The amplitudes of these cases after noise reduction are different as can be seen by Fig. 6. In order to avoid the influence of other factors on the fault quantification, the three sets of data are normalized. In addition, from Eq. (16), the more cycles of the signal involved in the calculation of the index J, the greater the difference between the working conditions. In this paper, one signal period is chosen here, and the values of index J are listed in Table 5. Table 5. The values of index J.

Values

J(S 1 , S 1 )

J(S 1 , S 2 )

J(S 1 , S 3 )

0.000

1.052

2.158

Table 5 shows that the differences between the adjacent indexes J are 1.052 and 1.106, respectively, and it can be said that the two differences are approximately the same, i.e., the variation of index J is closer to a straight line with a slope of 1. Therefore, the improved MMEP can better quantify the rotor system rub-impact faults.

-0.02 0

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Fig. 6. Comparison before and after noise reduction. a. Healthy condition b. Fault condition 1 c. Fault condition 2 (fault severity case 2 is greater than case 1)

5 Conclusion To calculate the number of effective singular values of the signal more accurately, the parameter  is proposed to improve the differential spectrum method. Comparing the noise reduction results before and after the enhanced method, it can be seen that the improved method can reconstruct the components of the signal more accurately. Therefore, the collected experimental signals are performed noise reduction by the improved differential spectrum method; then the differences between the signals are calculated by MMEP; finally, the faults are quantified based on the differences. The experimental results show that the method used in this paper can effectively quantify the rub-impact faults of the rotor system.

References 1. Klionskiy, D., Kupriyanov, M., Kaplun, D.: Signal denoising based on empirical mode decomposition. J. Vibroeng. 19(7), 5560–5570 (2017) 2. Yuan, J., Xu, C., Zhao, Q., et al.: High-fidelity noise-reconstructed empirical mode decomposition for mechanical multiple and weak fault extractions. ISA Trans. 129, 380–397 (2022) 3. He, K., Xia, Z., Si, Y., et al.: Noise reduction of welding crack AE signal based on EMD and wavelet packet. Sensors 20(3), 761 (2020) 4. Deng, K., Zhang, L., Luo, M.K.: A denoising algorithm for noisy chaotic signals based on the higher order threshold function in wavelet-packet. Chin. Phys. Lett. 28(2), 020502 (2011) 5. Yang, W.X., Peter, W.T.: Development of an advanced noise reduction method for vibration analysis based on singular value decomposition. NDT & E Int. 36(6), 419–432 (2003) 6. Golafshan, R., Sanliturk, K.Y.: SVD and Hankel matrix based de-noising approach for ball bearing fault detection and its assessment using artificial faults. Mech. Syst. Sig. Process. 70, 36–50 (2016) 7. Jiang, H., Chen, J., Dong, G., et al.: Study on Hankel matrix-based SVD and its application in rolling element bearing fault diagnosis. Mech. Syst. Sig. Process. 52, 338–359 (2015)

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8. Guo, M.J., Li, W.G., Yang, Q.J., Zhao, X.Z.: Purification of the axis trajectory of a rotor supported by sliding bearings based on the number rule of effective singular values. J. Vibr. Shock 38(22), 155–161 (2019). (in Chinese) 9. Lv, Y.L., Lang, R.L., Liang, J.C.: Decision of threshold for singular value decomposition filter based on SNR’s empirical value. Appl. Res. Comput. 26(9), 3253–3255 (2009). (in Chinese) 10. Zhao, X.Z., Ye, B.Y., Chen, T.J.: Difference spectrum theory of singular value and its application to the fault diagnosis of headstock of lathe. J. Mech. Eng. 46(1), 100–108 (2010). (in Chinese) 11. Li, J.T., Lu, H.H., Feng, K.P., Liu, Y., Zhao, Y.L.: Research on a new diagnosis index for fixed-point rub-impact of rotor system. Eng. Fail. Anal. 125, 105394 (2021) 12. Zhao, X., Ye, B.: Similarity of signal processing effect between Hankel matrix-based SVD and wavelet transform and its mechanism analysis. Mech. Syst. Sig. Process. 23(4), 1062–1075 (2009) 13. Zhao, X.Z., Nie, Z.G., Ye, B.Y., Chen, T.J.: Number law of effective singular values of signal and its application to feature extraction. J. Vibr. Eng. 29(3), 532–541 (2016). (in Chinese) 14. Zhao, X.Z., Ye, B.Y.: The influence of formation manner of component on signal processing effect of singular value decomposition. J. Shanghai Jiaotong Univ. 45(3), 368–374 (2011). (in Chinese)

Influence of Slight Gravitational Effect on the Characteristics of Onset Speeds of Instability and Stability in the Vertical Rotating Shaft Supported by Journal Bearing Li Fan(B) , Tsuyoshi Inoue, and Akira Heya Nagoya University, Nagoya 464-8603, Aichi, Japan [email protected], [email protected], [email protected]

Abstract. In rotating machinery such as vertical pumps supported vertically by sliding bearings, it is difficult to predict the dynamic characteristics because of the strong influence of the nonlinearity of the sliding bearings. In addition, because the direct stiffness coefficient is small in a vertical shaft, the effect of unbalance causes large whirling amplitude and its effect on the dynamic behaviors becomes large. The authors have recently shown these characteristics in a vertical rotor system. The stability of the vibration response changes qualitatively between destabilizing and stabilizing conditions depending on the magnitude of the unbalance, and a characteristic curve that represents the change of stability was obtained. This paper investigates the effect of a small amount of gravity acting on the vertical rotor system. In particular, the effect of this small gravity effect on the characteristic curve representing the change in stability of the vibration response is investigated. Keywords: Dynamics · Nonlinear vibration · Rotor dynamics · Stability

1 Introduction Journal sliding bearings (JSB) are widely used in rotating machineries to provide additional supporting stiffness and increase load capacity. The nonlinearity of JSB [1] may introduce various nonlinear phenomena and cause stability problems in the whirling motion of these machineries. Most studies have been conducted on horizontally supported rotating machineries [2–4], but some of these analyzing techniques are difficult to apply to vertical rotating machines. The typical difference between horizontally and vertically supported rotating systems is that the weight of shaft does not act much on the fluid film of bearing in a vertical case, which may result in weaker radial force and larger whirling amplitude relative to the bearing clearance. As a result, the small amplitude approximation of whirling orbits around equilibrium position applicable to horizontal supported bearings cannot be used in the stability analysis of vertical shaft systems. Orbital analyzing techniques such as the shooting method [5] could be adopted instead to research the nonlinear characteristics specific to vertical rotor-bearing systems. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 273–281, 2024. https://doi.org/10.1007/978-3-031-40455-9_23

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Some new destabilization and stabilization phenomena of a vertical shaft system under unbalanced excitation were found by Watanabe and Inoue [6]. A characteristic curve representing the change of stability was obtained with different amount of unbalance, by using shooting method. However, the influence of the variation of gravitational effect on the nonlinear characteristics of this vertical shaft system has not been considered by authors. On the other hand, some scholars have included gravitational effect in their researches as follows: Elia and Jürg [7] validated the quasi-linear unbalance method and the transient state space approach experimentally, through analyzing the forced response of rigid rotors, supported by herringbone-grooved gas journal bearings (HGJBs). They found that the influence of static loading due to gravitational force on the forced response could be neglected. Similarly, Rong and Wu [8] et al. conducted a gravity-independent experimental study on the performance of a high-speed rotor supported by aerostatic bearings, and discovered that the dynamic behaviors of such a system was independent of the effect of gravity. In the meanwhile, Miyake and Ikemoto et al. [9] focused on a coupled analysis of shaft vibration and fluid film force in the annular plain seal, and the stability limit where self-excited vibration occurred was found to be affected by gravity. Bhattacharyya and Dutt [10] also found that there existed a threshold value of gravity above which the stability limit decreased drastically for most rotordynamic systems. Furthermore, the influence of weight on free and forced vibration of an overhung rotor was researched by Tiaki and Hosseini et al. [11, 12]. And the gravity of rotor had a direct influence on the beat frequency of nonlinear free vibration, and the gravity also decreased the hardening nonlinearities of the rotor system near the primary resonances of forced vibration. Moreover, gravitational force might also introduce various nonlinear phenomena and change bifurcation behaviors of rotor systems, found by Philip and Itzhak et al. [13] and Inayat-Hussain and Kanki et al. [14], respectively. In this paper, the influence of slight gravitational effect on the characteristics of onset speeds of instability and stability of a vertical rotating shaft system has been researched, by performing shooting analysis with parametric study. First, the equations of motion (EOM) of a vertical two-disk simple rotor supported by one ball-bearing (BB) and one JSB under the excitation of gravitational, unbalanced and nonlinear journal bearing forces are derived. Second, several characteristic curves which indicate the change of stability in a specified range of unbalance are calculated by using shooting method, under different values of inclined angle of the shaft. Finally, the effect of gravity on the change of stability is explained by analyzing these characteristic curves.

2 Theoretical Analysis 2.1 Model and Equations of Motion (EOM) A two-disk simple rotor system with a drift is shown in Fig. 3. The drift x 20 is defined as the initial x-direction displacement of shaft (disk 2) in bearing clearance due to slight inclination of shaft at the equilibrium position. The inclination angle α of shaft can thus be approximately calculated as: α = x 20 /l. The axes of journal and bearing are assumed to be parallel, and the mass of shaft is neglected. The EOM of this rotor system under

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275

Fig. 1. A two-disk simple vertical shaft system with drift

the excitation of unbalance Fu (t), gravity Fg and nonlinear journal bearing forces Fjb can be written as: ˙ + Fg + Fu (t) Mq¨ + Cq˙ + Kq = Fjb (q, q)

(1)

M = diag[m1 m1 m2 m2 ],C = diag[c1 c1 c2 c2 ], ⎧ ⎤ ⎡ x20 a 1 ⎫ −m 0 −k1 al 0 g x − k1 1 1 ⎪ l a⎪ ⎪ ⎪ ⎬ ⎨ y1 a⎥ ⎢ 0 0 −k −m g k 1 1 1 l ⎥, F = a , K=⎢ g ⎣ −k2 l 0 ⎪ k2 0 ⎦ −m2 g(x2 − x20 ) 1l ⎪ ⎪ ⎪ a ⎭ ⎩ k2 −m2 g yl2 0 −k2 al 0  T ˙ = 0 0 fjbx (x2 , y2 , x˙ 2 , y˙ 2 ) fjby (x2 , y2 , x˙ 2 , y˙ 2 ) , Fjb (q, q)  T 3EIl 3EI Fu (t) = U ω2 cos ωt U ω2 sin ωt00 , k1 = 2 2 , k2 = 2 , a b lb q = {x1 , y1 , x2 , y2 }T , q˙ = {˙x1 , y˙ 1 , x˙ 2 , y˙ 2 }T , q¨ = {¨x1 , y¨ 1 , x¨ 2 , y¨ 2 }T

(2)

where

The physical meanings and values of these parameters are summarized in Table 1. The nonlinear journal bearing force is calculated numerically by using finite difference method (FEM) with successive over-relaxation (SOR) [15, 16] method. 2.2 Shooting Method [5, 6] In this paper, shooting method with Floquet multipliers analysis is mainly used to find periodic whirling orbits and to detect their stability. An unstable synchronous whirling orbit indicates the occurrence of self-excited vibration in the rotor system under unbalanced excitation. The brief procedure of shooting method is described as follows. For a periodic solution of Eq. (1), the following condition should be satisfied. R(U(t0 )) = U(T + t0 ) − U(t0 ) = 0

(3)

 T is the periodic solution in state space, and R(U(t 0 )) is a residual where U = q q˙ function during one period T = 2π/ω. Expanding Eq. (3) into the first-order Taylor series

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Rotating shaft

JSB

l: Shaft length [mm]

944

Bearing diameter [mm]

40

a: BB to disk 1 [mm]

734

Bearing length [mm]

40

m1 : Mass of disk 1 [kg]

9.46

Bearing clearance [μm]

70

m2 : Mass of disk 2 [kg]

0.45

Pressure in bearing [Pa]

1800

ω: Rotational speed [rpm]

200–1500

Lubricant (25°C)

water

E: Young’s modulus [GPa]

208

U: Unbalance [kgm] × 10–5

2.5–100

I: Moment of inertia [mm4 ]

2485

yields the following equation at the k-th iteration as:   ∂R(U(t ))    0  k R U(t0 ) + U(t0 )k+1 − U(t0 )k  0  ∂U(t0 ) k where k is the index of iteration. Substituting Eq. (3) into (4) gives:     ∂U(t + T )   0  − I U(t0 )k+1 − U(t0 )k = 0 U(t0 + T )k − U(t0 )k +  ∂U(t0 ) k

(4)

(5)

By solving Eq. (5) iteratively, U(t0 )k+1 will be updated, and the periodic solution can be obtained until the tolerance of R(U(t 0 )) is satisfied.

3 Results and Discussion The characteristic curve when x 20 = 0.8C r (α≈0) has already been calculated by Watanabe and Inoue [6], and the values of destabilization and stabilization rotational speeds were also verified experimentally. As a result, the characteristic curve of 0.8 C r drift was chosen as a reference, and multiple characteristic curves pictured in this paper were compared with it, to illustrate the gravitational effect. 3.1 Characteristic Curves Under Different Values of Drift 3.2 Discussion Multiple characteristic curves under different values of drift (inclination angle) have been drawn as shown in Figs. 2 and 3. For a given value of unbalance, the red circular points and lines shown in these figures represent the onset speed of instability (OSI1 or OSI2) where the synchronous whirling orbit changes from stability to instability (self-excited vibration occurs), and the black circular points and lines represent the onset speed of stability (OSS) where the unstable whirling orbit changes into a stable orbit (self-excited

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Fig. 2. Characteristic curves under drift values of (a) 0 C r and (b) 0.1 C r (Black color: Onset speed of stability (OSS) Red color: Onset speed of instability (OSI1/OSI2) Lines: x 20 = 0.8 C r Circular points: x 20 = 0 or 0.1 C r )

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Fig. 3. Characteristic curves under drift values of (a) 100 C r , (b) 300 C r and (c) 500 C r (Black color: Onset speed of stability (OSS) Red color: Onset speed of instability (OSI1/OSI2) Lines: x 20 = 0.8 C r Circular points: x 20 = 100, 300, or 500 C r )

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Fig. 3. (continued)

vibration disappears), as rotational speed of shaft increases gradually. And the value of unbalance acts as a parameter to control the characteristics of these destabilization and stabilization speeds. The characteristic curve representing the change of stability when x 20 = 0.8 C r (black and red lines) as shown in Figs. 2 and 3 indicates the following features: (1) The self-excited vibration may occur below the first critical speed ωcr1 (around 820 rpm) in this vertical rotor system, and OSI1 (about 500 rpm) which can be predicted by conventional eigenvalue analysis around equilibrium positions is approximately independent of amount of unbalance. (2) As rotational speed increases, there is a range where there is no occurrence of self-excited vibration (between OSS and OSI2), and the larger unbalance results in a larger width of this range. (3) As magnitude of unbalance increases, the OSS (smaller than ωcr1 ) will approach the OSI1. Then, when unbalance reaches a certain value, OSS and OSI1 coincides with each other. If unbalance increases further, OSS and OSI1 disappear, and self-excited vibration will not appear below ωcr1 . On the other hand, by comparing Fig. 2a with b, it can be found that by varying drift value smaller than bearing clearance from zero to 0.8 C r , the characteristic curve does not change significantly, which shows that the nonlinear characteristics and stability thresholds of self-excited vibration of this rotating shaft system are insensitive to the variation of gravitational effect when the gravitational (constant) force is relatively small. However, by comparing Fig. 3a with b and c, as x 20 increases larger than 100 C r (α > 0.42°), the increment of inclination angle (gravitational force) causes a slight decrement of OSI2. In the meanwhile, the range of unbalance where OSI1 and OSS exist simultaneously will decrease drastically. Thus it can be inferred that when the

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inclination angle (gravitational force) exceeds a certain value, the OSI1 and OSS will disappear, and the self-excited vibration will not appear below ωcr1 in the whole specified range of unbalance. This phenomenon implies that relatively larger gravitational force (larger inclination angle of shaft) benefits the stability of this rotor system below ωcr1 .

4 Conclusions The slight gravitational effect on the characteristics of onset speeds of instability and stability in the vertical rotating shaft supported by JSB has been investigated by using shooting analysis, and some results can be obtained as follows. The nonlinear characteristics and stability thresholds of self-excited vibration of this rotating shaft system are less responsive to the change of gravitational force while it is small. The increment of gravity can help stabilize this rotating shaft system under ωcr1 when the gravitational force is relatively larger. In the next step, the influence of greater gravity on the nonlinear characteristics and stability of the rotor system will be researched, by changing the inclination angle continuously from vertical shaft to horizontal shaft, to illustrate the (large) gravitational effect on the destabilization and stabilization speeds qualitatively.

References 1. Hori, Y., Kato, T.: Earthquake-induced instability of a rotor supported by oil film bearings. ASME J. Vibr. Acoust. 112(2), 160–165 (1990) 2. Zhao, S.X., Dai, X.D., Meng, G., Zhu, J.: An experimental study of nonlinear oil-film forces of a journal bearing. J. Sound Vib. 287(4), 827–843 (2005) 3. Meruane, V., Pascual, R.: Identification of nonlinear dynamic coefficients in plain journal bearing. Tribol. Int. 41(8), 743–754 (2008) 4. Muszynska, A.: Whirl and whip-rotor/bearing stability problems. J. Sound Vib. 110(3), 443– 462 (1986) 5. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995) 6. Watanabe, Y., Inoue, T.: Characteristics of self-excited vibration of vertical rotating shaft system considering amplitude dependent nonlinearity of sliding bearing. ASME J. Vibr. Acoust. 145(2), 021008 (2022) 7. Elia, I., Jürg, S.: Experimental and numerical investigation of the unbalance behavior of rigid rotors supported by spiral-grooved gas journal bearings. Mech. Syst. Signal Process. 174, 109080 (2022) 8. Rong, C., Wu, H., Li, Y., Lian, H., Xu, X., Yu, X.: Gravity-independent experimental study on a high-speed rotor supported by aerostatic bearings. Microgravity Sci. Technol. 32(6), 1077–1086 (2020). 9. Miyake, K., Ikemoto, A., Uchiumi, M., Inoue, T.: Coupled analysis of the rotor-dynamic fluid forces in the annular plain seal and the shaft vibration. In: Proceedings of the ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition. Volume 7A: Structures and Dynamics, Oslo, Norway (2018) 10. Bhattacharyya, K., Dutt, J.K.: Unbalance response and stability analysis of horizontal rotor systems mounted on nonlinear rolling element bearings with viscoelastic supports. ASME J. Vibr. and Acoust. 119(4), 539–544 (1997)

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11. Tiaki, M.M., Hosseini, S.A.A., Zamanian, M.: Nonlinear free vibrations analysis of overhung rotors under the influence of gravity. In: Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 234, no. 2, pp. 575–588 (2020) 12. Tiaki, M.M., Hosseini, S.A.A., Zamanian, M.: Nonlinear forced vibrations analysis of overhung rotors with unbalanced disk. Arch. Appl. Mech. 86(5), 797–817 (2015) 13. Philip, V., Itzhak, G.: Nonlinear phenomena, bifurcations, and routes to chaos in an asymmetrically supported rotor–stator contact system. J. Sound Vib. 336, 207–226 (2015) 14. Inayat-Hussain, J.I., Kanki, H., Mureithi, N.W.: On the bifurcations of a rigid rotor response in squeeze-film dampers. J. Fluids Struct. 17(3), 433–459 (2003) 15. Koondilogpiboon, N., Inoue, T.: Nonlinear Rotordynamics Investigation on Self-Excited Vibration of Rotors Supported by Fixed Profile and Tilting Pad Journal Bearings. Ph.D. thesis, Nagoya University, Tokai, Japan (2021) 16. Koondilogpiboon, N., Inoue, T.: Investigation of turbulence effects on the nonlinear vibration of a rigid rotor supported by finite length 2-lobe and circular bearings. ASME J. Comput. Nonlinear Dyn. 14(12), 121003 (2019)

Investigation of Journal Gas Foil Bearing Characteristics with Foils Prestress from Assembling Taken into Account M. Yu. Temis(B) and A. B. Meshcheryakov Central Institute of Aviation Motors, Moscow, Russia [email protected]

Abstract. Investigation of journal gas foil bearing characteristics with foils prestress due to its installation into the bearing race with pretension is performed on the base of two-dimensional finite element model with contact interaction between bearing elements taken into account and verified versus static experimental data for nonrotating shaft taken from the open sources. Finite element simulation results show a good agreement with experimental and analytical data. Influence of top foil prestress due to its installation into the bearing race with pretension on bearing static elastic characteristics is demonstrated on the base of finite-element simulations considering light (top foil radius is close to the shaft journal radius) and strong (top foil radius is much greater than the shaft journal radius) top foil prestress. Keywords: gas foil bearing · elastohydrodynamic contact

1 Introduction Gas foil bearing (GFB) characteristics for constant gas film layer parameters and rotor rotational frequency are determined by shaft journal displacement with respect to the bearing race and foil deformations. Foil bearing structure comprises top foil, supported by the corrugated damper that is attached with one end to the bearing race (Fig. 1). Hence foil bearing structure comprises several elastic elements that affect its stiffness characteristics. Foil bearing elastic properties depend on its elastic element parameters (top foil and corrugated damper geometry, number of bumps and pads). Foils may be installed prestressed within the race to provide desired bearing load and proper foildamper contact over the whole bearing with the peculiarities of the assembly process and foil-race attachment features taken into account. Herewith initial foil curvature may not be concentric with the race and shaft journal. First models describing GFB elastic deformations have been proposed by H. Heshmat and R. Ku [1, 2] and I. Iordanoff. [3]. Both models were analytical (based on approximating equations) and considered top foil supported by the uniform [1, 2] or linearly non-uniform [3] elastic foundation. Later and R. Ku [4] and H. Heshmat [4, 5] studied flat bump strip and complete GFB static elastic characteristics experimentally

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 282–290, 2024. https://doi.org/10.1007/978-3-031-40455-9_24

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and verified suggested analytical models with experimental data. Several GFB characteristic features were indicated as a result of those experimental investigations, such as elastic anisotropy and elastic hysteresis due to the friction produced in the contact zones between GFB elastic elements. A decade later D. Rubio and L. San Andres [6], S. Le Lez and M. Arghir [7], K. Feng and S. Kaneko [8] and recently J. Viera and S. Diaz [9], J. Larsen and A. Varela [10] developed more accurate equivalent spring, link-spring and FE models for theoretical prediction of GFB static elastic characteristics and carried out similar experiments that proved suggested models accuracy. These studies indicated that bump interaction not taken into account by analytical models is of a considerable influence on GFB elastic properties. Recently A. Fatu and M. Arghir [11] evaluated manufacturing errors impact on the GFB structural stiffness using 2D FE bearing model. The obtained results demonstrated another valuable aspect that affects GFB performance. All these and many other studies made through the last half-a-century provided a wide range of theoretical and experimental data for GFB performance analysis. However there are still some problems remaining understudied. Influence of GFB elastic elements prestress on its elastic characteristics due to the top foil installation into the bearing race with pretention is one of them. M. Mahner et al. [12, 13] studied prestress effect in 3-pad air foil journal bearing using 1D beamshell model based on Reissner finite-strain beam theory. The obtained results demonstrated considerable influence of the prestress condition on bearing static elastic characteristics. Presented manuscript investigates preload effect in 1-pad 38.1 mm diameter air foil journal bearing, which is considered repeatedly in a number of studies [6–8, 14], using two-dimensional plane strain FE model.

Fig. 1. Gas foil bearing

Gas foil bearing elastic properties are anisotropic and depend on shaft journal displacement direction. That anisotropy is provided not only by the foil attachment location with respect to the shaft journal displacement direction, but also by the foils prestress provided during its assembly process. That effect is confirmed by the results of the bearing static stiffness experimental investigations [4, 5, 11–13]. Foils deformation and contact interaction math model determine fluid film thickness accuracy during calculations and hence has an influence on the accuracy of bearing elastic characteristic calculations. Therefore verification of foil contact interaction models with their prestress determined by the foils geometry before and after shaft journal installation into the bearing is one of

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the relevant problems of gasdynamic bearing simulations. Foil bearing elastic characteristics are determined on the base of the GFB model schematically shown in Fig. 2 that considers static contact interaction between the shaft journal and GFB elastic elements.

2 Model of Foil Deformations Gasdynamic bearing design is a multidisciplinary problem that requires the coupled calculation of the gas film flow parameters in the gap and foil elastic deformations [1–11]. Multidisciplinary gas bearing model used for its elastic characteristics calculation was represented previously in [15–17]. Contact interaction between shaft journal and bearing through the gas film layer may be simulated with different models (2D, 3D) depending on elastic elements geometry and structure. Foil deformations are determined by plane (2D) foil bearing model based on the plane strain theory and taking contact interaction between shaft journal, top foil, corrugated damper and bearing race into account (contact zones 1, 2 and 3, respectively – see Fig. 2). Elastic Coulomb friction is assumed in both contact zones with 0.1 friction coefficient value [7].

Fig. 2. Foil bearing finite element model

Bearing static elastic characteristics are determined for different shaft journal displacement directions. Calculations are carried out for two shapes of corrugated damper: arc bumps (the most widespread form of the bumps) [1–11] (Figs. 2c, 3a) and sinusoidal bumps [18] (form of the bump, that is easier for manufacturing – see Fig. 2b, 3b). GFB dimensions are taken from [7] and represented in Table 1. Sinusoidal bump parameters are taken with respect to the ones of the arc bump (See Fig. 3). Foils prestress is provided by simulation of the bearing assembly process. At that radius of foil curvature is different from shaft journal and bearing radius. Foil prestress due to its installation into the bearing is modelled with special loading applied to the row of the top foil damper side elements (Fig. 4b). Two special loading cases are considered

Investigation of Journal Gas Foil Bearing Characteristics

Fig. 3. Bump geometry

Table 1. GFB parameters Bearing axial length, mm

38.1

Shaft journal diameter, mm

38.1

Nominal radial gap, μm

31.8

Number of bumps

26

Top foil thickness, mm

0.1016

Corrugated damper thickness t b , mm

0.1016

Bump height hb , mm

0.508

Bump pitch s0 , mm*

4.572

Arc bump half-length l 0 , mm

1.778

Distance between adjacent arc bumps l s , mm 1.016 * - bump pitch for particular damper type is determined as per Fig. 3a, b.

Fig. 4. Top foil prestress modelling

285

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for modelling strong and light top foil prestress. Top foil initial profiles are shown in Fig. 4. Terms “light prestress” and “strong prestress” are terms introduced to describe two marginal top foil shapes: – light prestress: top foil initial curvature is almost the same as bearing radius; – strong prestress: top foil initial curvature is close to straight line. Foil deformations are calculated in two loadsteps. On the first loadstep special loading is applied to the row of the top foil damper side elements with the shaft remaining fixed in its central position with respect to the bearing race. Thus top foil special loading simulates its prestress due to installation into the bearing race. On the second loadstep the prestressed bearing is loaded by the contact interaction with the shaft journal. Load is applied to the shaft in the certain direction. That simulates bearing elastic element deformations from the action of the supported shaft halfweight.

3 Calculation Results and Verification with Experimental Data Bearing elastic characteristics are determined experimentally and represented in [6, 7] for 38.1 mm diameter bearing. Finite element model simulations presented hereafter are carried out for the same bearing structure basing on that experimental data. Foil bearing static elastic characteristics are represented in Fig. 5 in comparison with prediction and experimental data for the corresponding shaft displacement direction in the GFB. Developed finite element model shows a good agreement with prediction and experimental data for the full shaft displacement range. Nominal radial clearance is clearly identified both on experimental and calculation data. One can see that FE simulations [15–17] provide the most accurate result and the best agreement with experimental data (Fig. 5). However, FE simulation results strongly depend on contact interaction accuracy parameters and hence the required calculation time is much greater in comparison with analytical models [2, 3].

Fig. 5. GFB elastic characteristics obtained with various prediction models

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At the same time, non-zero bearing reaction exists for horizontal shaft journal displacements in the bearing according to experimental data [6, 7], that may be achieved due to the mismatch in shaft and bearing race curvature values. This effect proves foils prestress existence and necessity of its consideration in the process of calculations. Calculation results show slightly worse agreement with experimental data when foil bearing is loaded and unloaded in horizontal direction. That is caused by the smooth foil prestress during bearing assembly. At the same time good agreement between calculation and experimental data is achieved in zone of considerable loads. Foil bearing characteristics calculated with top foil prestress (both light and strong) taken into account are represented in Fig. 6a, b. Strong prestress model shows a good agreement with the experimental data taken from [7] for leftward and rightward force direction – see Fig. 6b. Light prestress model shows a good agreement with the experimental data for downward and upward force direction – see Fig. 6a. Obtained results confirm proposed top foil prestress model usability. However, more accurate prestress modelling is required, but cannot be provided within the presented investigation due to the absence of the top foil initial geometry (undeformed shape prior to its installation into the bearing race) in [6, 7]. Total shaft journal radial displacements calculated for different top foil prestress cases are represented in Table 2. Slight GFB static rigidity growth is noted for shaft journal displacement direction varying from leftward (α≈90°) to rightward (α≈270°).

Fig. 6. Static GFB elastic characteristics calculated for different top foil prestress cases

Figures 7a-d demonstrate the considerable influence of the top foil prestress on its deflections from round shape and, hence, bearing static stiffness characteristics. Top foil deflections in the area of the force application show a good agreement between the models being considered. At the same time, the considerable difference is noted in the areas, located farther from the force application one. Top foil deflections for the upward force direction considerably depend on the top foil configuration in the area of its attachment to the bearing race. Top foil stiffness in that area is much higher in

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Shaft journal displacement direction

No prestress

Light prestress

Strong prestress

Leftward (α≈90°)

63

62

61

Downward (α≈180°)

62

62

62

Rightward (α≈270°)

58

56

54

Upward (α≈360°)

66

64

62

Fig. 7. Top foil radial deflections calculated for different top foil prestress cases

comparison with the rest of the top foil length, so relatively poor deflection distributions (see Fig. 7d) and deviations in static stiffness characteristics (see Fig. 6a) are detected.

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It also may be noted that both prestress and non-prestressed models show acceptable agreement with experimental data in terms of static stiffness characteristics in general, especially within the maximum load value areas. But particular top foil deflections distribution (eventually the gap between top foil and shaft journal) will evidently affect gasdynamic pressure and, hence, bearing load when it comes to the shaft rotation. Such a problem is considered by the authors as a subject of further investigations.

4 Conclusion Gas foil bearing prestress effect due to the top foil installation into the bearing race with pretention is investigated using FE model. Simulation results demonstrate considerable GFB prestress influence on its elastic characteristics and are of a good agreement with experimental data. Developed FE model provides multiple prestress level simulations ranging from light (top foil radius is close to the shaft journal radius) to strong (top foil radius is much greater than the shaft journal radius). Top foil deflections and, hence, gap distribution between top foil and shaft journal in the prestressed GFB considerably differs from the classic symmetrical one, that is usually may be observed between rigid rotating surfaces, and serves the base for elastohydrodynamic contact problem solution.

References 1. Heshmat, H., Walowit, J.A., Pinkus, O.: Analysis of gas-lubricated foil journal bearings. ASME J. Lub. Tech. 105, 647–655 (1983) 2. Roger Ku, C.-P., Heshmat, H.: Compliant foil bearing structural stiffness analysis: part i— theoretical model including strip and variable bump foil geometry. J. Tribol. 114(2), 394–400 (1992). https://doi.org/10.1115/1.2920898 3. Iordanoff, I.: Analysis of an aerodynamic compliant foil thrust bearing: method for a rapid design. ASME J. Tribol. 121, 816–822 (1999) 4. Roger Ku, C.-P., Heshmat, H.: Compliant foil bearing structural stiffness analysis—part ii: experimental investigation. J. Tribol. 115(3), 364–369 (1993). https://doi.org/10.1115/1.292 1644 5. Heshmat, H.: Advancements in the performance of aerodynamic foil journal bearings: high speed and load capability. ASME J. Tribol. 116, 287–295 (1994) 6. Rubio, D., San Andres, L.: Bump-type foil bearing structural stiffness: experiments and predictions. In: Proceedings of ASME Turbo Expo 2004, Power for Land, Sea and Air, June 14–17, 2004, Vienna, Austria (2004) 7. Le Lez, S., Arghir, M., Frene, J.: A new bump-type foil bearing structure analytical model. J. Eng. Gas Turbines Power 129(4), 1047–1057 (2007). https://doi.org/10.1115/1.2747638 8. Feng, K., Kaneko, S.: Link-spring model of bump-type foil bearings. In: Proceedings of ASME Turbo Expo 2009: Power for Land, Sea, and Air, June 8–12, 2009, Orlando, Florida, USA (2009) 9. Viera, J., Diaz, S.: A new predictive model for the static and dynamic analysis of gas foil bearings. In: Proceedings of ASME Turbo Expo 2014: Turbine Technical Conference and Exposition, June 16-20, Dusseldorf, Germany (2014) 10. Larsen, J.S., Varela, A.C., Santos, I.F.: Numerical and experimental investigation of bump foil mechanical behavior. Tribol. Int. 74(2014), 46–56 (2014)

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11. Fatu, A., Arghir, M.: Numerical analysis of the impact of manufacturing errors on the structural stiffness of foil bearings. In: Proceedings of ASME Turbo Expo 2017, Turbomachinery technical conference and exposition, June 26–30, Charlotte, NC, USA (2017) 12. Mahner, M., Li, P., Lehn, A., Schweitzer, B.: Elastogasdynamic model for air foil journal bearings: hysteresis prediction including preloading effects. In: Society of Tribologists and Lubrication Engineers (STLE), 71st Annual Meeting and Exhibition, Las Vegas, NV, May 15–19 (2016) 13. Mahner, M., Li, P., Lehn, A., Schweizer, B.: Numerical and experimental investigations on preload effects in air foil journal bearings. J. Eng. Gas Turbines Power 140(3), 032505 (2018). https://doi.org/10.1115/1.4037965 14. Zhang, W., Alahyari, A.A., Chiappetta, L.: A fully coupled fluid–structure interaction model for foil gas bearings. J. Eng. Gas Turbines Power 139(2), 022501 (2017). https://doi.org/10. 1115/1.4034343 15. Temis, J.M., Temis, M.J., Mescheryakov, A.B.: Elastohydrodynamic contact theory in foil gas bearing. In: Proceedings of 4th International Symposium on Stability Control of Rotating Machinery, Calgary, Canada, 27–30 August 2007 16. Temis, J.M., Temis, M.J., Meshcheryakov, A.B.: Gas-dynamic foil bearing model. J. Friction and Wear 32(3), 212–220 (2011) 17. Temis, J.M., Temis, M.J., Egorov, A.M., Meshcheryakov, A.B.: Dynamics of compact gas turbine rotor supported by gas bearings. In: 8th IFToMM International Conference on Rotor Dynamics 2010, Seoul, Korea, 12–15 September 2010, vol. 2, pp. 678–684. Curran Associates (2012) 18. Branagan, M., Griffin, D., Goyne, C., Untaroiu, A.: Compliant gas foil bearings and analysis tools. J. Eng. Gas Turbines Power 138(5), 054001 (2016). https://doi.org/10.1115/1.4031628

Investigation on Feature Attribution for Remaining Useful Life Prediction Model of Cryogenic Ball Bearing Byul An1,2 , Yunseok Ha1,2 , Yeongdo Lee1,2 , Wonil Kwak1,2 , and Yongbok Lee1,2(B) 1 Clean Energy Research Division, Korea Institute of Science and Technology, 5, Hwarangno

14-gil, Seongbuk-gu, Seoul 02792, South Korea [email protected] 2 Division of Energy and Environment Technology, University of Science and Technology, 217, Gajeong-ro, Yuseong-gu, Daejeon 34113, South Korea

Abstract. This paper investigates the feature attribution in remaining useful life (RUL) prediction model of cryogenic ball bearing. The RUL prediction model is constructed based on artificial neural network (ANN) by using the TensorFlow platform for training the degradation curve of bearing. To train the models, 5 runto-failure (RTF) data of cryogenic ball bearings were used. The experiment was driven to 3,600rpm with 20kN axial load and 2.5kN radial load for accelerated life test (ALT) of bearing. 6 sensor data (motor input current, bearing outer race torque, test bearing temperature, and support bearing top and bottom temperature) were used in each case. Before training, min-max scaler was used to avoid biased toward a specific range of values. The model has 3 hidden layers with 0.25 dropout for each. Mean absolute percentage error (MAPE) and Root mean square error (RMSE) were used for evaluating the model. By applying SHapley Additive exPlanations (SHAP), it was confirmed that the current is the most attributing feature for the RUL prediction model, then the torque. Temperature also attributes to the model in order of distance away from the test bearing. Keywords: Cryogenic ball bearing · ALT · RUL · feature attribution · SHAP

1 Introduction Bearing is one of the most basic mechanical elements, with 45% of total equipment failures [1]. Accordingly, research in the diagnosis and prediction of bearings’ failure through machine learning is being actively conducted for equipment maintenance. Most studies use vibration data for ordinary bearings [2], so these have a limitation that cannot be immediately applied to bearings with special driving conditions. Ding et al. [3] demonstrated the effectiveness of using multiple sensors measuring vibration, torque, and temperature for predicting the degradation of slewing bearing which extremely low rotational speed with heavy load. In the case of ball bearings in cryogenic environment, solid lubricants are used in the cryogenic ball bearing cage instead of the oil or grease that are typically used to lubricate ordinary ball bearings. Kwak et al. [4] proposed a friction © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 291–299, 2024. https://doi.org/10.1007/978-3-031-40455-9_25

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model that incorporates the hydrodynamic effect of cryogenic fluids, which showed a significant difference compared to existing models that did not consider such effects. As the instability of the ball bearings’ cage is a major cause of bearing damage [5], Choe et al. [6] investigated the effect of different cage clearances and rotation speeds on the cryogenic ball bearing cage’s dynamic behavior, which the design parameters are significantly different compared to ordinary bearings. Furthermore, considering the experiment by Yang et al. [7] which showed that combining temperature and vibration signals can highly suspect ball bearing failure due to cage damage, it is expected that thermal signals may have different feature importance in revealing degradation of cryogenic ball bearings compared to ordinary bearings. It is possible to construct a practical life prediction model for cryogenic ball bearing when the features that reflect the characteristics of the bearing are learned. However, cryogenic ball bearings are difficult to obtain a sufficient amount of data due to the large time and cost of data acquisition. So it is difficult to define the data to be acquired for building fault diagnosis and life prediction models except for the bearing experts. Therefore, this study evaluates the features attribution for predicting remaining useful life (RUL) of cryogenic ball bearings. Run-to-failure (RTF) data were obtained from the reliability evaluation experiment of the bearing used for constructing the RUL model. The feature rankings will be discussed for predicting RUL of cryogenic ball bearing.

2 Feature Evaluation Process 2.1 Experimental Setup Figure 1a shows the cryogenic ball bearing test rig [8], which has the form of a piping and instrumentation diagram (P&ID) as shown in Fig. 2 [9]. Each Pneumatic cylinder is installed at the upper and center adjust constant loads in axial and radial direction. A cryogenic environment can be simulated by filling the chamber with liquid nitrogen (LN2). The bearing of RTF data is a deep groove ball bearing (model 6314) for cryogenic use. It consists cage made of PTFE and others are SUS440C with satisfying the same precision class 5 according to the ISO(International Standards Organization) standard 492, internal clearance grade C4 (Fig. 1b) [6]. Axial and radial loads are measured by load cells (SLS-2T, curiosity technology, Korea). The LN2’s inlet mass flowrate is measured by Coriolis-type mass flow meter (CMF025M, Emerson Electric, USA), and the LN2 inlet, outlet and chamber pressure is measured by pressure transmitter (A-10, WICA, Korea). To measure the test bearing’s outer race temperature, support bearing top and bottom temperature, K-type thermocouples are applied. Current data is acquired from inverter (SOHO55VD4Y, Seoho Electric, Korea) analog output port. Bearing torque is obtained through a load cell (DBCM-30, Bongshin, Korea) with the same principle as [6]. All data was collected by DAQ (NI-9215, National Instruments, USA) with a sampling of 1. The experiment was conducted under the condition of a constant load of 20 kN in the axial direction and 2.5 kN in the radial direction with 3,600 rpm. RTF data was collected from 5 bearings for each, and the failure threshold was set based on a microphone sensor and the characteristic frequency [10]. After the experiment, a spall on the outer race can be seen, as shown in the red box of Fig. 1c.

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Fig. 1. Cryogenic ball bearing reliability test set-up and result (a) Cryogenic ball bearing test rig (b) test bearing (model 6314) (c) test bearing outer race after experiment

Fig. 2. Piping and instrumentation diagram (P&ID) of the cryogenic facility for the rolling bearing and tribological test apparatus

2.2 Model Definition Artificial neural network (ANN) was applied for RUL prediction with TensorFlow platform and Python (Fig. 2). The model consist of three hidden layers with rectified linear unity (ReLU) activations function. The ReLU function is defined such that it outputs 0 for input values less than 0, and for input values greater than or equal to 0, the output is equal to the input value. The ReLU function is as follows: h(x) = max(0, x)

(1)

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The function h(x) in the above equation represents the ReLU function, and x represents the input value. Additionally, to prevent overfitting, 25% dropout is applied between each layers of the ANN model. The specification of constructed model is shown in Table 1, and the output value of a node in a layer is expressed as follows:   n (2) yout = h ( wij · xi ) + bi i=1

Here, xi is the i-th input value, wij is the weight between the i-th input value and the j-th hidden layer, bi is the bias of the i-th hidden layer, h is the activation function (ReLU), and yout represents the output value of the i-th hidden layer.

Fig. 3. Deriving process of RUL prediction results

Table 1. Specifications of the applied model architecture Layer

Parameters

Input

1 × 6 matrix

Dense

6 × 16 networks activation = ReLU

Dropout

0.25

Dense

16 × 32 networks activation = ReLU

Dropout

0.25

Dense

32 × 16 networks activation = ReLU

Dropout

0.25

Dense

16 × 1 networks activation = ReLU

To build the RUL prediction model, 6 sensors’ data (motor input current, test bearing outer race temperature, outlet temperature, support bearing top/bottom temperature and test bearing torque) are used as the features as shown in Fig. 3. Min-max normalization was applied for each features to prevent from getting biased toward a specific range of

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values. Min-max normalization is expressed as follows: x =

xi − min(x) max(x) − min(x)

(3)

The min-max normalization defined above was applied individually to each sensor data. In this case, the max and min values of the sensor data were extracted from the dataset used for training, and x represents the normalized data. Of the 5 cases of RTF data, 4 cases were used as training data, and the remaining 1 case was used as test data for model evaluation. 20% of the training data was used as validation data. To label the RUL result for each data point, first predicting time (FPT) to the failure threshold point was divided at regular intervals. The model was trained under the conditions of 20 batch size and 100 iterations. The learning rate was 0.001 and the weight of each node was updated by applying adaptive moment estimation (Adam).

Fig. 4. RTF data of cryogenic ball bearing (FPT-2,000s to RTF)

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2.3 Feature Attribution Theoretically, the more features a model learns, the higher the complexity and the better its performance. However, if it learns actual data, the adoption of too many features can increase the learning time and the performance is also reduced by the noise of data (curse of dimensionality). Therefore, SHapley Additive exPlanations (SHAP) framework [11] was used to confirm each feature attribution for RUL prediction. SHAP is a technique for evaluate the importance of feature based on Shapley value (Eq. 4). When all features sets are F, a feature subset S which satisfies S ⊆ F is required in all cases. If fS∪{i} is a model currently learned by feature, and fS is a model learned by suppressing the feature, the Shapley value is as follows. φi =

 S⊆F{i}

|S|!(|F| − |S| − 1)! [fS∪{i} (xS∪{i} ) − fS (xS )] |F|!

(4)

3 Results Figure 4 presents the actual and predicted RUL using the same data as in Fig. 3, which was not used for training. The FPT seems to be almost exactly defined, but it can be seen that the predicted RUL value rapidly decreases after FPT. The mean absolute percentage error (MAPE) was 5.478 and the root mean square error (RMSE) was 3.974. This model has room for improvement in terms of predictive accuracy, however, as this study focused on feature selection through feature attribution evaluation, the methods to enhance the model’s predictive accuracy were not included.

Fig. 5. Prediction of the RUL of cryogenic ball bearing (RTF-1,000s to RTF)

When the Shapley value is positive, it has a positive effect on deriving prediction results and vice versa for negative [11]. In the case of the RUL prediction model, where

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the result value decreases according to the system’s degradation, the negative value means more important feature for RUL prediction. It was confirmed that the motor’s input current is the most dominant feature in building the cryogenic ball bearing RUL prediction model (Fig. 5). Table 2 shows the minimum Shapley value for each feature (Fig. 6).

Fig. 6. Visualizing of Shapley values

Table 2. Minimum Shapley value of each feature Rank

Feature

Shapley value (min.)

1

Current

−70.904

2

T.B torque

−39.504

3

T.B temp

−11.631

4

S.B-t temp

−4.231

5

Out. temp

−3.522

6

S.B-b temp

−0.479

4 Discussion Figure 4 shows the section where the prediction result changes rapidly after FPT. It is considered as lack of significant train data in the failure section due to low sampling [12]. Therefore, the sampling or sampling technique itself must be utilized for more accurate RUL prediction. In Table 2, it can be seen that the influence of current and torque, which can immediately affected by bearing movement, were the strongest. In addition, the order of

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temperature attribute can be seen that the value is different depending on the distance from the data acquisition point to test bearing. The Shapley value of T.B. temp. which is measured directly by the test bearing was the highest, followed by Out. temp., the LN2 outlet from the chamber where the test bearing was installed. Then S.B-t temp., the temperature of support bearing located just below the chamber. S.B-b temp. was the last which is furthest from test bearing. In other words, there is a difference in the attribution score depending on the distance from test bearing, which can be considered as the failure didn’t arise due to other elements in test rig. Considering that temperature showed relatively low attribution compared to current and torque, it can be inferred that features related to the mechanical properties, such as current and torque, contribute significantly to the RUL prediction of both cryogenic and ordinary ball bearings. However, in real industrial settings with high noise levels and system complexity, adding meaningful features such as T.B. temp. can be useful for enhancing the practicality of the RUL prediction model. Therefore, future research will consider the dynamics of the mechanical system affected by cryogenic ball bearings and select more sensors in proper locations. Additionally, features such as vibrations, similar to those used in [7], can be added to evaluate attribution to the model. By selecting appropriate features through further study, it will be possible to construct a more practical data-based RUL prediction model.

5 Conclusions The study was conducted to evaluate the importance of each feature to predict RUL of the cryogenic ball bearing by comparing the Shapley value. An ANN-based RUL prediction model was constructed using RTF data of 5 cases obtained for cryogenic ball bearing reliability experiments. There are a total of 6 features used, all of which are acquired with sampling of 1. In the order of current, torque, T.B. temp., S.B-t temp., Out. temp., and S.B-b-temp. has attributed to the RUL prediction result. In particular, in the case of temperature feature, the attribution of the RUL prediction result differs depending on the physical distance from test bearing, so it can be judged that the test bearing failed due to the fault itself, rather than failure of other elements. In actual industries may produce fault signals depending on the complex relationship between each element, so the selecting data to be acquired should be in consideration of the dynamics of the system. Acknowledgments. This study was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) with the project title “Development of platform technology and operation management system for design and operating condition diagnosis of fluid machinery with variable devices based on AI/ICT” (No. 2021202080026D). The authors thank them for their contribution to this study.

Nomenclature A To τt

Motor input current (Current) Outlet temperature (Out. temp.) Test bearing torque (Torque)

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Test bearing temperature (T.B. temp.) Support bearing top temperature (S.B-t temp.) Support bearing bottom temperature (S.B-b temp.) Remaining useful life Run-to-failure

References 1. Zheng, Z., Wang, F., Gong, G., Yang, H., Han, D.: Intelligent technologies for construction machinery using data-driven methods. Autom. Constr. 147, 104711 (2023) 2. Zhang, H., et al.: An unsupervised intelligent fault diagnosis research for rotating machinery based on NND-SAM method. Meas. Sci. Technol. 34(3), 35906 (2022) 3. Ding, P., Wang, H., Dai, Y.: A clustering-based framework for performance degradation prediction of slewing bearing using multiple physical signals. ASCE-ASME J Risk Uncertainty Eng. Syst. Part B: Mech. Eng. 5(2), 020908 (2019) 4. Kwak, W., Lee, J., Lee, Y.B.: Theoretical and experimental approach to ball bearing frictional characteristics compared with cryogenic friction model and dry friction model. Mech. Syst. Signal Process. 124, 424–438 (2019) 5. Kannel, J.W., Snediker, D.K.: The hidden cause of bearing failure (analytical modeling for performance prediction). Mach. Des. 49, 78–82 (1977) 6. Choe, B., Lee, J., Jeon, D., Lee, Y.: Experimental study on dynamic behavior of ball bearing cage in cryogenic environments, part I: effects of cage guidance and pocket clearances. Mech. Syst. Signal Process. 115, 545–569 (2019) 7. Yang, Y., et al.: Experimental study on vibration characteristics due to cage damage of deep groove ball bearing. Tribol. Int. 185, 108555 (2023) 8. Lee, Y.B., Choe, B., Lee, J.K., Ryu, S.J., Lee, B.: Bearing test apparatus for testing durability of bearing. U.S. Patent No. 9714883, 201 (2017) 9. Kwak, W.: Failure Criteria and Life Model to Verify the Reliability of Cryogenic Rolling Bearings, pp. 70–72. Ph.D. University of Science and Technology, Daejeon, Republic of Korea (2023) 10. Park, J., Kim, S., Choi, J.H., Lee, S.H.: Frequency energy shift method for bearing fault prognosis using microphone sensor. Mech. Syst. Signal Process. 147, 107068 (2021) 11. Lundberg, S.M., Lee, S.I.: A unified approach to interpreting model predictions. In: Advances in Neural Information Processing Systems 30 (2017) 12. Demir, S., Mincev, K., Kok, K., Paterakis, N.G.: Data augmentation for time series regression: applying transformations, autoencoders and adversarial networks to electricity price forecasting. Appl. Energy 304, 117695 (2021)

Numerical Investigation on Leakage Characteristics of a Novel Honeycomb Seal with Wall Holes Huzhi Du, Yinghou Jiao(B) , Xiang Zhang, and Renwei Che School of Mechanical Electronic Engineering, Harbin Institute of Technology, Harbin 150001, Heilongjiang Province, China [email protected]

Abstract. Honeycomb seals are a critical component to reduce leakage flow and improve system stability for turbomachines. In this work, a novel single-wallhole-honeycomb seal (S-WHHCS) is proposed, which is built by the traditional honeycomb seal (HCS) with single hole drilling in the honeycomb sidewall. The computational fluid dynamics (CFD) method was used to investigate the leakage characteristics of the S-WHHCS. A series of working conditions of rotation speeds and operating pressures were considered in simulations. The influences of the operating parameters on the leakage characteristics of the S-WHHCS were studied and analysed. The mechanism of the leakage reduction effect of the S-WHHCS was revealed compared with a typical HCS. Numerical results show that the newly added hole influences the pressure drop and turbulence kinetic energy distribution for the seals. Moreover, the numerical results also show that the leakage rate of the S-WHHCS can be reduced in a wide range of operating conditions compared with the typical HCS. The current studies on single-wall-hole-honeycomb seals provide an option to enhance honeycomb seal construction. Keywords: Honeycomb seal · Leakage characteristics · honeycomb wall hole · Numerical simulation

1 Introduction Seal structures are widely used in turbine machinery to restrict leakage flow through rotor–stator clearances [1]. A successful seal structure can significantly improve the stability and efficiency of rotating machinery [2]. There are various annular seal types used in turbomachinery, such as the labyrinth seal (LS) [3], finger seal, honeycomb seal [4], and scallop seal [5]. As a typical representative of damping seals with outstanding performance in both improving the dynamic characteristics and restricting the leakage flow of rotor systems, Honeycomb seals have been extensively used in turbomachinery in recent years [6]. The operating parameters and the structure of the HCS have a significant effect on the sealing and dynamic performance. Childs [7] conducted a series of studies on the leakage characteristics and rotordynamic characteristics of honeycomb seals, focusing © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 300–308, 2024. https://doi.org/10.1007/978-3-031-40455-9_26

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on the effect of seal clearance and geometrical parameters of the honeycomb cavity. The results show that the honeycomb hole structure has a significant impact on the leakage characteristics. Yan [8] and Li [9] investigated the effects of the pressure ratio, honeycomb diameter and depth on the leakage characteristics of honeycomb seals and proposed an optimal hole diameter and depth to achieve minimum leakage. Zhang [10] tested the leakage characteristics of the interlaced hole honeycomb seal (IHHCS) and the noninterlaced hole honeycomb seal (NIHHCS) by the CFD method. The study shows that the IHHCS possesses a better leakage performance than the NIHHCS. Jiang [11] proposed a novel hole-pattern damping seal with dovetail-like diversion grooves (DHPDS). The results show that the leakage rate and the sealing performance of D-HPDS are significantly better than those of traditional hole-pattern damping seals. This paper proposes a new type of single-wall-hole-honeycomb seal (S-WHHCS). The three-dimensional model of annular seals used in the HCS and S-WHHCS was established. The leakage characteristics of the HCS and S-WHHCS in different situations were compared and analysed by ANSYS Fluent, the leakage reduction mechanism of the S-WHHCS was revealed, and the influence of operation parameters on the leakage rate of the S-WHHCS was investigated.

2 Numerical Models and Calculation Method 2.1 Geometrical Model of the Single-Wall-Wole-Honeycomb Seal The new kind of seal structure designed in this paper is based on the traditional HCS, as shown in Fig. 1a. The design values of the HCS are the same as the model established by Zhang’s research [10]. On this basis, a new type of honeycomb damping seal structure named the singlewall-hole-honeycomb seal (S-WHHCS), which is shown in Fig. 1b, was proposed by drilling a single hole in the center of the right hole wall with an axial honeycomb cavity number of 14. The geometric parameters used in the HCS and S-WHHCS are listed in Table 1. The seal length L, hole depth H, wall thickness t, hole width w, rotor radius R, Radial clearance Cr and Axial hole numbers are all same for both seals, with the exception of some central holes with a diameter of 0.5 mm. The length of the outlet was extended appropriately for the full development of turbulence. 2.2 Numerical Method and Grid Meshing In this work, the commercial software ANSYS Fluent was employed to simulate the three-dimensional compressible flow in the seals. Table 2 lists the working parameters and boundary conditions in this paper. Many studies have proven the reliability of the standard k−ε turbulence model in simulating the flow characteristics in seals [12, 13]. Therefore, the standard k-ε turbulence model with a scalable wall function is adopted. The working substance is set as the ideal air gas, and the SIMPLE method is used to solve the pressure distribution in the sealing gap. The inlet and outlet boundary conditions are set to the inlet total pressure boundary and outlet static pressure boundary, respectively. The calculation is considered to meet requirements when all the residuals of the turbulence

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(a) HCS

Inlet

Outlet h

d

Φ

H

Outlet Extension Inlet Extension

(b) S-WHHCS

Fig. 1. Schematic geometry of the HCS and S-WHHCS.

equation, continuity equation, and momentum equation are lower than 10–6 , and the relative error between inlet and outlet mass flow rate is less than 0.01%. The full three-dimensional models of HCS and S-WHHCS were meshed through ICEM with sensible approaches. The meshing details of the seals are shown in Fig. 2. Since the calculations in this paper belong to hydrodynamics, the grid independence was verified by drawing different numbers of meshes. The grid independence verification was calculated using the parameters in Table 2. (Pin = 0.5 Mpa, ω = 5000 rpm), and the results are shown in Fig. 3. The number of the grids adopted in the HCS are 3.61, 5.02, 5.80 and 9.59 million, which means that the maximum volume of control volumes decreased from 0.0382 mm3 to 0.0258 mm3 . The number of the grids adopted in the S-WHHCS are 4.61, 6.13 and 9.93 million, which is more than the HCS. As wall holes are added, the meshes around the holes are correspondingly encrypted. The leakage flow rate sharply declines with increasing grid number from 3.61 to 5.02 million for the HCS and 4.61 to 6.13 million for the S-WHHCS. However, the leakage flow rate only

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decreased 0.47% and 0.36% when the number of the grids increased from 5.8 to 9.6 million for the HCS and 6.13 to 9.93 million for the S-WHHCS. Table 1. Geometric parameters of HCS and S-WHHCS. Parameters of two seals

Value

Seals length, L/mm

38.0

Inlet extension, Li /mm

10.0

Outlet extension, Lo /mm

40.0

Rotor radius, R/mm

30.0

Radial clearance, Cr /mm

0.20

Hole depth, H /mm

3.30

Hole width, w/mm

2.50

Wall thickness, t/mm

0.20

Axial hole numbers, N1

14/13

Hole diameter of S-WWHCS, φ/mm

0.50

Distance between hole and bottom, h/mm

1.65

Distance between hole and side, d /mm

0.72

Fig. 2. Mesh details of two different honeycomb seals.

3 Results and Discussion In this paper, a range of numerical simulations were carried out based on the working conditions, which are already shown in Table 2. The influence of the pressure and rotational speed on the leakage rate of the HCS and S-WHHCS is discussed. The pressure

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Fig. 3. Grid independence verification (Pin = 0.5 Mpa, ω = 5000 rpm) Table 2. The working parameters and boundary conditions. Parameters

Value

Working substance

Air (Ideal gas)

Turbulence model

Standard k − ε

Wall properties

Adiabatic, smooth wall

Inlet pressure, Pin /MPa

0.2, 0.3, 0.5, 0.8

Outlet pressure, Pout /MPa

0.1

Rotating speed, ω/rpm

0, 1000, 3000, 5000, 7000, 10000

field and the turbulence kinetic energy of the HCS and S-WHHCS are obtained, and their leakage characteristics are compared and analysed. Figure 4 displays the leakage flow rate of the HCS and S-WHHCS at different rotating speeds and operating pressures. The leakage flow rates of both the HCS and SWHHCS increase with increasing operating pressure. With the same operating pressure, the leakage flow rate of the S-WHHCS does not change significantly when the rotational speed is increased. This indicates that the leakage flow rate of S-WHHCS is insensitive to the growth of the rotating speed. However, the leakage flow rate of the HCS declined as the rotating speed growing, which means that the HCS is more sensitive to the variation in rotational speed than the S-WHHCS. Meanwhile, it can be seen that the leakage flow rate of the HCS is always larger than that of the S-WHHCS, regardless of the operating pressure and rotating speed. This phenomenon demonstrates that the S-WHHCS has a stronger capacity for fluid leakage control and exhibits excellent sealing performance over the HCS under a wide range of operating conditions. To investigate the sealing performance of the S-WHHCS in detail, the influence of operating conditions on the flow leakage rate was plotted, as shown in Fig. 5. It is apparent that the S-WHHCS has a certain degree of leakage reduction effect when the operating

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pressure ranges from 0.2 MPa to 0.8 MPa, and the leakage reduction amplitude increases as the operating pressure increases. What’s more, the reduction ratio of flow leakage shows a declining trend with increasing rotation speed. This shows that the S-WHHCS is more appropriate for application in high operating pressure and low rotating speed conditions, where the sealing performance of the S-WHHCS can be better exploited. Figure 6 shows the turbulence kinetic energy along the axial direction for the HCS and S-WHHCS when the inlet pressure is 0.3 MPa and the rotating speed is 3000 rpm. It is obvious that the turbulence kinetic energy of both seals reaches its maximum at the exit, which indicates that the turbulent dissipation effect occurs when the fluid flows through the seals. At the midstream and downstream of the seals, the turbulent kinetic energy growth rate of the S-WHHCS is higher than that of the HCS. At the outlet, the S-WHHCS has a significantly larger peak turbulent kinetic energy, while the HCS has a more uniform turbulent kinetic energy distribution. These phenomena indicate that the holes in the honeycomb wall have a significant effect on the turbulent kinetic energy distribution of the seal and the S-WHHCS is more helpful in dissipating the energy of the fluid.

Fig. 4. Flow leakages of the HCS and S-WHHCS under different operating conditions.

To reveal the flow leakage reduction mechanism of the S-WHHCS, the static pressure distribution in the neutral plane for two seals (Pin = 0.3 Mpa, ω = 3000 rpm) is drawn in Fig. 7. In each honeycomb cavity, the value and the increase in the amplitude of the static pressure in the S-WHHCS are greater than those in the HCS, which means that the vortex dissipation in the cavity of the S-WHHCS is more pronounced. When the fluid flows out of the cavities, the convergent gas compressed in the sealing gap due to the influence of fluid inertia, leads to a sudden decrease in static pressure. It is clear that

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Fig. 5. Influence of operating conditions on the flow leakage rate.

the static pressure drop in the sealing gap is greater for S-WHHCS, indicating a more obvious throttling contraction effect. In conclusion, the S-WHHCS has stronger vortex dissipation strength and outlet throttling shrinkage effect than the HCS, which is the main reason why the S-WHHCS exhibits excellent leakage control capacity.

Fig. 6. The turbulence kinetic energy along the axial direction for the two kinds of seals.

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Fig. 7. The static pressure distribution in the meridian plane for two types of seals.

4 Conclusions In this paper, a kind of single wall hole honeycomb seal is proposed, and its leakage characteristics are studied under different operating conditions by a numerical method. The results show that the flow leakage of the HCS and S-WHHCS shows a positive correlation with the static pressure changes. The S-WHHCS is insensitive to rotating speed changes, and it can reduce the flow leakage rate under a large range of working conditions. Meanwhile, the S-WHHCS has the best leakage reduction effect under high pressure and low speed situations compared to the HCS. The hole in the honeycomb walls has significant effects on the turbulence kinetic energy distribution and pressure drop. Compared with the HCS, the turbulent kinetic energy in the S-WHHCS is more irregularly distributed and has larger peaks. Moreover, the S-WHHCS has a higher intracavity static pressure and outlet pressure drop, which results in a stronger vortex dissipation intensity and outlet throttling contraction effect. This is the main reason why S-WHHCS has superior leakage control ability. Acknowledgements. This work was supported by the National Natural Science Foundation of China (No. 12072089, No. 11972131) and the National Science and Technology Major Project (2017-IV0010-0047).

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References 1. Lakshminarayana, B.: Fluid Dynamics and Heat Transfer of Turbomachinery, pp. 339–347. Wiley, New York (1996) 2. Yucel, U., Kazakia, J.Y.: Analytical prediction techniques for axisymmetric flow in gas labyrinth seals. J. Eng. Gas Turbines Power-Trans. Asme. 123(1), 255–257 (2001) 3. Gamal, A.J.M., Vance, J.M.: Labyrinth seal leakage tests: tooth profile, tooth thickness, and eccentricity effects. J. Eng. Gas Turbines Power 130(1), 012510 (2008) 4. K.K. Nielsen, K. Jnck, H. Underbakke: Hole-pattern and honeycomb seal rotordynamic forces: validation of CFD-based prediction techniques. J. Eng. Gas Turbines Power 134, 1e10 (2012). https://doi.org/10.1115/1.4007344 5. Zahorulko, A.V., Lee, Y.B.: Computational analysis for scallop seals with sickle grooves, part II: rotordynamic characteristics. Mech. Syst. Signal Process. 147, 107154 (2021). https://doi. org/10.1016/j.ymssp.2020.107154 6. Lisyanskii, A.S., Gribin, V.G., Sakhnin, Yu.A., Fat’kov, O.V., Gorlitsyn, K.V., Ushinin, S.V.: Practical experience with the introduction of honeycomb shroud seals on 250–800 MW supercritical pressure units. Power Technol. Eng. 47(6), 440–445 (2014) 7. Childs, D.W., Kim, C.H.: Analysis and testing for rotordynamic coefficients of turbulent annular seals with different, directionally homogeneous sur-face-roughness treatment for rotor and stator elements. ASME J. Tribol. 107(3), 296–306 (1985) 8. Yan, X., Li, J., Feng, Z.: Effects of inlet Preswirl and cell diameter and depth on honeycomb seal characteristics. ASME J. Eng. Gas Turbines Power 132(12), 122506 (2010) 9. Li, J., Kong, S., Yan, X., et al.: Numerical investigations on leakage performance of the rotating labyrinth honeycomb seal. ASME J. Eng. Gas Turbines Power 132(6), 062501 (2010). https:// doi.org/10.1115/1.4000091 10. Zhang, W., Gu, C., Yang, X., et al.: Effect of hole arrangement patterns on the leakage and rotordynamic characteristics of the honeycomb seal. Propul. Power Res. 11(2), 181–195 (2022). https://doi.org/10.1016/j.jppr.2022.03.002 11. Jiang, J., Zhang, X., Zhao, W., et al.: Leakage characteristics of a novel hole-pattern damping seal with dovetail-like diversion grooves: numerical simulation and experiment. J. Eng. Gas Turbines Power 144(7), 071004 (2022) 12. Zhang, X., Jiang, J., Peng, X., Li, J.: Leakage and rotordynamic characteristics of labyrinth seal and hole-pattern damping seal with special-shaped 3D cavity. Ind. Lubr. Tribol. 73(2), 396–403 (2021). https://doi.org/10.1108/ILT-07-2020-0262 13. Li, Z., Li, J., Feng, Z.: Numerical investigations on the leakage and rotordynamic characteristics of pocket damper seals—Part I: effects of pressure ratio, rotational speed, and inlet Preswirl. J. Eng. Gas Turbines Power 137(3), 032503.1-032503.15 (2015). https://doi.org/ 10.1115/1.4028373

Dynamic Performance of Spacecraft Flywheel Ball Bearing with Different Type and Distribution of Cage Pocket Shape Shuai Gao1,2(B)

, Lanyu Liu3 , and Qinkai Han1

1 Tsinghua University, Beijing 100084, China

[email protected]

2 Politecnico di Milano, 20156 Milano, Italy 3 Wuhan University of Technology, Wuhan 430070, China

Abstract. The dynamic behaviors of self-lubricating cage, which includes cage whirling state, ball-pocket collision force and skidding ratio, are strongly concerned by aerospace and military industry. In this paper, cages with different kinds of pocket shape and distribution mode are experimentally and theoretically investigated. A high-speed photograph technology is applied on the bearing test bench to obtain the instantaneous motion state of cage. The model is validated by the results of cage whirling orbits. The results found that the cage with rectangular pockets shows advantages in anti-skidding performance and stable whirling. The bearing with rectangular-circular combination pocket cage shows good antiskidding performance, but worst cage revolution stability. The collision force of the cage with circular pockets is lowest. Keywords: Rolling Bearing · Dynamics · Cage Pocket · Collision Force · Skidding

1 Introduction Self-lubricating polymer cages are widely applied in high-end equipment manufacturing, aerospace and other fields due to its independent “self-circulation” micro-oil supply features. Such as, angular contact ball bearing in spacecraft flywheel system, and highspeed grinder spindle [1, 2]. However, the dynamic stability and skidding of the selflubricating cage as it is driven by the rolling element, which lacks the drag effect of the lubricant, is an important factor affecting the performance and service reliability of the bearing [3, 4]. In recent years, scholars have carried out some studies on the whirling characteristics [5–7], contact deformation [8, 9] and wear rate [10] of the cage. There are a variety of influence factors which leads to bearing skidding and cage unstable whirling. It is demonstrated that the unstable behavior of bearing can be restrained by optimizing operating conditions [11–13], cage structure [7, 14, 15]and bearing design [16–19] and. Kannel [20] pointed out that the friction coefficient between the rolling element and the cage and the coefficient of restitution of the cage material are © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 309–319, 2024. https://doi.org/10.1007/978-3-031-40455-9_27

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two important factors that determine the stability of the cage. The higher the coefficient of restitution, the worse the stability of the cage. In the cryogenic environments, the parameter of the cage mass unbalance and bearing guide clearance are positively related to the cage whirl instability and pocket wear loss [7, 15]. Liu et al. [21, 22] experimentally and theoretically discussed the influence of pocket shape on skidding rate and pocket collision force, and found that the spherical pocket of cage has advantages over the cylindrical pocket in improving cage stability. Ye et al. [23] designed and developed a cage with pocket in half circular rectangle shape, which creates a higher pocket clearance in circumferential direction, and shows higher stability and impact resistance for heavy radial load bearing. Cui et al. [16] investigated the influence of the unbalance of the rolling element, as the unbalance mass increased to 2.4 g·mm, the vibration level of the cage increased by 8 dB. In addition to rolling elements, the surface integrity of the raceway also has an impact on the dynamic characteristics of the cage [18, 24], it is found that reducing the waviness of the raceway can improve the stability of the cage, but it also leads to an increase in the skidding degree of the rolling element. For the effect of operating conditions, Tu et al. [11] found that the pocket collision force becomes more heavy and frequent as the amplitude and frequency of rotating speed fluctuation increases. Due to the limitation of bearing structure and installation position, cage whirl measurement has always been a thorny problem. The traditional method of measuring cage rotation speed and radial whirl through eddy current sensor and optical fiber has greatly damaged the integrity of bearing structure [7, 25, 26]. It is difficult to accurately evaluate the radial whirling of the cage by the emerging method of triboelectric nanogenerator [19, 27]. Recently, with the development of high-speed cameras, the non-contact measurement method of capturing cage motion characteristics through photography technology has gradually attracted scholar’s attention [4, 28]. Although the above research has made in-depth discussion on cage performance, there are still many problems to be addressed on the influence of cage pocket shape and distribution mode.

2 Dynamic Model and Test Bench Description 2.1 Cage Dynamic Model To discover the differences of cage whirling state, collision force and bearing skidding under different pocket shape and distribution mode, an improved bearing dynamic model was proposed based on author’s previous studies [4, 29]. The force analysis of circular, rectangular, rhombic pocket shape are illustrated (Fig. 1), and the relationship between the coordinate systems and the geometry paramters are shown in Fig. 2. The pocket normal collision force can be calculated based on the Hertzian contact theory. For low stiffness and self-lubricating cages, the damping effect caused by the collision restitution of the pocket is considered [30]: Fp = Kc · δ 3/2 + Cp · δ˙

(1)

Cp = 1.5αp · Kc · δ 3/2

(2)

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where K c is the contact stiffness, δ is the deformation of contact point, C p is the damping coefficient, α p is the coefficient of restitution. The pocket friction force corresponding to the contact normal stress can be calculated by: fp = ∫(μ(x, y) · U (x, y) · A(x, y))/hp dS

(3)

where the μ is the viscosity of the oil film, U is the relative velocity of the contact area, A is the contact area, hp is the thickness of the oil film. The force analysis of the whole cage in radial and axial direction can be summarized based on the shape of the cage pocket, the collision force F g and friction force f g between cage surface and guide ring has been taken into consideration. For the rectangular pocket shape: Fcx = (−(1 − λ) · Fp · cosθ + λ · fpx · cosθ − fpy · sinθ ) + Fg · sinγ + fg · cosγ (4) Fcy = ((1 − λ) · Fp · sinθ − λ · fpx · sinθ − fpy · cosθ ) + Fg · cosγ + fg · sinγ Fcz = (λ · Fp · cosψ − (1 − λ) · fpx ) + mc g

(5) (6)

where f px and f py are the components of the friction force f p in Eq. 3, θ is the attitude angle of the center of pocket, γ is the collision position of the guide ring, ψ is the ball-pocket collision point angle in pocket local coordinate system, mc is the mass of the cage, λ is the indicator for the driven mode, g is the gravity constant. For the circular pocket shape: Fcx = (−Fp · sinψ · cosθ + fpx · cosψ · cosθ − fpy · sinθ ) + Fg · sinγ + fg · cosγ (7) Fcy = (Fp · sinψ · sinθ − fpx · cosψ · sinθ − fpy · cosθ ) + Fg · cosγ + fg · sinγ Fcz = (Fp · cosψ − fpx · sinψ)

(8) (9)

correspondingly, for the rhombic pocket shape: Fcx = (Fp · cosθ + λ · fpx · cosθ · cos(π/4) − fpy · sinθ) + Fg · sinγ + fg · cosγ Fcy = (Fp · sinθ · cos(π/4) − λ · fpx · sinθ · cos(π/4) − fpy · cosθ) + Fg · cosγ + fg · sinγ

Fcz = (λ · Fp · cos(π/4) + λ · fpx · cos(π/4)) + mc g

(10) (11) (12)

If the ball collides with the left wall of the rhombic pocket, λ = 1, otherwise, λ = − 1. Based on the force analysis, the ordinary differential equations for solving cage whirling motion speed (U cx U cy U cz ) and displacements (x c yc zc ) can be expressed by: [dUcx /dt; dUcy /dt; dUcz /dt] = [Fcx ; Fcy ; Fcz ]

(13)

[dxc /dt; dyc /dt; dzc /dt] = [Ucx ; Ucy ; Ucz ]

(14)

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Fig. 1. Schematic diagram of ball-pocket interaction for rectangular, circular, rhombic pocket shape.

Fig. 2. Schematic diagram of the coordinate system and geometry parameters.

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2.2 Test Bench A non-contact cage whirl test platform based on high-speed camera technology is established (Fig. 3). The cages with all circular pocket, all rectangular pocket, circularrectangular combination and rectangular-rhombic combination were tested. The axial force was applied on bearing outer raceway by lever. The axis of high-speed camera coincides with the axis of bearing. After the camera captures the instantaneous image of the cage, the coordinate of the mark point on the image is recognized by software to fit the centroid of the cage. The sampling frequency of high-speed camera is 10000 frame per second. The acquisition starts after the bearing runs for 1 min and lasts for 0.5s. The structural parameters and operating parameters of the cage and bearing are shown in Table 1. Table 1. Structural and operating parameters of the bearing and experiment Parameter

Value

Parameter

Value

Bearing type

7005C

Axial load F a

30 N

Number of balls N b

14

Rotating speed ωi

3000 rpm

Ball Material

G102Cr18Mo

Cage width

12 mm

Cage Material

Polyimide

Sampling frequency

10000 fps

Lubricant

4129

Cage-rotor speed ratio

0.423

Guide clearance cg

0.49 mm

Pocket clearance cp

0.35 mm

3 Discussion on Test and Calculation Result The whirling trajectories of cage center with circular, rectangular, rectangular-circular combination and rectangular-rhombic combination pockets are illustrated in Fig. 4. The orange line represents the test results, and the gray line represents calculation results of the model, the cage whirling orbit and skidding degree can be calculated by the parameters of x c , yc , zc which mentioned in Eqs. 1–14, and the collision force between ball and cage pocket can also be found. The black dot circle represents the guide clearance between the cage and the guide ring. If the orange line coincides with the black dot circle, it represents that the cage rubs to the guide ring. From the comparison of model results and test results, it can be found that the model is effective in predicting cage dynamic behavior. The whirl stability of the cage can be evaluated by the radius and fluctuation of the cage center trajectory. In general, compared with metal cages in cryogenic environments [7], self-lubricating cage made by polyimide shows higher whirling stability. Due to space limitations, the stability analysis method for the whilring trajectory of the cage can be referred to in Ref. [7]. Under the test conditions, the cage with rectangular-circular combination pocket shows intensive whirling, and the cage with rectangular pocket is slightly.

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Fig. 3. Schematic diagram of the cage motion test bench and cage samples.

The skidding degree of the bearing S r is another crucial indicator for evaluating the performance of bearing and cage stability. The macroscopic sliding of the rolling bearing can be calculated by the theoretical cage speed ωth as the bearing in pure rolling state: Sr = ωth − ωc /ωth

(15)

where ωc is the test results of cage rotation speed, which can be obtained by dividing the variation of the angle of marking point to the sampling time interval. The instantaneous bearing skidding ratio of the cage with circular, rectangular, rectangular-circular combination and rectangular-rhombic combination pockets are illustrated in Fig. 5. Based on the instantaneous skidding ratio information, the 95% confidence interval of the instantaneous cage skidding value is presented in Fig. 5. This represents the degree to which the instantaneous rotational speed of the cage deviates from the average stable speed. The average value of the skidding ratio represents the anti-skidding performance of the cage, and the range of the confidence interval represents the fluctuation of the cage revolution speed, which reflects the revolution stability of the cage. Under the test conditions, it can be seen that the cages with rectangular pockets and rectangularrhombic combination pockets shows weak anti-skidding performance. However, the speed fluctuation range of cage with rectangular pockets is the smallest, and that of cage with rectangular-circular combination pocket is the largest, which is consistent with the

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Fig. 4. Comparison of the cage whirling orbit between experiment results and model results of the four types of cages.

whirling stability performance shown in Fig. 4. In addition, the rotation speed fluctuation of the cage shows obvious periodic features, and its frequency is relevant to the rotation frequency of the cage. The instantaneous collision force of the ball pocket collision of the cage with circular, rectangular, rectangular-circular combination and rectangular-rhombic combination pockets are illustrated in Fig. 6. Due to the inability of the experimental test bench to directly measure the impact force of the cage, the model is used to calculate this indicator. For cages with a single pocket type, Fig. 6 shows the collision forces of a selected pocket. For cages with combined pocket type, Fig. 6 shows the collision forces of two adjacent pockets. A positive collision force represents that the rolling element drives the cage, and a negative value represents that the cage drives the rolling element. Under the test condition, it can be seen that the collision force of the cage with rectangular

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Fig. 5. Instantaneous skidding ratio and its 95% confidence interval of the four types of cages.

pockets is highest, and that of the cage with circular pocket is the lowest. The force of the combined pocket cage with circular one is lower than that with rhombic one. From the perspective of collision force, the cage with rectangular pockets has no advantage, which is different from the phenomenon reflected by the two indicators in Figs. 4 and 5.

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Fig. 6. A figure caption is always placed below the illustration. Short captions are centered, while long ones are justified. The macro button chooses the correct format automatically.

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4 Conclusions An improved bearing dynamic model for predicting the whirling and skidding of cage with different pocket shape and distribution mode was proposed in this paper. A noncontact cage whirling test platform for spacecraft flywheel rolling bearing is established, the cage transient state is captured by high-speed camera. It is found that the cage with rectangular pockets shows advantages in anti-skidding performance and stable whirling. The bearing with rectangular-circular combination pocket cage shows good anti-skidding performance, but worst cage revolution stability. The collision force of the cage with circular pockets is lowest. Acknowledgement. This research was supported in part by a scholarship from the China Scholarship Council (CSC) (No. 201806880007), the National Natural Science Foundation of China (No. 11872222), and the State Key Laboratory of Tribology (No. SKLT2021D11). The Italian Ministry of Education, University and Research is acknowledged for the support provided by the Project “Department of Excellence LIS4.0 – Lightweight and Smart Structures for Industry 4.0”.

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High Speed Rotor Drop-Downs in Different Planetary Touch-Down Bearings Differing in the Number of Bearing Units Benedikt Schüßler(B) and Stephan Rinderknecht Institute for Mechatronic Systems, Technical University of Darmstadt, Darmstadt, Germany {schuessler,rinderknecht}@ims.tu-darmstadt.de

Abstract. The design of touch-down bearings (TDB) for vertical, magnetically levitated outer rotor flywheels with high rotational speeds and operation under vacuum conditions is a challenging task. Conventional TDB are not suited for big diameters with high rotational speeds resulting in surface speeds of above 230 m/s. For such systems, the planetary design can be applied, consisting of several TDB units distributed circumferentially around the stator. This design has the advantages of a decoupling of the rotor diameter from the rolling element bearing diameter as well as a whirl-suppressing characteristic due to the polygonalshaped clearance. The influence of the number of TDB units on the drop-down behavior is rarely investigated in the literature. However, it is expected that the number of TDB units influences the drop-down behavior and consequently the bearing service life heavily. Therefore, this paper investigates in simulations and experiments drop-downs with six, four, and three TDB units. The simulations are conducted with the simulation software ANEAS. The experiments are conducted with the TDB test rig, which is a robust test rig designed for rotor drop-downs in TDB. The results of both simulation and experiments indicate that the forces on the TDB are lower in configurations with less than six TDB units. However, with three units, the TDB was destroyed the fastest within less than twelve drop-downs. Keywords: touch-down bearing · drop-down test · drop-down simulation · high-speed drop-down · backup bearing

1 Introduction Flywheels are energy storage systems with high power in relation to stored energy. Therefore, one possible application is in the electricity grid for grid stabilization and power quality improvement [1]. This becomes even more relevant with more and more renewable energy sources in the electricity grid. However, flywheels can be used not only in the electricity grid but also in industrial applications for example for peak shaving or energy flexibilization [2, 3]. In these applications, money is saved by requiring a lower maximum power or having other systems in a more efficient operating point. The advantages of flywheels are their high lifetime, low maintenance, and applicability in different environments [4–6]. However, for all applications, the efficiency of the system © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 320–337, 2024. https://doi.org/10.1007/978-3-031-40455-9_28

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is of high relevance. To have low losses, modern flywheels are often operated under vacuum conditions and use magnetic bearings to reduce mechanical friction [7, 8]. Up to now, three flywheel systems have been built at the Institute of Mechatronic Systems at TU Darmstadt. Figure 1 shows a section view of the CAD model of the rotor-statorsystem for the newest system SWIVT290. The main components of the system are labeled and described in the following. stator rotor

1 electric machine

3 active radial magnetic bearing

2 passive axial magnetic bearing

4 planetary touch-down bearing

4 contact zone to rotor

rolling element bearing

3

roller

2 1 3 4 stator part of the lower planetary touch-down bearing

section of touch-down bearing unit

Fig. 1. Section view of the rotor-stator system of the flywheel SWIVT290 without the containment.

To increase the energy stored in the system, it is built as an outer rotor system. The rotor is a hollow cylinder made of fiber-reinforced plastic with metal inserts for the active components. Compared to steel, fiber-reinforced plastic has a higher yield strength and lower weight, which allows higher rotational speeds and consequently a higher stored energy. A comparison of different rotor materials is given in [4]. The rotor is accelerated and decelerated by a permanent synchronous machine. As Fig. 1 shows, it is placed in the middle of the system. This type of electric machine is chosen due to its high conversion efficiency. On top of the electric machine, a passive magnetic bearing for the axial levitation is placed. These two components are surrounded by the active magnetic bearings (AMB) for the radial direction. Since AMB are not fail-safe, touchdown bearings (TDB) are needed to bear the rotor in case of overload or malfunction of the AMB. The TDB are located at the lower and upper end of the system. Figure 2 shows schematically on the left a conventional TDB and on the right a planetary TDB for outer rotor systems. The planetary TDB consists of several small bearing units distributed circumferentially around the stator. Each bearing unit consists of one and two rolling element bearings at the upper and lower end of the roller (see Fig. 1). The outer rings of the rolling element bearings are mounted in the stator. During a drop-down, the roller

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gets into contact with the rotor. For both TDB designs in Fig. 2, the rotor is in contact with TDB on the upper right side.

stator

stator

rotor Fig. 2. Schematic representation of a TDB (left: conventional; right: planetary) for an outer rotor system that is in contact with the TDB on the upper right side (not true to scale)

This planetary TDB design has certain advantages compared to conventional TDB. In a conventional TDB for an outer rotor system, the bearing diameter is only slightly smaller than the rotor diameter (see left side of Fig. 2). In a planetary TDB, the bearing diameter is significantly smaller than the rotor diameter and the exact size of the bearings can be chosen independently from the rotor diameter. The rotational speed of the rolling element bearing can thus be influenced by the roller diameter. Vertical rotor systems like the flywheel SWIVT290 have a higher tendency to enter a backward whirl. Hence, the whirl-suppressing characteristic of planetary TDB, which was shown in [9, 10] is a further advantage especially relevant for vertical flywheels. However, planetary TDB have the disadvantage that the used rolling element bearings are comparatively small and consequently have lower load ratings. As a result, it is necessary to reduce the loads as far as possible to reach a high service life of the TDB. The loads on the planetary TDB are influenced by several parameters. One parameter, which is rarely investigated in the literature, is the number of bearing units in the planetary TDB. In [10], the number of TDB units is investigated for low drop-down speeds on an outer rotor flywheel system. The investigated configurations are 8-unit and 6-unit TDB. The results indicate that a whirling motion is more likely to occur with the 8-unit TDB. Consequently, the loads are lower for the 6-unit TDB. In comparison to that, this paper aims to investigate a lower number of TDB units at higher drop-down speeds. The system used for the investigation is the TDB test rig instead of an outer rotor flywheel.

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2 Method The investigation of the influence of the number of units on the drop-down behavior and loads on the TDB is conducted in two steps: firstly, with a simulation study and secondly, with experiments. The test rig used for the experiments is also the basis for the modeling of the simulation study. Therefore, the test rig is described in more detail in Sect. 3. In contrary to the overall goal to build a robust TDB for outer rotor flywheels, the test rig used for this investigation is an inner rotor system. However, it has similar rotodynamic properties to outer rotor flywheels [11, 12]. The effort to investigate a wide range of different numbers of TDB units in planetary TDB with simulations is low compared to real experiments. Therefore, models with 3, 4, 5, 6, 7 and 8 units are derived in the simulation study. Because some parameters are not exactly known or vary from drop-down to drop-down, 24 simulations were conducted for each different TDB configuration. The 24 simulations vary in the initial conditions and some uncertain parameters are varied as well. For the experimental investigation, test series with three different numbers of units were analyzed. The test rig was built originally with 6-unit TDB. The results of this TDB configuration have already been described in [13]. However, they are also used in this paper for comparison. In addition to the 6-unit TDB, drop-down test series for 3-unit and 4-unit TDB were performed.

3 Test Rig The TDB test rig is dedicated to drop-down tests. For this purpose, it is built as a robust inner rotor system with an easy exchangeability of components. Another focus of the TDB test rig is the measurement of different values concerning the TDB like forces, temperatures, and rotational speeds. However, it is also designed to mimic the rotodynamic properties of the outer rotor flywheels. Therefore, the system is highly gyroscopic, vertically oriented, and has surface velocities of 230 m/s at the TDB. Since the purpose of the test rig is to test the TDB which may destroy it, the test rig has a secondary TDB for a save spin down even after the primary TDB failed. 3.1 Mechanical Design Figure 3 shows a model of the test rig, which is used for the investigation in this paper. In the following, the main components are explained and compared to the outer rotor flywheel shown at the beginning in Fig. 1. Like the flywheels, the rotor is levitated magnetically, and the electric machine is a permanent magnetic synchronous machine. In contrast to the flywheels, also the axial magnetic bearing is an AMB because efficiency is not the main objective of this test rig. The primary TDB plane has similar surface velocities like the flywheels and is made to mount a planetary TDB. Figure 4 shows a more detailed partial section view of the planetary TDB. On the left side of the section view, one can see the TDB unit with structurally integrated strain gauges (strain gauges themselves are not visible). The right side of the section of the planetary TDB shows the TDB unit where the forces are

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B. Schüßler and S. Rinderknecht containment secondary touch-down bearing radial position sensors planetary touch-down bearing rotor stator permanent magnetic synchronous machine axial active magnetic bearing radial active magnetic bearing anti-vibration mount

Fig. 3. Partial section view of the TDB test rig.

measured with a piezoelectric force sensor. As in the flywheels, the roller in the TDB unit gets into contact with the rotor during a drop-down. On the upper and lower end are two angular contact bearings. For the experiments shown in this paper, hybrid spindle bearings of type 6001 with vacuum grease were used. TDB unit with strain gauges

TDB unit with piezoelectric force sensor

rolling element bearing contact surface of the rotor to the TDB

piezoelectric force sensor roller

Fig. 4. Partial section view of the planetary TDB and the rotor. Detailed view of the TDB unit for the piezoelectic force measurement.

The secondary TDB is conventional with a single rolling element bearing in each plane and in comparison to the primary TDB, it has much lower surface velocities, due to the smaller radius. The TDB test rig has an inertia of around 0.073 kg m2 while the SWIVT290 has an inertia of around 5.1 kg m2 . Despite the differences the spin-down

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times after a drop-down are comparable since the flywheel decelerates with a braking resistor which is not used during a normal drop-down scenario in the TDB test rig. The TDB test rig decelerates mainly due to friction in the TDB resulting in spin-down times of 3 to 6 min depending on the condition of the TDB. To test the different numbers of TDB units, two different TDB housings have been used, which are shown in Fig. 5: one with 6 pockets and one with 8 pockets. The TDB housing with 6 pockets was used for the test series with the 6-unit TDB and for the test series with the 3-unit TDB in which only every second pocket was equipped with a TDB unit. The TDB housing with 8 pockets was used for the test series with the 4-unit TDB, again only every second pocket was equipped with a TDB unit. The pockets in the 8-pocket housing are bigger than the pockets in the 6-pocket housing. This is because space is needed for the measurement of the forces with strain gauges, which were only used in the test series with 4 TDB units.

Fig. 5. Two different designs for the housing of the planetary TDB in the TDB test rig (left: 8-pocket housing (real system), right: 6-pocket housing (CAD model)).

3.2 Measurement System Since the system uses AMB, the position of the rotor is always measured. For the radial direction, four eddy current sensors in each plane are used, enabling a high frequent position measurement. For the axial direction, two inductive position sensors are used. In addition, the TDB test rig is equipped with multiple sensors measuring different values concerning the TDB, for example roller speed, bearing temperature and forces on the TDB. With these measurements, a deeper analysis of the drop-downs is possible. At one TDB unit in each TDB plane, the radial contact force is measured with a piezoelectric force transducer. For the drop-down series with 4-unit TDB, a new TDB housing had to be built. For this revision, the force measurement was extended so that the force is measured at every TDB. The piezoelectric force sensor at one TDB unit in each TDB plane was kept whereas the forces on the remaining TDB units are measured by structurally integrated strain gauges. Both the TDB units with the piezoelectric force sensor and the TDB units with the strain gauges are made of aluminum and have a similar stiffness. At every TDB unit, the rotational speed of the roller is measured as

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well as the temperature of the bearing housing. The measurement of the rotational speed is performed by hall sensors giving a signal twice per revolution. The same principle and the same sensors are used to measure the rotor speed. The temperature measurement is performed with PT100 sensors. In the TDB unit with piezoelectric force transducers, the temperature is measured directly at the outer ring of the upper rolling element bearing. In the case of TDB units for force measurement with strain gauges, the temperature is measured by PT100 mounted at the outside of the TDB unit. Besides these measurements, also the temperatures of the raceways on the rotor in the TDB planes are measured with infrared temperature sensors. Table 1 gives an overview of the installed sensors and their manufacturers. Table 1. Measured values in the TDB test rig with measurement principle and producer. measured value

sensor type

manufacturer (type)

radial position

eddy current

eddylab (T2)

axial position

inductive

Balluff (BAW0033)

TDB temperature PT100

Thermo Sensor and Heraeus + transducer: LKM (Typ264)

rotor temperature

infrared

Optris (CT 3ML)

force

piezoelectric PCB (211B); transducer: Kistler (5073A411)

force

strain gauge

Omega (SGD-2/350-XY13)

rotational speed

hall-effect

Allegro (TS667LSH)

For the data acquisition, a measuring system from National Instruments is used. For the acquisition of the signal from the hall sensor, counter inputs of measurement cards of type NI PCIe-6612 are used. The other signals are measured with cards of type NI PCI 6251, NI PCI 6225 and NI PCI 6229. The configuration was changed for the test series with 4 TDB units due to the additional signals of the strain gauge measurement. Therefore, also the assignment of the data acquisition cards to the signals was changed. The position and force measurements with the piezoelectric sensor were always measured with a frequency of 8 kHz. The forces based on the strain gauges were measured with 6 kHz. The measurement frequency of the temperature was at least 100 Hz.

4 Modeling The simulations are conducted with the MATLAB-based simulation software ANEAS (Analysis of Nonlinear Active Magnetic Bearing Systems), which has been developed at the Institute for Mechatronic Systems at TU Darmstadt. The software can be used to investigate the behavior of systems levitated with AMB. Especially the case of dropdowns can be simulated. First, Orth [14] developed the software to investigate the dropdown behavior of inner rotor system in conventional bearing systems. Later, it was extended to also investigate the drop-down behavior in planetary TDB, firstly for inner

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rotor systems but later on also for outer rotor systems [12, 15]. However, only radial movements can be investigated with the simulation software. The model consists of a flexible rotor and a flexible stator model. The models are finite element models consisting of 1D Timoshenko beam elements. Figure 6 shows on the left the FE model of the rotor and on the right of the stator. The light grey elements are the elements that have stiffness as well as mass. The dark grey elements in comparison have only mass and do not contribute to the stiffness. For example, the motor windings and the potting around them are not expected to contribute significantly to the overall stiffness. Nodes are marked as circles. The yellow x marks the node that is placed in the center of gravity. The green diamond marks the node where the position sensor is placed. The red diamond marks the node where the TDB is placed and which is therefore important for the following calculations.

Fig. 6. Schematics of finite element model of rotor (left) and stator (right) of TDB test rig.

The rotor model has 32 nodes. For the rotor, every node has 5 degrees of freedom. Two lateral movements and three rotational movements are possible for the rotor nodes. The stator model has 25 nodes and every node has only four degrees of freedom. The rotational degree of freedom around the longitudinal axis is neglected since the stator does not rotate and the small rotations resulting from contacts are neglected. The properties obtained from the finite element model are then used as parameters for the equation of motion. Equation 1 shows the general equation of motion, which is used for the modeling. The rotational speed of the rotor is given by . q are the displacements and tilting angles in the absolute frame. Furthermore, M is the mass matrix, K the stiffness matrix, G the gyroscopic matrix. The matrix for the inner and outer Damping are expressed by Din , D∗in and Dout . The external forces are on the right side of the equation. M q¨ + (Dout + Din + G)˙q + (K − D∗in )q = f

(1)

For both the rotor and the stator, a model is built based on the equation of motion shown in Eq. 1. The model of the stator is simpler because it is not rotating and hence there are no gyroscopic effects for the stator. The boundary conditions for the overall system

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are that all four degrees of freedom of the node at the bottom of the stator model are fixed, while the rotor is only supported by external forces. The two equations of motion for the stator and the rotor are coupled mainly by contact forces during a drop-down or by magnetic forces during levitation. In addition to these forces, there are minor forces resulting from the permanent magnets of the electric machine which lead to a negative stiffness in the system. During a drop-down, the nonlinear contact forces have the highest relevance and therefore are described in more detail. On the left in Fig. 1, the stiffnesses considered for the stiffness calculation are marked in the section view of the TDB unit. On the right, the equivalent diagram of the stiffnesses is shown. The stiffness of the TDB unit is mainly determined by the stiffness of the rolling element bearing kbearing and the contact stiffness kcontact between the rotor-roller contact. The deformation of the TDB unit resulting from the normal contact force FN is calculated based on the rotor position xrotor and stator position xstator (Fig. 7).

Fig. 7. Left: section view of the TDB unit with strain gauges with considered stiffnesses in the modeling. Right: equivalent diagram of the stiffnesses.

The bearing stiffness is based on the approximation given by Gargiulo and shown in Eq. 2 [16]. FN _bearing is the normal force acting on the bearing, Z the number of balls, Dm the mean bearing diameter, β the contact angle of the angular contact bearing and δbearing the deflection of the bearing. kbearing =

 dFN _bearing = 3.312 · 1010 Z Dm (cos(β))5 δbearing d δbearing

(2)

The contact stiffness between the rotor and the roller results from the deformation of the two concave bodies during contact. For the approximation of this stiffness, the deformation-force relation from Goldsmith is used and shown in Eq. 3 [17]. This equation gives a slightly nonlinear relation between the contact deformation δcontact and the acting normal force. The radius of the contacting bodies is given by R and the contact length by L. The calculation for the equivalent Young’s modulus E ∗ is given in Eq. 4 where ν is the Poissons ratio and E the Young’s modulus.     L π E ∗ (Rrotor + Rroller ) FN ln +1 (3) δcontact = L π E∗ FN Rrotor Rroller E∗ =

1 2 1−νrotor Erotor

+

2 1−νroller Eroller

(4)

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For the overall stiffness calculation, it is assumed that the stiffness is equally distributed on both bearings in the TDB unit. With this assumption, the overall TDB stiffness kstatic is calculated as a sequence of the bearing stiffnesses and the contact stiffness. For the dissipative part of the contact force, the generic model of Hunt and Crossley is applied, which is shown in Eq. 5. δ is the deformation of the overall contact and respectively δ˙ is the contact velocity of the overall contact. The advantage of this damping model is that it is continuous and produces only positive contact forces for low and medium damping factors [18]. ˙ FN = kstatic δ(1 + α δ)

(5)

The damping coefficient α is based on the coefficient of restitution (COR) and the initial contact velocity δ˙− of the overall contact. Its calculation is shown in Eq. 6. α=

3(1 − COR) 2 − δ˙

(6)

5 Results In this chapter, the investigation results for the influence of the number of TDB units on the loads acting on the TDB during drop-downs are presented. First, the results of a simulation study based on the previously described model in ANEAS are presented. Afterward, the experimental results from the test series with three different TDB designs are presented. In each test series, a planetary TDB configuration was tested until failure. 5.1 Simulation Results In the simulation study, TDB configurations between three and eight units are investigated. Because some model parameters or initial conditions are not exactly known, these parameters are varied as well. The varied parameters and their values are given in Table 2. The selected values for the COR are based on literature values given in [19, 20]. The unbalance values are in the range calculated from the unbalance orbit in the measurements. For the drop-down simulations, only the maximum speed of the test rig of 20,000 rpm is investigated because for this speed the highest loads are expected. Therefore, for every number of TDB units, 24 simulations were conducted. At the beginning of a drop-down, all bearings are at standstill. The rotor accelerates the rollers in the TDB units during the contacts until a synchronous rotational speed of the rotor and the rollers is reached. Due to the initially high relative velocity between the rotor surface and the roller surface, these contacts result in high tangential friction forces which lead to a fast rotor movement. When the bearings are accelerated to the synchronous rotor speed, the friction forces decrease due to the equal surface velocity. Because this synchronization happens in the first seconds of the drop-down, a simulation time of 10 s is used for all simulations. For the comparison and evaluation of the different simulations, severity indicators are used. The rolling element bearings in the planetary TDB are comparatively small

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B. Schüßler and S. Rinderknecht Table 2. Varied parameters in the simulation study.

parameter

values

number of TDB units

3; 4; 5; 6; 7; 8

COR

0.6; 0.7; 0.8

friction coefficient in the bearing

0.0039; 0.0117

unbalance

0.1124 kg mm; 0.2248 kg mm

initial translational velocity

0.01 m/s; 0.05 m/s

to the overall system. It is likely that the amplitude of forces during a drop-down is in the range of the load rating of the bearing and that a single contact damages the bearing permanently. As a result, the maximal normal force Fmax on the TDB is used as one severity indicator. Another severity indicator not only taking a single value into account but the whole drop-down duration is the calculated bearing service life L10 . The bearing service life is based on the rotational speed of the bearing and the normal force in every time step. The TDB consists of multiple rolling element bearings for which the bearing service life is calculated. The bearing service life of the overall TDB L10,min is the minimal bearing service life out of all bearings. The bearing service life calculated for drop-downs is much higher than the actual service life of the TDB. However, even if the calculated value itself is not realistic, it is expected that the bearing service life can be used to compare different drop-downs to each other. If more bearing units are used in the TDB, the costs are higher. Therefore, this paper uses the relative bearing service life L10,min,rel , which is the bearing service life in relation to the deployed bearings in the configuration. In Fig. 8, the results of the simulation study are shown in boxplot diagrams for the two severity indicators maximum force and relative bearing service life. On the left side of Fig. 8, one can see that there is a tendency that the maximum force increases with the number of units in the TDB. The median value of the maximum force is the lowest for

Fig. 8. Box plot diagrams of maximal normal force and relative bearing service life in relation to the number of bearing units in the TDB. The boxes contain 50% of the values in the group, while the whiskers span at maximum the range of 1.5 of the box width. Values outside these ranges are visualized with circles.

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the 3-unit TDB with a value of 3,600 N. The highest median value of the maximum force of 11,538 N is reached for the 7-unit TDB but is similar to the 8-unit TDB. Similarly, the median value of the maximum force for the 4-unit TDB is 5,031 N, which is slightly higher than for the 5-unit TDB with 4,571 N. If one looks at the median of the relative bearing service life, the highest value is reached for the 5-unit TDB. However, for the 4-unit TDB, there are some parameter combinations with the highest bearing service life values of all simulations. 5.2 Experimental Results The test series were performed for TDB with 6, 4, and 3 units. Each test series consists of drop-downs between 5,000 rpm and 20,000 rpm. The drop-down speeds in the different test series were similar. At the beginning of the test series, the drop-down speed was stepwise increased, while later mainly drop-downs at 13,000 rpm and 20,000 rpm were performed. Only if during a drop-down an irregularity occurred, the next drop-down was made at a lower speed. As described in Sect. 3, the forces are measured at every TDB unit only in the test with the 4-unit TDB. Since only the piezoelectric force sensors were present in all test series, the following evaluations are based on these force measurements for a better comparability. Results for 6-Unit Touch-Down Bearing In Fig. 9, the maximum force measured with the piezoelectric force transducer and the drop-down time duration for the different drop-downs of the test series are shown. It can be seen that the forces are the lowest at the beginning of the test series. In the middle of the test series, the maximum forces rise above 10,000 N. In the second half of the test series, there is a tendency of falling maximum forces. Even at the end of the test series, no high values can be seen in the maximum forces. On the contrary, the drop-down time duration shown on the right of Fig. 9 reaches the highest values at the beginning of the test series where the run down in the TDB needs more than 5 min. After the 8th drop-down, the drop-down duration is between 100 s and 200 s till the 56th drop-down. After that, the drop-down duration decreases to 50 s, which indicates increased friction in the bearings.

Fig. 9. Maximum force and drop-down duration for the different drop-downs of the test series with the 6-unit TDB.

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Results for 4-Unit Touch-Down Bearing In the test series with the 4-unit TDB, 25 drop-down tests were performed. During the test series, some trouble occurred with the measurement system and the communication with the AMB. As a result, during some drop-downs, the data was not saved or for other drop-downs, also the axial AMB was switched off. For the evaluation, these drop-downs are skipped. However, especially the drop-downs where the axial AMB was switched off led to high forces which may have damaged the bearings considerably. Therefore, the overall number of drop-downs of this test series cannot be compared to the test series with the 6-unit TDB, where no problems occurred. Either way, the number of drop-downs a TDB withstands cannot be determined based on one test series because of probabilities. Nevertheless, Fig. 10 shows the maximum force measured with the piezoelectric force transducer and the drop-down duration for the different drop-downs. The behavior is different from the test series with 6-unit TDB. In the case of the 4-unit TDB, the forces are generally lower. Furthermore, they drastically increase for the last drop-downs of the test series. This has not been noticed for the 6-unit TDB. Moreover, the drop-down durations are different. For the 4-unit TDB, there is a run-in phase where the drop-down duration increases at the beginning of the test series, until it decreases at the last drop-downs before the TDB was destroyed. Generally, the drop-down durations in this test series are higher than in the one shown previously.

Fig. 10. Maximum force and drop-down duration for the different drop-downs of the test series with the 4-unit TDB.

It can be remarked that the maximum forces measured with the strain gauges which were available in this test series have as expected similar values to the one measured with a piezoelectric force sensor. The highest forces are reached at the beginning of the drop-down when a fast rotor movement occurs and all TDB get into contact with the rotor multiple times. After this phase, the rotor stays more often in one corner of the clearance in this test series. However, during this steady behavior, the forces on the bearings are low and hence this phase does not influence the maximum. Results for 3-Unit Touch-Down Bearing To test a TDB with only 3 units, the same housing as in the test series with the 6-unit TDB is used but only every second pocket is equipped with a TDB unit. Due to the increased clearance in the corners, the air gap had to be reduced to half of the one used in the

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test series with the 6-unit TDB. In this test series, the TDB was destroyed during the 11th drop-down. This drop-down was the second drop-down from full speed. Figure 11 shows the maximum force measured and the drop-down duration for the different dropdowns with the 3-unit TDB. Again, the forces are lower than in the 6-unit TDB, but they are slightly higher than in the 4-unit TDB. The drop-down durations are similar to the ones seen before. However, there is no run-in phase. From the beginning, the drop-down durations are decreasing and the maximum forces are increasing.

Fig. 11. Maximum force and drop-down duration for the different drop-downs of the first test series with the 3-unit TDB.

The TDB unit which failed during the 11th drop-down was the one equipped with the piezoelectric force sensor. Such an early failure of the TDB has not been expected. To exclude assembly errors or an already damaged bearing, the test series was partly repeated. Therefore, the test rig has been disassembled, the destroyed TDB unit has been changed, and the test rig was assembled again. With the changed TDB unit, the test series was repeated. This time, the TDB was destroyed during the 7th drop-down, which has been the first drop-down from full speed. The same TDB unit as before failed, which was the only one with new rolling element bearings. The maximum measured force was below 5500 N. The situation that both times the same TDB unit failed can be explained by the fact that this TDB unit had the most contact with the rotor. This results from a not aligned geometric and magnetic center. Therefore, the rotor tends to fall more often in one TDB unit, which as a result failed the first.

6 Discussion In the following, the results from the simulation study and the experiments are discussed. After that, a short comparison of the simulation results with the experimental results is shown. 6.1 Discussion Of Simulation Results The simulative investigation shows the tendency that lower number of TDB units lead to lower loads on the planetary TDB. These results are in accordance with the literature. In [15, 21], an outer rotor flywheel was dropped in two different planetary TDB

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at low speeds. For an 8-unit TDB instead of a 6-unit TDB, the rotor had a higher tendency towards a whirling motion and consequently higher TDB loads. Also in [21], the author showed with simulations that for a planetary TDB, the contact forces rise with an increasing number of TDB units. However, in that investigation, plain bearings were used instead of rolling element bearing and for more than 12 TDB units, the effect reversed. For the present investigation, it was expected that higher number of TDB units lead to higher forces and therefore, it was expected that the highest forces occur for the 8-unit TDB. The results from the simulation study in Fig. 8 show that the highest forces occurred for the 7-unit TDB. This deviation from the tendency that higher numbers of TDB units lead to higher forces can be explained by the relatively low number of simulations and the nearly chaotic behavior of the system. Generally, the analysis of the maximum normal force of the conducted simulations leads to the conclusion that a lower number of bearing units are preferred. However, if one looks at the relative bearing service life (see right side of Fig. 8), which is a severity indicator taking the whole drop-down into account, the 3-unit TDB is not the preferred one. Instead, the most cost-efficient TDB should have 4 or 5 TDB units, depending on if one looks at the median or the highest reached bearing service life values. 6.2 Discussion of Experimental Results In the test series with the 4-unit TDB, much higher drop-down durations occur than in the test series with the 6-unit TDB. This can be explained by the lower number of bearings, leading to the fact that energy is only dissipated by friction in 4 TDB units instead of 6 TDB units. Apart from this, the rotor stayed more often in one corner of the clearance. As a result, not all TDB units were rotating the whole time, which decreased the losses further and therefore increased the drop-down duration. The same explanation applies to the fact that the first test with 13,000 rpm with the 3-unit TDB had the highest drop-down duration of all experiments at that speed. The experiments after that had shorter dropdown durations. However, this can be explained by already damaged rolling element bearings and therefore higher friction in the bearings which leads to shorter drop-down durations. The simulations showed that the lifetime of the 3-unit TDB should only be slightly lower than the lifetime of the 4-element TDB. However, the first time the 3-unit TDB failed was at the second experiment from full speed. With the replaced bearing it failed at the first experiment from full speed. One possible explanation for the early failure is the acute angle in the TDB clearance, which results in constraining forces if the rotor surface is uneven. Figure 12 shows a rotor with bumps on the surface in a 3-unit TDB. When the rotor rotates, it can have continuous contact with two bearings. If the bump on the rotor reaches under rotation the first (related to direction of rotation) TDB unit it is in contact with, the bump wants to penetrate the TDB unit as shown in Fig. 12. To avoid this, the rotor has to move away from the TDB unit which hits the bump. However, this movement is partly constrained by the second TDB unit with which the rotor has continuous contact. The bumps can result from material adhesion from the roller material to the rotor material due to previous contacts. The constraining forces resulting from the bumps are not visible in the force measurement due to the high frequency. If such a bump is assumed with a width of 3 mm, the force peaks can reach frequencies up to

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70 kHz. However, the force measurement has a low-pass filter with a cutoff frequency of 2 kHz. Therefore, this high frequent force will not be measured with the piezoelectric force sensor. TDB housing herpolhode bump on rotor rotor center TDB clearance TDB unit rotor

Fig. 12. Rotor with bumps in a 3-unit TDB.

6.3 Comparison of Simulation and Experimental Results In the simulation study, unknown parameters like the COR or the friction were varied in an expected range. As the boxplot diagram on the left side in Fig. 8 shows, that these parameters have a great influence on the maximum force. For example for the 6-unit, TDB the maximum force varies between 3000 N and 18,500 N. However, this wide range of contact forces is also seen in the experiments. In the experiments with the 6unit TDB and a drop-down speed of 20,000 rpm, the maximum force varies between 3000 N and 14,425 N. Also for the other test series with the 4- and 3-unit TDB, the measured maximum force is within the simulated range. From the simulations, it was expected that the 3-unit TDB has only a slightly lower lifetime than the 4-unit TDB. However, the experimental results showed that the 3-unit TDB had a very short lifetime and failed both times in less than two drop-downs from full speed. That difference between simulation and experiment can be explained by the bumps on the rotor in the experiments from previous contacts. In the simulation, the rotor and the roller surface are considered even, hence bumps on the rotor are not modeled. Consequently, in the simulation no constraining forces occur and a higher lifetime is calculated.

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7 Conclusion In the paper, TDB for outer rotor flywheels are investigated. For the investigation of different planetary TDB designs the TDB test rig is used which was described in detail. The analysis consists of a simulation study as well as an experimental analysis. The simulations are analyzed with two severity indicators, the maximum normal force during the drop-down and the relative bearing service life of the TDB. The simulation results indicate: • that a lower number of units in TDB lead to lower loads on the bearing. • the highest relative bearing service life can be reached with TDB consisting of 4 or 5 TDB units. With the TDB test rig, three TDB designs with 3, 4, and 6 units have been investigated. During the test series, the rotor was dropped repeatedly in the planetary TDB until the bearings were destroyed. This led to the following results: • the highest number of drop-downs without replacing the TDB was reached with the 6-unit TDB. • the measured forces were lower for the 3- and the 4-unit TDB than for the 6-unit TDB. • the 3-unit TDB had a short service life. The TDB was destroyed within the second respectively the first drop-down from full speed. • this was explained with the acute angle in the TDB clearance of the 3-unit TDB. If the surface of the rotor or the TDB is not perfectly round the rotation in one corner of the clearance leads to constraining forces on the TDB units. As a result, acute angles in the TDB clearance should be avoided. Comparing the simulation results with the experimental results, it can be concluded that both show that forces on the TDB decrease for lower numbers of TDB units. However, in the simulation, a perfectly even rotor surface is assumed, and in the experiments, high frequent force peaks are not measured due to the low pass filtering. Therefore, the simulation model should be extended to uneven rotor surfaces. In addition, for further drop-down tests, the sampling frequency of the force measurement should be increased and an anti-aliasing low-pass filter with a higher cutoff frequency should be used. Acknowledgment. This research was funded by the German Federal Ministry for Economic Affairs and Energy, grant numbers 03ET6064A and 03EI3000A.

References 1. Lazarewicz, M.L., Rojas, A.: Grid frequency regulation by recycling electrical energy in flywheels. In: IEEE Power Engineering Society General Meeting, 2004. 2004 IEEE Power Engineering Society General Meeting, pp. 2038–2042. IEEE, Denver, CO, USA, 6–10 June 2004. ISBN 0-7803-8465-2 2. Hebner, R., Beno, J., Walls, A.: Flywheel batteries come around again. IEEE Spectr. 39, 46–51 (2002). https://doi.org/10.1109/6.993788

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3. Mouratidis, P., Schüßler, B., Rinderknecht, S.: Hybrid energy storage system consisting of a flywheel and a lithium-ion battery for the provision of primary control reserve. In: 2019 8th International Conference on Renewable Energy Research and Applications (ICRERA). 2019 8th International Conference on Renewable Energy Research and Applications (ICRERA), pp. pp 94–99. IEEE Brasov, Romania, 03–06 Nov 2019. ISBN 978-1-7281-3587-8 4. Bolund, B., Bernhoff, H., Leijon, M.: Flywheel energy and power storage systems. Renew. Sustain. Energy Rev. 11, 235–258 (2007). https://doi.org/10.1016/j.rser.2005.01.004 5. Liu, H., Jiang, J.: Flywheel energy storage—An upswing technology for energy sustainability. Energy Build. 39, 599–604 (2007). https://doi.org/10.1016/j.enbuild.2006.10.001 6. Hawkins, L., Mcmullen, P., Larsonneur, R.: Development of an AMB Energy Storage Flywheel for Commercial Application (2005) 7. Li, X.: Evaluation and Design of a Flywheel Energy Storage System. TU Darmstatd (2019) 8. Pena-Alzola, R., Sebastian, R., Quesada, J., Colmenar, A.: Review of flywheel based energy storage systems. In: 2011 International Conference on Power Engineering, Energy and Electrical Drives. 2011 International Conference on Power Engineering, Energy and Electrical Drives (POWERENG), pp. 1–6. IEEE, Malaga, Spain, 11–13 May 2011. ISBN 978-1-4244-9845-1 9. Schüßler, B., Hopf, T., Rinderknecht, S.: Simulative investigation of rubber damper elements for planetary touch-down bearings. Bull. Polish Acad. Sci.: Tech. Sci. 69, e139615 (2021). https://doi.org/10.24425/bpasts.2021.139615 10. Schüßler, B., Hopf, T., Rinderknecht, S.: Drop-downs of an outer rotor flywheel in different planetary touch-down bearing designs. Actuators 11(2), 30 (2022). https://doi.org/10.3390/ act11020030 11. Quurck, L., Schaede, H., Richter, M., Rinderknecht, S.: High speed backup bearings for outerrotor-type flywheels – proposed test rig design. In: Proceedings of ISMB 14. International Symposium on Magnetic Bearings, pp. 109–114. Linz, Austria, 11–14 Aug 2014 12. Quurck, L., Franz, D., Schuessler, B., Rinderknecht, S.: Planetary backup bearings for high speed applications and service life estimation methodology. Mech. Eng. J. 4(5), 17-00010– 00017-00010 (2017). https://doi.org/10.1299/mej.17-00010 13. Quurck, L., Viitala, R., Franz, D., Rinderknecht, S.: Planetary backup bearings for flywheel applications. In: Proceedings of ISMB 16. International Symposium on Magnetic Bearings, Beijing, China, 13–17 Aug 2018 14. Orth, M., Nordmann, R.: ANEAS: a modeling tool for nonlinear analysis of active magnetic bearing systems. IFAC Proc. Vol. 35, 811–816 (2002). https://doi.org/10.1016/S1474-667 0(17)34039-9 15. Schüßler, B., Hopf, T., Rinderknecht, S.: Drop-downs of an outer rotor flywheel in different planetary touch-down bearing arrangements. In: Gomes, Alfonso Celso, D.N., Santisteban, J.A., Stephan, R.M. (eds.) Proceedings of ISMB 17. The 17th Internatial Symposium on Magnetic Bearings, Virtual, pp. 255–266. Rio de Janeiro, Brazil (2021) 16. Gargiulo, E.P., Jr.: A simple way to estimate bearing stiffness. Mach. Des. 52, 107–110 (1980) 17. Goldsmith, W.: Impact: The Theory and Physical Behaviour of Colliding Solids. Edward Arnold, London (1960) 18. Hunt, K.H., Crossley, F.R.E.: Coefficient of restitution interpreted as damping in vibroimpact. J. Appl. Mech. 42, 440 (1975). https://doi.org/10.1115/1.3423596 19. Brake, M.R.W., Reu, P.L., Aragon, D.S.: A comprehensive set of impact data for common aerospace metals. J. Comput. Nonlinear Dyn. 12, 515 (2017). https://doi.org/10.1115/1.403 6760 20. Marinack, M.C., Musgrave, R.E., Higgs, C.F.: Experimental investigations on the coefficient of restitution of single particles. Tribol. Trans. 56, 572–580 (2013). https://doi.org/10.1080/ 10402004.2012.748233 21. Simon, U.: Rotor–Stator–Kontakt in polygonförmigen Fanglagern. Dissertation (2002)

Model Based Identification of the Measured Vibration Multi-fault Diagnostic Signals Generated by a Large Rotating Machine Tomasz Szolc(B)

, Robert Konowrocki , and Dominik Pisarski

Institute of Fundamental Technological Research of the Polish Academy of Sciences, Ul. Pawi´nskiego 5B, 02-106 Warsaw, Poland [email protected]

Abstract. Large rotating machines are usually affected by more or less severe vibrations excited simultaneously by various manufacturing errors and operational defects. In order to identify the causes of these adverse effects on the basis of measured diagnostic signals registered during a regular operation, it is necessary to obtain a theoretical basis regarding possible dynamic responses of the monitored machine to its most likely failures. This paper shows how to achieve this target on the example of monitoring results of a large blower used in the mining industry. In the advanced structural hybrid model of the rotor shaft system of this blower, in addition to the impact of static unbalance, there is included simultaneous interaction of dynamic unbalance of the blower overhung rotor, parallel and angular misalignments of the shaft sections, inner anisotropy of the couplings, pressure pulsation of the working medium caused by incorrect stagger angles of the blower rotor blades, and electromagnetic pull of rotors of the driving electric motors. The contribution of the above-mentioned imperfections to the dynamic behavior of the system will be identified by means of a multi-fault model-based identification method using the harmonic excitation approach, where vibratory motions are described in the space of modal coordinates and malfunction effects are modelled by the use of equivalent external loadings. Computational examples will be devoted to demonstrating the influence of a faulty setting of the stagger angles of the blower blades on lateral vibrations of the entire rotating system with the simultaneous influence of the aforementioned imperfections. Keywords: Rotor Machine · Model-Based Multi-Fault Identification · Monitoring of Vibration Signals · Rotor-Shaft Hybrid Model with Imperfections

1 Introduction Many manufacturing and operational defects of modern rotating machines are the cause of various types of vibrations of these objects. Such oscillations, on the one hand, can pose a serious threat to the correct operation of these machines, but on the other hand, these vibrations can be used to identify the above-mentioned defects. The most common imperfections in rotor-shaft systems of rotating machines include residual static and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 338–357, 2024. https://doi.org/10.1007/978-3-031-40455-9_29

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dynamic unbalances, mutual parallel and angular misalignments of rotor-shaft segments connected by couplings or joints, internal anisotropy of these shafts and couplings connecting them, shaft bows and transverse cracks, rotor-stator rub impact, various types of damages of bearings supporting these shafts and many others. The imperfections listed above are most often the cause of bending/lateral vibrations of the rotor-shaft systems of these machines. In contrast to the often occurring torsional and axial vibrations of shaft lines of the rotating machines, bending/lateral oscillations are relatively easy to measure using sensors usually mounted on housings of rotor-shaft bearing supports. Therefore, monitoring of these vibrations and proper processing of the measurement signals generated by these oscillations can be a source of valuable diagnostic information enabling an effective identification of the above-mentioned defects affecting the majority of typical rotating machines. According to the above, the problem of identifying defects in rotating machines has been an ambitious research challenge for the last few decades, engaging many outstanding researchers from leading scientific and industrial centers around the world to solve it. Apart from diagnosing the condition of a given machine, basing on the current monitoring of its vibrations, an extremely important aspect that was and still is an investigation of the sensitivity of this machine to excitations of certain types of vibrations by certain types of imperfections in its rotor-shaft system. Analyses of bending/lateral vibrations of rotating systems induced by unbalances, mainly static, are already a classic in the field of dynamics of rotating machines, which is reflected in numerous monographs, such as in [1, 2]. Recently, studies of the impact of transverse cracks in rotor shafts have gained a similar rank, as evidenced by so many publications, even an attempt to select the most representative ones is extremely difficult. Although the phenomenon of various cases of parallel and angular misalignments of shafts and rotors has been observed since the beginning of the drive systems of various types of machines, devices and vehicles, scientific research on these imperfections was intensified at the very end of the 20th century and especially in the two decades of the current century. This is confirmed by numerous publications from that period, for example [3–8]. Based on reviews of the available literature, it can be concluded that the problem of inner anisotropy of the dynamic properties of rotor systems has not been fully investigated so far. The inner anisotropy of rotor shafts, distributed continuously along their length, was investigated by means of the finite element method for cognitive purposes in the dissertation [9], and for the diagnosis of imperfections in [7]. Different values of shaft stiffness in mutually perpendicular directions to the axis of rotation as a result of a transverse crack in a given rotor shaft section, as e.g., in [10–13], or as a consequence of bad coupling assembly can be considered as the local anisotropy, which was the subject of research in [14]. It should be noted that in the above-mentioned work [7], a synthetic summary of theoretical models of the types of faults considered here was made, where the sensitivity of the rotor-shaft system to these imperfections acting simultaneously was examined by treating them as effects with stochastically distributed uncertain parameters. The sensitivity of the rotor-shaft system to the simultaneous action of the defects under consideration was also analyzed in [14], where an interval approach was used to take into account the uncertainty of parameters of these imperfections.

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With regard to examination of the sensitivity of rotor-shaft systems to various types of imperfections using various methods, their identification from the viewpoint of the fundamentals of the dynamics of mechanical systems boils down to solving the so-called inverse problems. Due to the great importance of this issue, many methods have been developed over the past two decades to identify faults in machines, including rotating machines, based on measured vibrations induced by these imperfections. In the work [15], the most important methods of fault identification applied in machines, vehicles and flying objects were classified. In turn, in the typical review paper [16], advantages and disadvantages of various commonly used methods for identifying defects in rotating machines are specified. Based on the considerations made in these works, as well as in the papers [10, 11], the fault identification methods based on recorded vibration signals can be divided into two main groups: The first one includes statistical methods that use empirically collected cause-and-effect relationships, where stochastic approaches, neural networks, fuzzy clusters and various methods of transformation of measured vibration time courses are most often used to guess the type of defect, its location and magnitude estimation. The second group includes methods based on physical and mathematical models of the vibrating objects themselves, in our case – the rotor-shaft systems, and on models of particular types of imperfections affecting these objects. It should be remembered that, in contrast to the methods belonging to the first group, the model-based fault identification methods require the most accurate knowledge of technical parameters and dynamic properties of the tested vibrating object and the development of a reliable theoretical model of it. The advantages of this group of identification methods were emphasized, among others in the work [16], justifying them with the use of physical fundamentals of the vibrating objects under study and the analyzed dynamic processes. Identification of defects in rotor-shaft systems using model-based methods can be carried out in a variety of ways. For example, in the works [10, 11] and [17], simultaneously interacting different imperfections are localized and identified, which have been interpreted as external forces exciting bending/lateral vibrations of the rotor-shaft line of a steam turbogenerator. The inverse problem for unbalanced systems of rotor shafts with a transverse crack was solved in [12, 13] by means of stochastic methods using the results of Monte-Carlo simulations of coupled bending-torsional-longitudinal vibrations of the tested objects. In turn, in [18], a regression approach was applied to solve the inverse problem to identify simultaneously acting various defects by the use of a model of rigid rotors. The cause-and-effect relationships resulting from an operation of individual types of imperfections were collected by means of numerical simulations carried out in [19] using the structural FEM model of the rotor system. Then, they could be used to identify these defects by analyzing Fourier, wavelet and Hilbert-Huang transformations of time courses of bending/lateral vibrations recorded on the real object. However, in the work [20], a transverse crack of an unbalanced rotor shaft with misalignments was identified theoretically and experimentally by observation in the time domain of transient resonances under unsteady operating conditions. Many model-based multi-fault identification methods enable more or less effective determination of the type of defect, estimation of its magnitude, and even localization. However, the majority of them require results of dynamic responses of the rotor-shaft system registered at variable rotational speed, as in [19, 20], e.g., in the conditions of

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start-ups or run-downs of the machine, at several rotational speed values, [10, 11, 18], and even for opposite directions of these speeds, as in [18]. But it should be remembered that obtaining such results is not always possible, especially in the conditions of continuous operation of a given rotating machine working at a constant, e.g., nominal, rotational speed. And just such a case of operation of a real rotor machine will be considered in the presented paper. Namely, the subject of research will be the model-based multi-fault identification carried out for a high-power blower used in the mining industry. The main purpose of considerations performed in this work is the current assessment of the condition of the rotor-shaft system of this machine by means of the detection and identification of simultaneously affecting faults during its normal operation. The most likely types of imperfections are considered to be static and dynamic unbalance of the overhung rotor of this blower, static unbalances of the drive motor rotors, parallel and angular misalignment of the shaft segments, internal anisotropy of the couplings and excessive pulsation of the blower working medium causing additional vibrations of the entire rotorshaft system. It should be emphasized that the above-mentioned types of imperfections should be considered the most probable due to structure properties of this object and the way of manufacturing and mutual assembly of its elements. Identification of these defects will be carried out on the basis of current measurements of lateral vibrations of bearing housings recorded in steady-state, nominal machine operating conditions.

2 Modelling of the Blower Rotor-Shaft System with Imperfections The object of considerations in this paper is a heavy industrial blower driven by two asynchronous motors mutually connected in series with a power of 3.55 MW each at the rated speed of 992 rpm. The scheme of the rotor-shaft system of this machine is shown in Fig. 1. The drive shaft of this blower is driven by the motors via two lamella couplings C1 and C2 interconnected by an intermediate shaft, and the motors are connected with each other via two lamella couplings C3 and C4. The rotor of this blower is characterized by the outer diameter of 4.6 m and the total mass 5.74 times greater than the mass of its drive shaft. This shaft is suspended on two oil-journal bearings #1 and #2 mutually distant by 0.81 m. The spans of the rolling element bearings #3 – #4 and #5 – #6 supporting the both motor rotors are equal to 1.76 m. Taking into account the structure of the entire drive system of this blower and the results of the routine, ongoing monitoring of the condition of the machine, the most probable imperfections affecting it are a static and dynamic unbalance of the blower rotor, static unbalance of the rotors of both drive motors, parallel and angular misalignments of all four lamellar couplings C1, C2, C3 and C4, an internal anisotropy of these couplings and a possibility of wrongly set-up stagger angles of selected blades of the blower rotor causing unfavorable pulsation of the working medium, which results in excitation of additional dangerous lateral vibrations. In order to perform an effective multi-fault identification of the above-mentioned types of imperfections, it is necessary to adopt a suitably reliable and computationally efficient physical and mathematical model of the tested object. To achieve this goal, a hybrid model will be used, the structure of which is analogous to the commonly used

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Fig. 1. Scheme of the rotor-shaft system of the industrial blower.

beam finite element models of rotor shaft systems. Namely, the hybrid model differs from the analogous FEM model in that the individual cylindrical segments of real rotor shafts are not discretized, but they are treated naturally as finite beam macro-elements with continuously distributed viscous-inertial-elastic properties. In this model, the flexural motion of cross-sections of each viscoelastic continuous macro-element is governed by the partial differential equation derived using the Rayleigh or Timoshenko rotating beam theory. Such equations contain gyroscopic forces mutually coupling rotor-shaft bending vibrations in the horizontal and vertical plane. The analogous coupling effect caused by the system rotational speed dependent shaft material damping, described by the use of the standard body model, is also included. With an accuracy that is sufficient for practical purposes, in the proposed hybrid model of the rotor-shaft system, some heavy rotors or coupling disks can be represented by rigid bodies attached to the macro-element extreme cross sections, as shown in Fig. 2a. Each bearing support is represented by the use of a dynamic oscillator of two degrees of freedom, where apart from the oil-film or rolling element interaction, also the visco-elastic properties of the bearing housing and foundation are taken into consideration, see Fig. 2b. This bearing model makes it possible to represent with relatively high accuracy kinetostatic and dynamic anisotropic and anti-symmetric properties of the oil-film or rolling elements in the form of constant or variable stiffness and damping coefficients.

a)

b)

Fig. 2. The continuous finite macro-element a), the oscillator representing a bearing support b).

As in the works [12–14, 21], mutual connections of the successive macro-elements creating the stepped shaft as well as their interactions with the bearing supports and

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rigid bodies representing the heavy rotors are described by equations of compliance conditions. These are the equations of geometrical conditions of equality for translational and rotational displacements of extreme cross-sections of the continuous macro-elements x = L i = l 1 + l 2 +… + l i-1 of the adjacent (i − 1)-th and the i-th elastic macro-elements: vi−1 (x, t) = vi (x, t),

∂vi (x, t) ∂vi−1 (x, t) = , ∂x ∂x

(1)

where vi (x,t) = ui (x,t) + jwi (x,t), ui (x,t) being the lateral displacement in the vertical direction and wi (x,t) the lateral displacement in the horizontal direction, and j denotes the imaginary number. The second group of compliance conditions are dynamic ones, which generally contain linear, parametric and nonlinear equations of equilibrium for concentrated external forces, static and dynamic unbalance forces and moments, inertial, elastic and external damping forces, support reactions and gyroscopic moments. For example, the dynamic compliance conditions formulated for the rotating Rayleigh beam, and describing a simple connection of the mentioned adjacent (i − 1)-th and the i-th elastic macro-elements, have the following form: ∂ 2 vi ∂ 3 vi ∂ 3 vi ∂ 3 vi−1 ∂ 3 vi−1 + EI − ρI − EI + ρI + i i i−1 i−1 ∂t 2 ∂x3 ∂x∂t 2 ∂x3 ∂x∂t 2 ∂ 2 vi ∂ 2 vi−1 +jΩ(t)ρI0i − jΩ(t)ρI0,i−1 = Yi (t), (2) ∂x∂t ∂x∂t ∂ 3 vi ∂ 2 vi ∂ 2 vi−1 ∂ 2 vi −Ji = Zi (t), + EIi 2 − EIi−1 + jΩ(t)J0i 2 2 ∂x∂t ∂x ∂x ∂x∂t

−mi

where the symbols mi , J i denote respectively the mass and diametric mass moment of inertia of the rigid disk, I i and I 0i are the cross-sectional diametric and polar geometric moments of inertia, E, ρ denote the shaft material constants, Ω(t) is the current average, i.e., corresponding to rigid body motion, shaft rotational speed, Y i (t) and Z i (t) denote the concentrated external excitations in the form of transverse force and bending moment, respectively, i = 1, 2, …, n, and n is the total number of macro-elements in the hybrid model. By means of the dynamic compliance conditions there are described shaft interactions with discrete oscillators representing shaft bearing supports. As it follows from [12, 13], such compliance conditions contain anti-symmetrical terms with crosscoupling oil-film stiffness components, which couple shaft bending vibrations in two mutually perpendicular planes. In these equations the stiffness and damping coefficients can be constant or variable, where non-linear properties of the oil-film are taken into consideration. The mathematical model of the coupling with an inner anisotropy and characterized also by the parallel and angular misalignment comes down to a description of the connection of the extreme cross-section of the rotating Rayleigh or Timoshenko beam, representing in this model the k-1-st coupling flange, with the extreme cross-section of the analogous beam, representing the k-th flange, by means of a massless spring with the given shear stiffness G0k and bending stiffness H 0k . This description is a condition of equilibrium for viscoelastic, inertial and gyroscopic transverse forces and bending moments, which in the case of applying the Rayleigh beam bending theory takes the

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following form containing concentrated harmonic excitations oscillating with a single1X and double-synchronous 2X frequency:   ∂ ∂ 3 vk−1 (x, t) ∂ 3 vk−1 (x, t) ∂ 2 vk−1 (x, t) − + ρIk−1 − 2jΩ(t)ρIk−1 −EIk−1 1 + e 3 2 ∂t ∂x ∂x∂t ∂x∂t   − G0k + GVk ej(2(t)−k ) · (vk−1 (x, t) − vk (x, t) + Dk ((t) − k )) = 0,   ∂ ∂ 3 vk (x, t) ∂ 3 vk (x, t) ∂ 2 vk (x, t) + − ρI + 2jΩ(t)ρI EIk 1 + e k k ∂t ∂x3 ∂x∂t 2 ∂x∂t   + G0k + GVk ej(2(t)−k ) · (vk−1 (x, t) − vk (x, t) + Dk ((t) − k )) = 0,    ∂ ∂ 2 vk−1 (x, t)  j(2(t)−k ) + H + H e EIk−1 1 + e 0k Vk ∂t ∂x2   ∂vk−1 (x, t) ∂vk (x, t) − − Fk ((t) − k ) = 0, · ∂x ∂x   2  ∂ ∂ vk (x, t)  j(2(t)−k ) EIk 1 + e + H + H e 0k Vk ∂t ∂x2   k−1  ∂vk−1 (x, t) ∂vk (x, t) li . · for x = − − Fk ((t) − k ) = 0 ∂x ∂x i=1

(3) Here, the all symbols above have been already defined for relationships (2), and the explicit time functions of the shaft current rotation angle Θ(t), which occur in (3), i.e., Dk (Θ(t) − Ψ k ), F k (Θ(t) − Φ k ) and C·exp( j(2Θ(t) − Ξ k )), where C = GVk or H Vk and Ξ k = Δk or Γ k , can be treated as concentrated external excitations applied to both flanges of the coupling.

3 Mathematical Solution of the Problem The complete mathematical formulation and solution for the rotor-shaft system hybrid model applied here can be found e.g., in [21] and [12, 13]. Namely, the solution for simulations of the forced lateral vibrations has been obtained using the analytical–computational approach described in the papers mentioned above. In the first step, by solving the differential eigenvalue problem for the linear orthogonal system, the set of bending eigenmode functions is determined. Next, all anti-symmetric, gyroscopic and parametric terms omitted to solve the eigenvalue problem are regarded here as response-dependent external excitations. Finally, for the hybrid model of the rotor-shaft system, the Fourier solution in the form of series in the orthogonal eigenfunctions is applied in the following form for the each i-th macro-element: vi (x, t) =

∞  m=1

Vim (x) ξm (t),

i = 1, 2, ..., n,

(4)

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where V im (x) = U im (x) + jW im (x) denote the orthogonal complex eigenfunctions and ξ m (t) are the unknown modal coordinates. Here, the eigenfunction real part U im (x) corresponds to rotor-shaft lateral displacements in the vertical plane and the imaginary part W im (x) corresponds to lateral displacements in the horizontal plane. This approach leads generally to an infinite number of known separate ordinary differential equations in modal coordinates. But, in the case considered here, the above mentioned responsedependent external excitations and gyroscopic forces mutually couple these equations. Thus, consequently and similarly as in [14] one obtains the following set of parametric ordinary differential equations in the modal coordinates:   (5) M0 r¨ (t) + D(Ω) r˙ (t) + K(Ω) + Ka (exp(j2(t))) r(t) = F(Ω 2 , (t)), where: D(Ω) = D0 + Dg (Ω) and K(Ω) = K0 + Kb + Kd (Ω), (t) =

t

Ω(τ )d τ.

0

The symbols M0 , K0 are the diagonal modal mass and stiffness matrices, respectively, Ka is the symmetrical matrix of parametric excitation with a double-synchronous frequency 2X due to anisotropic properties of the couplings, D0 denotes the symmetrical damping matrix and Dg (Ω) is the skew-symmetrical matrix of gyroscopic effects. Skew- or non-symmetrical elastic properties of the bearings are expressed by matrix Kb (Ω). Anti-symmetrical effects due to the standard body material damping model of the rotating shaft are described by the skew-symmetrical matrix Kd (Ω). The symbol F(Ω 2 (t), Θ(t)) denotes the vector of external excitations. The modal coordinate vector r(t) consists of the unknown time functions ξ m (t) standing in the Fourier solution (4). The mathematically proven quick convergence of the Fourier solution allows for limiting the number of Eqs. (5) to solve to the number of bending eigenmodes taken into consideration in the frequency range of interest.

4 Modelling of the Rotor-Shaft System Imperfections As mentioned above, the identification of faults in the rotor-shaft system of the blower under consideration can be carried out only in steady-state, nominal conditions of its operation. Owing to this, relations (1)–(3) and (5) can be solved for Ω(t) = Ω = const. Similarly to the papers [10, 11, 17], it was assumed that the bending/lateral vibrations of the tested system are induced by time-varying forces and moments caused by the imperfections sought. In the further considerations, the horizontal (marked with upper indices “H”) and vertical components (marked with upper indices “V”) of these forces and moments will constitute dynamic models of individual types of the most probable imperfections expected in the object under study. These are: 1. Static unbalances of the rigid rotor-disks with masses mi and eccentricities εi : YiH (t) = mi εi Ω 2 cos(Ωt − αi ), where α i are the unbalance phase angles;

YiV (t) = mi εi Ω 2 sin(Ωt − αi ),

(6a)

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2. Static unbalances of the cylindrical shaft segments with unit masses ρAi and eccentricities ei : yiH (t) = ρAi ei Ω 2 cos(Ωt − βi ),

yiV (t) = ρAi ei Ω 2 sin(Ωt − βi ),

(6b)

where β i are the unbalance phase angles; 3. Dynamic unbalances of the i-th rigid rotor-disks, [14]: 

  ZiH (t) = Ω 2 21 Iξ i − Iηi sin(2αi ) cos(Ωt − δi ), 

  ZiV (t) = Ω 2 21 Iξ i − Iηi sin(2αi ) sin(Ωt − δi ),

(6c)

where I ξ i , I ηi are the two out of three central, main mass moments of inertia of the rigid disk, α i denote the small angles of rotation of its central principal axes of inertia with respect to the disk centre of mass, so that one of these axes does not coincide with the axis around which this disk rotates, and δ i are the unbalance phase angles; 4. Parallel misalignments of the k-th coupling, [3, 5–8, 14]: DkH (t) = G0k δk cos(Ωt − k ),

DkV (t) = G0k δk sin(Ωt − k ),

(6d)

where G0k denotes the lamella coupling shear stiffness, δ k is the mutual shaft misalignment off-set and Ψ k denotes the parallel misalignment phase angle; 5. Angular misalignments of the k-th coupling, [7, 8, 14]: FkH (t) = F0k βk cos(Ωt − k ), FkV (t) = F0k βk sin(Ωt − k ),

(6e)

where F 0k denotes the lamella coupling bending stiffness, β k is the angular misalignment due to a coupling flange machining error and Φ k denotes the angular misalignment phase angle. It should be noted that all forces exciting vibrations, which result from the types of imperfections listed above and follow from physical fundamentals and practical observations, oscillate harmonically in time with a synchronous frequency of 1X. 6. Inner anisotropy of the couplings: The local inner anisotropy of the couplings is the cause of parametric effects in the adopted hybrid model, which was expressed in the system of Equations (5) by the components of the stiffness matrix fluctuating with a double synchronous frequency of 2X. Since these effects are analogous to the effects of the occurrence of a breathing transverse crack in the rotor-shaft, a similar model can be adopted for them as in works [10, 11]. Namely, the stiffness matrix in (5) is periodic and its Fourier expansion can be truncated at the third harmonic component: Ka (Ωt) = Kav + K1 ej(Ωt−1 ) + K2 ej(2Ωt−2 ) + K3 ej(3Ωt−3 ) ,

(6f)

where Kav is the coupling inner anisotropy average term, Kl , l=1,2,3, are the stiffness matrix fluctuation components to be determined by means of the multi-fault identification. Owing to this, it will be possible to determine the amplitudes and phase angles of stiffness fluctuation caused by the inner anisotropy of the couplings.

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7. Pulsation of the working medium As mentioned above, incorrectly set stagger angles of selected blades of the blower rotor can cause unfavorable pulsation of the working medium pressure. In the case of a huge overhung rotor of the tested blower with a relatively large diameter of more than 4 m, this pulsation induces unbalanced bending moments acting on the rotor shaft line. When all rotor blades are set correctly, i.e., when their stagger angles are mutually the same, the pulsation of the working medium is characterized by a relatively small amplitude compared to the average pressure value with the so-called blade-passing frequency equal to N × Ω, where N denotes the number of blades in the rotor rim, as defined e.g., in [22]. However, when the stagger angle of one or more blades changes, the pulsation of the working medium pressure, and thus the unbalanced bending moment acting on the rotor-shaft, will indicate significant components of 1X, 2X, 3X, 4X and higher, depending on how many and which blades are incorrectly set up. Therefore, for the purpose of identifying this type of imperfection, a simple model was adopted in the form of an external excitation bending moment, which allows obtaining qualitatively very similar results compared to the analogous findings achieved in the work [23] using a three-dimensional FEM model of a multi-blade impeller and working chamber and a three-dimensional simulation of the working medium flow. The function describing a time course of this moment has the following form: ZP (Ωt) =

N 

  Aj fjMM Ωt, (j − 1)2π N ,

(6g)

j=1

⎧     ⎨ sin Ωt + (j − 1)2π   if sin Ωt + (j − 1)2π N > 0, N  fj Ωt, (j − 1)2π N =  ⎩0 if sin Ωt + (j − 1)2π N ≤ 0, Aj are the bending moment amplitudes per blade to be identified and the natural exponent MM is properly selected in order to obtain possibly the best similarity of fluctuation courses in time as these achieved in [23] by means of the advanced three dimensional models of the fan and flow. It should be emphasized that in the case of correct angular positioning of all blades of the rim, the following can be assumed: Aj = A0 for j = 1,2,…N. Then, the amplitude spectrum of the respective time course Z P (Ωt) is characterized by a single component with a frequency corresponding to the blade-passing frequency N × Ω. However, if at least one amplitude Aj is different from A0 , the amplitude spectrum of Z P (Ω·t) will usually have numerous components with fundamental successive frequencies of 1X, 2X, 3X,…, which results in corresponding external excitation bending moments in the analogous form as in (6f). where:

5 Multi-fault Identification Procedure Since the identification tests of the blower under consideration can be carried out under steady-state operating conditions, i.e., at constant rotational speed Ω, and the effect of internal anisotropy of the couplings can be described by the external moments (6f), the modal equation of motion (5) is simplified to a typical form for a linear model subjected to harmonic loadings (6a)–(6g): M0 r¨ (t) + D r¨ (t) + K r(t) = F(Ωt),

(7)

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where all symbols retain their meaning as in the case of Eq. (5). Such harmonic loadings can be expressed as F(nΩt) = Q + P(nΩ) cos(nΩt) + R(nΩ) sin(nΩt),

(8)

where vectors P(nΩ), R(nΩ) contain the modal components of fault excitation amplitudes, vector Q contains the modal components of the rotor-shaft static gravitational load and n = 1,2,3,…, denotes the multiple of the synchronous frequency 1X. Then, in order to obtain the system’s harmonic response, an analytical solution of Eqs. (7) will be applied. For the above-mentioned harmonic excitation (8) the induced steadystate vibrations are also harmonic with the same multi-synchronous circular frequency nΩ. Thus, the analytical solutions for the successive modal functions ξm (t) contained in vector r(t) can be assumed in the following form: r(t) = G + C cos(nΩt) + S sin(nΩt),

(9)

where vectors C = [c1 ,c2 ,…]T , S = [s1 ,s2 ,…]T contain, respectively, the modal cosineand sine-components of forced vibration amplitudes and vector G contains the modal components of the rotor-shaft static deflection due to the gravitational load. Then, by substituting (8) and (9) into (7) one obtains the following systems of linear algebraic equations: K · G = Q,   K − (nΩ)2 M0 · C + nΩ · D · S = P(nΩ),   K − (nΩ)2 M0 · S − nΩ · D · C = R(nΩ).

(10)

In these equations the unknown components of vectors C, S and G are easy to determine if the excitation force vectors Q, P(nΩ) and R(nΩ) are known. However, the target of this work is to solve the inverse problem, i.e., to determine the modal components of fault excitation amplitudes contained in vectors P(nΩ) and R(nΩ) based on the dynamic response of the real system under study registered by measurements. It should be noted here that the first Eq. (10) determines the static component of the system response caused by the action of constant gravitational forces only. This response can be further treated as a reference in relation to the harmonically oscillating responses which are determined by solving the second and third Eqs. (10). Therefore, achieving such a goal comes down to solving Eqs. (10)2 and (10)3 , in which the left-hand sides should be treated as known and the right-hand sides as unknown. Since in the proposed multi-fault identification method all imperfections are described by means of external excitation forces and moments contained in vectors P(nΩ) and R(nΩ), and owing to this all components of matrices M0 , D and K matrix can be considered known, in the first step, it is necessary to determine the components of vectors C and S based on the measurement results. Let us assume that in the actual rotor-shaft system of the tested machine there are M measurement points, usually located on the housings of the bearing supports. When measurements are typically taken in the horizontal “H” and vertical “V” direction at these

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points, monitoring is carried out with 2M simultaneous diagnostic signals. Then, each of these signals can be expanded into a series with respect to orthogonal eigenfunctions of the hybrid dynamic model of the object under study in accordance with the Fourier Solution (4): ϕmH (t) ∼ =

2M  i=1

Wim ςi (t) and ϕmV (t) ∼ =

2M 

Uim ςi (t),

m = 1, 2, ..., M ,

(11)

i=1

where W im = W i (x m ) and U im = U i (x m ) denote the numerical values of respectively horizontal and vertical components of system eigenfunctions in the measurement point with the spatial coordinate of x m . In turn, in the case of periodic dynamic responses of the monitored object caused by the above-mentioned types of imperfections, each harmonic component of the temporarily unknown modal time function ς i (t) = ς i (nΩt) can be expressed in the form of: ςi (nΩt) = ci cos(nΩt) + si sin(nΩt),

(12)

where: i = 1,2,…,2M, m = 1,2,…,M and n = 1,2,3,…. In an analogous way, one can express each n-th harmonic component of the dynamic response in natural coordinates measured in the horizontal “H” and vertical direction “V”: H H H ϕmH (nΩt) = H m sin(nΩt + m ) = αm cos(nΩt) + βm sin(nΩt)

(13) V V V ϕmV (nΩt) = V m sin(nΩt + m ) = αm cos(nΩt) + βm sin(nΩt),    D 

 D D 2 + β D 2 , D = arc tg αm D , D = H,V, m = 1, 2, ..., M . αm where: m = m m β and

m

It should be emphasized here that numerical values of the amplitudes α m D , β m D or Φ m D can be easily determined on the basis of the results of the FFT analysis of the measured time signals. Then, considering α m D and β m D , D = H,V, as known, upon substituting (12) into (11), and then equating the corresponding signals in (11)–(13), the following two systems of algebraic equations are obtained by means of the harmonic balance method: V · C = A and

V · S = B,

(14)

where vectors C = [c1 ,c2 ,…,c2M ]T , S = [s1 ,s2 ,…,s2M ]T , A = [α 1 H , α 1 V , α 2 H , α 2 V ,…, α M H , α M V ]T , B = [β 1 H , β 1 V , β 2 H , β 2 V ,…, β M H , β M V ]T and the 2M × 2M matrix V contains numerical values of the eigenfunction imaginary parts W im , i.e., corresponding to the horizontal direction, in its successive odd rows and numerical values of the eigenfunction real parts U im , i.e., corresponding to the vertical direction, in its successive even rows, i = 1,2,…,2M, m = 1,2,…,M. In this way, separately for each n-th harmonic component of the monitored dynamic response, by solving both Eqs. (14), it is possible to experimentally determine 2M components of vectors C and S in Eqs. (10)2 and (10)3 corresponding to the 2M first eigenmodes of natural vibrations of the blower rotor-shaft system being tested. Next, by substituting them into these equations, their left-hand side values Pi L and Ri L , i = 1,2,…,2M, can be determined and contained respectively in vectors PL (nΩ) and RL (nΩ).

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It should be noted that the all external excitations (6a)–(6g) resulting from the types of imperfections considered can be presented uniformly as: TkH (t) = T0k cos(nΩt − θk ), TkV (t) = T0k sin(nΩt − θk ), k = 1, 2, ..., K, (15) where T 0k and θ k denote respectively the corresponding amplitudes and their phase angles, as defined in the successive relationships (6), and K is the total number of imperfections simultaneously sought by means of the proposed multi-fault identification procedure. These excitations are contained in vector F(Ωt) of the system of modal equations of motion (7), where the successive components of this vector have the form of following sums with components properly weighted by the modal coefficients following from dynamic properties of the real system: Fi (t) =

K  

 ˜ ik T H (t) + U˜ ik T V (t) , W k k

i = 1, 2, ...

(16)

k=1

˜ ik = W ˜ i (xk ) and U˜ ik = where respectively for the horizontal and vertical direction W ˜ Ui (xk ) denote the modal weight coefficients which are equal to: the modal displacements, ˜ ik = Wi (xk ) = Wik and U˜ ik = Ui (xk ) = Uik , if the k-th excitation has a form i.e., W of concentrated force, as in the cases of (6a), (6d), (6f) and (6g), the derivatives of the modal displacements with respect of the rotor-shaft line spatial coordinate x, i.e., ˜ ik = W  (xk ) = W  and U˜ ik = U  (xk ) = U  , if the k-th excitation has a form of W i i ik ik concentrated bending moment, as in the case of (6c) and (6e), or the integrals of the l

˜ ik = ∫ Wij (x)dx modal displacements with respect of the spatial coordinate x, i.e., W 0

l

and U˜ ik = ∫ Uij (x)dx, when the k-th excitation has a form of uniformly distributed 0

force along the j-th continuous macro-element of length l, as in the case of (6b). Then, taking into account formula (8), components of the external excitation vectors P(nΩ) and R(nΩ) in Eqs. (10)2 and (10)3 take the form: Pi =

K    ˜ ik T C − U˜ ik T S , W 0k 0k k=1

Ri =

K    ˜ ik T S + U˜ ik T C , W 0k 0k

(17)

k=1

C = T cos θ S where: T0k 0k k and T0k = T0k sin θk , i = 1, 2, ..., 2M . Then, the successive components Pi and Ri in (17) of vectors P(nΩ) and R(nΩ) can be equated to the respective previously determined values Pi L and Ri L , i = 1,2,…,2M, of the left-hand sides of Eqs. (10)2 and (10)3 contained in vectors PL (nΩ) and RL (nΩ), which leads to the following systems of 4M algebraic equations, i.e.: K  K      C S L ˜ ˜ ˜ ik T S + U˜ ik T C = RLi , Pi = Wik T0k − Uik T0k = Pi and Ri = W 0k 0k k=1

k=1

(18)

i = 1, 2, ..., 2M . One should be aware that under the proposed multi-fault identification method, the number of imperfections K searched for cannot be greater than the number of simultaneously measured 2M vibration signals. However, often the number of these imperfections

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may be smaller than the number of measured signals, i.e., K < 2M. In such cases, the number of unknown components in series in the relationships (17) and (18) is smaller than the total number of Eqs. (18) to be solved. The system therefore has no single solution for all the equations and similarly as in [10, 11] one has to use the least-squares approach in order to find an unambiguous solution of the identification problem that minimize the differences between the calculated and measured results of the system’s dynamic response. Another alternative is to artificially increase the number of imperfections sought, e.g., in the form of usually unavoidable static unbalances, which can be applied to practically every element of the rotor-shaft system, so as to achieve K = 2M. Then, Eqs. (18) can be rearranged into the following matrix form:  · T = ,

(19)

where vector T = [T 01 C , T 02 C ,…, T 0,2M C , T 01 S , T 02 S ,…, T 0,2M S ]T ,  = [ P1 L , P2 L ,…, P2M L , R1 L , R2 L ,…, R2M L ]T , and the successive rows of 4M × 4M matrix  contain:   ˜ i1 , ... , W ˜ i,2M , −U˜ i1 , −U˜ i2 , ... , −U˜ i,2M , i = 1, 2, ... , 2M , ˜ i1 , W W   ˜ i1 , W ˜ i2 , ... , W ˜ i,2M , i = 2M + 1, 2M + 2, ... , 4M . and U˜ i1 , U˜ i2 , ... , U˜ i,2M , W Solving the system of Eqs. (19) separately for successive multiples of the synchronous system rotational speed nX, n = 1,2,3,…, we identify the amplitudes of the sought types of imperfections by relationships (6a-g) and their phase angles,   described  S 

S 2 2 C bearing in mind that: T0k + T0k = T0k and arc tg T0k T C = θk . 0k

6 Exemplary Results of Identification The fundamental step in realizing the proposed model-based, multi-fault identification procedure is to determine the basic parameters of the structural hybrid model of the blower rotor-shaft system under study initially treated as free of imperfections. In this model, a total number of the continuous finite macro-elements substituting individual cylindrical segments of the real stepped shaft line, their geometrical dimensions and material constants have been set up basing on the technical documentation of the blower under study. As a result, the hybrid model of this object consists of n = 66 finite macroelements. In turn, numerical values of the bearing stiffness and damping coefficients were provided by manufacturers of the individual components of the blower drive system, i.e. the blower overhung rotor-shaft and the both electric motors. To determine stiffness coefficients of the bearing housings and their foundations, the three-dimensional finite element technique was applied, the introductory results obtained using which were verified next using proper experimental tests. As mentioned above, the entire rotor-shaft system of the tested industrial blower is suspended by 6 bearing supports. Accelerometers are installed on each of them to measure their lateral vibrations in the horizontal and vertical directions. Owing to this, 2M = 12 signals are being recorded simultaneously during regular operation of the machine at nominal, steady-state conditions. These vibration signals in time domain are

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properly filtered, integrated with regard of time, and their selected time windows are then transformed into frequency domain using the fast Fourier transformation technique (FFT). In Fig. 3 there are presented amplitude spectra of the signals measured by the sensors attached to the individual bearing supports, i.e., starting from bearing #1 in the vicinity of the blower rotor to bearing #6 at the non-driven end of the second driving motor “1” (see Fig. 1). The frequency values on the abscissa axes of plots in Fig. 3 are referred to the rotor-shaft’s nominal rotational speed, so that the most significant amplitude peaks in all these graphs correspond to successive multiples of the synchronous frequencies 1X, 2X, 3X, … According to formula (13), each peak maximal value corresponds to the respective partial amplitudes α m D and β m D , D = H,V, where H denotes the horizontal direction, and V denotes the vertical direction.

Fig. 3. Amplitude characteristics of the measured lateral vibrations at the bearing housings.

To obtain the modal response amplitudes ci and si , i = 1,2,…,2M, as defined by Eq. (12), the components of matrix V need to be determined first by means of the bending/lateral eigenvibration analysis of the blower rotor-shaft system model assumed to be free of imperfections. This goal is achieved by solving both systems of algebraic Eqs. (14). According to the principles of the proposed model-based multi-fault identification procedure, the causes of the successive peaks 1X, 2X and 3X shown in Fig. 3 are going to

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be found, where the system static response due to the gravitational loading will be used here as a reference. In the blower rotor-shaft system being tested, static unbalances of the blower overhung rotor and two electric motor rotors, dynamic unbalance of the blower rotor as well as parallel and angular misalignments of the four couplings C1, C2, C3 and C4 (see Fig. 1) are suspected to be the causes of 1X oscillation components, which are predominant in each amplitude-frequency characteristics presented in Fig. 3. Here, the total number K of these imperfections is equal to 12, what enables us to solve Eq. (19) under condition K = 2M = 12. Then, using expressions (6a)–(6e), parameters of these imperfections exciting the 1X vibration components can be obtained. The parameter values of these imperfections are demonstrated in Table 1. Based on the numerical values of the imperfection parameters inducing the vibration components with a frequency of 1X, it can be concluded that the vanishingly small values of β k prove negligible angular misalignments of all four couplings in the tested system. Table 1. Identification results for the 1X vibration components. Blower rotor dynamic unbalance

Blower rotor static unbalance

Rotor static unbalance of Motor 1

Rotor static unbalance of Motor 2

α = 0.0653 [deg]

mε = 0.592 [kgm]

e1 = 1.248·10–4 [m]

e2 = 0.536·10–4 [m]

δ = 0.784 [rad]

α = 0.586 [rad]

β 1 = 0.002 [rad]

β 2 = −1.912 [rad]

Coupling C1 parallel misalignment

Coupling C2 parallel misalignment

Coupling C3 parallel misalignment

Coupling C4 parallel misalignment

δ 1 = 0.00051 [m]

δ 2 = 0.000969 [m] δ 3 = 0.001139 [m]

δ 4 = 0.000991 [m]

Ψ 1 = 2.567 [rad]

Ψ 2 = 4.111 [rad]

Ψ 3 = −3.989 [rad]

Ψ 4 = 2.753 [rad]

Coupling C1 angular misalignment

Coupling C2 angular misalignment

Coupling C3 angular misalignment

Coupling C4 angular misalignment

β 3 = 1.526·10–4 [rad]

β 4 = 8.004·10–5 [rad]

β 1 = 2.013·10–5 [rad] β 2 = 7.315·10–6 [rad] Φ 1 = 4.872 [rad]

Φ 2 = −0.273 [rad] Φ 3 = 1.694 [rad]

Φ 4 = −3.116 [rad]

In the next step of this identification procedure, parameters of imperfections responsible for excitation of the 2X oscillation components are going to be determined. These are the inner anisotropies of all couplings C1, C2, C3 and C4, and wrongly set-up stagger angles of selected blades of the blower rotor causing unfavorable pulsation of the working medium pressure, and consequently unbalanced oscillatory bending moment imposed to the rotor-shaft. Then, the number of these faults is equal to 5. However, it should be remembered that, according to the dynamic equilibrium conditions (3), the inner anisotropy of each of these couplings is affected by two unknown parameters, i.e., the fluctuation amplitudes of shear and flexural stiffness GVk and H Vk , respectively. Consequently, K = 1 + 4·2 = 9 < 12 = 2M. Because here a greater number of Eqs. (19) is available than the number of unknowns, the abovementioned least-square approach

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had to be applied to obtain the most accurate solutions compared to those determined using the results of experimental measurements. The same approach was used to identify the 3X vibration components suspected to be induced by the same K = 9 parameters of imperfections. Then, as a result of inverse Fourier transformation of expression (6f), amplitudes of bending stiffness fluctuation due to the inner anisotropy of the four couplings as well as rates of the stagger angles of two blades in the blower rotor have been determined. Numerical values of the fluctuation amplitudes of shear and flexural stiffness GVk and H Vk , k = 1,2,3,4, are contained in Table 2. Comparing these stiffness values shows that GVk and H Vk for k = 1,2,3 are much smaller than GV 4 and H V 4 . This means that only coupling C4 (see Fig. 1) indicates a noticeable internal anisotropy. In turn, the determined amplitudes of time-histories of the unbalanced bending moment caused by excessive fluctuations of the pressure of the blower working medium have shown wrong stagger angles of two opposite blades out of a total of 20 blades in the rotor rim, as a result of which the flow rate for these two blades decreased by approximately half. Table 2. Identification results for the 2X vibration components. Coupling C1 shear stiffness fluct. ampl.

Coupling C2 shear stiffness fluct. ampl.

Coupling C3 shear stiffness fluct. ampl.

Coupling C4 shear stiffness fluct. ampl.

GV 1 = 5.1·105 [N/m] GV 2 = 1.13·104 [N/m]

GV 3 = 7.81·104 [N/m]

GV 1 = 3.75·107 [N/m]

Δ1 = 2.742 [rad]

Δ2 = −3.251 [rad]

Δ3 = −1.001 [rad]

Δ4 = 5.135 [rad]

Coupling C1 bending stiffness fluct. ampl.

Coupling C2 bending stiffness fluct. ampl.

Coupling C3 bending stiffness fluct. ampl.

Coupling C4 bending stiffness fluct. ampl.

H V 1 = 3.12·106 [Nm/rad]

H V 2 = 6.01·105 [Nm/rad]

H V 3 = 9.73·106 [Nm/rad]

H V 4 = 5.04·108 [Nm/rad]

Γ 1 = −1.783 [rad]

Γ 2 = 2.124 [rad]

Γ 3 = 4.944 [rad]

Γ 4 = −0.626 [rad]

Finally, in order to verify an accuracy of the results of the multi-fault identification procedure, and in this way to compare the real and identified fault parameters, a numerical simulation of bending/lateral vibrations of the hybrid model of the blower rotor-shaft system has been performed, taking into consideration simultaneous interactions of the all imperfections considered above. In Fig. 4 there are plotted time histories of the horizontal and vertical vibration velocities of the six bearing housings #1 – #6 obtained theoretically by solving Eqs. (5) and experimentally, where the amplitude-frequency spectra of these measured ones are presented in Fig. 3. From the comparison of the respectively corresponding plots a fairly good similarity of the measured and calculated results is visible. This proves a reliability of the proposed model-based multi-fault identification procedure proposed in this paper. It should be noted that the vibration velocity waveforms obtained both experimentally and computationally, especially in the case of bearing housings #3–#6 supporting both drive motors, are characterized by an additional relatively fast-fluctuating component with a frequency of 100 Hz caused by the typical magnetic pull of the rotors of these motors.

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time [s]

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time [s]

Fig. 4. Measured and simulated horizontal (left column) and vertical (right column) vibration velocities at the bearing supports #1–#6.

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7 Final Remarks The paper proposes a multi-fault identification method for rotating machines based on a structural continuous finite macro-element model of the tested object. In this method, the maximum number of simultaneously identified, the most probable imperfections is limited to the number of courses of lateral/bending vibrations simultaneously monitored on a real object. Here, subsequent peaks of the amplitude spectra of measured dynamic responses of the real object are the basis for determination of parameters of individual types of system’s imperfections with frequencies respectively corresponding to these peaks. In this paper, there were identified residual static and dynamic unbalances, parallel and angular misalignments and local inner anisotropies affecting simultaneously the rotor shaft system of the high-power blower used in the mining industry. Their mutual contributions within the numerical values of the amplitudes of the respective peaks of the frequency spectra recorded by individual sensors result from system’s sensitivity to the excitations caused by these imperfections, which was determined by the modal description of motion of the model of the tested object. Moreover, by means of this method wrong stagger angles of the two opposing blades in the rim of the blower rotor, which resulted in additional lateral/bending vibration components, were detected. It should be emphasized that the practical efficiency and accuracy of the proposed method depends on the credibility of the theoretical model of the rotor-shaft system of the tested machine as well as on sampling densities of the measured signals and the accuracy of FFT analyses performed for them.

References 1. Childs, D.: Turbomachinery Rotordynamics: Phenomena, Modeling, and Analysis. Wiley (1993). ISBN: 978-0-471-53840-0 2. Genta, G.: Dynamics of Rotating Systems. Springer US, New York, NY (2005) 3. Al-Hussain, K.M., Redmond, I.: Dynamic response of two rotors connected by rigid mechanical coupling with parallel misalignment. J. Sound Vib. 249(3), 483–498 (2002) 4. Al-Hussain, K.M.: Dynamic stability of two rigid rotors connected by a flexible coupling with angular misalignment. J. Sound Vib. 266, 217–234 (2002) 5. Lees, A.W.: Misalignment in rigidly coupled rotors. J. Sound Vib. 305, 261–271 (2007) 6. Redmond, I.: Study of a misaligned flexibly coupled shaft system having nonlinear bearings and cyclic coupling stiffness – Theoretical model and analysis. J. Sound Vib. 329, 700–720 (2010) 7. Didier, J., Sinou, J.-J., Faverjon, B.: Study of the non-linear dynamic response of a rotor system with faults and uncertainties. J. Sound Vib. 331, 671–703 (2012) 8. Pennacchi, P., Vania, A., Chatterton, S.: Nonlinear effects caused by coupling misalignment in rotors equipped with journal bearings. Mech. Syst. Signal Process. 30, 306–322 (2012) 9. Malta, J.: Investigation of anisotropic rotor with different shaft orientation. Doctoral Thesis, Darmstadt University of Technology, Department of Machinery Construction, D 17, Darmstadt (2009) 10. Bachschmid, N., Pennacchi, P., Vania, A.: Identification of multiple faults in rotor systems. J. Sound Vib. 254(2), 327–366 (2002) 11. Pennacchi, P., Bachschmid, N., Andrea Vania, A., Zanetta, G.A., Gregorib, L.: Use of modal representation for the supporting structure in model-based fault identification of large rotating machinery: part 1—theoretical remarks. Mech. Syst. Signal Process. 20, 662–681 (2006)

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12. Szolc, T., Tauzowski, P., Stocki, R., Knabel, J.: Damage identification in vibrating rotor-shaft systems by efficient sampling approach. Mech. Syst. Signal Process. 23, 1615–1633 (2009) 13. Szolc, T., Tauzowski, P., Stocki, R., Knabel, J.: Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom using stochastic methods. Nonlinear Dyn. 57, 533–557 (2009) 14. Szolc, T., Konowrocki, R.: Research on stability and sensitivity of the rotating machines with overhung rotors to lateral vibrations. Bull. Polish Acad. Sci. Tech. Sci. 69(6), e137987 (2021) 15. Isermann, R.: Model-based fault detection and diagnosis – status and applications. In: Proceedings of Automatic Control in Aerospace, pp 49–60. Elsevier IFAC Publications, Saint-Petersburg, Russia (2004) 16. Lees, A.W., Sinha, J.K., Friswell, M.I.: Model-based identification of rotating machines. Mech. Syst. Signal Process. 23, 1884–1893 (2009) 17. Bachschmid, N., Pennacchi, P., Vania, A.: Use of modal representation for the supporting structure in model based fault identification of large rotating machinery: part 2 – application to a real machine. Mech. Syst. Signal Process. 20(3), 682–701 (2006) 18. Lal, M., Tiwari, R.: Multi-fault identification in simple rotor-bearing-coupling systems based on forced response measurements. Mech. Mach. Theory 51, 87–109 (2012) 19. Harish Chandra, N., Sekhar, A.S.: Fault detection in rotor bearing systems using time frequency techniques. Mech. Syst. Signal Process. 72–73, 105–133 (2016) 20. El-Mongy, H.H., Younes, Y.K.: Vibration analysis of a multi-fault transient rotor passing through subcritical resonances. J. Vib. Control 24(14), 2986–3009 (2018) 21. Szolc, T.: On the discrete-continuous modeling of rotor systems for the analysis of coupled lateral-torsional vibrations. Int. J. Rotating Mach. 6(2), 135–149 (2000) 22. Xie, F., Li, Z., Ganeriwala, S.: Vibration signal analysis of fan rotors. Technote, SpectraQuest Inc. SQi-14A-052007 (2007) 23. Ye, X., Ding, X., Zhang, J., Li, C.: Numerical simulation of pressure pulsation and transient flow field in an axial flow fan. Energy 129, 185–200 (2017)

Vibration Reduction in an Unbalanced Rotor System Using Nonlinear Energy Sinks with Varying Stiffness Harikrishnan Venugopal(B) , Kevin Dekemele, and Mia Loccufier Department of Electromechanical Systems and Metal Engineering, Ghent University, Tech Lane Ghent Science Park- Campus A 125, 9052 Ghent, Belgium {harikrishnan.venugopal,kevin.dekemele,mia.loccufier}@ugent.be

Abstract. The study of dynamics of rotating machinery has risen to be of great importance in recent decades, with the industry vying for higher operating speeds, higher loads, and lower weight. This trend tends to place the system in the vicinity of its critical speeds, limiting its operational potential, and the traditional approach to reduce vibrations is to attach a tuned mass damper to the system. However, these are tuned based on 1:1 resonance of the system and fail to work effectively otherwise. Accordingly, recent developments have seen the use of the Nonlinear Energy Sink (NES) as their substitute, providing a more robust means of vibration mitigation, by being functional for a broad range of frequencies. In this paper, a passive vibration absorber with nonlinear stiffness is attached to a simple Jeffcott rotor. A speed-dependent force due to mass eccentricity is used here, as is common in rotordynamic systems. The conventional hardening (cubic) stiffness and a newly introduced softening stiffness are studied for their feasibility and compared against each other based on their behaviour and performance. The system’s frequency response is obtained using first-order harmonic balancing and the stability of the solution branches is studied using the multiple time-scales method. Furthermore, a parametric study of the response behaviour and NES performance for various stiffness characteristics is presented. With these methods, one can effectively prove the feasibility of using NESs for vibration mitigation in rotor systems. Keywords: Nonlinear Energy Sink · Harmonic balancing · Jeffcott rotor · Tuning Methodology · Nonlinear Stiffness

1 Introduction The field of rotordynamics has seen a burst of development in the recent decades due to general industry trends towards lower weight, higher power density and higher operating speeds. While understandable in terms of increasing productivity, these developments worsen the issues regarding vibration and stability of rotor systems. In this regard, it may not always be possible to restrict the rotational speed of the system within its first critical speed, and a stable operating speed would need to be obtained by running through one or more critical speeds [1, 2]. Accordingly, it becomes a priority to mitigate the vibrations of the system at these resonances. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 358–373, 2024. https://doi.org/10.1007/978-3-031-40455-9_30

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Nonlinear Energy Sink(s) (NES) come across as a potential solution. They are passive vibration control devices with nonlinear stiffness or damping characteristics, connected to a host system under consideration. NESs are becoming increasingly popular due to their capability to tackle multiple resonant frequencies and act as a broadband vibration absorber [3]. The NES is incapable of completely suppressing the host system resonant response like the TMD but it is more effective in the vicinity of the resonance. Furthermore, NES can perform better in case of perturbations in the host system parameters, in comparison to the traditional Tuned Mass Damper(s) (TMD) [4]. It should be noted that the design of the NES is more challenging than for a TMD, as here the host system’s response is dependent on the forcing magnitude and can have detached bifurcations and regions of quasi-periodic, ‘beating’ oscillations [3, 5]. Recent research has shown that NES is perfectly capable of tackling a myriad of issues in rotordynamics. Ghasem et.al. [6] used NES to deal with the problem of rotorstator contact in an unbalanced Jeffcot rotor. Here, a Jeffcot rotor mounted on journal bearings is subjected to rotor-stator contact force and mass unbalance forces, and thereafter, different configurations of NESs and TMDs are evaluated. The suitability of NES for a more representative primary system with a flexible/rigid rotor with flexible blades is investigated by Bab et al. [7]. Apart from rotor-stator interactions, the mass unbalance force has also been studied by Hongliang et al. [8] using magnetic NES, with both cubic and linear stiffness. The results point favourably towards implementing higher nonlinearity, however its effects on system stability are yet to be observed. As for realizing various nonlinearities, a nonlinear stiffness is modelled in [3] by running the linear spring over a nonlinear path. A similar idea for obtaining custom nonlinear forces is also expounded in [9]. A summary of the former technique is presented in the appendix. In the current study, an NES with nonlinear stiffness is attached to a rotor system with mass unbalance forcing. Two different types of nonlinearities are considered, namely hardening and softening [10]. A parametric study is also conducted to explain the bifurcating and quasi-periodic behaviour of the responses. Finally, both stiffness types are compared on their ability to reduce peak response of the primary system.

2 Theoretical Formulation A representation of a 4-degree of freedom Jeffcott rotor is shown in Fig. 1 above. The rotor rotates with speed ω, and has polar moment of inertia Ip , diametrical moment of inertia Id and a mass m concentrated at a distance e from the axis of its rotation. This eccentricity creates a centrifugal force which is dependent on the rotational speed of the system. The rigid rotor shaft is connected to two bearings A and B, each possessing a stiffness and damping in x and y direction. The rotor is located at a distance h from bearing A. Apart from the linear motion in the x and y coordinates, the rotor can also have rotatory motion in ϕ and θ axes. The equations of motion for this system is as follows: m¨y + kAy (y + hϕ) + kBy (y − (L − h)ϕ) + cAy (˙y + hϕ) ˙ + cBy (˙y − (L − h)ϕ) ˙ = meω2 sin(ωt)

    m¨x + kAx (x − hθ ) + kBx (x + (L − h)θ ) + cAx x˙ − hθ˙ + cBx x˙ + (L − h)θ˙ = meω2 cos(ωt)

(1) (2)

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H. Venugopal et al.   Id θ¨ + Ip ωϕ˙ − hkAx (x − hθ ) + (L − h)kBx (x + (L − h)θ) − hcAx x˙ − hθ˙ + (L − h)cBx   x˙ + (L − h)θ˙ = 0

(3)

Id ϕ¨ − Ip ωθ˙ + hkAy (y + hϕ) − (L − h)kBy (y − (L − h)ϕ) + hcAy (˙y + hϕ) ˙ − (L − h) ˙ =0 cBy (˙y − (L − h)ϕ) (4) In many cases, it is reasonable to assume that the system is symmetric; both in terms of its bearings’ properties (equal stiffness and damping) and the location of the rotor (h = L/2). This decouples the dynamic equations Eqs. (1) and (2). Furthermore, this makes it also reasonable to neglect Eqs. (3) and (4) as the force due to eccentricity doesn’t excite the respective resonances in θ and ϕ coordinates. Hereby we obtain a simplified system as shown in Fig. 2 below. Note that a NES with mass mna , nonlinear stiffness (Fig. 3) kna , and linear damping cna is shown attached to the primary system.

Fig. 1. Representation of the 4-DoF Jeffcot Rotor model, with front view (a) and side view (b)

The dynamics of the simplified Jeffcott rotor with the NES can be expressed by the following equations: m¨x + c˙x + kx + cna (˙x − x˙ na ) + kna F(x − xna ) = meω2 cos(ωt)

(5)

mna x¨ na + cna (˙xna − x˙ ) + kna F(xna − x) = 0

(6)

F(xna − x) = (xna − x)3 , for hardening stiffness

(7)

F(xna − x) = arctan(ks (xna − x)), for softening stiffness.

(8)

where,

Note that the decoupled state of the Eqs. (1) and (2) allows us to treat each√coordinate separately. In this case, they share the same natural frequency at ω1,2 = k/m, and hence the same tuned NES can be used in both coordinates. Realizations of different.

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Fig. 2. Representation of the simplified Jeffcott rotor with NES attachment

Fig. 3. Comparison of stiffness functions

2.1 Application of Harmonic Balancing The principle of Harmonic Balancing (HB) rests on the idea that a periodic solution of an ordinary differential equation can be approximated by its truncated Fourier series. While linear ordinary differential equations require only a single harmonic to represent their solution, nonlinear ones lead to solutions with multiple dominant harmonics. An in-depth explanation of the method is provided in [11]. Examples of its implementation for both transient and forced systems are also seen in [3, 12, 13]. Eqs. (5) and (6) are modified as follows: x¨ + μω0 ξ x˙ + ω02 x + μ¨xna = μω2 Pcos(ωt)

(9)

μ¨xna + μω0 ξna (˙xna − x˙ ) + μω02 γ F(xna − x) = 0

(10)

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where, mω02 = k μω0 ξ m = c ω0 ξna mna = cna mμ = mna mna γ ω02 = kna

μP = e

Here the solution of the equations via HB is represented as a first order harmonic. Thus we introduce the following expression: x˙ + x = 2A(t)eiωt iω

(11)

z˙ + z = 2B(t)eiωt iω

(12)

Both A and B are time dependent complex variables containing the amplitude and phase information of the response and z = xna − x. The time dependence is not shown further explicitly. These variables can be used to describe a slow moving envelope of vibration [14]. A major advantage of complexification is that by separating the oscillations term (eiωt ) and the envelope (A and B), the second-order equations of motion can be modified to a first-order equations for the envelope motion whereafter the fixed points and their stability can be studied with ease. From Eqs. (11) and (12) we get:  ∗ x˙ x˙ (13) +x+ + x = 2(Aeiωt + A∗ e−iωt ) ⇒ x = Aeiωt + A∗ e−iωt iω iω  ∗ x˙ x˙ +x− + x = 2(Aeiωt − A∗ e−iωt ) ⇒ x˙ = iω(Aeiωt − A∗ e−iωt ) (14) iω iω Similarly, z = xna − x = Beiωt + B∗ e−iωt

(15)

z˙ = iω(Beiωt − B∗ e−iωt )

(16)

The superscript * represents complex conjugate. Differentiating Eq. (11) we get: x¨ ˙ iωt + 2iωAeiωt + x˙ = 2Ae iω

(17)

Substituting Eqs. (14) and (13) in Eq. (17) we get the following relation: ˙ iωt x¨ + ω2 x = 2iωAe

(18)

˙ iωt z¨ + ω2 z = 2iωBe

(19a)

Similarly,

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For first order HB, the Fourier series coefficient of the first harmonic of the nonlinear stiffness function F(z) is also calculated as follows: 2π

F1 (z) = B.f (B, B∗ ) =

  ω ω ∫ kna F B, B∗ e−iωt dt 2π 0

(19b)

  f B, B∗ = 3B.B∗ Equation for hardening stiffness   f B, B∗ =



(20)

4BB∗ ks2 + 1 − 1 Equation for softening stiffness 2BB∗ ks

(21)

Substituting Eqs. (11)–(19b) in Eqs. (9) and (10) we get: ˙ − ω2 A + 2Biω ˙ − ω2 B) = 2iωA˙ + (ω02 − ω2 )A + iμω0 ωξ B + μ(2Aiω

μω2 P 2

˙ − ω2 B + 2iωA˙ − ω2 A + iξna ω0 ωB + ω02 γ Bf (B, B∗ ) = 0 2iBω

(22) (23)

In order to find the response at the state of steady oscillation, we assume A˙ = B˙ = 0 . Thus, the above equations become: √ (24) σ A + iξ X A − XA − XB = PX /2 √ iξna X B − XA − XB + γ f (B, B∗ ) = 0

(25)

where, X =

ω2 and 1 − X = μσ ω02

Squaring the real and imaginary parts of Eq. (25) respectively we obtain:   2    2 X 2 A2 = B2 ξna X + γ f B, B∗ − X

(26)

(27)

Multiplying Eq. (24) by X and substituting in Eq. (25) for XA we get:  √ √  √     i σ ξna X + ξ X γ f B, B∗ − X − X ξna X B       PX 2 + (σ − X ) γ f B, B∗ − X − ξ ξna X − X 2 B = 2

(28)

Squaring the real and imaginary part of Eq. (28) ,we get: 

 2     2 2    B (X − σ ) γ f B, B∗ − X + ξ ξna X + X 2 + X ξna (X − σ ) + ξ X − γ f B, B∗ P2 X 4 = 4

(29)

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Since the variables A and B are complex in nature, they can be expressed in their polar form as follows: A=

a iα b e B = eiβ 2 2

Substituting Eq. (30) in Eqs. (27) and (29) we get:   2 X 2 a2 = b2 ξna X + (γ f (b) − X )2

(30)

(31)

  2 (X − σ )(γ f (b) − X ) + ξ ξna X + X 2 + X (ξna (X − σ ) + ξ (X − γ f (b)))2 b2 = P 2 X 4 (32)

For a given value of X, Eq. (32) can be solved for b. Equation (31) is the Slow Invariant Manifold (SIM) relating a and b. It should be noted that the relation is also dependent on the frequency ratio X and forcing P. Dividing Eq. (31) by Eq. (32) we get the expression for the nonlinear frequency response of the primary system:   2 X + (γ f (b) − X )2 X 2 ξna a = (33) P D(X , b) where, D(X , b) =



(X − σ )(γ f (b) − X ) + ξ ξna X + X 2

2

 + X (ξna (X − σ ) + ξ (X − γ f (b)))2

2.2 Estimation of Solution Stability From the equations above we have found the fixed points a and b of the system by assuming steady state. To characterize the stability of these points, we use the multiple time-scales theory [3, 12, 13]. Here we interpret the total time spent as the sum of two time scales, one moving across the time domain faster than the other: τ1 = ω0 t τ2 = μω0 t

(34)

Here τ1  τ2 , implying that τ1 is the faster time scale. The main advantage of this method becomes evident, as we are more interested in the long-term evolution of the response envelope. The time derivative is modified as shown below: d ∂ ∂ = ω0 + μω0 dt ∂τ1 ∂τ2

(35)

Substituting the modified derivative into Eqs. (22) and (23) and reorganizing the terms according to the power of μ we get: μ0 : 2iω0 ω

2iω0 ω

∂A =0 ∂τ1

∂B − ω2 A − ω2 B + iω0 ωξna B + ω02 γ B.f (B, B∗ ) = 0 ∂τ1

(36a) (36b)

Vibration Reduction in an Unbalanced Rotor System

μ1 :

2iω0 ω

P ∂A + ω02 σ A + iω0 ωξ B − ω2 A − ω2 B = ω2 ∂τ2 2 2iω0 ω

∂B ∂A + 2iω0 ω =0 ∂τ2 ∂τ2

365

(37a) (37b)

Rewriting the Eq. (36b) in terms of X we get: √    ∂B 1  = A, B, B∗ , X = √ iXB + iXA − iω02 γ Bf (B, B∗ ) − ξna X B ∂τ1 2 X

(38)

Equation (38) is linearized at the fixed points previously found: B = Beq + B

(39a)

  ∂B ∂B =0+ ∂τ1 ∂τ1

(39b)

Combining the relation from Eq. (38) and expanding, we get:     ∂ ∂ ⇒  = B + B∗ ∂B B=Beq ∂B∗ B=Beq     ∂ ∂ ∗ ∗ a11 = a22 = a12 = a21 = ∂B B=Beq ∂B∗ B=Beq 





a11 a12 B  = a21 a22 B∗  ∗

(40) (41)

(42)

The Eq. (42) is the linearized state equation of the system and the eigenvalues of the Jacobian matrix dictate the stability. Positive real eigenvalues indicate instability of the chosen fixed point.

3 Parametric Analysis on Response Behaviour In this analysis, a parametric study is conducted by varying the stiffness parameters, keeping other parameters constant. For hardening stiffness, this means changing the factor γ and for softening stiffness both ks and γ are to be varied (see Eqs. (5)–(8)). The system is analysed close to its resonance, by varying frequency ratio (ω/ω0 ) from 0.9 to 1.1. Firstly, a numerical simulation of the system is performed by using the Runge-Kutta (RK) method, implemented by the ODE 45 function in MATLAB. The envelope of time signal is obtained, and the upper and lower values of the envelope are plotted in blue and orange circles respectively (Fig. 4). Thereafter, the solution from the HB is compared with the numerical solution and with a primary system without an NES. Stability of the solution branches are also evaluated. Stable branches are shown in black and the unstable ones are shown in red.

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Value

ξna

0.25

ξ

1

γ

1–400

μ

0.01

ω0 [rad/s]

1

e[m]

0.001

3.1 Primary system with hardening NES The parameter values for the primary system with hardening NES are given in Table 1 below. Note that the dimensionless parameters of Eq. (9) and (10) can be used as inputs, so that a general class of system can be defined. In Fig. 4, we can see the evaluation of the primary system frequency response w.r.t the stiffness factor γ . It is observed that for all γ the numerical simulations agree closely with the HB-solutions, except in the regions of HB-unstable. Increasing this constant increases the degree of nonlinearity of the stiffness force. This means that at higher γ the displacement will get to traverse regions of higher nonlinearity (i.e. Here, as γ increases, the response at the primary system’s resonance is indeed being suppressed, but with the introduction of certain unconventional behaviour. At γ = 55 (Fig. 4b and c), we see regions of instability arise near the resonance. However, it should be noted that the stability here only means that the envelope magnitude is not steady as assumed (A˙ = B˙ = 0); as from simulations it is seen that the response resembles that of a quasi-periodic response (Fig. 8). This phenomenon is also referred to as NeimarkSacker bifurcation [15]. This is also shown in the plot, as the upper and lower envelope values don’t coincide in this region. At γ = 185 (Fig. 4d), we see the inception of a bifurcation, detached from the main curve. This is referred to as an Isolated Resonance Curve (IRC), and is often considered to be disadvantageous for NES tuning. This is because of the possibility that the response can fall into to these resonance curves when started at certain initial conditions. As γ increases, the IRCs become larger in size and merge with the main curve at γ = 200 (Fig. 4e). At this point we can see how the numerical solution (with zero initial displacement and velocity) follows the now attached part of the IRC. As γ increases further, this results in the formation of two separate bifurcations (Fig. 4f), one on either side of the resonance peak. Additionally, the peak of the almost-attached IRC, becomes the new response peak. The two bifurcations diverge from each other as γ increases. It should be noted that increasing the damping via ξna or ξ , only raises the lower limit of values of γ where a particular phenomenon is found. The behavior remains just the same.

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Fig. 4. Frequency response of the primary system for various kna for NES with hardening stiffness

3.2 Primary System with Softening NES The arctangent function of the softening stiffness force has two main factors namely γ controlling the ceiling of the restoring force, and ks controlling the degree of nonlinearity of the stiffness function. Figure 5 below shows the influence of these factors. Table 2 details the input values chosen for analysis.

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H. Venugopal et al. Table 2. Parameter values for primary system with softening NES. Parameter

Value

ξna

0.25

ξ

1

γ

0.01–1

μ

0.01

ω0 [rad/s]

1

ks

10–60

e[m]

0.001

Fig. 5. Arctangent function with factors C and γ

In the case of softening stiffness, both γ and ks are varied as shown in Fig. 6 below. In general for all values of γ and ks , the numerical simulations agree well with the HB solution, except in the region of HB-unstable. This is because of the previously mentioned Neimark-Sacker bifurcations. For a given ks , there is an optimal γ above and below which the response of the primary system approaches to that of without having the NES (see Fig. 6a–c). This is because changing γ also changes the slope of the initial, approximately-linear region of the arctan function. Thus, it can also be the case that the response purely lies within the quasi-linear region of the stiffness function. Changing ks also causes the same effect, but with the additon of changing the nonlinearity of the function. Therefore, it causes the typically nonlinear behaviour of having

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Fig. 6. Frequency response of the primary system for various γ and C for NES with softening stiffness

bifurcations and regions of quasi-periodic responses. In this case, the response displacement stays below the limit where the stiffness force becomes constant (see Fig. 5), and thus linearity is still prominent. Thus at higher ks the linear part of the stiffness curve has shifted from its optimum. This explains the effect shown in Fig. 6d–f. Additionaly, the lack of IRCs should also be noted, as it is a significant advantage in terms of tuning.

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As seen for hardening stiffness, the damping has no effect on the possible behaviour that can be observed and only affects the range of values where a behaviour is observed.

4 Comparison of Stiffness Characteristics Both stiffness models are compared based for their vibration suppression capabilities and their behaviour qualities for the primary system given in Tables 1 and 2. It should be noted that bifurcations and unstable regions (i.e. the typically nonlinear behaviour) arise only when the response goes through a highly-nonlinear region in the stiffness function. This can be the case when there is either high forcing (via eccentricity e) or light damping or when the nonlinearity constants are high. Since there is only a single factor to control in hardening stiffness (i.e. γ ), which decides its degree of nonlinearity, increasing γ to lower the response makes this aforementioned behaviour unavoidable. This is in contrast to softening stiffness, where the nonlinearity factor ks can be modified separately from saturation factor γ . Additionally, from the parametric analysis it is evident that for the given range of values IRCs are absent in the system with the softening stiffness, which is greatly advantageous for tuning applications. The comparison with optimal values is visualized in Fig. 7. Here the HB solutions and the upper envelope values from numerical simulation are presented for both cases. Based on the parametric analysis, for hardening stiffness, increasing γ leads to lower response at primary system resonance. However, γ = 183 is chosen as the optimal value for hardening stiffness, as higher values lead to IRCs of high peak response. Similarly,

Fig. 7. Optimal response curve for both hardening and softening NES.

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  Fig. 8. Comparison of scaled time responses at ωω0 = 1, for both softening (a) and hardening (b) NES attachment.

for softening stiffness an optimal γ and ks is found based on its response reduction. The Hence, value of ks is also selected so as to include a reasonable degree of nonlinearity.   ω γ = 0.1 and ks = 14 is taken as the optimum. A time simulation at ω0 = 1 is shown in Fig. 8. The response plot above shows a clear advantage for the softening stiffness. It should be noted that at the instability regions in the HB-hardening, the numerically calculated envelope limits are better estimates of the response magnitude. The NES with softening stiffness is able to achieve 79% response reduction at the primary system resonance when compared to 62% with the hardening NES. This can also be verified from the time simulation in Fig. 8. Furthermore, the amplitude modulation of the quasi-periodic response for the hardening NES could also be considered undesirable. For modelling purposes, the optimized stiffness function can be used to obtain the nonlinear profile f (x) for the realization shown in Appendix A.

5 Conclusion A simplified Jeffcott rotor model coupled to an NES with stiffness nonlinearity is explored in this paper for its feasibility, primarily through the Harmonic Balancing method. The HB method has been successfully implemented in this regard and validated using response envelopes from time simulations; save for the regions detected by HB as unstable, where a quasi-periodic response is observed. An intuitive comparison between hardening and softening stiffness of the NES has been made based on their behaviour and their vibration absorption capability. It has been found that the softening NES has a distinct advantage over the conventional hardening NES, both in terms of avoiding IRCs and in terms of optimal response attenuation. The current model relies on the simplification that the rotor system is symmetric, removing the cross-coupling of coordinates. In the future, a more generalized (non-symmetric) rotordynamic system will be considered, with inclusion of coupling and gyroscopic effects. The robustness of the NES, i.e. usefulness of de-tuning, would also be explored in detail.

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Appendix A: Realization of Stiffness Nonlinearity This section details the method used in [3] to create nonlinear stiffness for the NES attachment. Here the NES is attached to a linear spring with a rolling element on the other side. The rolling element is made to follow a nonlinear guided path designed according to the optimized parameters of the nonlinear stiffness function. Figure 9 below shows the principle.

Fig. 9. Realization of custom nonlinear stiffness using a nonlinear profile f (x) (a) with its force balance (b) [3]

The NES mass mna , with motion along x, is attached to a linear spring with spring constant kl , which is compressed according to a nonlinear profile f (x). The reaction forces Fx and Fy are defined as follows: Fx = 2Rsin(θ ), Fy = Rcos(θ ),

(43)

⇒ Fx = 2Fy tan(θ ),

(44)

where tan(θ ) = ∂f∂x(x) and Fy = kl f (x) The given stiffness function Fx will be equated according to Eq. (44) to obtain the profile f (x).

References 1. Friswell, M.I., et al.: Dynamics of Rotating Machines. Cambridge University Press (2010). https://doi.org/10.1017/CBO9780511780509 2. Genta, G.: Dynamics of Rotating Systems. Springer US, New York, NY (2005) 3. Dekemele, K.: Performance measures for nonlinear energy sinks in mitigating single and multi-mode vibrations: theory, simulation and implementation. Diss. Ghent University (2021) 4. Den, H., Pieter, J.: Mechanical Vibrations. Courier Corporation (1985)

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5. Kuether, R.J., et al.: Nonlinear normal modes, modal interactions and isolated resonance curves. J. Sound Vib. 351, 299–310 (2015). https://doi.org/10.1016/j.jsv.2015.04.035 6. Tehrani, G.G., Dardel, M., Pashaei, M.H.: Passive vibration absorbers for vibration reduction in the multi-bladed rotor with rotor and stator contact. Acta Mech. 231(2), 597–623 (2019) 7. Bab, S., et al.: Vibration attenuation of a continuous rotor-blisk-journal bearing system employing smooth nonlinear energy sinks. Mech. Syst. Signal Process. 84, 128–157 (2017) 8. Yao, H., Zheng, D., Wen, B.: Magnetic nonlinear energy sink for vibration attenuation of unbalanced rotor system. Shock Vib. 2017, 1–15 (2017) 9. Zou, D., et al.: A device capable of customizing nonlinear forces for vibration energy harvesting, vibration isolation, and nonlinear energy sink. Mech. Syst. Signal Process. 147, 107101 (2021) 10. Dekemele, K., Habib, G.: Inverted resonance capture cascade: modal interactions of a nonlinear energy sink with softening stiffness. Nonlinear Dyn. 111, 9839–9861 (2023) 11. Krack, M., Gross, J.: Harmonic Balance for Nonlinear Vibration Problems, vol. 1. Springer International Publishing, Cham (2019) 12. Dekemele, K., Habib, G., Loccufier, M.: The periodically extended stiffness nonlinear energy sink. Mech. Syst. Signal Process. 169, 108706 (2022) 13. Dekemele, K., Habib, G., Loccufier, M.: Vibration mitigation with a nonlinear energy sink having periodically extended stiffness. In: ISMA: International Conference on Noise and Vibration Engineering (2022) 14. Smirnov, V.V., Manevitch, L.I.: Complex envelope variable approximation in nonlinear dynamics. Rus. J. Nonlin. Dyn. 16(3), 491–515 (2020) 15. Kuznetsov, Y.A., Sacker, R.J.: Neimark-Sacker bifurcation. Scholarpedia 3(5), 1845 (2008)

Modelling and Validation of Rotor-Active Magnetic Bearing System Considering Interface Contact Yang Zhou1 , Jin Zhou1(B) , Jarir Mahfoud2 , Yue Zhang1 , and Yuanping Xu1 1 College of Mechanical and Electrical Engineering,

Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China {zhouyang0216,zhj,zhangyue08,ypxu}@nuaa.edu.cn 2 University of Lyon, Lyon, France

Abstract. Rotor-active magnetic bearing (rotor-AMB) system is increasingly applied in rotating machinery such as blowers, compressors, vacuum pumps, etc. However, due to the coupling of interface contact (such as bolt-joint surface and cylindrical-joint surface) formed by impeller assembly and AMB supporting, the rotor will vibrate at a modal frequency once levitated, which causes system instability and equipment breakdown. Therefore, it is essential to reveal the principle of this vibration. In rotor modelling, interface contact is equivalent to an additional stiffness matrix based on a massless spring unit. The value of the matrix is obtained by energy law. Subsequently, interface contact’s influence is considered an internal disturbance in the rotor-AMB system. This modelling method is validated by different test rigs. Through rotor response and stability analysis of different test rigs, it is found that rotor modal resonance mode depends on contact interface parameters (such as bolt pre-tightening force). Based on these results, stability analysis is carried out to study the general relationship between system stability and contact interface parameters, which will further reveal the vibration mechanism and guide the following maglev machine structural design to avoid this vibration. Keywords: Interface Contact · Contact stiffness · Active Magnetic Bearing · Mechatronics Modelling · Modal Resonance

1 Introduction Active magnetic bearings (AMBs) have been widely used in centrifugal gas compressors and other high-speed rotating machinery applications [1]. AMBs do not need lubrication, and their non-contact environment significantly reduces friction, making the system efficient. Another attractive advantage of AMBs is that the electromagnetic force is controlled and changed actively through AMBs, making the rotor work stably at high speed. As an indispensable part of rotating machinery, the impeller is also needed in the magnetic levitation compressor. However, rotor assembled with the impeller constantly vibrates intensely in high frequency during levitation which significantly influences © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 374–390, 2024. https://doi.org/10.1007/978-3-031-40455-9_31

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the system’s stability. This phenomenon is successively found in magnetic levitation machines like blowers, compressors, vacuum pumps, etc. This problem is caused by the coupling of the influence of assembly interface contact and complex electromechanical characteristics of AMB system. Few types of researches on these two aspects have been carried out. On the aspect of the influence of assembly interface contact, Tan [2] modeled the shrink-fit interface of the rotor as the internal damping force and analyzed it’s influence on the system stability by decomposing the asymmetric stiffness component from the system equations. Liu [3] adopted the spring unit to simulate the interface contact formed by rod-fastened structure and analyzed the instability factor caused by the odds of the pretightening force of the rod. Liao et al. [4] and Liu et al. [5] adopted thin-layer elements to simulate the complicated interface contact based on the finite element method. The interface contact significantly influences the system stability from the researches above. However, researches are all based on the excitation of rotation, and a complex model of interface contact is not suitable in AMB system. On the other aspect of the complicated electromechanical characteristics of the AMB system, Yannick Paul [6] mentioned that the bad connection between the rotor and the impeller may cause high-frequency vibration during levitation. Wei [7] equaled the influence of shrinkage fit in AMB-rotor to a disturbance in the direction of rotation considering that the rotor is rigid. Simon [8, 9] analyzed the influence of material internal damping in the direction of translation based on modelling of a flexible rotor and design a robust controller based on the model above. The interface contact can only be equivalent to disturbance rather than the influence on system stability due to the limitation of rotor modelling. However, the modelling of electromechanical characteristics of the AMB system can be combined with the modelling of interface contact. The researches above lack the study of the influence of interface contact on internal system and fail to obtain the clear relationship between disturbance and property of contact interface such as magnitude of interference. In this paper, we present the modelling of the magnetic bearing-rotor system considering interface contact based on the modelling of interface contact and modelling of the magnetic bearing-rotor system. The spring unit with specific contact stiffness is used to simulate the interface contact of the rotor and impeller in the form of an additional stiffness matrix added to the system dynamics equation. The dynamic characteristic of magnetic bearing-rotor system considering interface contact is obtained by comparing the response and frequency of vibration of different pre-tightening torque based on simulation and experiment results. This paper proceeds as follows. Section 2 presents the modelling of magnetic bearingrotor system and interface contact. In Sect. 3, the numerical simulation and the experimental validation are described. Section 4 presents the stability analysis. Section 5 concludes the paper.

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2 Modelling of the Rotor-AMBs System Considering Interface Contact 2.1 Modelling of the Rotor-AMBs System As shown in Fig. 1(a), the rotor-AMBs system is divided into three parts: the rotor model, the model of the electronic control system, and the model of interface contact. The structure of a radial magnetic bearing is shown in Fig. 1(b). A sensor measures the displacement of the rotor from its reference position, a microprocessor as a controller derives a control signal from the measurement, a power amplifier transforms this control signal into a control current, and the control current generates a magnetic field in the actuating magnets, resulting in magnetic forces in such a way that the rotor remains in its hovering position.

(a) Structure of the rotor-AMBs system

(b) Structure of the radial magnetic bearing Fig. 1. Schematic diagram of rotor-AMB mechatronic system considering interface contact.

The theory of the Bernoulli-Euler beam is applied to the finite element modelling of the rotor. The equation of the rotor-AMB mechatronic model takes the following form: MR q¨ + (CR + ωGR )˙q + KR q = TaT fa + TuT fu

(1)

where M R , C R , GR , and K R respectively represent the mass matrix, the damping matrix, the gyroscopic matrix, and the stiffness matrix of the rotor, ω is the rotation frequency, f a

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= [F ax F ay F bx F by ], represents the electromagnetic force vector in the x and y directions generated by the AMBs A and B, T a is the transfer matrix of the AMB nodes, and f u represents the unbalanced force vector in the x, y directions, T u is the transfer matrix of the unbalance mass nodes. q are system displacements and angles vector under the Cardan description. The radial magnetic force is modeled as a generalized external force. Near the equilibrium point, the magnetic force can be approximated as Eq. (2) T  (2) fa = FAMB,Ax FAMB,Ay FAMB,Bx FAMB,By = kx qa + ki ia where k x and k i are displacement stiffness and current stiffness, respectively. qa and ia are the radial displacements of AMBs nodes and control currents. The relationship between control currents and the radial rotor displacement qs detected by sensors is given as Eq. (3)   KD s ka (2π fz ) KI + qs ia = S(s)A(s)C(s)qs = ks · · KP + (3) s+(2π fz ) TI s + 1 TD s + 1 where S(s), A(s), and C(s) is the transfer function of the sensor, amplifier, and the controller, respectively. k s is the gain of the sensor. k a is the gain of the power amplifier, and f z is the cut-off frequency. K P , K I and K D respectively represent the proportional gain, the integral gain, the derivative gain. T D is the derivative time constant. 2.2 Modelling of Interface Contact In the modelling of interface contact, the interface contact between the disk and the rotor is equivalent to spring units uniformly distributed over the contact interface, as shown in Fig. 2(a). The spring unit connects any point A (x A , yA , zA ) on the rotor contact interface and its corresponding point B (x B , yb , zB ) on the disk contact interface. The stiffness of the massless spring unit is referred to as contact stiffness. The contact stiffness can be subdivided into the normal contact stiffness k f and the tangential contact stiffness k q . The directions of k f and k q are perpendicular and parallel to the contact interface, respectively. There is relative deformation between the contact interfaces when the rotor vibrates. The energy generated by the spring deformation is calculated to study the influence of interface contact. As shown in Fig. 2(b), floating coordinates oi x 3 y3 z3 and oj x 3 y3 z3 are established by taking the contact interface centers i and j as the coordinate origin and coordinate transformation in order to obtain the relative deformations of the spring unit in x, y, z directions. The generalized coordinates of centers i and j are (x i ,yi ,α i ,β i ,zi ,ωt) and (x j ,yj ,α j ,β j ,zj ,ωt) in the absolute coordinate system oxyz, where x i , yi , and zi (x j , yj , and zj ) are the translations of i ( j) in the x, y and z directions and α i , β i, and ωt (α j , β j, and ωt) are the rotations of i ( j) around the x, y and z axis, x 3 y3 plane coincides with the contact interface and the orientation of z3 is perpendicular to the contact surface. In a floating coordinate system, the coordinates of points A and B can be determined by the polar coordinates (r,θ ), where r represents the distance between point A and center i or the distance between point B and center j, θ represents the angle between line Ai, Bj and x 3 axis, counterclockwise.

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(a) Dynamic model of interface contact

(b) Diagram of coordinate transformation Fig. 2. Modelling of interface contact.

The total energy generated by the spring unit is obtained by summing up the unit spring energy u as Eq. (4). ¨  1 kq x2 + kq y2 + kf z 2 d A u= A 2 ¨ 

2

2  = 0.5kq d A xj − xi + yj − yi (4) A ¨ ¨

2

2 y2 d A αj − αi + 0.5kf x2 d A βj − βi +0.5kf A

A

where x = x B -x A , y = yB -yA , z = zB -zA , R is the radius of the contact interface, Rb is the radius of the bolt hole. The total energy of the spring unit can be expressed as the elastic potential energy Eq. (5).   T 1 qi qj Ke qi qj 2



diag kp , kp , kc , kc −diag kp , kp , kc , kc

Ke = −diag kp , kp , kc , kc diag kp , kp , kc , kc     kp = kq π R2 − R2b , kc = 0.25kf π R4 − R4b u=

(5)

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where K e is the additional stiffness matrix generated by interface contact. Based on the Lagrange equation, the force generated by the interface contact is expressed Eq. (6).  T Fc = Ke qi qj = Ke Tc q

(6)

The disturbance introduced by the disk assembly arises from the slight fluctuation between the contact interfaces during the suspension. To sum up, the closed-loop model of the rotor-AMBs system considering interface contact can be expressed as Eq. (7). MR q¨ + (CR + ωGR )˙q + KR q = kx TaT Ta q + ki TaT ia + TcT Ke Tc q

(7)

where T c is the transfer matrix of the interface contact nodes m and n. The disturbance F c acts on the system as the generalized force and its value is related to the rotor’s vibration amplitude. The response of the closed-loop system is influenced by the disturbance F c . The disturbance will excite the unstable modal vibration. When the modal frequency is within the control bandwidth of the magnetic bearing, the system is able to control the disturbance. But when the modal frequency is out of the control bandwidth of the magnetic bearing, the control effect will decrease as the disturbance increase (Fig. 3).

Fig. 3. Diagram of the closed-loop mathematical model.

3 Numerical Simulation and Experiment Validation of Different Magnetically Suspended Rotors In this section, two different magnetically suspended rotors are modelled and the influences of interface contact are investigated. The numerical simulation results are verified by experiments.

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3.1 Magnetic Levitation Blower The structure of the magnetic levitation blower is shown in Fig. 4. It can be seen that the impeller is connected to the rotor by bolted joint, which forms the interface contact between the rotor and the impeller.

Fig. 4. Typical structure of the magnetic levitation blower.

Based on the modelling method, the mechatronic model of the magnetic levitation blower is established. The rotor is modeled with 74 beam elements for a total mesh of 296 DOF for the lateral analysis (see Fig. 5). Based on the modal test, the experimental and theoretical bending mode frequencies of the rotor with bolted joint are shown in Table. 1. The simulation of the mechatronic model considering interface contact is shown in Fig. 6(a), (b). The system is unstable and the rotor vibrates at the second bending mode when the disturbance F c caused by interface contact acts on the system. The vibration of the second bending mode is excited by interface contact, the displacements of the rotor are increased rapidly.

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Fig. 5. Theoretic mode shapes of the free-free rotor (magnetic levitation blower)

Table. 1. The bending mode frequencies of the rotor in magnetic levitation blower. Order

Experimental frequency (Hz)

Theoretical frequency (Hz)

1st mode

735.12

2nd mode 3rd mode

1245.66

1234.7

1892.25

2057.3

730.05

Figure 7(a), (b) presents the experimental results when the disk is connected to the rotor by bolt fastening. The rotor displacement in the Ax direction is in a similar tendency to simulation results. From spectral analysis (fast Fourier transform, FFT), the second bending mode frequency exists when interface contact exists in the rotor. The influences of interface contact and order of vibration mode agree with the simulation, which verifies the model’s accuracy. The influence of the normal contact stiffness on the rotor displacement is shown in Fig. 8. Varying the value of the normal contact stiffness from 6 × 1013 N/m3 to 10 × 1013 N/m3 , as shown in Fig. 8(a), time domain response shows that, as k f increases, the amplitude of the response increases and instability becomes more apparent. In the frequency domain shown in Fig. 8(b), the second bending mode frequency vibration increases as the normal contact stiffness increases.

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(a) Rotor time-domain response

(b)Rotor frequency-domain response Fig. 6. Rotor response in simulation (magnetic levitation blower).

3.2 Magnetic Levitation Motor The structure of the magnetic levitation motor is shown in Fig. 9. The coupling is connected to the rotor by bolted joint, which forms the interface contact between the rotor and the coupling. The mechatronic model of the magnetic levitation blower is established based on the modeling method. The rotor is modeled with 76 beam elements for a total mesh of 304 DOF for the lateral analysis (see Fig. 10). Based on the modal test, the experimental and theoretical bending mode frequencies of the rotor with bolted joint are shown in Table. 2.

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(a) Rotor time-domain response

(b)Rotor frequency-domain response Fig. 7. Rotor response in experiment (magnetic levitation blower).

The simulation of the mechatronic model considering interface contact is shown in Fig. 11(a), (b). The system is unstable and the rotor vibrates at the first bending mode when the disturbance F c caused by interface contact acts on the system. The vibration of the first bending mode is excited by interface contact, the displacements of the rotor are increased rapidly. Figure 12(a), (b) presents the experimental results when the disk is connected to the rotor by bolt fastening. The rotor displacement in the Ax direction is in a similar trend as simulation results. From spectral analysis, there is the first bending mode frequency when interface contact exists in the rotor. The influences of interface contact and order of vibration mode agree with the simulation, which verifies the model’s accuracy.

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(a) Comparison of rotor time-domain responses (different normal contact stiffness)

(b) Comparison of rotor frequency-domain responses (different normal contact stiffness) Fig. 8. Comparison of dynamic responses in magnetic levitation blower (different normal contact stiffness).

Fig. 9. Typical structure of the magnetic levitation motor.

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Fig. 10. Theoretic mode shapes of the free-free rotor (magnetic levitation motor).

Table. 2. The bending mode frequencies of the rotor in magnetic levitation motor. Order

Experimental frequency (Hz)

1st mode 2nd mode

Theoretical frequency (Hz)

720.87

705.9

1503.89

1511.13

The influence of the normal contact stiffness on the rotor displacement is shown in Fig. 13. Varying the value of the normal contact stiffness from 6 × 1013 N/m3 to10 × 1013 N/m3 , as shown in Fig. 13(a), time domain response shows that, as k f increases, the amplitude of the response increases and instability becomes more obvious. In the frequency domain shown in Fig. 13(b), the first bending mode frequency vibration increases as the normal contact stiffness increases.

4 System Stability Analysis For linear time-invariant (LTI) systems, root locus analysis is applied in this study. The stability analysis of the magnetic levitation blower system and the magnetic levitation motor system are depicted in Fig. 14, respectively. From the root locus of magnetic levitation blower system, varying the value of the normal contact stiffness from 1 × 1013 N/m to 10 × 1013 N/m, the root locus of the closed-loop system is shown in Fig. 14(a). As k f increases, the characteristic roots of the third bending modes move towards the negative direction of the real axis and the real parts of these roots are all negative. On the contrary, as k f increases, the characteristic roots of first and second bending modes move towards the positive direction of the real axis and the real parts of the second bending mode roots turn from negative to positive. As the real part of a system’s characteristic root becomes positive, the system becomes unstable. As k f increases, the real parts of the second bending mode roots become positive, which leads to high-frequency vibration. Because the first bending mode roots also move towards the positive direction, the vibration of the first bending mode is also excited, as shown in Fig. 6 and Fig. 7. But these roots do not cross the imaginary axis,

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(a) Rotor time-domain response

(b) Rotor frequency-domain response Fig. 11. Rotor response in simulation (magnetic levitation motor).

the vibration amplitude of the first bending mode is smaller than the vibration amplitude of the second bending mode. From the root locus of the magnetic levitation motor system, varying the value of the normal contact stiffness from 1 × 1013 N/m to 10 × 1013 N/m, the root locus of the closed-loop system is shown in Fig. 14(b). As k f increases, the characteristic roots of the second and third bending modes move towards the negative direction of the real axis and the real parts of these roots are all negative. On the contrary, as k f increases, the characteristic roots of first bending modes move towards the positive direction of the real axis and the real parts of the first bending mode roots turn from negative to positive. As the real part of a system’s characteristic root becomes positive, the system becomes unstable. As k f increases, the real parts of the first bending mode roots become positive, which leads to high-frequency vibration.

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(a) Rotor time-domain response

(b) Rotor frequency-domain response Fig. 12. Rotor response in experiment (magnetic levitation motor).

The contact stiffness is related to the pre-tightening torque [10], the normal and tangential contact stiffness increase as the increase of the pre-tightening torque. To sum up, when the pre-tightening torque of the bolted joint increases, the contact stiffness increases, which will make the roots of the unstable mode move towards the positive direction and cross the imaginary axis. As a result, unstable modal vibration will be caused.

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(a) Comparison of rotor time-domain responses (different normal contact stiffness)

(b) Comparison of rotor frequency-domain responses (different normal contact stiffness) Fig. 13. Comparison of dynamic responses in magnetic levitation motor (different normal contact stiffness).

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(a) Magnetic levitation blower system

(b) Magnetic levitation motor system Fig. 14. Root locus of the closed-loop system.

5 Conclusion This paper studies the modal vibration caused by interface contact in rotor-AMB systems when the rotor is levitated without rotating by establishing the mechatronic rotor-AMB model considering the interface contact. The mechanism of this high-frequency vibration and the contributing influences of the pre-tightening torque are revealed by combining the results of dynamic analysis and experimental validation. There is a slight fluctuation of the rotor when being levitated by AMBs because rotorAMB systems are active systems. Therefore, there are multivariant slight vibrations in the contact interfaces formed by the rotor and the disk. Such vibrations are amplified into an internal disturbance of the system by the effect of the contact stiffness matrix of the contact interface. The internal disturbance caused by the interface contact and the energy input from the AMB supporting increases as the pre-tightening torque increases. From the results of root locus analysis, as the internal disturbance increases, the roots of

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the unstable modes move towards the positive direction of the real axis and the system stability weakens. When the real part of a root turns from negative to positive, the system turns from stable to unstable, and modal vibration is excited.

References 1. Schweitzer, G., Maslen, E.H.: Magnetic bearings. Springer, Berlin (2009) 2. Tan, D., Chen, J., Liao, M., Wang, S.: Instability caused by cylindrical surface fit in rotor system. Mechan. Sci. Technol. Aeros. Eng. 33, 1786–1790 (2014) 3. Liu, Y., Liu, H., Yi, J., Jing, M.: Investigation on the stability and bifurcation of a rod-fastening rotor bearing system. J. Vib. Control 21, 2866–2880 (2014) 4. Liao, H., Zang, C., et al.: Dynamic modelling and updating for contact interface of rod fastening rotor based on thin-layer element. J. Aeros. Power 34(9), 1927–1935 (2019) 5. Liu, Y., Liu, H.., et al.: Image method for measuring normal interfacial stiffness. J. Eng. Tribolo. 0(0), 1–14 (2016) 6. Paul, Y., Smithanik, J.: New API 617 standards applied to magnetic bearings turbo-expanders. Chinese Journal of Turbomachinery 058(005), 6–18 (2016) 7. Wei, T., Fang, J.: Self-excited vibration depression of high-speed rotor in magnetically suspended control moment gyroscope. Journal of Astronautics (02), 291–296 (2006) 8. András S., Flowers, G.: Suppression of internal damping-induced instability using adaptive techniques. In: ASME International Design Engineering Technical Conferences & Computers & Information in Engineering Conference, Las Vegas (2007) 9. András, S., Flowers, G.T.: Adaptive disturbance rejection and stabilization for rotor systems with internal damping. In: ASME International Design Engineering Technical Conferences & Computers & Information in Engineering Conference. San Diego (2009) 10. Shi, W., Zhang, Z.: Contact characteristic parameters modelling for the assembled structure with bolted joints, Tribology International 165 (2022)

Experiment and CFD Analysis of Plain Seal, Labyrinth Seal and Floating Ring Seal on Leakage Performance Yunseok Ha1,2 , Yeongdo Lee1,2 , Byul An1,2 , and Yongbok Lee1,2(B) 1 Clean Energy Research Division, Korea Institute of Science and Technology, 5, Hwarangno

14-Gil, Seongbuk-Gu, Seoul 02792, South Korea [email protected] 2 Division of Energy and Environment Technology, University of Science and Technology, 217, Gajeong-Ro, Yuseong-Gu, Daejeon 34113, South Korea

Abstract. This paper investigates the leakage characteristic of a non-contact cryogenic seal applied through experiment and CFD analysis. Plain seal (PS), labyrinth seal (LS), and floating ring seal (FRS) were designed for leakage performance test of cryogenic environment. ANSYS Fluent (18.2) was used for creating the flow field of the sealing unit for leakage performance analysis. In order to validate the CFD analysis results, the leakage experimental performance test was also conducted in a cryogenic environment. To create a cryogenic environment, liquid nitrogen (83K) was injected into the test chamber at 0.5 bar, and the test time for each seal was 5,000 s. The results show that among the three types of seals, the FRS had the smallest leakage and the amount of leakage converged quickly. It was confirmed that the largest turbulent kinetic energy was generated when an FRS was used. Also, FRS effectively reduced leakage compared to PS and LS due to the characteristic that clearance changes according to pressure difference. After the leakage performance test, the wear characteristic according to the seal shape was analyzed through surface roughness measurement. In the case of PS, the wear of the surface progressed considerably more than the FRS. Inadequate fluttering adversely affects the wear and leakage performance of the seal. It was found that the wear of PS due to thermal contraction accelerated the friction, and in contrast, the FRS had less friction even with thermal contraction. Keywords: Cryogenic · Seal · Leakage · CFD · Wear

Nomenclature: PS LS FRS Tseal Hseal TLS HLS

Plain seal Labyrinth seal Floating ring seal Seal thickness Seal height Labyrinth seal thickness Labyrinth seal height

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 391–405, 2024. https://doi.org/10.1007/978-3-031-40455-9_32

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Floating ring seal thickness Total force acting on floating ring seal Top side force acting on floating ring seal Bottom side force acting on floating ring seal Top side area of floating ring seal Bottom side area of floating ring seal Reynolds-averaged flow velocity and pressure Fluid density Dynamic viscosity Additional body forces Reynolds stresses

1 Introduction In turbomachinery applied to cryogenic environments, the sealing unit is a crucial part that directly affects the stability and operating efficiency of the system. Unlike room temperature, the cryogenic sealing unit has many factors to consider. The material of the cryogenic seal is non-isotropic, brittleness, thermal contraction, thermal conductivity, etc., to prevent damage caused by the low-temperature effect. Materials should be selected by considering many factors [1]. In addition, a rapid temperature change in the cryogenic fluid can cause a phase change in the working fluid, which can change the leakage and dynamic characteristics of the seal [2]. In this way, various studies have been conducted to understand the flow characteristics of the working fluid inside the seal in a cryogenic environment, such as the cavitation phenomenon due to the two-phase occurrence according to the seal type as well as the damage due to the thermal contraction of the cryogenic seal. Ma, Y., et al. [3] presented a CFD model that comprehensively considered heat transfer, cavitation, and evaporation to identify the accurate flow characteristics of the cryogenic fluid and applied it to a two-stage mechanical sealing unit. Through this, it was possible to accurately predict the trend of turbulent flow and phase change occurring in the seal cavity, and it was found that the performance of the impeller had a direct effect depending on the seal’s performance. Han, L., et al. [4, 5] analyzed the cavitation phenomenon occurring in the flow inside the labyrinth seal (LS) cavity through the pressure model of the liquid-phase flow. A plain seal (PS) in shape to a plain journal bearing is a mechanism that simply generates flow turbulence and reduces leakage, whereas the LS can effectively reduce leakage by confining the working fluid in the cavity and increasing the turbulence effect. However, since the LS has the possibility of causing dynamic instability of the turbomachinery, a dynamic characteristic analysis must be accompanied. It was confirmed that the cavitation phenomenon occurring in the cryogenic environment greatly affected the LS leakage flow and the geometry parameters such as tooth length and height of the LS. Anbarsooz, M., et al. [6] and Bae, J.-H., et al. [7] conducted a CFD analysis on the floating ring seal (FRS), which can reduce the leakage of working fluid by actively finding the optimal eccentric position according to the rotor position. In the clearance between the FRS and the rotor surface, the same hydrodynamic force as in the fluid film bearing is generated in the radial direction, so

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the FRS moves toward the center of the rotor, and the FRS moves until the frictional force of the support ring surface and the hydrodynamic force are balanced. It is fixed in that position and acts like a PS with an arbitrary eccentricity. Huo, C., et al. [8] and Jin, Z., et al. [9] analyzed the dynamic characteristics of the LS and damper seal, which change according to the operating condition, as well as the leakage flow rate of the cryogenic working fluid, using the rotor whirling model of CFD. In the case of the LS, the effective damping ratio of the seal changes greatly depending on the number of teeth, which affects the system’s ability, resulting from the change in the swirl ratio of the downstream. However, lack of experimental results of the sealing unit in the cryogenic environment for validating the analysis model because it is challenging to construct experimental apparatus for testing the sealing unit in a cryogenic environment. Needs of research validating analysis model using experimental results under various operating conditions. Therefore, this paper performed the seal leakage performance test in a liquid nitrogen environment by changing the existing cryogenic ball bearing test apparatus [10] to validate the seal experiment data and CFD model. The cryogenic seal unit consists of three seals: PS, LS, and FRS, and conducts CFD analysis and leakage experiments. After the experiment, surface roughness measurement was conducted to determine the wear according to the shape of the seal.

2 CFD Modeling of Sealing Unity in Cryogenic Environment 2.1 Descriptions of Test Apparatus and Seal Geometry The test apparatus for the cryogenic seal leakage performance test is shown in Fig. 1. Liquid nitrogen in the chamber leaks between the test bearing part and the lubrication system and a stepped labyrinth seal was applied to reduce leakage in this part. According to the seal shape suggested above, the leakage performance test was conducted using the first cavity part. In order to confirm the leakage flow reduction effect according to the seal shape, a downstream hole was made to measure the temperature and pressure, and the temperature of the supporter bearing of the chamber lubrication part was measured. If the leakage reduction effect of the seal is not achieved, a large amount of liquid nitrogen flows into the lubrication system and the supporter bearing temperature (SBT) bearing drops, so the SBT was also used as data to check the seal performance. The seals used in the cryogenic leakage experiment were designed PS, LS, and FRS, as shown in Fig. 2. In the case of FRS, cryogenic seals were designed by dividing the thickness into two cases (1.5T, 2.0T) because, as shown in (1), the total force acting on the FRS depends on the area. Ftotal = Ftop − Fbottom = Pmixed × Atop − Pup × Abottom

(1)

The geometry information on the three types of non-contact seals is depicted in Table 1.

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Fig. 1. Schematic description of cryogenic performance test apparatus and seal: (a) cryogenic test rig (b) partial magnifying view [10].

2.2 CFD Results of Sealing Unit in the Cryogenic Environment 2D CFD analysis was performed using commercial software ANSYS Fluent (18.2) to analyze the flow characteristics of cryogenic sealing unit leakage. Figure 3 is the mesh for three cases configured using ANSYS meshing. The mesh was constructed to apply the wall function so that the y + value was 30–300 [11]. The mesh was formed using the multi-zone Quad/Tri method, and the mesh was more densely constructed between the seal and the rotor to represent the viscous flow properties better. The Fluent turbulence model determination method is expressed as Eqs. (2), (3) based on the Reynolds Averaged Navier-Stokes (RANS) equation [11]. ∇ ·u =0

(2)

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Table 1. Information on design parameters of cryogenic test sealing unit Parameter

Value

Seal Thickness Tseal ,

4 mm

Seal Height,Hseal

3.5 mm

LS Thickness,TLS

3 mm

LS Height,HLS

3 mm

FRS Thickness,TFRS

1.5 mm, 2 mm

Material

Copper

Fig. 2. Modeling of cryogenic seal for leakage performance test: (a) PS (b) LS (c) FRS.

∂u + ρ(u · ∇)u = −∇p + η · u − ∇ · τ RS + f D ∂t

(3)

This paper utilized the standard k-ε model as the turbulence model, and the pressurebased solver and ideal gas equation were applied. Liquid nitrogen from the Fluent

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Fig. 3. Mesh generation of cryogenic seal for CFD analysis.

Table 2. Mesh and boundary conditions of CFD model Mesh Information

Value

Method of meh generation

Multi zone Quad / Tri

Geometry of element

Hexahedral

Boundary conditions

Value

Solver type

Pressure-based / Axisymmetric Swirl

Working fluid

Liquid nitrogen

Inlet pressure

0.5bar

Inlet temperature

83K

Outlet pressure

0.1bar

Rotating speed

3,600rpm

database was used as the working fluid, and the effect of the rotational speed of the rotor was also considered by applying an axisymmetric swirl. A total energy heat transfer model and adiabatic no slip walls were set, and the analysis convergence was determined when the residual terms of k and e and the velocity in the x, and y direction were less than 10−5 the detailed mesh boundary conditions listed in Table 2. Figure 4 shows the cryogenic seal pressure contour and streamline together. It is noteworthy that Fig. 4(c), (d), the flow field of the down streamline in the cavity is greatly affected by the FRS thickness change. The turbulence kinetic energy contour

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Fig. 4. Graphic contours of pressure and streamline according to cryogenic seal type: (a) PS (b) LS (c) FRS(1.5T) (d) FRS(2T).

is shown in Fig. 5 to confirm the effect of flow change. As seen in Fig. 5, when LS is applied, it can be confirmed that the turbulence kinetic energy is the smallest in the downstream cavity. On the other hand, in the case of FRS (1.5T), the most considerable amount of downstream kinetic energy occurred, which means that the amount of leakage can be reduced by increasing the flow resistance due to energy dissipation [12]. The data summarizing

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Fig. 5. Graphic contours of turbulence kinetic energy according to cryogenic seal type: (a) PS (b) LS (c) FRS(1.5T) (d) FRS(2T).

the outlet flow rate is listed in Table 3, and as expected in Fig. 5, the FRS (1.5T) case has the lowest outlet flow rate.

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Table 3. CFD analysis results of leakage flow rate for sealing unit Seal Type

Leakage Flow Rate

PS

0.012 kg/s

LS

0.015 kg/s

FRS(1.5T)

0.010 kg/s

FRS(2.0T)

0.011 kg/s

3 Experimental System for Leakage Performance and Roughness Measurement 3.1 Description of Experiment Leakage Performance Test for Sealing Unit Figure 6 shows a piping and instrumentation diagram capable of conducting a seal leakage performance test in a cryogenic environment. The pressure of liquid nitrogen flowing into the chamber from the liquid nitrogen tank is controlled by a manual valve. To confirm that the liquid nitrogen flowing through the chamber inlet flows at constant pressure and temperature, a liquid nitrogen flow meter, pressure gauge, and thermometer are installed on the inlet pipeline, and an outlet pipeline that can discharge liquid nitrogen inside the chamber due to vaporization is installed. Figure 7 depicts a photograph of a test apparatus for the seal leakage performance test in a cryogenic environment.

Fig. 6. Piping and instrumentation diagram of the cryogenic facility for the seal on leakage performance [13].

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Fig. 7. Photograph of experimental test apparatus for leakage performance test of cryogenic seal type.

Table 4. Cryogenic experimental operating conditions Parameter

Value

Inlet Pressure

0.5bar

Rotating Speed

3,600rpm

Test Time

5,000s

A hole was made in the seal test part to measure the temperature and pressure according to the seal type. As described above, the SBT of the lubrication system was measured in real-time. As explained in Sect. 2.1, the SBT was used as a criterion of the leakage reduction effect according to the seal shape. This is because the SBT may vary depending on the performance of the test seal. After all, the leak of the seal inserted into the first cavity of the test apparatus is connected to the passage leading to the supporter bearing. The experiment was conducted after filling the test chamber to a certain level by adjusting the valve so that a constant pressure and flow rate of liquid nitrogen flowed from the LN2 tank. The measurement time was 5,000 s according to the seal type, and the measured data was collected through NI (National Instrumental) DAQ (Data Acquisition) device. The measured data was monitored in real-time through the Labview monitoring program, and the experimental conditions were summarized in Table 4. 3.2 Experimental Results of Leakage Performance Test Figure 8 shows the temperature measured at the outlet hole according to the seal type. It can be seen that all four cases converge to a specific temperature after 4,000 s.

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Fig. 8. Leakage experimental results of outlet temperature depending on seal type.

Fig. 9. Leakage experimental result of SBT depending on seal type.

Among them, in PS, it was confirmed that the temperature fluctuated after the test time was 2,000 s. It is thought that the fluctuation of the temperature was caused by the friction between the rotor and the seal due to the thermal contraction of the seal, and the SBT was checked. Figure 9 is a graph of the SBT data of four cases, and it can be seen that the slope of the SBT changes significantly after 2,000 s, just like the outlet temperature change. In the case of PS, the SBT dropped significantly, while in the case of

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FRS (2T), there was a temperature fluctuation in the outlet hole, but the SBT increased. In the case of PS and LS, the temperature convergence tendency is different, but it can be seen that the SBT converges similarly at about 2C˚. Through the SBT experimental data, it can be seen that the FRS effectively reduced the leakage compared to PS and LS, and in the case of FRS, there was a difference from the CFD analysis result. As a result of CFD analysis, there was no significant difference in leakage flow rate, but FRS 2.0T converged to 10C˚ and FRS 1.5T converged to 4 C˚. It is thought that FRS 2.0T effectively reduced leakage compared to 1.5T because the clearance is changed by pressure balance depending on the thickness of FRS. The clearance was fixed and analyzed in the CFD analysis, so it is different from the experimental results. 3.3 Roughness Measurement of Seal Surface for Investigating Wear Roughness measurement was conducted to check seal surface wear due to friction generated during the leakage performance test. Figure 10 describes the measurement equipment used Taylor HOBSON I60 and the test seal.

Fig. 10. Description of a roughness measurement device for investigating seal surface wear.

In the case of LS, the teeth are so thin that it is impossible to measure with current equipment, so the seal surface before the leakage performance test and the surface of PS and FRS after the test were measured, and the details are summarized in Table 5 of the measurement conditions.

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Table 5. Specification of measurement device and method Specification of measurement device

Value

Vertical range

1mm

Resolution

16nm

Horizontal traverse

200mm

Straightness error

0.5um

Data spacing

0.125um

Measurement method

Value

Measurement range

2mm

Data interval

1.25um

Fig. 11. Results of surface roughness according to seal type.

Figure 11 is the roughness measurement result, and it can be confirmed that the roughness Ra of PS was measured larger than FRS. The Ra value of PS was measured to be about 70% larger than that of FRS and the detailed information listed in Table 6. As shown in Fig. 8, the fluctuation of outlet temperature during the measurement period is thought to have caused continuous friction on the seal surface. In the case of plain seal, clearance change between the rotor and seal due to thermal contraction may occur, and

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wear may progress. In the case of FRS, even if thermal contraction occurs, clearance is changed due to pressure balance, so it can be seen that less friction of roughness has occurred. Table 6. Results of roughness measurement for sealing unit Parameter

Ra Value

Before experiment

0.1063um

PS

1.3558um

FRS(1.5T)

0.7994um

FRS(2.0T)

0.5297um

4 Conclusion In this study, CFD analysis and experiments were conducted to understand the leakage characteristics of the sealing unit in a cryogenic environment. For the sealing unit, PS, LS, and FRS, 3 types of seals were used, and FRS changed the thickness, so a total of 4 types of sealing units were used. As a result of CFD analysis, it was confirmed that FRS reduced the amount of leakage by effectively generating turbulent kinetic energy in the cavity space compared to PS and LS. In the experiment, the leakage reduction effect of the seal was used as a standard by using the SBT data of the lubrication system. As a result of the experiment, as in the CFD analysis, the SBT was measured to be high in the two cases of FRS, and it was confirmed that the SBT of FRS (2.0T) effectively reduced leakage compared to FRS (1.5T). To confirm the outlet temperature fluctuation, the sealing unit roughness was measured after the experiment, and it was confirmed that PS had greater wear than FRS. This is thought to be wear caused by thermal contraction due to low temperature changing the clearance of the seal, and it was confirmed that the characteristics of FRS can reduce the wear caused by thermal contraction. In future studies, we plan to analyze the flow characteristics generated in the cryogenic environment by conducting an analysis that considers the clearance change of FRS, which needs to be reflected in this study. Acknowledgments. This study was supported by Korea Institute of Energy Technology Evaluation and Planning (KETEP) with the project title “Development of platform technology and operation management system for design and operating condition diagnosis of fluid machinery with variable devices based on AI/ICT” (No. 2021202080026D). The authors thank them for their contribution to this study.

Appendix. Sensor Uncertainty In order to confirm the reliability of the results of this cryogenic leakage experiment, an uncertainty analysis of the sensor was performed. For the uncertainty of the sensor, the same sensor used in the experiment of Choe, B., et al. [14] was used, and Eq. (4)

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represents the uncertainty of the entire measurement system.  U ={

∂A u1 ∂x1

2

 +

∂A u2 ∂x1

2

 + ··· +

∂A un ∂x1

2

1

}2

(4)

where U is the uncertainty of the entire system A = (x1 ; x2 ; · · · ; xn ) and u1 ; u2 ; · · · ; un were uncertainty of each sensor. The uncertainty of cryogenic leakage experiment is 1.14% and is the result obtained by using Eq. (4) (pressure sensor: ± 0.15%, thermocouple sensor: ± 1.50%, flow meter: ± 0.05%).

References 1. Jia, L.X.: Experimental and theoretical studies of cryogenic sealing. Florida Atlantic University (1996) 2. San Andres, L., Yang, Z., Childs, D.W.: Thermal effects in cryogenic liquid annular seals— Part II: Numerical solution and results (1993) 3. Ma, Y., et al.: Numerical investigation on sealing behaviors of an extremely high-speed twostage impellers structure in cryogenic rockets. Asia-Pac. J. Chem. Eng. 13(5), e2229 (2018) 4. Han, L., et al.: Theoretical modeling for leakage characteristics of two-phase flow in the cryogenic labyrinth seal. Int. J. Heat Mass Transf. 159, 120151 (2020) 5. Han, L., et al.: Theoretical leakage equations towards liquid-phase flow in the straight-through labyrinth seal. Journal of Tribology 144(3) (2022) 6. Anbarsooz, M., et al.: Effects of the ring clearance on the aerodynamic performance of a CO2 centrifugal compressors annular seal: A numerical study. Tribol. Int. 170, 107501 (2022) 7. Bae, J.-H., et al.: Numerical and experimental study of nose for LOx floating ring seal in turbopump. Aerospace 9(11), 667 (2022) 8. Huo, C., et al.: Influence of tooth geometrical shape on the leakage and rotordynamic characteristics of labyrinth seals in a cryogenic liquid turbine expander. Int. J. Refrig 145, 105–117 (2023) 9. Jin, Z., et al.: A comparison of static and rotordynamic characteristics for three types of impeller front seals in a liquid oxygen turbopump. J. Eng. Gas Turbines Power 145(3), 031025 (2023) 10. Lee, Y.B., Choe, B.S., Lee, J.K., Ryu, S.J., Lee, B.K.: Bearing Test Apparatus for Testing Durability of Bearing. U.S. Patent No. 9714883 (2016) 11. Manual, U.: ANSYS FLUENT 12.0. Theory Guide (2009) 12. Greifzu, F., et al.: Assessment of particle-tracking models for dispersed particle-laden flows implemented in OpenFOAM and ANSYS FLUENT. Engineering Applications of Computational Fluid Mechanics 10(1), 30–43 (2016) 13. Kwak, W.I.: Failure criteria and life model to verify the reliability of cryogenic rolling bearing. University of Science and Technology (2023) 14. Choe, B., et al.: Experimental study on dynamic behavior of ball bearing cage in cryogenic environments, Part I: Effects of cage guidance and pocket clearances. Mech. Syst. Signal Process. 115, 545–569 (2019)

Rolling Element Bearing Fault Diagnosis Using Hybrid Machine Learning Models Mario Antunovi´c1 , Sanjin Braut2(B) , Roberto Žiguli´c2 , Goranka Štimac Ronˇcevi´c2 , and Mario Lovri´c3 1 Ascalia Ltd, Ulica Trate 16, Cakovec, ˇ Croatia

[email protected]

2 Faculty of Engineering, University of Rijeka, Vukovarska 58, Rijeka, Croatia

{sbraut,zigulic,gstimac}@riteh.hr

3 Faculty of Electrical Engineering Computer Science and Information Technology Osijek,

Josip Juraj Strossmayer University of Osijek, Kneza Trpimira 2B, 31000 Osijek, Croatia [email protected]

Abstract. Rolling element bearings are widely employed in a variety of mechanical equipment. An effective fault diagnosis can limit the occurrence of accidents, maximize the operating profile of the bearing, and save maintenance costs. Faults in a rolling element bearing commonly include the faults in inner race, outer race, rolling elements, and cage. In this work, an approach based on machine learning algorithms was used. For testing, a public data set from the Case Western Reserve University bearing center (CWRU) was used. After testing different combinations for feature extraction and selection, the best method of processing the original data according to the machine learning model was chosen to be used for the multiclass classification problem. The developed diagnostic methods demonstrated the ability to classify and detect damage to roller bearings. The logistic regression, support vector machine, and random forest algorithms were tested, and the best classification result was obtained using the random forest algorithm on selected variables using the permutation importance of features from the time, frequency, and time-frequency domains. The algorithm based on the Random Forest method achieved the highest average value of accuracy in predicting faults in the roller bearing. Keywords: Fault diagnosis · Rolling element bearing · Machine learning · Random forest · Support vector machine · Logistic regression

1 Introduction Rolling element bearings (REB) are key components for several rotating machines. In general, the rolling bearing - rotor system usually includes one or more rotors supported by (REB). Rolling bearings are important parts of rotating machinery, which can be divided into ball bearings and roller bearings. The feasibility of bearing fault diagnosis using vibration signals is based on the presence of series of repeating impacts which are generated after the appearances of faults at the early stages [1]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 406–421, 2024. https://doi.org/10.1007/978-3-031-40455-9_33

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These repeating impacts considered signals are often analyzed by means of signal decomposition. Commonly used methods for decomposition of signals include bandpass filtering, wavelet transform (WT) or wavelet packet transform (WPT) [2], empirical mode decomposition (EMD) [3], variational mode decomposition (VMD) [4, 5] and singular value decomposition (SVD) [6]. Fast kurtogram for bearing damage detection as a combination of 1/2-binary tree bandpass filtering and time domain kurtosis were presented in [7]. In [8], the authors propose a method for diagnosing bearing defects called “Protrugram” based on fixed bandpass filtering and SES kurtosis, while in [9], the authors applied several previously fine frequency bands for filtering and combined the peak index to obtain a peaks map and realize the optimal frequency band for bearing fault diagnosis. Machine learning techniques also found their ways into fault diagnosis. They are widely used in the field of condition monitoring as a data manipulation tool that enables inference from complicated data distributions in multidimensional spaces. Artificial neural networks (ANN) [10], support vector machines (SVM) [11, 12], k-means clustering [13], Gaussian mixture models [14], and hidden Markov models [15, 16] are some examples of approaches that have been used for bearing diagnostics. However, the latest trends show that combining expert knowledge by means of signal processing and mathematical techniques (first-principle models) with machine learning, also called data-driven models, can yield higher accuracies in prediction and are termed theory-inspired or hybrid models [17, 18]. In this work three machine learning algorithms, driven by their differences in explainability and complexity, namely logistic regression, support vector machine, and random forest algorithms were tested combined with feature extraction techniques from signal processing instead of using purely raw time series. Signal processing was conducted utilizing decomposition based on time, frequency, and time-frequency domains. Such a synergy, referred to as hybrid models were then further inspected based on their classification result and using permutation importance to assess the benefit of using signal processing in hybrid models.

2 Machine Learning Algorithms Classification algorithms determine the class of input data if it’s a categorical or binary variable. During the learning process in classification supervised learning, the algorithm creates a description of each of the given classes, using a set of already labelled examples. If the classification problem is simple and linear and does not use many features, it is advisable to apply simpler algorithms such as logistic regression (LogReg). On the other hand, when dealing with more complex problems with a larger amount of data, the probability of random correlation between the features that describe certain classes increases, therefore simple methods in such cases prove to be quite inferior [19]. More complex machine learning which might more suitable for this type of problem are, for example, the random forest (RF), the support vector machines (SVM). For the classification of roller bearings faults, three machine learning algorithms were used in this work: logistic regression, random forests, and support vector machines. These three were chosen based on their explainability, fast training using CPUs, both factors relevant for

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application in the industry, but also based on their different algorithms and complexity levels. 2.1 Logistic Regression Logistic regression is mainly applied when the observed classes represented in a multidimensional space are separated linearly. The simplest machine learning algorithm is linear regression whose model can be represented by the equation: fw,b (x) = wx + b,

(1)

where the output value is determined by the values of the feature x i and the parameters W and B. If it is a binary classification, i.e., the data set {(x i , yi ), i = 1…N} contains two classes, Y ∈ {0, 1} the goal of logistic regression is to create a function that for any real number x i gives a value yi that can take the value 0 or 1. One such function shown in is called a standard logistic function or sigmoid function: f (x) =

1 . 1 + e−x

(2)

In this case, the value given by the model for the input value x is passed to the logistic function, which takes the value 0 if the passed value is close to zero, and otherwise returns the value 1. Logistic regression is therefore a probabilistic model because the output value (2) is the probability of belonging to some of two classes for given input values of features x i . Finally, the logistic model looks like: fw,b (x) =

1 1 + e−(wx+b)

.

(3)

When learning a logistic regression model, it is necessary to optimize the hyperparameters of the model w and b, by maximizing the belonging value of the features to a particular class according to the following expression: Lw,b =



fw,b (xi )yi (1 − fw,b (xi ))(1−yi ) .

(4)

i=1,...,N

In practice, it is more convenient to use the logarithm of expression (4), since it is necessary to ensure that the maximum value appears at the same point as the original probability function:   LogLw,b = ln Lw,b (x) . (5) The optimization procedure of the function (5) depends on the implementation of the algorithm, but the so-called Gradient descent algorithm is most often used. The multiclass logistic regression problem is solved by converting it into a binary classification by using the One-vs-rest classifier, whereby a binary classification problem is constructed and solved for each class.

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2.2 Random Forest The random forest (RF) algorithm is based on the idea that, instead of trying to learn a single model it trains many low accuracy models – decision trees, which are then collected into a forest to obtain a high accuracy meta-model. The RF model has many “weak learners” or decision trees that are combined to make a final prediction. Decision trees are used for decision-making by splitting data based on features and following branches. When reaching leaf-nodes, the example is classified into a specific class. The RF algorithm uses an ensemble of decision trees with different feature subsets to ensure low correlation among trees for maximum impact. Strong predictors lead to feature selection, causing correlated trees that don’t improve prediction accuracy and can lead to agreement among bad models during decision making. [19]. Overlearning occurs when a model explains small variances due to limited data. The RF model avoids overlearning by using bootstrap aggregation to reduce variance and improve predictability. Optimizing hyperparameters like tree number, depth, and variables included leads to impressive results for various problems. 2.3 Support Vector Machines Support vector machines (SVM), as well as LogReg and RF, is a supervised machine learning algorithm that is used in many fields today due to its efficiency when dealing with high-dimensional data. SVM frames each observation as a point in multi-dimensional space and finds the hyperplane that best separates classes in the given data set. A hyperplane is a multi-dimensional plane [11]. The optimization goal when training the model is the distance between the separating hyperplane and the closest data sample to that plane, which are also called support vectors. The hyperplane is given by equation: wx + b = 0,

(6)

where w is a vector of real values of the same dimension as the input vector x, and b is the displacement from the center of the coordinate system. In a binary classification problem, the input data must satisfy one of the two following conditions: wxi − b ≥ 1, for yi = +1and

(7)

wxi − b ≤ −1, for yi = −1.

(8)

The goal of the algorithm is to optimize the parameters given by the hyperplane so that positive and negative samples are separated by the largest margin. The margin is the smallest distance between two classes, and the larger it is, the better the model will generalize, i.e. the more robust it will be, wxi − b = 1

(9)

wxi − b = −1.

(10)

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The planes described by Eqs. (9) and (10) define two parallel hyper-planes whose mutual distance is equal to: ρ=

2 . w

(11)

It is evident from Eq. (11) that by minimizing the Euclidean norm ||w|| the maximization of margin width ρ is achieved. When dealing with problems that are not linearly separable, the SVM algorithm maps the input data into a new space of higher dimensions, while the probability for linear separation increases.

3 Diagnosis of Rolling Element Bearing Faults The diagnosis of rotary machine faults is of great practical importance in industry to avoid economic losses and increase machine availability. The contact force of the normal bearing is continuous in time in the normal state. However, if the bearing is worn, it will generate periodic impulsive vibration signals that differ from those in normal operation. These vibration signals can be measured by sensors and analyzed using signal processing techniques for feature extraction. For vibration signal processing, methods have been developed that extract features from the time, frequency, and time-frequency domains, but also by combining physical and mathematical laws, which gives an edge to pure data-driven models by utilizing such theory-informed processes, hence the name hybrid modeling. This work covers damage diagnostics on rolling element bearings based on machine learning methods and using features from the time, frequency, and time-frequency domains. 3.1 Data Set To verify the effectiveness of the feature extraction and classification of individual algorithms, in this paper vibration data available on the website of Case Western Reserve University (CWRU) [20] are used. Figure 1 shows the test rig from which the data were collected. The test rig consists of the motor, two accelerometers, the torque converter, accelerometer - drive end

accelerometer - fan end

torque converter and encoder

electric motor

bearing

dynamometer

bearing

Fig. 1. The sketch of CWRU test rig

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the encoder and the dynamometer. The analysis was performed on two bearings that support the motor shaft, one of which is on the drive side of the motor, and the other bearing is located near the fan. The type of tested rolling bearings are 6205-2RS JEM and 6203-2RS JEM both deep groove ball bearing, SKF. Using the electrical discharge machining (EDM) on the correct bearings, errors were made at one point on one of the 3 parts of the ball bearing – the inner ring, the outer ring, and the rolling element. The bearings were damaged in three diameter sizes: 0.007 inch (0.178 mm), 0.014 in. (0.356 mm) and 0.021 in. (0.533 mm). With a normal bearing without any defects, this makes a total of 10 different bearing classes (Table 1). Data were collected at 48000 samples per second. Table 1. Fault specification – class names Fault diameter [inch]

Fault depth [inch]

Fault location

Class name

0.007

0.011

Inner Raceway

IR_007

0.007

0.011

Ball

Ball_007

0.007

0.011

Outer Raceway

OR_007

0.014

0.011

Inner Raceway

IR_014

0.014

0.011

Ball

Ball_014

0.014

0.011

Outer Raceway

OR_014

0.021

0.011

Inner Raceway

IR_021

0.021

0.011

Ball

Ball_021

0.021

0.011

Outer Raceway

OR_021

0

0

No fault - normal

Normal

3.2 Data Processing and Feature Extraction All data manipulations and model building was conducted using Python with the following packages: NumPy, SciPy, Pandas, Scikit-Learn, Matplotlib and Seaborn as tested in previous work [17] The original data includes recorded signals with about 480000 samples, for 10 bearing conditions, and each file contains data for the bearing on the drive side and on the fan side of the electric motor. The recorded signals were divided into 230 signals samples (vectors) with 2048 data points (Fig. 2). As a result of shaping the raw data, a total of 4600 signal samples with a length of 2048 data points were obtained. In addition to signal segmentation, the division into three sets of data was performed. 70% of the data was used as a set for learning the model (train set), or in this case 3220 signal samples. The rest is divided into a validation set and a test set. The validation set therefore contained 690 signal samples, and served for an unbiased assessment of the accuracy of the model during model optimization. The test set was the same size as the validation set, and served to inspect the accuracy of the model, i.e., to evaluate the generalization. Furthermore, it was necessary to extract features to

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Fig. 2. Signal sample extraction

obtain input data that the algorithm can receive. High-quality feature extraction is the key to the system for diagnosing faults in rotary machines and is most often performed by a combination of methods in the time, frequency, and time-frequency domains [21, 22]. Time Domain. Temporal features are statistical descriptors extracted from the time domain [23]. In this paper, several statistical features from the time domain were used, such as mean value, standard deviation, mean square and others, in order to characterize different signals. In addition, due to the non-stationary nature of the vibration signal of the faulty bearing, more advanced statistical features such as kurtosis and skewness were used (Table 2). Frequency Domain. The extraction of features from the signal was done by first applying the Butterworth band-pass filter, then using the Hilbert transformation to extract the envelope of the signal, and finally by frequency spectrum of the envelope obtained by applying the fast Fourier transformation. Features at characteristic frequencies were also used:   d nfr 1 + cosα . (12) BPFO = 2 D Table 2. Time domain features max(xi ) N  μ = N1 xi

min(xi )  (xi −μ)2 σ = N

i=1

RMS μ max(xi ) RMS

N 4 i=1 (xi −μ) N σ4

−3

max(xi ) − min(xi )  RMS =

N 3 i=1 (xi −μ) N σ3

xi2 N

Rolling Element Bearing Fault Diagnosis

  nfr d BPFI = 1 − cosα . 2 D

2  d Dfr cosα . 1− BSF = d D

413

(13)

(14)

For the theoretical characteristic frequencies, Eqs. (12) – (14), the maximum amplitudes in the range of ±30 Hz were determined, thus obtaining three features for each signal. In addition to the listed features, the features listed in Table 3 were calculated based on the signal in the frequency domain.

Table 3. Frequency domain features max(Xk ) N  μ = N1 Xk

min(Xk )  (Xk −μ)2 σ = N

i=1

RMS μ max(Xk ) RMS



N 4 i=1 (Xk −μ) N σ4

N  i=2 Xk Xk N 2π i=1 Xk2

−3

max(Xk ) − min(Xk )  RMS =

Xk2 N

N 3 i=1 (Xk −μ) N σ3



N   2 i=2 Xk  2 4π 2 N i=1 Xk

 N  2 N   2 i=2 Xk i=2 Xk Xk −   N 2 4π 2 i=1 Xk2 2π N i=1 Xk

Time-Frequency Domain. Considering the non-stationarity of the signals obtained by measuring the vibrations caused by bearing faults, it is necessary to use appropriate signal processing techniques to extract features that contain useful information about the condition of the bearing elements. In general, stationarity implies variability of statistical features over time. It is therefore necessary to identify non-stationarities to be able to relate them to the cause. This paper covers the use of energy and entropy features using wavelet transformation when extracting features for the classification of vibration signals from the time-frequency domain. Wavelet packet transform (WPT) [22] is a generalization of the classical continuous wavelet transform. During packet transformation, the signal representation is obtained using digital filters. When passing through the filters, the signal is split into a low-frequency band A1 and a high-frequency band D1. The decomposition can be continued until the desired level is reached, and the result of the decomposition is a tree-shaped structure. The PyWavelets package with basic function sym8 from the group of functions called Symlets was used for wavelet transformation. After the wavelet transformation up to the third level is performed, eight signals were obtained from the original signal (Fig. 3).

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Fig. 3. Tree structure when using WPT up to the third level

Furthermore, from each of the eight obtained signals, the energy values were calculated according to expression: +∞

Ei = ∫ xji (t)dt. −∞

(15)

The total signal energy can be calculated as: j

E=

2 

Ei .

(16)

i=1

It is also possible to calculate the entropy values. The concept of entropy is often used as a measure of system disorder [22]. Using the expression proposed by Shannon, the entropy values for each of the eight obtained signals will be calculated as: S=−

 Ei E

ln

Ei . E

(17)

3.3 Training The goal of a machine learning algorithm is to make successful predictions on new, unseen data (model validation) leading to so-called model generalization. To improve predictions on an external set, model hyperparameters (e.g. tree-depth in RF) must be optimized. This requires adjusting values to find the best generalization on a validation set. Manual searching is time-consuming, so the Grid search method examines all combinations of defined search space to find the best result. The parameters spaces are given in Table 4. Cross-validation is used to prevent model accuracy from depending on the learning set. It divides the learning set into k equal parts and repeats the process k times using different validation sets. For training logistic regression algorithms, support vector machines and random forests, in this work 10xCV is used. To prevent overlearning and speed up the algorithm, important variables are selected after optimizing hyperparameters. Permutation importance measures a variable’s importance by calculating its impact on classification results when changed or permuted on the train set. If a random change of a variable

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significantly changes the results of the model, this feature is given high importance. The same procedure was carried out several times, to increase the confidence interval of the change of results due to the permutation of the value of a particular descriptor. After selecting the most important variables, the grid search of optimal hyperparameters and cross validation are once again carried out to adapt the model to the new feature set. Table 4. Lists of parameters grids and chosen parameters in the best model Algorithm

Parameter Grid

Chosen Parameters

Random Forest

criterion: gini max_depth: 7, 9, 11 max_features: 3, auto, log2 min_samples_leaf: 1, 2, 3 min_samples_split: 5, 7 n_estimators: 100, 150, 200, 250, 300

bootstrap: True class_weight: None criterion: gini max_depth: 9 max_features: 3 max_leaf_nodes: None max_samples: None min_impurity_decrease: 0.0 min_samples_leaf: 1 min_samples_split: 5 min_weight_fraction_leaf: 0.0 n_estimators: 250

Support vector machines

C: 0.1, 100, 500, 1000, 10000 gamma: 0.001, 0.01, 0.5, 1 kernel: rbf

C: 10000 break_ties: False cache_size: 200 class_weight: None coef0: 0.0 decision_function_shape: ovr degree: 3 gamma: 0.01 kernel: rbf max_iter: -1 shrinking: True tol: 0.001

Logistic Regression

C: 0.1, 1, 10, 100, 1000, 10000

C: 100.0

4 Results and Discussion Three theory-informed feature extraction techniques were applied to a total of 4,600 signals obtained by vibration measurement to enable training of machine learning models. Figure 4 shows 10 signal samples, each of which contains 2048 data points and describes the different health status of the bearing. The first three displayed signals show three sizes of damage to the rolling element (0.007 inch, 0.014 inch and 0.021 in.), then the signals with equal damage on the inner ring of the bearing and on the outer ring of the bearing are shown, respectively. The last signal shows the correct bearing.

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No. of signal samples

Fig. 4. 10 signal samples of 10 different bearing conditions

Fig. 5. Learning curve for RF algorithm, case with all three domain features

The generalization ability of the algorithms was tested on a test data set that the model has not yet encountered. Accuracy of classification is defined as the proportion of correctly classified examples in the set of all examples. The test data set, just like

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the validation data set, contained of 690 observations on which the generalization was inspected. The learning curve for RF algorithm with all three domain feature groups is shown in Fig. 5. In search of the best group of features and the algorithm with the highest accuracy, all combinations of algorithms and groups of features were tested. For simplicity, the notations for the feature groups listed in Table 5 will be used in further text. Table 5. Groups of features Group of features

Label

Time domain features

P1

Frequency domain features

P2

Time – frequency domain features

P3

Table 6 shows the prediction results combinations of the feature sets and the used algorithm on the validation data set containing 690 observations. It can be seen that the best results were models with the RF algorithm for the feature set that includes all three feature extraction methods. Table 6. Classification accuracy Random forest [%]

SVM [%]

Logistic regression [%]

P1

82.8

81.6

78.0

P2

91.2

87.1

88.6

P3

96.5

87.2

85.9

P1 + P2

95.6

96.8

94.1

P1 + P3

97.2

95.5

94.6

P2 + P3

97.1

96.3

94.3

P1 + P2 + P3

98.8

97.3

97.0

The results shown in Table 6 are also presented in Figs. 6 and 7. It can be concluded that the best results are given by the random forest algorithm and a combination of features that includes all three domains, i.e. P1 + P2 + P3. The combination of multiple feature sets from different domains results in higher accuracy and more robust models, as indicated by the low variance in scored accuracy while also improving accuracy of the predictions. Similar results were obtained in our previous work where hybrids based on random forest showed best results [17].

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Fig. 6. Contribution of the algorithm

Based on the classification of the signals on the entire validation set using the best model, it is possible to display the confusion matrix, i.e., the model confusion matrix. The confusion matrix is used to determine the performance of the classification model for a given set of test data. The validation set contains 10 classes to be classified, each class containing 69 signals. The confusion matrix is shown in Fig. 8. 8 signals out of a total of 690 signals were incorrectly classified by the model, however it is crucial to note that the algorithm properly detected all non-faulty bearings, which is crucial in the context of health management.

Fig. 7. Contribution of feature groups

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Fig. 8. Confusion matrix of the best model trained with the random forest algorithm with P1 + P2 + P3 features.

5 Conclusions Inner race, outer race, rolling elements, and cage defects are frequently present in rolling element bearings. Hybrid or theory-informed algorithms were applied in this study driven by theory-informed feature extraction (signal processing) and machine learning. An open data collection from the Case Western Reserve University bearing center (CWRU) was utilized for testing the approach. The optimum way of processing the original data according to the machine learning model was chosen to be used for the multi-class classification problem after evaluating several combinations for feature extraction and selection. The established diagnostic techniques proved their ability for identifying and classifying roller bearing damage for each of the damage classes. The logistic regression, support vector machine, and random forest algorithms were tested. The best classification result was obtained using the random forest algorithm on extracted features which identified by the permutation importance of features from the time, frequency, and time-frequency domains. Acknowledgements. This work has been fully supported by the University of Rijeka under the project number uniri-tehnic-18–225.

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References 1. Antoni, J., Randall, R.B.: Differential diagnosis of gear and bearing faults. J. Vib. Acoust. Trans. ASME (2002). https://doi.org/10.1115/1.1456906 2. Wang, D., Tse, P.W., Tsui, K.L.: An enhanced Kurtogram method for fault diagnosis of rolling element bearings. Mech. Syst. Signal Process. 35, 176–199 (2013). https://doi.org/10.1016/ J.YMSSP.2012.10.003 3. Huang, N.E., et al.: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A.454, 903–995 (1998) 4. Dragomiretskiy, K., Zosso, D.: Variational mode decomposition. IEEE Trans. Signal Process. (2013). https://doi.org/10.1109/TSP.2013.2288675 5. Braut, S., Žiguli´c, R., Skoblar, A., Štimac Ronˇcevi´c, G.: Partial Rub Detection Based on Instantaneous Angular Speed Measurement and Variational Mode Decomposition. J. Vibr. Eng. Technol. 8(2), 351–364 (2019). https://doi.org/10.1007/s42417-019-00177-2 6. Xu, L., Chatterton, S., Pennacchi, P.: Rolling element bearing diagnosis based on singular value decomposition and composite squared envelope spectrum. Mech. Syst. Signal Process 148 (2021) https://doi.org/10.1016/j.ymssp.2020.107174 7. Antoni, J.: Fast computation of the kurtogram for the detection of transient faults. Mech. Syst. Signal Process (2007). https://doi.org/10.1016/j.ymssp.2005.12.002 8. Barszcz, T., Jabłonski, A.: A novel method for the optimal band selection for vibration signal demodulation and comparison with the Kurtogram. Mech. Syst. Signal Process (2011). https:// doi.org/10.1016/j.ymssp.2010.05.018 9. Chatterton, S., Pennacchi, P., Vania, A., Borghesani, P.: A Novel Procedure for the Selection of the Frequency Band in the Envelope Analysis for Rolling Element Bearing Diagnostics. In: Pennacchi, P. (ed.) Proceedings of the 9th IFToMM International Conference on Rotor Dynamics. MMS, vol. 21, pp. 421–430. Springer, Cham (2015). https://doi.org/10.1007/9783-319-06590-8_33 10. Samanta, B., Al-Balushi, K.R., Al-Araimi, S.: Artificial neural networks and support vector machines with genetic algorithm for bearing fault detection. Eng. Appl. Artif. Intell. 16(7–8), 657–665 (2003) 11. Gryllias, K.C., Antoniadis, I.A.: A support vector machine approach based on physical model training for rolling element bearing fault detection in industrial environments. Eng. Appl. Artif. Intell. 25(2), 326–344 (2012) 12. Fernández-Francos, D., Martínez-Rego, D., Fontenla-Romero, O., Alonso-Betanzos, A.: Automatic bearing fault diagnosis based on one-class m-SVM. Comput. Ind. Eng. 64(1), 357–365 (2013) 13. Yiakopoulos, C.T., Gryllias, K.C., Antoniadis, I.A.: Rolling element bearing fault detection in industrial environments based on a K-means clustering approach. Expert. Syst. Appl. 38(3), 2888–2911 (2011) 14. Marwala, T., Mahola, U., Nelwamondo, F.V.: Hidden markov models and gaussian mixture models for bearing fault detection using fractals. In: International Joint Conference on Neural Networks, Vancouver, Canada, pp. 3237–3242 (2006) 15. Miao, Q., Makis, V.: Condition monitoring and classification of rotating machinery using wavelets and hidden Markov models. Mech. Syst. Signal Process. 21(2), 840–855 (2007) 16. Ocak, H., Loparo, K.A.: HMM-based fault detection and diagnosis scheme for rolling element bearings. J. Vib. Acoust. 127(4), 299 (2005) 17. Lovri´c, M., et al.: Parasitic resistance as a predictor of faulty anodes in electro galvanizing: a comparison of machine learning, physical and hybrid models. Advanced Modeling and Simulation in Engineering Sciences 7(1), 1–16 (2020). https://doi.org/10.1186/s40323-02000184-z

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18. Hoffer, J.G., Ofner, A.B., Rohrhofer, F.M., et al.: Theory-inspired machine learning—towards a synergy between knowledge and data. Weld World 66, 1291–1304 (2022). https://doi.org/ 10.1007/s40194-022-01270-z 19. Guo, L., Ma, Y., Cukic, B., Singh, H.: Robust prediction of fault-proneness by random forests. In: 15th International Symposium on Software Reliability Engineering, Saint-Malo, France, pp. 417–428 (2004). https://doi.org/10.1109/ISSRE.2004.35 20. Smith, W.A., Randall, R.B.: Rolling element bearing diagnostics using the Case western reserve university data: a benchmark study. Mech. Syst. Signal Process, 64–65 (2015) https:// doi.org/10.1016/j.ymssp.2015.04.021 21. Xiao, Z., Boyang, Z., Yun, L.: Machine learning based bearing fault diagnosis using the case western reserve university data: a review. IEEE Access. 9, 15559–155608 (2021) 22. Varanis, M., Pederiva, R.: Wavelet Packet Energy-Entropy Feature Extraction and Principal Component Analysis for Signal Classification, Aug. (2015) 23. Thelaidjia, T., Moussaoui, A., Chenikher, S.: Feature extraction and optimized support vector machine for severity fault diagnosis in ball bearing. Engineering Solid Mechanics 4(4) pp. 167–176 (2016)

Fluid-Structure Analysis of a Hybrid Brush Seal Joury Temis(B) and Alexey Selivanov Central Institute of Aviation Motors, Moscow, Russia [email protected]

Abstract. A multidisciplinary mathematical model of a hybrid brush seal is developed. Gas flow pressure distributions and lifting forces acting on the seal compliant elements depend on current positions of the latter ones and vice versa. Seal operating clearance and leakage are obtained by means of the iterative approach for solving static fluid-structure interaction problem. A model comprising rigid pad attached to elastic support with the equivalent stiffness matrix is used for calculation of seal compliant element displacements. A model of flow in the clearances between the pads and the shaft based on 2D Reynolds equation is used for calculation of gas lifting force acting on pads. Both simplified models are verified by comparing with results obtained by means detailed 3D simulations. It is shown that the circumferential inclination of pads has significant influence on the gas lifting force generated under the pads. Keywords: Hybrid Brush Seal · Fluid-Structure Interaction · Reynolds Equation · Radial Clearance · Mathematical Simulation

1 Introduction Seals using to reduce gas leakage through the gaps between rotating and stationary parts of turbomachines. The most common art labyrinth seal, which due to their shape they create increased hydraulic resistance, which prevents the flow of gas from the highpressure area to the low-pressure one. The amount of gas leakage through the labyrinth seal is proportional to the clearance, and the installation of a minimum gap allows for a high degree of sealing. However, the deformation of the parts of the sealing unit as part of gas turbine engine under the influence of variable thermal inertial loads can lead to a change in the working gap by up to 1.0 mm [1]. As a result, during the engine’s operating cycle, the labyrinth seal can operates with an excessive clearance, which reduces its efficiency. Reduction of parasitic secondary flows in gas turbine engines allows to improving engine performance. Unlike rigid labyrinth seal, a feature of promising seals is to ensure a guaranteed small gap and small gas leakage at various engine operating modes. In such constructions, the external change in the gap is compensated by the displacement of elastic elements under the action of gas forces. Lately new high-efficient seal technologies have been developed for air-to-air leakage reducing – brush seals, finger seals, foil seals, and etc. [2]. One of the promising contactless seals is a hybrid brush seal [3–5], and all-metal compliant HALO seal [6]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 422–429, 2024. https://doi.org/10.1007/978-3-031-40455-9_34

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2 Hybrid Brush Seal Hybrid brush seal consists of a number of elastically supported pads arranged along the shaft circumference (Fig. 1). The pads are attached to the housing by means of an elastic support, which has low rigidity in the radial direction. Axial stiffness of the support is large, which makes it possible to use this seal between areas with a large pressure drop. The brush seal acts like a secondary seal preventing the direct leakage through the stationary seal parts. At the same time, the surfaces between which the brush seal is installed have zero relative rotation speed. Gas lifting forces acting on the pads are determined by the gas flow pressure and depend on the current positions of the pads and vice versa. That provides a potential for the hybrid brush seal to adjust clearance and maintain its minimal value with respect to an external relative motion of rotor and stator parts. Gas-dynamic balancing of the pads on a thin gas film above the shaft surface is investigated by means of numerical simulations. It is assumed that all pads deflect independently of each other. Then the model of the hybrid brush seal can be built based on the consideration of one supported pad. The results of bench tests of a combined brush seal are known, for the manufacture of the suspension of which the method of electroerosion treatment was used [3]. The tests carried out confirmed the contactless operation mode and the reduction of gas leakage by two or more times compared to the labyrinth seal.

Fig. 1. Hybrid brush seal: 1 – housing; 2 – pads; 3 – elastic support; 4 – shaft; 5 – secondary brush seal.

The level of detail of mathematical models should allow considering a large number of design options while maintaining an acceptable complexity of calculations. At the same time, a large number of simplified approaches and calculation methods have already been developed for the brush seal [7–9], which is secondary in the design under consideration. Therefore, the main attention is paid to the construction of a mathematical model for the analysis of balancing pads on an elastic suspension under the action of gas-static and gas-dynamic lifting forces. The brush seal is considered as an elastic porous base, for the calculation of which well-known models can be used [9]. When choosing the design scheme, the following assumptions are made: the main contribution to balancing is made by the flow of gas in the gap between the pads and

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the rotor; radial movements of the pads do not lead to contact between them; the contact of the pads and the brush seal is considered dense and uniform. The precession of the rotor and the asymmetry of the outer casing were not considered in this work. Under these conditions, all pads will occupy the same deformed position, and circumferential cycling of the working gaps will be performed. Therefore, the main characteristics of the seal (flow rate, the size of the working gap, etc.) can be determined by the results of the calculation of one pad. In case of violation of the rotational symmetry of the arrangement of pads or gas loads, it is necessary to consider a set of pads, each of which can be calculated according to the algorithm given below. Note that the independence of the pad movements allows the seal to correct the asymmetry of the radial gap. Thus, the design scheme of the seal includes: an arbitrarily selected pad on an elastic suspension cantilevered along the outer diameter (Fig. 2); a thin gas layer in the gap between the rotor surface and the inner surface of the pad; a sector of the brush seal above the pad.

Fig. 2. Hybrid brush seal and elastically supported pad.

3 Fluid-Structure Interaction Model The representation of a seal compliant element in the form of a rigid cylindrical pad and an elastic support is applied as results a preliminary analysis. The brush seal is considered as a simple elastic foundation with the certain stiffness and leakage rate. An equivalent fluid-structure interaction (FSI) model based on the spatial motion of the padsupport interface is proposed. To obtain the steady fluid-structure interaction solution iterative approach incorporating simplified time-efficient structure stress-strain and gas flow models is used, see Fig. 3. The position of the pad is determined by six degrees of freedom (generalized coordinates): displacements δτ , δυ , δβ and rotation angles θτ , θυ , θβ of section F, in which the pad is associated with the support, see Fig. 2. The validity of this assumption is confirmed by the results of three-dimensional calculations.

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Fig. 3. Static fluid-structure interaction model.

The interdependence of the underpad pressure and the pad position leads to the nonlinear equilibrium equations introducing the balance of gas loads and elastic forces of support and brush: Ksup · U = Rgas (U),

(1)

where Ksup is matrix of the equivalent stiffness of the elastic support (taking into account the stiffness of the secondary brush seal); U = {δτ , δυ , δβ , θτ , θυ , θβ }T is vector of generalized coordinates; Rgas (U) is the vector of the equivalent gas load, composed of the components of the summary force and moment, reduced to the center of the section F. The pad displacements are obtained by multiplying the compliancy matrix (inverse to stiffness matrix) of the elastic support with brush and the gas load vector. The corresponding compliances are obtained by applying unit loads with respect to the model DOFs during the initial 3D structural analysis performed only once. Simplified model is verified by comparing with the results of the 3D numerical simulations for complex loading (Fig. 4).

Fig. 4. 3D numerical simulations: model, radial displacements.

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During the iterations the gas pressure distributions are obtained by solving 2D Reynolds equation for a thin gas layer, which is widely used for hydrodynamic seals and bearings analysis [10]:     ∂ ∂(ph) ∂ 3 ∂p 3 ∂p ph + ph = 6μωRsh , (2) ∂z ∂z ∂s ∂s ∂s where p(z, s, t) is gas flow pressure; h(z, s, t) is the gap thickness (clearance); μ is dynamic viscosity; ω and Rsh are an angular velocity and a radius of the rotor; z and s are «axial» and «circumferential» local Cartesian coordinates on the gap involute Dinv . Gas inertia effects are neglected. Flow is treated to be compressible, adiabatic and laminar. Static pressures values are used as the boundary conditions: p = pup in the inlet; p = plow in the outlet and circumferential sides. In 2D case with accounting circumferential flow solution is found numerically by using the in-house unsteady nonlinear FEA program based on implicit Euler scheme of the in time integration and Newton-Raphson method [11]. The Reynolds equation was made dimensionless to get good convergence of each time step solution. Developed simplified model based on Reynolds equation is verified by comparing with 3D CFD results for different pressure drops and shaft velocities. Pressure distributions under rigid lift pads for circumferentially convergent and divergent clearances are shown in Fig. 5. It allows us to calculate aerodynamic lifting force, acting on the seal pad. This model has also been previously successfully used to calculate gas flow in the finger seals [11, 12].

Fig. 5. Hybrid brush seal loading and underpad pressure distributions.

The results of solving the fluid-structure interaction problem for the hybrid brush seal with pads initially located concentrically to the rotor surface (installation gap 10 μm) are shown in Fig. 6. The same figure shows the results of the verification 3D simulation. Such calculations were made for various initial gaps in the seal. It is shown that the circumferential inclination of the pads demonstrate significant influence on the gas lifting force generated under the pads. At small clearances either a gas-dynamic build-up in a circumferentially converging film, or a «suction» force otherwise is developed. Then several design features (e.g. position of the support relatively to the pad) are investigated

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to assure a converging operating clearance. This is to be the main contributor to the efficient seal operating, i.e. being non-contacting and with low leakage.

Fig. 6. Static fluid-structure interaction results for the hybrid brush seal.

The simplified approaches can be used also for dynamic analysis of the hybrid brush seal. For example, the dynamic response of the pad to modeling radial excursions of the shaft and gas pressures drop changing is shown in Fig. 7 for case with reduced numbers of the DOFs.

Fig. 7. Dynamic response of the pad of hybrid brush seal to modeling radial excursions y of the shaft and gas pressures drop changing.

4 Conclusions Multidisciplinary mathematical simulation technique based on two-way FSI coupling is developed for performance evaluation of the hybrid brush seal. The models are based on the simplified approaches: the model comprising rigid pad attached to elastic support

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with the equivalent stiffness matrix is used for calculation of seal compliant element displacements; the model of gas flow in the clearance between a pad and the rotor based on the 2D Reynolds equation is used for calculation of gas lifting force. Both simplified models are verified by comparing with the results obtained by means of detailed 3D simulations. Calculation time for fast models is significantly less than for 3D detailed models meanwhile their accuracy is sufficient for hybrid brush seal design features preliminary investigation. Developed models allow to introduce simple iterative algorithm to solve inverse problem of mathematical simulation and to calculate seal geometric design parameters according to the specified restrictions on the radial displacement of the pads. This result would have been difficult to obtain using time-consuming traditional numerical 3D approaches. Acknowledgement. The authors are grateful to engineer I.J. Dzeva (ex-employee of Central Institute of Aviation Motors, Moscow) for his significant contribution to this work.

References 1. Temis, J.M., Selivanov, A.V., Yakushev, D.A.: “Virtual Engine” Approach For the Coupled Analysis of Engine Structure. Proceedings of 23rd Int. Symposium on Air Breathing Engines (ISABE 2017). Economy, Efficiency and Environment. Manchester, UK. P. 2431–2439. Paper 22645, p. 9 (3–8 Sept. 2017) 2. Chupp, R.E., Hendricks, R.C., Lattime, S.B., Steinetz, B.M.: Sealing in Turbomachinery. NASA/TM-2006-214341, p. 60 (2006) 3. Justak, J.F., Crudgington, P.F.: Evaluation of a Film Riding Hybrid Seal. Proceedings of 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, AIAA 2006-4932, p. 9. Sacramento, California (9–12 July 2006) 4. San Andres, L., Baker, J., Delgado, A.: Measurements of leakage and power loss in a hybrid brush seal. ASME Journal of Engineering for Gas Turbines and Power 131(1), 012505 (2009) 5. Justak, J.F., Doux, C.: Self-acting clearance control for turbine blade outer air seals. Proceedings of ASME Turbo Expo 2009: Turbine Technical Conference and Exposition, GT2009-59683, p. 9. Orlando, Florida, USA (June 8–12, 2009) 6. San Andres, L., Anderson, A.: An all-metal compliant seal versus a labyrinth seal: a comparison of gas leakage at high temperatures. Proceedings of ASME Turbo Expo 2014: Turbine Technical Conference and Exposition: June 16–20, , Vol. 5C: Heat transfer. GT2014-25572, Dusseldorf, Germany. p. 9 (2014) 7. Chew, J.W., Hogg, S.I.: Porosity Modeling of brush seals. ASME Journal of Tribology 119, 769–775 (1997) 8. Dogu, Y.: Investigation of brush seal flow characteristics using bulk porous medium approach. ASME Journal of Engineering for Gas Turbines and Power. 127, 136–144 (2005) 9. Demiroglu, M., Gursoy, M., Tichy, J.A.: An investigation of tip force characteristics of brush seals. Proceedings of ASME Turbo Expo 2007: Turbine Technical Conference and Exposition. Montreal, Canada. Paper GT2007-28042, p. 12 10. Constantinescu, V.N.: Gas Lubrication. American Society of Mechanical Engineers, New York (1969)

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11. Temis, J.M., Selivanov, A.V., Dzeva, I.J.: Dynamic analysis of a non-contacting finger seal. Proceedings of 9th IFToMM International Conference on Rotor Dynamics, Vol. 21, pp. 2031– 2042. Springer. Mechanisms and Machine Science (2015) 12. Temis, J.M., Selivanov, A.V., Dzeva, I.J.: Finger seal design based on fluid-solid interaction model. Proceedings of ASME Turbo Expo 2013: Turbine Technical Conference and Exposition, June 3–7, Paper GT2013-95701, p. 9. San Antonio, Texas, USA (2013)

Effects of Relief Hole on the Static Characteristics of Externally Pressurized Steam-Lubricated Hybrid Journal-Thrust Bearing Zhansheng Liu, Jinlei Qi(B) , Xiangyu Yu, Peng He, Bing Han, Yishun Yang, and Jiaqi Wang Harbin Institute of Technology, Harbin 08544, NJ, China [email protected]

Abstract. In order to improve the static performance of externally pressurized steam hybrid journal-thrust bearing (EPSHJTB), this study proposes a novel relief hole structure. First, based on the classical bearing experimental data, the correctness of simulation method is verified. The detailed flow analysis of hybrid bearing is conducted using the computational fluid tool. Flow field variables are presented which include pressure, temperature and etc. Second, the relief hole structure is designed to reduce the radial and axial film outlet pressure. This study compares the static load capacity of EPSHJTB with and without relief holes. It’s found that the relief holes actually improve the load capacity of bearing. The numerical results also show that the position of relief holes influence the pressure distribution. Keywords: Externally-Pressurized Steam Lubrication · Hybrid Journal-Thrust Bearing · Static performance · Load capacity

1 Introduction With the development of high power density and closed-cycle steam turbines in deep-sea exploration missions, studies on process fluid lubrication represent a growing field due to its numerous advantages [1, 2]. Externally pressurized steam-lubricated bearing is one of the key components of process fluid lubrication, which plays a crucial role in oil-free and closed-cycle steam turbines. Understanding how bearing geometrical structure and parameters affect the bearing overall performance is in importance in light of practical application. A considerable amount of literature has been published on improving the load capacity of externally pressurized gas bearing. Structure design is the fundamental direction for bearing optimization. Mori [17] conducted a detailed theoretical analysis of aerostatic thrust bearing and pointed out that the conical cavity structure could improve the load capacity of aerostatic thrust bearing. Du et al. [14] considered the effect of pressure equalizing grooves on aerostatic journal bearing. Li et al. [15, 16] proposed a backflow © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 430–446, 2024. https://doi.org/10.1007/978-3-031-40455-9_35

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channel aerostatic thrust bearing with shunt injection to increase the load capacity and stiffness. In order to further reduce the structure size and improve machine power density, numerous researchers have tried to combine thrust bearings and journal bearings. The previous patent by Yates [4] demonstrated a journal-thrust hybrid bearing which made use of leakage flow from journal bearing to supply thrust bearing. This novel design has obvious advantages such as decreasing the total mass flow and simplifying the whole system [3]. Zhang [11] designed a new gas flow path for aerostatic hybrid bearing in ultraprecision spindle and concluded that hybrid bearing could achieve effective lubrication. Lu et al. [9, 10] researched the fluid-structure interaction effect on the hybrid bearing characteristics and also found that aerostatic spindle could be supported readily. Taken together, these studies support the notion that improving the load capacities of hybrid journal-thrust bearing is fundamental for machine reliability. However, there is a relatively few studies on the relief hole. Work is needed to fully understand the implications and effects of relief hole. This paper researches the EPSHJTB based on CFD-numerical method. On the basis of flow and heat transfer of the EPSHJTB, a novel relief hole structure and hollow shaft concept are proposed to improve the load capacity. A further study on relief hole geometrical parameters is also conducted.

2 Numerical Method 2.1 Geometry Model and Mesh Externally pressurized steam hybrid journal-thrust bearing (EPSHJTB) makes use of high temperature and high pressure superheated steam as the lubricating medium. Superheated steam flows through 3 rows of radial throttle orifices and 1 row of axial throttle orifices and finally forms continuous lubricating film, as shown in Fig. 1. The axial relief holes are distributed uniformly on the shaft circular plate. And the radial relief holes are distributed uniformly on the shaft surface. The thickness and pressure distribution of the lubricating film will be nonuniform due to shaft eccentricity. Geometric parameters of the EPSHJTB are presented in Table 1.

Fig. 1. Externally pressurized steam journal-thrust bearing sketch

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Z. Liu et al. Table 1. Geometric parameters of EPSHJTB

Parameter

Value

Diameter of the journal D1

109.924 mm

Diameter of the bearing D2

110 mm

Average radial film thickness c

0.038 mm

Axial film thickness h1 /h2

0.0075 mm/0.0225 mm

Outer diameter D5

220 mm

Distribution Circle Diameter D4

165 mm

Length of the bearing L1

110 mm

Diameter of the orifice d1

2 mm

Diameter of the orifice d2

2 mm

Radial Position L5

25 mm

Axial Position L4

10 mm

Axial Position L3

40 mm

Axial Position L2

55 mm

Diameter of the relief hole d3

3 mm

Eccentricity ratio /Eccentricity distance e

0.5/0.019 mm

Number of radial orifices

3x16

Number of axial orifices

1x16 each thrust face

Number of axial relief holes

1x16 each thrust face

Number of radial relief holes

2x16

Groove depth H

1 mm

Groove outer diameter D3

120 mm

Groove inner diameter D6

111 mm

The relief hole structure is demonstrated in Fig. 2. Radial relief holes and thrust relief holes are marked. Model A is one sixteenth of the whole fluid domain and model B is one half of the radial bearing part. There are 2 radial relief holes and 2 thrust relief holes in model A. Besides, there exist 2 relief grooves in model A. And there are 18 radial relief holes in model B. The fluid domain and mesh details are displayed in Fig. 3. Based on the ICEM CFD 19.0, structured multiblock grids are adopted to decrease the total mesh number of the computational fluid domain, considering the large dimensional difference between film thickness direction and circumferential/axial direction. O-blocks are created in order to improve the mesh quality. Meshes are locally refined around throttle orifices. The mesh convergence test result is shown in Table 2. The node number in the film height direction is more important than the aixal node number and circumferential node number, which affects significantly the load capacity result. When the node number in the film

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Fig. 2. EPSHJTB fluid domain with relief hole structure: a Model A; and b Model B

height direction reaches 14, the load capacity changes slowly and tends towards stable. Therefore, an appropriate node distribution is adopted and there are 13 layers in film thickness direction. With regard to model A, the total nodes number is 1589772 and the total elements number is 1670526. The axial node number is 360 and the circumferential node number is 60. The minimum determinant 3x3x3 value is above 0.77 and mesh quality is above 0.60 for model A. With regard to model B, the total nodes number is 6958980 and the total elements number is 7304888. The axial node number is 320 and the circumferential node number is 640. The minimum determinant 3x3x3 value is above 0.60 and mesh quality is above 0.55 for model B.

Fig. 3. EPSHJTB fluid domain mesh details: a Model A; and b Model B

2.2 Properties of Steam Figure 4 presents the pressure-temperature diagram of steam with various pressures and temperatures. This graph shows that the real properties of steam according to Coolprop

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Mesh Model Mesh Number Aixal Node Circumferential Film Hegiht Load Capacity Number Node Number Node Number Model A

Model B

1431438

360

60

4

1060.333 N

1527198

360

60

8

728.995 N

1670526

360

60

14

516.187 N

1790538

360

60

19

474.169 N

1862358

360

60

22

472.291N

4830258

320

640

4

842.325 N

1432460

160

320

8

981.865 N

4775560

210

400

8

1043.873 N

5820110

320

640

8

1009.769 N

6562499

320

640

11

1250.012N

7304888

320

640

14

1396.605 N

8542203

320

640

19

1476.220 N

9284592

320

640

22

1384.252 N

database [20]. It is apparent that the inlet steam is superheated and the degree of superheat is 66.15 K.

Fig. 4. Pressure-temperature diagram of steam

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Table 3 lists the thermodynamic properties of steam. The dynamic viscosity and thermal conductivity are assumed to be constant which don’t vary with temperature and pressure changing. Redlich-Kwong Equation is adopted as the steam equation of state. Table 3. Properties of steam Lubricant-Steam

Value

Temperature

573.15K

Pressure

3 MPa

Dynamic viscosity

1.34x10 -5 Pa·s

Thermal conductivity

26.1 W/(m·K)

Standard Redlich-Kwong Equation: P=

α0 RT − V − b V (V + b)T 0.5

α0 = 0.42748

(1)

R2 Tc2.5 Pc

(2)

RTc Pc

(3)

b = 0.08664

where R is gas constant, T c is critical temperature and Pc is the critical pressure. 2.3 Numerical Model and Verification The numerical simulations in this article are conducted on ANSYS FLUENT 19.0. Based on previous numerical studies, we use realizable k-epsilon turbulence model and standard wall function. The convergence residual criteria for continuity, momentum and energy are all set to 10–6. The solution algorithm for pressure-velocity coupling is SIMPLE. Steady absolute pressure-based solver is adopted. The governing equations are discretized with the second-order upwind scheme in density/momentum/energy, with the second-order scheme in pressure, while with the first upwind scheme in turbulent kinetic energy and turbulent dissipation. The gradient is discretized by least squares cell-based method. Continuity equation: ∂ρ Dρ + ρ∇ · u = + ∇ · (ρu)=0 Dt ∂t

(4)

 → ∂ ρ− u + ∇ · (ρuu )= −∇p + ∇ · τ ∂t

(5)

Momentum equation:

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Energy equation:   ∂(ρh∗ ) ∂p − + ∇ · ρUh∗ =∇ · (λ∇T ) + ∇ · (U · τ ) ∂t ∂t

(6)

As shown in Table 4, no-slip velocity condition and adiabatic condition are imposed on all walls. The inlet total pressure is equal to 3 MPa and inlet total temperature is equal to 573.15 K. The inlet turbulent intensity is set to 5% and the inlet turbulent viscosity ratio is set to 10. The outlet pressure is assumed to be 0 and outlet temperature is assumed to 300 K. Table 4. Boundary conditions Position

Boundary Conditions

Inlet

Gauge total pressure is 3 MPa, Total temperature is 573.15 K, Supersonic and initial pressure is 0;

Outlet

Pressure is 0, Temperature is 300 K

Wall

No slip and static wall without heat flux

The externally pressurized steam-lubricated journal bearing static performance is characterized by load capacity and mass flow rate. In this study, such an approximate method is adopted to calculate the load capacity of EPSHJTB: calculate the axial load capacity by using the thrust gas film pressure distribution of model A and calculate the radial load capacity by using the radial gas film pressure distribution of model B. Besides, we calculate the mass flow rate by using the inlet flow of model A. Axial Load Capacity:  P · dS (7) Faxial = 16 mod el−A

Radial Load Capacity:  Fradial = 2

P · dS

(8)

mod el−B

Mass Flow Rate:

 ρu · dS

m= ˙ 16

(9)

mod el−A

In this study, we compare the distribution of gas film pressure [7]. Based on the Fig. 5, we find that Realizable k-epsilon turbulence can capture the pressure depression phenomenon and the pressure distributions are similar. The reason for the deviations is the influence of throttle orifice inlet length and structure difference. The orifice inlet geometry will change the gas flow field such as pressure and velocity distribution, so the final gas film distribution will be influenced.

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Fig. 5. Curve of radial pressure distribution: a Whole radius; and b Enlarged view

3 Results and Discussion 3.1 Traditional Externally Pressurized Steam Hybrid Journal-Thrust Bearing This section focuses on investigating the flow and heat transfer characteristics of traditional hybrid journal-thrust steam bearing (EPSHJTB), which includes pressure distribution and temperature distribution. The purpose of this section is to show load capacity generation mechanism. Figure 6 depicts the pressure distribution of whole fluid domain. It can be seen that the radial gas film pressure is uniform due to high outlet back pressure. Therefore, the radial load capacity is nearly zero. Figure 7 indicates the pressure distribution of main thrust face and auxiliary thrust face along the radial direction. The pressure of thrust gas film fluid domain near the radial gas film domain has the higher pressure than near the outlet. The pressure of main thrust face gas film is 2.9961 MPa. And the pressure of auxiliary thrust face gas film is 2.8803 MPa. It can be seen that the pressure suddenly drops near the throttle orifice of the auxiliary thrust face from 3 MPa to 2.2917 MPa and 2.5562 MPa. This pressure depression phenomenon is related to the clearance value [17, 18]. When the clearance is 7.5 µm, the pressure depression phenomenon is disappeared such as the main thrust face. The axial load capacity is generated due to the pressure distribution difference between main thrust face and auxiliary thrust face. Figure 8 shows the temperature distribution of main thrust face and auxiliary thrust face. It can be seen that the temperature distribution is similar to pressure distribution. The temperature of main thrust face gas film near the radial part is 572.95 K. And the temperature of auxiliary thrust face gas film near the radial part is 571.85 K. The similar temperature depression phenomenon is also observed near the throttle orifice of the

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auxiliary thrust face. The temperature drops from 573.15 K to 557.74 K near the outlet and drops from 573.15 K to 564.49 K near the radial gas film.

Fig. 6. Pressure distribution contour of traditional EPSHJTB

Fig. 7. Curve of thrust film pressure distribution

3.2 Externally Pressurized Steam Hybrid Journal-Thrust Steam Bearing with Relief Holes This section focuses on investigating the flow characteristics of EPSHJTB with relief holes, which includes pressure distribution. The purpose of this section is to show the difference between traditional EPSHJTB and EPSHJTB with relief holes. Radial relief holes and thrust relief holes are all opened in EPSHJTB with relief holes.

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Fig. 8. Curve of thrust film temperature distribution

Figure 9 presents the pressure distribution contour of model A. It is apparent from this figure that thrust relief hole and groove structure can effectively reduce the outlet back pressure. Figure 10 presents the pressure distribution contour of model B. This figure illustrates how radial relief holes lower the radial gas film outlet pressure. Figure 11 compares the pressure distribution between traditional EPSHJTB and EPSHJTB with relief holes. It can be seen that the new EPSHJTB’s pressure difference between thrust faces is larger than traditional EPSHJTB’s. The pressure of thrust relief hole and groove structure domain is nearly zero. As shown in Table 5, the load capacity and mass flow rate of EPSHJTB with relief holes is higher than tranditional

Fig. 9. Pressure distribution contour of EPSHJTB with relief holes-model A

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EPSHJTB. Besides, the radial load capacity of EPSHJTB with relief holes is produced by eccentricity. Figure 12 displays the circumferential pressure distribution of the model B, which is in the axial middle position of the radial gas film. When the angle equals 0 and 180 degrees seperately, the clearance film thickness value is the biggest and the smallest accordingly. Therefore, the gas film pressure reaches the lowest and highest respectively. In order to improve the load capacity of thrust bearing part and radial bearing part, the thrust bearing and radial bearing outlet back pressure should be lowered.

Fig. 10. Pressure distribution contour of EPSHJTB with relief holes-model B

3.3 Effects of Relief Holes Distribution This section focuses on investigating the effects of relief hole distribution. The purpose of this section is to show the static characteristic variations between EPSHJTB with different relief hole distributions. Type A refers to EPSHJTB with radial relief holes only, while type B refers to EPSHJTB with thrust relief holes only. Figure 13 demonstrates the pressure distribution contour of model A with type A and type B. Figure 14 presents the pressure distribution curve in the thrust gas film. It is found that the type B owns the lower thrust gas film outlet back pressure, which achieves the higher pressure difference between main thrust bearing surface and auxiliary thrust face. Therefore, the axial load capacity of type B is larger, which is 8244.17 N, as shown in Table 6. Figure 15 compares the temperature distribution of main thrust face type A and type B. Figure 16 displays the pressure distribution contour of model B with type A and type B. Figure 17 shows the pressure distribution curve in the axial middle position of the radial gas film. It is found that the maximum average gas film pressure of type A is about 2.825 MPa and the minimum pressure is about 2.59 MPa. Besides, maximum average gas film pressure of type B is about 2.915 MPa and the minimum average gas

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Fig. 11. Curve of thrust film pressure distribution-EPSHJTB with relief holes

Fig. 12. Circumferential distribution of journal bearing film pressure-EPSHJTBwith relief holes Table 5. Comparison of load capacity and mass flow Bearing Type

Axial Force/N

Radial Force/N

Traditional

3332.40

0.00

Relief Holes

8259.00

2875.90

Mass Flow/(kg/s) 0.0056512 0.043208

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film pressure is about 2.8 MPa. The pressure difference of type A is larger. Therefore, the radial load capacity of type A is larger, which is 1843.89 N, as shown in Table 6. Figure 18 illustrates the temperature distribution curve in the axial middle position of the radial gas film.And This temperature distribution is also similar to pressure distribution. Taken together, EPSHJTB with radial relief holes only-type A has the smaller axial force and the larger radial force compared with type B.

Fig. 13. Pressure distribution contour of EPSHJTB with relief holes: a Type A-with radial relief holes only; and b Type B-with thrust relief holes only

Fig. 14. Curve of thrust film pressure distribution

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Fig. 15. Curve of thrust film temperature distribution

Fig. 16. Pressure distribution contour of EPSHJTB with relief holes: a Type A-with radial relief holes only; and b Type B-with thrust relief holes only

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Fig. 17. Circumferential distribution of journal bearing film pressure

Fig. 18. Circumferential distribution of journal bearing film temperature

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Table 6. Comparison of load capacity and mass flow Bearing Type

Axial Force/N

Radial Force/N

Mass Flow/(kg/s)

All relief holes

8259.00

2875.90

0.043208

Type-A

4685.41

1843.89

0.037284

Type-B

8244.17

1162.84

0.029134

4 Conclusion This study sets out to study the flow and load capacity characteristics of externally pressurized steam hybrid journal-thrust bearing (EPSHJTB). It provides advice for improving the load capacity of EPSHJTB. 1. Traditional EPSHJTB has nearly zero radial load capacity due to the high radial gas film outlet back pressure. And the limited axial load capacity is caused by the pressure equalization of high axial gas film outlet back pressure. 2. EPSHJTB with relief holes can improve the load capacity of the traditional EPSHJTB by reducing the radial and axial gas film outlet back pressure. 3. The distribution of relief holes will influence the static performance of EPSHJTB owing to the change of outlet back pressure.

References 1. Olumide, O., Meihong, W., Greg, K.: Closed-cycle gas turbine for power generation: A state-of-the-art review. Fuel, 694–717 (2016) 2. Orcutt, F.K., Dougherty, D.E., Malanoski, S.B., et al.: Investigation of externally pressurized steam-lubricated journal bearing. J Lubr Technol 90(4), 723–730 (1968) 3. Rowe, W.B.: Hydrostatic, aerostatic and hybrid bearing design. Elsevier, New York (2012) 4. Yates, H.G.: Combined journal and thrust bearings. UK Patent 639, 293, June,1963 (1950) 5. Lund, J.W.: Static stiffness and dynamic angular stiffness of the combined hydrostatic journalthrust bearing. Mechanical Technology Inc. Report Number MTI 63 TR45 (1963) 6. Colombo, F., Lentini, L., Raparelli, T., et al.: The challenges of oil free bearings in micro-turbomachinery. In: Proceedings of I4SDG Workshop 2021: IFToMM for Sustainable Development Goals 1. Springer International Publishing, pp. 403–411 (2022) 7. Belforte, G., Raparelli, T., Trivella, A., et al.: Discharge coefficients of orifice-type restrictor for aerostatic bearings. Tribol Int 40, 512–521 (2007) 8. Chen, S., Zhang, Q., Fu, B., et al.: Analysis of static and dynamic characteristics of aerostatic bearing with reflux orifices. Ind Lubr Tribol 73(6), 961–970 (2021) 9. Liu, L., Lu, L., Yu, K., et al.: A steady modeling method to study the effect of fluid–structure interaction on the thrust stiffness of an aerostatic spindle. Eng. Appli. Computat. Fluid Mechan. 16(1), 453–468 (2022) 10. Lu, L., Gao, Q., Chen, W., et al.: Investigation on the fluid–structure interaction effect of an aerostatic spindle and the influence of structural dimensions on its performance. In: Proceedings of the institution of mechanical engineers, part J: journal of engineering tribology 231(11), 1434–1440 (2017) 11. Zhang, Z.: Research on static and dynamic characteristics of aerostatic spindal in ultraprecision grinding machine. PhD Thesis. Harbin Institute of Technology, China (2019)

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12. Kobayashi, T., Yabe, H.: Numerical analysis of a coupled porous journal and thrust bearing system. ASME J Tribol 127(1), 120–129 (2005) 13. Ise, T., Osaki, M., Matsubara, M., et al.: Vibration reduction of large unbalanced rotor supported by externally pressurized gas journal bearings with asymmetrically arranged gas supply holes (Verification of the effectiveness of a supply gas pressure control system). ASME J Tribol 141(3), 031701 (2019) 14. Du, J., Zhang, G., Liu, T., et al.: Improvement on load performance of externally pressurized gas journal bearings by opening pressure-equalizing grooves. Tribol Int 73, 156–166 (2014) 15. Li, W., Zhang, Y., Cai, S., et al.: Numerical and experimental investigation on the performance of backflow channel aerostatic bearings with shunt injection. Tribol Trans 2022, 1–22 (2022) 16. Li, W., Wang, G., Feng, K., et al.: CFD-based investigation and experimental study on the performances of novel back-flow channel aerostatic bearings. Tribol Int 165, 107319 (2022) 17. Mori, H.: A theoretical investigation of pressure depression in externally pressurized gaslubricated circular thrust bearings. J Basic Eng 83, 201–208 (1961) 18. Mori, H., Miyamatsu, Y.: Theoretical flow-models for externally pressurized gas bearings. J Lubr Technol 91, 181–193 (1969) 19. Chunyang, N., Yulong, L., et al.: Research on bearing and lubrication technology of heat pump compressor under microgravity. Manned Spacecraft 25(01), 50–55 (2019) 20. Bell, I.H., Wronski, J., Quoilin, S., et al.: Pure and pseudo-pure fluid thermophysical property evaluation and the open-source thermophysical property library CoolProp. Ind. Eng. Chem. Res. 53(6), 2498–2508 (2014)

Simulation Analysis of Main Bearing Vibration Characteristics of Wind Turbine Liang Xuan1,2(B) , Ao Shen1,2 , Xiaochi He1,2 , Shuai Dong1,2 , and Jiaxin Dong1,2 1 State Key Laboratory of Precision Blasting, Jianghan University, Wuhan 430056, China

[email protected] 2 School of Smart Manufacturing, Jianghan University, Wuhan 430056, China

Abstract. The operating condition of the main bearing of the wind turbine directly affects the stability of the whole wind turbine, and it is crucial to analyze the vibration characteristics of the main bearing. In this paper, not only the theoretical analysis of the main bearing’s inherent frequency was carried out, but also the vibration characteristics under different constraints are studied by the finite element analysis method, and the first six orders of inherent frequency and vibration pattern of the bearing are listed, and it is found that the deformation is mainly concentrated in the axial direction. Further, the axial vibration simulation experiments of the constrained radial direction were carried out, and the maximum deformation in the axial direction was obtained as 24.9% of the thickness of the main bearing. It provides a theoretical basis for the selection of sensor parameters (range and signal sampling frequency) for further main bearing operation condition monitoring. Keywords: Wind power spindle bearing · Modal Analysis · Vibration characteristics · Finite element simulation

1 Introduction As one of the key components of the wind turbine [1], the main bearing is subjected to dynamic loads of different directions and magnitudes in complex environments, and its stable operation directly affects the performance of the whole wind turbine unit [2]. During the operation of the main bearing of the wind turbine, different vibration characteristics are generated under different loads. To ensure the stable operation of the main bearing and to improve its vibration resistance, its vibration characteristics must be analyzed to ensure its stable and reliable operation [3]. The assembly diagram of the main bearing of the wind turbine is shown in Fig. 1. There are many theoretical analysis methods to study the vibration pattern of wind turbine main shaft bearings, such as the theoretical force state analysis method, dynamics analysis method, finite element modal analysis method, etc. can obtain the vibration characteristics of the main bearing [4]. In this paper, the modal analysis of the main bearing of the wind turbine is carried out by the finite element analysis method, and its corresponding inherent frequency and deformation are obtained according to the vibration pattern of the main bearing under different load states [5]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 447–461, 2024. https://doi.org/10.1007/978-3-031-40455-9_36

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Fig. 1. Wind turbine assembly diagram

In this paper, not only the theoretical analysis of the main bearing inherent frequency was carried out, but also the vibration characteristics under different constraints are studied through the finite element analysis method to provide a theoretical basis for further development of the selection of the main bearing operating condition monitoring sensor parameters.

2 Theoretical Analysis of the Inherent Frequency of the Main Bearing During the movement of a bearing, an impact may be generated between the rolling element and the inner and outer rings, which in turn causes vibration of the bearing itself. The size of the inherent frequency of the bearing is determined by the shape and quality of the component itself as well as the stiffness of the bearing component itself, independent of the service conditions and operating speed of the rolling bearing. The formula for calculating the inherent frequency of a rolling element made of steel is as follows:  0.424 E (1) fball = db 2ρ where: db is the diameter of the rolling body, ρ is the density of the rolling body material, E is the modulus of elasticity of the rolling body. The inner ring inherent frequency calculation formula is as follows:    n n2 − 1 5 2 EI × √ fin = 9.40 × 10 × Ri (2) M n2 + 1 The inherent frequency of the outer ring is calculated by the following formula:    n n2 − 1 5 2 EI × √ (3) fout = 9.40 × 10 × Ro M n2 + 1 where Rx (x = i,o) is the radius from the inner and outer ring rotation axis to the neutral axis, n is the order of the intrinsic frequency, n = 1,2,3,… I is the moment of inertia

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of the circular section around the neutral axis, and M is the mass per unit length of the circular [6]. The same bearing E, I, M and other parameters of the same value, take the same inherent frequency order n, because the outer ring radius Ro is greater than the inner ring radius Ri, so the same order of the outer ring inherent frequency is greater than the inner ring inherent frequency, the stability of the outer ring of the bearing more need to monitor.

3 Establishment of Finite Element Model 3.1 Model Building In this paper, a bearing with an inner diameter of 300 mm, an outer diameter of 420 mm, a thickness of 56 mm, and model number 61960 was used for the simulation experiments. As Fig. 2 Fig. 3 shows the 3D model of the main bearing of the wind turbine established by Solidworks software, the cage was removed for modal analysis in the simulation experiment [7].

Fig. 2. Dimensional drawing of main bearing

Fig. 3. 3D model of main bearing

The 3D assembly drawing model of the main bearing in Solidworks was imported into ANSYS Workbench 18.0 in x_t format. Since the main bearing of the wind turbine is an assembly, contact settings are required between the parts, and the bonding method between the inner and outer rings of the main bearing and the rollers is selected as bound (bonded) contact [8]. 3.2 Adding Material Properties The main bearing of the wind turbine is made of common bearing steel AISI52100 (GCr15), where Young’s modulus E is 2 × 105 MPa, Poisson’s ratio is 0.3, and density is 7850 kg/m3 . 3.3 Mesh Division In the process of finite element simulation, the quality of meshing directly affects the accuracy of the solution [9]. In this paper, automatic meshing is used, the global cell size of the meshing is set to 5mm and the smoothness is set to Medium. Following the above

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steps, the 3D solid cell model of the main bearing of the wind turbine is transformed into a 3D finite element cell model, which contains a total of 134,544 nodes and 310,043 cells. Figure 4 shows the 3D finite element cell model after meshing.

Fig. 4. 3D finite element cell model

3.4 Force Analysis The main bearing of the wind turbine is mounted on the main shaft, the main shaft is fixed with the inner ring of the main bearing, the shoulder of the main shaft, and the end face of the end cap clamp both sides of the inner ring of the main bearing, the housing restricts the radial movement of the outer ring of the main bearing, the end face of the end cap and the side of the housing restricts the axial movement of the outer ring of the main bearing. In summary, the main bearing needs to bear the load in both radial and axial directions, this paper uses the main bearing inner ring fixed, and the main bearing outer ring to apply different sizes of axial force and radial force restraint methods.

4 Modal Analysis 4.1 No-Load Modal Analysis Determination of Working Conditions and Loads The no-load modal analysis of the main bearing fixes only the inner surface of the inner ring of the main bearing, and its no-load simulation experimental model constraint is shown in Fig. 5. Simulation Results and Analysis In this paper, the first six orders of intrinsic frequencies and vibration patterns are selected for analysis and comparison, and since no load acts on the main bearing, the vibration patterns are only the relative values associated with free vibration [11]. Table 1 shows the simulation data, and the simulation results are shown in Fig. 6. As shown in Fig. 6, the first-order mode is the torsional vibration along the axial direction of the bearing; the second, third, and fourth orders are the bending vibration along the radial direction of the bearing; the fifth and sixth orders are the bendingtorsional coupling vibration. As can be seen from the first six-order mode vibration

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Fig. 5. No-load simulation experimental model constraint diagram

Table 1. No-load simulation experimental inherent frequency and maximum displacement Number of steps

Frequency /Hz

Maximum displacement /mm

Vibration diagram

First step

2867.8

10.428

Figure 6-(a)

Second step

2994.4

23.689

Figure 6-(b)

Third step

3021.0

19.787

Figure 6-(c)

Fourth step

3120.6

19.909

Figure 6-(d)

Fifth step

3129.6

20.548

Figure 6-(e)

Sixth step

3224.6

17.424

Figure 6-(f)

diagram, the maximum displacement is produced in the second-order mode, and the outer ring of the bearing is twisted in the radial direction, with a maximum displacement of 23.689 mm, corresponding to a frequency of 2994.4Hz. RATIO (ratio) indicates the ratio of the vibration mode participation coefficient to the first-order vibration mode participation coefficient. In the PARTICIPATION FACTOR CALCULATION information table, the order with the value of 1 is the direction in which the vibration change of the model at that intrinsic frequency occurs mainly. From the information of the no-load simulation experimental solution scheme in Table 2, it can be seen that the first-order mode has the largest deformation in the Ydirection, Z-direction, and around the X-axis, and the corresponding fixed frequency is 2867.82 Hz. The sixth-order mode has the largest deformation in the X-direction and around the Y-direction, and the corresponding fixed frequency is 3224.64 Hz. The fifthorder mode has the largest deformation around the Z-axis, and the corresponding fixed frequency is 3129.63 Hz.

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᧤a᧥First-step modal results᧤b᧥Second-step modal results᧤c᧥Third-step modal results

᧤d᧥Fourth-step modal results᧤e᧥Fifth-step modal results᧤f᧥Sixth-step modal results

Fig. 6. The first six orders of mode vibration diagram

4.2 Loading Modal Analysis Determination of Working Conditions and Loads After fixing the inner surface of the inner ring of the model bearing, different magnitudes of radial force Fr and axial force Fa need to be applied according to the actual load situation. The simulation experimental value of the equivalent dynamic load (P/N) is determined by combining the actual load on the main bearing of the wind turbine and the bearing load range of the model. Thus, the radial force and axial force are allocated according to Eq. (4) and the determined equivalent dynamic load. P = fd (XFr + YFa )

(4)

where the equivalent dynamic load P is in N, X, and Y are the radial dynamic load coefficient and axial dynamic load coefficient, Fr is the radial load and Fa is the axial load, both in N. For the steady state bearing without shock, check the table to take the load coefficient of 1. Check the table to take the radial dynamic load coefficient X as 0.56, axial dynamic load coefficient Y as 2.0, and judgment coefficient e as 0.22.

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Table 2. No-load simulation experiment solving scheme information

(continued)

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The basic dynamic load rating Cr of the bearing model 61960 is 270 KN and the basic static load rating Cor is 370 KN, so this simulation experiment uses radial force 240 KN and axial force 32.8 KN, i.e., the equivalent dynamic load P is 200 KN group simulation experiment as a comparison test for analysis. The model constraint diagram when loading radial force and axial force simultaneously is shown in Fig. 7.

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Fig. 7. Applied load constraint diagram

Simulation Results and Analysis The analysis of the sixth-order modal results before the intercept loading simulation experiment, Table 3 shows the simulation data and the simulation results are shown in Fig. 8. Table 3. Loading simulation experimental inherent frequency and maximum displacement Number of steps

Frequency /Hz

Maximum displacement /mm

Vibration diagram

First step

2865.3

10.418

Figure 8-(a)

Second step

2991.1

23.465

Figure 8-(b)

Third step

3019.1

19.217

Figure 8-(c)

Fourth step

3120.3

19.179

Figure 8-(d)

Fifth step

3124.6

20.934

Figure 8-(e)

Sixth step

3221.6

17.237

Figure 8-(f)

As shown in Fig. 8, compared with the no-load simulation results, the first sixthorder mode vibration pattern of the loading simulation experiment is the same as that of the no-load simulation experiment. As can be seen from the sixth-order mode vibration diagram of the former loading simulation experiment, the maximum displacement is generated in the second-order mode, and the outer ring of the bearing is distorted in the axial direction, with the maximum displacement of 23.465 mm, corresponding to a frequency of 2991.1 Hz. The strength of the outer ring surface of the main bearing will directly affect the performance of the main bearing of the wind turbine.

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᧤a᧥First-step modal results᧤b᧥Second-step modal results᧤c᧥Third-step modal results

᧤d᧥Fourth-step modal results᧤e᧥Fifth-step modal results᧤f᧥Sixth-step modal results

Fig. 8. The first six orders of modal vibrations

From the information of the loaded simulation experimental solution scheme in Table 4, it can be seen that the first-order mode has the largest deformation variables in the Y-direction, Z-direction, and around the X-axis, corresponding to a fixed frequency of 2865.32 Hz. The sixth-order mode has the largest deformation variables in the X-direction and around the Y-direction, corresponding to a fixed frequency of 3221.60 Hz. The fifth-order mode has the largest deformation variables around the Z-axis, corresponding to a fixed frequency of 3124.65 Hz. 4.3 Data Analysis Comparison and Conclusion The data from the unloaded modal analysis and the loaded modal analysis are compared as shown in the folded lines Fig. 9 and Fig. 10. It can be concluded that the maximum difference between the applied restraint and no-load state is 5 Hz, and the maximum difference of the maximum deformation is 0.73 mm. And according to the first six orders of the main bearing modal vibration, the winner bearing will produce deformation in a both no-load and loaded state, but the deformation mainly occurs in the axial direction, and the radial deformation is not obvious. According to the actual installation process, because there will be some dimensional constraints in the radial direction, the next axial vibration analysis simulation experiments of the constrained radial direction.

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Table 4. No-load experiment solving scheme information

(continued)

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L. Xuan et al. Table 4. (continued)

Fig. 9. Comparison of inherent frequency

Fig. 10. Comparison of maximum deformation

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5 Constrained Radial Axial Vibration Simulation Experiment 5.1 Determination of Working Conditions and Loads The simulation experiment mainly observes the deformation of the main bearing of the wind turbine along the axial direction in different orders under no load, so the inner ring of the main bearing is fixed constrained, while the displacement of the outer ring of the main bearing excluding the axial direction is set to 0. The axial simulation experimental model constraint is shown in Fig. 11.

Fig. 11. Axial simulation experimental model constraint diagram

5.2 Simulation Results The first six orders of modal results are intercepted and analyzed, Table 5 shows the simulation data, and the simulation results are shown in Fig. 12. Table 5. Axial simulation experimental inherent frequency and maximum displacement Number of steps

Frequency /Hz

Maximum displacement /mm

Vibration diagram

First step

5340.9

10.065

Figure 12-(a)

Second step

5661.1

13.931

Figure 12-(b)

Third step

5678

13.524

Figure 12-(c)

Fourth step

6553.9

13.682

Figure 12-(d)

Fifth step

6573

13.664

Figure 12-(e)

Sixth step

7837.6

13.557

Figure 12-(f)

As can be seen from the first six orders of mode vibration diagram, in the second order mode, the displacement of the main bearing along the axial scuttle is the largest, and the deformation is 13.931 mm, accounting for 24.9% of the main bearing thickness,

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（a）First-step modal results（b）Second-step modal results（c）Third-step modal results

（d）Fourth-step modal results（e）Fifth-step modal results（f）Sixth-step modal results

Fig. 12. Sixth-order mode oscillation before the axial simulation experiment

corresponding to a frequency of 5661.1 Hz. The impact of axial deformation on the whole bearing is not negligible. During the actual use of the main bearing of the wind turbine, the trend of its axial deformation will be translated into the restraining effect of the main shaft shoulder and end cover on it, and act on the bolt preload force, and the change of its axial force will affect the change of the bolt preload force. According to the first six orders of the main bearing modal vibration pattern and vibration frequency, it is known that the sampling frequency needs to reach about 10000 Hz, and the preload force of the bearing is inferred from the magnitude of the deformation.

6 Conclusion Through the comparative analysis of the sixth-order vibration, pattern results under different modes of the wind power main bearing, it is obtained that the main vibration deformation of the wind power main bearing radially and axially in the operating condition occurs mainly in the axial direction, and the deformation in the radial direction has less influence. Further, the axial vibration experiments in the constrained radial direction are carried out, and the vibration characteristics of the main bearing along the axial scuttle are obtained, and the maximum deformation can reach 24.9% of the bearing thickness. This deformation will be restrained by the interaction force of the bearing with the end cap and spindle shoulder under actual working conditions. By monitoring the deformation of the main bearing in the axial direction, the operating condition of the main bearing can be monitored, and the magnitude of the preload force of the axial end cap bolts of the main bearing can be further analyzed to provide a theoretical basis for the selection of the parameters (range and signal sampling frequency) of the main bearing operating condition monitoring sensor.

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Funding. This research was funded by China Postdoctoral Science Foundation, grant number 2021M692452, and Jiangsu Provincial Bureau of Market Regulation in China, grant number 2021SC64–4-35.

References 1. Xiao, X., Liu, J., Liu, D., Tang, Y., Qin, S., Zhang, F.: A normal behavior-based condition monitoring method for wind turbine main bearing using dual attention mechanism and BiLSTM. Energies 15(22), 8462 (2022) 2. Yang, X., Zhang, T., Li, L., Wang, Y.Q.: Mechanical loading model of main bearing of wind turbine based on test bench. Eng. Appl. Sci. 6(3), 55–65 (2021) 3. Li, Z.: A review of data-driven techniques for early bearing fault diagnosis. Mech. Trans. 47(03), 165–176 (2023) 4. Wang, C., Chen, C., Wei, W.: Analysis of contact stress characteristics of wind turbine bearings under multivariable loads. Noise Vibr. Control 38(S1), 108–111 (2018) 5. Huilan, L., Wan, Y.: Modal analysis of deep groove ball bearings based on ANSYS. Intern. Combust. Engine Accessories 323(23), 56–57 (2020) 6. Hu, Q., Yang, X.: Multi-objective optimization design of angular contact ball bearings using sensitivity analysis. J. Huaqiao Univ. (Nat. Sci. Ed.) 41(05), 555–560 (2020 7. Hu, Y., Dong, H., Mei, G., et al. Mechanical analysis of a new single-point mooring spherical plain bearing. Bearings 476(07), 11–15 (2019) 8. Liu, Z., Zou, H., Miao, H., Chen, D., Zang, C.: Confirmation of kinetic model of on-load tap-changer components based on super model. Mach. Manufact. Autom. 48(05), 131–135 (2019) 9. Zhao, T., Ma, P., Xu, N.: Optimization analysis of response surface of shallow buried pipe after dam based on ANSYS Workbench. Hydroelectr. Power, 44(04), 29–32+115 (2018) 10. Wu, Z., Liu, J., Ran, L., Ye, J.: Dynamic stress analysis and structural optimization of horizontal sieve body based on ANSYS Workbench. Equip. Manufactur. Technol. 09, 43–47 (2020) 11. Zhao, Z.: Application of ANSYS workbench-based software in bearing housing modal analysis. Explosion-proof Electr. Mach. 57(01), 27–28+42 (2022)

Study of Multiplicative Load on the Misaligned Rotor-AMB System Atul Kumar Gautam(B) and Rajiv Tiwari Department of Mechanical Engineering, Indian Institute of Technology, Guwahati, India {katul,rtiwari}@iitg.ac.in

Abstract. This study examines the dynamic response of a two-rotor system under misalignment, imbalance and residual bow. The impact of bow and unbalance on the system’s behavior is analyzed through a parametric study with varying bow amplitude relative to unbalance, while keeping another parameters constant. This innovative approach to analyzing the compound effect of bow and misalignment on the system’s dynamics is a unique contribution to the field of rotor dynamics. To examine the system’s dynamic behavior, a basic model of a rotor-AMB system is considered with certain assumptions, and an active magnetic bearing (AMB) is utilized to mitigate vibration brought on by faults. To simulate misalignment behavior, an excitation function is selected, which generates both odd and even harmonics in the spectrum plot. The existence of a bow in a misaligned rotor system causes multiplicative loads that primarily influence multi-harmonics other than the 1x harmonic, making it challenging to interpret the system’s behavior. The steadystate reactions are numerically simulated and analyzes the time domain, orbit, and displacement spectra in order to draw attention to the harmonic components arises from the combination of faults. The results show that the interaction between bow and misalignment can significantly impact the system’s behavior, leading to increased vibration amplitudes and potential failure modes. The findings of this study can aid in the design and maintenance of rotating machinery to mitigate the effects of bow and misalignment and ensure safe and reliable operation. Keywords: Rotor-bearing- system · Active magnetic bearing · Misalignment · Residual bow · Multiplicative load

1 Introduction Misaligned rotor systems are commonly found in rotating machinery including turbines, compressors, and pumps. These can occur because of manufacturing tolerance, thermal growth, or wear and tear. It can lead to excessive vibration, which adversely affects the performance, reliability, and longevity of the system. It is common for a rotor to exhibit multiple faults; thus, a rotor system can be considered as a complex system, and necessitating the analysis and isolation of faults from vibration signals. In addition to misalignment, a rotor system can be subjected to individual fault or combination of fault. Unbalance and bow are the other common issues that can cause significant vibration and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 462–481, 2024. https://doi.org/10.1007/978-3-031-40455-9_37

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can lead to mechanical failure [1]. In one of the earliest studies on topic [2] conducted research on the shaft bow and its impact on a Laval-Jeffcott rotor system, did the analysis to investigate how bow and unbalance affect the behaviour of rotor system. Then, [3] Compared the imbalance reactions of the Jeffcott rotor system to shaft bow and shaft runout numerically and experimentally, and discovered that the experimental results were almost identical to the numerical ones. With the advancement in technology, trend is shifting towards complex rotor system that incorporate multiple rotors, which make the analysis more challenging. The complexity of the rotor system can arise from the simultaneous occurrence of multiple faults (i.e., unbalance and bow, crack with bow, misalignment and bow along with unbalance, rub and bow etc.) at a time which changes the dynamics of the rotor and impacts the various components, ultimately leading to degraded performance, therefore it becomes crucial to proactively anticipate and detect the development of various faults at an early stage of operation [4]. Over the past decade multiple studies have focused on the identification and localization of fault parameters. Researchers have made a significant progress in this field, and research is still ongoing with regard to the analysis of various faults and their implication on other faults [5], this paper primarily focuses on the simultaneous presence of misalignment and bow and their interaction with each other, particularly in terms of their influence on the dynamics of the system. The dynamics of rotary machines with shaft misalignment and imbalance are explained by [6, 7] emphasizing the rotor system under misalignment, equipped universal joint coupling, can rise to resonance phenomenon close to the natural frequency, with dominant peaks at harmonics of 1x and 2x motor speed [8] further analyzed the dynamic responses of rotor systems experiencing parallel misalignment, highlighting the potential for coupling misalignment to induce resonance during transient states. Bow in the rotor system can caused by uneven heating or cooling of the symmetrical rotor, [9] proposed the different sources of bow and coupling misalignment in the rotor system and classified the bow as local and expanded, depending on whether it is centered on a signal element or distribute throughout the rotor. The bowed rotor’s dynamics with a transverse surface fracture were examined by [10] and observed the residual bow affects stiffness properties and alters the behavior of the system, magnitude and phase are the main parameter of bow affecting the dynamics, [11] investigated the rotor’s behavior using analytical and experimental method, examine with different bow magnitude and its phase relative to unbalance and noticed that bow factor rises with decreasing imbalance eccentricity and mass. Then, [12] examined the transverse rotor responses subjected to wrapped shaft, constructed a dynamic model of a wrapped rotor, did the experimental analysis. The finding revealed that the presence of shaft bow had a notable impact on both the natural frequencies and vibration amplitudes of the rotor. In industries machine misalignment and shaft bending account for 35% of the machine’s down time, [13] conducted an experimental analysis on the dynamics of the bent rotor system with the aim of detecting its unique vibration signature. The study involved extracting the responses and utilizes the FFT analyzer to generate the vibration spectrum, and observed that primarily exhibited the 1x and 2x running speed harmonic in FFT. Correlation analysis plays a crucial role in developing algorithm for fault identification, in this context [14, 15] put forth two approaches to identify imbalance and bow.

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The first approach employed analytical modelling utilizing FEM to analyze a Jeffcott rotor, meanwhile, the other one involved correlation analysis and SEREP (sequential estimation of rotodynamic parameters) for a two-disk rotor, the study showed that both techniques can reliably predict the fault parameter from collected responses and are valuable methods for detecting and balancing bowed rotors. [16] conducted a study on the rotor responses experiencing angular misalignment, utilized the structural analysis of the flexible coupling through numerical simulation and experiment, and observe that the proposed approach has the potential to accurately model the misaligned metal disc coupling. In order to enhance the performance and lifespan of rotating machines, it is essential to control the vibrations that can significantly degrade system performance. [1] highlights the importance of identifying fault factors and controlling vibrations to achieve these objectives. One effective method for vibration control is the utilization of active magnetic bearings (AMBs), which are electromechanical devices. AMBs play a crucial role in enhancing operating conditions and minimizing potential humans’ injury. They offer advantages such as faster rotating speeds and no need for lubrication when compared to conventional bearings [17]. Due to their capabilities and benefits, AMBs have gained significant attention from researchers and are being employed in various research and commercial applications to actively control vibrations in rotating systems. [18] carried out the investigation to identify the various fault parameters associated with misaligned rotor system levitated by an AMB (active magnetic bearing) using the unique trial misalignment approach, performed the FE modelling based on the Timoshenko beam theory, and verified the algorithm’s robustness against various measurement noise percentage [19] reviewed of several diagnosis and balancing approach for wrapped rotor system, discussed the time domain, frequency-domain, and time-frequency-domain analyses to deal with the vibration responses obtained from the system. To identify the various parameters affecting the dynamics of rotor in the presence of crack, [20] proposed a modeling strategy based on a frequency domain algorithm. The active magnetic bearing was employed to minimize the vibration’s amplitude, and the acquired displacement and current responses were utilized to estimate the fault parameters. The behavior of 4-DOF, comprising two laval rotor equipped with AMBs, were investigated by [21, 22]. They conducted a comparative analysis of system responses under different conditions, including pure misalignment, pure unbalance, or a combination of both, and concluded that the shape of the orbit plot was influenced by factors such as the degree of misalignment, stiffness parameters, and operating speed. Furthermore, the study was expanded to develop techniques for identifying misalignment, imbalance, damping, and additive coupling stiffness, thereby providing a direct assessment of the severity of misalignment. In order to control the multi-harmonic vibration generated by parallel misalignment, unbalance, and load-torque operating conditions in the rotor-AMB system, researchers [23] devised a hybrid control technique combining proportional derivative and multi-harmonic adaptive vibration control feedforward control. The PD feedback law was employed to ensure system stability, while the MHAVC feedforward control was utilized to minimize steady-state vibration. [24] investigated the rotor vibrations

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under misalignment with various types of current-biased radial Active Magnetic Bearings (AMBs), revealing the prevalence of 2× and 4× harmonic components influenced by the electromagnet’s quantity and gap between the stator and rotor. In recent years, a considerable amount of research has been conducted on the dynamics on the rotor system subjected to misalignment and their identification. Several studies have investigated the individual effect of unbalance and bow on the system vibration response, but most of studies do not account for the presence of combination of fault (i.e., bow and misalignment, crack and misalignment etc.) while investigating the dynamics of rotor system [25]. However, there is still lack of understanding of how these two parameters interact with each other and with misalignment to affect the rotor system’s response. In this regard [26], conducted a comprehensive study involving numerical and experimental analysis to investigate the two rotor system dynamics with multiplicative faults, utilized mathematical models and simulation techniques to conduct numerical analyses, while experimental tests were performed on a physical two-rotor setup to validate the findings. This paper presents a thorough analysis of the system dynamics of a rotor train with an AMB and coupled through a flexible coupling assisted by conventional bearing. The study focusses on the examining the variation of fault parameters and their effect on the dynamic response of a rotor system when multiple faults occurred simultaneously, including coupling angular misalignment, residual bow in the shaft, and unbalance. To analyze the system dynamics, the governing equations incorporating the effect of multiplicative fault are formulated using LaGrange’s equation. To simulate the misalignment behavior and capture its impact on the system, an excitation function is introduced in the modeling, which generate harmonics in both side of the displacement spectrum. To simplify the analysis, the study assumes weight dominance allowing the system equations to be linearized and facilitating their solution. The study then proceeds to explore the influence of bow on misalignment and its consequent effect on the system dynamics. Since bow and unbalance display similar behavior when occurring individually, so to make the difference the analysis focusses on varying the magnitude of bow relative to unbalance, which is the crucial parameter affecting the system dynamics. Five different cases are considered, by varying the bow magnitude while retaining other parameters constant, such as the location (phase) of bow and the magnitude and phase of imbalance. This variation allows for a comprehensive examination of the system responses under different situation. The analysis is performed for three different conditions: when all faults are present, when a multiplicative force (generated from misalignment and bow) is absent, and when misalignment is not considered in the system. By conducting this analysis, the study aims to gain insights into the combined effects of these faults and their interaction, providing a better understanding of the system’s dynamic response in real-world scenarios.

2 System Modeling and Analysis This section presents a mathematical model used to analyze a two bowed-rotor system with angular misalignment and equipped with Active Magnetic Bearings (AMB). The model assumes weight dominance, making the system equations linear, and considers a flexible coupling between the rotors and conceptualized as a helical torsional spring. The

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AMB is utilized to mitigate vibration due to the combination of faults. The study’s goal is to examine the interaction among misalignment and residual bow in the system. The coupling’s mass and damping are neglected, and equivalent coupling angular stiffness in transverse directions are assumed to facilitate system modeling. 2.1 System Configuration and Governing Equation Figure 1 illustrates the two misaligned residually wrapped rotor system equipped with Active Magnetic Bearing (AMB), where a rigid disc is located at the middle of the shaft and assisting bearings are present at end. An AMB is installed close to disc of rotor 2 to suppress vibration responses by using the electromagnetic force generated by the magnetic actuator. The disc on each rotor is positioned at a distance of 0.25 m from the bearing support, and each rotor is modeled with stiffness and damping parameters represented by kiab and ciab , respectively, which are assumed to be equal in the orthogonal direction with a, b = x, y.

Fig. 1. A simple misaligned rotor system with AMB

The flexible coupling between the two shafts allows only for transverse moments and is modeled with identical angular stiffness in the lateral orthogonal directions, denoted c and k c . Transverse displacement responses at the disc, are selected as generalized as kϕx ϕy coordinates. To comprehensively describe the motion of the model, four coordinates are required. These coordinates are represented by the total displacement vectors of rotor 1 and rotor 2, denoted as u1 (t) and u2 (t) respectively. These vectors encompass multiple components of displacement, including dynamic displacement, displacement resulting from static weight, and displacement induced by the residual bow in the shaft. They can be mathematically expressed as follows: ui (t) = ui (t) + δusi + ubi (t); i = 1, 2

(1)

 T T T   ui (t) = xi yi ;δusi = δxsi δysi ;ubi (t) = xbi ybi

(2)

where, ui (t), δusi and ubi (t) are the vector for dynamic displacement, displacement from static weight, and displacement caused by residual bow in the shaft, respectively. xi and yi are the vibratory transverse displacement, xbi and ybi represents the orthogonal displacement resulting from residual bow in the vertical and horizontal direction, and δxsi indicates the displacement from static weight. The coupling in this system is designed to be flexible and is represented by a transverse angular spring [22], as depicted in Fig. 2;

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here it is assuming that the coupling can twist or deform specifically in response to angular misalignment and residual bow between the bearing centers. This flexibility allows the coupling to accommodate and compensate for the angular misalignment and bow only. Since, the analysis assumes weight dominance, which linearize the EOM. Therefore, a linear relationship occurred among the slope at the coupling and the disc transverse displacement. The existence of residual bow and misalignment in the shaft might can influence the overall angular displacement (slope) at the coupling location. This relationship can be expressed as the sum of angular displacement from misalignment and wrapped shaft, as illustrated in Fig. 2

Fig. 2. A schematic diagram illustrating the slope at the coupling location

In the figure, the angular displacement due to misalignment and residual bow at the coupling location are represented as m ϕ1c , b ϕ1c , m ϕ2c and b ϕ2c , and can be expressed mathematically in orthogonal directions as follows:         m ϕyc1 m ϕyc2 b ϕyc1 b ϕyc2 + ;ϕ 2c = + (3) ϕ 1c = m ϕxc1 m ϕxc2 b ϕxc1 b ϕxc2 By employing the weight dominance, the angular displacement at the coupling locaT T   tion m ϕ1c + b ϕ1c and m ϕ2c + b ϕ2c for both the rotors can be expressed with a transverse displacement by factor ε1 and ε2 written as, 

 

 x1 xb1 x2 xb2 + ;ϕ 2c = ε2 + (4) ϕ 1c = ε1 y1 yb1 y2 yb2 The factors ε1 = 2a31 and ε2 = 2a32 , may be calculated using the deflection formulae at any point along the simply supported beam [27]. The governing equations can be derived by employing the energy equation and subsequently applying Lagrange’s equations. The kinetic energy of the disc can be written as, T=

1

1 1 1 2 m1 x˙ 1 + y˙ 12 + Ip1 ω2 + m2 x˙ 22 + y˙ 22 + Ip2 ω2 2 2 2 2

(5)

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The potential energy for the rotor system can be expressed as follows:

(6)

Upon substituting Eq. (4) into Eq. (6), the potential energy can be modified to incorporate the impacts of coupling misalignment and shaft bow in relation to transverse and angular displacements. The system assumes that the shafts possess equal stiffness in two perpendicular directions, i.e., k1xx = k1yy and k2xx = k2yy , The coupling, on the other hand, is characterized by equivalent transverse angular stiffness in both perpendicular directions, represented as kϕcx = kϕcy . By performing the substitution, the potential energy can be expressed in a more detailed and comprehensive form, and can be written as,

1 1 1 1 1 k1xx x12 + k1yy y12 + k2xx x22 + k2yy y22 + kϕcx ε12 y12 + ε22 y22 + 2ε1 ε2 y1 y2 2 2 2 2 2

1 c 2 2 2 2 + kϕy ε1 x1 + ε2 x2 + 2ε1 ε2 x1 x2 2 (7)

V =

Equation (7) represents the potential energies of the shaft and coupling with respect to translational displacement responses at the discs. Considering the influence of external viscous damping, the dissipative energy of the rotor system can be expressed as follows: D=

1 1 1 1 c1xx x˙ 12 + c1yy y˙ 12 + c2xx x˙ 22 + c2yy y˙ 22 2 2 2 2

(8)

The Lagrange’s equation has been applied to the current rotor model with four degrees of freedom (x1, y1, x2, y2) to obtain the governing equation. The equation, which describes the behavior of the rotor system, can be expressed as follows:

  c m1 x¨ 1 + k1xx + ε12 kϕy x1 + ε1 ε2 kϕcy x2 + c1xx x˙ 1 = m1 eu1 ω2 cos ωt + βu1 + k1xx δxs1   +k1xx eb1 cos ωt + βb1

  c m1 y¨ 1 + k1yy + ε12 kϕx y1 + ε1 ε2 kϕcx y2 + c1yy y˙ 1 = m1 eu1 ω2 sin ωt + βu1 + k1yy δys1   +k1yy eb1 sin ωt + βb1

  c m2 x¨ 2 + k2xx + ε22 kϕy x2 + ε1 ε2 kϕcy x1 + c2xx x˙ 2 = m2 eu2 ω2 cos ωt + βu2 + k2xx δxs2   +k2xx eb2 cos ωt + βb2

  c m2 y¨ 2 + k2yy + ε22 kϕx y2 + ε1 ε2 kϕcx y1 + c2yy y˙ 2 = m2 eu2 ω2 sin ωt + βu2 + k2yy δys2   +k2yy eb2 sin ωt + βb2 (9)

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2.2 Modelling of Coupling Misalignment The coupling stiffness fluctuates with time as the shaft rotates, essentially changing the intact coupling stiffness and generates the static and dynamic coupling stiffness (function of time). Several investigations have shown that misaligned rotor system exhibit both odd and even multiple harmonics component in frequency spectrum plots. In order to capture all harmonics and mimic the rotor’s dynamic behavior under misalignment, [22] employed a rectangular wave with a 40% duty cycle as the excitation function, and can be written as, σ (t) = ao +

∞ 

an cos(nωt)

(10)

  tp 2A 2A sin nπ = sin(0.4nπ ) nπ T nπ

(11)

n=1

ao = 0.4 ; an =

where A, tp , and T represents the magnitude, pulse width, and time period, then the excitation function for different an can be written as: σ (t) = 0.4 + 0.6055 cos(ωt) + 0.1871 cos(2ωt) − 0.1247 cos(3ωt) − 0.1514 cos(4ωt) +0.1009 cos(6ωt) + 0.0535 cos(7ωt) − 0.0468 cos(8ωt) − 0.0673 cos(9ωt) + 0.055 cos(11ωt)

(12)

+0.0312 cos(12ωt) − 0.0288 cos(13ωt) − 0.0432 cos(14ωt) + 0.0378 cos(16ωt) + 0.022 cos(17ωt)

2.3 AMB Force Modeling To support the system and reduce vibration, an 8-pole radial active magnetic bearing with an isotropic actuator was employed. It maintains the spinning shaft free of physical contact by suspending the rotor in the air using magnetic force. The electro-magnetic forces generated by AMB are given [1].

−ksx2 x2 + kix2 ix2 (13) 2 fAMB = −ksy2 y2 + kiy2 iy2 Herein ksx2 , ksy2 , kix2 , and kiy2 represents the AMB displacement and current stiffness and determine the stiffness characteristic of AMB system, ix2 and iy2 are the AMB control current in x and y directions are the resulting outputs obtained through the utilization of controller. Control is of utmost importance for ensuring the stability, precession, performance and robustness in the operation of the AMB. The study employs a PID controller to accomplish the control current and achieve the desired control objectives in the AMB system. By utilizing the PID controller, the researchers aim to ensure that the AMB system functions reliably and efficiently. The PID controller continuously monitors the system’s feedback signals, such as the displacement and current stiffness coefficients, and computes the appropriate control currents in the x and y directions, for the transverse movement of rotor 2 it can be expressed as,  ix2 (t) = kp2 x2 (t) + kI2

x2 (t)dt+kD2

dx2 (t) ;iy2 (t) = kp2 y2 (t) + kI2 dt

 y2 (t)dt+kD2

dy2 (t) dt

(14)

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where kp2 , kI2 and kD2 represents the proportional, integral, and derivative gain parameters of the PID controller, and have a significant impact on the resonant speeds and dynamic responses of the system. However, When the rotor-AMB system exhibits negative displacement stiffness, it inherently becomes unstable. Therefore, to stabilize the rotor system, it is necessary to appropriately adjust the control parameters of the AMB. The determination of these control parameters is carried out using the Routh-Hurwitz criteria, as outlined by [1]. Consequently, by incorporating the AMB system, the EOMs for rotor 1 and rotor 2 are modified and can be written as,

M1 u¨ 1 + (CE )1 u˙ 1 + K1 + ε12 Kϕc u1 + ε1 ε2 Kϕc u2 = 1 funb + 1 fbow + 1 fmis − 1 fcons

M2 u¨ 2 + (CE )2 u˙ 2 + K2 + ε22 Kϕc u2 +ε1 ε2 Kϕc u1 = 2 funb + 2 fbow + 2 fmis − 2 fAMB − 2 fcons (15) With;





j ωt+βu1 j ωt+βu2 j ωt+βb 1 ; 2 funb = m2 eu2 ω2 e ; 1 fbow = eb k1xx e 1

j ωt+βu2 2 2 c c ; 1 fcons = ε1 kϕ δxs1 + ε1 ε2 kϕ δxs2 ; 2 fcons = ε22 kϕc δxs2 + ε1 ε2 kϕc δxs1 2 fbow = m2 eu2 ω e





1 2jωt ε 2 δx + ε ε δx jωt e cos β + e cos β 1 2 s2 + 2e 1 fmis = Kξ σ (t) 1 + e b1 b1 b2 b2 1 s1 2 2 1 funb = m1 eu1 ω e

2 fmis =







1 Kξ σ (t) 1 + e2jωt ε22 δxs2 + ε1 ε2 δxs1 + 2ejωt eb cos βb + eb cos βb 2 2 1 1 2

3 Forced Response Analysis To investigate the rotor’s dynamic behavior considering angular misalignment and residual bow, the governing EOM are expressed in a complex form. To accomplish this, the equations of motion (EOMs) in the y-direction are multiplied by the imaginary unit j and then added to the EOM in the x-direction. This process combines the equations and allows for a unified treatment of the system dynamics., as, q1 = x1 + jy1 ;q2 = x2 + jy2 . This manipulation allows for a comprehensive representation of the system’s dynamics, considering both the x and y directions simultaneously. By combining these equations, the final EOM can be expressed as:

    m1 q¨ 1 + (c1xx )q˙ 1 + k1xx + ε12 kϕc q1 + ε1 ε2 kϕc q2 = m1 eu1 ω2 ej ωt+βu1 + k1xx eb1 ej ωt+βb1

i=∞     i=∞ + kξ ε12 δxs1 + ε1 ε2 δxs2 ri ejiωt + kξ eb1 cos βb1 + eb2 cos βb2 si ejiωt 



i=−∞

rotor 1 misalignment force

− ε12 kϕc δxs1 + ε1 ε2 kϕc δxs2   







i=−∞



rotor 1 multiplicative force

rotor 1 constant force

(16)

Study of Multiplicative Load on the Misaligned Rotor





j ωt+βu2 j ωt+βb 2 m2 q¨ 2 + (c2xx )q˙ 2 + k2xx + ε22 kϕc − ks2 q2 + ε1 ε2 kϕc q1 = m2 eu2 ω2 e + k2xx eb e 2

471



i=∞

i=∞   −ki2 ic2 + kξ ε22 δxs2 + ε1 ε2 δxs1 ri ejiωt + Kξ eb cos βb + eb cos βb si ejiωt 2





i=−∞



2



rotor 2 misalignment force

1



1

i=−∞



(17)

rotor 2 multiplicative force



− ε22 kϕc δxs2 + ε1 ε2 kϕc δxs1    rotor 2 constant force

The variables ri and si denotes the participating factors associated with the misalignment and residual bow forces, respectively. These factors contribute to the generation of multi-harmonics due to the influence of misalignment and multiplicative forces. The specific values of the participating factors corresponding to each harmonic are provided in Table 1. The participating factors quantify the relative influence or contribution of the misalignment and multiplicative fault on the generation of each harmonic. Table 1. The magnitudes of the participation factors resulting from the misalignment and bow Multiple Harmonics

Participation factor si (due to the residual bow)

ri (due to the misalignment) 0× 1× 2× 3× 4× 5× -1× -2× -3× -4× -5×

0.24678 0.30275 0.24678 0.12020 0.00893 -0.03118 0.12020 0.00893 -0.03118 -0.01263 0.01338

0.30275 0.40000 0.30275 0.09355 -0.06235 -0.07570 0.09355 -0.06235 -0.07570 0 0.05045

3.1 Numerical Simulation The study’s numerical simulation attempts to analyze the rotor’s dynamic behavior when multiple faults, including misalignment, residual bow, and unbalance, occur simultaneously. There are two key parameters, namely the magnitude and location (phase), which directly influence the effect of bow on the system. In particular, the study examines the influence of bow on the system’s dynamics by varying the magnitude. Previous literature suggests that bow and unbalance exhibit similar behavior when both are present in the system. They predominantly manifest the 1x harmonic in the frequency spectrum with the highest amplitude. The interaction between bow and misalignment is a key focus of

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the research, and the study aims to investigate how changes in bow magnitude affect the overall behavior of the system. Numerical analysis has been performed with variation of bow parameter and keeping the unbalance constant. The amplitude of the bow is systematically varied while maintaining the other factors, such as the amplitude and phase of the imbalance, as well as the phase of the bow, constant. For the parametric study, five specific cases are considered: the first case represents a scenario with zero bow magnitude, followed by cases where the bow magnitude is reduced by 30%, 60%, and 90% from the magnitude of the unbalance, respectively. The final case examines the situation where the bow magnitude is equal to that of the unbalance. To examine the influence of bow on misalignment, the study considers three different scenarios, firstly, the system is analyzed when both bow and misalignment coexist along with unbalance. Secondly, the multiplicative load resulting from the combination of bow and misalignment is removed, isolating the individual effects of bow and misalignment. Lastly, the scenario without misalignment is investigated, where only bow and unbalance are present in the system. These three scenarios are examined to gain insights into how bow affects the behavior of the rotor system under the presence or absence of misalignment. The suggested rotor-AMB model, incorporating misalignment, bow, and unbalance, is simulated using a dedicated SIMULINKTM block within the MATLAB platform, as depicted in Fig. 3. The simulation is conducted by employing the ODE45 RungeKutta solver with a constant step size of 0.0001. The duration of the simulation is set to 5 s, utilizing the parameters provided in Table 2. To obtain reliable and meaningful results, the analysis focuses on the signals recorded between 2 to 3 s of the simulation. This range is selected to exclude any initial numerical transient signals and ensure the analysis captures the steady-state behaviour of the system. Table 2. The system parameter of a proposed rotor model Parameters

c1xx

Values 80 N-s/m

Parameters

eb2

c2 xx

80 N-s/m

βb

ch

30 N-s/m

βb

δ xs

2.6×10-5 m

eu1

1

δ xs

-5

1

2

Values 3×10-5 m

15 18 2.4×10-4 m -4

Parameters

Δ kξ

Values 5.8×103 N-m/rad

kP

5000 A/m

kI

500 A/(m-s)

ks

105210 N/m

3×10 m

eu2

2.4×10 m

ki

42.1 N/A

k1xx

5

6×10 N/m

βu1

30

kD

4 A-s/m

k2 xx

6105 N/m

β u2

45

eb1

3×10-5 m

kϕc

9×103 Nm/rad

2

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473

Fig. 3. MATLAB Simulink model to generate responses

3.2 Result and Discussion Displacement responses play a crucial role in assessing the dynamic behavior of a rotor system, these responses are captured at disc location when the rotor is spinning at a speed of 32 Hz and analyzed using different techniques (Time-domain, frequency-domain, and orbit plot). In order to simplify the illustrations and minimize the number of figures, the similar vibration pattern of input rotor is omitted, and only responses of the output rotor are displayed. The time domain analysis provides a graphical representation, as shown in Fig. 4, of the actual displacement values of the rotor system over time. This figure illustrates the system’s behavior in transverse direction under different fault conditions. Figure 4(a) and Fig. 4(b) displays the responses when all three faults are present simultaneously and affecting the rotor system. On the other hand, Fig. 4(c) to 4(f) show the responses without multiplicative load and without misalignment in both vertical and horizontal direction. By examining Fig. 4, it becomes evident that the responses vary across the five different cases analyzed. It is observed that the rotor system experiences higher vibration when the magnitude of bow matches the magnitude of unbalance. As the reduction percentage increases (from 30% to 60% to 90%), the vibration decreases. When the magnitude of bow is zero, the system experiences very low vibration. In Fig. 4(a) to Fig. 4(d) the response curves appear non-sinusoidal, indicating the presence of misalignment. Non-sinusoidal behavior in the time domain plot suggests the occurrence of various harmonics of speed multiples, which commonly arise when misalignment is present in the system. Notably, the non-sinusoidal curve is observed particularly when the magnitude of bow is relatively small compared to the unbalance. As the magnitude of bow increases, it dominates over the misalignment effect, resulting in a nearly sinusoidal pattern while increasing the vibration amplitude.

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In Figs. 4(a) to 4(d), when the bow magnitude matches the unbalance magnitude, the horizontal direction exhibits higher vibration amplitudes (1.3178e–04, 1.3050e–04) compared to the vertical direction (1.2399e–04, 1.2414e–04) in the presence of all three faults and absence of the multiplicative fault. This difference in amplitudes is attributed to the presence of misalignment in the vertical direction, which rises the effective stiffness in that direction relative to the horizontal direction. Consequently, the horizontal direction experiences greater vibration. Figures 4(e) and 4(f), however, show equal amplitudes (1.2389e–04) in both directions when misalignment is not considered, this confirms that the increased vibration amplitude in the horizontal direction (as shown in Fig. 4(b) and Fig. 4(d)) is specifically caused by misalignment.

Fig. 4. Dynamic responses at rotor speed of 32 Hz (a–b) With misalignment, residual bow, and unbalance (c–d) Without multiplicative load (e–f) Without misalignment

To actively mitigate the vibrations in both directions of rotor 2, an Active Magnetic Bearing (AMB) is strategically positioned near the disc. The AMB operates by leveraging the current responses to counteract the transverse vibration. Figure 5 illustrates the AMB current responses necessary for effectively suppressing the transverse vibration. Specifically, Fig. 5(a) and Fig. 5(b) display the current component when all faults occur simultaneously, representing a comprehensive approach to addressing multiple faults. On the other hand, Fig. 5(c) to 5(f) demonstrate the responses without considering the multiplicative component resulting from bow and misalignment, with the additional consideration of the system without misalignment. These figures provide valuable insights into the optimal current amplitude required to mitigate the vibrations in the transverse direction, enabling effective control and stabilization of rotor system. Figure 5(a) and Fig. 5(b) clearly demonstrate that a higher amount of current, specifically 0.7022A, is necessary to control the vibration in the horizontal direction compared to the vertical direction, which has an amplitude of 0.6596A. These figures correspond to the scenario where all three faults are present simultaneously. The higher current requirement in the horizontal direction indicates that the vibrations in that direction are more severe and demand greater control efforts from the Active Magnetic Bearing (AMB) system. This

Study of Multiplicative Load on the Misaligned Rotor

475

finding aligns with the previous observation that the horizontal direction experiences higher vibration amplitudes due to the presence of misalignment. Therefore, the AMB system must exert a stronger corrective force to effectively suppress the vibrations in the horizontal direction in comparison to the vertical direction. The effectiveness of active magnetic bearings can be observed in Fig. 6, where it is evident that they reduce vibrations and can be utilized to mitigate them. In Fig. 6(a) to 6(d), the impact of varying bow magnitude is also observed. When the bow magnitude is negligible (Case-I) or significantly smaller than the unbalance (Case-IV), the influence of misalignment is clearly noticeable. However, as the bow magnitude becomes equal to the unbalance magnitude (Case-V), it surpasses the effect of misalignment and exhibits an almost sinusoidal curve with a higher amplitude.

Fig. 5. AMB current responses at rotor speed of 32 Hz (a–b) With misalignment, residual bow, and unbalance (c–d) Without multiplicative load (e–f) Without misalignment

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Fig. 6. Dynamic responses with and without AMB at rotor speed of 32 Hz (a–b) Case-I (c–d) Case-IV (e–f) Case-V

Orbit plots are a valuable tool for analyzing rotor dynamics, providing a visual representation of the rotor shaft’s motion in a two-dimensional plane. In order to gain further clarity on the dominant nature of bow, an orbit plot is employed, as depicted in Fig. 7. In Figs. 7(a), 7(b), and 7(c), orbit plots are presented for different conditions, revealing noncircular patterns indicative of a looping nature associated with misalignment. Notably, in Figs. 7(a) and 7(b),when the magnitudes of bow and unbalance are nearly equal, the effect of misalignment is reduced, with bow dominating the behavior, as observed in case (V). Conversely, cases (I), (II), (III), and (IV) clearly exhibit the presence of misalignment, as evidenced by the non-circular patterns. Unbalance and bow show the almost similar pattern, therefore when the misalignment is not present in the system, the orbit plot is observed to be pure circular as can be seen from Fig. 7(c).

Fig. 7. Orbit plot for rotor 2 at rotor speed of 32 Hz (a) when all three faults are present (b) when multiplicative load is absent (c) when misalignment is absent

The presence of bow in a misaligned rotor system introduces an extra load that has a multiplicative effect on the system, directly influencing its dynamic behavior. However,

Study of Multiplicative Load on the Misaligned Rotor

477

this multiplicative force is not easily discernible in the time domain and orbit plot analysis. To accurately determine how changes in bow magnitude, in relation to unbalance, impact the dynamic behavior of a misaligned rotor system, the Full-spectrum tool is employed. It involves transforming the displacement responses from the time domain to the frequency domain using techniques like Fourier analysis. It shows both forward and backward whirl, enabling the assessment of external and inherent forces. Figure 8 illustrate the full-spectrum plot for the responses of rotor 2 for all five cases. Figures 8(a) and 8(c) exhibit various harmonics under simultaneous occurrence of multiple faults (misalignment, unbalance, and bow), and excluding the additional load component caused by bow and misalignment. The absence of the multiplicative load results in similar harmonics amplitude for all the cases, except for 1x. However, the presence of the multiplicative load leads to clear variations, as indicated in Table 3. Therefore, Figs. 8(a) and 8(c) suggest that the multiplicative load arising from the combination of bow and misalignment primarily affects harmonics other than 1x. Figure 8(e) demonstrates the full-spectrum plot when only bow and unbalance are present, excluding misalignment. It shows that only 1x occurs, attributed to bow and unbalance, and emphasizing the role of misalignment in the harmonic generation. From Table. 3, following conclusion can be made; a. All the harmonics for case (I) i.e., when the magnitude of bow is zero, are appears to be equal when the system is subjected to all three faults and when the multiplicative load is not present. b. 1x harmonic for each case is appeared to almost equal for the particular case, and can be concluded that multiplicative load does not affect the 1x harmonic when the magnitude of bow is changes. c. The impact of multiplicative load can be clearly observed from the harmonics other than 1x, from Table 3, it is observed that vibration amplitude at harmonics 2x, 3x, 5x, –1x, –3x, –5x are appears to be higher as compared with the condition when multiplicative is not considered. Thus, it can be said the multiplicative load generally affect the harmonics other than 1x. These findings shed light on the interplay between harmonics, the existence of a multiplicative load, system faults, and the impact of bow and misalignment on amplitudes of vibration within the system.

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Fig. 8. Complex responses in full-spectrum for rotor 2 at rotor speed of 32 Hz (a–b) Magnitude and phase of translational displacement with misalignment, residual bow, and unbalance (c–d) Magnitude and phase of translational displacement without multiplicative load (e–f) Magnitude and phase of translational displacement without misalignment

Study of Multiplicative Load on the Misaligned Rotor

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Table. 3. Harmonic amplitude obtained from numerical simulation for different scenario involving different cases Harmonics

Case-I With bow & mis

0×

1×

2×

3×

4×

5×

-1×

-2×

-3×

-4×

-5×

1.0214e-05

Case-II 9.9236e-05

Case-III

Case-IV

Case-V

1.0048e-05

1.0172e-05

9.7991e-05

Without multi

1.0214e-05

1.0203e-05

1.0207e-05

1.0212e-05

1.0198e-05

Without mis

1.0709e-07

1.0770e-07

1.0731e-07

1.0711e-07

1.0831e-07

With bow & mis

2.0731e-05

9.3501e-05

6.1295e-05

3.0098e-05

1.2594e-05

Without multi

2.0731e-05

9.3249e-05

6.1165e-05

3.0078e-05

1.2556e-05

Without mis

2.0549e-05

9.1659e-05

5.9748e-05

2.9261e-05

1.2389e-05

With bow & mis

2.8498e-06

3.2566e-06

3.0822e-06

2.9079e-06

3.4390e-06

Without multi

2.8490e-06

2.8445e-06

2.8468e-06

2.8490e-06

2.8423e-06

Without mis

0

0

0

0

0

With bow & mis

2.3448e-06

2.5578e-06

2.4665e-06

2.3752e-06

2.6491e-06

Without multi

2.3448e-06

2.3426e-06

2.3436e-06

2.3445e-06

2.3427e-06

Without mis

0

0

0

0

0

With bow & mis

5.7726e-07

1.0500e-07

3.074e-07

5.0979e-07

9.7398e-08

Without multi

5.7726e-07

5.7973e-07

5.7867e-07

5.7761e-07

5.8079e-07

Without mis

0

0

0

0

0

With bow & mis

3.3836e-07

4.3355e-07

3.9275e-07

3.5196e-07

4.7345e-07

Without multi

3.3836e-07

3.3686e-07

3.3750e-07

3.3814e-07

3.3622e-07

Without mis

0

0

0

0

0

With bow & mis

1.0590e-06

1.1589e-06

1.1161e-06

1.0732e-06

1.2017e-06

Without multi

1.0590e-06

1.0618e-06

1.0606e-06

1.0594e-06

1.0630e-06

Without mis

0

0

0

0

0

With bow & mis

1.0324e-07

2.0380e-08

5.5863e-08

9.1394e-08

1.5394e-08

Without multi

1.0324e-07

1.0516e-07

1.0433e-07

1.0351e-07

1.0599e-07

Without mis

0

0

0

0

0

With bow & mis

6.1329e-07

7.8707e-07

7.1259e-07

6.3811e-07

8.6155e-07

Without multi

6.l329e-07

6.1166e-07

6.1236e-07

6.1305e-07

6.1096e-07

Without mis

0

0

0

0

0 7.8737e-07

With bow & mis

7.8688e-07

7.8722e-07

7.8708e-07

7.8693e-07

Without multi

7.8688e-07

7.8718e-07

7.8705e-07

7.8693e-07

7.8730e-07

Without mis

0

0

0

0

0

With bow & mis

1.4515e-07

2.0879e-07

1.8152e-07

1.5424e-07

2.3607e-07

Without multi

1.4515e-07

1.4435e-07

1.4469e-07

1.4504e-07

1.4515e-07

Without mis

0

0

0

0

0

4 Conclusion This paper presents a comprehensive analysis of a rotor train system considering multiple faults, including misalignment, bow, and unbalance. The study formulates governing equations incorporating the effect of multiplicative fault and uses LaGrange’s equation

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to analyze system dynamics. By varying the magnitude of bow relative to unbalance, the study explores the influence of bow on misalignment and its impact on system behavior. Different cases are examined, including the presence and absence of faults. The analysis highlights the effect of the multiplicative load on harmonics other than the fundamental frequency (1x). The results demonstrate that the presence of the multiplicative load, resulting from the combination of misalignment and bow, leads to higher vibration amplitudes at harmonics such as 2x, 3x, 5x, −1x, −3x, and −5x. The analysis provides detailed insights into the dynamics of the rotor train system and the combined effects of these faults, enhancing our understanding of the system’s dynamic response in realworld conditions. The adoption of AMB enables a substantial reduction in vibration at different harmonics, corresponds to multiple faults. Consequently, the implementation of AMB effectively regulates undesirable vibrations, leading to smooth and efficient system operation. Future research can be expand upon the current work by analyzing the dynamics of the rotor system under the influence of a multiplicative load, considering changes in the location of bow in relation to the location of unbalance. Additionally, there is potential for identifying various fault parameters of the rotor system including, residual bow, unbalance, damping, coupling additive stiffness (CAS), multiplicative force, and the AMB coefficients such as displacement and current stiffness.

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12. Song, G.F., Yang, Z.J., Ji, C., Wang, F.P.: Theoretical-experimental study on a rotor with a residual shaft bow. Mech. Mach. Theory 63, 50–58 (2013) 13. Khaire, M.P.: Experimental study to identify the vibration signature of bent shaft. Int. J. Eng. Res. Technol. 3, 214–216 (2014) 14. Sanches, F.D., Pederiva, R.: Theoretical and experimental identification of the simultaneous occurrence of unbalance and shaft bow in a Laval rotor. Mech. Mach. Theory 101, 209–221 (2016) 15. Sanches, F.D., Pederiva, R.: Simultaneous identification of unbalance and shaft bow in a two-disk rotor based on correlation analysis and the SEREP model order reduction method. J. Sound Vib. 433, 230–247 (2018) 16. Tuckmantel, F.W.S., Castro, H.F., Cavalca, K.L.: Investigation on vibration response for misaligned rotor- bearing-flexible disc coupling system - theory and experiment. J. Vibr. Acoust. Trans ASME 142, 1–13 (2020) 17. Gouws, R.: A review on active magnetic bearing system limitations, risks of failure and control technologies. Int. J. Eng. Technol. 7, 6615–6620 (2018) 18. Kumar, P., Tiwari, R.: Finite element modelling, analysis and identification using novel trial misalignment approach in an unbalanced and misaligned flexible rotor system levitated by active magnetic bearings. Mech. Syst. Signal Process. 152, 107454 (2021) 19. Rezazadeh, N., De Luca, A., Lamanna, G., Caputo, F.: Diagnosing and balancing approaches of bowed rotating systems: a review. Appl. Sci. 12(18), 9157 (2022) 20. Sarmah, N., Tiwari, R.: Analysis and identification of the additive and multiplicative fault parameters in a cracked-bowed-unbalanced rotor system integrated with an auxiliary active magnetic bearing. Mech. Mach. Theory 146, 103744 (2020) 21. Srinivas, R.S., Tiwari, R.: Interaction between unbalance and misalignment responses in flexibly coupled rotor systems integrated with AMB. In: Gas Turbine India Conference, vol. 58516 (2017) 22. Siva Srinivas, R., Tiwari, R., Kannababu, Ch.: Identification of coupling parameters in flexibly coupled jeffcott rotor systems with angular misalignment and integrated through active magnetic bearing. In: Cavalca, K.L., Weber, H.I. (eds.) IFToMM 2018. MMS, vol. 62, pp. 221–235. Springer, Cham (2019). https://doi.org/10.1007/978-3-319-99270-9_16 23. DeSmidt, H.A., Wang, K.W., Smith, E.C.: Multiharmonic adaptive vibration control of misaligned driveline via active magnetic bearings. J. Dyn. Syst. Measur. Control Trans. ASME 130, 0410061–04100613 (2008) 24. Bouaziz, S., Belhadj Messaoud, N., Mataar, M., Fakhfakh, T., Haddar, M.: A theoretical model for analyzing the dynamic behavior of a misaligned rotor with active magnetic bearings. Mechatronics 21, 899–907 (2011) 25. Kumar, P., Tiwari, R.: A review multiplicative faults and model - based condition monitoring strategies for fault diagnosis in rotary machines. J. Braz. Soc. Mech. Sci. Eng. (2023) 26. Gautam, A.K., Tiwari, R.; Dynamic response characteristics of multiplicative faults in a misaligned-bowed rotor-train system integrated with active magnetic bearings. Int. J. Dyn. Control (2023) 27. Gere, G.: Mechanics of material. Cengage learning (2012)

Intelligent Fault Classification of a Misaligned Geared-Rotor Machine Equipped with Active Magnetic Bearings Pantha Pradip Das(B) , Rajiv Tiwari, and Dhruba Jyoti Bordoloi Indian Institute of Technology Guwahati, Assam, India {p.pantha,rtiwari,djb}@iitg.ac.in

Abstract. Gear can be considered as one of the most vital components in any rotating machine. Like any mechanical system, geared-rotor systems can experience various types of faults or failures. One of the most commonly experience faults is shaft misalignment, which can cause due to improper installation or wear. In recent years, machine learning and deep learning techniques have sparked great interest in accurately diagnosing geared-rotor faults. Therefore, in this paper a fault detection intelligent method is implemented to predict the type of fault class provided with a labeled set of input vibration and current data. A multi-class classification artificial neural network (ANN) model is developed with statistical features extracted from time-domain vibration. The vibration dataset is built by conducting an experiment on a geared-rotor test rig where angular misalignment is deliberately introduced with four fault conditions: no misalignment and three severity levels of misalignment. Transverse vibration data are recorded with proximity probes. The set-up is also equipped with two active magnetic bearings (AMBs) mounted on the input and output rotors, which are used for vibration suppression. The control current signal running through the AMB coils and time-domain vibration signal at three different speeds are used for misalignment diagnosis. Features like mean, root mean square and entropy have been found to perform the best. The optimum tuning of the hyper parameters of the ANN model is done to achieve about 98.33% prediction accuracy. Keywords: Fault Diagnosis · Geared-Rotor System · Misalignment · Active Magnetic Bearings · Artificial Neural Network (ANN)

1 Introduction Gears are widely used in various applications like automobiles, wind turbines, power industries, etc. A rotor system with gearbox is susceptible to transverse vibration vastly caused by faults in the machine. One serious fault is shaft misalignment which is responsible for generating additional dynamic forces on the gear teeth and thereby leading to excessive vibration during gear engagement [1]. Therefore, intelligent condition monitoring of such geared-rotor systems where artificial intelligent (AI) techniques are © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 482–494, 2024. https://doi.org/10.1007/978-3-031-40455-9_38

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used to accurately and automatically predict the machine health condition [2]. Traditional machine faults identification includes advanced signal processing techniques on the recorded data which requires specialized engineers. However, around the past two decades, accurate diagnosis of machine faults using machine learning and deep learning techniques, have generated utmost curiosity [3]. Samanta [4] implemented an ANN and a support vector machine (SVM) based classifiers to predict two classes of a faulty gearbox. The author selected the features based on genetic algorithm (GA) from vibration signals, and studied the efficiency of the proposed model at different loads and sampling rates. Both the algorithms were compared and with use of GA, the testing was highly accurate. Zang et. al. [5] also used GA for feature selection on an ANN classifier from 9 fault conditions of the gear test setup. The fault classes comprised combination of various faults like structural faults like unbalance, looseness, and misalignment, and gear failure like broken tooth, tooth crack, etc. Classification accuracy using GA combined with back propagation (BP) was found to be improved when compared with BP alone. Bansal et.al. [6] proposed a novel interpolation/extrapolation method for SVM classifiers, where in the absence of input data for training at all operating speeds, the required training data is estimated at speeds near the speed of the test data. A healthy gear and three faulty gear conditions were used as four classes and the time domain and frequency domain signals were used for prediction. The experiment was performed on a bevel gear rotor system and signals were recorded using tri-axial accelerometers at various speeds. Bordoloi et. al. [7] implemented SVM techniques with algorithms like grid-search, GA and artificial bee colony (ABC) to estimate the optimum parameters. The test rig was same as mentioned in [6]. They also used interpolation and extrapolation for performing training and testing at speeds other than measured ones. Other than ANN based and SVM based algorithms, different classifiers like Knearest neighbors (KNN), convolution neural network (CNN), etc. are also used for gear fault problems. Wang [8] examined three experiments on gear setup to establish that the developed methodology delivers superior fault detection accuracies compared to KNN based approaches in identifying various levels of gear crack under varying motor operating frequencies and loads. Principal component analysis (PCA) was employed to decrease the dimension of the insignificant statistical characteristics and produce the valuable ones. Khan et. al. [9] performed axial, radial and angular misalignment fault experiments and predicted the types and severity of the fault using regression models with proportional change in the RMS condition indicator. The dataset was constructed by using vibration signals and sound signals measured using accelerometer and microphone respectively. Li et. al. [10] implemented one dimensional CNN using vibration signal and combined with a gated recurrent unit (GRU) which was trained by acoustic emission signals. Seven gear pitting fault conditions were used to test the proposed model. The performance of single CNN and GRU model was also compared. Zuber et. al. [11] used gear fault dataset including faults like tooth crack, wear, missing, and chipped tooth at various frequencies and loads to test the performance of ANN classifier. They also showed how dimension reduction by using PCA with eigenvalues, there was a large increase in prediction accuracy. Habbouche et. al. [13] made a relative analysis of two machine learning (ML) models with signal processing techniques. The one proposed

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model used long short-term memory (LSTM) for classification, and other used CNN and LSTM for feature extraction and classification respectively. The dataset was constructed by using multiple sensors on a test rig with four heath conditions. Le et. al. [12] used shaft misalignment in a rotating system to capture vibration time-domain data. The time-domain data was converted to frequency domain for feature extraction using PCA method for reducing the dimension. For defect classification SVM was used. Lee et. al. [14] used novelty class detection method to predicts better with lesser frequency of fault data which is generally seen in any rotating machine industry. For feature selection GA and for extraction PCA was implemented. The dataset was recorded from a gearbox system with various faults like misalignment, backlash, tooth breakage, and SVM was used for fault classification. It is seen from the literature that shaft misalignment fault on gear rotor system is rarely used to build a dataset for intelligent fault prediction. Therefore, in this paper vibration data are recorded along with control current signal from AMB coils which is proposed to be used as an integral component of the geared-rotor test rig, to benefit vibration suppression and precise prediction of various alignment conditions. Artificial neural network (ANN) was used for learning from the selected features and for classification. Angular misalignment is introduced in the setup and measurements were taken at three different speed which are below the fundamental natural frequency of the machine. The raw time-domain vibration and current signals are used to extract features from and feed them to the different classifiers for predicting the fault classes.

2 Experimental Test Rig The experimental geared-rotor AMB system (Fig. 2) was developed in the advanced dynamics and vibration lab, IIT Guwahati. The schematic of the physical system is depicted in Fig. 1, which exhibits the spur gear pair with the pinion on the input shaft

Fig. 1. A schematic diagram of the experimental test rig with the spur gear pair, four conventional rolling element bearings, two AMBs (each mounted on the input and output shaft), electric motor and load.

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and gear wheel mounted on the output shaft. Conventional rolling element bearings are used to support both the shafts.

Fig. 2. (a) Spur geared-rotor experimental test rig (b) Zoomed view of the gear pair with shim plates installed under conventional bearing 2 for inducing the angular misalignment fault in the geared-rotor experimental test rig.

Two magnetic bearings are positioned in between the conventional bearings which actively suppress the transverse vibration of the shafts. Active magnetic bearings (AMBs) are electromechanical devices that use magnetic fields to diminish the transverse disturbances of the core attached to a rotating shaft, allowing it to spin without any contact with the stator core [15]. Figure 2(a) demonstrates the physical test rig developed to study the dynamic behaviour of a spur gear pair rotor machine equipped with active magnetic bearings. The centre distance between the input and the output shaft is 56.25 mm. A pinion of 25 teeth is mounted on the input shaft and the gear wheel with 50 teeth is mounted on the output shaft. As shown in the figure (Fig. 2), the physical deviation of the shaft centreline with respect to the bearing centreline is captured by the proximity probes in terms of voltage. These Bently Nevada manufactured probes are non-contact sensors that are sensible to any change in the gap between the sensor head and the conductive

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surface of the shaft. In order to obtain a linearized output, this gap should be maintained at −5.75 ± 0.5VDC and the probe has a sensitivity of 1.27 V/mm. A total of four eddy current probes are used, two for each shaft that measures horizontal and vertical transverse displacement response. The recorded signal run through the sensor box as it amplifies and then it gets fed to the dSpace MicroLab Box data acquisition system. The designed PID (proportional, integral and derivative) controller is executed in real-time in the DSpace-SIMULINK interface and the response is displayed in the DSpace software (DS2102). 2.1 Principal of Operation of Active Magnetic Bearings The operational concept of an active magnetic bearing depends on the closed-loop arrangement of the AMB components. In Fig. 3(a) the schematic design of the closedloop system is represented where a slight shift of the rotor core is recorded by the sensor and sent to the controller which then generate the control current. The required control current is amplified by the power amplifiers and then it passes through the AMB actuator coils to produce the magnetic flux for minimizing the vibration. This control current is measured by current probes as shown in Fig. 2(a). The implemented control law PID, is designed and implemented through MATLAB Simulink. The parameters of the PID (proportional gain, derivative gain and integral gain) are adjusted with trial and error method in order to obtain a stable operation of rotor. The tuning of the PID parameters are done until a desired vibration suppression is achieved. Depending on the control parameters, the required coil current is produced in accordance to which magnetic force is generated in the actuator. Since, control current time domain data is also used for training the model, therefore the model performance will in turn depend on the control law. The probe used in our research is KEYSIGHT TECHNOLOGIES (1146B) made, that measures both AC and DC signals (100 mA to 100 A RMS) using hall-effect sensing

Fig. 3. (a) A schematic of the working principal of an 8-pole active magnetic bearing depicting the stator and the rotor core, and the direction of the flow of control current (b) The actual AMB showing the windings of the 8-pole actuator, and the rotating core.

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technology. The probes are attached with the wire through it that is directly connected to the poles of the AMB actuators. As such, the control current flowing through the electromagnet windings to minimize the vibration gets captured by the current probe. The probes directly connect to DSpace controller input channel through insulated BNC and the current signal is displaced in the monitor in real time. The system parameters of the experimental test rig and the PID parameters are mentioned in Table 1. 2.2 Introduction of Angular Misalignment Fault in the Test Rig In this paper, we have developed a machine learning model to classify the four levels of angular misalignment defects purposefully induced in the geared-rotor test rig. To mimic angular misalignment in a practical scenario, we have use precision shim plates of three thickness value; 0.1 mm, 0.2 mm, and 0.4 mm under the conventional bearing 2 (refer Fig. 1 & 2(b)). This will create four defect conditions; no misalignment, 0.1 mm misalignment, 0.2 mm misalignment, and 0.4 mm misalignment in the input shaft.

Fig. 4. Transverse displacement and control current signals for no misalignment (M0) fault condition (left column) and for 0.4 mm misalignment (M3) fault condition (right column), where (a, b) depicts the input shaft horizontal displacement (c, d) input shaft vertical displacement (e, f) output shaft horizontal displacement (g, h) output shaft vertical displacement (i, j) control current through the coil of poles in vertical direction.

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Moreover, the data is recorded at three different speed values; 15 Hz, 23 Hz, and 30 Hz. Figure 4 shows the raw vibration and control current signals obtained from the experiment. From these raw signals, statistical features were extracted to be fed to the classification algorithm. The fault conditions and operating parameters are mentioned in Table 2. Table 1. Parameters of the Experimental Test Rig Parameters

Values

Number of teeth (Pinion & Gear)

25 & 50

Module (mm)

1.5

Tooth width (mm)

15

Shaft diameter (mm)

12

Shaft Length (mm)

600

Gear Pair Material & Type

Stainless Steel & KHK

KP (Proportional Gain for Pinion & Gear)

2750

KD (Derivative Gain for Pinion & Gear)

1.5

KI (Integral Gain for Pinion & Gear)

500

Table 2. Experimental parameters and fault conditions. Parameters

Values

Fault Classes (Misalignment Severities)

M0 (No Misalignment), M1 (0.1 mm Misalignment), M2 (0.2 mm Misalignment), M3 (0.4 mm Misalignment)

Operating Frequencies

15 Hz, 23 Hz, 30 Hz

Total quantity of defect conditions

4 × 3 = 12

Sampling rate

10,000 samples/sec

3 Intelligent Fault Classification Framework The general structure of machine learning fault class prediction of a given problem comprises of: dataset building, pre-processing of the data for transforming the data into a suitable format for analysis, feature engineering, dataset splitting, classification model selection and training, model testing [16]. For building the dataset, the test rig is run for 50 s with a sampling rate of 10,000 samples/sec for each fault condition.

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3.1 Construction of Suitable Features Feature extraction strategies aid in lowering the dimensionality of the data through the selection of a more condensed set of valuable features and thereby eliminating any redundancy in the dataset. These are health indicators which are used as input attributes to boost the learning algorithms’ efficacy and efficiency [17]. Features are domain specific so in this paper features that are suitable for time-domain signals are utilised. Widely used such features [3, 18] are: mean, standard deviation, root mean square which are dimensional ones and dimensionless features that can be used are skewness, kurtosis, entropy. These six features are used in our research. Table 3 gives a brief explanation of all the selected features, where Xi denotes individual data points (ith number data point), N denotes the total number of data points in the dataset. X X’ denotes mean of all the data points. Table 3. Feature definition and formula Features

Definition

Formula

Mean

In statistics, mean represent the central average of samples of a dataset [6]

X = N1

Standard Deviation Standard deviation is a statistical measure of value variance. It measures the deviation from the mean [6] Root mean square

Kurtosis

Skewness

RMS values can describe signal amplitude and energy. It is not sensitive to the sign of the samples [9] Kurtosis measures distribution shape statistically. It compares a distribution’s peak prominence or flatness to a normal distribution [9]

N 

Xi

i=1

    N 1   2 XSD = N (Xi − X ) i=1

    N 1  2  XRMS = N Xi i=1

N  N (Xi −X )4 i=1 Xk =  2 N  2 (Xi −X ) i=1

N  3  Skewness is an indicator of statistics that Xi −X N characterises the lack of symmetry of a i=1 Xskw = ⎡   ⎤3   probability distribution [9]  N

⎢ ⎢ ⎢ ⎣

Shannon Entropy

i=1

(Xi −X )2 N −1

N The Shannon entropy, which is also  H = − Xi2 log(Xi2 ) referred to as simple entropy, is a i=1 statistical metric that quantifies the level of uncertainty or randomness present in a given dataset [19]

⎥ ⎥ ⎥ ⎦

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3.2 Performance of Various Features Among various supervised learning techniques for fault classification of gear systems, SVM, decision tree, random forests are widely popular [3]. In this paper, after extracting suitable attributes from the raw data, the above three and Naïve Bayes (NB) classification was implemented to find the best performing feature. NB classifier is used since it is quite simple to implement and performs well in large datasets [21].

Fig. 5. Bar chart depicting prediction accuracy score of the four different classifiers fed by the extracted attribute at three different operating frequencies with all the fault classes combine using 80:20 training testing ratio.

As seen in the Fig. 5, mean, RMS, and entropy proved to perform excellent (all close to 100%) for all the classification algorithms. The bar chart in the figure shows the performance of these three condition indicators to be the best in all the speeds using all the sensor data (vibration and current data). However, these algorithms have some demerits which motivated us to implement deep learning algorithm like ANN for the best performance in various given situations. For instance, when we have large datasets, it can pose challenges while fitting for Support Vector Machines (SVMs), which can demand on higher computation time and power [3]. 3.3 Implementing ANN Based Multilayer Perceptron ANN allows us to implement a multilayer perceptron [3] with input layer, one or more hidden layers and output layer which are fully connected, i.e. every neuron in one layer

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is linked to every neuron in the layer next to it [20]. This gives high flexibility and also allows handling large datasets. The number of neurons in the output layer depend on the fault class used in the study, in our research it is four. In this study, an ANN model was developed using Keras library in Tensorflow Python.

Fig. 6. Bar diagram depicting classification accuracy of the developed artificial neural network model, using four different activation functions, namely linear, tanh, ReLU and exponential linear unit (ELU) with 7 hidden layers and 80:20 training testing dataset, showing maximum prediction accuracy of 98.33%.

The features that were found to perform the best in the Sect. 3.2 were used for training and testing the ANN model. Since our prediction involved 4 fault classes with misalignment defect, so the dataset comprised of four misalignment classes each with a single feature, hence 4 files for a single feature. Since the output column has 4 defect conditions as such one hot encoder is used as a data pre-processing technique to create 4 binary columns, so that each row with the four columns represents a fault category. Now, the training and testing data was split as 80:20. For improving the convergence of the algorithm, feature scaling is done by performing standardization. The transformed training and testing data is now fed to the ANN model input layer. The model has 7 hidden layers with 65 perceptions in each hidden layer. For weight initialization kernel-initializer is used and 4 activation functions for each hidden layer ere used and with 200 mini batch size and 100 number of epoch, the overall prediction accuracy was found to be 98.33% (comparison shown in Fig. 6). In the output layer, there are 4 neurons representing 4 defect classes and the activation function used was softmax. Softmax activation function was used since it gives probabilistic output, which is common in multi-class classification problem. The optimizer variable has been configured to utilise the Adam optimisation algorithm, with a specified learning rate of 0.001. As mentioned in the introduction section, many researchers have used ANN or combined ANN with genetic algorithm or principal component analysis (PCA) for gearbox

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fault classifications. In [4], the prediction accuracy with ANN was found to be in the range 48.61% to 100% and ANN with genetic algorithm (GA), in range 97.22–100%. Similarly, in [5] overall accuracy with ANN with the application of GA was found to be 95%. And in [11] an overall accuracy of about 98.67% was found by using PCA dimensional reduction with ANN. Whereas in our study, an overall prediction accuracy of 98.33% was achieved by using ANN and three statistical features (mean, entropy and RMS) as the input (Fig. 6).

Fig. 7. Confusion matrix to check the performance of the classification algorithm for batch size of 200 and epoch 100, with activation function exponential linear unit (ELU).

4 Conclusion In this paper, we have highlighted the potential faults that can occur in a gear rotor machine, with a specific focus on shaft misalignment. The paper emphasizes the use of machine learning and deep learning techniques for accurately diagnosing faults classes. The data was recorded from an experimental test rig with AMB as an integral component. An ANN model was developed that takes input vibration and current data and predicts the type of fault class. The raw time-domain vibration data and AMB control current data was pre-processed for significant feature extraction, where 6 statistical condition indicators were selected and their performance were tested by popular machine learning based classifiers like SVM, Decision Tree, Random Forest, and Naïve Bayes. Among the six features, mean, RMS, and Shannon entropy performed the best with accuracy very close to 100%. These three features were then used for classification in the ANN model. The hyper-parameters namely activation function, mini batch size, epoch, etc. of the ANN model were optimized to achieve a high accuracy rate of approximately 98.33%.

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In future, a different deep learning model like 1-dimensional convolution neural network can be incorporated, and a dataset with larger number of faults and operating frequencies may be used.

References 1. Saxena, A., Parey, A., Chouksey, M.: Effect of shaft misalignment and friction force on time varying mesh stiffness of spur gear pair. Eng. Fail. Anal. 49, 79–91 (2015) 2. Stetco, A., et al.: Machine learning methods for wind turbine condition monitoring: a review. Renew. Energy 133, 620–635 (2019) 3. Lei, Y., Yang, B., Jiang, X., Jia, F., Li, N., Nandi, A.K.: Applications of machine learning to machine fault diagnosis: a review and roadmap. Mech. Syst. Signal Process. 138, 106587 (2020) 4. Samanta, B.: Gear fault detection using artificial neural networks and support vector machines with genetic algorithms. Mech. Syst. Signal Process. 18(3), 625–644 (2004) 5. Yang, Z., Hoi, W.I., Zhong, J.: Gearbox fault diagnosis based on artificial neural network and genetic algorithms. In: Proceedings 2011 International Conference on System Science and Engineering, pp. 37–42. IEEE (2011) 6. Bansal, S., Sahoo, S., Tiwari, R., Bordoloi, D.J.: Multiclass fault diagnosis in gears using support vector machine algorithms based on frequency domain data. Measurement 46(9), 3469–3481 (2013) 7. Bordoloi, D.J., Tiwari, R.: Optimization of support vector machine based multi-fault classification with evolutionary algorithms from time domain vibration data of gears. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 227(11), 2428–2439 (2013) 8. Wang, D.: K-nearest neighbours based methods for identification of different gear crack levels under different motor speeds and loads: revisited. Mech. Syst. Signal Process. 70, 201–208 (2016) 9. Khan, M.A., et al.: Gear misalignment diagnosis using statistical features of vibration and airborne sound spectrums. Measurement 145, 419–435 (2019) 10. Li, X., Li, J., Qu, Y., He, D.: Gear pitting fault diagnosis using integrated CNN and GRU network with both vibration and acoustic emission signals. Appl. Sci. 9(4), 768 (2019) 11. Zuber, N., Bajri´c, R.: Gearbox faults feature selection and severity classification using machine learning. Eksploatacja i Niezawodno´sc´ 22(4), 748–756 (2020) 12. Lee, Y.E., Kim, B.K., Bae, J.H., Kim, K.C.: Misalignment detection of a rotating machine shaft using a support vector machine learning algorithm. Int. J. Precis. Eng. Manuf. 22, 409–416 (2021) 13. Habbouche, H., Benkedjouh, T., Amirat, Y., Benbouzid, M.: Gearbox failure diagnosis using a multisensor data-fusion machine-learning-based approach. Entropy 23(6), 697 (2021) 14. Yu, H.T., Park, D.H., Lee, J.J., Kim, H.S., Choi, B.K.: Novelty class detection in machine learning-based condition diagnosis. J. Mech. Sci. Technol. 37(3), 1145–1154 (2023) 15. Tiwari, R.: Rotor Systems: Analysis and Identification. CRC Press, Boca Raton (2017) 16. Koukoura, S., Carroll, J., McDonald, A.: A diagnostic framework for wind turbine gearboxes using machine learning. In: Annual Conference of the PHM Society, vol. 11, no. 1 (2019) 17. Guyon, I., Elisseeff, A.: An introduction to feature extraction feature extraction: foundations and applications, 1–25 (2006) 18. Sánchez, R.V., Lucero, P., Vásquez, R.E., Cerrada, M., Cabrera, D.: A comparative feature analysis for gear pitting level classification by using acoustic emission, vibration and current signals. IFAC-PapersOnLine 51(24), 346–352 (2018)

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19. Karaca, Y., Baleanu, D., Zhang, Y. D., Gervasi, O., Moonis, M. (Eds.).: Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems. Academic Press (2022) 20. Liu, R., Yang, B., Zio, E., Chen, X.: Artificial intelligence for fault diagnosis of rotating machinery: a review. Mech. Syst. Signal Process. 108, 33–47 (2018) 21. Yuan, Y., Li, S., Zhang, X., Sun, J.: A comparative analysis of svm, naive bayes and gbdt for data faults detection in wsns. In: 2018 IEEE International Conference on Software Quality, Reliability and Security Companion (QRS-C), pp. 394–399. IEEE (2018)

Eigenvalues of the Free Rotation Mode of the Multi-bladed Rotor Chao Peng(B)

and Alessandro Tasora

Department of Engineering and Architecture, University of Parma, 43124 Parma, Italy {chao.peng,alessandro.tasora}@unipr.it

Abstract. The eigenvalues of the free rotation mode of multi-bladed rotor systems are of crucial importance when designing controllers. An unstable free rotation mode of the rotor is unfavorable. In this paper, the Differential Algebraic Equations (DAEs) of the multi-flexible-body system are linearized after having introduced a corotational formulation with respect to the floating frame of reference of the rotating center. The Multi-Blade Coordinate (MBC) transformation is performed to obtain the time-invariant system. The proposed formulations are verified through the numerical experiments carried out on the rotors of increasing complexity: a single rigid body, a rigid rotor, and a flexible rotor. Results reveal that the geometric stiffness matrix and the tangent stiffness matrix of constraints exhibit a centrifugal stiffening effect, moving the eigenvalues toward pure imaginary numbers; in contrast, the inertial stiffness matrix introduces a centrifugal softening effect, pushing the eigenvalues toward real numbers. Generally one has to discard the geometric stiffness matrix, the inertial stiffness matrix, and the tangent stiffness matrix of constraints in the linearized DAEs to obtain the zero eigenvalues for the free rotation mode, whereas the inertial damping matrix can be involved. Keywords: free rotation mode effect · softening effect

1

· multibody linearization · stiffening

Introduction

A multi-bladed rotor system always involves a free degree of freedom (DOF) corresponding to the rotation around its axis. For instance, wind turbines and helicopter rotors can rotate freely to exchange kinetic energy between the rotor structure and the surrounding airflow. This rotational DOF introduces the rigid body mode, also called free-free mode, which is commonly considered to have zero eigenvalues [4]. However, the eigenvalues are unnecessary to be zero; hence we prefer to call it free rotation mode. El-Absy [3] investigated the stability of the rigid body mode of a rotating rotor. Using time-marching simulation, the stability of simple rotors built through three different models with or without the consideration of the effect of the longitudinal displacement due to bending in the inertia and elastic forces c The Author(s), under exclusive license to Springer Nature Switzerland AG 2024  F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 495–514, 2024. https://doi.org/10.1007/978-3-031-40455-9_39

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are studied. The numerical results demonstrated that the geometric centrifugal stiffening term in the inertia forces is unnecessary to be included to obtain a stable solution. However, only the time series of the beam displacements were plotted to indicate the stability, the eigenvalues were not discussed. Many researches investigated the stability of multi-bladed rotor systems, however most of them [2,4,5,7,17] focused on the higher order modes, typically the bending modes of the rotor, which are relevant for the design of rotor systems to avoid potential resonances and instabilities. The centrifugal stiffening effect in the rotating beams was realized and its influence on the modal dynamics of rotor systems was well inspected [8,9]. The centrifugal softening effect was also explored [5,20,21]. In the controller design phase, the linearized equations of motion are required to represent the linear controlled systems. The accuracy of the linearization of the original highly nonlinear dynamics system could affect the controller’s performance and robustness. Taking the example of a wind turbine, the near-zero frequency bandwidth, which corresponds to the overall rotational characteristics of the rotor system, is the most important range for the controller design to stabilize the rotor speed and maximize the power capture [6]. The transfer functions within the near-zero frequency bandwidth exhibit a direct correlation with the eigenvalues of the free rotation mode in the rotor dynamics system. In this paper, the analytical linearization of the three-bladed rotor system is derived. The time-invariant system is obtained through Multi-Blade Coordinate (MBC) transformation. The eigenvalues of the free rotation mode of rigid and flexible three-bladed rotors are investigated to verify the proposed linearized model. The influences of the geometric stiffness matrix, the inertial stiffness matrix, and the tangent stiffness matrix of constraints on the eigenvalues of the free rotation mode are examined.

2

Methodology

We express the equations for the motion of rigid bodies and finite elements using the Newton-Euler formulation [15]. In this context, we adopt a mixed basis approach for the states of rigid bodies and beam nodes: translational DOFs are expressed in the inertial frame whereas, in contrast, rotational DOFs are expressed in the local frame. The coordinates of beam nodes in the mixed basis are denoted as T  (1) δq m = δr Ta , θ δT l where r a is the position vector in the inertial frame, θ δl is the virtual rotation vector in the local frame. 2.1

Corotational Formulation

To simplify the problem, a three-dimensional Euler– Bernoulli beam element is utilized in this study. The material stiffness matrix K m and the geometric stiffness matrix K g expressed in the local reference frame of the beam element can

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be found in a finite element textbook, such as in [14]. The structural damping matrix generally stabilizes rotor dynamic systems. However, this paper focuses on the linearization and the influences of different terms in the linearized models, so the structural damping matrix is omitted to prevent its impact on the eigenvalues of the free rotation mode of the rotor systems. A simplified corotational formulation is used to deal with the large deflection of beam elements. A rotation transformation matrix R♦ ∈ R12×12 is introduced [19]:   T T (2) RF , RF , RB RF R♦ = diag RF , RA where RA , RB ∈ R3×3 are the rotation tensors of two nodes A, B of the beam element, RF ∈ R3×3 is the rotation tensor of the corotated frame F located in the middle of the centerline. The tangent stiffness matrix of the Euler– Bernoulli beam element K 0t ∈ 12×12 is obtained via R K 0t = R♦ (Km + K g ) RT♦

(3)

Transformation to the Floating Frame of Rotation Center. The configuration of the rotating rotor is periodically dependent on time, which can be dealt with by the Lyapunov-Floquet (L-F) transformation. If the rotor is isotropic, MBC transformation is a special case of the L-F transformation [16], can be employed to obtain a time-invariant system. Thanks to its simplicity and computational efficiency, MBC transformation is preferred where possible. Since the mixed basis δq m is chosen, the position vectors of beam nodes at the same spanwise of the blades are not equal; thus, the current formulation does not satisfy the prerequisite of MBC transformation. To address this limitation, the coordinates of blade nodes should be transformed with respect to the rotating center. The scheme of a three-bladed rotor is shown in Fig. 1. Three auxiliary reference frames H1 , H2 , H3 are introduced for three blades, respectively, which are rotated by the corresponding azimuth angles ψi , i = 1, 2, 3 about the rotation axis X from the stationary hub center frame H. The transformation relations are RHi = RH Rψi , i = 1, 2, 3, where RH , RHi are the rotation tensors of the reference frames H, Hi , and Rψi is the transformation matrix of the rotation about X axis by the azimuth angle ψi . The coordinate of blade nodes q m can be transformed to the floating reference frame Hi as the variation relation [13] δq m = B h δq h + B b δq b

(4)

where q h represents the coordinate of the floating reference frame Hi , q b is the relative coordinate of blade nodes with respect to its corresponding floating reference frame Hi . Moreover, B h , B b ∈ R6×6 are two auxiliary transformation matrices     I −RHi r RHi 0 N , Bb = (5) Bh = 0 0 0 I

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B1

ψ1

N1

rN

z

RH1 y H1

x z H

H2

B2

B3

N3

y

RH

x H3

ωx

N2

z y O

x

Fig. 1. The scheme of a three-bladed rotor system. The three auxiliary reference frames H1 , H2 , H3 are located at the same position as H, but shown separately to avoid mess. The three blades are rotated by the corresponding azimuth angles ψ1 , ψ2 , ψ3 . The angles ψ2 , ψ3 are not shown.

where r N is the position vector of the blade node N with respect to its corresponding floating reference frame Hi . Extending the coordinate transformation relation Eq. (4) to two nodes A, B of the beam element, a larger transformation matrix Bfc ∈ R12×18 is introduced   Bh A Bb A 0 Bfc = (6) Bh B 0 Bb B Using the principle of virtual work, the tangent stiffness matrix K 0t in the original mixed basis δq m can be transformed into the auxiliary mixed basis  T T δq h , δq Tb as T K 0t Bfc K 1t = Bfc

(7)

where K 1t ∈ R18×18 , in which the top-left block is the term projected onto the hub center, the bottom-right block is the term of the blade beam element, the off-diagonal blocks are the coupling terms between blade and hub.

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The hub block in K 1t is respect to the floating reference frame Hi which is different for three blades. One needs to rotate K 1t back to the consistent reference frame H, as T K 2t = RXi K 1t RX i

(8)

where RXi = diag [I 3×3 , Rψi , I 12×12 ] ∈ R18×18 is a block diagonal matrix. The tangent stiffness matrix K 2t is finally assembled in the system stiffness matrix according to its corresponding coordinate index. 2.2

Constraints

The method of Lagrange multipliers is used to formulate the equations of motion for dynamics systems involving multiple constraints, which represent the joint connections among rigid bodies and beam nodes. The change of the orientation of the reaction forces and torques in joints would generate the geometric ∂C T

stiffness term K c = ∂qq γ [1], where C q is the Jacobian matrix of the constraint, γ is the Lagrange multipliers which are associated with the reaction forces and torques. The Jacobian matrix C q and the tangent stiffness matrix of holonomic constraints K c can be evaluated in closed-form analytical expressions in the original mixed basis δq m , which have been implemented in the open-source multibody library chrono [18]. Similar transformations in Eqs. (7) and (8) are performed for C q and K c to achieve a consistent basis. These two matrices are used in the DAE of the rotor system. 2.3

Inertial Forces and Torques

In this work, the lumped mass model is assumed for the beam elements. The inertial forces F Ia and the inertial torques M Il , when expressed in the original mixed basis δq m , are: l c r a + νR˜ cT ω˙ l + νR ωl ω F Ia = ν¨ l J ω l cRT r¨ a + J ω˙ l + ω M Il = ν˜

(9)

where ν is the lumped mass at the node, c is the vector of the mass center offset in the local frame of the node, J is the tensor of the moments of inertia in the local frame of the node, r¨ a is the translational acceleration in the inertial frame, ω l is the angular velocity in the local frame, R is the rotation tensor of the local frame of the node. Using the coordinate transformation in Eq. (4) and its derivatives, after long algebraic manipulations, the inertial forces F Ia and the inertial torques M Il T  can be transformed to the auxiliary mixed basis δq Th , δq Tb , then linearized through variation techniques, leading to the inertial mass matrix M i , the inertial damping matrix Ri , and the inertial stiffness matrix K i [13]. These three inertial matrices are rotated back to the consistent reference frame H with a similar transformation as in Eq. (8), then scattered into the system matrices.

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2.4

MBC Transformation

After transforming all the mass, damping, and stiffness matrices to the consistent T  auxiliary mixed basis δq Th , δq Tb , the MBC transformation is performed on both the structural generalized coordinates δq and the Lagrange multipliers δγ, leading to the linearized time-invariant DAE [13]: M z δ¨ z + Rz δ z˙ + K z δz + Cq Tz δγ z = 0 Cq T δz = 0

(10) (11)

where Rz consists of only the inertial damping matrix Ri , which includes the gyroscopic damping term; K z is composed of the structural material stiffness matrix of beam elements Km , the geometric stiffness matrix of beam elements K g , the inertial stiffness matrix K i , and the tangent stiffness matrix of constraints K c . 2.5

Eigenvalue Analysis

The generalized descriptor form E y δ y˙ = Ay δy of the linearized DAE Eqs. (10) T  and (11) is built by introducing the state vector δy = δz T δ z˙ T δγ Tz and two system matrices ⎤ ⎡ ⎡ ⎤ 0 I 0 I 0 0 Ay = ⎣ −K z −Rz −Cq Tz ⎦ Ey = ⎣ 0 M z 0 ⎦ , (12) 0 0 0 −Cq T 0 0 The corresponding eigenvalues can be evaluated using different numerical methods, among these we endorse a recent embodiment of the Krylov-Schur iteration [11] because of its robustness when dealing with clustered and nearzero eigenvalues. Using this generalized descriptor, one obtains twice the number of eigenvalues as the number of degrees of freedom, plus spurious eigenvalues corresponding to the rows of the constraint jacobian C q T , if any. For instance, a rigid body freely rotating in 3D space will generate six eigenvalues relative to the three rotations. The eigenvalues of the free rotation mode always have the smallest magnitude, therefore they can be extracted easily.

3

Numerical Experiments

The proposed approach is verified through a series of numerical experiments on rotors with different complexities. In all the numerical experiments discussed in these pages, gravity is not considered.

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3.1

501

Single Rigid Body

The first model we investigated is a single rotating rigid body. The intermediate axis theorem, also called as tennis racket theorem, deals with a freely rotating rigid body with three distinct principal moments of inertia: the rotation of the rigid body around its minor and major principal axes is stable, while the rotation around its intermediate principal axis is unstable [10]. The intermediate axis theorem provides a reference result for the eigenvalue analysis of a rotor. For a single rotating rigid body, the mass matrix is M = diag [m, m, m, Jxx , Jyy , Jzz ]

(13)

where m is the mass, Jxx , Jyy , Jzz are the moments of inertia about its three principal axes, respectively. Supposing the rotating center is located at the mass center, the cross term Jyz = 0. The damping matrix is   0 0 R= (14) l J − J 0ω ωl where J = diag [Jxx , Jyy , Jzz ] is the tensor of the moments of inertia, ω l = T [ωx , ωy , ωz ] is the angular velocity vector expressed in the local frame of the rigid body. The stiffness matrix is K = 0. The eigenvalues of two cases are investigated: free rotation about three axes, and constrained body with free rotation about the X axis. Free Rotation About Three Axes. The rigid body rotates about three axes freely. The eigenvalues are listed in Table 1. In case the three moments of inertia are distinct as Jxx > Jyy > Jzz , the eigenvalues for rotation around X (which is the minor principal axis) and Z (which is the major principal axis) are a pair of pure imaginary numbers, implying a constant-amplitude oscillation, whereas the eigenvalues for rotation around Y (which is the intermediate principal axis) are a pair of real numbers, indicating an unstable movement. This result is identical to the intermediate axis theorem. In case the three moments of inertia hold a relation Jxx = 2Jyy = 2Jzz which mimics an isotropic three-bladed rigid rotor, the eigenvalues for rotation around X (which is the major principal axis, also is the rotation axis of the rigid rotor) are a pair of pure imaginary numbers, and the natural frequency is equal to the angular velocity π. The eigenvalues for rotation around Y and Z axes are zero. There is no unstable mode. Constrained Body with Free Rotation About the X Axis. The rigid body is allowed to rotate freely about the X axis only. The rotational DOFs about Y and Z axes are constrained by a revolute joint. The eigenvalues are zero, as listed in Table 2.

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Table 1. Eigenvalues of a single rigid body in free rotation about three axes. Only two eigenvalues are listed since the other four eigenvalues are zero. Subcase

m Jxx Jyy [kg] [kg m2 ]

Jzz

ωx ωy ωz λ [rad s−1 ]

Jxx > Jyy > Jzz

1.3

1.9

π 0 0

0 π 0

0 0 π

±3.79i ±1.75 ±1.67i

5.7 2.85 2.85 π 0 0

0 π 0

0 0 π

±πi 0, 0 0, 0

5.7 3.3

Jxx = 2Jyy = 2Jzz 1.3

Table 2. Eigenvalues of a single rigid body in free rotation about X axis. Subcase

m Jxx Jyy [kg] [kg m2 ]

Jzz

ωx ωy ωz λ [rad s−1 ]

Jxx > Jyy > Jzz

1.3

1.9

π

0

0

0, 0

5.7 2.85 2.85 π

0

0

0, 0

5.7 3.3

Jxx = 2Jyy = 2Jzz 1.3

3.2

Rigid Rotor

An ideal isotropic rigid rotor is investigated as the second model. As shown in Fig. 2, the rigid rotor consists of four rigid bodies: the hub H located at the rotating center, and three bearings P1 , P2 , P3 located at the same radius but equally distributed azimuth positions. The three bearings are linked to the hub by three fixed joints C1 , C2 , C3 , respectively. The hub is linked to the ground by the joint CR , of which the rotational DOFs can be adjusted to free the rotation about three axes or only the X axis. P1

z

C1 y

z C2

x

y

x H

C3

P2

P3 CR

Fig. 2. The scheme of the three-bladed rigid rotor.

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503

The moments of inertia of the hub and bearings are set as zero to facilitate the comparison with the single rigid body model. The masses of the hub and the bearing are 629 kg and 2222 kg, respectively. The radii between the hub and bearings are 0.2 m. Since the rigid rotor has no flexible finite element, the material stiffness matrix Km and the geometric stiffness matrix K g are absent. The influence of the inertial stiffness matrix K i , the inertial damping matrix Ri , and the tangent stiffness matrix of constraints K c on the eigenvalues are investigated through including or excluding them in Eq. (10). Similarly, two cases with different rotational constraints are investigated. Table 3. Eigenvalues of a three-bladed rigid rotor. The ”×” symbol means the corresponding matrix is included. For the case of free rotation about three axes, the other four eigenvalues are zero, thus not listed. Subcase

K i R i K c ωx λ ×

Free rotation about three axes × Free rotation about X axis

×

×

× × ×

×

×

×

×

×

0 π 0 π

0, 0 ±πi 0, 0 ±πi

0 π 0 π 0 π 0 π

0, 0 0, 0 0, 0 ±πi 0, 0 ±π 0, 0 0, 0

Since the ideal rigid rotor can be lumped to a single rigid body with the relation Jxx = 2Jyy = 2Jzz , its eigenvalues should be identical with the results in Tables 1 and 2. As shown in Table 3, if the rotational velocity around the rotation axis X is zero, the centrifugal forces and gyroscopic torques are zero, thus the matrices K i , Ri , K c = 0, leading to zero eigenvalues for the two different constraint conditions. When the rotational velocity is π, in case of free rotation about three axes, if the inertial damping matrix Ri is included, the eigenvalues are ±πi, thus the natural frequency is equal to the rotational velocity, which is identical with the result in Table 1. In this case, the matrices K i , K c must be absent or present at the same time, otherwise the eigenvalues deviate from ±πi, leading to erroneous results which are not listed in Table 3 for the sake of conciseness.

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When the rotational velocity is π, in case of free rotation about X axis, if Ri is included and if K i , K c are absent or present at the same time, the eigenvalues are zero, which is identical with the result in Table 2. If only K c is involved on the basis of Ri , the eigenvalues change to ±πi; if only K i is involved on the basis of Ri , the eigenvalues change to ±π. The tangent stiffness matrix of constraints K c moves the eigenvalues away from zero along the imaginary axis, implying a stiffening effect; in contrast, the inertial stiffness matrix K i moves the eigenvalues away from zero along the real axis, implying a softening effect. For the rigid rotor, the stiffening effect of K c and the softening effect of K i can counteract completely. The identical results between Table 3 and Table 1, 2 indicate that the linearized DAEs established using the proposed corotational formulation with respect to the floating frame of reference of the rotating center and the MBC transformation are applicable on the eigenvalue analysis of three-bladed rotors. 3.3

Flexible Rotor

Three flexible blades are linked to the three bearings of the rigid rotor model via fixed joints to make the system closer to the wind turbine rotor. The blades are straight with constant rectangle cross section in 0.6 m×0.2 m. The blade length is 6.0 m. The blade material is assumed as steel with elastic modulus E = 2.1 × 1011 Pa, Poisson’s ratio μ = 0.3, and density ρ = 7800 kg m−3 . The scheme of the flexible rotor is shown in Fig. 3.

B1

z x

P1 P2 B2

z x

y

H

CR

y

P3 B3

Fig. 3. The scheme of the three-bladed flexible rotor.

Every blade is discretized to 10 Euler– Bernoulli beam elements. The eigenvalues of the flexible rotor are solved using the proposed method. Only the eigenvalues of the free rotation mode of the rotor are listed in this study.

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No External Forces. When no external forces act on the blades, the eigenvalues of the two different constraint conditions are reported in Table 4. Table 4. Eigenvalues of a three-bladed flexible rotor without external forces. The ”×” symbol means the corresponding matrix is included. The material stiffness matrix Km is included in all cases. Subcase

K g K i R i K c ωx λ ×

Free rotation about three axes ×

×

×

×

×

Free rotation about X axis ×

× ×

×

×

× ×

×

×

×

0

0, 0

0, 0

0, 0

π 0 π

±πi ±0.0063 ±0.0139i 0, 0 0, 0 0, 0 ±πi ±0.0108 ±0.0141i

0

0, 0

π 0 π 0 π 0 π 0 π

±0.0081 0, 0 ±3.0598i 0, 0 ±3.1380 0, 0 ±0.6964i 0, 0 ±0.0072i

When the rotational velocity is π, in case of free rotation about three axes, the eigenvalues ±πi appear as expected, but the other four eigenvalues deviate to two pairs of real numbers and pure imaginary numbers with small magnitudes, which tend to be due to numerical errors. When the rotational velocity is π, in case of free rotation about X axis, if only Ri is included, the eigenvalues change to a pair of small real numbers ±0.0081; in contrast, if the four matrices K g , K i , Ri , K c are all included, the eigenvalues change to a pair of small pure imaginary numbers ±0.0072i. This result is considered to be due to numerical errors. If K g or K c are involved based on Ri individually, the eigenvalues change to a pair of imaginary numbers, which implies that K g and K c contribute a stiffening effect; in contrast, if K i is involved based on Ri , the eigenvalues become to a pair of real numbers, which implies that K i contributes a softening effect. For the flexible rotor without external forces, the stiffening effect of K g , K c and the softening effect of K i can counteract completely. With External Forces. The out-of-plane forces F x = 725630.0 N and the inplane forces F y = 439010.0 N are equally distributed at blade nodes to mimic

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the aerodynamic loadings on the rotor. The forces follow the orientation of blade nodes when the rotor deflects, which are called follower forces. The hub is locked firstly in the quasi-static equilibrium analysis, then the rotational DOF about X axis is freed to perform the eigenvalue analysis. Since the wind turbine rotor has only one rigid motion DOF: the rotation about X axis, the case of free rotation about three axes is not discussed here. The out-of-plane and inplane deflections of the blade tip in case of rotational velocity 0 rad s−1 are 0.24 m and 0.016 m, respectively. The ratio of deflections to the blade length is less than 5 %, thus it is still in the range of small deflections. The eigenvalues of the free rotation mode of the flexible rotor with external follower forces are listed in Table 5. Table 5. Eigenvalues of a three-bladed flexible rotor with external follower forces. The ”×” symbol means the corresponding matrix is included. The material stiffness matrix Km is included in all cases. Subcase

K g K i R i K c ωx λ

Free rotation about X axis

× ×

× ×

×

×

× ×

×

×

×

0 π 0 π 0 π 0 π 0 π

0, −0.0002 ±0.0002 ±0.7034 ±2.9768i ±0.0001 ±3.1379 ±0.2800 ±0.6388i ±0.7570 ±0.7598

If only Ri is included, the eigenvalues are near zero. The deviation away from zero is due to numerical errors. If K g is involved based on Ri , in case of ωx = 0, the eigenvalues change from near zero to a pair of real numbers ±0.7034, which implies that the geometric stiffness term due to transverse external follower forces F x , F y has a softening effect; in case of ωx = π, the eigenvalues change to a pair of pure imaginary numbers ±2.9768i. The increase of the rotational velocity moves the eigenvalues from the real axis to the imaginary axis, which implies that the geometric stiffness term due to the centrifugal forces has a stiffening effect. If K i is involved based on Ri , in case of ωx = 0, the eigenvalues are still near zero, that is because the quadratic velocity terms of the inertial forces are zero, leading to K i = 0; in case of ωx = π, the eigenvalues change to a pair of real numbers ±3.1379, which confirms the softening effect of K i . If K c is involved based on Ri , in case of ωx = 0, the eigenvalues change from near zero to a pair of real numbers ±0.2800, which implies that the tangent

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stiffness term of constraints due to transverse external follower forces F x , F y has a softening effect; in case of ωx = π, the eigenvalues change to a pair of pure imaginary numbers ±0.6388i. The increase of the rotational velocity moves the eigenvalues from the real axis to the imaginary axis, which confirms that the tangent stiffness term of constraints due to the centrifugal forces has a stiffening effect. If the four matrices K g , K i , Ri , K c are all included, the eigenvalues change from near zero to a pair of real numbers, which means the rotor cannot rotate stably. The eigenvalues of the free rotation mode of the rotor, which are the poles of the system matrix in the state-space representation of the linearized dynamics system, should be zero or at least near zero, otherwise it is impractical to design the controller. In view of a proper controller design, one has to discard the geometric stiffness matrix K g , the inertial stiffness matrix K i , and the tangent stiffness matrix of constraints K c , whereas the inertial damping matrix Ri can be included. Interpretation of the Stiffening and Softening Effects. The stiffening and softening effects of the geometric stiffness matrix K g , the tangent stiffness matrix of constraints K c , and the inertial stiffness matrix K i are interpreted, respectively. Geometric Stiffness Matrix. The geometric stiffness matrix K g derived based on the linear assumption of the finite beam element [14], is proportional to the internal axial forces of the beam elements. If the internal axial forces tend to stretch the beams, like the guitar string, K g exhibits the stiffening effect to increase the natural frequencies. In contrast, if the internal axial forces tend to compress the beams, like the cabled tower of a wind turbine, K g contributes the softening effect to decrease the natural frequencies, leading to the buckling instability in extreme cases. If the transverse external forces follow the orientation of the nodes when the beam is deflected, which are follower forces, as shown in Fig. 4a, the force F j acting on the node Nj results in an axial compression force F p = F j sin (θj − θi ) within the inner node Ni . In this case, the geometric stiffness matrix K g contributes the softening effect. If the transverse external forces maintain a consistent direction relative to the inertial frame, irrespective of the beam deflection, which are called constantdirectional forces in this paper, as shown in Fig. 4b, the force F j acting on the node Nj results in an axial stretching force F p = F j sin (θj ). In this case, the geometric stiffness matrix K g exhibits the stiffening effect. Another numerical experiment is conducted for the flexible rotor with external constantdirectional forces, in which only the out-of-plane forces F x = 725630.0 N are applied and the inplane forces F y = 0 to reserve the isotropy of the rotor. As listed in Table 6, the eigenvalues move from zero to pure imaginary numbers after involving K g , thus the stiffening effect is confirmed.

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Since the centrifugal forces always tend to stretch the blades, the geometric stiffness matrix K g from the centrifugal forces generates the stiffening effect for the rotating blades. Tangent Stiffness Matrix of Constraints. The tangent stiffness matrix of constraints K c arises from the change of the orientation of the reaction forces and torques in the joints. Minaker [12] derived the closed-form expressions for several joints, and revealed its impact on the eigenvalues of the vehicle suspension systems. K c is proportional to the reaction forces and torques. A simple example to demonstrate the effects of K c is the pendulum. If K c is neglected, the eigenvalues of the pendulum under gravity are two zeros which are incorrect.

Fp

θj Nj

θj Fj

Fp θi

Ni z

Fj

Nj

Ni z

y x (a) Follower forces.

ωx

y x

ωx

(b) Constant-directional forces.

Fig. 4. Two distinct external forces exert different effects on the deflected beam.

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Table 6. Eigenvalues of a three-bladed flexible rotor with external constant-directional forces. The ”×” symbol means the corresponding matrix is included. The material stiffness matrix Km is included in all cases. K g K i R i K c ωx λ

Subcase Free rotation about X axis

× ×

×

0 π 0 π

0, 0.0001 ±0.0001 ±0.9740i ±3.2068i

For a normal pendulum shown in Fig. 5a, gravity generates a pulling force F c at the root joint, which is similar to the axial stretching force in the beam. In this case, the tangent stiffness matrix of constraints  K c provides the stiffening effect, moving the eigenvalues from zero to ± g/Li. For an inverted pendulum shown in Fig. 5b, gravity generates a pushing force F c at the root joint, which is similar to the axial compression force in the beam. In this case, the tangent stiffness matrix of constraints  K c provides the softening effect, changing the eigenvalues from zero to ± g/L which implies an unstable motion. When the transverse external follower forces are applied, similar to the axial compression force F p shown in Fig. 4a, a pushing force F c will be generated at the root joint. Analogy to the inverted pendulum, the tangent stiffness matrix of constraints K c provides the softening effect.

y O

x

m

Fc L

L

y g m

g

x

(a) Normal pendulum.

Fc O (b) Inverted pendulum.

Fig. 5. Two pendulum models. g is the gravity acceleration, L is the pendulum length.

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The centrifugal forces always generate a pulling force F c at the root joint. Analogy to the normal pendulum, the tangent stiffness matrix of constraints K c provides the stiffening effect. Inertial Stiffness Matrix. The inertial stiffness matrix K i involves an important  term k22 = −r N f c in the hub block of the detailed expression [13], where   f c = νω lH ω lH r N is the centrifugal forces of the nodes of the rotating beams. ν is the lumped mass at the node, and ω lH is the angular velocity of the rotating center H in the local frame. Since the centrifugal forces consistently point towards the tip of the rotating beams, and because of the presence of the minus sign in k22 , negative stiffness values accumulate at the hub block of the system stiffness matrix. Ultimately, the inertial stiffness matrix K i consistently contributes a softening effect. The stiffening and softening effects of K g , K c , K i are summarized in Table 7. Table 7. The stiffening and softening effects of the geometric stiffness matrix K g , the tangent stiffness matrix of constraints K c , and the inertial stiffness matrix K i from several different forces. Matrix

Force

Effect

Geometric stiffness matrix Kg

Transverse external follower forces Transverse external constant-directional forces Centrifugal forces

Softening

Tangent stiffness matrix of constraints K c

Transverse external follower forces Centrifugal forces

Softening

Inertial stiffness matrix K i

Centrifugal forces

Softening

Stiffening Stiffening

Stiffening

Parameter Sweeping Analysis. The parameter sweeping analysis is performed to investigate the relationship of the eigenvalues of the free rotation mode with respect to the rotor rotational speed and external forces. Only the case of free rotation about X axis is studied. The four matrices K g , K i , Ri , K c are all included in the parameter sweeping analysis. Rotational Speed. The constant external follower forces F x = 725 630.0 N, F y = 439 010.0 N are applied on blade nodes. The rotor rotational speed is swept from 0 to 600 r min−1 . As shown in Fig. 6, when the rotor speed increases, the eigenvalues move from a pair of real numbers toward zero and reach zero at approx. 290 r min−1 , then deviate away from zero to a pair of pure imaginary numbers. The stiffening effect of K g , K c due to the centrifugal forces and the softening effect of K i compete with each other. In the low rotational speed

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511

range, the softening effect of K i dominates, leading to real eigenvalues. In the high rotational speed range, the stiffening effect of K g , K c dominates, leading to imaginary eigenvalues. At a certain rotational speed, zero eigenvalues are obtained when they are canceled out.

Fig. 6. Eigenvalues of the free rotation mode of the rotor with respect to the rotor rotational speed.

External Forces. The rotor rotational speed is set as πrad s−1 . The external follower forces distributed on the blade nodes are swept from 0 to F x = 725 630.0 N, F y = 439 010.0 N. As depicted in Fig. 7, when the external follower forces increase, the eigenvalues disperse away from zero along the real axis proportionally to the force amplitude, which is because the softening effect of K g induced by transverse external follower forces grows correspondingly. The eigenvalues at zero forces are ±0.0072i, which is due to numerical errors as explained in Table 4.

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Fig. 7. Eigenvalues of the free rotation mode of the rotor with respect to the external follower forces applied on the blade nodes.

4

Conclusions

Eigenvalues of the free rotation mode of three-bladed rotor systems are investigated. A corotational formulation with respect to the floating frame of reference of the rotating center is applied to the rotor dynamics system. An analytical linearization of the constrained multi-flexible-body DAE is performed, including the geometrical stiffness matrix due to internal forces K g , the inertial stiffness matrix K i , the inertial damping matrix Ri and the tangent stiffness matrix of constraints K c . The MBC transformation is implemented to obtain the time-invariant system, followed by the eigenvalue analysis. Different numerical experiments are carried out on rotors of increasing complexity: a single rigid body, a rigid rotor, and a flexible rotor. It is demonstrated that the linearized DAE formulated using the proposed approach is applicable to the eigenvalue analysis of three-bladed rotors. We discussed the influence of the four matrices K g , K i , Ri , K c on the eigenvalues of the free rotation mode of the rotor. The terms in the geometrical stiffness matrix K g due to transverse external follower forces, the tangent stiffness matrix of constraints K c due to transverse external follower forces and the inertial stiffness matrix K i due to centrifugal forces exhibit a softening effect, moving the eigenvalues toward real numbers. The terms of the geometrical stiffness matrix K g due to centrifugal forces, the tangent stiffness matrix of constraints K c due to centrifugal forces, introduce

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a stiffening effect that pushes the eigenvalues toward pure imaginary numbers. Aiming at a proper controller design, in order to obtain the zero eigenvalues of the free rotation mode, generally one has to discard the geometric stiffness matrix K g , the inertial stiffness matrix K i , and the tangent stiffness matrix of constraints K c in the linearized DAE, whereas, the inertial damping matrix Ri can be used anyway. The parameter sweeping analysis demonstrates that the zero eigenvalues could be probably reached at a certain rotational speed even if the four matrices K g , K i , Ri , K c are all considered. The blade external follower forces tend to move the eigenvalues toward real numbers, and thus are unfavorable for the stability of the free rotation mode of the rotor. Further research should consider a rigorous corotational formulation for the beam finite elements, as well as a new finite element based on the Geometrically Exact Beam (GEB) theory. It is interesting to find the conditions of zero eigenvalues when all the geometric nonlinear terms K g , K i , K c are considered.

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15. Shabana, A.A.: Dynamics of Multibody Systems. Cambridge University Press, Cambridge, England, fourth edition (2013) 16. Skjoldan, P., Hansen, M.: On the similarity of the Coleman and Lyapunov-Floquet transformation for modal analysis of blade rotor structures. J. Sound Vibr. 327, 424–439 (2009) 17. Skjoldan, P.,Hansen, M.: Implicit Floquet analysis of wind turbines using tangent matrices of a nonlinear aeroelastic code. Wind Energy, 15,275–287 (2012) 18. Tasora, A., et al.: Chrono: an open source multi-physics dynamics engine. In: ˇıstek, J., Rozloˇzn´ık, M., Cerm´ ˇ Kozubek, T., Blaheta, R., S´ ak, M. (eds.) HPCSE 2015. LNCS, vol. 9611, pp. 19–49. Springer, Cham (2016). https://doi.org/10.1007/ 978-3-319-40361-8 2 19. Tasora, A., Masarati, P.: Analysis of rotating systems using general-purpose multibody dynamics. In: Pennacchi, P. (ed.) Proceedings of the 9th IFToMM International Conference on Rotor Dynamics. MMS, vol. 21, pp. 1689–1701. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-06590-8 139 20. Thomas, O., S´en´echal, A., De¨ u, J.-F.: Hardening/softening behavior and reduced order modeling of nonlinear vibrations of rotating cantilever beams. Nonlinear Dyn. 86, 1293–1318 (2016) 21. Zhao, G., Du, J., Wu, Z.: A geometric softening phenomenon of a rotating cantilever beam. Arch. Appl. Mech. 87, 1049–1059 (2017)

The Passive Vibration Control in Bridge Configured Winding Cage Rotor Induction Motor: An Experimental Analysis Rakesh Deore(B) , Bipul Brahma, Shahrukh, and Karuna Kalita Indian Institute of Technology, Guwahati 781039, Assam, India {rdeore,bipul.brahma,shahrukh1995,karuna.kalita}@iitg.ac.in Abstract. In electrical systems, very small eccentricity creates an asymmetric magnetic flux distribution in the air gap. Due to the eccentricity, these additional magnetic fields exert an extra pulling force called Unbalance Magnetic Pull (UMP), which severely causes unwanted vibrations in the system. However, it is essential to mitigate the effect of the UMP in the system. This paper demonstrates the effect of the new winding-based configuration called the Bridge Configured Winding (BCW) on the squirrel cage induction motor with the help of experimental analysis. Hence, the modified system’s parameters (rotor) have been scrutinized experimentally using the impulse test and the half-power bandwidth method. The BCW is an effective winding scheme that acts as a damper winding in the squirrel cage induction motor. The experimental analysis found that the BCW induction can serve as the built-in force actuator and significantly reduces the effect of additional magnetic fields due to the P ± 1 pole pair by flowing equalizing current through the bridge winding. The modified analytical rotor dynamic model for the coupled electromechanical forces due to static eccentricity has been proposed for the BCW cage induction motor. The FFT of the experimentally measured rotor response of the modified BCW induction motor has been compared for bridge ON and bridge OFF conditions. The study shows that BCW can effectively minimize the oscillating component of the UMP due to static eccentricity. Keywords: Vibration Control · Unbalance Magnetic Pull (UMP) · System Identification · Sub-harmonics · Electromechanics · Rotordynamics · Electrical Machines

1 Introduction Rotating electric machines have many applications, especially induction machines, which are extensively used in industries at heavily loaded conditions and operated at various speeds. Infelicitously, 66% of the faults are due to mechanical-related faults like mass unbalance, manufacturing defects, assembly tolerances, damaged bearings, bending of the shafts, etc. However, these faults result in rotor eccentricity, which creates an uneven air gap between the stator and rotor, resulting in nonuniform flux, which produces an extra P ± 1 pole pair (P is the fundamental pole pair). This extra pole pairs with the fundamental one can create the net lateral force toward the minimum air gap, called Unbalance Magnetic Pull (UMP) [1]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 515–527, 2024. https://doi.org/10.1007/978-3-031-40455-9_40

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UMP’s effects rapidly increase if the motor is operated from the no-load condition to the loaded condition [2]. This UMP in the system has many undesirable effects like a rubbing of stator and rotor, wear and tear of the bearings, bending in flexible shafts, an increase in nasty vibrations of the system, and mainly again, an increase of the eccentricity in the system. So, minimizing the effect of the UMP is necessary for the flawless operation of the machines. UMP in the system cannot vanish entirely but can be diminished. This can be minimized by mechanical balancing of the rotor or introducing electrical damping in the system. Burakov and Arkkio investigated the effects of series winding and parallel winding in the stator and rotor, where it was found that two Parallel stator winding connections in an induction motor can most effectively mitigate UMP than rotor cage winding and series stator winding [3]. A.Laiho and A. Sinervo tried to eliminate the UMP in two-pole squirrel cage induction motor, by introducing an additional four-pole stator winding [4, 5]. A. Lahio used this four-pole winding as a built-in force actuator for rotor position control or self-bearing application [4]. Khoo and Kalita developed a single set of winding arrangement based on the Wheatstone bridge configuration, which can control the phenomenon of lateral force production in the electric machines and practically implemented BCW as a stator winding in a 4-pole permanent magnet machine [6, 7]. In this paper, the effect of the BCW scheme on the rotor vibrations has been experimentally analyzed for the squirrel cage induction motor. The rotor response is analyzed for a wide range of speeds and compared for the Bridge ON and Bridge OFF conditions. The rotor parameters like stiffness and damping factor have been estimated using the impact method and can be used in the rotordynamic analysis of the system. The modified analytical rortordynamic model is also proposed for the Bridge Configured Winding cage induction motor based on the existing rotordynamic model. D. Guo et al. formulated the analytical method to estimate the UMP for any pole three-phase generator’s axially constant static and dynamic eccentricity based on the airgap permeance method. This method is used to calculate the air gap flux density (product of the airgap permeance and fundamental MMF) at no load state, which can be expressed in terms of the Maxwell stress tensor, where the components of the UMP are obtained by integrating the Maxwell stress tensor over a rotor surface. This analysis can be extended for the four-pole squirrel cage induction motor. Thus, the x and y components of the UMP are given by Eq. (1), [8]. The component of UMP contains the terms one is independent of time having maximum magnitude f1 and other terms depend on time having the oscillating frequency twice the supply frequency (ωs ) and maximum magnitude f2 , f3 . The terms f1 , f2 , f3 are nonlinear functions of relative eccentricity. And θ represents the angle of the relative eccentricity.  ump    f1 cos θ + f2 cos(2ωt − 3θ ) + f3 cos(2ωt − 5θ ) fx for P = 2 (1) ump = f1 sin θ + f2 sin(2ωt − 3θ ) + f3 sin(2ωt − 5θ ) fy This transverse electromagnetic force can be coupled with the mechanical unbalance; the system’s rotor dynamic equation due to coupled electromechanical effect can be modeled as Eq. (2), ump

2 cos(ω t) + f m¨x + cx x˙ + kx x = meωm m x 2 sin(ω t) + f ump m¨y + cy y˙ + ky y = meωm m y

(2)

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where m is the rotor mass, where kx , & ky are the stiffness coefficients, cx , & cy is the damping coefficients of the system in the x, and y direction, respectively. 1.1 Bridge Configured Winding (BCW) BCW is a single-set parallel winding configuration based on the Wheatstone Bridge Configuration (WBC), used as the built-in actuator to generate controllable torque and transverse force without affecting the fundamental magnetic field. The equalizing currents are supplied between the mid of parallel branches called the bridge supply current. The schematic representation of the BCW connection is shown in Fig. 1. The main features of BCW are compact area occupancy and less power loss compared to double-set winding configuration for the same torque generation, and it is an economical solution for induction motors. a1

P aa1

a2

aa2

a3

aa3

a4

aa4

a5

aa5 A2

A1 aa6

a6

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a7 aa8

a8 aa9

a9 aa10

B1 bb6 bb7 b5

bb8 bb9 bb10 w8

b4 b3

w5

w7

b2 w6

b1

Q

bb1 bb2

bb3 bb4 bb5

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B2

C1

c9

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c7

c6

C2 cc5

w11

cc4 w12

w10

cc3 cc2

w9

cc1

R c5

c4

c3

c2

c1

Fig. 1. Schematic representation of the BCW connection

In Fig. 1, each phase has been connected into two parallel paths. For phase A, the main supply current flows through ten coils connected in two parallel branches (PA1 O & PA2 O). Where branches PA1 & A2 O are connected in opposite directions with a span of 1800 but have the same magnitude and direction of currents. Branch A1 O and PA2 have reverse polarity w.r.t branch PA1 & A2 O. Thus, the branches PA1 & A2 O form a group as they have similar features; hence PA1 & A2 O form another group. They are placed in the stator slots diametrically opposite to each other. The branch A1 A2 connected in mid of branches PA1 O & PA2 O is called the bridge supply branch used to produce levitative forces; hence, the current flowing through branch A1A2 is called the bridge current or levitative current. The bridge currents are small compared to the current flowing across A (main supply current). Based on the same principle of production of the UMP, BCW works. A two-pole additional magnetic field purposefully embedded with the four-pole fundamental magnetic field by supplying equalizing currents through bridge branches or by connecting the bridge points, which leads to superimposing both fundamental as well

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as additional magnetic fields, and levitation force (transverse force) can be generated to counteract the UMP.

2 Experimental Test Setup A 50 HP three-phase, four poles, star-connected squirrel cage induction motor is used to demonstrate the experiments. To investigate the effect of the BCW winding, the original winding of the motor has been modified to the three-phase, double-layered, four-pole, distributed BCW configuration. The BCW winding arrangements are placed in the stator slots according to the connections discussed in Sect. 1.1. The other system parameters and winding specifications are detailed in Table 1.

Fig. 2. The dimension of the Rotor.

Table 1. Designed specifications of machines and winding Sr. No Parameters

Values

1

Number of poles

4

2

Number of stator slots

60

3

Number of rotor bars

48

4

Supply frequency (Hz)

50

5

No of turns

11

6

No. of strands

3

7

Coil diameter (m)

1.22 × 10–3

8

Stator inner diameter (m)

0.223

9

Rotor core outer diameter (m) 0.221

10

Length of the rotor core (m)

0.214

11

Rotor weight with shaft (kg)

96.56

12

The total length of rotor (m)

1.8

The length of the cage rotor core mounted on shaft is 214 mm, and the total span of the rotor is 1.622 m. The detailed dimensions of the rotor are shown in Fig. 2, where a rotor core is asymmetrically placed over the span of the shaft. A perforated disc is installed on the rotor to create a mechanical unbalance by attaching the known mass to

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the disk’s holes. The rotor is supported by the bearings at both ends, enclosed in the bearing housings, and ascended on a rigid support. An auxiliary bearing with a radial clearance of 0.5 mm is placed in the original stator housing of the induction motor to provide extra support to the greater span of the rotor shaft in chaotic operations. This clearance is also helpful while demonstrating active rotor vibrations due to the effect of equalizing currents in bridge supply conditions. The side view of the assembled experimental test rig for the cage induction motor is shown in Fig. 3,

Fig. 3. A detailed side view of the rotor-stator assembly (Rear Side)

This complete rotor-stator assembly is installed on the large rigid cast iron bed without any external dampening effects. The very rigid bearings support is placed at both rotor- ends. The rotor response in two orthogonal directions has been measured with the eddy current-based proximity displacement sensors (Bently Nevada proximity probe), having a 7.88 V/mm standard scaling factor. The physical mounting of displacement sensors at different shaft locations is shown in Fig. 4.

Fig. 4. Placement of the displacement sensors at two locations (Front Side).

NI PXI e 1050 data acquisition system stores and processes data with LabView 14 environment. The sensors are connected to the data acquisition system through SBN

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68-A Channel. To acquire the data, 10000 samples are recorded with a sampling rate of 1000 Hz. The FFT and orbit plot data are also recorded through DAQ assistance (FPGA block) in the LabView 14 software. 2.1 Rotor Modal Parameter Identification The rotor model parameter identification is essential for the mechanical analysis of the system. The modal analysis of the rotor bearing system is done with a non-random excitation method called impulse excitation with a mechanical pinpoint hammer. This is a quick method and requires less equipment. The higher frequencies quickly get excited for a shorter pulse because of the directional striking force without other transverse forces. The displacement of the rotor center in horizontal and vertical directions due to impact excitation w.r.t time is shown in Fig. 5-a). The combined orbit plot of the rotor response due to impulse excitation is shown in Fig. 5-b). The rotor frequency is estimated using the peak-picking method in the frequency domain, which investigates the frequency locally but is applied only if the frequency peaks are separable. The halfpower bandwidth method is used to estimate the stiffness and damping of the system at a particular frequency. A sample frequency bandwidth plot for the horizontal response of the rotor due to impulse excitations is explained in Fig. 6.

Fig. 5. a) Response of the rotor due to impact w.r.t time in the X and Y directions, b) Rotor response in X-Y plane.

The maximum amplitude of horizontal response of the system due to impact force is excited at 76.66 Hz (481.67 rad/sec), so this is nothing but the natural frequency of the system. So, the first critical bending frequency of the rotor is nearer to 76 Hz. Thus, the static stiffness of the rotor with bearing on its ends is 22.402 MN/m, and the damping coefficient of the system is given as 533.901 N.sec/m (damping ratio 0.574%). However, 481.67 rad/sec is the resonant mechanical spinning frequency of the rotor. The 4-pole cage induction motor has a nominal mechanical spinning speed of 25 Hz (1500 rpm) at a rated electrical frequency of 50 Hz. The rotor response of the proposed built-in actuator system at Bridge OFF & Bridge ON conditions at synchronous speed & sub synchronous speeds is analyzed experimentally (speed is varied by VFD).

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Fig. 6. FFT of the horizontal response of the system with the half-power bandwidth method.

3 Result and Discussions There are three bridge branches in the circuit, as shown in Fig. 1; each phase has one Bridge branch, Bridge ON condition means all three branches are short-circuited. Whereas, in Bridge OFF condition, it is a simple parallel winding induction motor with disconnected Bridge points. The rotordynamic response of the BCW cage induction motor in Bridge OFF and Bridge ON conditions for variable electrical supply frequency (ω) between 0 Hz to 75 Hz over a range of 5 Hz has been analyzed experimentally. All investigations have been done at no load running condition for the squirrel cage induction motor. The rotor response is recorded at 3 locations, near the front and rear end of bearing support and near the Auxiliary bearing on the front side of the cage induction motor, as shown in Fig. 3 & Fig. 4 in the vertical and horizontal directions. These responses are recorded for all data sets at each location and frequency; among that, some of the orbit plots for the rotor response at sub-synchronous and synchronous speeds of the supply frequency are shown in Fig. 7. The orbital rotor response in Fig. 7 shows the effective reduction in the amplitude of the rotor response if switched from the Bridge OFF condition to the Bridge ON condition in both directions. This phenomenon is observed in the case of the voltage supply at a very low frequency (20 Hz), rated frequency (50 Hz), and high frequency (60 Hz and 75 Hz). The rotor vibrations in the induction motor have many causes but ultimately reflect in the form of eccentricity. This eccentricity in the air gap creates unbalance forces on the rotor responsible for the vibrations. These rotor vibrations are mitigated due to the bridge-configured winding. These vibrations are comprised of the amplitude at multiple frequencies of the excitations in both directions, called the higher harmonics or sub-harmonics. In investigations, it has been noticed that there is a significant reduction in the component of the rotor response excited at twice the supply frequency in the Bridge ON condition compared to the Bridge OFF condition. The vibration amplitude at twice the

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supply frequency significantly contributes to the rotor vibration component in either direction. Figure 8 compares the FFT of the horizontal and vertical response in the Bridge OFF and Bridge ON conditions at a rated voltage supply frequency (50 Hz).

Fig. 7. Comparison of rotor orbit for Bridge OFF and Bridge ON conditions at supply frequency (ω) equal to a) 35 Hz, b) 38 Hz, c) 50 Hz, d) 60 Hz, e) 75 Hz.

Table 2 shows the percentage reduction in the amplitude of the oscillating component of horizontal rotor response (Ax2ω ) at twice the supply frequency in the bridge ON condition. From Fig. 8 and Table 2, it can be seen that the amplitude of the horizontal response at 100 Hz (for a supply frequency of 50 Hz) is reduced by 88%. In contrast, the amplitude of the vertical response at 100 Hz is also completely mitigated. Table 2. Percentage reduction in amplitude of oscillating component of the horizontal rotor response at 2ω frequency in bridge ON condition. Voltage Supply Frequency

Ax2ω (mm) in Bridge OFF Condition

Ax2ω (mm) in Bridge ON Condition

% Reduction of Ax2ω in Bridge ON Condition

35 Hz

0.024

0.0035

85.42%

38 Hz

0.252

0.0016

99.3%

40 Hz

0.021

0.0022

89.52%

50 Hz

0.0115

0.00137

88.09%

60 Hz

0.00315

0.00023

92.70%

75 Hz

0.00327

0.000085

99.99%

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A similar pattern has been seen in the experimental investigations at multiple supply frequencies. The components of the response amplitude at other frequencies have also been mitigated, but the dominant effect has been seen at twice the supply frequency. The oscillating component in the rotor system at twice the supply frequency is prevalent when there are extra pole pair forces due to static eccentricity [8]. Another dominant amplitude of the rotor vibration at the whirling frequency of the rotor (here at 50 Hz supply frequency, whirling frequency is 25 Hz) due to the dynamic motion of the rotor or dynamic eccentricity can be effectively reduced by the bridge supply condition or the active control.

Fig. 8. Comparison of a) horizontal rotor response and FFT b) vertical rotor response for Bridge OFF and Bridge ON conditions at a rated supply frequency of 50 Hz

Table 3 and Fig. 9 indicate the amplitude of the oscillating component of the horizontal (ax2ω ) and vertical (ay2ω ) rotor response at twice the supply frequency for Bridge OFF, Bridge ON for one phase, Bridge ON for two phases, and Bridge ON conditions. The observations show that this technique can effectively mitigate the rotor vibrations even at a single Bridge ON condition. Figure 9 shows that the rotor response at 38 Hz is much greater than that at 35 Hz and 40 Hz supply frequency. It is because, for a supply frequency of 38 Hz, the rotor rotates at 19 Hz, which is nothing but the sub-harmonic frequency of the rotor (the natural frequency of the rotor is 76 Hz).

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Table 3. The amplitude of oscillating component of the horizontal (ax2ω ) and vertical (ay2ω ) rotor response at 2ω frequency. Voltage Supply Frequency

Bridge OFF

One Bridge On

Two Bridge on

Bridge ON

Ax2ω

Ay2ω

Ax2ω

Ay2ω

Ax2ω

Ay2ω

Ax2ω

Ay2ω

35 Hz

0.024

0.0075

0.0199

0.0068

0.012

0.0053

0.0035

0.0018

38 Hz

0.252

0.033

0.21

0.026

0.0973

0.0124

0.021

0.0016

40 Hz

0.021

0.0092

0.010

0.0065

0.0059

0.0036

0.0038

0.0022

Fig. 9. Comparison amplitudes of a) horizontal rotor response, b) vertical rotor response at 2ω under different operating conditions w.r.t supply frequency (ω)

The rotor response in the horizontal and vertical direction with FFT and orbit plot is shown in Fig. 10 for the supply frequency of 38 Hz. Even though the amplitude of the rotor response is high at 38 Hz compared to the amplitude at consecutive supply frequencies 35 and 40 Hz, the BCW is effectively (about 99%) mitigating the oscillating component at 2ω frequency of 38 Hz (76 Hz). The rotor behaves abnormally at 38 Hz frequency, possibly because it is rotating at one of its sub-harmonic frequencies. A higher harmonic is present at four times the supply frequency, and surprisingly, this component also vanishes in the Bridge ON condition. 3.1 Proposed Analytical Rotordynamic Model Due to the effect of oscillating UMP, the FFT of the rotor response also shows an oscillating vibratory response in the system at twice the supply frequency. The experimental investigations for the BCW cage induction motor show a significant reduction in the system’s response at twice the supply frequency if switched from bridge OFF to Bridge ON condition. As discussed earlier, if the equalizing current is not flown through the bridge points, the system’s behavior is like the simple parallel winding type motor, so the existing rotordynamic model can be used to analyze the system. But, as soon as bridge points are connected through the external controlling switches, there is a flow of bridge currents due to a voltage difference between the bridge points. However, it reflects as a

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passive control effect in the response of the rotor vibrations, where it has been seen that the BCW effectively mitigates the oscillating component of the rotor vibrations.

Fig. 10. Comparison of a) horizontal response of the rotor w.r.t time, b) vertical response of rotor w.r.t time, c) FFT of the horizontal response of the rotor, d) FFT of the vertical response of rotor, e) rotor orbit for Bridge OFF, One Bridge ON, Two Bridge ON, Bridge ON conditions at supply frequency (ω) equals to 38 Hz.

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The existing model given by Eqs. (1) & (2) is needs to be modified to analyze the effect of the Bridge ON in the analytical study of rotor vibrations. According to the physics of the BCW [9], it can create transverse forces opposite to UMP. Thus, the modified rotordynamic equations of the motion for the system equipped with BCW (ON condition) are given by Eq. (3) & (4). ump

bridge

2 cos(ω t) + f − fx m¨x + cx x˙ + kx x = meωm m x 2 sin(ω t) + f ump − f bridge m¨y + cy y˙ + ky y = meωm m y y

(3)

The transverse forces produced in the airgap due to Bridge ON conditions, f x bridge , and f y bridge can be modeled by experimental investigations as,     bridge fx fb1 cos θ + fb2 cos(2ωt − 3θ ) for P = 2 (4) = bridge fb1 sin θ + fb2 sin(2ωt − 3θ ) fy f b1 and f b2 depend on the relative eccentricity and the equalizing current flowing through the control winding. So, for the existing eccentricity condition, the current flowing through the bridge branches can be controlled by varying the potential difference (applying the variable voltage) between the bridge points through the external controllable AC voltage source.

4 Conclusions The parameters of the rotor used in the experimental setup, such as mass, stiffness, and damping ratio, have been estimated, which can be later used in numerical as well as analytical rotordynamic analysis of the system. The critical bending frequency of the rotor is 76 Hz (481.67 rad/sec), and the damping ratio is 0.574%. The experimental setup for the bridge-configured winding squirrel cage induction motor has been developed, and the effect of the Bridge ON condition has been investigated for a range of voltage supply frequencies. The response of the rotor in horizontal and vertical directions has been analyzed. However, it has been seen that the BCW in induction motors passively suppresses the rotor vibrations. It is because after connecting the bridge points, the bridge branch acts as the control winding, and there is the flow of the equalizing currents due to potential differences at the bridge points and suppresses the effect of the rotor vibrations due to forces like UMP. A significant reduction in rotor orbit has been seen in the bridge ON condition due to the passively controlled rotor vibrations. The response of the rotor is enormously high at its one-fourth natural frequency, which is the sub-harmonic frequency of the rotor. Still, the BCW can effectively mitigate the rotor vibrations. The passively controlled condition of the BCW scheme effectively mitigates the oscillating component of the rotor amplitude at twice the supply frequency case due to static eccentricity. The whirling component of the rotor vibrations may also be mitigated by applying the voltage through an external power source to bridge points. Based on these observations, a new analytical

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rotordynamic model is proposed for the BCW electric systems based on the passive controllability of the BCW induction motor. Future Work The complete system parameter identification by including the coupled effect of the electromagnetic force production in the BCW induction motor due to bridge ON and Bridge Supply conditions along with the mechanical unbalance force needs to be identified numerically and experimentally. Later, these Electromechanical interactions for the BCW cage induction motor for different conditions are used for the rotor dynamic analysis of the system. The vibratory response of the BCW motor due to the effect of UMP and the Bridge ON condition needs to be investigated by the proposed analytical rotor dynamic model.

References 1. Friswell, M.I., Penny, J.E.T., Garvey, S.D., Lees, A.W.: Dynamics of rotating machines. Cambridge University Press (2010) 2. Dorrell, D.G., Thomson, W.T., Roach, S.: Analysis of airgap flux, current, and vibration signals as a function of the combination of static and dynamic airgap eccentricity in 3 phase induction motors. IEEE Trans. Indus. Appl. 33(1), 24-34 (1997) 3. Burakov, A., Arkkio, A.: Comparison of the unbalanced magnetic pull mitigation by the parallel paths in the stator and rotor windings. IEEE Trans. Magn. 43(12), 4083–4088 (2007). https:// doi.org/10.1109/TMAG.2007.906885 4. Laiho, A., Tammi, K., Zenger, K., Arkkio, A.: A model-based flexural rotor vibration control in cage induction electrical machines by a built-in force actuator. Electr. Eng. 90(6), 407–421 (2008). https://doi.org/10.1007/s00202-007-0091-1 5. Sinervo, A., Arkkio, A.: Eccentricity related forces in two-pole induction motor with fourpole stator damper winding analyzed using measured rotor orbits. IEEE Trans. Magn. 49(6), 3029–3037 (2013). https://doi.org/10.1109/TMAG.2013.2238637 6. Khoo, W.K.S.: Bridge configured winding for polyphase self-bearing machines. IEEE Trans. Magn. 41(4), 1289–1295 (2005). https://doi.org/10.1109/TMAG.2005.845837 7. Khoo, W.K.S., Kalita, K., Garvey, S.D.: Practical implementation of the bridge configured winding for producing controllable transverse forces in electrical machines. IEEE Trans. Magnet. 47(6) part 2, 1712–1718 (2011). https://doi.org/10.1109/TMAG.2011.2113377 8. Guo, D., Chu, F., Chen, D.: The unbalanced magnetic pull and its effects on vibration in a three-phase generator with the eccentric rotor. J. Sound Vibr. 254(2), 297–312 (2003). https:// doi.org/10.1006/jsvi.2001.4088 9. Kumar, G., Kalita, K., Tammi, K.: Analysis of bridge currents and UMP of an induction machine with bridge configured winding using coupled field and circuit modelling. IEEE Trans. Magn. 54(9), 1–16, 8104416 (2018). https://doi.org/10.1109/TMAG.2018.2854666

Identification Method for Cage Rubbing Faults of Flywheel Bearings Based on Characteristic Frequency Ratio and Convolutional Neural Network Jianwen Wang1 , Hong Wang2 , Tian He1(B) , and Tao Qing2 1 School of Transportation Science and Engineering, Beihang University, Beijing 100191, China

[email protected] 2 Beijing Key Laboratory of Long-Life Technology of Precise Rotation and Transmission

Mechanisms, Beijing Institute of Control Engineering, Beijing 100194, China

Abstract. Bearings are the core components of space inertia actuators such as flywheels. Their running status is directly related to the performance of the machine. Currently, cage rubbing faults are commonly seen during the operation of flywheel bearings and their vibration signals are particularly complex, making it difficult to identify the rubbing status. In response to this issue, this article proposes a process of identification method for cage rubbing faults by characteristic frequency ratio (CFR) and one-dimensional convolutional neural network (1DCNN). This method can fully utilize the expert experience and classification advantages of the neural network model to accurately identify the operating status of flywheel bearings. Firstly, the raw data is obtained through bearing vibration experiments. Then, these data are processed for preliminary state identification based on demodulated resonance technique and CFR to separate the abnormal bearings and the normal bearings. On this basis, the 1DCNN model is constructed to automatically extract fault features and classify states, realizing fault localization. The results indicate that this method is feasible and effective in identifying the rubbing faults of bearing cages. Keywords: Flywheel Bearing · Fault Diagnosis · Cage Rubbing Faults · Characteristic Frequency Ratio · One-dimensional Convolutional Neural Network

1 Introduction As a key component of aerospace high-speed rotating equipment, flywheel bearings directly affect the safety and operation efficiency of spacecraft. Therefore, it is of great significance to develop a diagnostic method for flywheel bearing faults. Among many diagnostic methods, vibration analysis is widely used for detecting bearing defects due to its sensitivity and effectiveness. Currently, the flywheel bearing requires strict ground vibration testing before product delivery for health screening. In these tests, it is necessary to develop a scientific and effective method to monitor and identify the operation of bearings. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 528–538, 2024. https://doi.org/10.1007/978-3-031-40455-9_41

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At present, data-driven methods have received widespread attention in the field of bearing fault diagnosis. The mainstream data-driven methods for bearing fault diagnosis include signal processing technology and machine learning [1, 2]. Yu [3] used a timefrequency analysis method to characterize and extract transient components in bearing fault signals. Jalayer [4] et al. enhanced the accuracy of bearing fault diagnosis by constructing a feature engineering model which combines fast Fourier transform (FFT), continue wavelet transform (CWT), and statistical features of raw signals. In recent years, many new ideas have been brought to this field with the rapid development of deep learning [5]. Xu [6] et al. developed a hybrid deep learning model based on convolutional neural network (CNN) and gcForest for bearing fault diagnosis, which can extract fault features from the time-domain waveform. Zhang [7] et al. used the scaled exponential linear unit (SELU) and hierarchical regularization to obtain better training results when using deep learning models for bearing fault diagnosis. Jin [8] et al. proposed a bearing diagnosis model for bearing diagnosis based on decoupling attention residual network. The results show that the diagnostic model can obtain better results with fewer fault samples. He [9] et al. proposed a new model that combined wavelet packet transform, simulated annealing algorithm, and CNN to achieve better performance in bearing diagnosis. Wang [10] et al. proposed a multi-attention one-dimensional convolutional neural network (MA1DCNN) to diagnose wheelset-bearing faults. The above methods are mainly aimed at diagnosing early bearing faults in ground experiments and mainly focus on local faults such as pitting corrosion or spalling. However, cage rubbing faults caused by unstable or abnormal lubrication are more common in flywheel bearings with light load. During the high-speed rotation process of bearings, rubbing often occurs between balls and cage. If the cage is guided by the inner or outer ring, rubbing may also occur between the guiding surfaces and the bearing sleeve. These two types of rubbing processes are often accompanied by balls or inner and outer ring coupling faults, resulting in complex vibration signal information and difficult identification. In addition, due to the light load and numerous structural components of the flywheel bearings, similar fault characteristics may also occur during normal operation. These factors all increase the difficulty of diagnosing and identifying cage rubbing faults. At present, the false negative rate and false alarm rate are usually high when applying traditional diagnostic methods. To solve these problems, this article proposes a fault diagnosis method that combines characteristic frequency ratio (CFR) and 1DCNN classifier, which can fully utilize the expert experience and classification advantages of neural networks to accurately identify flywheel bearing cage faults.

2 Fault Identification Method Based on CFR and CNN 2.1 The Basic Principle of CFR Demodulated resonance technique is the most classic and commonly used method for bearing fault diagnosis in applications. The bearing can be judged whether there is any fault by observing and analyzing the envelope spectrum obtained by resonance demodulation. Ref. [11] studied the failure mode of flywheel bearing cages. It is pointed out that when the cage friction fault occurs, the cage revolution frequency f c and the

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ball spin frequency f b will be coupled in the envelope spectrum. A sideband with f b and its frequency multiplication as carrier frequency, and f c or 2f c as modulation frequency is formed. Table 1 lists the characteristic frequencies of cage rubbing faults when the bearing speed is 4600 rpm. Table 1. Characteristic frequency involved in cage rubbing faults Parameter

Value

Parameter

Value

fb

124.7

2f b

249.4

fc

29.7

2f c

fb – fc fb + fc

95.0

2f b – f c 2f b + f c

219.7

154.3

59.4 528.5

f b – 2f c

65.3

2f b – 2f c

190.0

f b + 2f c

184.1

2f b + 2f c

308.8

Based on this, this article conducts diagnostic work with characteristic frequencies. In order to quantitatively evaluate fault information, the envelope spectrum CFR calculation method in Ref. [12] is adopted, and the specific process is as follows. Let f E be the analysis bandwidth in the envelope spectrum. In this article, f E is set to three times max( f d ), where f d is the characteristic frequency related to cage rubbing fault listed in Table 1. f i is the ith frequency point and Ae ( f i ) is the spectrum value at f i in the envelope spectrum. Aea =

Ne −1 1  Ae (fi ) Ne

(1)

i=0

where N e is the number of spectral lines of the analysis bandwidth in the envelope spectrum, and Aea is the average value of the spectrum. Due to the rotating speed instability caused by the skidding fault and the error between the calculated parameters and the actual parameters, the frequency difference f is introduced as the difference between the theoretical calculation f d and the actual. Hence, Aed is calculated as ⎞ ⎛ Ae (ifd − f ) ⎟ ⎜ ⎜ Ae (ifd − f + fss )⎟ n  ⎟ ⎜ 1 ⎟ ··· max⎜ (2) Aed = ⎟ ⎜ n ⎟ ⎜ i=1 ⎝ Ae (ifd + f + fss )⎠ Ae (ifd + f ) i where f ss is the frequency resolution. The CFR δA can be defined as δA =

Aed Aea

(3)

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2.2 The Basic Principle of CNN Deep learning enables the fault diagnosis model to adaptively learn the feature representation from the signal to complete the fault recognition. However, most of the classic two-dimensional convolutional neural network (2DCNN) models are designed for recognizing the features of two-dimensional images. In the fault diagnosis of bearings, because the input signal is one-dimensional, the classic 2DCNN model needs to first convert the one-dimensional signal into a two-dimensional image for deep learning. Then the selection of coordinate axes, image stretching, displacement, and image resolution will interfere with the representation of fault features on the image, which will directly affect the effect of deep learning. Therefore, it is more advantageous to use the 1DCNN model for the diagnosis of one-dimensional vibration signals of bearings [13]. The 1DCNN can directly process the one-dimensional vector of the vibration signal. Using the adaptive learning characteristics of CNN, the original signal is processed directly, which avoids the interference of the signal transformation process and realizes end-to-end fault diagnosis. Figure 1 shows a typical 1DCNN structure, which consists of input layer, convolution layers, pooling layers, fully connected layer, and output layer. As the first layer of 1DCNN, the input layer is used to receive training and prediction samples. The convolution layer can effectively extract image features. One-dimensional convolution operation refers to the use of one-dimensional convolution kernel to perform local convolution calculation on the input one-dimensional signal to generate onedimensional convolution feature map. After passing through the convolution layer, the parameters tend to be more, so the pooling layer is needed to reduce the parameters of the neural network. Then, the data needs to be flattened into feature vectors, and these vectors are input into the fully connected layer. Finally, the classification is performed by the ‘softmax’ classifier in the output layer.

Fig. 1. Structure diagram of a typical 1DCNN model

2.3 Diagnostic Process Based on CFR and 1DCNN The intelligent diagnosis of flywheel bearing faults based on CFR and 1DCNN model can be divided into the following 8 steps, as shown in Fig. 2.

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(1) Collect vibration signals of flywheel bearing working status through sensors; (2) Perform ensemble empirical mode decomposition (EEMD) on the raw signal to obtain intrinsic mode functions (IMFs); (3) Reconstruct the signal based on kurtosis normalization to extract fault features from the original signal; (4) Perform envelope analysis on the reconstructed signal base on Hilbert transform (HT) and calculate the CFR based on Eqs. (1), (2) and (3); (5) The fault status is initially diagnosed based on the distribution of CFR. Then, the data with unrecognized status is used as input samples for subsequent identification process based on 1DCNN model; (6) Distribute samples into test data, validation data, and training data in a certain proportion; (7) Construct an 1DCNN model, and train this model using training data sets. The trained diagnostic model is then obtained by iteration; (8) Input the identified samples into the network and combine the outputs with the results in step (5) to obtain the final diagnostic results for all sample sets.

Construction of data sets

Data acquisition

Condition monitoring

Data with labels

Fault Feature Extraction

EEMD

Kurtosis K

IMF Selection

If K>3

Signal reconstruction

Yes

Step 1: Diagnosis based on characteristic frequency ratio the envelope spectrum

>δAlim

No

Normal

Yes

Characteristic frequency ratio

Fault

Step 2: Diagnosis based on 1DCNN Training Set Build 1DCNN model

Validation Set The trained model

Test Set

Fault types

Bearing fault diagnosis

Fig. 2. Flow chart of the diagnostic method based on CFR and 1DCNN

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3 Experiment and Analysis 3.1 Vibration Experiment Figure 3 shows the vibration experimental device of flywheel bearings. The rotor of the bearing assembly is supported by a diagonal contact ball bearing. The inner ring of the bearing is fixed, and the outer ring is exported from the motor assembly at 4600 rpm. The accelerometer is installed on the flywheel support base. The sampling frequency of the vibration signal is 25600 Hz. So far, batches of flywheel bearings have been tested. To find out abnormal bearings, all bearings are monitored online by the method proposed in Ref. [12]. In addition, the faulty bearings are disassembled after diagnosis to verify their real states. These data with state labels will be used for subsequent research on fault identification methods and verification.

Fig. 3. Vibration experimental device of the flywheel bearing.

According to the status of bearings, the vibration data collected from vibration experiments is divided into 6 groups, including three states: normal, abrasion on cage pockets, and abrasion on guiding surfaces. Obtain vibration signals with a length of 60 s in each state, and divide the collected data into samples with a length of 1024, as shown in Table 2. 3.2 Primary Diagnosis Based on EEMD, HT, and CFR In this article, 80% of samples are randomly selected as training data, and the remaining 20% are used as test data. Firstly, the original signal is reconstructed to extract fault information based on EEMD decomposition and kurtosis criterion. Then, envelope demodulation analysis is performed on the reconstructed training samples by HT. After that, the CFR distribution can be obtained by calculating through Eqs. (1), (2), and (3). Figure 4 shows the mean and standard deviation of CFR in training data. The lower limit of all fault ratios is above the upper limit of the CFR of all normal state data, so the judgment threshold can be well determined at this time. The threshold can be determined as needed, such as the upper bound of CFR under normal conditions (2.09) or the lowest lower bound under fault conditions (3.04). In this article, the threshold δAlim is determined as the upper bound of CFR under normal conditions, which is 2.1. In this way, the CFR can be used for automatic monitoring of bearing faults. Subsequently, the CFR distribution of test data can be calculated using the same way. By comparing

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specific CFR values, the status of bearings can be preliminarily identified. When δAtest < δAlim , the test data is considered to be in a normal state. When δAtest > δAlim , the test data is considered to be abnormal, and these samples will be put into the subsequent CNN model for fault localization. Table 2. Information of sample data sets Number

Bearing state

Sample size

Sample length

Label

1#

Normal

1500

1024

F1

2#

Normal

1500

1024

F1

3#

Abrasion on cage pockets

1500

1024

F2

4#

Abrasion on cage pockets

1500

1024

F2

5#

Abrasion on guiding surfaces

1500

1024

F3

6#

Abrasion on guiding surfaces

1500

1024

F3

Fig. 4. CFR distribution map of cage friction fault

3.3 Further Diagnosis Based on 1DCNN Model For further diagnosis, the training data is directly put into the 1DCNN to construct the classification model. When constructing the 1DCNN model, the recognition accuracy and efficiency can be improved by selecting appropriate network structures and parameters. Table 3 shows the optimal structure and parameters obtained through the trial-and-error method. A total of 10 layers of the network are set up, including an input layer, 2 convolutional layers, 2 batch normal (BN) layers, 2 pooling layers, a dropout layer, a fully connected layer, an output layer, etc.

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Table 3. The structure of 1DCNN Layer type

Kernel size / Stride

Channel

Activation function

Shape

Input layer

−

−

−

1024 × 1

Conv1D

(3 × 1)/1

16

ReLU

16 × 1024 × 1

BN

−

−

−

16 × 1024 × 1

Max-Pooling

(2 × 1)/2

−

−

16 × 512 × 1

Conv1D

(3 × 1)/1

32

ReLU

16 × 512 × 1

BN

−

−

−

32 × 512 × 1

Max-Pooling

(2 × 1)/2

−

−

32 × 256 × 1

Dropout(0.2)

−

−

−

8192

FC

−

−

ReLU

60

Output layer

−

−

Softmax

3

3.4 Analysis Figure 5 shows the changes in the accuracy and loss of the training and validation data sets when the epoch increases. It can be seen that when the epoch reaches 30, the curve has stabilized, and the accuracy of training and validation data has reached 100%. It is thus clear that the 1DCNN model constructed this time has good efficiency. Then, the test data is put into the trained model to obtain the positioning results. Combined with the results of primary diagnosis obtained in Sect. 3.2, the operating status of bearings can be finally determined. Figure 6 shows the confusion matrix of test data. From Fig. 6, each state can be accurately classified, with an accuracy rate of up to 100%.

Fig. 5. Training and validation accuracy

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Fig. 6. The confusion matrix of test data

4 The Application Effect and Discussion To further verify generalization performance, the proposed method is applied to new data sets of bearings that have been monitored in ground tests. The basic information of the data is shown in Table 4. Each type of data contains 750 samples, each with a length of 1024. In addition, the methods that directly put the original signal into a 1DCNN model for feature classification are used to demonstrate the effectiveness of the proposed method. In order to evaluate the method identification ability, three common evaluating indicators in fault diagnosis, such as Accuracy, Recall, and Precision, are introduced. At the same time, a total of 10 experiments are conducted to avoid errors caused by one experiment. After calculation, the average values of evaluating indicators are shown in Table 5. The average diagnostic accuracy of the proposed method in this article is 98.13%, indicating that the method has good diagnostic promotion ability. The diagnostic Accuracy, Recall, and Precision of the direct method are significantly lower than the accuracy of the proposed approach, which further proves that the method based on CFR and 1DCNN can effectively improve the accuracy of cage rubbing faults diagnosis. In conclusion, a deep convolutional model that combines manual diagnostic experience has better diagnostic adaptability. Table 4. Dataset for promotional validation Number

Bearing state

Sample size

Sample length

Label

7#

Normal

750

1024

F1

8#

Abrasion on cage pockets

750

1024

F2

9#

Abrasion on guiding surfaces

750

1024

F3

Figure 7 shows the confusion matrix diagnosed by two methods. It can be seen that when using the direct method, 42 normal samples were mistakenly diagnosed as faults. When using the proposed approach, only 8 normal samples were mistakenly diagnosed, which means the false alarm rate is greatly reduced. In addition, the number

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of fault samples whose states are diagnosed mistakenly as normal in the direct method is 42, compared to the proposed approach in this article, which is only 3. This result proves that the proposed method also has advantages in reducing the false negative rate. In conclusion, after the primary diagnosis based on EEMD decomposition and CFR method, the normal data and fault data can be effectively separated, thereby significantly reducing the false alarm rate and missed alarm rate. Table 5. The average values of evaluating indicators Metric

Proposed approach

1DCNN

Accuracy

0.9813

0.9516

Recall (Macro-average)

0.8189

0.7994

Precision (Macro-average)

0.9815

0.9516

Fig. 7. The confusion matrix of generalized samples: (a) proposed approach in this article; (b) direct method.

5 Conclusion To solve the difficulties in diagnosing and identifying cage rubbing faults, this paper proposes a fault diagnosis process that combines CFR and CNN. It can fully utilize the expert experience and classification advantages of deep learning models to accurately identify flywheel bearing cage rubbing faults. Through the verification and application of experimental samples, the following conclusions can be drawn: (1) Defining the CFR can quantify the classical resonance demodulation method, which can effectively distinguish between normal bearings and bearings with cage rubbing faults;

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(2) The diagnostic results show that the proposed fault identification method can effectively diagnose cage rubbing faults, and accurately distinguish status such as normal, abrasion on cage pockets, and abrasion on guiding surfaces. (3) Compared with the direct method, the proposed identification process by combining CFR and CNN shows a significant increase in diagnostic accuracy and demonstrates higher generalization performance in applications.

References 1. Zhou, H., Huang, X., Wen, G.: Construction of health indicators for condition monitoring of rotating machinery: a review of the research. Expert Syst. Appl. 117297 (2022) 2. Cheng, Y., Lin, M., Wu, J.: Intelligent fault diagnosis of rotating machinery based on continuous wavelet transform-local binary convolutional neural network. Knowl.-Based Syst. 216, 106796 (2021) 3. Yu, G.: A concentrated time–frequency analysis tool for bearing fault diagnosis. IEEE Trans. Instrum. Meas. 69(2), 371–381 (2019) 4. Jalayer, M., Orsenigo, C., Vercellis, C.: Fault detection and diagnosis for rotating machinery: a model based on convolutional LSTM, fast fourier and continuous wavelet transforms. Comput. Ind. 125, 103378 (2021) 5. Li, J., Liu, Y., Li, Q.: Intelligent fault diagnosis of rolling bearings under imbalanced data conditions using attention-based deep learning method. Measurement 189, 110500 (2022) 6. Xu, Y., Li, Z., Wang, S.: A hybrid deep-learning model for fault diagnosis of rolling bearings. Measurement 169, 108502 (2021) 7. Zhang, Y., Xing, K., Bai, R.: An enhanced convolutional neural network for bearing fault diagnosis based on time–frequency image. Measurement 157, 107667 (2020) 8. Jin, Y., Qin, C., Huang, Y.: Actual bearing compound fault diagnosis based on active learning and decoupling attentional residual network. Measurement 173, 108500 (2021) 9. He, F., Ye, Q.: A bearing fault diagnosis method based on wavelet packet transform and convolutional neural network optimized by simulated annealing algorithm. Sensors 22(4), 1410 (2022) 10. Wang, H., Liu, Z., Peng, D.: Understanding and learning discriminant features based on multiattention 1DCNN for wheelset bearing fault diagnosis. IEEE Trans. Industr. Inf. 16(9), 5735–5745 (2019) 11. Chen, C., Deng, Z., Wang, H.: Simulation of friction fault of lightly loaded flywheel bearing cage and its fault characteristics. Sensors 22(21), 8346 (2022) 12. Wu, D., Wang, J., Wang, H.: An automatic bearing fault diagnosis method based on characteristic frequency ratio. Sensors 20(5), 1519 (2020) 13. Shao, H., Xia, M., Han, G.: Intelligent fault diagnosis of rotor-bearing system under varying working conditions with modified transfer convolutional neural network and thermal images. IEEE Trans. Industr. Inf. 17(5), 3488–3496 (2020)

Unbalance Measurement Traceability Weijian Guo(B) Beijing SYTH Testing Co., Ltd., Room 101, Huacheng Building, No. 4 Shangdi Street, Haidian District, Beijing 100085, China [email protected]

Abstract. Rotor unbalance measurement is not traceable with existing technology. By acquiring the mass-center of unbalance weight, achieving the unbalance increment traceable, by decomposing unbalance of a rotor and unbalance outside of the rotor, making the zero point of the unbalance traceable. Based on the two steps, unbalance measurement traceability is realized. Keywords: Balance · rotor unbalance · balancing machine · Unbalance traceability

1 Introduction Rotating mechanical part, also referred to as a rotor, is widely used in aerospace, automotive, energy, and general industries. The rotor will possess unbalance when the mass of the rotor is asymmetrically distributed with respect to the rotational axis. The rotor will generate inertial centrifugal force when rotating, and will apply alternating force on supporting structure, thus causing vibration on mechanical equipment. A balancing machine is used to measure unbalance of a rotor. Current standard for testing balancing machines is ISO 21940–21:2012 Mechanical vibration - Rotor balancing - Part 21: Description and evaluation of balancing machines [1]. Two terms are used to describe accuracy of a balancing machine: Minimum Achievable Residual Unbalance Umar and Unbalance Reduction Ratio URR. Testing of these two terms is using “proving rotor” and following procedures described in the standard; but the proving rotor does not possess a definite unbalance value. Rising problem is that when using two balancing machines to measure a proving rotor, two different values may got, and it is not sure, nor able to verify, which value is correct. With the existing technology, it is not possible to make a “standard rotor” which possesses definite reference unbalance value. The root of the problem is that the current unbalance measurement method is not with traceability, or it cannot be traced to SI units [2]. The pursuit of unbalance measurement with traceability is done in the past decades. A study of the traceability of unbalance measurement was carried out by the Chinese Academy of Metrology in 2000, and the traceability route was: the mass of the rotor and mass of test weight are traced to SI Unit Mass (kg), the geometric dimensions of the rotor and the angle of the location of the screw holes of the proving rotor were traced to © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 539–545, 2024. https://doi.org/10.1007/978-3-031-40455-9_42

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SI Unit Length (m) [3]. However, this traceability method does not determine or verify the residual unbalance of the rotor, so it does not realize traceability. Schenker RoTec GmbH carried out the traceability of unbalance measurement around 2010 by building a “MAMA” machine and a precision BAR. The traceability route is as follows: The BAR is precisely machined, and its shape and size are measured to calculate the unbalance of the BAR as a reference value of unbalance, then the BAR is placed on the MAMA balancing machine, and the balancing machine is adjusted so that the measured value of the balancing machine is the same as the reference unbalance value of the BAR, then the machine is regarded as being calibrated correctly [4]. However, the prerequisite of this traceability route is with the assumption that the material of the BAR is homogeneous; but this assumption cannot be verified. Therefore, this traceability route has a theoretical gap in the comparison chain. The traceability of unbalance measurement is an open technical point. With this paper, this technical issue is solved by establishing traceable increment of unbalance and by decomposing unbalance of the rotor and unbalance outside the rotor.

2 Rotor Unbalance and Unbalance Measurement 2.1 Rotor Unbalance As shown in Fig. 1, when a rotor rotates around its axis of rotation (N-N), each mass point in the rotor will generate an unbalance vector U i , U i = mi ∗ ri .

Fig. 1. Rotor Unbalance

For the entire rotor, most of the mass point is offsetting with each other, the unbalance of the rotor is generated by the mass point which does not offset. Unbalance of a rotor can be expressed by any two planes perpendicular to the axis of rotation, as U I on plane I and U II on plane II shown in Fig. 1. The rotor discussed herein is limited to a rigid rotor, i.e., it is considered that the mass on the rotor do not make displacement or shifting in the measuring speed range and correspondingly, the unbalance of the rotor does not change with measuring speed. 2.2 Unbalance Measurement Unbalance of a rotor is measured by a balancing machine. Figure 2 shows the measurement diagram of a typical universal horizontal balancing machine. The balancing machine has two pedestals (1, 2 in Fig. 2), on which the rotor is supported via rollers. A motor drives the rotor via a belt to accelerate the rotor to measure speed; two vibration

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signal sensor (5, 6) are installed on each of the two pedestals (1,2). A phase sensor (7) detects the angle reference point on the rotor and generates a pulse signal in each revolution of the rotor. The vibration signal and the phase pulse signal are processed by the measurement system (8) of the balancing machine to display the amount and angle of the rotor unbalance.

Fig. 2. Unbalance Measurement Diagram

In Fig. 2, the driving of the rotor is by belt. It has no rigid mechanical part constantly in contact with the rotor and rotates at the same speed with the rotor during the measurement. Similar driving methods include support roller drives, air drive, etc. For such driving methods, an angle reference point is made on the rotor commonly in the form of a small strip of reflective tape, as shown in in Fig. 2.

3 Traceable Unbalance Measurement Method The unbalance traceability is achieved by two steps: Step 1: make traceable unbalance increment; Step 2: Decompose unbalance of the rotor and unbalance outside of the rotor. 3.1 Unbalance Weight The amount of unbalance generated by an unbalance weight is the product of the mass of the weight and the distance of the weight mass-center to the axis of rotation. The existing international standard assumes that the material of the unbalance weight is homogeneous and that the geometric center of the weight is the mass-center of the weight. In practice, the mass of the weight is not always uniformly distributed and the mass-center is not always located at the geometric center. Therefore, it is necessary to obtain the exact position of the mass-center of the weight. The following is an illustration of a method for obtaining the mass-center of weight [5]. As shown in Fig. 3. The weight is cubic in shape and a stud is mounted on each of its six faces at the geometrically symmetrical center.

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B1

C1

A1

A1

A2

C2

B2

C1 B2

(a)

B2

(b)

Fig. 3. Weight

R N

N

(a)

(b)

Fig. 4. Rotor

Figure 4 shows a rotor R with a screw hole machined in the outer circle surface of the rotor to fit the stud of the weight. The weight shown in Fig. 3 is screwed into the thread hole of the outer circle of the rotor with the stud on face A1, then the rotor with the weight will be put on a balancing machine to measure unbalance, as shown in Fig. 5. The measured value is recorded as Uf 1 ; then the weight is unscrewed from the rotor and re-screw into the thread hole of the outer circle of the rotor with the stud on the face A2, the unbalance of the rotor will be measured again and is recorded as Uf 2 . If the above two measurements Uf 1 and Uf 2 are not the same, it means that the mass-center of the weight is not located in the middle of face A1 and A2. According to the amount of unbalance Uf 1 and Uf 2 , make a mass correction on the stud end on the face A1 or A2 by milling or hand filing. Repeat the above procedure until the first measurement of the unbalance Uf 1 and the second measurement of the unbalance Uf 2 are the same or less than a set value. Now the masscenter of the weight is located in the middle plane of faces A1 and A2 (or deviates from the middle plane by less than a set value). The same operation is performed for faces B1 and B2, C1, and C2. Then a weight whose mass-center lies at the center of symmetry of the outer surfaces (or deviates from the center by less than a set value) is made. Now the mass-center of the weight coincides with the center of the geometry. In the process of measuring the mass-center of the weight described above, it is not required that the unbalance value of the balancing machine be traceable, because the measurement process is a relative measurement and relative comparison process. Measuring the mass of the weight on a scale, and records as m1 , measuring the distance between faces A1-A2, recorded as 2 ∗ r 1 . The radius of the outer circle of the rotor where the weight is added is r2 . When the weight is added to the outer circle of the rotor, the amount of unbalance generated by the weight is: U f = m1 ∗(r1 + r2 )

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A2

A2

r1 A1

A1

r2 N

N S1

S1 (a)

(b)

Fig. 5. Measure the mass-center of weight

Thus the unbalance increment amount is traced to SI base unit mass and length. The increased unbalance direction is pointing from an intersecting point, which is by a surface perpendicular to the axis of rotation and through the mass center of the weight, to the mass-center of the weight. So the angle of the unbalance is traced to SI derived unit radian. 3.2 Decompose Unbalance of Rotor and Unbalance Outside of the Rotor First, use a balancing machine to measure unbalance of the rotor R, as shown in Fig. 4, and get an unbalance U 1 . U 1 is not necessarily the unbalance of the rotor only. It may contain unbalance outside the rotor, such as the electrical unbalance compensation applied in the measuring system, or zero point drifting in the measuring system; Separate the unbalance of the rotor from the unbalance outside the rotor by the following method [6]: removing the reference point P1 on the rotor and making a new reference point P2 on the rotor at an angle away from the original reference point position with A degree (90° in this example), see to Fig. 6, in which P1 is the original reference point, P2 is the changed reference point, the rotor is rotating counter clockwise, the unbalance increase angle is clockwise. The reference point is made by a thin and narrow reflecting tape with neglectable unbalance.

Fig. 6. Change reference point

Then measuring the unbalance of this rotor again and get unbalance U 2 . In unbalance U 2 and U 1 , the amount of rotor unbalance is not changed, but the angle of rotor unbalance is changed by A with respect to the new angle reference point on the rotor. While the

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amount and angle of unbalance outside the rotor have no change because this unbalance is not related to the position of the angle reference point. Figure 7 illustrates the vector relationship between the unbalance vectors U 2 , U 1 : connecting vector end point A of U 1 and the vector end point B of U 2 to form a line AB, making an isosceles triangle ACB with AB as the bottom side and a A angle as the top angle, thus get a vertex C. Connecting the coordinate origin O to the vertex C of the isosceles triangle, the vector OC (U M ) is a calculating unbalance which is outside the rotor, the vector CA (U R ) is a calculating unbalance which is of the rotor before the angle reference point is changed, vector CB is a calculating unbalance of the rotor after the angle reference point is changed, and the point C is the measurement zero point of the rotor unbalance [7]. By the above method, the unbalance of the rotor and the unbalance outside of the rotor are decomposed.

Fig. 7. Decompose the unbalances

After obtaining the unbalance U M outside the rotor, the zero point of the balancing machine can be adjusted to zero by digital compensation so that the vertex C point returns to the coordinate zero point O, then the measured unbalance of the balancing machine is only the unbalance of the rotor, the unbalance outside of the rotor is zero. When both traceable unbalance increment and traceable unbalance zero point are realized, unbalance traceability is achieved, i.e., the unbalance amount is traced to SI units mass and length, unbalance angle is traced to SI derived unit radian.

4 Application With the traceable unbalance measurement method in this paper, by verifying the zero point and the unbalance increment, it is possible to judge which machine(s) is not correct when getting different unbalance values for one rotor to be measured on two balancing machines. Several such verification have been done successfully in balancing machine users in recent years in China. With the technology in this paper, it is applicable to make Standard Rotor (Standard of Instruments) with specific reference value, for example, to make “zero” unbalance rotor,

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and combined with different unbalance weights to generate desired specific unbalance amount. In the past years, quite number of this kind of standard rotor and mass weights have been measured and produced to serve balancing machine users in various industries.

5 Conclusion Innovative methods have been created to obtain traceable unbalance increment, and true unbalance of a rotor, as well as true zero point of a balancing machine. The technical problem of the existing international standard which is lack of unbalance traceability is solved. Based on the methods introduced in this paper, new standard for balancing machine accuracy testing and rotor unbalance measurement can be constructed.

References 1. ISO 21940–21:2012 Mechanical vibration –Rotor balancing – Part 21: Description and evaluation of balancing machines 2. Guo, W., Guo, X., Wen, J., Zhenfeng, S.: Analysis and judgement to national standard GB/T 9239.21–2019. Eng. Test. 60(3), 10–15 (2020) 3. Yu, M., Yu, Z.: Balancing machine and its metrology characteristic. Mod. Metrol. Test. 5, 3–7 (2000) 4. Schenck, , RA1047e 5. Guo, W., Guo, X.: One type Unbalance Weight.CN, 202022124858.2. 2020–05–11 6. Weijian, G., Jiang, F.: Method to determine the unbalance of a rotor. US 10101235 B2. 2018– 10–16 7. Weijian, G., Jiang, F.: Method for acquiring unbalance amount of rotor. US 10928267 B2. 2021–02–23

Author Index

A Alcantar, Andy 78 Amirabadizadeh, Hadi 25 An, Byul 291, 391 Andrés, Luis San 173 Antunovi´c, Mario 406 B Bao, Yifan 54 Bondarenko, Maxim 211 Bordoloi, Dhruba Jyoti 482 Brahma, Bipul 515 Braut, Sanjin 406 Buscaglia, Gustavo C. 201 C Che, Renwei 36, 71, 300 Chen, Lihui 111 Chen, Zhaobo 261 Choudhury, Tuhin 190 D Das, Pantha Pradip 482 Dekemele, Kevin 358 Deore, Rakesh 515 Dong, Jiaxin 447 Dong, Shuai 447 Dong, Xinxin 253 dos Anjos, Luis F. 201 Du, Huzhi 71, 300 F Fan, Li 273 Feng, Shaobao 159 Firouz-Abadi, R. D. 25, 46 G Gao, Shuai 309 Gao, Xinyu 54 Gautam, Atul Kumar 462 Goswami, Giota 149 Guo, Weijian 539

H Ha, Yunseok 291, 391 Han, Bing 430 Han, Qinkai 309 He, Peng 430 He, Tian 528 He, Xiaochi 447 Heikkinen, Janne 149 Heya, Akira 273 Hong, Jie 1, 159 Hong, Ling 253 Hu, Xilong 130 Hu, Xiuli 36 I Inoue, Tsuyoshi 273 J Jaramillo, Alfredo 201 Jiang, Feilong 236 Jiang, Jun 253 Jiao, Yinghou 36, 71, 300 K Kalita, Karuna 515 Kazakov, Yuri 224 Koltsov, A. Yu. 245 Konowrocki, Robert 338 Koo, Bonjin 173 Korneev, A. Yu. 245 Kwak, Wonil 291 L Lee, Yeongdo 291, 391 Lee, Yongbok 291, 391 Li, Chao 1 Li, Hui 15 Li, Jintao 261 Li, Shengbo 245 Li, Zhenping 15 Li, Zigang 253 Lian, Peiyuan 111

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 139, pp. 547–548, 2024. https://doi.org/10.1007/978-3-031-40455-9

548

Liu, Lanyu 309 Liu, Zhansheng 430 Loccufier, Mia 358 Lovri´c, Mario 406 Lu, Xueliang 173 M Ma, Yanhong 1, 159 Ma, Yongbo 159 Mahfoud, Jarir 374 Martikainen, Iikka 149 Mehralian, Fahimeh 25, 46 Meshcheryakov, A. B. 282 Mishchenko, E. V. 245 N Narsakka, Juuso 190 Nicoletti, Rodrigo 201 Nutakor, Charles 149 P Pan, Chenghui 111 Peng, Chao 495 Pisarski, Dominik 338 Polyakov, Roman 211 Q Qi, Jinlei 430 Qin, Fan 130 Qing, Tao 528 R Ranjan, Gyan 190 Rinderknecht, Stephan 320 Rodríguez, Bryan 93 S San Andrés, Luis 78, 93 Savin, L. A. 245 Savin, Leonid 211, 224 Schüßler, Benedikt 320 Selivanov, Alexey 422 Shahrukh, 515 Shen, Ao 447 Shutin, Denis 211, 224 Sikanen, Eerik 149 Song, Shiping 1 Sopanen, Jussi 149, 190 Stebakov, Ivan 211, 224

Author Index

Štimac Ronˇcevi´c, Goranka Szolc, Tomasz 338

406

T Tasora, Alessandro 495 Temis, Joury 422 Temis, M. Yu. 282 Tiwari, Rajiv 462, 482 V Venugopal, Harikrishnan W Wang, Chongyang 130 Wang, Congsi 111 Wang, Hong 528 Wang, Jianwen 528 Wang, Jiaqi 430 Wei, Bin 36 X Xu, Qian 111 Xu, Yuanping 374 Xuan, Liang 447 Xue, Song 111 Y Yang, Lihua 130 Yang, Qin 236 Yang, Yishun 430 Yao, Hongliang 15 Yao, Jianfei 54 Yousefi, Masoud 25, 46 Yu, Dong 261 Yu, Xiangyu 430 Z Zhang, Haijun 236 Zhang, Jian 1 Zhang, Xiang 71, 300 Zhang, Yinwei 111 Zhang, Yue 374 Zhang, Zeliang 54 Zhao, Wei 236 Zhao, Wulin 111 Zhou, Jin 374 Zhou, Yang 374 Žiguli´c, Roberto 406

358