Proceedings of the 11th IFToMM International Conference on Rotordynamics: Volume 2 (Mechanisms and Machine Science, 140) [1st ed. 2024] 3031404580, 9783031404580

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Mechanisms and Machine Science

140

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 2

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 978-3-031-40458-0 ISBN 978-3-031-40459-7 (eBook) https://doi.org/10.1007/978-3-031-40459-7 © 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

Numerical Analysis of Seal Force Under Brush Seal Hysteresis Effect . . . . . . . . Yuan Wei, Xuhe Ran, Zhaobo Chen, and Yinghou Jiao

1

Investigation on Nonlinear Behavior of a Rotor System with Friction Effect Due to End-Face Seal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Li Hou, Pingchao Yu, and Cun Wang

11

Analysis and Safety Design of Aero-Engine Rotor Dynamic Response with Multiple Loads Due to Fan Blade off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jie Fu, Chao Li, Jie Hong, and Yanhong Ma

34

Numerical Simulation of Aero-Engine Rotor-Blade-coating Coupling System with Rub-impact Fault and Its Dynamic Response . . . . . . . . . . . . . . . . . . . Jiewei Lin, Bin Wu, Xin Lu, Jian Xu, Junhong Zhang, and Huwei Dai

56

Modeling and Simulation Analysis of Dual-Rotor System in the Early Stage of Bearing Pedestal Looseness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cai Wang, Jing Tian, Yan-ting Ai, Feng-ling Zhang, Zhi Wang, and Ren-zhen Chen Similarity Design and Behavior Prediction of Rotor Systems Subject to Non-uniform Preloads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Runchao Zhao, Yeyin Xu, Zhitong Li, Zhaobo Chen, and Yinghou Jiao

76

91

Vibration Reduction Optimization Design of an Energy Storage Flywheel Rotor with ESDFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Dafang Lin, Siji Wang, Chengyang Wang, Zhoudian Chen, Yuan Liu, and Jinqi Zhang Sensitivity of Spline Self-excited Vibration to Structure Parameters . . . . . . . . . . . 117 Yingjie Li, Guang Zhao, Zexin Zhang, Yunbo Yuan, Jian Li, and Yongquan Wang Parametric Optimization of BNES in Torsional Vibration Suppression of Rotor Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Jinxin Dou, Rui Xue, Hongliang Yao, Hui Li, and Jianlei Li Aviation Spline Wear Test Bench and Fretting Wear Measurement . . . . . . . . . . . . 147 Xiangyang Zhao, Guang Zhao, Yunbo Yuan, Fanrong Kuang, Mei Guo, and Haofan Li

viii

Contents

Dynamic Analysis of the Finger Seal-Rotor System . . . . . . . . . . . . . . . . . . . . . . . . 156 Sai Zhang, Xiuli Hu, Renwei Che, and Yinghou Jiao Investigation on Information Assessment for Vibration Sensor Locations Installed in Aero-Engine Based on Unbalance Response Analysis . . . . . . . . . . . . 168 Alexander A. Inozemtsev, Konstantin V. Shaposhnikov, Sergey A. Degtyarev, Mikhail K. Leontiev, and Ivan L. Gladkiy Torsional Vibration Modelling of a Two-Stage Closed Differential Planetary Gear Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Guanghe Huo, Yinghou Jiao, Miguel Iglesias Santamaria, Xiang Zhang, Javier Sanchez-Espiga, Alfonso Fernandez-del-Rincon, and Fernando Viadero-Rueda Research on Robustness Analysis and Evaluation Method of Bearing-Support System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Fangming Liu, Jie Hong, Yanhong Ma, and Xueqi Chen Investigation on the Transient Lateral Vibration of a Flexible Rotor System with Substantial Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Pingchao Yu, Zihan Jiang, Cun Wang, and Li Hou Dynamic Behaviors of a Bolted Joint Rotor System Considering the Contact State at Mating Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Yuqi Li, Zhimin Zhu, Zhong Luo, Chuanmei Wen, Lei Li, and Long Jin Identification of High-Speed Gear Traveling Wave Resonance Based on Phase Space Reconstruction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Ziyang Xu, Jing Wei, Haibo Wei, Zhirou Liu, Yujie Zhang, and Hao Lin Rotordynamics of a Vibroflot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Florian Tezenas du Montcel, Sébastien Baguet, Marie-Ange Andrianoely, Régis Dufour, Stéphane Grange, Laurent Briançon, and Piotr Kanty Complex Stable and Unstable Subharmonic Vibrations of a Nonlinear Brush-Seal Rotor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Wenbo Ma, Yeyin Xu, Yinghou Jiao, and Zhaobo Chen Multi-objective Optimization of Active Dry Friction Damper-Rotor Systems Based on Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Minghong Jiang, Wengheng Li, Xianghong Gao, and Changsheng Zhu

Contents

ix

Applying Central Manifold Theory in the Definition of Active Gas Foil Bearing Configurations for High-Speed Stability of Rotors . . . . . . . . . . . . . . . . . . 304 Ioannis Gavalas, Emmanouil Dimou, and Athanasios Chasalevris Locating Period Doubling and Neimark-Sacker Bifurcations in Parametrically Excited Rotors on Active Gas Foil Bearings . . . . . . . . . . . . . . . . 324 Emmanouil Dimou, Ioannis Gavalas, Fadi Dohnal, and Athanasios Chasalevris Dynamic Design of the High-Speed Rotor System Considering the Distribution of Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Cong Liu, Yongfeng Wang, Ruiqi Jia, and Jie Hong Optimization of Journal Bearings Considering Their Adjustable Design and Rotor Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Denis Shutin, Alexander Fetisov, and Leonid Savin Stability Margin Optimization for Unsymmetrical Rotor/Stator Dynamic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Yaqun Jiang Remaining Useful Life Prediction for Anti-friction Bearings Based on Envelope Spectrum and Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . 390 Haobin Wen, Long Zhang, and Jyoti K. Sinha New Comprehensive Approach for Torsional Analyses of Industrial Powertrains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 Timo P. Holopainen and Tommi Ryyppö Modeling of the Divergently Worn Annular Seal for the Two-Way Coupled Fluid–Structure Interaction Analysis of Shaft Vibration and Clearance Flow . . . 409 Shogo Kimura, Tsuyoshi Inoue, Hiroo Taura, and Akira Heya Vibration Control of Rotor Bearing Systems Using Electro and Magneto Rheological Elastomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Mnaouar Chouchane and Faiza Sakly A New Type of Inerter Nonlinear Energy Sink Using Chiral Metamaterials . . . . 429 Hui Li, Hongliang Yao, and Yangjun Wu Vibration Characteristic Analysis and Optimization of the Propulsion Shaft in the Underwater Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 Yuchen An, Jing Liu, Chiye Yang, and Guang Pan

x

Contents

Computation of Components System Stiffness for Variable Stator Vane Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 Jing Chang and Zhong Luo An Unbalance Identification Method of a Whole Aero-Engine Based on the Casing Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 Weimin Wang, Jiale Wang, and Qihang Li Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

Numerical Analysis of Seal Force Under Brush Seal Hysteresis Effect Yuan Wei1(B)

, Xuhe Ran1 , Zhaobo Chen2 , and Yinghou Jiao2

1 School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444,

China [email protected] 2 School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China

Abstract. Brush seal has certain advantages over traditional labyrinth seal. As the hysteresis characteristics of the brush seal is a key factor affecting its sealing performance, it is particularly important to conduct in-depth study on the seal force. This paper takes hysteresis as the premise of research, and establishes the model of brush seal force with whole circle bristles under the hysteresis considering the 3D bending of bristles, and set up the dynamic equation of brush seal force for whole circle bristles. This work further studies the influence of system parameters on the brush seal force under the hysteresis, and compares the influence of each system parameter on seal force. And then compared with the model without hysteresis to study the influence degree of hysteresis effect on seal force. The results show that the hysteresis has a great influence on the variation trend of brush seal force, and the seal force is much smaller when the hysteresis effect is taken into account than that not considered. The increase of bristle inclination is beneficial to increase the stability of the rotor system, but for the seal force on the rotor, the greater lay angle is not better. It is of great significance to study the seal force under the lag of brush seal to further understand the sealing characteristics of brush seal. Keywords: Brush Seal · 3D Bending · Hysteresis · Brush Seal Force · Rotor System

1 Introduction The main influencing factors of the seal force of brush seal include the fluid flow of brush seal, the main structural parameters such as the diameter of bristle, and the assembling angle between bristles and rotor surface, etc. Therefore, it is necessary to study the seal force of brush seal from the interaction between bristles, airflow. Therefore, in order to more truly reflect the force of brush seal on the rotor under the influence of mechanical properties such as hysteresis effect, hardening effect and friction heat effect, it is necessary to establish a fluid–structure coupling numerical model that can truly simulate the brush seal, so as to study the mechanism of the seal force of brush seal and its influencing factors. Many scholars have studied the complex nonlinear problem of rotating subsystem seal force for a long time. The seal force model has also experienced a gradual evolution © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 1–10, 2024. https://doi.org/10.1007/978-3-031-40459-7_1

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from simple to complex and from linear to nonlinear. In terms of the study of nonlinear seal force, Black deduced the theoretical formula of seal force for short sealing power system by using short bearing theory [1, 2]. Later, Childs and Nelson et al. [3–6] discussed the flow field in sealing device in detail by using Hirs turbulent lubrication equation. Childs et al. [3] derived the dynamic coefficient expression, and Muszynska et al. [7] proposed the Muszynska nonlinear seal force model through a series of tests. Based on the Muszynska nonlinear seal force model, many scholars conducted a series of studies. Li et al. [8] determined the empirical parameters of airflow excitation force in Muszynska model through the calculation of fluid dynamics. Li et al. [9] analyzed the dynamic characteristics of an unbalanced rotor-seal system with sliding bearings through Floquet and bifurcation theory. Hua et al. [10] established a nonlinear rotor-seal system model, and studied the nonlinear behavior of the unbalanced rotor-seal model through an effective and high-precision direct integration method. Ding et al. [11] studied a symmetric rotor-seal system and analyzed the Hopf bifurcation of the system. Wang et al. [12] established a nonlinear mathematical model, which was used to analyze the motion of a rotor system under the influence of leakage of interlocking seals. Considering the complexity of rotor system structure and the complexity of bristle force, the linear model has great limitations to describe the system dynamics under large whirling state. Modi [13] calculates the contact force without considering the friction between the brush and the friction. Basu et al. [14] verified the existence of the bristle rigidness effect of brush seal through experiments, and the rigidness effect has a serious impact on the service life of brush seal. Akist et al. [15] established a three-dimensional finite element model of brush seal taking into account the friction between bristles. The study found that when the bristle was subjected to uniform pressure load, the contact force would increase continuously. Stango et al. [16] proposed a cantilever beam model by considering the amount of interference between the brush and the rotating shaft, and numerically calculated the frictional contact force. Crudgington et al. [17] studied the strength semi-empirical calculation method of blowdown effect of brush seal by combining test and finite element method. Lelli et al. [18] established a solution model of three-dimensional cylindrical brush and studied the deformation rule of the brush under different aerodynamic conditions. Crudgington etc. [19] was studied by numerical and experimental method of combining the considering bristle installation angle, the results show that with the increase of installation angle, bristle and the contact force obviously increases with the increase of the pressure difference. Pekris et al. [20] studied the mechanical properties of the traditional brush seal structure and the pressure-balanced chamber structure, the results showed that the pressure-balanced chamber brush seal structure could reduce the friction between the bristle and the backplate, leading to the reduction of its sealing performance and service life. At present, there are still some deficiencies in investigating brush seal force. Most of these studies do not consider the hysteresis effect of brush seal, which makes the force analysis of brush seal not perfect. In this paper, the hysteresis degree of the bristle is quantified based on the previous classification research results of single bristle [21, 22], and the bristle is divided into three parts to establish the seal force model under hysteresis effect. The hysteresis effect and the dynamic influence of rotor-brush seal

Numerical Analysis of Seal Force Under Brush Seal Hysteresis Effect

3

system parameters on brush seal force under different rotor eccentricity and eccentric direction angle are studied.

2 Seal Force Analysis 2.1 Hysteresis Effect When the rotor has radial eccentricity or the load increases, due to the action of fluid gas pressure, friction will occur between the bristle and the backplate in the brush seal. It will prevent the bristle from moving with the rotor, and similarly increase the instantaneous stiffness of the bristle. This phenomenon is called the rigid effect of the bristle. Bristle with eccentric rotor radial motion to the largest, the rotor started from radial eccentricity or load falls, because the bristle around the tailgate and the friction of the role, and fail to follow the rotor to restore the initial state, lead to produce certain radial spacing and rotor surface, the contact force and rotor radial eccentricity or load increases, this phenomenon is called the hysteresis effect of the brush. The process of hysteresis is shown in Fig. 1. In Fig. 1 (a), the rotor is in a state of equilibrium position. When the rotor due to the increase in eccentric load and maximum eccentric position, namely, Fig. 1 (b). Then the status of the load drop, after the rotor began to recover, because the brush friction between the bristles and the effect of backplate friction, bristles will lag the recovery speed of rotor speed of recovery, at this time, a gap between the bristles and the rotor surface (yellow area) is created, as shown in Fig. 1(c).

(a) Balancing rotor

(b) Maximum eccentric rotor

(c) Rotor recovery

Fig. 1. Brush seal hysteresis effect

2.2 Seal Force Model In Fig. 2, there has a certain gap at this time. In order to calculate the seal force, this paper adopts the method of integrating the force on the bristle, and splits the bristle pack into three portions. In the process of rotor eccentricity, there is no interaction between the brush with hysteresis effect and the rotor. Therefore, it is necessary to integrate the

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Fig. 2. Schematic of rotor under hysteresis effect

supporting force of the rotor. Expression of supporting force F Ns of the bristle s is Eq. (1): ⎧ qf L3 ⎪ 2EI (θB1 + 6EI ) ⎪ ⎪ , ϕ ∈ [0,  − ϒ] ∪ [ + ϒ, 2π ] ⎪ ⎪ ⎨ L2 (cos θB1 +μ cos θ sin θB1 ) s     ϕs ∈  −  2,  +  2 FNs = 0, ⎪ ⎪ qf L3 ⎪     ⎪ ⎪  ∪  +  ,  + ϒ ⎩ 2 2EI (θB1 − 6EI ) , ϕ ∈  − ϒ,  − s 2 2 L (cos θ +μ cos θ sin θ ) B1

B1

(1) And the bristle is subjected to friction force of the rotor surface, as shown in Fig. 3. And then, the friction force is decomposed into two directions: Ff 2 = μFNs sin θ (2) Ff 1 = μFNs cos θ Suppose the rotor rotates counterclockwise, the force of bristles at different positions is shown in Fig. 4. The first type direction of friction force deviates from the recovery direction of bristles. The second type of bristles are affected by the friction force as a whole, and the end is also affected by the friction and support force of the rotor surface. Since the bristles with hysteresis effect have been out of contact with the rotor, the force of the bristles within the hysteresis angle are not needed to be calculated when analyzing the seal force of the brush seal on the rotor. The absolute coordinate system X 0 O0 Y 0 was established with the rotor center as the origin and the cross section as the coordinate plane.

Numerical Analysis of Seal Force Under Brush Seal Hysteresis Effect

Fig. 3. Stress on single bristle

Fig. 4. Bristles under three conditions

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According to the partial force of single bristle, the supporting reaction force of all the bristles on the rotor is integrated, and the whole circle bristles acting force can be obtained. Equation (3) is the brush seal force: ⎧

+ϒ

−φ/2

2π

−ϒ ⎪ ⎪ Fx = p( FNx d ϕs + FNx d ϕs + FNx d ϕs + FNx d ϕs ) ⎪ ⎪ ⎪ +φ/2 −ϒ +ϒ 0 ⎪ ⎪ ⎪

+ϒ

−φ/2

2π

−ϒ ⎨ Fy = p( FNy d ϕs + FNy d ϕs + FNy d ϕs + FNy d ϕs ) ⎪ +φ/2 −ϒ +ϒ 0 ⎪ ⎪ ⎪

+ϒ

−φ/2

2π

−ϒ ⎪ ⎪ ⎪ ⎪ FN d ϕs + FN d ϕs + FN d ϕs + FN d ϕs ) ⎩ Fz = p( +φ/2

z

−ϒ

z

+ϒ

z

0

z

(3) where p is bristle layer number.

3 Result and Discussion 3.1 Influence of Hysteresis Effect on Brush Seal Force In this section, the effect of hysteresis on the magnitude of the seal force is investigated when axial force is considered. When hysteresis effect of bristle is not considered or considered, the component force of brush seal force changes with rotor eccentricity and rotor eccentricity direction, as shown in Fig. 5 and Fig. 6, respectively. Figure 5 shows the case without hysteresis effect, and Fig. 6 shows the case with hysteresis effect. When hysteresis effect is not considered, it is found through the analysis of Fig. 5 that the seal force component also exists when the rotor does not have eccentricity. This is due to the axial force, resulting in the seal force at the equilibrium position. In other eccentric directions, the seal force will also increase with the increase of eccentricity.

a) Change trend of brush seal force on x0

b) Change trend of brush seal force on y0

Fig. 5. Change of brush seal force without considering hysteresis effect

When hysteresis effect is considered, it is found through the analysis of Fig. 6 that in any eccentricity direction, the seal force will first decrease with the increase of eccentricity, and then increase with the increase of eccentricity after the rotor reaches the

Numerical Analysis of Seal Force Under Brush Seal Hysteresis Effect

a) Change trend of brush seal force on x0

7

b) Change trend of brush seal force on y0

Fig. 6. Variation of brush seal force when hysteresis effect is considered

equilibrium position. In the case of hysteresis effect, the balance position of the rotor will change, and with the change of the balance position, the seal force will also change, and the direction of the seal force will change. According to the comparative analysis of Fig. 5 and Fig. 6, when hysteresis effect is considered, the seal force will become smaller, and the changing trend of seal force will become more complex. At the same time, the zero rotor eccentricity angle and the maximum seal force on x 0 and y0 do not appear at 0º, 90º, 180º, and 270º due to the presence of the bristle installation angle, resulting in some deviation. Whether hysteresis is considered or not, x 0 has a zero seal force near 70º and 250º. Zero seal force occurs near 160º and 340º of y0 . And the maximum seal force on x 0 corresponds to the zero value of seal force on y0 ; And the maximum seal force on y0 corresponds to the seal force zero value on x 0 . 3.2 Influence of Brush Inclination Angle on Seal Force Under Hysteresis Effect This section studies the influence of brush inclination angle on seal force considering hysteresis effect, as shown in Fig. 7(a) and Fig. 7(b), that is, the size of brush seal force F x0 and F y0 when the brush inclination angle is 30°. Figure 7(c) and Fig. 7(d) show the size of brush seal force F x0 and F y0 when the brush inclination angle is 45°. As can be seen from Fig. 7, the smaller the brush inclination angle is, the larger and more unbalanced the brush force will be. This is because the smaller the brush inclination angle is, the greater the rotor contact force will be on the brush when the rotor is eccentric, thus leading to the greater the seal force. The influence of brush inclination angle on the rotor force balance point is consistent with the hysteresis angle and other parameters. In any eccentric direction, the seal force decreases first with the increase of eccentricity, and then increases with the increase of eccentricity when the rotor balance is reached, that is, when the seal force is 0. In addition, the seal force decreases significantly as the angle increases from 30º to 45º, while the seal force decreases only slightly during 45º to 60º. Combined with the force and seal force analysis of single bristle, it is found that the inclination angle of bristle is reasonable to set at about 45º.

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According to the analysis, it is found that the increase of brush angle will make the seal force gradually decrease, in addition, it will also make the seal force change range decrease, which is conducive to improving the stability of the rotor system. However, too large brush angle will lead to too small seal force, reduce the seal performance of brush seal, so the bristle angle is not the smaller the better, but there is a reasonable choice of interval.

a) Change trend of Fx0 when γ is 30º

b) Change trend of Fx0 when γ is 30º

c) Change trend of Fx0 when γ is 45º

d) Change trend of Fx0 when γ is 45º

e) Change trend of Fx0 when γ is 60º

f) Change trend of Fx0 when γ is 60º

Fig. 7. Change trend of F x0 and F y0 at different brush inclination angles

Numerical Analysis of Seal Force Under Brush Seal Hysteresis Effect

9

4 Conclusions 1. When hysteresis effect is considered, the seal force component is significantly smaller than that without hysteresis effect, and its variation rule becomes more complex. At the same time, the zero rotor eccentricity angle and the maximum seal force on x 0 and y0 do not occur at 0º, 90º, 180º, and 270º due to the presence of the bristle mounting angle, resulting in some deviation. Whether hysteresis effect is considered or not, x 0 has zero seal force near 70º and 250º. y0 has zero seal force near 160º and 340º. And the maximum seal force on x 0 corresponds to the zero value of seal force on y0 , the maximum seal force on y0 corresponds to the seal force zero value on x 0 . 2. With the increase of brush angle, the seal force decreases gradually, and the variation range of seal force decreases, which is conducive to improving the stability of the rotor system. However, too large brush angle will lead to too small seal force, reduce the sealing performance of brush seal. So the bristle angle is not the smaller the better, but there is a reasonable choice of interval. Acknowledgements. The research was supported by the National Natural Science Foundation of China (No. 11802168, No. 52075310).

References 1. Black, H.F., Jenssen, D.N.: Effects of high-pressure ring seals on pump rotor. ASME Paper, 71-WA/FF-38 (1971) 2. Nordmann, R., Diewall, W.: Dynamic analysis of turbo-pump rotors with fluid-mechanical interactions. In: Proceedings of International Conference on Mechanical Dynamics (1987) 3. Childs, D.W.: Dynamic analysis of turbulent annular seals based on Hirs’ lubrication equation. J. Lubr. Technol. 105(3), 429–436 (1983) 4. Childs, D.W.: Finite-length solutions rotor dynamic coefficients of turbulent annular seals. J. Lubr. Technol. 105(3), 437–444 (1983) 5. Nelson, C.C.: Rotor dynamic coefficients for compressible flow in tapered annual seals. J. Tribol. 107(3), 318–325 (1985) 6. Nelson, C.C., Nguyen, D.T.: Comparison of Hirs’ equation with moody’s equation for determining rotor dynamic coefficients of annular pressure seals. In: The 4th Workshop on Rotor Dynamic Instability Problems in High Performance Turbo Machinery, Taxas A&M University, pp. 189–203 (1986) 7. Muszynska, A., Bently, D.E.: Frequency-swept rotating input perturbation techniques and identification of the fluid force models in rotor/bearing/seal systems and fluid handling machine. J. Sound Vibr. 143(1), 103–124 (1990) 8. Li, Z.G., Chen, Y.S.: Research on 1:2 subharmonic resonance and bifurcation of nonlinear rotor-seal system. Appl. Math. Mech. (English Edition) 33(4), 499–510 (2012). https://doi. org/10.1007/s10483-012-1566-7 9. Li, S.T., Xu, Q.Y., Zhang, X.L.: Nonlinear dynamic behaviors of a rotor-labyrinth seal system. Nonlinear Dyn. 47(4), 321–329 (2007). https://doi.org/10.1007/s11071-006-9025-0 10. Hua, J., Swaddiwudhipong, S., Liu, Z.S., Xu, Q.Y.: Numerical analysis of nonlinear rotor-seal system. J. Sound Vibr. 283(3–5), 525–542 (2005)

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11. Ding, Q., Cooper, J.E., Leung, A.Y.T.: Hopf bifurcation analysis of a rotor/seal system. J. Sound Vibr. 252(5), 817–833 (2002) 12. Wang, W.Z., Liu, Y.Z., Meng, G., Jiang, P.N.: Nonlinear analysis of orbital motion of a rotor subject to leakage air flow through an interlocking seal. J. Fluids Struct. 25(5), 751–765 (2009) 13. Modi, V.: Modeling bristle lift-off in idealized brush seal configurations. In: Seals Flow Code Development Workshop, Honolulu (1992) 14. Basu, P., Datta, A., Loewenthal, R.: Hysteresis and bristle stiffening effects in brush seals. J. Propul. Power 10(4), 569–575 (1994) 15. Aksit, M., Tichy, J.: A computational model of brush seal bristle deflection. In: Joint Propulsion Conference and Exhibit, New York (1996) 16. Stango, R.J., Zhao, H., Shia, C.Y.: Analysis of contact mechanics for rotor bristle interface of brush sea. ASME J. Tribol. 125(2), 414–421 (2003) 17. Crudgington, P., Bowsher, A.: Brush seal blow down. In: AIAA Joint Propulsion Conference and Exhibit, New York, vol. 4697, pp. 20–23 (2003) 18. Lelli, D., Chew, J.W., Cooper, P.: Combined 3D fluid dynamics and mechanical modelling of brush seals. ASME J. Turbomach. 128(1), 188–195 (2006) 19. Crudgington, P., Bowsher, A., Walia, J.: Bristle angle effects on brush seal contact pressures. In: 45th AIAA Joint Propulsion Conference and Exhibit, vol. 5168, pp. 2–5 (2009) 20. Pekris, M.J., Franceschini, G., Gillespie, D.R.H.: An investigation of flow, mechanical, and therma performance of conventional and pressure-balanced brush seals. J. Eng. Gas Turbines Power 136(6), 062502 (2014) 21. Wei, Y., Chen, Z., Dowell, E.H.: Nonlinear characteristics analysis of a rotor-bearing-brush seal system. Int. J. Struct. Stab. Dyn. 18(5), 1850063 (2018) 22. Wei, Y., Liu, S.: Numerical analysis of the dynamic behavior of a rotor-bearing-brush seal system with bristle interference. J. Mech. Sci. Technol. 33(8), 3895–3903 (2019). https://doi. org/10.1007/s12206-019-0733-z

Investigation on Nonlinear Behavior of a Rotor System with Friction Effect Due to End-Face Seal Li Hou1 , Pingchao Yu1(B) , and Cun Wang2 1 College of Civil Aviation, Nanjing University of Aeronautics and Astronautics,

Nanjing 211106, People’s Republic of China [email protected] 2 Beijing Power Machinery Institute, Beijing 100074, People’s Republic of China

Abstract. The end-face seal is widely used in rotating machinery to minimize leakage between the rotating shaft and the housing, and it requires consideration in the system dynamics for successful design. In this paper, a nonlinear friction model of end-face seal considering rotor swing motion is proposed. By introducing this model, the nonlinear dynamic equations of an offset-disk rotor system with end-face seal are established. The effects of seal structure on rotordynamics are studied in detail from the perspective of mode and response. The results show that the response behavior of rotor system depends on the relative relationship between the input energy of sealing load and the energy consumed by system damping. Through the analysis of the key parameters, it is found that one can enhance the stability of the rotor-seal system by reducing contact radius, friction coefficient, and axial stiffness of seal structure or improving the damping ratio of rotor system. Keywords: end-face seal · offset-disk rotor system · Nonlinear excitation · vibration response · instability

1 Introduction Aero-engine is a kind of high-speed rotating power machinery. There are a large number of moving-static contact surfaces in the structure. The gap between the contact surfaces not only brings losses to the engine performance, but also worsens the working conditions and environment, directly affects the life and reliability of the components, and even causes damage to the components. Therefore, an efficient sealing device is needed to perform moving-static sealing on the mechanical interface. The sealing performance of the sealing structure has a great influence on the working efficiency of the engine components and the whole machine [1–5]. Domestic and foreign scholars have carried out many studies on the design, mechanical properties and sealing performance of the sealing structure. Zhang [6] studied the influence of sealing ring structure size on the deformation of sealing contact end face. Jacobs [7] proposed a simulation design and optimization method for microstructured mechanical end face seals. Xie [8] studied the relationship between the surface tensile stress, the main interface © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 11–33, 2024. https://doi.org/10.1007/978-3-031-40459-7_2

12

L. Hou et al.

shear stress, the side interface shear stress and the normal tensile stress of the sealing end face and the performance of the coating end face. Li [9] studied the influence of spring stiffness change on the end face contact mechanical seal. Considering the change of contact pressure caused by different wear states, Magyar [10] studied the contact behavior of seal based on finite element method. Long [11] studied the influence of different friction states on the performance of mechanical seals. Sun [12] analyzed the influence of spring pressure on balancing the mechanical sealing frictional characteristics and sealing ability. The above research is mainly aimed at the study of the seal itself. In practice, the seal structure is used in conjunction with the rotor/stationary components and works together. The end face seal can produce a non-negligible excitation load on the rotor, resulting in complex dynamic behavior of the rotor. Li [9] studied the influence of the sealing system on the vibration of a compressor rotor. The results show that the correlation coefficient of the seal will cause the rotor instability. In [13, 14], the effects of rotor inertia, liquid film stiffness and damping on rotor dynamics in a double flexible rotor end face seal system were studied. Liu [15] pointed out that in the rotor-bearing-seal system, oil film force, fluid excitation force in the seal system, rubbing, dry friction and many other factors affect the stability of the rotor system. Sun [16] established a two-degree-of-freedom friction vibration system model of dry gas seal under dry friction condition, and revealed the friction vibration law between the end faces of dry gas seal. Zhang [17] established the bending-torsion coupling dynamic equation of the rotor considering the friction of the contact surface for the turbine pump rotor-graphite contact seal system, and analyzed the influence of the nonlinear force of the seal on the vibration response of the rotor. The results show that the friction of the graphite seal enhances the coupling of the bending and torsional vibration of the rotor and may cause the instability of the rotor system. Although many studies have been carried out on the dynamic characteristics of the rotor-seal system, there are many types of seals involved in the actual rotating machinery, and the seal structures are different. Their working principles and effects on rotor dynamics will also be quite different. The contact end face seal of graphite structure is a widely used sealing device in aero-engines. It usually contains one or more pairs of graphite contact ends perpendicular to the rotating axis. The graphite structure is under the elastic force of the compensation mechanism and cooperates with the auxiliary sealing structure to prevent fluid leakage [18–20]. In the working process of the rotor, the sealing structure which relies on the preload of the contact end face will produce the excitation force to the transverse vibration of the rotor due to the uneven distribution of the pressing force with the vibration of the rotor, thus inducing the complex vibration behavior of the rotor. However, the relevant dynamic mechanism has not been revealed, and the dynamic model of the rotor-sealing system is lacking. In view of this, this paper takes the end face seal of graphite-rotor system in aero-engine as the background, and takes the simplified single-disk biased rotor as the object. The nonlinear excitation force model of end face of graphite seal is proposed, and the dynamic model of rotor system under end face seal is established. On this basis, the influence of end face seal of graphite and its key parameters on rotor dynamic behavior and stability is analyzed.

Investigation on Nonlinear Behavior of a Rotor System

13

2 Modeling of Offset Disc Rotor-End Seal System As shown in Fig. 1, it is a typical end face seal of graphite structure in aero-engine, including graphite static ring, sealing moving ring, wave spring, O-ring, etc. The graphite end face sealing static ring is installed in the stator mounting seat, the sealing moving ring is installed on the rotor shaft to rotate with the rotor, and the wave spring behind the graphite static ring provides the axial compression force of the graphite static ring and the sealing moving ring to realize contact seal.

Static subdivisions

O-ring Seal the moving ring

Mounting bracket Graphite rings

Wave springs

Fig. 1. Sealing structure diagram of rotating/static end face

When the rotor works, relative motion occurs between the static ring and the moving ring, and the contact surface produces contact friction. Because of the uniform pressing force, the total friction force is zero, which does not affect the lateral vibration of the rotor. However, when the rotor vibrates under the action of unbalance, the end face seal of graphite will produce transverse excitation load on the rotor. Due to the rotor swing, there is axial relative motion between the graphite static ring and the sealing moving ring, and the positive pressure between them presents non-uniform distribution, and at the same time it will produce non-uniform friction. This non-uniform friction will form a transverse excitation force on the rotor, and the excitation has nonlinear characteristics, which significantly changes the dynamic characteristics of the rotor and may induce rotor vibration instability. 2.1 Mechanical Model of Rotary/Static End Seal In order to analyze the influence of end face seal of graphite on rotor dynamics, a rotor system model of offset disk with end face seal is established, as shown in Fig. 2. The non-rotating component of graphite sealing structure is simplified as the axial spring-rigid graphite ring unit shown in Fig. 2. The axial springs are evenly distributed along the circumferential direction and the total axial stiffness is k s , which represents the axial stiffness generated by the graphite ring and its pressing device. The graphite ring is simplified as a rigid ring with infinite stiffness and its mass and inertia are ignored. The transverse and torsional stiffness of the non-rotating assembly of graphite sealing structure is considered infinite, that is, it is considered that the transverse and torsional

14

L. Hou et al. ks e

x

y

z y

m Jp Jd

N0

Graphite rings

x o

rs

Fig. 2. Model of Offset Disc Rotor System with End Face Seal

motion does not occur during the contact with the rotating structure, and the graphite static ring only has axial freedom. In the figure, the wheel disc represents the moving ring of the sealing structure, which is considered to be rigid and integrated with the rotor. The rotor speed in the working process is ω, only considering the transverse vibration of the rotor, so the degree of freedom of the moving ring includes x, y, θ x , θ y . The static ring is pressed in contact with the moving ring (disk) under the initial pressing force N 0 . It is assumed that the contact between the static ring and the moving ring is linear contact, and the radius of the contact circle in the static state is r s . The mechanical parameters of the mechanical model of rotary/static end face seal include axial stiffness k s , friction coefficient of contact surface μ, initial pressing force N 0 and contact radius r s . In practice, these parameters are determined by the specific structure and material properties of graphite seal. When the rotor does not vibrate, the friction force between the moving ring and the static ring is evenly distributed, and the resultant force is 0. Subjected to initial pressing force N 0, the initial compression displacement of the static ring is as shown in Eq. (1): z0 =

N0 ks

(1)

When the rotor vibrates, the moving ring will move and swing, and the pressing force between the static ring and the moving ring shows a non-uniform distribution (Fig. 3). For any circumferential position α of the infinitesimal d α, the axial displacement caused by the vibration of the moving ring is expressed as Eq. (2): z = θx (rs sin α − y) − θy (rs cos α − x)

(2)

Among them, r s is the average radius of the static ring. Then the positive pressure of the corresponding infinitesimal at the circumferential position α is expressed as Eq. (3):  ks (z + z)d α (z0 + z) ≥ 0 dN (α) = 2π 0 (3) 0 (z0 + z) < 0 Due to the relative motion between the moving ring and the static ring, based on the Coulomb friction law, the friction expression at the contact surface is obtained as shown in Eq. (4): d fs = μdN (α)t(α)

(4)

Investigation on Nonlinear Behavior of a Rotor System

15

Among them, t(α) is the unit direction vector, and its direction is opposite to the relative velocity direction of the moving ring and the static ring. For the infinitesimal at the circumferential position α, the relative motion velocity of the moving ring and the stationary ring is expressed as Eq. (5): v = (˙x − ω(rs sin α − y))i + (˙y + ω(rs cos α − x))j

(5)

where, i and j are unit vectors of x direction and y direction respectively. The t(α) is expressed as Eq. (6): t(α) = − = +

v v

−(˙x − ω(rs sin α − y))

i

(˙x − ω(rs sin α − y))2 + (˙y + ω(rs cos α − x))2 −(˙y + ω(rs cos α − x))

j

(6)

(˙x − ω(rs sin α − y))2 + (˙y + ω(rs cos α − x))2

Substituting Eqs. (5) and (6) into Eq. (4), and decompose the friction force to obtain the expression of the friction force of the infinitesimal in the x and y direction as shown in Eqs. (7) and Eq. (8): −μ(˙x − ω(rs sin α − y))dN (α) dfsx =  (˙x − ω(rs sin α − y))2 + (˙y + ω(rs cos α − x))2

(7)

−μ(˙y + ω(rs cos α − x))dN (α) dfsy =  (˙x − ω(rs sin α − y))2 + (˙y + ω(rs cos α − x))2

(8)

In addition to the friction force, the non-uniform positive pressure also generates a bending moment excitation to the rotor. The bending moment generated by the infinitesimal at the circumferential position α is expressed as Eqs. (9) and (10): dMsx = −(rs sin α − y)dN (α)

(9)

dMsy = (rs cos α − x)dN (α)

(10)

The resultant force and resultant moment of the graphite sealing ring to the moving ring are obtained by integrating the Eqs. (7) and (8) along the circumferential direction, as shown in Eq. (11): ⎧   −μ(˙x−ω(rs sin α−y))dN (α) Fsx = s dfsx = s √ ⎪ ⎪ ⎪ (˙x−ω(rs sin α−y))2 +(˙y+ω(rs cos α−x))2 ⎪   ⎨ −μ(˙y+ω(rs cos α−x))dN (α) Fsy = s dfsy = s √ 2 (11) y+ω(rs cos α−x))2 x −ω(r (˙ s sin α−y)) +(˙   ⎪ ⎪ ⎪ M = dM = −(r sin α − y)dN (α) sx sx s ⎪ s s ⎩ Msy = s dMsy = s (rs cos α − x)dN (α)

16

L. Hou et al. y Moving rings

dα

ω

y α x

x

Non-uniform positive pressure

Fig. 3. Distribution of pressing force between moving ring and static ring when rotor vibrates

2.2 Dynamic Equation of Offset Disc Rotor-End Seal System Section 2.1 gives the mechanical model of the end face seal. In this section, the dynamic equation of the offset disc rotor-end seal system is established. For the offset disk rotor system in Fig. 2, the mass of the shaft section is neglected, the mass of the disk (moving ring) is m, the polar moment of inertia is J p , the moment of inertia of the diameter is J d , and the mass bias distance is e; for the offset disk rotor, the disk has both translation and swing during the whirling process of the rotor, and the translation degree of freedom x and y, the swing degree of freedom θ x and θ y are used. Based on the Lagrange energy method, the dynamic equation of the bias disk rotor under the action of unbalance and graphite sealing friction is derived. The generalized coordinate of the rotor is that

T q = x y θx θy , the kinetic energy of the rotor is the translational and swing kinetic energy of the disk, as shown in Eq. (12): E=

1 

m 2 x˙ + y˙ 2 + Jd θ˙x2 + θ˙y2 + Jp ω2 − 2Jp ωθ˙y θx 2 2

The generalized force acting on the rotor system is shown in Eq. (13): ⎧ ⎪ Q1 = −k11 x − k14 θy − c11 x˙ + Fsx + Fux ⎪ ⎪ ⎨ Q2 = −k22 y + k23 θx − c22 y˙ + Fsy + Fuy ⎪ Q3 = −k32 y − k33 θx + Msx ⎪ ⎪ ⎩ Q = −k x − k θ + M 4 41 44 y sy

(12)

(13)

Among them, k ij (i, j = 4) is the equivalent stiffness of the rotor shaft at the position of the disk, k 11 is the force along the x-direction required for the unit displacement in the x-direction of the center of the disk, k 22 is the force along the y-direction required for the unit displacement in the y-direction of the center of the disk, k 33 is the torque of the ox-axis required for the unit rotation angle of the disk around the ox-axis, k 44 is the torque of the oy-axis required for the unit rotation angle of the disk around the oy-axis, k 14 is the force along the x-direction required for the unit rotation angle of the disk around the oy-axis, k 23 is the force along the y-direction required for the unit rotation angle of the disk around the ox-axis, k 32 is the torque of the ox-axis required for the

Investigation on Nonlinear Behavior of a Rotor System

17

unit displacement in the y-direction of the center of the disk. k 41 is the torque applied to the oy-axis when the unit displacement occurs in the x-direction of the disc center. The above unit displacement or unit rotation angle is based on the condition that the displacement or rotation angle in other directions is limited to 0. c11 and c22 are the damping coefficients of the rotor along the x-axis and the y-axis respectively. Fsx , Fsy , Msx and Msy are the load acting on the disk by the sealing structure, Fux , Fuy is the component of the unbalance force of the rotor in x-axial and y-axial directions. According to the Lagrange equation, as Eq. (14): 

∂T d ∂E − = Qj , j = 1, 2, · · · n (14) dt ∂ q˙ j ∂qj The dynamic equation of the offset disc rotor is shown in Eq. (15): ⎧ ⎪ m¨x + c11 x˙ + k11 x + k14 θy = Fsx + Fux ⎪ ⎪ ⎨ m¨y + c22 y˙ + k22 y − k23 θx = Fsy + Fuy ⎪ Jd θ¨x + Jp ωθ˙y − k32 y + k33 θx = 0 ⎪ ⎪ ⎩ J θ¨ − J ωθ˙ + k x + k θ = 0 d y p 41 44 y x

(15)

It is written as a matrix form, as shown in the following Eq. (16): Mq¨ + (C + ωG)q˙ + Kq = F

(16)

In the above dynamic equations, the equivalent stiffness of the shaft can be obtained by Euler beam theory. For the shaft hinged at both ends, the equivalent stiffness is shown in Eq. (17):  2 ⎧ a −ab+b2 ⎪ k = k = 3lEI ⎪ 22 ⎨ 11 a 3 b3  2 a −ab+b2 (17) = k = k = k = 3lEI k 14 41 23 32 3 3 ⎪ a b ⎪ ⎩ k33 = k44 = 3lEI ab where, a, b and l are the length parameters of the shaft.

3 Vibration Response Analysis of Rotor System Based on the mechanical model established in the previous section, this section analyzes the dynamic characteristics of the rotor system under the action of end face seal. 3.1 Rotor System Parameters and Calculation of Inherent Characteristics Firstly, the critical speed and vibration mode of the rotor system are calculated by ignoring the effect of the end face seal, so as to grasp the inherent characteristics of the rotor system. In the vibration response analysis of the rotor system, the mechanical parameters of the rotor are shown in Table 1. Calculate the critical speed and vibration mode of the rotor system and summarize the results, as shown in Table 2. According to the results in the table, the first-order forward/backward precession critical speeds of the rotor are 2337 r/min and 2268 r/min respectively, and the vibration mode is first-order bending. The critical speed of rotor second-order backward precession is 8074 r/min, and the vibration mode is second-order bending.

18

L. Hou et al. Table 1. Design conditions

parameter

numeric value

Roulette parameters

Hinge parameters

Graphite sealing parameters

quality md (kg)

20

damping ratio ε

0.01

Polar moment of inertia Jp (kg*mm^2)

1.44e5

Diameter moment of inertia J d (kg*mm^2)

7.20e4

Eccentric distance e (mm)

5.00e−3

Total length l (mm)

750

The length of the disc with the left support a (mm)

250

The length of the disc with the right bearing b (mm)

500

diameter d (mm)

30

Elastic modulus E (GPa)

205.8

rs (mm)

50

ks (N/mm)

2e3

N0

50

μ

0.3

Table 2. Summary of rotor critical speed

Order

Critical speed

Backward precession

2268r/min/37.8Hz

Forward precession

2337r/min/38.9Hz

Backward precession

8074r/min/134.6Hz

Mode description

3.2 Response Characteristics of Rotor System Under Excitation Force of End Face Seal In this section, the classical Newmark numerical integration method is used to solve the dynamic Eq. (15). The time step is set to 5e−5 s, and the total calculation time is set to 15 s. According to Sect. 3.1, the first-order critical speed of the rotor system is 2340 r/min. The speed less than 2340 r/min is called subcritical speed, and the speed more than 2340 r/min is called supercritical speed.

Investigation on Nonlinear Behavior of a Rotor System

19

3.2.1 Analysis of Rotor System Response Characteristics

x-directional displacement

x-directional displacement

mm)

mm)

The vibration response of the rotating speed system between 500–4500 r/min is calculated, and the steady-state response of the 1/4 time period is extracted and represented by bifurcation diagram and three-dimensional waterfall diagram, as shown in Fig. 4 and Fig. 5. It can be seen from the figure that the vibration response of the rotor system is a single-period response without considering the seal, and the response frequency only includes the speed frequency, and the rotor system resonates at 2340 r/min. When the sealing effect is considered, the response of the rotor system is a quasi-periodic response. The response frequency includes the speed frequency composition, the first-order forward precession modal frequency component and the second-order reverse precession modal frequency component. In the whole speed range, the amplitude corresponding to the first-order forward precession frequency component is much higher than other frequency components. In the supercritical speed range, the second-order reverse precession modal frequency component is relatively significant, and the higher the speed, the higher the amplitude.

rotate speed r/min)

rotate speed r/min

(a) The effect of sealing is not considered

(b) The effect of sealing is considered

mm

Amplitude(mm)

Fig. 4. Bifurcation diagram of rotor vibration displacement in x direction with rotational speed

Amplitude

2340 r/min

1×f

rotate speed(r/min)

1×f fn1

Frequency Frequency(HZ)

(a) The effect of sealing is not considered

fn2-

HZ rotate speed

r/min

(b) The effect of sealing is considered

Fig. 5. Three-dimensional waterfall diagram of rotor system (x-axis vibration displacement)

20

L. Hou et al. Stabilizati on phase Divergent phase

(a) Time-domain curve and its wavelet transform plot at 0-15s stage

fn1 f

(b) Response characteristics within 0 to 1 s of the initial divergence phase

fn1+ Precession direction

(c) Response characteristics within 14 to 15 s of the stabilization phase Fig. 6. Time-frequency response characteristics of rotor system at subcritical speed of 1000 r/min

The rotor time-frequency response and axis orbit at typical speed are further extracted. The response characteristics of the rotor system at the subcritical speed of 1000 r/min are analyzed, as shown in Fig. 6. It can be seen that the response of the rotor system diverges in the initial time period, and diverges faster in 5–10 s, and gradually tends to be stable after 10 s, and the vibration response amplitude is about 0.5 mm. The response evolution is dominated by the first-order forward precession mode. In the initial stage, the system response contains the speed frequency and the significant first-order

Investigation on Nonlinear Behavior of a Rotor System

21

positive precession mode frequency composition. The amplitude of the first-order forward precession gradually increases with time. After 5 s, its amplitude is much higher than the speed frequency composition, which has become the dominant rotor response. After 10 s, the amplitude of the first-order positive precession mode frequency tends to be stable, and the rotor response is basically stable. The response characteristics of the rotor system at the critical speed of 2340 r/min are shown in Fig. 7. It can be seen from Fig. 7 that the system response is diverging first and then stable, and the response frequency is 39 Hz. At the critical speed, the response characteristics of the seal-rotor system are consistent with the response characteristics of the rotor system without seal, but the response amplitude of the rotor is higher when considering the seal effect, and the seal load can significantly increase the vibration of the rotor at the critical speed.

(a)Time domain curve for the 0-15s phase

(b)Wavelet transform plot at the 0-15s stage

Fig. 7. Time - frequency response characteristics of rotor system at critical speed of 2340 r/min

The vibration response of the rotor system at supercritical speed of 3600 r/min is shown in Fig. 8. It can be seen from the figure that the vibration amplitude of the rotor system gradually increases within 0–9 s, and reaches the maximum at about 9 s. In this stage, the vibration response of the rotor system is dominated by the first-order forward precession mode, and the amplitude corresponding to the first-order forward precession mode frequency gradually increases, resulting in a gradual increase in the vibration response of the system. After 10 s, the amplitude of the rotor vibration decreases first, reaches the minimum after about 14 s, and then shows an increasing trend. After 10 s, the amplitude corresponding to the first-order forward precession modal frequency in the system gradually decreases, resulting in the decrease of the response amplitude of the system. In this stage, the second-order backward precession modal frequency component gradually increases and gradually dominates the response of the rotor system. Taking the response characteristics of 14–14.2 s as an example, the amplitude of the second-order backward precession mode frequency component is the highest in the system frequency component, and the axis orbit indicates that the whirling direction of the rotor system is opposite to the speed direction, that is, the system response is dominated by the second-order backward precession mode.

22

L. Hou et al.

fn2-:

f:

fn1+ :

(a) Time-domain curve and its wavelet transform plot at 0-15s stage fn1+ Afn1+=0.01 3

f Af=0.009

(b) Response characteristics within 0 to 1 s in the initial stage

fn1+ Afn1+=0.38

Precession direction fn2-

(c) Response characteristics within 8 to 9 s

fn2-

Precession direction

fn1+ Afn1+=0.02 f Af=0.009

(d) Response characteristics within 14 to 14.2 s

Fig. 8. Time-frequency response characteristics of rotor system at supercritical speed of 3600 r/min

Investigation on Nonlinear Behavior of a Rotor System

23

3.2.2 Friction Force of Face Seal and the Work Analysis The analysis in Sect. 3.2.1 shows that the end face seal makes the rotor system present complex vibration response. This section focuses on the analysis of the friction load of the end face seal, and further reveals the influence of the end face seal on the vibration response of the rotor system from the perspective of the work of the seal load. Firstly, the calculation method of the work done by the load is given. For the excitation load acting on the vibration system, the work done by the system in one speed cycle is shown in Eq. (18).  WF =

Tr

 F(t)dq =

0

Tr

F(t)˙q(t)dt

(18)

0

Friction force (N)

x y

Displacement y (mm)

where, q(t) and q˙ (t) represent the displacement and velocity of the vibration system respectively. Firstly, the load characteristics of end face seal at sub-critical speed of 1000 r/min are analyzed, as shown in Fig. 9. It can be seen that when the sub-critical speed is 1000 r/min, the friction direction of the end face seal is always consistent with the rotor precession direction, indicating that the friction force of the end face seal does positive work on the rotor and inputs vibration energy into the rotor system.

Time (s)

Whirl direction

Friction force direction

Displacement x (mm)

(a) Frictional loads within 2-2.12s and their relationship to the axial trajectory

Time (s)

Friction force direction Displacement y (mm)

Friction force (N)

x y

Whirl direction

Displacement x (mm)

(b) Frictional loads within 14-14.12s and their relationship to axial trajectories Fig. 9. The friction force and work of end face seal at speed of 1000 r/min

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For the rotor system, the friction force inputs the vibration energy to the rotor system, and the initial damping of the rotor plays a role in dissipating the energy of the system. The relationship between the two determines whether the vibration energy or vibration amplitude of the system increases or decreases. Calculate and summarize the work done by the friction force of the end face seal at different times and the work done by the system damping force, as shown in Table 3. In order to facilitate comparative analysis, the work of sealing friction force is taken as the benchmark for normalization. The value in the table is positive, indicating positive work, and the value is negative, indicating negative work. It can be seen that in the initial stage, the work of the sealing friction force is higher than that of the rotor damping force, indicating that the work dissipated by the damping force is not enough to balance the positive work done by the friction force, so the vibration energy of the system gradually increases, which is shown by the increase of the vibration amplitude in Fig. 6. After 14 s, the work of the sealing friction force and the work of the rotor damping force are basically balanced, and the vibration response of the system is basically stable, that is, the stable stage of the rotor vibration response in Fig. 7. Table 3. Work on friction force and damping force of end face seal (1000 r/min) Time (s)

Average work in a rotational cycle (10-3J)

Normalized value of work

Seal the friction

Rotor damping force

Seal the friction

Rotor damping force

2–2.12

4.65E−03

−3.60E−03

1

−0.77

14–14.12

101.7

−101.5

1

−1.00

The friction load and its work at supercritical speed of 3600 r/min are further analyzed, as shown in Fig. 10. From the direction of the friction force and the direction of the rotor precession, the friction force always does positive work to the rotor system at this speed. It can be seen from the time domain curve of friction that the frequency component of friction load gradually changes from low frequency to high frequency over time, which is consistent with the evolution of rotor vibration response. The work of friction and damping force at different time periods is calculated, and the calculation results are shown in Fig. 11. It can be seen from the figure that in general, the positive work done by the friction force is always greater than or equal to the negative work done by the damping force, indicating that the system energy has been increasing. Specifically, it can be divided into two stages: The first stage is within 0–10 s. In this stage, the work of the rotor damping force relative to the work of the friction force shows a trend of decreasing first and then increasing. In the 0–5 s stage, the work of the rotor damping force relative to the work of the friction force gradually decreases. At this stage, the energy of the rotor system increases rapidly, which is manifested by a rapid increase in amplitude (according to the above analysis, it is mainly the amplitude of the first-order forward precession modal frequency). In the 5 s–9 s stage, the work of the rotor damping force relative to the work of the friction force gradually increases. At this time, the system energy/amplitude still shows an increasing trend, but the increase rate slows down. The amplitude is relatively stable. After 10 s, the relative balance between

Friction force (N)

x y

Displacement y (mm)

Investigation on Nonlinear Behavior of a Rotor System

25

Whirl direction Friction force direction

Time (s)

Displacement x (mm)

(a) Frictional loads within 2-2.08s and their relationship to axial trajectories

Friction force (N)

Displacement y (mm)

x y

Time (s)

Friction force direction Whirl direction

Displacement x (mm)

(b) Frictional loads within 5-5.08s and their relationship to the axial trajectory

Friction force (N)

Displacement y (mm)

x y

Time (s)

Whirl direction Friction force direction

Displacement x (mm)

(c) Frictional loads within 14-14.08s and their relationship to axial trajectories Fig. 10. The friction force and work of end face seal at speed of 3600r/min

the work of the friction force and the work of the damping force is broken again, and the work of the damping force decreases with time, so the vibration energy of the rotor system increases again and the growth rate increases gradually. Although the vibration energy of the rotor increases at this stage, the amplitude does not increase, but mainly the increase of the vibration frequency, that is, the frequency component of the second-order backward precession mode increases continuously. The specific evolution process of the response is shown in Fig. 8.

L. Hou et al. Do work (normalized value)

26

The work of friction The work of the damping force

Time(s)

Fig. 11. The work of friction and damping force at different times

4 The Influence of Key Parameters on Rotor Vibration Response On the basis of the previous section, this section studies the influence of the key parameters of the system on the dynamic characteristics of the rotor, including rotor parameters and sealing parameters. The rotor parameters include unbalance of rotor and rotor damping. The sealing parameters include sealing contact radius, pressing force, axial stiffness and contact surface friction coefficient. 4.1 Influence of Rotor Parameters 4.1.1 Unbalance of Rotor The influence of the unbalance of rotor on the vibration response of the rotor is calculated. The results are represented by the rotor bifurcation diagram and the three-dimensional waterfall diagram of the rotor system, as shown in Fig. 12. From Fig. 12, it can be seen that when the rotor unbalance is 10 g*mm, the response of the rotor system is dominated by the first-order modal vibration in the whole speed range, and the firstorder forward precession modal frequency is dominant in the frequency domain, and a certain second-order backward precession modal frequency component is also included in the supercritical speed. When the rotor unbalance is increased to 1000 g*mm, the rotor system is characterized by the vibration of the first-order forward precession mode at the subcritical speed. At the supercritical speed, it is characterized by the quasiperiodic vibration formed by the superposition of the first-order positive precession mode component, the unbalanced response composition and the second-order backward precession mode composition. Near the critical speed, the rotor is a periodic vibration response dominated by the speed frequency. At this time, the graphite sealing load does not excite the unstable rotor modal frequency composition. The reason is that the rotor vibration energy is increased, which can improve the modal stability.

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27

fn2fn1+ rotate speed rotate speed

r/mi n

r/mi n

(a) Unbalance amount is 10g*mm

f

fn2-

fn1+ Frequency

rotate speed

HZ

rota te speed

r/min

r/min

(b) Unbalance amount is 1000g*mm Fig. 12. Effect of Unbalance on Vibration Response of Rotor

4.1.2 Rotor Damping The rotor vibration response under different rotor damping is calculated, and the results are shown in Fig. 13. Combined with the results when the damping ratio is 0.01 in Figs. 4 and 5, it can be seen that the vibration response of the rotor decreases significantly with the increase of the damping ratio of the rotor. When the damping ratio is 0.05, the vibration response of the rotor at subcritical speed and critical speed is a periodic oscillation dominated by the rotational speed frequency, and the vibration response at supercritical speed is a quasi-periodic oscillation caused by the superposition of the second-order backward precession composition and the unbalanced response composition. When the damping ratio is 0.1, the rotor is dominated by the frequency composition in the whole speed range. The increase of rotor damping ratio is beneficial to improve the modal stability of each order.

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Amplitude(mm )

Amplitude(mm)

28

fn2f

rota te speed

rotate speed

r/min

frequency

r/min

HZ

Amplitude(mm )

Amplitude(mm)

(a) Rotor damping ratio is 0.05

f rota te speed r/min

rotate speed

r/min

frequency

HZ

(b) Rotor damping ratio is 0.1 Fig. 13. Influence of Rotor Damping Ratio on Rotor Vibration Response

4.2 The Influence of Sealing Mechanical Parameters 4.2.1 Sealing Contact Radius The rotor vibration response under different sealing contact radius is calculated, and the results are shown in Fig. 14. When the sealing contact radius is 25 mm, the rotor is a periodic oscillation dominated by speed frequency in the whole speed range. With the increase of the sealing contact radius, the vibration of the rotor is transformed from the periodic oscillation dominated by the speed frequency composition to the vibration dominated by the modal frequency composition, and the higher the sealing radius, the higher the amplitude of the rotor system. This is because the smaller the sealing contact radius of the seal is, the less easily the static ring is separated from the rotor disk under the same vibration degree, and the higher the modal stability is.

Amplitude(mm )

Amplitude(mm)

Investigation on Nonlinear Behavior of a Rotor System

f

rota te speed

rotate speed

29

r/min

frequency

r/min

HZ

Amplitude(mm )

Amplitude(mm)

(a) Sealing contact radius is 25mm

rotate speed

r/min

fn1+

frequency frequency

HZ HZ

fn2-

rota te speed r/min

(b) Sealing contact radius is 75mm Fig. 14. Influence of Sealing Contact Radius on Vibration Response of Rotor

4.2.2 Pressing Force The rotor vibration response under different pressing forces is calculated, as shown in Fig. 15. When the pressing force is 10 N, the vibration of the rotor at the subcritical speed is dominated by the first-order forward composition mode. At the supercritical speed, it is a quasi-periodic oscillation superimposed by the first-order forward precession mode composition, the unbalanced response composition and the second-order backward precession mode composition. Near the critical speed, it is a periodic oscillation dominated by the speed frequency composition. With the increase of the pressing force, the vibration amplitude of the rotor increases significantly, and the vibration is dominated by the modal frequency composition in the whole speed range. The greater the compression force, the more difficult it is to separate the static ring from the rotor disk under the same vibration degree, but the static ring and the rotor disk are always compressed, and the friction force does not change with the increase of positive pressure. The axial pressure has no effect on the stability of the rotor.

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Amplitude(mm )

Amplitude(mm)

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f

fn2-

fn1+ frequency

rotate speed

rota te speed r/min

HZ

r/min

Amplitude(mm )

Amplitude(mm)

(a) Compression force is 10N

rotate speed

r/min

f

fn2-

fn1+ frequency

HZ

rota te speed r/min

(b) Compression force is 100N Fig. 15. Effect of Compression Force on Vibration Response of Rotor

4.2.3 Axial Stiffness The rotor vibration response under different axial stiffness is calculated, as shown in Fig. 16. When the axial stiffness is 1e3 N/mm, the rotor is a periodic oscillation dominated by the speed frequency composition. With the increase of the axial stiffness, the firstorder forward mode and the second-order backward mode of the rotor are unstable, resulting in the transition of the rotor vibration response to the vibration dominated by the modal frequency composition, and the rotor vibration amplitude increases significantly with the increase of the axial stiffness. This is because the lower the axial stiffness of the seal under the same vibration degree, the less likely the static ring is to separate from the rotor disk, the lower the axial stiffness, the higher the stability of the seal rotor.

Amplitude(mm )

Amplitude(mm)

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f

rota te speed

rotate speed

31

r/min

frequency HZ

r/min

Amplitude(mm )

Amplitude(mm)

(a) Axial stiffness is 1e3N/mm

rotate speed

r/min

fn1+ frequency HZ

fn2rota te speed r/min

(b) Axial stiffness is 3e3N/mm Fig. 16. Effect of Axial Stiffness on Vibration Response of Rotor

4.2.4 Friction Coefficient of Contact Surface The rotor vibration response under different friction coefficient of contact surface is calculated, as shown in Fig. 17. When the friction coefficient is 0.1, the rotor vibration response is a periodic oscillation dominated by the speed frequency composition. When the friction coefficient is 0.2, the vibration response of the rotor under supercritical conditions changes, which becomes a quasi-periodic oscillation superimposed by the unbalanced response and the second-order backward modal frequency composition. When the friction coefficient is increased to 0.3, the rotor shows a quasi-periodic motion formed by the superposition of modal composition and speed frequency composition in the whole speed range, and the first-order forward precession modal frequency composition is the main one. The larger the friction coefficient is, the more energy is input to the rotor under the same conditions, which is equivalent to more negative damping, and the modal motion will be unstable. Reducing the friction coefficient can increase the stability of the sealed rotor.

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Amplitude(mm )

Amplitude(mm)

32

f

frequency HZ

rotate speed r/min

rotate speed r/min

Amplitude(mm )

Amplitude(mm)

(a) Friction coefficient is 0.1

f

rotate speed r/min

rotate speed r/min

fn 2-

frequency HZ

(b) Friction coefficient is 0.2 Fig. 17. Effect of friction coefficient on rotor vibration response

5 Conclusion Aiming at the influence of the end face seal of graphite on the dynamic characteristics of rotor system, the nonlinear excitation load of the end face seal of graphite is proposed, and the dynamic model of offset disk rotor with seal effect is established. Based on numerical simulation, the dynamic characteristics and stability evolution rule of seal rotor system are analyzed in detail. The main conclusions are as follows. The swing of the rotor system causes the friction on the sealing contact surface to be nonhomogeneous distributed, forming a transverse excitation load on the rotor. The load always does positive work to the rotor and inputs vibration energy to the system. The response behavior of the rotor system depends on the relationship between the input energy of the sealing load and the energy consumed by the system damping. When the system damping is not enough to dissipate the input energy of the seal load, the steady-state response of the rotor is no longer a periodic response dominated by the rotation frequency, and the vibration response shows a divergent trend in the whole speed range. However, in terms of specific characteristics, it is also related to the rotor speed, at subcritical speed, it is mainly based on the first-order forward precession modal

Investigation on Nonlinear Behavior of a Rotor System

33

frequency composition, while at supercritical speed, it can also excite the second-order backward precession modal frequency composition of the rotor. Through the analysis of the influence law of parameters, it is found that reducing the friction coefficient, sealing radius, axial stiffness of graphite seal and improving the damping of the rotor system itself can significantly increase the stability of the sealed rotor. The axial compression force has no effect on the stability of the rotor.

References 1. Fang, C.: Prospect of aero-engine development. Aero-engine 2004(01), 1–5 (2004) 2. Koop, W.: The Integrated High Performance Turbine Engine Technology (IHPTET) Program. 13th ISABE, Chattanooga (1997) 3. General Editorial Board of ‘Engine Design Manual’. Aeroengine Design Manual, vol. 10, pp. 248–252. Huang Qingnan Sub-Volume Editor-in-Chief. Aviation Industry Press, Beijing (2001) 4. Mayhew, E., Bill, R., Voorhees, W.: Military engine seal development - potential for dual use. In: 30th Joint Propulsion Conference and Exhibit (1994) 5. Alan, O.: Principle and Design of Mechanical Seal, Huang, W., et al. (ed.), pp. 1–13. Mechanical Industry Press, Beijing (2016) 6. Zhang, Y., Zhang, X., Jiang, Z., et al.: Structural design and coupled deformation analysis of ultra-high pressure dynamic seal. Mech. Sci. Technol. 32(11), 1708–1711+1716 (2013) 7. Neumann, S., Jacobs, G., Feldermann, A., et al.: Reducing friction and leakage by means of microstructured sealing surfaces – example mechanical face seal. In: Internationales Fluidtechnisches Kolloquium (2016) 8. Xie, H., Chen, D., Huang, J.: Contact stress analysis of double coating interface. Surf. Technol. 43(02), 1–5+17 (2014) 9. Li, J., He, L., Shen, W., et al.: Effect of sealing system on vibration of a compressor rotor. Turbine Technol. 2005(04), 289–291+312 (2005) 10. Frölich, D., Magyar, B., Sauer, B.: A comprehensive model of wear, friction and contact temperature in radial shaft seals. Wear 311(1–2), 71–80 (2014) 11. Gu, L.W.: Friction mechanism and friction state of end faces of mechanical seals. Lubr. Eng. 2008(06), 38–41 (2008) 12. Sun, J.J., Qin, G.B., Wei, L.: Influence of spring pressure on basic performance of balanced mechanical seals. Petrochemical Equipment 06, 9–12 (2005) 13. Wileman, J., Green, I.: Parametric investigation of the steady-state response of a mechanical seal with two flexibly mounted rotors. J. Tribol. 121(1), 69–76 (1999) 14. Wileman, J., Green, I.: Steady-state analysis of mechanical seals with two flexibly mounted rotors. J. Tribol. 119(1), 200–204 (1997) 15. Liu, Y., Zhu, H., Fan, S., et al.: Nonlinear vibration characteristics of rotor-bearing-seal coupling system. Lubr. Sealing 34(01), 32–35+66 (2009) 16. Sun, B., Ding, X., Yan, R., et al.: Study on high-frequency micro-amplitude friction vibration of sliding seal face of dry gas seal. Lubr. Seal 46(01), 10–18 (2021) 17. Zhang, N., Qian, D., Liu, Z., et al.: Vibration characteristics of turbopump rotor-graphite seal system. J. Southeast Univ. (Nat. Sci. Ed.) 40(03), 512–516 (2010) 18. Yu, Q., Sun, J., Tu, Q., et al.: Research progress in basic properties of contact mechanical seals. Fluid Mach. 43(02), 41–47 (2015) 19. Chen, Z., Cai, Y., Gu, C.: Study on the end face contact characteristics of mechanical seals under dry friction based on fractal theory. Eng. Sci. Technol. 53(03), 188–196 (2021) 20. Wei, L., Gu, B., Feng, X., Feng, F.: Fractal model of end face contact of friction pair of mechanical seal. J. Chem. Eng. 60(10), 2543–2548 (2009)

Analysis and Safety Design of Aero-Engine Rotor Dynamic Response with Multiple Loads Due to Fan Blade off Jie Fu1 , Chao Li2 , Jie Hong1,2(B) , 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

Abstract. After the blade losing, the aero-engine rotor system is subjected to multiple loads, including sudden unbalance excitation, angular acceleration excitation caused by time-varying speed, and rub-impact excitation due to the fan blade off (FBO), which results in a complex dynamic characteristic of the rotor system. At present, the dynamic response mechanism analysis and high-precision simulation results in this case are not complete, which seriously restricts the safety design and verification in the engine development process. This paper analyzes the load characteristics of the rotor when FBO occurs. Then the dynamic equation of inertia asymmetric rotor is carried out considering the combined action of multiple loads. the dynamic response analysis model of the low-pressure rotor of the turbofan engine with the condition of blade loss is established, the dynamic response of inertia symmetric rotor and inertia asymmetric rotor are compared and studied. A safety design idea of the support structure is proposed according to the characteristics of the inertia asymmetric rotor structure. This research shows that after the FBO occurs, the vibration response of the rotor increases sharply when it decelerates passing the critical speed, and the rotor system appears backward whirl frequency with the action of multiple loads. Compared with the inertia symmetric rotor, the inertia asymmetric rotor is more prone to appear backward whirl characteristics. According to the characteristics of inertia asymmetric rotor load with multiple loads, this paper changes the dynamic characteristics of rotor system by reducing the support stiffness, and increases the support damping to improve the energy dissipation, so as to reduce the vibration response of the inertia asymmetric rotor and to ensure the integrity of the support structure after blade loss. Keywords: Rotor dynamics · Fan blade off · Inertia asymmetry · Rub-impact · Safety design

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 34–55, 2024. https://doi.org/10.1007/978-3-031-40459-7_3

Analysis and Safety Design of Aero-Engine Rotor Dynamic Response

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1 Introduction Fan Blade Off (FBO) is a severe safety accident in the aero-engine. The study of FBO has always been an important part of aero-engine safety design. After FBO occurs, the diameter moments of inertia of the blade-disk structure are no longer equal in different directions due to the FBO, and show strong inertia asymmetry. In the meanwhile, the rotor system is subject to the combined action of multiple loads (sudden unbalance excitation, angular acceleration excitation caused by speed change, and rub-impact excitation) [1]. The action mechanism of various loads on the dynamic response of asymmetric rotor system and the interaction mechanism of various loads under strong nonlinear conditions make it very difficult for researchers to accurately obtain the rotor dynamic characteristics with the FBO. For the study of rotor load excitation with FBO, the early Genta [2] took Jeffcott rotor as the research object, and studied the vibration response of the rotor with sudden impact load instead of the load excitation after FBO occurs. For the complex rotor system, Ren Xingmin [3], Gu Jialiu [4] calculated and analyzed the sudden unbalance response by using the improved transfer matrix method. With the in-depth research, it is found that the inertia asymmetry of the disk will occur after the blade is lost. Yu [1] established the dynamic model of the inertia asymmetric disk, and found that the mass matrix and gyroscopic matrix of the inertia asymmetric disk are time-varying. Liu Di [5] established the modal characteristic analysis method of inertia asymmetric rotor through Hill infinite determinant to analyze the dynamic response characteristics of inertia asymmetric rotor considering time-varying speed. Ishida et al. [6] established a complex rotor model and analyzed the impact of inertia asymmetry characteristics on the dynamic characteristics of the rotor system when FBO events occurs. After the blade is lost, the vibration amplitude of the rotor system will increase sharply, which will cause rub-impact between rotor and stator. Rub-impact will produce impact, friction and constraint effects on the rotor system, and make complex vibration phenomena to the rotor system [7]. Beatty [8] established a piecewise linear rub-impact force model for the first time, and studied the typical rub-impact characteristics. Zhang Siqi [9] gave a detailed description of the process of rotor–stator rub-impact, and found that the rotor–stator rub-impact presented nonlinear characteristics through numerical simulation. Liu Di [10] studied the rub-impact characteristics of rotor under sudden unbalance excitation from the perspective of complex nonlinearity, indicating that severe rub-impact will lead to backward whirl. The excessive reaction forces of the support after FBO will seriously threaten the structural integrity of the rotor system [11]. Li Chao [12] designed a buffer damping support structure to reduce the reaction forces of the support of the rotor system with FBO; Hong Jie [13] proposed a safety design strategy, which can effectively reduce the vibration of the fan disk by changing the bearing position and the support stiffness. The above research on the dynamic characteristics of rotor system when FBO occurs is mostly focused on the impact of sudden unbalance load generated by FBO events on the rotor system, while the specific research on the load characteristics of rotor system after blade loss is less. In addition, there is little research on the inertial asymmetry of the rotor with the combined action of multiple loads (sudden unbalance excitation, angular acceleration excitation caused by time-varying speed and rub-impact excitation) after

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FBO occurs, which results that there is no clear concept about the dynamic response of the inertia asymmetry rotor with the excitation of multiple loads after FBO occurs. In this paper, when FBO occurs, the load characteristics of the rotor system are described with the low-pressure rotor structure of a high bypass ratio turbofan engine. The dynamic model of the inertial asymmetric rotor system is established considering the action of multiple loads, the dynamic response of the inertial symmetric rotor and the inertial asymmetric rotor with the action of multiple load excitation are compared and analyzed. The safety design strategy of inertia asymmetric rotor is proposed, which provides certain theoretical and technical support for the safety research of aeroengine.

2 Description of Rotor Load Characteristics After FBO After FBO occurs, the motion process of the engine rotor system is variable and the load situation is complex. Take the high bypass ratio turbofan engine as an example. After the large diameter fan blade losing, the rotor motion state and load characteristics of aero-engine can be divided into three stages: 1) Before FBO occurs, the rotor system with fine assembly and balance can be considered that the center of mass (G point) coincides with the center of rotation (O point). During stable state, the rotational inertia excitation load is at a low level. 2) At the moment of FBO, the rotor system is impacted by sudden unbalance. Due to the lack of blade, the diameter moment of inertia of the blade-disk structure shows asymmetry, resulting in the inertia asymmetry (Jx = Jy ). 3) After the blade is lost, the speed of the rotor system will change in a short time, and the rotor system will be affected by the angular acceleration caused by the time-varying speed; The center of mass (G point) will no longer coincide with the center of rotation (O point), which will cause the center of mass deviation, and the rotor system will also subject to large unbalanced rotation excitation. The rotor unbalance vibration makes the rotor amplitude exceed the blade-casing clearance. The blade will contact

f (t )

Jy

Jy

Jx

J x m0 h 2

Jp

me h 2

Jp o

M

F

M

G

2

e

Ip

Je

e

Fig. 1. Variation of rotor load characteristics after FBO.

Analysis and Safety Design of Aero-Engine Rotor Dynamic Response

37

and rub the casing. Therefore, the fan casing will generate additional rubbing force on the rotor system, as shown in Fig. 1.

3 Action Mechanism of Multiple Loads with FBO 3.1 Sudden Unbalance Load Before FBO occurs, the initial unbalance of the rotor system is small (about 100g · mm), and the rotation excitation is small. After the blade is lost, the unbalance increases significantly (the unbalance can reach 106 g · mm ), and the center of mass will suddenly deviate at the moment of FBO, which results in a sudden unbalance load. After FBO occurs, the rotation speed will rapidly reduce to the windmill speed. With the rapid reduction of speed, the rotor system will be affected by the tangential load (FT = me eθ¨ ) caused by the time-varying speed (Fig. 2). The expression is given in Eq. (1):  {0, 0}T   t < tB−off (1) P(t) = me e θ¨ sin θ + θ˙ 2 cos θ, −θ¨ cos θ + θ˙ 2 sin θ t ≥ tB−off wherein, me represents the mass of the fan disk after FBO, e represents the eccentricity, θ represents the phase angle of the fan disk, and tB−off represents the moment of FBO.

Fr = me eω 2 G FT = me eθ

Or

O

ω θ

Fig. 2. The load of unbalanced rotor during time-varying speed.

3.2 Inertial Asymmetric Disk FBO makes the blade disk structure mass eccentricity e. The method proposed in reference [1] to analyze the inertial asymmetric disk. According to the theorem of parallel

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axis, the polar moment of inertia around Or point is Je = Jp −mh2 , the diameter moment of inertia around inertia main shaft are Jx = Jp /2−mh2 , Jy = Jp /2. the diameter moment of inertia in both directions are not equal, which are called inertia asymmetric, as shown in Fig. 3.

Fig. 3. The inertial asymmetric characteristics of the fan disk.

The control equation of the rotor system in the absolute coordinate system is established with the Lagrange energy method, as shown in Eq. (2):   ∂T ∂U d ∂T − + =Q (2) dt ∂ q˙ ∂q ∂q q = [x, y, θx , θy ]T is the radial and angular freedom of the disk. The kinetic energy expression of the asymmetric disk is shown in Eq. (3): 2 1  2 1 1  Td = m(˙x2 + y˙ 2 ) + Jx θ˙x cos ωt + θ˙y sin ωt + Jy θ˙x sin ωt − θ˙y cos ωt 2 2 2   2 

1 + Jx + Jy ω + ω θ˙x θy − θ˙y θx (3) 2 Equation (4) defines the average diameter moment of inertia J and Eq. (5) defines the inertia unsymmetrical J . J = (Jx + Jy )/2

(4)

J = (Jy − Jx )/2

(5)

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39

Substitute Eq. (4) and Eq. (5) to obtain the fan disk kinematics equation, as shown in Eq. (6): (Mdisc + Mdisc−c cos 2ωt + Mdisc−s sin 2ωt)¨q +(Gdisc + Gdisc−c cos 2ωt + Gdisc−s sin 2ωt)˙q = 0

(6)

wherein, Mdisc and Gdisc are the mass matrix and gyro matrix respectively, Mdisc−c cos 2ωt and Mdisc−s sin 2ωt are the time-varying mass matrix of fan disk, Gdisc−c cos 2ωt, Gdisc−s sin 2ωt are the time-varying gyro matrix of the disk. ⎡ ⎢ Mdisc = ⎢ ⎣

⎤

m

⎡

0

⎢ ⎥ ⎥, Gdisc = ⎢ 0 ⎣ ⎦

m J

⎤ ⎥ ⎥. 2J ω ⎦

0 −2J ω 0 ⎡ ⎡ ⎤ ⎤ 0 0 ⎢ 0 ⎢ ⎥ ⎥ ⎥, Gdisc−c = 2ωJ ⎢ 0 ⎥. Mdisc−c = J ⎢ ⎣ ⎣ −1 ⎦ 0 −1 ⎦ 1 −1 0 ⎡ ⎡ ⎤ ⎤ 0 0 ⎢ 0 ⎢ ⎥ ⎥ ⎥, Gdisc−S = 2ωJ ⎢ 0 ⎥. Mdisc−s = J ⎢ ⎣ ⎣ ⎦ ⎦ 0 −1 1 −1 0 −1 J

3.3 Rub-Impact Excitation After the blade is lost, the fan casing and the blade collide with each other, resulting in mutual force. In this paper, the method proposed in reference [14] will be used to obtain the rubbing force between the blade and the casing. The origin (i.e. the rotating center of the disk) is O, the geometric center is Or , the distance from the blade tip to the geometric center Or is r, and the displacement of geometric center Or in the x and y directions is recorded as x1 and y1 , respectively. Take the i-th blade for analysis. At the moment of t, the phase angle ϕi of the i-th blade is calculated as shown in Eq. (7): ϕi = ωt + 2π(i − 1)/n

(7)

n is the number of blades and ω is the rotor speed. Equation (8) defines the distance from the i-th blade tip to the center of rotation (O point):   (8) ri = (x1 + r cos ϕi )2 + (y1 + r sin ϕi )2 φi is the included angle between ri with the x direction.

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As shown in Fig. 4, the velocities of the disk geometric center in the x and y directions are respectively defined as vx1 and vy1 . Therefore, the velocities of the i-th blade tip in the x and y directions are shown in Eq. (9):  vxi = −vx1 + ωr sin ϕi (9) vyi = −vy1 + ωr cos ϕi The speed of the i-th blade tip relative to the casing is calculated as follows in Eq. (10): vτ i = vyi cos i + vxi sin i

The i-th blade

y1 Fan disk

Or

o

v y1

x1

v yi

vxi

r

v i x1

vi

(10)

i

ri

Fan case

Fig. 4. Speed of the blade tip relative to fan case.

The rubbing force between fan blade and casing is decomposed into rubbing normal force and rubbing tangential force, as shown in Fig. 5. The rubbing normal force adopts linear elastic model, and the rubbing tangential force adopts Coulomb friction model. Establish rubbing criteria: Pi = ri − r − r0 . If Pi ≤ 0, it means that the fan blade is not in contact with the fan casing, and there is no rotor–stator rubbing; If Pi > 0, it means that there is no clearance between the fan blade and the casing, and there has been friction. If rub-impact occurs, the fan casing will generate radial rubbing force Fni = kc Pi and tangential rubbing force Fτ i = μFni on the rotor system, which are decomposed in the x and y directions shown in Eq. (11):   Fxi = kc fxi f = −H (Pi )(Pi cos i − sign(vτ i )μPi cos i ) (11) , xi Fyi = kc fyi fyi = −H (Pi )(Pi sin i + sign(vτ i )μPi sin i ) ⎧  ⎨ 1 vτ i > 0 1 P>0 In Eq. (11): H (Pi ) = , sign(vτ i ) = 0 vτ i = 0 . ⎩ 0 Pi ≤ 0 −1 vτ i < 0 Translate the component force in the x and y directions to the geometric center Or of the disk, and obtain the friction force (shown in Eq. (12)) of the rotor blade disk in

Analysis and Safety Design of Aero-Engine Rotor Dynamic Response

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the x and y directions. In this paper, it did not consider the torque acting on the center of disk:  n n   Fx = kc fx , fx = fxi , fy = fyi (12) Fy = kc fy i=0

i=0

vi

The i-th blade

y1

r

Or

o

i

x1

Fyi

ri

Fi

Fni Fxi

i

Fan disk Fan case

Fig. 5. Rubbing force between the blade and fan case.

3.4 Rotor Dynamics Model with Multiple Loads After FBO occurs, the inertia symmetric rotor system becomes an inertia asymmetric one. The rotor system is subject to the combined action of multiple loads (sudden impact load, large unbalance rotating excitation, angular acceleration excitation and rub-impact excitation). Write multiple load excitation in the form of Eq. (11):

P(t) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

 T t < tB−off m0 e θ˙ 2 cos θ, θ˙ 2 sin θ  T 2 2 ˙ ˙ tB−off ≤ t < t1 me eθ cos θ + px , me eθ sin θ + py  T 2 ¨ ˙ ¨ t1 ≤ t < t2 me eθ sin θ + me eθ cos θ + px , −me eθ cos θ + me eθ˙ 2 sin θ + py  T 2 2 t2 ≤ t ≤ tend me eθ˙ cos θ + px , me eθ˙ sin θ + py

(13)

wherein, mo is the mass of the fan disk before the blade losing, me is the mass of the fan disk after FBO, e is the eccentricity, θ is the phase angle of the fan disk, px , py are the component of the friction force in the x and y directions, tB−off is the moment of FBO, t1 is the time when the speed starts to decrease, t2 is the time when the speed reaches the speed of the windmill, and tend is the end time. Therefore, the rotor dynamics equation of inertia symmetric rotor with multiple loads is shown in Eq. (14): M q¨ + (C + G)˙q + Kq = P(t)

(14)

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And the rotor dynamics equation of inertia asymmetric rotor with multiple loads after FBO is given in Eq. (15): (Mdisc + Mdisc−c cos 2ωt + Mdisc−s sin 2ωt)¨q +(C + Gdisc + Gdisc−c cos 2ωt + Gdisc−s sin 2ωt)˙q + Kq = P(t)

(15)

P(t) refers to multiple load excitation of rotor system, including initial unbalance excitation, sudden unbalance excitation, angular acceleration excitation, large unbalance excitation and rub-impact excitation.

4 Modeling and Analysis Method of Low-Pressure Rotor 4.1 Establishment of Low-Pressure Rotor Model Referring to the low-pressure rotor of a typical high bypass ratio turbofan engine, a finite element model is established. The low-pressure rotor is mainly composed of 30 fan blades, 4-stage compressor, 5-stage low-pressure turbine, fan disk and compressor structure, fan stub shaft, low-pressure turbine shaft, turbine disk and 3 fulcrums, as shown in Fig. 6.

Fig. 6. LP rotor finite element model.

The modal characteristics of the rotor system are analyzed. The first order critical speed of the rotor system is ω1 = 4900 rpm and the mode shape is the pitch vibration of fan; The second order critical speed is ω2 = 5964 rpm, and the mode shape is the translation motion of the turbine; The third order critical speed is ω3 = 18028 rpm, and the mode shape is the overall bending vibration. As shown in Fig. 7, the working speed of the rotor system is 9000 rpm. The critical speed of the overall bending vibration mode is designed above the working speed. The rotor system does not go through the overall bending critical speed during the acceleration and deceleration process, which is in accordance with the engineering design.

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200

Frequency Hz

150

100

50

4900

5964 9000 Rotational speed RPM

Fig. 7. The LP rotor resonance speed distribution.

4.2 Dynamic Response Analysis Method First, the three-dimensional solid model of LP rotor is established through finite element analysis software, and the established model is output to the [K], [C], [M], [G] matrix in HBMAT format. Use the simulation software to convert the matrix of HBMAT format into the rotor dynamics matrix. The dynamic equations of inertia symmetric rotor and inertia asymmetric rotor are constructed in the simulation software based on Sect. 3.4, and the mechanical boundary conditions of multiple loads are applied to the rotor system. Finally, the established rotor dynamics equation is solved by the Newmark-β method. The dynamic analysis method is shown in Fig. 8. Establishing three-dimensional solid finite element model of rotor support system

Forming HBMAT format rotor dynamics matrix [K] [C] [M] [G]

Rotor establishment

Unbalanced load

MATLAB format rotor dynamics matrix [K] [C] [M] [G]

Equation solution

Blade-casing rubbing

Rotor dynamics equation under complex load environment

Rotor dynamic response under complex load environment

Inertial asymmetric excitation

Fig. 8. Dynamic analysis method.

5 Analysis of Rotor Dynamic Response with Multiple Load Excitation The radius of the fan is 460 mm, and the rub-impact clearance is 2.5 mm (if the radial vibration displacement of the fan blade is greater than 2.5 mm, the rotor rub-impact occurs). The rigidity of the fan casing is K=2 × 107 N /m, and the friction coefficient is μ = 0.2.

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In the initial state, the rotor runs stably at the working speed. When FBO occurs, the rotor system speed will rapidly reduce to the windmill speed in a short time, and keep the windmill speed rotating continuously. The rotation speed-time diagram is shown in Fig. 9.

t B −off

working speed

t1

Windmill speed

t2

Fig. 9. Rotation speed-time diagram.

5.1 Inertia Symmetric Rotor From the Sect. 3.4, we can get the dynamic equation of inertia symmetric rotor, as shown in Eq. (14). In this paper, Newmark-β method is adopted to solve Eq. (14), and the dynamic response law of the inertia symmetric rotor with multiple loads is obtained. Time–Frequency Displacement Response. Figure 10(a) and Fig. 10(b) show the horizontal and vertical displacement response of the fan disk in time domain. It can be seen that the initial unbalance of the fan disk is small before the blade lost, and the vibration amplitude is small for the action of the initial unbalance load. After FBO occurs, the amplitude of the fan disk increases sharply. When passing the first critical speed, the amplitude of the fan disk reaches a peak. Figure 10(c) shows the amplitude of the fan disk in time domain. From the Fig. 10(c), we can see that the peak amplitude of the fan disk in time domain reaches 2.73 mm, which exceeds the clearance between the fan blade and the fan casing(2.5 mm). Rub-impact is a continuous process. Figure 10(d) is the vibration response of the fan disk in frequency domain. In Fig. 10(d), it can be seen that the main frequency 81.33 Hz is the frequency when the rotor system passes the first critical speed. A lot of clutter in the frequency 85 Hz–100 Hz are caused by the rub-impact. In the frequency domain figure (Fig. 10(d)), it can be seen that the first-order backward whirl frequency of the rotor system is 75.17 Hz, which is due to the backward whirl frequency caused by the action of the tangential rubbing force when rub-impact occurs.

Analysis and Safety Design of Aero-Engine Rotor Dynamic Response

2

2

Y displacement /mm

(b) 3

X displacement /mm

(a) 3

1

0

-1

-2

-3

45

1

0

-1

-2

0

1

2

3

4

5

-3

6

0

1

2

3

Time /s

4

5

6

Time /s

(c) 3

(d) 0.8 0.7

2.5

0.6

1

mm 1.5

0.5

Amplitude

Amplitude /mm

2

0.4 0.3 0.2

0.5

0.1 0

0

1

2

3

Time /s

4

5

6

0

0

20

40

60

80

100

120

140

160

Frequency Hz

Fig. 10. The vibration response of the inertia symmetric fan disk (a) horizontal displacement in time domain (b)vertical displacement in time domain (c)vibration amplitude in time domain (d)response in frequency domain.

The Reaction Forces of the Supports. Figure 11 shows the reaction forces of No.1, No.2 and No.3 supports of inertia symmetric rotor in time domain with the action of multiple loads. In the normal working conditions, the amplitude of the supports’ reaction forces is small due to the effect of initial unbalanced load. When FBO occurs, the supports’ reaction forces increase sharply. When passing the first critical speed, the reaction forces of supports reach the peak value, the reaction force of No.1 support reaches 9.65 × 104 N , the reaction force of No.2 support reaches 5.944 × 104 N , and the maximum the reaction force of No.3 support reaches 6.401 × 104 N ; The second mode shape of the rotor system is the translation motion of the turbine, which is small effect on the fan (including No.1 support and No.2 support). On the contrary, it has a large impact on the turbine rear support (No.3 support). When passing the second critical speed, the reaction force of No.3 support reaches a peak value of about 4.068 × 104 N . To sum up, when the rotor system passes the critical speed, the supports’ reaction forces of the rotor system reaches the peak value. Therefore, when the rotor decelerates beyond the critical speed point, the vibration response of the rotor is the largest and it is the most dangerous point, which needs to be focused on. Rub-Impact Characteristics. The rub-impact characteristics of the rotor system are analyzed, and axis orbit of inertia symmetric rotor during rubbing in Fig. 12 is obtained.

46

J. Fu et al. 104

10

9.65 104 N

9 8

6.401 104 N

6

5.944 104 N

Reaction force

N

7

5 4

4.068 104 N

3 2 1 0

0

1

2

3

Time

4

5

6

s

Fig. 11. The reaction forces of the supports of the inertia symmetric rotor in time domain.

Fig. 12. Axis orbit of inertia symmetric rotor during rubbing.

Figure 12 shows axis orbit of the fan disk of the inertia symmetric rotor system during rubbing. From the Fig. 12, it can be seen that the form of rubbing between the fan blade and the casing is full annular rub, and the rubbing is continuously.

5.2 Inertia Asymmetric Rotor The mutually orthogonal diameter moments of inertia are no longer equal due to the fan losing. At the rotating state, the mass matrix and gyro matrix of the rotor system are time-varying instead of constant. From the Sect. 3.4, we can get the dynamic equation of inertia asymmetric rotor, as shown in Eq. (15). In this paper, Newmark-β method is adopted to solve Eq. (14), and the dynamic response law of the inertia asymmetric rotor with multiple loads is obtained. Time–frequency displacement response. Figure 13(a) and Fig. 13(b) show the horizontal and vertical displacement response of the fan disk in time domain. It can be seen that the response of inertia asymmetric rotor with multiple load excitation is similar to that of inertia symmetric rotor. the initial unbalance of the fan disk is small before the

Analysis and Safety Design of Aero-Engine Rotor Dynamic Response

47

blade lost, and the vibration amplitude is small for the action of the initial unbalance load. After FBO occurs, the amplitude of the fan disk increases sharply. When passing the first critical speed, the maximum horizontal displacement reaches 2.921 mm, the maximum vertical displacement reaches 3.692 mm. Figure 13(c) shows the amplitude of the fan disk in time domain. From the Fig. 13(c), we can see that the peak amplitude of the fan disk in time domain reaches 3.797 mm, which exceeds the rotor-static clearance of 2.5 mm, and the rotor-static friction occurs between the fan blade and the casing. Figure 13(d) is the vibration response of the fan disk in frequency domain. In Fig. 13(d), it can be seen that the main frequency 82 Hz is the frequency when the rotor system passes the first critical speed. From the Fig. 10(d), it also can be seen that the first-order backward whirl frequency of the rotor system is 76 Hz, which is caused by the action of the tangential rubbing force when rub-impact occurs. Compared with inertia symmetric rotor, inertia asymmetric rotor is subject to more serious rub-impact and represents more obvious rub-impact characteristics. It can be seen from the Fig. 13(d) that there are two main frequencies in the rotor system with the excitation of multiple loads: 82 Hz and 76 Hz. They are the first critical speed (forward whirl) frequency and the first backward whirl frequency, which indicate that the inertia asymmetric rotor has serious rub-impact between rotor and stator with the excitation of multiple loads, directly resulting in the backward whirl in the rotor system. (a)

(b)

3

4 3

2

Y displacement/mm

X displacement /mm

2 1

0

-1

1 0 -1 -2

-2

-3 -3

0

1

2

3

4

5

-4

6

time/s

1

2

3

4

5

6

time/s

(c)

(d) 4

0.8

3.5

0.7

3

0.6

Amplititude mm

Amplitude /mm

0

2.5 2 1.5

0.5 0.4 0.3

1

0.2

0.5

0.1

0

0

0

1

2

3

Time /s

4

5

6

0

20

40

60

80

Frequensy

100

120

140

160

Hz

Fig. 13. The vibration response of the inertia asymmetric fan disk (a) horizontal displacement in time domain (b) vertical displacement in time domain (c) vibration amplitude in time domain (d) response in frequency domain.

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The Reaction Forces of the Supports. Figure 14 shows the reaction forces of No.1, No.2 and No.3 supports of inertia asymmetric rotor in time domain with the action of multiple loads. From the Fig. 14, the supports’ reaction forces of inertia asymmetric rotor are similar to the supports’ reaction forces of the inertia symmetric rotor. In the normal working conditions, the amplitude of the supports’ reaction forces is small due to the effect of initial unbalanced load. When FBO occurs, the supports’ reaction forces increase sharply. When passing the first critical speed, the reaction forces of supports reach the peak value, the reaction force of No.1 support reaches 1.349 × 105 N , the reaction force of No.2 support reaches 8.245 × 104 N , and the maximum the reaction force of No.3 support reaches 8.194 × 104 N ; The second mode shape of the rotor system is the translation motion of the turbine, which is small effect on the fan (including No.1 support and No.2 support). On the contrary, it has a large impact on the turbine rear support (No.3 support). When passing the second critical speed, the reaction force of No.3 support reaches a peak value of about 4.196 × 104 N .

1.349 105 N

8.194 104 N

No.1 support No.2 support No.3 support

8.245 104 N

4.196 104 N

Fig. 14. The reaction forces of the supports of the inertia asymmetric rotor in time domain.

Table 1 is obtained by comparing the peak of the supports’ reaction forces of the inertia asymmetric rotor system with the peak of the supports’ reaction forces of the inertia symmetric rotor system with multiple load excitation. From Table 1, compared with the peak value of the supports’ reaction forces of the inertia asymmetric rotor and the peak value of the inertia symmetric rotor, the peak value of the supports’ reaction forces increases. The supports’ reaction forces increase more obviously at the first critical speed. Therefore, at the first critical speed, the reaction force of No.1 support increased by 39.79%, about 38400N; The reaction force of No.3 support increased by 37.85%, about 22500N; The reaction force of No.2 support increased by 28.81%, about 18440N. It has little influence on the peak of supports’ reaction forces at the second critical speed.

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Table 1. Peak value comparison of the supports’ reaction forces with multiple loads Support number

No.1 support

No.2 support

No.3 support (the 1st order)

No.3 support (the 2nd order)

The inertia symmetric rotor system

96500N

64010N

59440N

40680N

The inertia asymmetric rotor system

134900N

82450N

81940N

41960N

Variation

+38400N

+18440N

+22500N

+1280N

Change rate

+39.79%

+28.81%

+37.85%

+3.15%

To sum up, the inertia asymmetry will seriously affect the reaction forces of the supports, make the rotor supports bear greater reaction forces, and seriously threaten the safety of the bearing support structure. Rub-Impact Characteristics The rub-impact characteristics of the rotor are analyzed, mainly the axis orbit at the rub-impact stage, as shown in Fig. 15(a). Figure 15(b) shows the backward whirl characteristics of the inertia asymmetric.

(a)

(b)

4 3 2

0.1

Y displacement /mm

Y displacement /mm

0.2 0.15

1 0 -1

0.05 0 1

-0.05

-2

-0.1

-3

-0.15

-4 -3

-2

-1

0

X displacement /mm

1

2

3

-0.2 -0.15

2

-0.1

-0.05

0

0.05

0.1

0.15

X displacement /mm

Fig. 15. Axis orbit of inertia asymmetric rotor (a) rub-impact stage, (b) backward whirl characteristics.

It can be seen from the Fig. 15(a) that the form of rubbing between the fan rotor blades and the casing is local rub. There are multiple contact-collisions and reboundseparation between blades and casing. From the Fig. 15(b), the inertia asymmetric rotor shows the backward whirl feature after the rub-impact. The whirl direction of the rotor system at the initial time is anticlockwise (forward whirl). After rub-impact, the whirl direction of the rotor system turns clockwise (backward whirl) with the action of rubimpact excitation, which is opposite to the initial whirl direction. The rotor system has the backward whirl characteristics.

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Compared with the inertia symmetric rotor, the inertia asymmetric rotor has more serious rub-impact. From the Figs. 12 and 15(a), the inertia symmetric rotor mainly rubs in the form of contact non-separation full-cycle rubs, while the inertia asymmetric rotor mainly rubs in the form of contact separation local rub with multiple load excitation. From the previous analysis, it can be seen that inertia symmetric rotor has only one backward whirl frequency with multiple load excitation, and there is no backward whirl characteristics; However, with the combined action of multiple loads, the degree of rub-impact of the inertial asymmetric rotor system is more serious, and the time of rub-impact is longer. The serous rub-impact directly leads to the backward whirl characteristics of the rotor system. Therefore, the inertial asymmetric rotor system is more prone to backward whirl than the inertial rotor system.

6 Safety Design of Support Structure 6.1 Safety Design Strategy With the action of multiple loads, the transient response peak of the inertia asymmetric rotor system can reach above 1.3 × 105 N , which can affect the completeness of the supporting structure, and in severe cases, it can cause secondary accidents such as bearing jamming, damage to mount, shaft breakage, etc. Therefore, it is necessary to design a safety structure of the aeroengine rotor to reduce the reaction forces of supports. (1) The low-pressure rotor is a flexible rotor with multiple fulcrums. After FBO occurs, the No.1 support of the rotor system is the closest to the fan and its load is the largest. Therefore, the load environment of the No.1 support of the rotor system is the most severe. It is necessary to carry out safety design for it. (2) For modern aero-engines, the No.1 support is a roller bearing, which mainly bears radial load. Figure 16(a) and (b) shows the safety design of typical support structures at home and abroad. As can be seen from Fig. 16(a) and (b), in order to avoid secondary failure caused by excessive reaction force of support, the safety design of the fuse structure (design of thinned conical shell (e.g. Fig. 17(a)) and design of fuse bolt (e.g. Fig. 17(b))) has been carried out at the No.1 support structure, so that the rotor system can be changed from 3 to 2 fulcrums to realize the variable stiffness design of the rotor. This will change the dynamic characteristics of the rotor system, reduce the critical speed, thereby reducing the reaction force of support when the rotor passes the critical speed. (3) From the perspective of energy, damping will absorb the energy of rotor vibration and reduce the reaction force of support. In order to reduce the peak of support’s reaction force of the inertia asymmetric rotor with the excitation of multiple loads and protect the safety of the support structure, the damping of the rotor can be appropriately increased to reduce the reaction force of support of the rotor system. Therefore, a support structure of roller bearing is designed based on the safety design characteristics of inertia asymmetric rotor, as shown in Fig. 17(a) and (b). The support structure mainly consists of two key structures, the fuse structure and the low stiffness structure. When the aero-engine rotor works normally, the vibration response of the rotor system is at a relatively low level. The reaction force of support

Analysis and Safety Design of Aero-Engine Rotor Dynamic Response

(a)

51

(b)

Fig. 16. Typical safety design of No.1 support structure (a) design of thinned conical shell (b) design of fuse bolt.

(a)

(b)

Fig. 17. A new safety design of support structure (a) working state. (b) after blade loss.

is transmitted to the left and right convex shoulder through the fulcrum bearing and the bearing outer ring (e.g. Fig. 17(a)). However, when FBO occurs, the reaction force of support is large, and the left and right convex shoulder are broken. The reaction force of support is transmitted to the elastic ring through the fulcrum bearing (e.g. Fig. 17(b)); The elastic ring is made of metal rubber with low stiffness. The metal rubber produces great damping effect and consumes vibration energy under the reaction force of support, reduces the external transmission of vibration load, and protects the support frame and mounting joint of the engine. 6.2 Effect of Reducing Support Stiffness Change the stiffness of the No. 1 support to obtain the critical speed of the rotor system under different support stiffness. As shown in Fig. 18, which shows the change of the critical speed of the rotor system with reducing the No. 1 support stiffness. The data in Fig. 18 is sorted out to draw Table 2. It can be seen from Fig. 18 and Table 2 that changing the stiffness of No. 1 support has little impact on the second critical speed of the rotor system (the translation motion of the turbine mode). However, it has a great impact on the first critical speed of the rotor system (the pitch vibration of fan mode). The No. 1 support relative stiffness is reduced by 50%, and the first critical speed is reduced by 15%. When No. 1 support stiffness becomes 1% initial support stiffness, the first critical speed of the rotor is significantly

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Fig. 18. Change of critical speed with reducing No. 1 support relative stiffness. Table 2 Change of critical speed with No. 1 support relative stiffness No. 1 support relative stiffness

1

0.75

0.5

0.3

0.1

0.01

0

First critical speed

4900

4549

4145

3778

3363

3156

3132

Percentage (%)

100

92.84

84.59

77.10

68.63

64.41

63.92

Second critical speed

5964

5928

5905

5892

5882

5879

5878

Percentage (%)

100

99.40

99.01

98.79

98.93

98.57

98.56

reduced by about 35.5%. When No. 1 support stiffness becomes 0, the support structure loses its restraint capacity, and the first critical speed decreases by 36%, which is almost the same as that of 1%. Therefore, when the No. 1 support stiffness is two magnitudes smaller than the initial support stiffness, the No. 1 support has basically lost its restraint capacity. By reducing the No. 1 support stiffness, the critical speed of the rotor can be reduced, which will change the dynamic characteristics of the rotor and reduce the supports’ reaction forces of the rotor system. 6.3 Effect of Increasing Support Damping According to the previous design idea (e.g. Fig. 17), after FBO occurs, increasing the support damping can increase the energy consumption of damping, reduce the peak value of the reaction force of the support, and reduce the vibration response of the rotor system. The reaction force of No. 1 support under different damping is obtained by changing the No. 1 support damping, as shown in Fig. 19.

Analysis and Safety Design of Aero-Engine Rotor Dynamic Response

(a)

(b)

(c)

(d)

53

Fig. 19. The reaction forces of supports under different No.1 support damping (a) 2 × support damping (b) 4 × support damping (c) 8 × support damping (d) 10 × support damping.

The peak value of reaction forces of supports under different No.1 support damping is arranged and drawn into Table 3. It can be seen from Fig. 19 and Table 3 that the change of No. 1 support damping can greatly change the reaction forces of rotor supports. When the No.1 support damping becomes twice the initial support damping, the peak value of No.1 support’s reaction force of the rotor system decreases by 16200N, about 12%. When the No.1 support damping is increased to 10 times the initial support damping, the peak value of No.1 support’s reaction force of the rotor system decreases by 82480N, 61.14%. Through the above analysis, the dynamic characteristics of the rotor can be changed through reducing support stiffness, so as to restrain the vibration response of the rotor; The peak value of supports’ reaction force of the inertia asymmetric rotor with multiple loads can be reduced by increasing the support damping, so as to reduce the vibration response of the rotor and ensure the structural integrity of the engine.

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Table 3 The peak value of each support’s reaction force under different No.1 support damping Support number

No.1 support

No.2 support

No.3 support (the 1st order)

No.3 support (the 2nd order)

Initial No.1 support damping

134900N

82450N

81940N

41960N

2 × support damping

118700N

75520N

72250N

41020N

Change rate (%)

−12.01

−8.41

−11.83

−2.24

4 × support damping

92930N

59590N

49890N

39510N

Change rate (%)

−31.11

−27.73

−39.11

−5.84

8 × support damping

61150N

39150N

33010N

36800N

Change rate (%)

−54.67

−52.52

−59.71

−12.30

10 × support damping

52420N

33490N

28030N

35820N

Change rate (%)

−61.14

−59.38

−65.79

−14.63

7 Conclusion In this paper, the following conclusions are obtained by analyzing the dynamic characteristics of inertia symmetric rotor and inertia asymmetric rotor with multiple loads after FBO occurs: (1) The load characteristics of the blade are described in detail after FBO occurs, and the rotor dynamic model of inertia symmetric rotor and inertia asymmetric rotor with multiple load excitation is established. (2) After the blade loss, the vibration response of aero-engine rotor system is the largest when it decelerates passing the critical speed, which is also the most dangerous point; With the action of multiple load excitation, the rotor system will appear backward whirl frequency for the excitation of rub-impact. (3) With the action of multiple load excitation, the vibration response of the inertia asymmetric rotor is greater than that of the inertia symmetric rotor. and the peak reaction forces of No.1, No. 2 and No. 3 supports increase by 39.79%, 28.81% and 37.85% respectively. (4) Compared with the inertia symmetric rotor system, the inertia asymmetric rotor system has more serious rub-impact between rotor and stator; The inertia symmetric rotor mainly rubs in the form of contact non-separation full-cycle rubs, while the inertia asymmetric rotor mainly rubs in the form of contact separation local rub with multiple load excitation; The inertia asymmetric rotor system shows backward whirl characteristics with the action of multiple loads.

Analysis and Safety Design of Aero-Engine Rotor Dynamic Response

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(5) A safety design idea of the support structure is proposed, and the vibration response of the rotor is effectively reduced by reducing the No. 1 support stiffness and increasing the No. 1 support damping. Acknowledgements. The authors would like to acknowledge the financial support from the National Science and Technology Major Project of the Ministry of Science and Technology of China (Grant Nos. J2019-VIII-0008-0169, Y2019-VIII-0011-0172 and 2017-I-0008-0009).

References 1. Yu, P., et al.: Dynamic modeling and vibration characteristics analysis of the aero-engine dual-rotor system with fan blade out. Mech. Syst. Signal Process. 106, 158–175 (2018) 2. Genta, G.: Dynamics-of-Rotating Systems. Springer, Heidelberg (2005). https://doi.org/10. 1007/0-387-28687-X 3. Ren, X.M., Gu, J.L.: Response of suddenly applied unbalance for aeroengine -bearing system. J. Vibr. Eng. 4(3), 75–82 (1991) 4. Gu, J., Ren, X.M.: A study on solution of suddenly applied unbalance response of a rotorsupport system. Chinese J. Appl. Mech. 8(4), 56–62 (1991) 5. Liu, D., Hong, J., Su, Z., et al.: Dynamic characteristics of inertia asymmetry rotor and safety design. J. Propul. Technol. 43(4), 200269 (2022) 6. Ikeda, T., Murakami, S.: Dynamic response and stability of a rotating asymmetric shaft mounted on a flexible base. Nonlinear Dyn. 20(1), 1–19 (1999). https://doi.org/10.1023/A: 1008302203981 7. Muszynska, A.: Rotordynamics. Taylor & Francis, New York (2005) 8. Beatty, R.F.: Differentiating rotor response due to radial rubbing. J. Vibr. Acoust. 107(2), 151–160 (1985) 9. Zhang, S.Q., Lu, Q.S., Wang, Q.: Analysis of rub-impact events for a rotor eccentric from the case. J. Vibr. Eng. 11(4), 492–496 (1998) 10. Liu, D., Li, C., Yang, H., et al.: Analysis of backward whirl for rubbing rotor caused by excitation of sudden unbalance. J. Aerosp. Power 36(7), 1509–1519 (2021) 11. Sinha, S.K.: Rotordynamic analysis of asymmetric turbofan rotor due to fan blade-loss event with contact-impact rub loads. J. Sound Vib. 332(9) (2013) 12. Li, C., Liu, D., et al.: Dynamic characteristics of flexible rotor and safety design of support structure with fan blade-off. J. Aerosp. Power 35(11), 2263–2274 (2020) 13. Hong, J., Xu, M.-L., et al.: Structure safety design strategy of rotor-support system due to fan blade loss. J. Aerosp. Power 31(11), 2273–2730 (2016) 14. Thiery, F., Gustavsson, R., Aidanpää, J.O.: Dynamics of a misaligned Kaplan turbine with blade-to-stator contacts. Int. J. Mech. Sci. 99, 251–261 (2015)

Numerical Simulation of Aero-Engine Rotor-Blade-coating Coupling System with Rub-impact Fault and Its Dynamic Response Jiewei Lin1 , Bin Wu1 , Xin Lu2 , Jian Xu3 , Junhong Zhang1(B) , and Huwei Dai1(B) 1 State Key Laboratory of Engines, Tianjin University, Tianjin 300072, China

{zhangjh,dhwmail}@tju.edu.cn

2 Aeronautical Engineering Institute, Civil Aviation University of China, Tianjin 301636, China 3 Laboratory of Aeroacoustics and Vibration Aviation Technology, Aircraft Strength Research

Institute of China, Xi’an 710065, China

Abstract. Rub-impact fault is a common problem in aircraft engines. To simulate the dynamic process and rubbing force, a finite element model of bladecoating rubbing was developed and verified experimentally through a rotor rig test. Considering the aero-engine’s structural characteristics and material properties, a simulation model for rub-impact fault was developed. It considered the blade-casing-coating material model selection and parameter settings, constraints, contact, and load settings of the components, which was calculated using explicitimplicit integration scheme. A ‘0–2-1’ supported aero-engine rotor-blade-coating coupling system was also developed to study the rub-impact fault’s influence on the system vibration response, combining the rubbing force obtained at different rotating speeds and invasion depths. Results show that the rotor vibration mainly concentrates on the fundamental frequency and its multiplications during blade-coating rubbing. Rotating speed significantly affects the vibration of the frequency multiplications. Moreover, the dynamic response of the rotor system varies considerably with the invasion depth. Keywords: rub-impact · dynamic response · numerical simulation · rotor system · vibration

1 Introduction To improve engine efficiency and reduce fuel consumption, the radial clearance between the blade and the casing is decreasing, making it easier for rubbing to occur between the rotor and stator [1]. To avoid the damage caused by blade-casing rubbing, the sealing coating is sprayed inside the casing to reduce the contact force caused by the rub-impact fault. Therefore, the effect of the sealing coating should be considered in the aero-engine rub-impact fault.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 56–75, 2024. https://doi.org/10.1007/978-3-031-40459-7_4

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Rub-impact fault is common in aero-engines. Many scholars have conducted simulation and test studies [2], which mainly includes two aspects, the blade-casing/coating rub-impact study and the rotor system vibration problem caused by the rub-impact fault. In the rub-impact research, early research was mainly based on experimental studies, rub-impact tests were carried out by simplifying the blade into a single pendulum and the casing into a flat plate [3]. In keeping with the real engine structure, the casing was simplified to an arc structure and the rubbing force was collected [4]. Rub-impact test rigs that can simulate the blade-casing rubbing fault had also been built [5, 6], which were used to study the relationship between the rub-impact force and the invasion depth, the rub-impact between the blade and the coating [7], verify the rub-impact force prediction model [8]. With the development in numerical analysis methods, the blade and casing were simplified to a cantilever beam and spring-damped system, and the rub-impact study was performed by deriving the dynamic equation [9], and gradually from the singlepoint and local rub development to the whole circle rub [10]. Due to the sophisticated finite element technology, the blade and the casing were built as solid models and the rub-impact process was reproduced in simulation software such as ANSYS and LSDYNA [11, 12]. The nonlinear dynamic characteristics of rubbing [13], contact dynamic characteristics [14], and system dynamic response under different invasion depths and different speeds [15] had been studied by the simulating calculation. However, less consideration was given to abradable sealing coatings in the above studies, to better study the rubbing characteristics of the abradable coating, the blade-coating rubbing problem was simulated in this paper. The dynamic response of the rotor system under the rub-impact fault has always been the focus of studies. The earlier study was carried out to analyze the dynamics under rotor-stator rub-impact fault, mainly focusing on the rotor-stator rubbing without the blade [16]. Subsequently, the study of the dynamics under the blade and casing rubbing fault started to emerge, and the effect of the blade-casing rubbing on the blade vibration, rotor vibration and casing response vibration was studied [17]. The blade-casing rubimpact-force model had also been derived and applied to the rotor system model for the study of dynamic characteristics [18]. The coating was located on the casing’s inner wall and its abradable ability reduced the energy loss of the rotor system [19], which will influence the dynamic characteristics of the rotor system with the rub-impact fault. The coating’s effect is often ignored in existing studies, so it is necessary to study the dynamic response of the rotor system under blade-coating rub-impact fault. Based on the contact dynamics theory, a blade-casing simulation model is established, a numerical simulation based on the coating is carried out, and the accuracy of the simulation is verified through an experimental bench. The rub-impact force characteristics under different seal coting materials and invasion depths were discussed. A rotor system dynamics model was established to study the dynamic response under the rub-impact fault.

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2 Rub-impact and Rotor Dynamic Modeling 2.1 Rub-impact Modeling A simulation model of coating-blade rub-impact was established using LS-DYNA software, based on a simplified low-pressure compressor casing-coating-blade structure of an aero-engine prototype. To perform numerical simulation of the rub-impact process, certain simplifications and assumptions were made to the structure based on the calculation requirements. Since the hub primarily experiences radial loading and possesses significantly greater radial stiffness than the blade and casing, the hub was simplified to a rigid shell in order to expedite calculations. To better analyze the rub-impact characteristics, only a single blade was involved in each impact, so the influence of the blades’ number on the calculation results was ignored. In addition, the calculation did not consider the effect of temperature and aerodynamic loads. The model is shown in Fig. 1, which was built based on the above assumptions. In this simulation model, the model parameters were derived from real engine data and given in Table 1.

Fig. 1. Blade-coating-casing rub-impact finite element model

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Table 1. Blade-coating-casing rub-impact finite element model parameters Parameters

Value/Unit

Parameters

Value/Unit

Casing diameter

300 mm

Blade thickness

3 mm

Casing thickness

3 mm

Rotational speed

5000 r/min

Coating thickness

2 mm

Unbalance mass

6700 g.mm

Blade-coating clearance

0.5 mm

Equivalent support stiffness

4e6 N/m

Blade height

40 mm

Equivalent support damping

500 N.s.m−1

Material Parameter: The casing was modelled elastically and was made of Q235 steel. Considering that the coating would be exposed to excessive load during the rub-impact process, the coating material properties were described by plastic kinematic model, which can consider the failure process. When the strain on the coating exceeded the failure strain of the material, the failure elements were removed. The seal coating was Al-Si polyphenyl ester M601, and its material parameters could be seen in Ref. [20]. The yield strength of the blade increases with the strain rate, so the plastic kinematic model was chosen to simulate the blade, and the material was Ti-6Al-4V. Material parameters are shown in Table 2.

Table 2. Material parameters of rub-impact components [20]. Components

Mass density (kg/m3 )

Elastic modulus (GPa)

Poisson’s ratio

Casing

7900

209

0.3

Coating

1270

2.08

0.25

Blade

4430

113

0.33

Rigid shell

4430

113

0.33

Contact Parameter: Proper contact pair setting is the key to getting accurate results. In the rub-impact process, blades and coating, the casing will be contacted and rubbed, combined with the characteristics of the finite element model, through the face-to-face contact type provided by the software, set the blade-coating and blade-casing contact relationship. The key word (Contact_Surface_To_Surface) was used to set blades and casing. The dynamic friction coefficient was 0.2, and the static friction coefficient was 0.28. The blade-coating contact type was set using Contact_Eroding_Surface_To_Surface with a coefficient of dynamic friction of 0.3 and a coefficient of static friction of 0.43. This contact type ensured that after the failure of the external elements of the coating, the rest elements can still perform normal contact calculations. The contact modeling is two-way treatment of contact, and the contact algorithm used is the symmetric penalty function method.

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Element and Mesh Division: The elements of blades, coating and casing were hexahedral solid elements; the rigid shell elements were shell elements; the discrete element was used for elastic support. The blade mesh size was 1 mm × 0.4 mm × 0.5 mm (length, width, and height), the number was 2480; the coating mesh size was 2 mm × 1 mm × 0.5 mm, total mesh 40000; the casing was discretized using 3000 elements (5 mm × 2.5 mm × 1 mm in size). Constraint Settings: The connection between the coating and the casing was a tied connection, via the Contact_Tied_Surface_To_Surface method provided by the software. The hub was elastically supported, and the blades were also connected to the hub using a tied connection in the form of Contact_Tied_Nodes_To_Surface. To limit the casingcoating movement, the endpoints on both sides were fully constrained. The hub and the elastic units were open to the translation in X and Y directions and rotation about the Z-axis, and the center point of the elastic unit was set with full constraint. Load Settings: The load contains two parts: the rotational speed and the unbalanced load. The unbalanced load was used to produce displacement of the rotor, when the displacement was larger than the initial gap between the blade and the coating, the touch can occur, and change unbalance can control the invasion depth. The unbalance was equivalent to the unbalanced force loaded on the hub, which can be decomposed into simple harmonic force in the X and Y direction, the expression is: Fx = eω2 cos(ωt + ϕ) Fy = eω2 sin(ωt + ϕ)

(1)

e is the rotor’s unbalance amount; ω is the rotor rational speed; ϕ is the phase angle. 2.2 Solution Process Analysis The rotor rub-impact basic equation is: M q¨ (t) + C q˙ (t) + [K + B(t)]q(t) = P(t)

(2)

M is the system mass matrix, C is the system damping matrix, K is the system static stiffness matrix, B(t) is the stiffness matrix brought about by the centrifugal load, P(t) is the rub-impact load vector, and t is time. Since the rotor system is affected by the centrifugal load during high-speed rotation, the centrifugal force brings about the stiffness matrix B(t) when solving this equation, see Eq. (2), so the centrifugal force in the blade-coating rub-impact process cannot be neglected. The blade produces a nonlinear response under the friction load, which is faster using the explicit method, but the centrifugal force produces a linear steady-state response, which is suitable for implicit calculation, so an explicit-implicit method is used to calculate the rub-impact problem. The centrifugal force of the rotor system in the rotating state is first calculated using the implicit solution and then imported into LS-DYNA as a preload for the explicit calculation. The rotational speed is given with an unbalanced load, and under the unbalanced force, the blade approaches the coating, and the touch occurs.

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2.3 Rotor System Modeling To analyze the rotor system dynamic response with blade-coating rubbing fault, a ‘0–2-1’ supported aero-engine rotor-blade-coating coupling system was established according to a real low-pressure rotor structure. The lumped-mass method was used to simplify the shaft and disk into several mass points, see Fig. 2. Meanwhile, for a better study of the rub-impact fault, the following assumptions were made: 1. The rubbing fault only occurred on the compressor disk. 2. Considered only radial vibration, ignore torsional vibration, axial vibration, and gyroscopic moment. 3. The mass of the rotor system was distributed on each mass point. 4. The connections between the blade, disk, and shaft were ignored. 5. The blades were only used in rub-impact simulation, their mass was counted into the mass of the Disk 1, regardless of their structure.

Fig. 2. The rotor system dynamical model

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According to Newton’s second law, the rotor system motion equations are: ⎧ 2 ⎪ ⎪ MD1 x¨ D1 + CD x˙ D1 + Kr (xD1 − xS1 ) = MD1 eD1 ω cos ωt + Fp,x ⎪ ⎪ ⎪ ⎪ MD1 y¨ D1 + CD y˙ D1 + Kr (yD1 − yS1 ) = MD1 eD1 ω2 sin ωt + Fp,y − MD1 g ⎪ ⎪ ⎪ ⎪ ⎪ MS1 x¨ S1 + CB x˙ S1 + Kr (xS1 − xD1 ) + Kr (xS1 − xS2 ) = FB1x ⎪ ⎪ ⎪ ⎪ ⎪ M ¨ S1 + CB y˙ S1 + Kr (yS1 − yD2 ) + Kr (yS1 − yS2 ) = FB1y − MS1 g S1 y ⎪ ⎪ ⎪ ⎪ ⎪ MS2 x¨ S2 + Kr (xS2 − xS1 ) + Kr (xS2 − xS3 ) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ MS2 y¨ S2 + Kr (yS2 − yS1 ) + Kr (yS2 − yS3 ) = −MS2 g ⎪ ⎪ ⎪ ⎨ MS3 x¨ S3 + CB x˙ S3 + Kr (xS3 − xS2 ) + Kr (xS3 − xS2 ) = FB2x ⎪ MS3 y¨ S3 + CB y˙ S3 + Kr (yS3 − yS2 ) + Kr (yS3 − yS4 ) = FB2y − MS3 g ⎪ ⎪ ⎪ ⎪ ⎪ MS4 x¨ S4 + Kr (xS4 − xS3 ) + Kr (xS4 − xD2 ) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ MS4 y¨ S4 + Kr (yS4 − yS3 ) + Kr (yS4 − yD2 ) = −MS4 g ⎪ ⎪ ⎪ ⎪ ⎪ MD2 x¨ D2 + CD x˙ D2 + Kr (xD2 − xS4 ) + Kr (xD2 − xS5 ) = MD2 eD2 ω2 cos ωt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ MD2 y¨ D2 + CD y˙ D2 + Kr (yD2 − yS4 ) + Kr (yD2 − yS5 ) = MD1 eD1 ω2 sin ωt − MD2 g ⎪ ⎪ ⎪ ⎪ ⎪ MS5 x¨ S5 + CB x˙ S5 + Kr (xS5 − xD2 ) = FB3x ⎪ ⎪ ⎩ MS5 y¨ S5 + CB y˙ S5 + Kr (yS5 − yD2 ) = FB3y − MS5 g (3) M D1 and M D2 are the masses of Disk 1 and Disk 2. M s1 to M s5 are the masses of the concentrated points, respectively. C R is the damping coefficient of the disk, C B is the bearing damping coefficient, and K r is the stiffness of the shaft. eD1 and eD2 are the eccentricities of the disk. F p,x and F p,x are the rub-impact forces in the X and Y directions. These two forces are calculated by the blade-coating rubbing in the simulation model. Since the study focuses on the response of the rotor, the structure of the blade is not considered in the Eq. (3). F Bix and F Biy (i = 1,2,3) are the bearing forces. ω is the rotational speed of the rotor. The parameters are shown in Table 3. Table 3. The parameters of the rotor system Symbol

Value/Unit

Symbol

Value/Unit

M D1

6.72 kg

M s5

1 kg

M D2

1.5 kg

CR

500 N·s/m

M s1

2 kg

CB

750 N·s/m

M s2

1 kg

Kr

2.5e7 N/m

M s3

0.5 kg

eD1

1e−3 m

M s4

2 kg

eD2

1e−4 m

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2.4 Model Validation To simulate the rubbing process of the blade coating in aero-engine and verify the accuracy of the numerical simulation setup, a test rig for rub-impact was constructed (as shown in Fig. 3) along with a rub-impact simulator (as shown in Fig. 4). The test rig was motor-driven and had a ‘0-2-1’ supported structure, which was identical to the dynamic model of the modeled rotor system. The rotor system was equipped with speed and displacement sensors to record the system signals. The rubbing plate was coated uniformly with M601 abradable coating and was mounted on a displacement table containing pressure sensors, allowing for control of the invasion depth. The pressure sensors were used to collect the rub-impact force signal.

Fig. 3. The rotor rub-impact test rig

Fig. 4. Blade-coating/casing rub-impact simulator

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Rub-impact experiments were conducted on the test rig with an invasion depth of 0.4 mm and a rotating speed of 1000 r/min. A corresponding simulation calculation group was built according to the experiment, following the finite element modeling process described in the previous paper. The curves of the rubbing force with time were obtained under both conditions, as shown in Fig. 5. The maximum rubbing force of the simulation was 20.3 N, while the maximum of the experiment was 21.1 N, with an error within 5%. In the simulation calculation, the coating was thoroughly scraped, so the peaks of the rubbing force were less pronounced compared to the test, but the magnitude of the force was close. As the rubbing force magnitude was the main influencing factor of the rubbing fault, the comparison of the force verifies the accuracy of the simulation calculation settings.

Fig. 5. Time-domain diagram of rubbing force.

To validate the accuracy of the rotor dynamics model, the vibration equations were solved using the Newmark-β method by combining the rub-impact forces obtained from the simulation. The rotational speed was 1000 r/min, eccentricity was 1e−3 m, and the coating material used was M601 in the simulation. The simulated rub-impact force was added to the solution of Eq. (3) by interpolation to simulate the rub-impact fault. The dynamic response of the rotor system was obtained both with and without the rub-impact fault, as shown in Fig. 6.

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c

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b

d

Fig. 6. The dynamic responses of rotor system without and with the rub-impact fault: (a) time history of displacement without the fault in the X direction, (b) time history of displacement without the fault in the X direction, (c) frequency spectrum without the fault, (d) frequency spectrum without the fault.

The displacement response of the rotor system in the experiment without and with rubbing was measured by the test rig, see Fig. 7. Without the rub-impact fault, in Fig. 7a the vibration waveform approximates periodic sinusoidal vibration, and the fundamental frequency amplitude is 7.9e−5 m in Fig. 7c, which is the main frequency. In Fig. 6a the rotor is moving periodically with an amplitude of 8.1e−5 m, the simulation results agree with the experimental results. After rubbing occurs, in Fig. 7b the system shows a clear peak at the trough, while the frequency of double and quadruple appear in the spectrum (see Fig. 7d). The same situation is also seen in Fig. 6b, and the double and quadruple frequency also appears, and the amplitude is near. There are more frequency components in the Fig. 7d, all harmonics show a decreasing trend, which is due to the use of coupling and bearings in the experiment. When the rub happened, the misalignment occurred in the coupling and the influence of rolling bearings also appeared. Under the effect of both, the frequency component was changed. The influence of the coupling and rolling bearings on the rotor system had been studied in Ref [21]. Because the main frequency components are evident, the results verify the accuracy of the rotor system model.

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a

b

c

d

Fig. 7. The dynamic responses of rotor system from the experiment without and with the rubimpact fault: (a) time history of displacement without the fault in the X direction, (b) time history of displacement without the fault in the X direction, (c) frequency spectrum without the fault, (d) frequency spectrum without the fault.

3 Results and Discussions 3.1 Analysis of Rub-impact Simulation Based on the established finite element model, the dynamics simulation of blade casing/coating rubbing is carried out in LS-DYNA. The rub-impact process is calculated for 0.036 s at a speed of 5000 rpm, an eccentricity of 1e−3 m. Figure 8 shows the axis orbit during the rub-impact simulation. The rotor system moves in the positive direction of the X and Y under the action of the initial unbalance force. The first rub with the coating occurs at 2 ms, when the rotor displacement is larger than the tip clearance for the first time, it ends at 11 ms. Under the combined effect of the rub-impact load, centrifugal load and elastic support, the motion orbit bends back, and the displacement increases. As the coating is continuously scraped and consumed, the rotor undergoes a second rub at 15 ms and ends at 22.5 ms. Subsequently, due to the reduction of the coating, the clearance between the blade and the coating is increasing, the rub no longer occurs. During the whole solving process, the blade and the coating are rubbed twice.

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Fig. 8. Axis orbit diagram

When the blade and the coating rub, the maximum Mises stress at the blade tip is 487 MPa, the stress distribution diagram is shown in Fig. 9a. From the stress diagram can be seen blade stress is mainly concentrated in the blade root, because the blade and disk using rigid connection leads to stress concentration here, so do not analyze. Figure 9b displays the blade tip stress when the blade rubs the casing, the maximum stress at the blade tip is 973 MPa. Comparing the magnitude of blade stress with and without the coating, the presence of the coating significantly reduces the blade tip stress and reduces the risk of blade damage. 3.2 Rub-impact Results Analysis Under Different Coating Materials Coating performance directly affects the rub-impact response, to research the coating material effect, the rub-impact process between the coating/casing and blade in the four cases of casing\M601\NiCrAl-Bentonit\AiSi-hBN was simulated by using the above finite element simulation method. The material parameters are shown in the Table 4. Compare the rub-impact force of four cases, see the Fig. 10, the blade and casing/coating rub is approximately a pulse process, and the pulse feature of the bladecasing rub force is more obvious because the radial runout of the blade occurs under the radial force. And when the coating exists, weakened the impact between the blade and the casing, so that this pulse is not clear, reflecting the existence of the coating weakened the impact. The maximum rub force in the four cases of Casing, M601, NiCrAl-Bentonit and AiSi-hBN are 1628 N, 108 N, 213 N and 605 N respectively, and the rub force generated by the rubbing with the casing is much larger than the presence of the coating. So, the sealing coating can effectively reduce the rub force magnitude. The peak magnitude increases with increasing density and hardness of the coating material for three coating materials.

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Fig. 9. Blade tip stress diagram when the blade rubs: (a) the coating, (b) the casing.

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Table 4. The common coating material’s parameters [22] Material

Elastic modulus GPa

Mass density kg/m3

Yield stress MPa

Tangent modulus GPa

Failure strain

M601

2.08

1.27

2.5

0.18

0.007

NiCrAl-Bentonit

0.75

2.05

1.3

0.076

0.0101

17.22

2.26

18.2

10

0.00236

AiSi-hBN

Fig. 10. Time-domain diagram of blade-casing/coating rub-impact forces

Figure 11 Shows the energy transferred from the blade to the coating in three cases. The comparison shows the casing absorbs more energy and the rub-impact fault is more dangerous when the coating is made by AiSi-hBN. Therefore, when choosing the coating, the softer material can reduce the rub force and the energy transferred to the coating. 3.3 Rub-impact Results Analysis Under Different Invasion Depth The invasion depth is determined by the initial eccentricity of the rubbing disk. Three different eccentricities of 1e−3 m, 1.5e−3 m and 2e−3 m were set, and the maximum invasion depths were calculated to be 1 mm, 1.5 mm and 1.75 mm, respectively. As the eccentricity increases, the unbalanced load attached to the rotor increases and therefore the invasion depth increases. Compare the rub forces in the three cases, see Fig. 12. The maximum rub forces are 108 N, 239.5 N and 276 N. The deeper the invasion, the more obvious the pulse characteristics, and due to the depth increase, in the latter two cases, the third rub happens.

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Fig. 11. Time-domain diagram of the energy transferred to the coating.

Fig. 12. Time-domain diagram of blade-coating rub-impact forces under different invasion depths.

Analyzing the energy transferred to the coating under different invasion depths, every time contacting the coating, the energy increases once. The greater the depth of invasion, the more energy the coating absorbs, and the greater the possibility of causing damage to the casing. Therefore, reducing the invasion depth, can effectively reduce the damage caused by the rub-impact fault.

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Fig. 13. The dynamic response of rotor system with the rub-impact fault: (a) time history of X displacement at 1000 r/min, (b) time history of X displacement at 5000 r/min, (c) axis orbit at 1000 r/min, (d) axis orbit at 5000 r/min, (e) frequency spectrum at 1000 r/min, (f) frequency spectrum at 5000 r/min.

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3.4 The Dynamic Response of Rotor System with Rub-impact Fault To study the dynamic response of the rotor when the blade-coating rub-impact fault occurs, combined with the low-pressure compressor speed range, the vibration characteristics with rub-impact fault at high speed (5000 r/min) and low speed (1000 r/min) were calculated. The results are shown in Fig. 13. Figure 6a shows the variations of the displacement in the X direction vs. time without fault, it is clear that the vibration waveform of the system is a periodic sinusoidal vibration. The system runs smoothly, and the maximum displacement is 8.1e−5 m. The rotor system vibration frequency is the fundamental frequency, the rotor system makes stable single periodic motion. When rubbing occurs, at low speed, the vibration waveform of the system shows several obvious peaks at the peaks and valleys, with obvious nonlinear characteristics. The vibration amplitude increases to 6.8e−5 m, and the axial trajectory becomes disordered under the action of rub-impact force, as shown in Fig. 13c. In the vibration spectrum (see Fig. 13e), there are crossover frequencies at integer multiples, mainly double and quadruple frequencies, where the frequency amplitude is less than the fundamental frequency. At 1000 r/min, the rub-impact fault can cause significant rotor system nonlinear vibrations. In Fig. 13b and Fig. 13d, at high speeds, the amplitude of the rotor increases, but the nonlinear characteristics are weakened, and the rotor moves in multiple periods. Only 1/3- and 2/3-times frequencies exist in the vibration spectrum, and the amplitude is much smaller than the fundamental frequency (see Fig. 13f). Comparing the vibration response at high speed and low speed, the system has a large change in the time-domain graph and frequency-domain graph, which shows that the rotational speed has a large influence on the vibration characteristics, the rub-impact fault is more pronounced at low speeds. To investigate the effect of invasion depth on the vibration characteristics of the rotor system, the same method was used to calculate the vibration response at a speed of 5000 r/min and eccentricities of 1e−3 , 1.5e−3 and 2e−3 m, respectively, and the calculated results are presented in Fig. 14. Under the three invasion depths, the rotor system vibration waveforms do not differ much, and the vibration spectrum is similar, but the amplitude varies significantly, and the maximum amplitude values are 7.1e−4 , 7.7e−4 and 1.46e−4 m. The invasion depth mainly affects the vibration amplitude and changes the vibration response, the greater the invasion depth, the more pronounced the rotor system vibration.

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Fig. 14. The dynamic response of rotor system at different invasion depth: time history of X displacement at eccentricity of (a) 1e−3 m, (b) 1.5e−3 m and (c) 2e−3 m; frequency spectrum at eccentricity of (d) 1e−3 m, (e) 1.5e−3 m and (f) 2e−3 m.

4 Conclusions The blade-coating rub-impact simulation calculation model and the ‘0–2-1’ rotor system dynamics model were established, the rub-impact characteristics of blade and casing and different material coatings were calculated in LS-DYNA, and the effect of invasion depth on the rub-impact fault was analyzed. The dynamic response with rub-impact fault at

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high and low speeds was compared, and the different invasion depths’ influence on the vibration response was investigated. The main conclusions are as follows: (1) During blade-casing/coating rubbing, the rub-impact force exhibits pulse characteristics. In the case of blade-casing rubbing, the rub-impact force and the blade tip’s Von-Mises stress are much higher than those with a blade-coating rub. Hence, the presence of a coating reduces the risk of blade damage. An analysis of different coating materials shows that the softer the coating material, the more effective it is in protecting the blade. (2) By adjusting the eccentricity of the rub-impact disk to vary the invasion depth, the rub-impact force increases as the invasion depth increases. Moreover, increasing invasion depth results in a higher vibration amplitude in the rotor system. (3) The rotor system displays clear nonlinear characteristics at low speeds, and the vibration frequency produced by the rub-impact fault exhibits a double frequency and a quadruple frequency, in addition to the fundamental frequency, which is consistent with the test data. As the rotational speed increases, the 1/3 and 2/3-time frequency also appear, with their amplitude being much lower than that of the fundamental frequency. The effect of rub-impact force on the rotor system decreases with the increase in speed, leading to obscure rub-impact-fault characteristics.

References 1. Ahmad, S.: Rotor casing contact phenomenon in rotor dynamics—literature survey. J. Vib. Control 16(9), 1369–1377 (2010) 2. Li, B., Zhou, H., Liu, J., et al.: Modeling and dynamic characteristic analysis of dual rotorcasing coupling system with rubbing fault. J. Low Freq. Noise Vibr. Active Control 41(1), 41–59 (2022) 3. Kennedy, F.E.: Single pass rub phenomena - Analysis and experiment. J. Tribol. 104(4), 582–588 (1982) 4. Ahrens, J., Ulbrich, J., Ahaus, H.: Measurement of contact forces during blade rubbing. Plos Genet. 8(12), e1003155-e1003155 (2000) 5. Stringer, J., Marshall, M.B.: High speed wear testing of an abradable coating. Wear 294–295, 257–263 (2012) 6. Padova, C., Barton, J., Dunn, M.G., et al.: Development of an experimental capability to produce controlled blade tip/shroud rubs at engine speed. J. Turbomach. 127(4), 281–289 (2005) 7. Padova, C., Jeffery, et al.: Experimental results from controlled blade tip/shroud rubs at engine speed. J. Turbomach. 129(4), 1165–1177 (2006) 8. Ma, H., Yin, F., Guo, Y., et al.: A review on dynamic characteristics of blade–casing rubbing. Nonlinear Dyn. 84(2), 437–472 (2016). https://doi.org/10.1007/s11071-015-2535-x 9. Tai, X., Ma, H., Tan, Z., et al.: Analysis of vibration response during blade-casing rub events. J. Vibr. Measur. Diagn. 34(2), 280–287 (2014) 10. Wang, H.F., Chen, G.: Casing response characteristics and its verification considering multiple blades-casing multiple point deformation rotor-stator rubbing model. J. Propul. Technol. 37(1), 128–145 (2016) 11. Liu, S.G., Hong, J., Chen, M.: Numerical simulation of the dynamic process of aero-engine blade-to-case rub-impact. J. Aerosp. Power 26(6), 1282–1288 (2011)

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12. Ma, H., Lu, Y., Wu, Z., et al.: A new dynamic model of rotor–blade systems. J. Sound Vib. 357, 168–194 (2015) 13. Garza, J.W.: Tip rub induced blade vibrations: experimental and computational results. The Ohio State University (2006) 14. Legrand, M., Batailly, A., Magnain, B., et al.: Full three-dimensional investigation of structural contact interactions in turbomachines. J. Sound Vib. 331(11), 2578–2601 (2012) 15. Hui, M.A., Zhi-Yuan, W.U., Tai, X.Y., et al.: Dynamic characteristic analysis of rotor-bladecasing system with rub-impact fault. J. Aerosp. Power 030(008), 1950–1957 (2015) 16. Vashisht, R.K., Peng, Q.: Dynamic modelling and diagnosis of transverse crack and rotor/stator rub in a flexible rotor system. In: ASME 2019 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (2019) 17. Jin, Y., Liu, Z., Yang, Y., et al.: Nonlinear vibrations of a dual-rotor-bearing-coupling misalignment system with blade-casing rubbing. J. Sound Vib. 497(6), 115948 (2021) 18. Prabith, K., Praveen Krishna, I.R.: The numerical modeling of rotor–stator rubbing in rotating machinery: a comprehensive review. Nonlinear Dyn. 101(2), 1317–1363 (2020). https://doi. org/10.1007/s11071-020-05832-y 19. Zhang, J.-H., Lu, X., He, Z.-P., et al.: Technique application and performance evaluation for abradable coating in aeroengine. J. Mater. Eng. 44(4), 94–109 (2016) 20. Zhang, J.H., Wang, J., Xin, L.U., et al.: Rubbing process of aero-engine blades considering seal coating. J. Zhejiang Univ. (Eng. Sci.) 52(5), 8 (2018) 21. Zhang, J., Lu, X., Lin, J., et al.: Dynamic characteristics analysis of blade-casing rubbing faults with abradable coatings. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 235(6), 975–987 (2021) 22. Johnston, R.E.: Mechanical characterisation of AlSi-hBN, NiCrAl-Bentonite, and NiCrAlBentonite-hBN freestanding abradable coatings. Surf. Coat. Technol. 205(10), 3268–3273 (2011)

Modeling and Simulation Analysis of Dual-Rotor System in the Early Stage of Bearing Pedestal Looseness Cai Wang, Jing Tian(B) , Yan-ting Ai, Feng-ling Zhang, Zhi Wang, and Ren-zhen Chen Liaoning Key Laboratory of Advanced Test Technology for Aeronautical Propulsion System, Shenyang Aerospace University, Shenyang 113006, China [email protected], [email protected] Abstract. In order to quickly capture and analyze the dynamic characteristics of the dual-rotor system at the beginning of bearing pedestal looseness, an 8degree-of-freedom dynamic model of the dual-rotor system including a gyroscopic moment and a loosening fault is developed. The Newmark-β method is used to solve the developed model. The dynamic characteristics of the dual-rotor system at different speeds when the bearing pedestal first starts to loosen is analyzed. A dual-rotor fault simulation test bench is built to simulate the bearing pedestal looseness fault. The comparison results show that the dynamic simulation results are in good agreement with the test results, and the fault frequency distribution pattern in the envelope spectrum of the simulation results is consistent with that of the test results, which proves the validity of the proposed dynamics model. It is also found that there is a clipping phenomenon in the rotor vibration signal when the bearing pedestal first starts to loosen. There are high pressure and low pressure rotor rotation frequency, multiplication frequency and sum frequency in the fault characteristic frequency of the dual-rotor system. Compared with the late stage of bearing pedestal looseness, the fractional frequency and 3-times frequency do not appear in the frequency spectrum of the loose rotor. Loose rotor axis orbit changes more obviously in the higher speed, showing a quasi ellipse. Keywords: Dual-rotor system · Bearing pedestal looseness · Dynamic characteristics · Clipping phenomenon

1 Introduction Dual-rotor systems usually generate vibrations due to insufficient assembly accuracy [1]. Prolonged vibrations are very likely to loosen the supports, which in turn can lead to more typical failures such as rotor-casing touching [2–4], rotor misalignment [5, 6], bearing damage [7, 8], and other mechanical failures [9, 10]. Such failures can lead to more complex nonlinear dynamical representations in the rotor system [11–13]. The occurrence of faults triggers periodic bearing pedestal runout, which leads to changes in the system stiffness and very complex dynamical representations especially when multiple faults are coupled [14–16]. Therefore, effective prediction and extraction of the vibration characteristics at the early stage of bearing pedestal looseness is essential for the safe operation of a dual-rotor system. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 76–90, 2024. https://doi.org/10.1007/978-3-031-40459-7_5

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At the early stage of exploration, scientists believed that the discrepancy between the results calculated by empirical equations and the actual test conclusions mainly originated from the insufficient amount of collected data, and the nonlinear influence factors did not attract the attention of relevant scholars at that time. In 1995 Muszynska et al. [17] proposed that rotor system nonlinear factors are related to intermittent contact between rotor and stators, accompanied by high harmonics in both shock and friction, and experimentally confirmed the existence of ordered cycles and chaotic responses in rotor systems when faults occur. Since the 21st century, nonlinear influences within the system have gradually become a research hotspot in the field. Chu and Luo et al. [18, 19] established the vibration differential equations of a rotor system with three-degree-offreedom bearing pedestal looseness fault respectively, obtained the periodic solution of the system using the hitting target method, analyzed the stability of the periodic solution using Fourier theory, and found the path law of the system from periodic to period-like to chaotic. Zhang et al. in reference [20] used Lagrange’s equation to model a rotor system with loose faults, used the Runge-Kutta method to solve the dynamic response of the rotor system, and verified the accuracy of the harmonic balance method by comparing the bifurcation diagram, the time-frequency curve, and the Poincaré diagram. Chen et al. [21] considered nonlinear Hertzian contact forces, bearing clearances, and other nonlinear factors in this model to establish a coupled fault dynamics model of rotor-support-static subsystem with loose touching and friction. The Runge-Kutta method was used to solve the model. It is found that the looseness of the bearing at low rotational speed can lead to the generation of multi-harmonic frequency components in the system. Frictional harmonic components can be excited at high rotational speeds. This finding is of great value for effective identification of friction faults. Yang et al. [22] further discussed the effectiveness of nonlinear vibration dampers in suppressing bearing pedestal looseness and rotor frictional vibrations based on the above model. It was shown that the nonlinear damper can effectively suppress such nonlinear vibrations and also mitigate the adverse effects from coupling faults. The literature [23] proposes a nonlinear measure for the evaluation of bearing pedestal looseness in a rotor system at constant speed. The Taylor expansion of the static equilibrium system converts the nonlinear term into a linear term, and the difference between the linear and nonlinear models is quantified by evaluation. The accuracy of the method is demonstrated by solving the dynamic response using the Runge-Kutta method and verifying the reliability of the method through experiments. The literature [24] derived a formula for calculating the time-varying stiffness of the contact surface during bolt looseness and calculated the effect of time-varying joint stiffness on the steady-state response of the rotor, and the findings of this study will provide a theoretical basis for the detection of bearing bolt looseness in rotating components in large rotating machinery. Ma et al. [25] proposed a finite element model of the rotor system. The variation law of the rotor bifurcation diagram when the bearing pedestal looseness displacement varies with the loosening clearance is discussed. The results show that when the rotor presents higher order harmonic components as the slack clearance decreases, and the combined frequency components present a continuous spectrum. Throughout the above research results, we find that scholars have some research results on the mechanism of bearing loosening failure of rotor system, but no research reports on the characterization of system dynamics at the early stage of bearing looseness.

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In this paper, we establish a dynamic model of the dual-rotor-mediated bearing system based on Newton’s law of motion, and simulate the early failure of bearing pedestal looseness at different rotational speeds based on this model. The Newmark-β method is used for numerical calculations to analyze the time-frequency dynamics characterization of high and low pressure rotors and their axial trajectories. A dual-rotor bearing looseness simulation test bench is mounted to test the initial bearing looseness failure. The accuracy of the model is investigated by comparing the test and model simulation results. In this paper, an 8-degree-of-freedom model of a dual-rotor system containing the early failure of bearing pedestal looseness is developed in Sect. 2. In Sect. 3.1, the transient dynamics simulation of the loosened bearing pedestal system with the low pressure rotor speed ni = 1500 rpm and high pressure rotor speed no = 300 rpm at low speed, and the low pressure rotor speed ni = 5300 rpm and high pressure rotor speed no = 15300 rpm at high speed is implemented. In Sect. 3.2, a dual-rotor bearing pedestal looseness fault simulation test bench is built to verify the accuracy of the simulation calculation. The main conclusions are given in the last section.

2 Dynamic Modeling of Bearing Pedestal Looseness in Dual-Rotor System In this paper, a simplified model of a dual-rotor inter-shaft bearing system with bearing looseness fault is established, which reduces the bearing support forces at both ends of the rotor to linear elastic forces and damping forces. The inter-shaft bearing is supported between the high and low pressure rotor, and the Hertz contact force and radial clearance of the intermediate bearing are considered in the model. The simplified model is shown in Fig. 1.

Fig. 1. Model of dual-rotor inter-shaft bearing system

Based on the nonlinear Hertz contact theory, the self-weight of the high and low pressure rotor, the radial load, the vibration of the high and low pressure rotor in both

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horizontal and vertical directions, and the rotation in both axial and radial directions are considered. Considering the centrifugal force caused by the rotor misalignment and the eccentricity of the inner and outer rings. Let bearing No. 1 be the loose bearing, and l1 − l5 be the distance from bearing No. 1 to the low-pressure rotor, the distance from bearing No. 2 to the low-pressure rotor, the distance from bearing No. 3 to the high-pressure rotor, the distance from the inter-shaft bearing to the high-pressure rotor, and the distance from the inter-shaft bearing to the low-pressure rotor, respectively. The eight-degree-of-freedom failure dynamics model is established according to Newton’s law of motion. ⎧ dα d 2x dx ⎪ m1 21 + Fkx + K2 (x1 + α1y l2 ) + C2 ( dt1 + dt1y l2 ) = m1 ω12 e1 cos(ω1 t) − Fx − m1 g ⎪ ⎪ dt ⎪ ⎪ 2 ⎪ d y dy dα ⎪ ⎪ m1 21 + Fky + K2 (y1 − α1x l2 ) + C2 ( dt1 − dt1x l2 ) = m1 ω12 e1 sin(ω1 t) − Fy ⎪ dt ⎪ ⎪ 2 ⎪ d α d 2 y1 d α1x d α1x 1y ⎪ ⎪ ⎪ ⎪ Jd 1 dt 2 + Jp1 ω1 dt + Fky l1 − K2 l2 (y1 − α1x l2 ) − C2 l2 ( dt 2 − dt l2 ) = Fy (l5 − l2 ) ⎪ 2 ⎪ 2 ⎪ ⎨ J d α1y − J ω d α1x − F l + K l (x − α l ) + C l ( d x1 + d α1y l ) = −Fx (l − l ) p1 1 dt 2 2 1 1y 2 2 2 dt 2 2 5 d 1 dt 2 kx 1 dt 2 ⎪ m d 2 x2 + K (x − α l ) + C ( dx2 − d α2y l ) = m ω2 e cos(ω t) + F − m g ⎪ x ⎪ 2 dt 2 3 2 2y 3 3 dt 2 2 2 2 2 dt 3 ⎪ ⎪ ⎪ d 2 y2 dy2 d α2x ⎪ 2 e sin(ω t) + F ⎪ + K (y − α l ) + C ( − l ) = m ω m ⎪ y 2 dt 2 3 2 2x 3 3 dt 2 2 2 2 dt 3 ⎪ ⎪ ⎪ d α2y ⎪ d 2 α2x d 2 y2 d α2x ⎪ J + J ω + K l (y + α l ) + C l ( − l ) = −Fy l4 ⎪ p2 2 3 3 2 2x 3 3 3 3 d 2 ⎪ dt dt dt 2 dt 2 ⎪ ⎪ ⎪ d 2 α2y d α2y d α2x d 2 x2 ⎩ Jd 2 2 − Jp2 ω2 dt − K3 l3 (x2 + α2y l3 ) − C3 l3 ( 2 − dt l3 ) = Fx l4 dt

(1)

dt

In Eq. 1, except for the constant length parameter, the parameter with subscript 1 represents the parameter of low pressure rotor, and the parameter with subscript 2 represents the parameter of high pressure rotor, x and y represent the displacement in vertical and horizontal directions, α is the angle, Jp and Jd represent the polar and radial rotational inertia, m is the mass, e is the eccentricity, ω is the angular velocity, K is the stiffness coefficient, C is the damping coefficient, F is the elastic recovery force in the corresponding direction, g is the acceleration of gravity, t is the vibration time, Fkx , Fky are the direction loosening additional force, Fx , Fy are the inter-shaft bearing x direction and y direction elastic recovery force, all the above parameters units are calculated in accordance with the international units of the scale. The simulated bearing pedestal early looseness dynamics model in this paper is shown in Fig. 2. In the above figure, the bolt on the left side of the bearing pedestal is loosened and the bearing deflects around the xy plane normal direction by an angle of α. Since the bearing pedestal is loosened at the early stage, the value of α tends to be close to 0. In the loosening failure model, the rotor center is set to 1 point, the center of the unloosened side is 2 points, the center of the loosened side is 3 points, the early coordinates of the bearing pedestal center of gravity are (zx , zy ), the contact stiffness and damping of the rotor and bearing are K1x , K1y , C1x , C1y , the loosening side occurs Bearing pedestal from the base upward movement to make the bolt stretch, corresponding to contact stiffness and damping are K2x , K2y , C2x , C2y , not loose side corresponding to contact stiffness and damping are K3x , K3y , C3x , C3y , bearing pedestal center of gravity to bearing pedestal bottom distance is h, rotor center to bearing pedestal bottom distance is H , bearing pedestal width is L, all the above parameters units are calculated in accordance with the international units of the scale.

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Fig. 2. Bearing pedestal early looseness dynamics model

According to the geometric parameter relationship in Fig. 2 d1x = Zx − Sx − (H − h)α d1y = Sy − Zy − (H − h)(1 − d2x = −Zx − hθ − d2y = Zy + H − h −

α2 ) 2

Lα 2 4

Lα 2 Lα + 2 2

d3x = −Zy − hθ + d3y = Zy + H − h +

(2)

Lα 2 4

Lα 2 Lα + 2 2

(3) (4) (5) (6) (7)

According to Hooke’s law F1x = K1x d1x F2x = K2x d2x F3x = K3x d3x Substituting the above equation to get

F1y = K1y d1y F2y = K2y d2y F3y = K3y d3y

(8)

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Fkx = (K3x − K1x − K2x )zx − (K1x + K2x )hα +

K2x − K1x 2 Lα − K3x [x1 + (H − h)α] 4

Lα 2 hα 2 + ) 2 2 Lα 2 α hα 2 +K2y (H − h + + ) + K3y [y1 − (H − h)(1 − )2 ] 2 2 2

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(9)

Fky = (K1y + K2y − K3y )zy + K1y (H − h −

(10)

For computational simplicity, Eq. 1 is converted into a matrix expression. ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x˙ o xo x¨ o ⎢ y˙ ⎥ ⎢y ⎥ ⎢ y¨ ⎥ ⎢ o⎥ ⎢ o⎥ ⎢ o⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x˙ i ⎥ ⎢ xi ⎥ ⎢ x¨ i ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ y˙ ⎥ ⎢y ⎥ ⎢ y¨ ⎥ ⎢ i⎥ ⎢ i⎥ ⎢ i⎥ [M]⎢ ⎥ + [C]⎢ ⎥ + [K]⎢ ⎥ = [Q], where the mass matrix [M], the damping ⎢ x˙ 1 ⎥ ⎢ x1 ⎥ ⎢ x¨ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ y˙ ⎥ ⎢y ⎥ ⎢ y¨ ⎥ ⎢ 1⎥ ⎢ 1⎥ ⎢ 1⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ x˙ 2 ⎦ ⎣ x2 ⎦ ⎣ x¨ 2 ⎦ y¨ 2 y˙ 2 y2 matrix [C], the stiffness matrix [K] and the external force array [Q] are ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ [M] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

m1

⎡

⎤ m1 ω12 e1 cos(ω1 t) − Fx − m1 g − Fkx ⎢ ⎥ ⎥ m1 ω12 e1 sin(ω1 t) − Fy − Fky m1 ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ Fy (l5 − l2 ) − Fky l1 Jd 1 ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ Jd 1 −F (l − l ) + F l ⎥ x 2 1 5 kx ⎥, ⎥, [Q] = ⎢ ⎢ ⎥ 2 ⎥ m2 ⎢ m2 ω2 e2 cos(ω2 t) + Fx − m2 g ⎥ ⎥ ⎢ ⎥ 2 ⎥ m2 m ω e sin(ω t) + F ⎢ ⎥ y 2 2 2 ⎥ 2 ⎢ ⎥ ⎦ Jd 2 ⎣ ⎦ −Fy l4 Jd 2 F x l4 ⎡ ⎤ C2 C 2 l2 ⎢ ⎥ C2 −C2 l2 ⎢ ⎥ ⎢ ⎥ 2 J ω −C l −C l 22 2 2 p1 1 ⎢ ⎥ ⎢ ⎥ 2 −Jp1 ω1 C2 l2 ⎢ C 2 l2 ⎥ [C] = ⎢ ⎥ ⎢ C3 −C3 l3 ⎥ ⎢ ⎥ ⎢ ⎥ C 3 C 3 l3 ⎢ ⎥ ⎣ C3 l3 C3 l 2 Jp2 ω2 ⎦ ⎤

−C3 l3 ⎡ K2 − K3x K 2 l2 ⎢ K + K −K l 2 3y 22 ⎢ ⎢ K3y l1 − K2 l2 K2 l22 ⎢ ⎢ K2 l22 ⎢ K l + K 2 l2 [K] = ⎢ 3x 1 ⎢ ⎢ ⎢ ⎢ ⎣

3

−Jp2 ω2 C3 l32

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ K3 −K3 l3 ⎥ ⎥ ⎥ K 3 K 3 l3 ⎥ ⎦ K3 l3 K3 l32 −K3 l3 K3 l32

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According to the mechanical properties of rolling bearings, the contact between rolling element and raceway is regarded as nonlinear Hertz contact [26, 27]. The dynamics of the actual bearing is more complex, and for the purpose of studying the vibration of the bearing, the bearing dynamics model established by Patil is referred to, and the actual bearing is equated to a mass-spring-damping system, and the effects of the cage and rolling element masses on the system are neglected. The simplified bearing dynamics model is shown in Fig. 3.

Fig. 3. Simplified model of inter-shaft bearing

According to the relationship between the nonlinear load and displacement of the bearing Q = Kδ n

(11)

The size of n is related to the type of bearing, when it is deep groove ball bearing, n is 3/2, when it is cylindrical roller bearing, n is 10/9. When the bearing is subjected to external load, the relative radial deformation displacement δn will be generated between the two raceways, its size is equal to the sum of radial displacement generated between the rolling body and the inner and outer raceways, so there is δn = δi + δo The total stiffness in Eq. (11) is

n 1 K= (1/Ki )1/n + (1/Ko )1/n

(12)

(13)

Ki and Ko in the above equation are the contact stiffness of the rolling element and the inner and outer raceway respectively. For ball bearings, Ki and Ko are calculated as Ki = Ko = 8.06 × 104 l 8/9

(14)

where l is the length of the roller, Ki and Ko into the formula (13) can be calculated in the non-linear load and displacement relationship between the total stiffness K. xi , yi

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and xo , yo for the inner and outer ring of the bearing in the X and Y direction of and displacement, so the inner and outer ring relative to a rolling body produced by the radial deflection δ for δ = (xi − xo ) cos θbi + (yi − yo ) sin θbi − Cr

(15)

Cr is the radial clearance of the bearing, θbi is the angle of the i rolling body relative to the x axis. From Eq. (2.45), the radial contact force Qi generated by a certain rolling body is  10/9 Qi = K (xi − xo ) cos θbi + (yi − yo ) sin θbi − (Cr + Ht)

(16)

The radial force Qi generated by each rolling body in the direction of X and Y is the sum of the component forces Fx and Fy . Fx = K

Z   10/9 (xi − xo ) cos θbi + (yi − yo ) sin θbi − Cr cos θbi

(17)

i=1

Fy = K

Z   10/9 (xi − xo ) cos θbi + (yi − yo ) sin θbi − Cr sin θbi

(18)

i=1

x and y are the displacement of the inner and outer ring in the direction of X and Y. λ1 is the switch quantity of whether the roller is in contact with the raceway, if contact λ1 = 1, not contact λ1 = 0.

3 Analysis of Dual-Rotor System in the Early Stage of Bearing Pedestal Looseness 3.1 Simulation Analysis of the Dynamics of the Dual-Rotor System at the Early Stage of Bearing Pedestal Looseness In this paper, based on the dynamics equations of the eight-degree-of-freedom dual-rotor system, the Newmark-β method is used to solve the established model and realize the transient dynamics simulation of the bearing pedestal early loosening system at different speeds. The parameters of the high and low pressure rotors and intermediate bearings in this paper are as follows. M1 = 2.5 kg, M2 = 3 kg, Mo = 0.1 kg, Mi = 0.05 kg, Jp1 = 2.5 kg · m2 , Jp2 = 4.5 kg · m2 , Jd 1 = 2 kg · m2 , Jd 2 = 4 kg · m2 , k1 = 6 × 107 N/m, k2 = 6 × 107 N/m, k3 = 6 × 107 N/m, C1 = 600 N · m/s, C2 = 600 N · m/s, C3 = 600 N · m/s, Co = 300 N · m/s, Ci = 300 N · m/s, e1 = 1 × 10−6 mm, e2 = 5 × 10−7 mm, Z = 11, Cr = 1.2 × 10−5 mm, Di = 1.5 × 10−3 mm, Do = 2.8 × 10−3 mm, D = 2.5 × 10−3 mm, d = 5 × 10−3 mm, α = 0,

The inter-shaft bearing is a cylindrical roller bearing of NSK-NU202EM type with the parameters shown in Table 1.

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Numerical value

Inner ring diameter

15 mm

Outer ring diameter

35 mm

Section circle diameter

25 mm

Roller diameter

5 mm

Number of rollers

11

Contact angle

0

Radial clearance

12 μm

The low pressure rotor speed is ni = 1500 rpm and high pressure rotor speed is no = 300 rpm at low speed, and the low pressure rotor speed is ni = 5300 rpm and high pressure rotor speed is no = 15300 rpm at high speed, and the corresponding rotational frequency is shown in Table 2. The time domain diagrams, envelope spectra and axial trajectory diagrams of the low speed bearing pedestal early looseness fault low pressure rotor, high pressure rotor and axial trajectory diagrams are shown in Fig. 4 and Fig. 5, and the time domain diagrams, envelope spectra and axial trajectory diagrams of the high speed bearing pedestal early looseness fault low pressure rotor, high pressure rotor are shown in Fig. 6 and Fig. 7. Table 2. Rotor frequency table Low speed high pressure rotor frequency

Low speed low pressure rotor frequency

High speed high pressure rotor frequency

High speed low pressure rotor frequency

Low speed frequency summation

High speed frequency summation

5 Hz

25 Hz

255 Hz

88.3 Hz

30 Hz

343.3 Hz

Fig. 4. Time domain diagram, envelope spectrum and axial trajectory diagram of low pressure rotor with early looseness of low speed bearing pedestal failure

The time domain signals of the simulation results show good periodicity as can be seen from Fig. 4 to Fig. 7. For the proposed dual-rotor system model, the clipping

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Fig. 5. Time domain diagram, envelope spectrum and axial trajectory diagram of high pressure rotor with early looseness of low speed bearing pedestal failure

Fig. 6. Time domain diagram, envelope spectrum and axial trajectory diagram of low pressure rotor with early looseness of high speed bearing pedestal failure

Fig. 7. Time domain diagram, envelope spectrum and axial trajectory diagram of high pressure rotor with early looseness of high speed bearing pedestal failure

phenomenon occurs at the early looseness of the bearing housing. The rotor envelope spectrum shows the high and low pressure rotor frequencies fo , fi , the sum of frequencies fo + fi and the double frequency corresponding to the loose bearing pedestal supported rotor. Figure 4 shows the difference between high and low pressure transient fo = 4.96 Hz, fi = 24.91 Hz and the theoretical value respectively 0.04 Hz, 0.09 Hz, Fig. 5 shows the difference between high and low pressure transient fo = 4.97 Hz, fi = 24.91 Hz and the theoretical value respectively 0.03 Hz, 0.09 Hz, Fig. 6 shows the difference between high and low pressure transient fi = 86.98 Hz, fo = 255.1 Hz and the theoretical value respectively 1.35 Hz, 0.1 Hz, and Fig. 7 shows the difference between high and low pressure transient fi = 86.06 Hz, fo = 254.8 Hz and the theoretical value respectively 2.27 Hz, 0.2 Hz. Due to the connection of the high and low pressure rotors through inter-shaft bearing in a dual-rotor system, the rotational frequency is modulated to produce the sum of the rotational frequencies fo + fi during system operation. The low pressure rotor time domain amplitude is significantly higher than the high pressure rotor

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time domain amplitude, which is because the low pressure rotor bearing pedestal looseness causes the dual rotor system to produce acceleration signals of higher amplitude. High and low pressure rotor envelope spectrum corresponding to the measured rotor fundamental frequency amplitude is higher, the fundamental frequency corresponding to the most obvious two-fold frequency characteristics, low pressure bearing pedestal looseness in the early envelope spectrum of low pressure rotor speed two-fold frequency. From the axial trajectory diagram, we can see that the low pressure rotor axial trajectory is more complex at the beginning of the bearing looseness, the high pressure rotor axial trajectory at low speed is clearer in an elliptical shape, and the low pressure rotor axial trajectory is more chaotic due to the bearing looseness. At high speed, the low pressure rotor shaft trajectory is more chaotic than the low speed shaft trajectory, while the high pressure rotor shaft trajectory has not changed much. 3.2 Experimental Analysis of the Dynamics of the Dual-Rotor System at the Early Stage of Bearing Pedestal Looseness According to the actual working condition of the double-rotor system, a double-rotor bearing pedestal looseness fault simulation test bench was built. The structure of the test bench is shown in Fig. 8. The inter-shaft bearing of this test system is installed between the high and low pressure rotors, and the rotors are driven by two sets of mutually independent control motors.

Fig. 8. Dual-rotor bearing pedestal looseness fault test bench

The LMS SCADAS Recorder SCR202 data acquisition system from Siemens was selected for this experiment to acquire vibration signals, as shown in Fig. 9. This acquisition system has 2 speed channels, 8 vibration signal acquisition channels, a maximum sampling frequency of 102400 Hz, and 24-bit sampling accuracy. The vibration signals can be accurately acquired, and the data can be processed and characterized. In this experiment, six acceleration sensors, all model 333B30, were used, three of which had sensitivities of 98.6 mV/g, 99 mV/g and 97.6 mV/g and were mounted on the bearing housing near the low-pressure rotor, while the remaining three were mounted on

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Fig. 9. LMS data collector

the bearing housing near the high-pressure rotor to detect the vibration of the bearing in three directions. The sensitivity is 99.3 mV/g, 98.3 mV/g and 100.4 mV/g, and the eddy current sensor sensitivity is 1 mV/mm and 0.99 mV/mm, respectively. The mounting position is shown in Fig. 10.

Fig. 10. Acceleration sensor, eddy current sensor arrangement schematic

The low pressure rotor bearing pedestal bolt is slightly loosened, the inner ring speed is taken as ni = 1500 rpm, the outer ring speed is taken as no = 300 rpm. Based on the fault dynamics simulation to simulate the low speed bearing pedestal early looseness fault low pressure rotor, high pressure rotor time domain diagram, envelope spectrum diagram as shown in Fig. 11 and Fig. 12. From Fig. 11 and Fig. 12, it can be seen that the test results show a series of errors due to the complex structure of the test stand, which leads to a class of periodicity in the time domain. Under the low speed condition, there is a clipping phenomenon at the early stage of bearing pedestal looseness in the double rotor test bench. The rotor envelope spectrum shows the high and low pressure rotor frequencies fo , fi , the sum of frequencies fo + fi , and the duplex frequency corresponding to the loose bearing pedestal supported rotor. Figure 11 and Fig. 12 show the high and low pressure rotor frequencies fo = 5 Hz, fi = 25 Hz, both of which are the same as the theoretical values.

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Fig. 11. Time domain diagram, envelope spectrum and axial trajectory diagram of low pressure rotor with early looseness of low speed bearing pedestal failure

Fig. 12. Time domain diagram, envelope spectrum and axial trajectory diagram of high pressure rotor with early looseness of low speed bearing pedestal failure

The difference with the simulated value is 0.04 Hz, 0.09 Hz respectively, due to the self-excited vibration of each component and structural part of the experimental table under the action of external forces, and the existence of force imbalance and dynamic imbalance factors in the experimental table, so the experimental time domain signal amplitude is larger than the simulated value. Due to the high pressure shaft rigidity, the high pressure rotor axis trajectory is more stable than the low pressure rotor axis trajectory. Compared with the simulation, it basically conforms to the simulation analysis results. The acceleration amplitude of the test time domain and spectrum is higher than that of the simulation time domain and spectrum because we choose the measurement point on the nearest bearing pedestal to the low pressure rotor and high pressure rotor, as shown in Fig. 10. The acceleration amplitude is high because of the strong signal feedback on the bearing pedestal due to the binding force generated by the bolt restraint. The conclusion of this test is reliable from the eigenfrequency point of view, which proves the accuracy of the model built in this paper.

4 Conclusion In this paper, an 8-degree-of-freedom dynamics model is established for the early looseness fault of the bearing pedestal of the dual-rotor system, and the vibration equations of the system are obtained. Simulation calculations are performed using the Newmark-β method to investigate the vibration characteristics of the high pressure rotor and low pressure rotor at different rotational speeds. The fault characteristic frequencies of time domain diagram, axial trajectory diagram and envelope spectrum diagram are analyzed, and the following conclusions are obtained.

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1. This paper establishes a dynamics model of early looseness fault of bearing pedestal in 8-degree-of-freedom dual-rotor system. The simulation results based on this model are basically consistent with the experimental values. The simulation results have the same law with the test time domain waveform and envelope spectrogram. It proves the accuracy of the dynamics model of the early looseness fault of the bearing pedestal of the dual-rotor system. 2. There is an obvious “clipped” signal in the time domain waveform of bearing pedestal early looseness. In the envelope spectrum, there are high and low pressure rotor rotational frequency, the sum of frequency and the characteristic frequency of 2-fold of rotational frequency. In case of early looseness of the low pressure bearing pedestal, 2-fold of rotational frequency of the low pressure rotor speed appears in the envelope spectrum. 3. Bearing pedestal early looseness low pressure rotor axis trajectory is more complex. At low speed, the high pressure rotor axis trajectory is clearer like elliptical, low pressure rotor because of loose bearing axis trajectory is more chaotic. In high speed, low pressure rotor axis trajectory compared to low speed axis trajectory more chaotic, while the high pressure rotor axis trajectory has not changed much.

References 1. Mu, X., Wang, Y., Yuan, B., Sun, W., Liu, C., Sun, Q.: A New assembly precision prediction method of aeroengine high-pressure rotor system considering manufacturing error and deformation of parts. J Manuf Syst 61, 112–124 (2021) 2. Yu, P., Hou, L., Wang, C., Chen, G.: Insights into the nonlinear behaviors of dual-rotor systems with inter-shaft rub-impact under co-rotation and counter-rotation conditions. Int J Nonlin Mech 140, 103901 (2022) 3. Wang, Q., Wu, W., Zhang, F., Wang, X.: Early rub-impact fault detection of rotor systems via deterministic learning. Control Eng Pract 124, 105190 (2022) 4. Yu, M., Chen, W., Lu, Y.: Fault diagnosis and location identification of rotor–stator rub-impact based on Hjorth parameters. Eng Fail Anal 138, 106299 (2022) 5. Kumar, P., Tiwari, R.: Dynamic analysis and identification of unbalance and misalignment in a rigid rotor with two offset discs levitated by active magnetic bearings: a novel trial misalignment approach. Propulsion and Power Research 10(1), 58–82 (2021) 6. 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) 7. Jiang, Y., Huang, W., Luo, J., et al.: An improved dynamic model of defective bearings considering the three-dimensional geometric relationship between the rolling element and defect area. Mech. Syst. Signal Process. 129(15), 694–716 (2019) 8. Niu, L., Cao, H., Hou, H., et al.: Experimental observations and dynamic modeling of vibration characteristics of a cylindrical roller bearing with roller defects. Mech. Syst. Signal Process. 138, 106553 (2020) 9. Powers, K., et al.: A new first-principles model to predict mild and deep surge for a centrifugal compressor. Energy 244, 123050 (2022) 10. Yoon, J., Wilailak, S., Bae, J., Lee, C., Kim, I.: Surge analysis in a centrifugal compressor using a dimensionless surge number. Chem. Eng. Res. Des. 164, 240–247 (2020) 11. Peng, G., Lei, H., et al.: Local defect modelling and nonlinear dynamic analysis for the inter-shaft bearing in a dual-rotor system. Appl. Math. Model. 68, 29–47 (2019)

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12. Jayakanth, J., Chandrasekaran, R., Pugazhenthi, R.: Impulse excitation analysis of material defects in ball bearing [J]. Materials today Proceedings. 39, 717–724 (2021) 13. Soualhi, A., Medjaher, K., Celrc, G., et al.: Prediction of bearing failures by the analysis of the time series. Mech. Syst. Signal Process. 139, 106607 (2020) 14. Yuqi, L., Zhong, L., Jiaxi, L., et al.: Dynamic modeling and stability analysis of a rotor-bearing system with bolted-disk joint. Mech. Syst. Signal Process. 158, 107778 (2021) 15. Sameera, M., Muzakkir, S., Sidra, K.: Theoretical and experimental analyses of vibration impulses and their influence on accurate diagnosis of ball bearing with localized outer race defect. J. Sound Vib. 513(24), 116407 (2021) 16. Qin, Y., Cheng, C.L., Xing, G.W., Yang, Y.W., Hai, Z.C.: Multiple-degree-of freedom dynamic model of rolling bearing with a localized surface defect. Mech. Mach. Theory 154(10), 40–47 (2020) 17. Muszynska, A., Goldman, P.: Chaotic responses of unbalanced rotor/bearing/stator systems with looseness or rubs. Chaos, Solitons Fractals 5(9), 1683–1704 (1995) 18. Chu, F., Tang, Y.: STABILITY AND NON-LINEAR RESPONSES OF A ROTOR-BEARING SYSTEM WITH PEDESTAL LOOSENESS. J Sound Vib 241(5), 879–893 (2001) 19. Luo, Y.G., Zhang, S.H., Wu, B., Hu, H.Y.: Stability Analysis of Nonlinear Stiffness RotorBearing System with Pedestal Looseness Fault. Appl. Mech. Mater. 483, 285–288 (2013) 20. Zhang, H., Lu, K., Zhang, W., Fu, C.: Investigation on dynamic behaviors of rotor system with looseness and nonlinear supporting. Mech Syst Signal Pr 166, 108400 (2022) 21. Chen, G., Li, C.G., Wang, D.Y.: Nonlinear Dynamic Analysis and Experiment Verification of Rotor-Ball Bearings-Support-Stator Coupling System for Aeroengine With Rubbing Coupling Faults. J. Eng. Gas Turbines Power 132, 022501–022511 (2010) 22. Yang, Y., Chen, G., Ouyang, H., Yang, Y., Cao, D.: Nonlinear vibration mitigation of a rotorcasing system subjected to imbalance–looseness–rub coupled fault. Int J Nonlin Mech 122, 103467 (2020) 23. Jiang, M., Wu, J., Peng, X., Li, X.: Nonlinearity measure based assessment method for pedestal looseness of bearing-rotor systems. J Sound Vib 411, 232–246 (2017) 24. Qin, Z., Han, Q., Chu, F.: Bolt loosening at rotating joint interface and its influence on rotor dynamics. Eng Fail Anal 59, 456–466 (2016) 25. Ma, H., Zhao, X., Teng, Y., Wen, B.: Analysis of Dynamic Characteristics for a Rotor System with Pedestal Looseness. Shock Vib 18(1–2), 13–27 (2011) 26. Hai, M.Y., Lei, H., Peng, G., et al.: Nonlinear resonance characteristics of a dual-rotor system with a local defect on the inner ring of the inter-shaft bearing. Chin. J. Aeronaut. 34, 110–124 (2021) 27. Hai, M.Y., Lei, H., Peng, G., et al.: Combined resonance characteristics of double rotor system with local defects in inner ring of intermediate bearing. Journal of Aeronautical power 35(09), 1964–1976 (2020)

Similarity Design and Behavior Prediction of Rotor Systems Subject to Non-uniform Preloads Runchao Zhao1 , Yeyin Xu2 , Zhitong Li1 , Zhaobo Chen1 , and Yinghou Jiao1(B) 1 Harbin Institute of Technology, Harbin 150000, China

[email protected] 2 Xi’an Jiaotong University, Xi’an 710049, China

Abstract. In the study of rotor dynamics, it is of great theoretical and practical significance to design a scaled rotor system that can reflect the dynamic characteristics and behavior of the prototype accurately. However, modern rotor systems are usually assembled by bolted structures, the discontinuity and complexity of its structure bring difficulties to the similarity design of scaled rotors. In this regard, this paper proposes a similarity design method that can accurately predict the dynamic behavior of discontinuous rotor systems. First, the rotor structure is treated continuously using the equivalent material modeling theory. Then, based on the similarity theory, the scaling factors of key structural and physical parameters are obtained. Finally, two kinds of scaled rotor systems under different preload conditions with different sizes are designed. In addition, the first-two critical speeds of the prototype are predicted. The calculated results show that the maximum deviation of the rotor model with different reduced scale dimensions in predicting the critical speed of the prototype is less than 4.03%. The reduced scale rotor model obtained by using this similarity design method can accurately predict the dynamic behavior of the prototype. This method provides theoretical guidance and reference for the design of the reduced scale model of the discontinuous rotor system. Keywords: Similarity design · Behavior prediction · Rotor system · Critical speed · Contact modeling

1 Introduction Rotor systems play important roles in the field of industrial generation and power drive. Modern large-scale rotor systems usually adopt bolted structures, the rotor is assembled by applying proper preload force on the tie rods. Therefore, the rotor cannot be regarded as a whole one, which brings difficulties to modeling, dynamic solution as well as scaling design. The schematic diagram of the discontinuous bolted rotor structure is shown in Fig. 1. In terms of modeling of discontinuous structures, Zhang et al. [1] introduced the concept of surface contact stiffness, then the relationship between contact stiffness and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 91–100, 2024. https://doi.org/10.1007/978-3-031-40459-7_6

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Tie Rods

Nuts

Bolted structure

Fig. 1. Schematic diagram of the discontinuous bolted rotor structure

preload was given. After that, the tie rod rotor was modeled by finite element method to model, and the effectiveness of the modeling method was verified through modal tests. Gao et al. [2] established the spring-hinge model of the contact surface by using the contact theory, this model takes the contact separation into account. Then they explored the influence of the contact effect on modal through and experiment. Wriggers et al. [3] modeled two elastomers by equivalent element method and verified the universality of the proposed method through a simulation case. Liu et al. [4] used the fractal model to model the contact interface, then they derived the expression of the lateral bending stiffness of the contact interface. Xu et al. [5] studied the influence of preload on the dynamic characteristics of the tie rod rotor, the results showed that the preload within a small range had a great impact on the natural frequency of the rotor system. Zhao et al. [6] modeled the rods fastened rotor system by contact theory and equivalent material layer, the influence of preload on the rotor structure was reflected by the adjust the parameters of equivalent material layer. In terms of similarity design of rotor system, based on dimensional analysis, Wu et al. [7] systematically presented the scaling laws of each unit of the rotor system for the first time, and they studied the response of the rotor-bearing system under forced vibration. Luo et al. [8] studied the scaling method of rotor system considering the influence of gravity, the effectiveness of the proposed method was verified through calculation and experimental results. Li et al. [9] considered the scaling design method of the dualrotor system with bolted structures, they predicted the dynamic characteristics of the aeroengine rotor. Zhao et al. [10] proposed a scaling design method for the rotor system with limited parameters through sensitivity analysis and optimization algorithm, the accuracy of the proposed method was verified through modal experiments. At present, there is little research on scaling design of rotor systems with discontinuous structures. This paper combines contact mechanics model and scaling laws to solve the similarity design problem of discontinuous rotors. The flow chart of the proposed method is shown in Fig. 2. This paper contains five sections. In Sect. 2, the contact mechanics relationship between bolt structures is derived, an equivalent material modeling method is introduced to establish continuous model of bolted rotors. In Sect. 3, the scaling factors of the rotor system are obtained based on the similarity theory, the prototype rotor system model studied in this paper is established, two scaled rotor models are obtained according to the similarity laws. In Sect. 4, the dynamic behaviors of the prototype are predicted by

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Discontinuous rotor system Contact mechanics model

Equivalent material method

Continuous rotor system Dynamic behavior prediction

Similarity theory

Scaling factor 1

Scaling factor 2

...

Scaling factor i

Scaled rotor system

Fig. 2. Flow chart for similarity design of rotor system with discontinuous structures.

scaled models, which verifies the effectiveness of the proposed design method. Finally, some conclusions are summarized in Sect. 5.

2 Continuous Structure Treatment Due to the discontinuity of actual gas turbine rotor structure, the traditional similarity design method is no longer applicable. If the contact characteristics between the components are ignored, the lateral stiffness and torsional stiffness of the rotor system will inevitably increase, the resulting scaled model will further magnify the error, which will greatly reduce the accuracy of scaled models. In order to establish the prototype model accurately, it is necessary to consider the contact effect between disks. 2.1 Contact Mechanics Modeling The gas turbine rotor system usually adopts bolted structure. Each disk is assembled by applying preload on the rods. Adjusting the preload will affect the lateral bending stiffness of the rotor structure. Based on the contact theory, the contact mechanics model between the disks considering the full deformation process is obtained, the relationship between the preload and the contact stiffness is given. For a disk with a given surface roughness, the relationship between contact force and contact deformation is [6]  +∞ 2 4 0.5 1 − (z−μ) ERm (z − h)1.5 e 2σ 2 dz (1) f = λA √ 3 2π σ h where, λ is the areal density of asperities, A is the contact area of disk, σ is the standard deviation of asperity height, E is the elastic modulus of asperity, Rm is asperity radius, h is the distance of two surfaces, μ is the average height of asperities. The contact stiffness between the disks can be expressed as [6]  +∞ 2 4 0.5 1 − (z−μ) ERm (z − h)0.5 e 2σ 2 dz (2) k = λA √ 3 2π σ h

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Furthermore, according to Eq. (1) and Eq. (2), the relationship between contact load and contact stiffness can be obtained as f (d ) ∼ k(d ) k˜ = a0 +

n 

ai f i

(3) (4)

i=0

2.2 Structural Continuity Modeling Method For rods fastened rotor, the disks are assembled together by applying a proper preload force on the tie rods. In this section, the modeling method based on equivalent material layer is introduced to realize the continuous modeling of rotor system with discontinuous structure. The idea of this modeling method is that according to the contact stiffness between two components, it is equivalent to a thin layer element with characteristic material parameters, the change of equivalent material layer parameters reflects the weakening effect of contact effect on the rotor stiffness. The length, mass and other physical structure parameters of the rotor remain unchanged before and after equivalence. According to the published research [6], the relationship between the elastic modulus of the equivalent material layer and the contact stiffness is:   li + lj Ei Ej k˜ (5) Eequ = li Ej k˜ + lj Ei k˜ + Ei Ej A where, Eequ is the elastic modulus of equivalent material layer, E i and E j are the elastic modulus of disk i and disk j respectively, li and l j are the thickness of disk i and disk j respectively, k˜ is the contact stiffness. The relationship between Poisson’s ratio of equivalent material layer and contact stiffness is     υi + υj A υj lj Eequ υi li   + + (6) υequ = ˜ Ei Ej l i + lj 2k where, υequ is the Passion’s ratio of equivalent material layer, υi and υj are the Passion’s ratio of disk i and disk j respectively. The density of equivalent material layer is ρequ =

ρi Vi + ρj Vj Vequ

(7)

where, ρequ is the density of equivalent material layer, ρi and ρj are the density of disk i and disk j respectively, Vi and Vj are the volume of disk i and disk j respectively, Vequ is the volume of equivalent material layer. The thickness of equivalent material layer is hequ = li + lj

(8)

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For a given working condition, the fitting relationship between the elastic modulus of the equivalent material layer and the preload can be obtained. Generally, within a certain preload range, the relationship can be expressed as a quadratic polynomial function 2 Eequ = p1 · frod + p2 · frod + p3

(9)

where, p1 , p2 and p3 are the coefficients of quadratic polynomial fitting respectively, f rod is the preload of rods.

3 Similarity Theory Buckingham PI similarity theory is often used in dimensional analysis. According to the Buckingham PI similarity theory and the differential equation of motion of the rotor system, the scaling law of the rotor system shaft element, disk element and support stiffness is derived in this section. The differential equation of motion of rotor shaft section before and after scaling is    2 2 3 2 ∂ ∂ ∂2 s s∂ δ s s∂ δ s ∂ δ s ∂ δ E + ρ + 2 i J J =0 (10) I A − ∂z 2 ∂z 2 ∂t 2 ∂z d ∂z∂t 2 ∂z d ∂z∂t

  s s s 2 P λρ λA λδ λsE λsI λsδ ∂2 P I P ∂ 2 δP E ρ p Ap ∂ Pδ 2 − 4 2 2 + 4 s s P P

(λl )λs λs∂ (z )  ∂ (z ) (λt ) λs λs λs ∂ (t ) (11)

3 δP Js δ Js δ ∂ ∂ P P i ∂ J P ∂ 2 δP d d J + 2 = 0  eml 2  eml 2 ∂z P d ∂z P ∂ (t P )2 d ∂z P ∂t P ∂z P (λsl )2 λst λl λt In the similarity design, the dimensions of each item in the equation shall be consistent, according to which the relationship between the scaling factors can be obtained as follows   s λs /λs 1/2 λ ρ D E λs = (12)  s 2 λl The scaling law of the disk is 2 λdD = λsl 

λdD

=

λdl

(13)

The scaling law of the support stiffness is λbk

 4 λsE λsD =  3 λsl

(14)

According to the relationship between the above scaling factors, the scaling design of rotor system supported by linear stiffness springs can be completed.

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4 Modeling and Verification The prototype model established in this study is shown in Fig. 3. The rotor system has a total length of 1700 mm and a total mass of 153.8 kg. The compressor section is assembled by 8 disks, the turbine section is assembled by 3 disks. The design working speed of the rotor is 3000 r/min. The rotor is supported by 3 bearings, the scaling factor of support stiffness is listed in Table 1. Equivalent material layer

Equivalent material layer

Bearing 2 Bearing 3

Bearing 1

Turbine

Compressor

Fig. 3. Diagram of prototype of the rotor system.

In this study, two scaled rotor models with different sizes were designed. λl is the scaling factor of rotor length. For M1 rotor, λl =0.5, which means that the length of M1 rotor (850 mm) is a half to prototype (1700 mm). Similarly, the length of M2 rotor (λl =0.33, 566.6 mm) is one third to prototype. According to the similarity theory in Sect. 2.2, the scaling factors of each model were obtained as shown in Table 1. Table 1. Scaling factors for different scaled rotor model. Rotor model

λ

λl

λD

λE

λk

Prototype

1

1

1

1

1

M 1 (850mm)

1

0.5

0.25

1

0.031250

M 2 (566.6mm)

1

0.33

0.11

1

0.004115

Three working conditions are set, Case1: the bolt preload of compressor section is 50 kN, the bolt preload of turbine section is 50 kN, the support stiffness is 1e7 N/m. Case 2: the bolt preload of compressor section is 30 kN, the bolt preload of turbine section is 100 kN, support stiffness is 5e7 N/m. Case 3: the bolt preload of compressor section is 100 kN, the bolt preload of turbine section is 30 kN, support stiffness is 1e8 N/m. Parameters in different cases are shown in Table 2.

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Table 2. Parameters of equivalent material layer under different preloads. Parameters

Case 1

Case 2

Case 3

f rod (comressor)

50 kN

30 kN

100 kN

f rod (turbine)

50 kN

100 kN

30 kN

E (comressor)

1.62e9 GPa

1.51e9 GPa

1.88e9 GPa

E (turbine)

1.62e9 GPa

1.88e9 GPa

1.51e9 GPa

Stiffness

1e7 N/m

5e7 N/m

1e8 N/m

1 BW 1 FW 2 BW 2 FW 1X

100

Frequency (Hz)

90 80 70 60

4247.3 r/min

50 40 30

1461.4 r/min

(b)

80 70 60 40 30 20

10

10 1000

2000

3000

4209.9 r/min

50

20 0

1 BW 1 FW 2 BW 2 FW 1X

100 90

Frequency (Hz)

(a)

0

4000

1444.6 r/min

1000

Rotation speed (r/min)

1 BW 1 FW 2 BW 2 FW 1X

100

Frequency (Hz)

90 80 70 60

4132.5 r/min

50 40 30

3000

4000

1412.1 r/min

20

(d)

1st 2nd

5000

Rotation speed (r/min)

(c)

2000

Rotation speed (r/min)

4247.3

4209.9

4132.5

4000 3000 2000 1461.4

1444.6

1412.1

1000

10 0

1000

2000

3000

Rotation speed (r/min)

4000

0

1 Prototype

2 M1

3 M2

Rotor system

Fig. 4. Campbell diagram of rotor systems in case 1, (a) prototype, (b) model 1, (c) model 2, (d) comparison of critical speeds.

4.1 Results of Case 1 Campbell diagram and critical speeds for different rotor models in case 1 are shown in Fig. 4. The first two critical speeds of the prototype are 1461.4 r/min and 4247.3 r/min. After 1:2 and 1:3 scale design, the first two critical speeds of M1 are 1444.6 r/min and 4209.9 r/min, and the first two critical speeds of M2 are 1412.1 r/min and 4132.5 r/min.

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In Fig. 4 (d), with the reduction of the scaled model size, the critical speed will decrease slightly, and the maximum error is −3.37% (1st for M2). The results show that when the preload of compressor section and turbine section is consistent, the scaling design method proposed in this paper can accurately reflect the critical speeds of the prototype. 4.2 Results of Case 2 Campbell diagram and critical speeds for different rotor models in case 2 are shown in Fig. 5. The first two critical speeds of the prototype are 1887.3 r/min and 5755.4 r/min. After 1:2 and 1:3 scale design, the first two critical speeds of M1 are 1869.5 r/min and 5624.2 r/min, the first two critical speeds of M2 are 1830.0 r/min and 5523.5 r/min. In Fig. 5 (d), when the preload of the compressor section is less than that of the turbine section, the change of the second critical speed is obvious, and the maximum error is − 4.03% (2nd for M2). The results show that the proposed method can accurately predict the first two forward whirls of the prototype. 1 BW 1 FW 2 BW 2 FW 1X

Frequency (Hz)

140 120

5755.4 r/min

100 80 60 40 1887.3 r/min

(b) 120

5624.2 r/min

100 80 60 40 1869.5 r/min

20 0

1 BW 1 FW 2 BW 2 FW 1X

140

Frequency (Hz)

(a)

20 1000

2000

3000

4000

5000

0

6000

1000

Rotation speed (r/min)

1 BW 1 FW 2 BW 2 FW 1X

Frequency (Hz)

140 120 5523.5 r/min

100 80 60 40

1830.0 r/min

20 0

1000

2000

3000

4000

5000

Rotation speed (r/min)

3000

4000

5000

6000

Rotation speed (r/min)

6000

1st 2nd

(d) 6000

Rotation speed (r/min)

(c)

2000

5755.4

5624.2

5523.5

5000 4000 3000 2000

1887.3

1869.5

1830

1000 0

1 Prototype

2 M1

3 M2

Rotor system

Fig. 5. Campbell diagram of rotor systems in case 2, (a) prototype, (b) model 1, (c) model 2, (d) comparison of critical speeds.

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4.3 Results of Case 3 Campbell diagram and critical speeds for different rotor models in case 3 are shown in Fig. 6. The first two critical speeds of the prototype are 2065.7 r/min and 6832.3 r/min. After 1:2 and 1:3 scale design, the first two critical speeds of M1 are 2049.5 r/min and 6663.2 r/min, the first two critical speeds of M2 are 2004.8 r/min and 6559.7 r/min. In Fig. 6(d), when the preload of compressor section is greater than that of turbine section, the change of second critical speed is greater than that of first critical speed, and the maximum error is −3.99% (2nd for M2).

Frequency (Hz)

150 6832.3 r/min

100

50

0

200 150 6663.2 r/min

100 50

2065.7 r/min

2000

4000

6000

1 BW 1 FW 2 BW 2 FW 1X

(b) Frequency (Hz)

1 BW 1 FW 2 BW 2 FW 1X

(a) 200

8000

0

10000 12000

2049.5 r/min

2000

Rotation speed (r/min)

Frequency (Hz)

200 150 6559.7 r/min

100 50 0

2004.8 r/min

2000

4000

6000

8000

10000 12000

Rotation speed (r/min)

(d)

6000

8000

10000 12000

1st 2nd

6832.3

7000

Rotation speed (r/min)

1 BW 1 FW 2 BW 2 FW 1X

(c)

4000

Rotation speed (r/min)

6663.2

6559.7

6000 5000 4000 3000 2000

2065.7

2049.5

2004.8

1000 0

1 Prototype

2 M1

3 M2

Rotor system

Fig. 6. Campbell diagram of rotor systems in case 3, (a) prototype, (b) model 1, (c) model 2, (d) comparison of critical speeds.

5 Conclusions In this paper, a scaling design method of rotor system considering the influence of inhomogeneous preload is proposed. First, the discontinuous rotor is treated continuously. Second, the scaled rotors of different sizes are obtained according to the scaling law. Finally, the effectiveness of the scaling design method is verified through the results of critical speeds. The main conclusions are as follows.

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(1) The equivalent material layer is introduced to realize the structural continuous treatment of the rod fastened rotor, the physical model that can be applied to the scaling treatment using PI similarity theory is obtained. (2) The scale factors of each unit of the rotor system are derived, the 1:2 and 1:3 scaled models considering the influence of preload are obtained. (3) The first two critical speeds of prototype rotor and scaled rotor are compared. The maximum critical speed deviation is -4.03% (2nd for M2 in Case 2). The results show that the proposed method can accurately predict the critical speeds of discontinuous rotor system.

References 1. Zhang, Y., Zhaogang, D., Shi, L., Liu, S.: Determination of contact stiffness of rod-fastened rotors based on modal test and finite element analysis. J. Eng. Gas Turbines Power 132, 094501 (2010) 2. Gao, J., Yuan, Q., Li, P., Feng, Z., Zhang, H., Lv, Z.: Effects of bending moments and pretightening forces on the flexural stiffness of contact interfaces in rod-fastened rotors. J. Eng. Gas Turbines Power 134, 102503 (2012) 3. Wriggers, P., Rust, W.T., Reddy, B.: A virtual element method for contact. Comput. Mech. 58, 1039–1050 (2016) 4. Liu, Y., Qi Yuan, P., Li, G.Z.: Modal analysis for a rod-fastened rotor considering contact effect based on double fractal model. Shock Vib. 2019, 1–10 (2019) 5. Xu, H., Yang, L., Xu, T.: Dynamic analysis of the rod-fastened rotor considering the characteristics of circumferential tie rods. Appl. Sci. 11, 3829 (2021) 6. Zhao, R., Jiao, Y., Chen, Z., Li, Z., Qu, X.: Nonlinear analysis of a dual-disk rotor system considering elastoplastic contact. Int. J. Non-Linear Mech. 141, 103925 (2022) 7. Wu, J.-J.: Prediction of lateral vibration characteristics of a full-size rotor-bearing system by using those of its scale models. Finite Elem. Anal. Des. 43, 803–816 (2007) 8. Luo, Z., Li, L., He, F., Yan, X.: Partial similitude for dynamic characteristics of rotor systems considering gravitational acceleration. Mech. Mach. Theory 156, 104142 (2021) 9. Li, L., Luo, Z., He, F., Qin, Z., Li, Y., Yan, X.: Similitude for the dynamic characteristics of dual-rotor system with bolted joints. Mathematics 10, 3 (2021) 10. Zhao, R., Jiao, Y., Qu, X.: Scaling design strategy for experimental rotor systems subjected to restricted support stiffness. Appl. Math. Model. 109, 265–282 (2022)

Vibration Reduction Optimization Design of an Energy Storage Flywheel Rotor with ESDFD Dafang Lin(B) , Siji Wang, Chengyang Wang, Zhoudian Chen, Yuan Liu, and Jinqi Zhang School of Power and Energy, Northwestern Polytechnical University, Xi’an 710129, China [email protected]

Abstract. To solve the excessive vibration of an energy storage flywheel rotor under complex operating conditions, an optimization design method used to the energy storage flywheel rotor with elastic support/dry friction damper (ESDFD) is proposed. Firstly, the dynamic model of the ESDFDs-energy storage flywheel rotor coupling system is established by using the finite element method. Secondly, through variables sensitivity analysis, support stiffness, support position, shaft stiffness and the position of flywheel are determined as design variables. Then, the optimization objective function is constructed by comprehensively considering critical speed constraint, influence factors of mode unbalance, proportion of strain energy and energy consumption rate of damper. Finally, the improved particle swarm optimization is used to optimize the design of the energy storage flywheel rotor with ESDFDs. The results show that the damping performance of the ESDFDs increase by 25%-40% and the unbalance sensitivity of rotor decreases compared with initial model, and it indicates the optimization design of the energy storage flywheel rotor with ESDFDs is effective. Keywords: energy storage flywheel rotor · elastic support/dry friction damper · vibration reduction optimization design · particle swarm algorithm

1 Introduction Energy storage has been taken as the important technology for the sustainable development. Flywheel energy storage, a physical energy storage technology, converts electric and kinetic energy through motors and generators. Because flywheel energy storage presents many notable merits such as high energy density, rapid response and prolonged lifespan, it has broadly applicated in energy storage, uninterruptible power supply and wind power frequency regulation [1, 2]. Nevertheless, the high-speed rotation of the flywheel under the vacuum environment, accompanied by the cyclic acceleration and deceleration, leads to alterations in the load size and direction on the rotor-support system, exacerbating the vibration issue. The stable operation of the rotor system is closely related to its support parameters and damper parameters. Hence, the multi-parameter optimization design of the flywheel rotor system assumes critical research significance. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 101–116, 2024. https://doi.org/10.1007/978-3-031-40459-7_7

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Many scholars have focused on the optimization design of rotor dynamics. Choi et al. [3, 4] considered the margin between the critical speed and operating speed, as well as the minimization of shaft system mass and vibration response as key design objectives and utilized a genetic algorithm to optimize the length, diameter of the shaft, and support stiffness. Jiao et al. [5] aimed to minimize the vibration response of the disk, employing the Multi-Island Genetic Algorithm and Sequential Quadratic Programming to optimize the stiffness of the elastic supports. Jin [6] established a dynamic model of the rotor system using the finite element method, and utilized the NSGA-II genetic algorithm to optimize the goal of minimizing the disk vibration response, the external force of the support, and the total mass of the shaft. Based on the sensitivity analysis of parameters, Wang [7] obtained optimization design variables and conducted the rotor dynamic optimization design with the goal of keeping the critical speed as far away from the operating speed as possible. Huang et al. [8] optimized the position of the disk to decrease its vibration response, utilizing a multi-island genetic algorithm. In general, the scholars mentioned above primarily focus on optimizing the crucial parameters of the rotor system to achieve the objectives of complying with the critical speed margin design criterion and reducing vibration response. Nonetheless, flywheel rotor regular traversals through the critical speed during the charging and discharging phases and need to rotate continuously at any speed, which make it impossible to satisfy the critical speed margin criteria. To further enhance the stability of the rotor system, the rotor needs to install dampers. Therefore, the damping performance of the damper must also be taken into consideration while optimizing the rotor system. Most dry friction dampers are commonly used for controlling vibrations in blades. However, the elastic support dry friction damper (ESDFD) is a novel vibration reduction device for rotors. Fan [9] initially introduced the ESDFD, based on the principle of redistributing the strain energy of the rotor-support system through elastic supports, which increased the strain energy of supports and dissipate it through friction dampers. Fan conducted several experiments to demonstrate the effectiveness of ESDFD for vibration reduction. Based on Fan’s work, an active elastic support/dry friction damper was proposed, which can achieve active control by using electromagnet and piezoelectric ceramic as actuators [10, 11]. The aforementioned research indicates that ESDFD possesses significant damping capabilities and can be utilized for active control. Nonetheless, the design method of ESDFD and rotor matching has not been investigated yet. The rest of the paper is organized as follow. Section 2 describes the structure of the ESDFD and establishes the dynamic model of the energy storage flywheel rotor with ESDFDs. Section 3 constructs the optimization objective function and shows the optimal results based on improved particle swarm optimization (IPSO) [12]. Section 4 discusses the damping performance of the ESDFDs and the unbalance sensitivity of rotor. Section 5 draws the conclusions to the above results and discussion. Section 6 is the acknowledgments.

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2 Dynamical Model The energy storage flywheel rotor with ESDFDs is proposed as shown in Fig. 1. The rotor system mainly consists of the flywheel rotor, bearing, and ESDFD. Specifically, the flywheel rotor is supported by two elastic supports, with each support incorporating an ESDFD. Elastic support Bearing

A

A

Upper support ESDFD

Flywheel rotor

Lower support ESDFD

A

A

Fig. 1. The structure of energy storage flywheel rotor with ESDFDs

The ESDFD located between the load-carrying and the elastic support is shown in Fig. 2a and consists of 3 key components: the elastic support, the friction pairs (consisting of fixed ring and moving ring) and the actuator. The moving ring, fixed ring, and mounting ring are depicted in Fig. 2b, c, and d, respectively. The moving ring is mounted on the end cross section of the elastic support and vibrates with the elastic support, but does not rotate with the rotor. The fixed ring is mounted on the mounting ring and can be moved in the axial direction by the actuator. When ESDFD works, the fixed ring moves toward the moving ring by the actuator and forms a contact aera. During the operation of rotor, the deformation of the elastic support is caused by the unbalanced load. As a result of the relative displacement between the moving ring and the fixed ring, friction occurs at the contact area. The damping introduced by friction can dissipate the vibration energy. 2.1 Two-Dimensional Friction Model A two-dimensional friction model has been developed to establish the mathematical relationship between the friction force and the relative displacement of the friction pairs.

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Piezoelectric ceramic actuators

Elastic support

Mounting ring

Bearing

Fixed ring

Casing

Moving ring ESDFD

a. Detailed information of ESDFD

b. Moving ring

c. Fixed ring

d. Mounting ring

Fig. 2. The structure of ESDFD

As showed in Fig. 3, the model comprises of two masses, mj and md , representing the fixed ring and the moving ring respectively. The connection between fixed ring and the mounting base is simulated by using two elastic elements and two viscous elements, represented by kj and cj respectively. The moving ring is mounted to the elastic support. Two elastic elements, kd , and two viscous elements, cd , are used to simulate the action of the squirrel cage elastic support on the moving ring. The friction force results from the relative motion between the moving ring and the fixed ring. Y

kj cj

cd

kd

O

X

md

mj

Moving ring Fixed ring

Fig. 3. Two-dimensional friction model

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Let the vector u describe the motion of the fixed ring and the vector w that of the moving ring. T T   and w = xd , yd u = xj , yj

(1)

then with this model, Eq. (2) describes the motion of the moving ring and fixed ring.  mj u¨ + cj u˙ + kj u = Ff (2) md w¨ + cd w˙ + kd w = F − Ff where F is centrifugal force generated by the rotation of the unbalance mass on the rotor and Ff is frictional force. The external force makes the damper to move in a circular motion. The external force calculated by Eq. (3). F = mεΩ 2 cos(Ωt − β)

(3)

where m is the unbalance mass. ε is the eccentricity distance of unbalance mass. β is the phase of unbalance mass. The friction force between the fixed ring and the moving ring is Coulomb’s friction force. So, the magnitude of the friction is related to the state of motion of the moving ring. If the acting force on the moving ring is less than frictional force, the moving ring is stationary relative to the fixed ring, namely the friction pairs is in the state of stick, the frictional force is Ff = kc (w − u)

(4)

where kc is the tangential contact stiffness of the contact interface. The tangential contact stiffness can be computed by Eq. (5) [13]. kc =

G ∗ 3.7 ·N · E∗ l

(5)

where 2 − νj 1 2 − νd = + , G∗ 4Gj 4Gd

l − νj2 l − νj2 1 = + , E∗ Ej Ej

(6)

where ν denotes Poisson’s ratio. E is elastic modulus. N denotes the pressing force. If |kc (w − u)| ≥ μN , the moving ring will slip, the friction pairs are in the state of slipping, and the frictional force is Ff = μN where μ is the frictional coefficient.

u˙ − w˙ |˙u − w| ˙

(7)

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2.2 Dynamics Model of the Flywheel Rotor with ESDFDs To investigate the dynamic characteristics of the flywheel rotor and the damping performance of the ESDFD, the model of the ESDFDs-energy storage flywheel rotor coupling system is established by the finite element method [14]. The dynamic equation for rotor system with ESDFDs can be expressed as Eq. (8) M q¨ + (C − G)˙q + Kq = Q + Qf

(8)

where q denotes the vector, the rows of which are components of two-dimensional translational degree and two-dimensional rotational degree of nodes. M , C, G, and K are the mass, damping, gyroscopic, and stiffness matrix of the flywheel rotor-support system, respectively. Q is the external sinusoidal forcing vector. Qf denotes the frictional force vector.

3 Optimization Design of Flywheel Rotor System with ESDFDs 3.1 Design Parameters There are mainly three categories of the parameters of rotor system: 1) Support parameters: position and stiffness, etc. 2) Disk parameters: position, mass, polar moment and diametral moment, etc. 3) Shaft parameters: length, inner diameter and outer diameter, etc. Figure 4 shows a dynamic model of an energy storage flywheel rotor equipped with ESDFDs used to optimize. The working speed range of the system is from 0 to 10 krpm. The model consists of two parts, the rotor and two ESDFDs (depicted by the dashed boxes). The flywheel rotor is a vertical construction consisting of a flexible shaft supported by two elastic supports located at the extremities. The two ESDFDs are installed at the indicated elastic supports, as illustrated by the dashed boxes in Fig. 4. L and D are respectively the length and diameter of the shaft, where l4 , l8 and l9 are the position of the flywheel, upper support and lower support, respectively. m, Ip and Id respectively denote the mass, the polar moment of inertia and the moment of inertia about a diameter of flywheel. E, ρ and u respectively represent the elastic modulus, density and Poisson’s ratio. ku and kl are respectively upper and lower support stiffness. du and dl denote respectively upper and lower support damping. The specific parameters of the rotor and damper are listed in Tables 1 and 2 [15] respectively. The Mean Impact Value (MIV) method [16] can assess the degree of the influence of individual independent variables on the dependent variable. The evaluation index Miv is computed by means of the MIV method to determine the extent of influence of each design parameter on the mode shape and critical speed. The dynamic characteristics of the rotor are greatly affected by the support stiffness, support position, and flywheel axial position. These parameters are considered as optimization design parameters listed in Table 3.

Vibration Reduction Optimization Design Elastic support

ku d u

D1 l8 l1

Moving disk Stationery disk

D2

Shaft

l2

E

ρ

l4

u

L l3

D3

Flywheel

D4 l5 l6

D5 kl dl

l9

l7

D6

Fig. 4. Dynamic model of flywheel rotor with ESDFDs

Table 1. Parameters of the Flywheel Rotor for Numerical Analysis Categories

Parameters

Values

Parameters

Values

Support

l8 /m

0.082

l9 /m

0.0775

ku /(N/m)

1.50 × 106

du /(N · s/m)

200

kl /(N/m)

14.9 × 106

dl /(N · s/m)

200

m/(kg)

2990.21

Id /(kg · m2 )

245.50

Ip /(kg · m2 )

373.78

l4 /m

1.5505

L/m

1.968

l1 /m

0.104

l2 /m

0.951

l3 /m

0.253

l4 /m

1.5505

l5 /m

0.2425

Disk Shaft

Material

l6 /m

0.065

l7 /m

0.110

E/1011 Pa

2.05

ρ/(kg/m3 )

7820

v

0.29

107

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D. Lin et al. Table 2. Parameters of ESDFD

Parameters

Values

Mass of fixed ring/kg

0.642

Mass of moving ring/kg

0.582

Stiffness of fixed ring/(N/m)

9.159 × 107

Tangential contact stiffness/(N/m)

5.08 × 107

Friction coefficient

0.16

Maximum pressing force/N

3000

Table 3. Optimization Range of Design Variables Parameters

Bound Lower

Upper

ku

1.00 × 106 N/m

3.00 × 106 N/m

kl

10.0 × 106 N/m

20.0 × 106 N/m

l8

0.060/m

0.120/m

l9

0.060/m

0.130/m

l4

1.400/m

1.600/m

3.2 Constraint Condition The conventional design constraint that critical speeds should be kept as far away from the operating speed as possible is inadequate for the flywheel rotor that needs to frequently traverse critical speeds during the charging and discharging phases. Research findings indicate that when the critical speed is maintained at a separation margin of about 8.3% from the critical speed in rigidly supported conditions, the critical vibration peak of the rotor can be reduced by 83.6% [17]. This can be attributed to the failure of the system damping when the rotating speed is close to the critical speed under rigidly supported conditions. So, the constrain is the 10% separation margin of the critical speed of rotor system, compared with the critical speed under the rigidly supported conditions. The constraint condition can be expressed as f (ωcri ) = (ωcri − 0.9ω˜ cri )(ωcri − 1.1ω˜ cri )

(9)

where ωcri is the elastic support critical speed. ω˜ cri is the rigid support critical speed. If the value of ωcri is in the range of (0.9–1.1) ω˜ cri , f (ωcri ) is less than or equal to 1. The constraint is not satisfied. Therefore, the critical speeds under the elastic supports within the working speed range are required to meet Eq. (10). Otherwise, the optimization function needs to be adjusted by the penalty factor. min(f (ωcri )) > 0

(10)

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3.3 Objective Function Influence Factors of Mode Unbalance In recent years, to enhance the energy storage capacity of the flywheel rotor system, the flywheel has undergone progressive development towards greater speeds and higher mass. This inevitably causes the problems such as uneven mass distribution and shaft tilting during the manufacturing and assembly process. These problems can easily lead to severe vibration of flywheel rotor. The design method that the sensitivity of rotor to unbalance is reduced by optimizing the modal shape of rotor is proposed. The influence factors of modal unbalance are R 

U˜ i =

j=1

2 ai,j ri,j

Ti i

, ai,j ∈ [0, 1], U˜ i ∈ [0, 1]

(11)

where R is the total number of nodes of rotor finite element model. ai,j denotes the weight coefficient of jth node of ith-order mode shape. ri,j is normalized displacements at the jth node in the ith-order mode shape. i is normalized displacements of ith-order mode shape. If the influence factors of modal unbalance becomes larger, the similarity between mass distribution and mode is higher which will increase unbalance response. Proportion of Strain Energy The elastic supports redistribute the strain energy of the whole rotor-support system and increase the elastic-supported strain energy. The ESDFD is affixed on the elastic supports to dissipation energy. If the elastic-supported strain energy is higher, the more energy could be consumed by the ESDFD. Therefore, the vibration of the rotor is easy to control. Proportion of strain energy is esdfd

Pi =

Ei

Ei + Eish

esdfd

+ Eirb

, Pi

∈ (0, 1)

(12)

where esdfd

Ei

=

m  z=1

2 0.5kz ri,z , Eirb =

n 

 2 0.5krb ri,rb m Eish = 0.5Ti K sh i

(13)

rb=1

where Eiesdfd is sum of strain energy of elastic supports, considering that the rotor may have multiple elastic supports. Eish denotes the shaft strain energy. kz is the zth elastic is the rbth rigidly supported stiffness. K sh is the stiffness matrix of support stiffness. krb shaft. Energy Consumption Rate of Damper The ESDFD shares the elastic support with the rotor. The variation of support parameters not only affects the dynamic characteristics of the rotor system but also has a significant impact on the damping performance of the damper. Therefore, in the optimization design

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process, the influence of support parameters on the damping performance of the damper should also be taken into account. Energy consumption rate of damper is the energy consumed per cycle divided by the total energy of the support with ESDFD installed, and the formula is as follows. ηi =

m 2  π df ,z ωcri ri,z df ,z ωcri Di,z = = , ηi ∈ (0, 1) 2 2π Ei,z kz 2π · 0.5kz ri,z z=1

(14)

where Dz,i is energy consumption in a single cycle. df ,z denotes the damping coefficient of the zth ESDFD [18]. Ei,z is the strain energy of elastic supports. Optimization Objective Function Intelligent optimization algorithms are widely employed in optimizing the rotor dynamics design. The objective function is crucial in optimization design, which can reflect the design objective and should be monotonic. The optimization objective function is constructed by comprehensively considering critical speed constraint, influence factors of mode unbalance, proportion of strain energy and energy consumption rate of damper. The objective function is where n is the number of modes in the working speed range. ζi denotes the weight coefficient, ζi ∈ [0, 1]. fpenalty is the constraint of critical speed. ⎛ ⎜ f = max(fre ) + fpenalty = max⎜ ⎝1 −

n  i=1

esdfd

ζi (1 − Pi

n 

)(1 − ηi )U˜ i

ζi

⎞ ⎟ ⎟ + fpenalty ⎠

(15)

i=1

3.4 Optimization Design Based on MPSO The fundamental principles for optimizing the design of flywheel rotor with ESDFDs are as follows: ensuring that the rotor system meets the critical speed constraint, the maximum value of the optimization objective function is obtained by MPSO within the range of design parameters. In the paper, the dynamics calculation program for the ESDFDs-energy storage flywheel rotor is written to calculate the optimization objective function, and the program is embedded into the MPSO. During each iteration, the MPSO adjusts the values of upper and lower support stiffness, upper and lower support stiffness positions, and the flywheel position within the recommended range to maximize the optimization objective function. The iteration terminates when the optimization objective function is no less than 0.85. The flowchart of the optimization design process based on MPSO is shown in Fig. 5. The initial and optimal values of design parameters are listed in Table 4.

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Start Setting optimisation of design parameters and MPSO parameters Substitute particles into the equation

Generate the position of particles

i

Update the particle best position and global best of swarm

= ( 1,

2

,

D

),

Solve the rotor system equation

Next iteration Update each particle velocity and positions of the particles

No

Caculate the objective function of every particle

Satisfy termination condition ? Yes Solution is global best of swarm End

Fig. 5. Flow chart of optimization design

Table 4. Optimization Result of Design Variables Model

kl /(N/m)

ku /(N/m)

l4 /m

l8 /m

l9 /m

Initial

1.50 × 106

14.9 × 106

1.5505

0.082

0.0775

Optimal

1.82 × 106

13.1 × 106

1.4742

0.095

0.0633

4 Results and Discussion The stiffness level of the upper and lower supports’ bearings is 108 N/m. Under the assumption of rigid support condition, the stiffness of the upper and lower supports is set to 1 × 108 N/m. Based the support stiffness, the dynamic characteristics of the initial and optimized flywheel rotor models are shown in Fig. 6. As shown in Fig. 6a and Fig. 6b, there are two 1st critical speeds in the working speed range, which are 5721 r/min and 5918 r/min, respectively. Figure 6c shows the mode shapes of the initial and optimized models under the rigid support condition, both of which are the axial bending mode shapes. The vibration displacement of the supports with ESDFDs are small, and the dampers are difficult to play good damping performance.

D. Lin et al. Natural vibration frequency/(rad/s)

Natural vibration frequency/(rad/s)

112

1000 800

Forward whirl Backward whirl

600 400

Critical speed

200 0

0

2000

4000 5721 8000 Speed of rotation (rpm)

10000

a. Campbell diagram of initial model

1000 800

Forward whirl Backward whirl

600 400

Critical speed

200 0

0

2000

4000 5918 8000 Speed of rotation (rpm)

10000

b. Campbell diagram of optimized model

Relative displacement

1 0.5 0 -0.5 -1

Flywheel Support

0

0.5

1

1.5

Initial Optimized

2

2.5

Axial location/m

c. Mode shape Fig. 6. The dynamic characteristics of the flywheel rotor under rigid supports

To evaluate the efficacy of the optimization design method, an amount of unbalance is deliberately introduced into the flywheel rotor to induce vibration. The upper and lower end faces of the flywheel are employed as the position of adding unbalance. Based on the dynamic balance grade of G2.5 under GB/T 9239.1–2006 standard and the calculation procedure in the literature [19], the permitted unbalance distribution of the rotor is computed and listed in Table 5. By utilizing the values in Tables 1, 4, and 5, the dynamic characteristics of the flywheel rotor equipped with ESDFDs are calculated. The results are listed in Table 6 and Figs. 7, 8, 9 and 10. Table 5. Distribution of Unbalance Upper end of flywheel

Lower end of flywheel

Unbalance/(g·cm)

Phase/°

Unbalance/(g·cm)

Phase/°

188.5

0

159.42

0

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Table 6. The Proportion of strain energy of Rotor System Mode

Model

1st

Initial

5.57%

41.91%

52.52%

Optimal

3.35%

40.35%

56.30%

13.15%

55.68%

31.17%

7.62%

57.39%

34.99%

2nd

Shaft

Initial Optimal

Upper support

Lower support

1000

Forward whirl Backward whirl

800 600 400

Critical speed

200 0

0 550

2120

4000 5149 6293 Speed of rotation (rpm)

a. Initial model

10000

Natural vibration frequency/(rad/s)

Natural vibration frequency/(rad/s)

The proportion of strain energy of rotor system is listed in Table 6. It is shown that compared with initial model, the proportion of strain energy of shaft of 1st and 2nd modes of optimized model are lower. Therefore, the more strain energy of rotor system is concentrated in the elastic support, which is favorable for the ESDFD. Figure 7 is the Campbell diagram of the flywheel rotor. Figure 7a shows that there are two critical speeds of initial model in the working speed range. Figure 7b shows there are two critical speeds of optimized model in the working speed range. Under rigidly supported conditions, the critical speed of initial and optimized model are 5721r/min and 5919r/min respectively. The gray shaded area of the figure shows the speed range that considers a separation margin of about 10% compared to the critical speed under rigidly supported conditions. Compared the initial model, the 1st and 2nd critical speed of optimized model increases slightly, but both meet the constraint condition of critical speed.

1000 800

Forward whirl Backward whirl

600 400 Critical speed 200 0

0 570

2230

4000 5327 6509 Speed of rotation (rpm)

10000

b. Optimized model

Fig. 7. Campbell diagram of the flywheel rotor

Figure 8 is the mode shape of flywheel rotor. The 1st-order-mode shape of optimized model is similar with initial model, but the 2nd-order-mode shape is slightly different. For 2nd-order-mode shape, the position of flywheel is closer to the intersection of mode shape and dotted line which means the sensitivity of flywheel to unbalance is lower.

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D. Lin et al. 1 Relative displacement

Relative displacement

1 0.5 0 -0.5

Initial Optimized

Flywheel Support

-1 0

0.5

1

1.5

2

2.5

0.5 0 -0.5

Flywheel Support

Initial Optimized

-1 0

0.5

1

(a) 1st mode

1.5

Axial location/m

Axial location/m

2

2.5

(b) 2nd mode

Fig. 8. The mode shape of flywheel rotor

Figure 9 is the unbalance response of the flywheel of initial and optimized model without ESDFDs. As can be seen in the figure, the mass unbalance response at the 1st and 2nd critical speed decrease by 70.42% and 75.16% compared with initial model, and it indicates the sensitivity of optimized model to unbalance is greatly reduced. 350

(550,346.40)

Amplitude/(μm)

300

Initial model Optimized model

250 200 150

(570,102.48)

100 50

(2120,96.75) (2230,24.03)

0 0

500

1000

1500 2000 2500 Speed of rotation (rpm)

3000

3500

4000

Fig. 9. Mass unbalance response of the flywheel without ESDFDs

Figure 10 shows the mass unbalance response of the upper support in the initial and optimized models, both with and without the implementation of ESDFDs. The damper parameters for the initial and optimized models are identical. In Fig. 10a, the vibration reduction efficacy of ESDFDs at the 1st and 2nd critical speeds is noted to reach 25.37% and 33.33%, respectively, for the initial model. In Fig. 10b, the vibration reduction efficacy of ESDFDs at the 1st and 2nd critical speeds is noted to reach 32.33% and 45.54%, respectively, for the optimized model. When compared to the initial model, the vibration reduction efficacy of ESDFDs in the optimized models is shown to increase by 27.43% and 36.63%, respectively. As it is obvious from the results, the optimization method has yielded a decrease in the flywheel rotor’s sensitivity to the unbalance, as well as a considerable improvement in the damping performance of the ESDFDs.

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1000 Without ESDFDs With ESDFDs

Amplitude/(μm)

550,860.91 800 550,642.53

600

2120,446.95

400 200 0 0

500

1000

1500 2000 2500 Speed of rotation (rpm)

3000

3500

4000

3500

4000

a. Initial model 400

Amplitude/(μm)

Without ESDFDs With ESDFDs

(2230,395.60)

300

(2230,215.43)

200

(570,180.10) (570,121.88)

100

0 0

500

1000

1500 2000 2500 Speed of rotation (rpm)

3000

b. Optimized model Fig. 10. Mass unbalance response of the upper support with ESDFDs

5 Conclusions The following conclusions are drawn from the work: (1) The energy storage flywheel rotor with ESDFDs designed by the optimization design method of this paper is less sensitive to the unbalance and the damping performance of ESDFDs is improved by 25% –40%. This indicates the optimization design of the energy storage flywheel rotor with ESDFDs is effective. (2) The optimization objective function constructed in this paper considers not only the dynamic characteristics of the rotor, but also the damping performance of damper. The constructed objective function is suitable for multi-parameter optimization design of energy storage flywheel rotor with ESDFDs. Acknowledgments. This work was supported by the National Science and Technology Major Project of China (Grant No. J2019-IV-0005-0072).

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References 1. Wu, X., Chen, Y.L., Liu, Y.B.: Structure optimization of matal rotor of grid-connected flywheel energy storage system. Acta Energiae Solaris Sinica 42(02), 317–321 (2021) 2. Yu, S.H., Guo, W.Y., Teng, Y.P., et al.: A review of the structures and control strategies for flywheel bearings. Energy Storage Sci. Technol. 10(05), 1631–1642 (2021) 3. Choi, B.G., Yang, B.S.: Optimum shape design of rotor shafts using genetic algorithm. J. Vibr. Control 6(2), 207–222 (2000) 4. Choi, B.K., Yang, B.S.: Optimal design of rotor-bearing systems using immune-genetic algorithm. J. Vib. Acoust. ASME 123(3), 398–401 (2001) 5. Jiao, X.D., Qin, W.Y., et al.: Dynamic response of double-disk rotor with squeeze oil-film dampers and optimization of its supports. Noise Vib. Control 33(05), 1–3 (2013) 6. Jin, L.: Research on Optimal Design Method of Aero-Engine Rotor System Dynamics. Northwestern Polytechnical University, Xi’an (2013). (in Chinese) 7. Wang, J.Y., Zhao, Y.C.: Optimization design of flexible rotor of SFD-sliding bearings. Noise Vib. Control 33(04), 103–106 (2013) 8. Huang, J.J., Zheng, L.X., Wang, Z.W., et al.: Dyanmica characteristics analysis and optimization of a two-disk rotor system. J. Propul. Technol. 35(11), 1530–1536 (2014) 9. Fan, T.Y., Liao, M.F.: Dynamic behavior of a rotor with dry friction dampers. Mech. Sci. Technol. (05), 743–745+760 (2003) 10. Wang, S.J., Liao, M.F., Yang, S.J.: Experimental investigation on rotor vibration control by elastic support/dry friction damper. J. Aerosp. Power 22(11), 1893–1897 (2007) 11. Song, M.B., Tan, D.L., Liao, M.F.: Experiment on vibration reduction by elastic support/dry friction damper with piezoelectric ceramic. J. Aerosp. Power 28(10), 2223–2227 (2013) 12. Wang, H.: Rotordynamic Characteristic of Variable Speed Turbo-shaft Engine. Northwestern Polytechnical University, Xian (2022) 13. Valentin, L.: Popov, Contact Mechanics and Friction-physical Principles and Applications. Springer-Verlag, Berlin (2010) 14. Liao, M.F.: Aero-Engine Rotor Dynamics, pp. 208–234. Northwestern Polytechnical University, Xi’an (2015) 15. Wang, S.J., Wang, C.Y., Lin, D.F., et al.: Integrated Configuration Design and Experimental Research on Vibration Reduction of an Active Elastic Support/Dry Friction Damper. J. Propul. Technol. 1–12 (2022) 16. Pi, J., Huang, J.B.: Aero-engine fault diagnosis based on IPSO-Elman neural network. J. Aerosp. Power 32(12), 3031–3038 (2017) 17. Huang, J.B., Liao, M.F., Lei, X.L., et al.: Workable mode design and experimental verification of aero-engine low-pressure rotor system. J. Aerosp. Power 37(05), 964–979 (2022) 18. Song, M.B.: Dynamic Design of Elastic Support/dry Friction Damper Matching Rotor. Northwestern Polytechnical University, Xian (2016) 19. GB/T 9239.1-2006, Mechanical vibration—balance quality requirements for rotor in a constant(rigid) state—Part 1: Specification and verification of balance tolerances

Sensitivity of Spline Self-excited Vibration to Structure Parameters Yingjie Li1 , Guang Zhao1 , Zexin Zhang1 , Yunbo Yuan2(B) , Jian Li3 , and Yongquan Wang4 1 School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024,

China 2 School of Control Science and Engineering, Dalian University of Technology, Dalian 116024,

China [email protected] 3 AECC Hunan Aviation Powerplant Research Institute, Zhuzhou 412002, China 4 AECC Shenyang Engine Research Institute, Shenyang 110015, China

Abstract. Many aviation splines have to work in grease-lubricated or nonlubricated environments because of structural, weight and space restrictions. When a spline rotor system works at speeds above the critical speed, a poor lubrication or a specific misalignment will lead the spline in such a rotor experiencing selfexcited vibration. In this paper, a constant stiffness and time-varying damping model is established for the spline coupling used in helicopter tail transmission shaft system. A multi-degree-of-freedom model for helicopter tail transmission shaft system is developed according to Timoshenko beam theory and lumped-mass method. The coupled differential equations are solved using Newmark-β method. According to the magnitude of low-frequency component, the self-excited vibration sensitivities to critical system parameters (i.e., spline tooth number, modulus, pressure angle and tooth width) are investigated for splines used in helicopter tail transmission shaft system. This study provides a reference for the mechanism research and suppression of the self-excited vibration of splines encountered in helicopter tail transmission shaft system. Keywords: Aviation Spline · Self-excited Vibration · System Parameters · Sensitivity

1 Introduction In aero-engines, spline couplings are widely used in aviation accessory transmission systems because of their lightweight, high balance potential at high speed, high load capacity, and good misalignment compensation capacity [1, 2]. Floating splines in loose fit often operate under various working conditions and bear mechanical loads such as centrifugal force, periodic torque, impact torque, and misalignment load [3]. Although lubrication can effectively extend aviation splines’ service life, many of them can only operate in grease-lubricated or non-lubricated environments due to structure, weight, and space limitations. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 117–132, 2024. https://doi.org/10.1007/978-3-031-40459-7_8

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When the rotor runs across the critical speed, the poor lubrication of the spline tooth surface and the large misalignment angle between the internal and external splines will cause an increase in the spline friction coefficient or the negative damping of the rotor system, which may induce the instability of the rotor, excite the rotor resonance, and generate the self-excited vibration of the floating spline. Because of the supercritical operation of the rotor at this time, the resonance frequency of the rotor is low-frequency vibration, that is, sub-synchronous whirl. The low-frequency amplitude is high and the energy is large, which will cause serious damage or accidents [4]. Many researchers have paid attention to the stability of rotating machinery with splines. Williams and Trent [5] considered the spline tooth surface sliding friction and unbalance, calculated the influence of nonlinear and asymmetric support stiffness on the asynchronous vorticity of the system, and confirmed for the first time that spline friction was the source of the asynchronous whirl of the rotor. However, this research was based on the assumption of a rigid rotor, and its engineering applicability was limited. Marmol et al. [6, 7] established the friction model of the tooth surface of splines, calculated the internal damping coefficient of splines, and then predicted the internal damping of splines with different structures. They observed the instability phenomena during the experiments but did not find instability in the simulation because their model had too many uncertain parameters. Based on Marmol’s research, Ku et al. [8] only studied the influence of spline angle stiffness and damping coefficient on rotor stability. Experimental test results showed that self-excitation vibration occurred when the speed exceeded the critical speed of the rotor. Park [9] found that the change of crossstiffness caused by angular damping generated by spline friction was the main source of asynchronous whirl instability of the rotor. On the basis of literature [6–9], Zhu et al. [10] put the spline model into the dynamic equation of the three-support double-span rotor system. The findings demonstrate that the spline-rotor system exhibits self-excited vibration once the friction coefficient of the spline tooth surface reaches a critical value and crosses the rotor’s critical speed. The low-frequency is the system’s initial natural frequency. Brommundt et al. [11] analyzed the nonlinear problem of gear coupling. Walton et al. [4] and Gao [12] deduced four key coefficients of the spline in the turbine using the mechanical analysis method. According to the energy method, it was judged that the stability of the spline-rotor system was greatly affected by the angular damping coefficient and the lateral damping coefficient. Zhao et al. [13, 14] found the self-excited oscillation phenomenon of the loose fit spline through the test. The results show that increasing the misalignment angle of the spline can improve the instability speed of the system. Kang et al. [15] found in the helicopter transmission system that the inner damping due to the friction in the spline and on the supporting surface of both ends of the spline is the springhead of self-excited vibration. Li et al. [16] analyzed the research and development trend of spline self-excited vibration. Wang et al. [17, 18] and Huang et al. [19] proposed the boundary conditions for rotor instability and carried out test verification for the stability analysis of the spline-rotor system. The above research models the spline stiffness and damping coefficient from the perspective of energy and mechanical analysis, and determines the boundary conditions of spline self-excited vibration from simulation calculation and experimental research. However, the influence of spline structural parameters on the self-excited vibration of the

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system has not been investigation. Therefore, aiming at the spline structure parameters (spline tooth number, modulus, pressure angle, tooth width), this paper explores the influence of various parameters on spline self-excited vibration from the aspects of rotational speed and amplitude produced by self-excited vibration.

2 Analytical Model 2.1 Spline Stiffness The conventional solving methods of spline stiffness are simulation, experimental tests, and theoretical derivation. Nevertheless, the simulation calculation ignores the microstructure of the spline tooth surface, and the test error is significant. Therefore, the material mechanics method is used in this paper [20]. According to the contact stiffness of involute spline, the method of material mechanics simplifies the single spline tooth to the variable cross-section cantilever beam on the elastic basis. It is considered that the comprehensive elastic deformation δ j of the meshing spline includes bending deformation δ Bj , shear deformation δ Sj , compression (tensile) deformation δ cj , additional elastic deformation δ Mj of the foundation and contact deformation δ Cj at the meshing point of the tooth surface.

Fig. 1. Deformation analysis of a single tooth

Cantilever Beam Deformation In the calculation, the gear tooth is divided into n segments, and the section i is the shadow part in the Fig. 1. Equation (1), (2) and (3) define bending deformation, shearing deformation and compression (tensile) deformation.      Ti3 + 3Ti2 Li + 3Ti L2i sj Ti2 /2 + Ti Li sj F 2 δBbi = cos βj − cos βj sin βj (1) Ee 3Ii 2Ii    12(1 + a)Ti F cos2 βj (2) δBsi = Ee 5Ai

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δBci



 Ti F 2 = sin βj Ee Ai

(3)

where, F is the force, T i is the thickness of the section i, Ii is the moment of inertia of the section i, sj is the tooth thickness of the load action point, si is the thickness of the section i, L i is the distance from the segment i to the load point along the x direction, L j is the equivalent meshing distance of the distance from the load point j to the tooth root along the x direction, β j is the angle between load F and y axis, Ai is the section area, B is the tooth thickness, s is the thickness of the pitch circle, α is the Poisson’s radio, E is the modulus of elasticity, E e is the equivalent modulus of elasticity, when B/s > 5, E e = E/(1 − α2 ), otherwise E e = E. Related parameters can be expressed as: Ti =

Li = Lj

Lj n

2n − 2i + 1 2n

(4) 

Ai = B · si Ii = IZc =

B · si3 12

(5) (6) (7)

βj = arccos(db /dj ) (db ≤ dj )

(8)

dj = df + 2Lj

(9)

ai = arccos(db /di )

(10)

si =

s · di − di [inv(ai ) − inv(β)] d

(11)

where, d b is the base circle diameter, when i = n, sj = sn , when i = 0, sf = s0 . If the interval from the j point on the spline to the tooth root is divided into small blocks i along the x direction as shown in Fig. 1, the bending, shearing and compression deformations of each small block are obtained and accumulated respectively, then the bending deformation, shear deformation and compression (tensile) deformation of the spline cantilever beam can be calculated as follows: δBj =

n

δBbi

(12)

δBSi

(13)

i=1

δSj =

n i=1

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δCj =

n

121

δBci

(14)

i=1

Additional Deformation Caused by the Root of the Tooth The spline tooth root is considered to be a rigidly fixed cantilever beam for the purposes of calculating the bending, shearing, and compression deformation of the spline above. However, since the spline’s root is elastic, it is important to take into account the additional deformation of the meshing point brought on by the elasticity of the tooth root. For the case of wide teeth (B/s > 5), it is treated as a plane strain problem. δM j =

F cos2 βj BEe

⎡ ⎣5.306



Lf Hf

2



1 − a − 2a2 +2 1 − a2



Lf Hf



 + 1.534 1 +

Lf = Lj − sj · tan(βj )/2

0.4167tg 2 (βj ) 1+a

⎤ ⎦

(15) (16)

where, H f = sf . Contact Deformation at Mesh Point According to the literature [21], the contact deformation δ C of the meshing point of the tooth surface can be calculated according to the following formula: δCj =

1.275 2Ee0.9 Be0.8 F 0.1

(17)

where, F is the unit normal load. It can be seen from the above formula that the contact deformation of the meshing point is only related to the elastic modulus of the material, force and equivalent tooth width. When the tooth widths of two meshing surfaces are equal, Be = B. When the tooth widths of the two meshing surfaces are not equal, Be takes a smaller tooth width value. In this way, by the superimposition of the above three deformations, the total elastic deformation of a single tooth calculated by the material mechanics method can be written as follows: δj = δBj + δMj + δCj

(18)

Therefore, the meshing stiffness of the single tooth: Kj = 1/δj =      

 n T 3 + 3Ti2 Li + 3Ti L2i sj Ti2 + 2Ti Li sj Ti F 12(1 + a)Ti 2 2 + cos βj − cos βj sin βj + sin βj + Ee 3Ii 5Ai 4Ii Ai i=1   −1   

2 Lf Lf 0.4167tg 2 (βj ) F · cos2 βj 1 − a − 2a2 1.275 + 1.534 1 + +2 5.306 + B · Ee Hf 1 − a2 Hf 1+a 2Ee0.9 Be0.8 F 0.1

(19)

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Then, Spline tooth pair contact stiffness: K=

1 δjext + δjint

(20)

where, δ ext j is the total elastic deformation of external spline, δ int j is the total elastic deformation of internal spline. Lateral stiffness of the spline [12]: kl =

z K 2

(21)

B2 z K 24

(22)

where, z is the number of the spline tooth. Angular stiffness of the spline [12]: kα =

2.2 Spline Damping According to the principle of conservation of energy, it is deduced that the lateral damping coefficient and angular damping coefficient of splines are [6]: cl = cα =

2μT ωc rπ δ0 cos2 ϕ

(23)

μTB 2ωc rπ α0 cos2 ϕ

(24)

where, μ is the contact friction coefficient of the spline tooth surface, T is the spline transmission torque, ωc is the difference between the system speed ω and the critical speed, r is the pitch radius, δ 0 is the difference between the internal and external spline displacement, α 0 is the difference between the internal and external spline angular displacement, ϕ is the pressure angle. 2.3 Spline-Rotor System Based on the Timoshenko beam element model, as shown in Fig. 2, the finite element model of the rotor system is constructed. The Timoshenko beam element model considers the influence of shear deformation based on the traditional Euler-Bernoulli beam element model. For a two-node element of 8 degrees of freedom, its generalized coordinates are assumed to be: T  (25) X = xi , yi , θxi , θyi , xi+1 , yi+1 , θxi+1 , θyi+1 , i = 1, 2, 3, ... A dynamic model is established for the helicopter tail transmission system (Fig. 3), in which nodes 6 and 11 represent floating spline coupling, nodes 3, 20, 26, and 30 are bearing supports, and nodes 16, 21, and 27 are diaphragm couplings.

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Fig. 2. Timoshenko beam element

Fig. 3. Spline-rotor system

Then the spline stiffness matrix in the fixed coordinate system is: ⎛

kl ⎜ −ωcl KS =⎜ ⎝ 0 0

ωcl 0 kl 0 0 kα 0 −ωcα

⎞ 0 0 ⎟ ⎟ ωcα ⎠ kα

(26)

where, the cross term in the equation is generated by transforming the damping matrix in the rotating coordinate system to the fixed coordinate system. The spline damping matrix in the fixed coordinate system is: ⎛

cl ⎜0 Ds =⎜ ⎝0 0

0 cl 0 0

0 0 cα 0

⎞ 0 0⎟ ⎟ 0⎠ cα

(27)

The rotor system is connected by diaphragm couplings, ignoring the damping of the diaphragm coupling, and its stiffness matrix is: ⎛

kc1 ⎜ 0 Kc =⎜ ⎝ 0 0

0 kc1 0 0

0 0 kc2 0

⎞ 0 0 ⎟ ⎟ 0 ⎠ kc2

(28)

where, k c1 and k c2 are the lateral stiffness coefficient and angular stiffness coefficient of the diaphragm couplings.

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The nonlinear characteristics of the four supporting bearings in the rotor system are neglected. The stiffness matrix and damping matrix are: ⎛

kb1 ⎜ 0 Kb =⎜ ⎝ 0 0 ⎛ cb1 ⎜ 0 Cb =⎜ ⎝ 0 0

⎞ 0 0 ⎟ ⎟ 0 ⎠ kb2 ⎞ 0 0 0 cb1 0 0 ⎟ ⎟ 0 cb2 0 ⎠ 0 0 cb2 0 kb1 0 0

0 0 kb2 0

(29)

(30)

where, k b1 and k b2 are the lateral stiffness coefficient and angular stiffness coefficient of the bearings, cb1 and cb2 are the lateral damping coefficient and angular damping coefficient of the bearings. In summary, the total equation of motion of the system is: ¨ + (C − ωJ s )X ˙ + KX = Fu (t) MX

(31)

where, M is the total mass matrix of the system, including shafting, splines, diaphragm coupling, bearing, and disk mass, C is the system damping matrix, including shafting damping (Rayleigh damping), spline damping, and bearing damping, K is the system stiffness matrix, including shafting stiffness, spline stiffness, bearing stiffness, and diaphragm coupling stiffness, Js is the system gyro matrix, Fu is the unbalanced force on the system. 2.4 System Parameter For the above system, the parameters of spline, the dynamic parameter of bearing and diaphragm coupling are given, as shown in Tables 1, 2 and 3.

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Table 1. Dynamic parameters of bearings and diaphragm couplings Lateral stiffness /(N/m)

Angular stiffness /(Nm/rad)

Lateral damping coefficient /(Ns/m)

Angular damping coefficient /(Ns/rad)

Bearing 1

2 × 107

1 × 106

750

350

Bearing 2

4 × 106

2 × 105

750

350

Bearing 3

2 × 107

1 × 106

750

350

Bearing 4

2 × 107

1 × 106

750

350

Diaphragm couplings 1

9.9 × 108

3.98 × 104

--

--

Diaphragm couplings 2

9.9 × 108

3.98 × 104

--

--

Diaphragm couplings 3

9.9 × 108

3.98 × 104

--

--

Table 2. Parameters of splines Number of teeth

Module /mm

Pressure /°

Tooth width /mm

Poisson’ ratio

Elastic module /GPa

16

1.25

30

14

0.33

80

Table 3. Parameter of the system Node

1–5

5–8

9–12

12–16

17–21

22–27

28–30

Outer diameter /m

0.06

0.035

0.021

0.03

0.03

0.03

0.03

Inner diameter /m

0

0.022

0.012

0.012

0.015

0.015

0.015

Shaft length /m

0.12

0.04

0.05

0.06

0.75

1

0.13

3 Influence of Each Parameter on the Self-excited Vibration of the System According to the parameters, Newmark- β is used to calculate the response of the system. Figure 4 is the response diagram of the system 0–6000 r/min. It can be found from the Fig. 4 that the first-order critical speed of the system is about 1800 r/min, and the secondorder critical speed is 3000 r/min. The modal shape of the system is drawn (Fig. 5). The first order mode is the bending vibration of the long shaft, and the second mode is mainly the bending vibration of the short shaft.

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Fig. 4. Spline-rotor system response

Fig. 5. Spline-rotor system vibration mode

3.1 Number of the Spline Teeth When the boundary conditions such as spline transfer torque and tooth surface friction coefficient remain unchanged, the variation of spline self-excited vibration with the number of spline teeth is shown in the Fig. 6. It is discovered that the self-excited vibration happens when the system’s rotational speed is higher than 50 Hz, taking into account dynamic friction damping and the spline’s stiffness. This phenomenon mainly manifests as follows: low-frequency vibration occurs when the system speed exceeds the rotor’s critical speed. Its amplitude will continue to increase with the increase of the speed. And its vibration frequency is about the first-order natural frequency of the short shaft. When the rotational speed of the system exceeds 3000 r/min, the system begins to appear in the phenomenon of self-excited vibration. And with the increase of rotational speed, the amplitude of low-frequency vibration of the system becomes larger. Because the damping coefficient of the spline is affected by the amplitude of spline vibration, when the amplitude of the system is large, the input energy of spline internal friction damping to the system is small. And when the amplitude is reduced, the input energy of spline internal friction damping to the system is increased. Thus, the self-excited vibration amplitude of the system decreases at the rotation speed of 4800–5000 r/min and then continues to increase (Fig. 6 a)).

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Fig. 6. The self-excited vibration of spline changes with the number of teeth

Furthermore, with the increase in spline teeth, the self-excited vibration amplitude becomes smaller. The system with 14 teeth has the most significant low-frequency vibration amplitude when the speed is 4600 r/min, and the system with 21 teeth has the smallest low-frequency vibration amplitude as the number of teeth increases. When the number of teeth is 21, the rotational speed of the system with self-excited vibration is about 3400 r/min, and when there are 14 teeth, the speed of the system with self-excited vibration is about 3100 r/min. The speed threshold of the system’s self-excited vibration will rise as the number of spline teeth increases. The system’s self-excited vibration’s amplitude varies only slightly when the spline is changed from 14 to 21 teeth; instead, the velocity value at which self-excited vibration appears is primarily affected. 3.2 Spline Module The variation of spline self-excited vibration with the spline module is shown in the Fig. 7. The amplitude of the spline self-excited vibration and the modulus do not have a proportionate connection. When the module of the spline changes between 0.75 mm– 2 mm, the self-excited vibration amplitude of the system also changes. The self-excited vibration amplitude of the system at 4600 r/min was extracted for four different modules, and it was found that the self-excited vibration amplitude of the system is the smallest when the module is 0.75 mm. The amplitude of the spline self-excited vibration increases

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Fig. 7. The self-excited vibration of spline changes with the spline module

initially and subsequently diminishes as the spline modulus increases. When the modulus fluctuates between 1.25 mm and 2 mm, the system’s overall self-excited vibration amplitude steadily lower. The calculation results show that the self-excited vibration of the system changes very little in amplitude and speed range when the spline modulus is changed from 0.75 mm to 2 mm. 3.3 Spline Pressure Angle The variation of spline self-excited vibration with the spline pressure angle is shown in the Fig. 8. With the increase of spline pressure angle, the amplitude of system self-excited vibration becomes smaller at the rotation speed of 4600 r/min. When the rotational speed is 4600 r/min, the system low-frequency amplitude is the maximum when the pressure angle is 28°, and the spline low-frequency vibration amplitude is the minimum when the pressure angle is 45°. When the pressure angle decreases gradually, the maximum low-frequency amplitude of the system in the 1600–6000 r/min speed range decreases gradually, but the speed range of self-excited vibration increases gradually and continuously. The speed range of spline self-excited vibration fluctuates as the pressure angle changes from 28° to 45°. The speed of self-excited vibration varies little, but the amplitude of low-frequency vibration varies significantly.

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Fig. 8. The self-excited vibration of spline changes with the spline pressure angle

3.4 Spline Width The variation of spline self-excited vibration with the spline width is shown in the Fig. 9. When the system’s rotational speed surpasses 3000 r/min, the system will appear the self-excited vibration. With the increase of rotational speed, the amplitude of lowfrequency vibration of the system becomes larger. However, with regard to the amplitude of 4600 r/min, the system’s low-frequency amplitude is most significant when an 8 mm tooth width is employed, and it becomes smaller as the tooth width is increased. Furthermore, with the increase of tooth width, the rotational speed of self-excited vibration in the system increases gradually, and the apparent range decreases gradually. The speed value of the system’s self-excited vibration gradually rises as tooth width grows, but the speed range visibly diminishes. When the tooth width varies from 8 mm to 16 mm, the speed range of the system’s self-excited vibration fluctuates wildly, as does the amplitude range of the low-frequency. In this paper, the influences of four spline parameters on the self-excited vibration of the system are discussed, which are spline tooth number, modulus, pressure angle, and tooth width. According to the spline contact stiffness model described above, it can be found that spline tooth number, modulus, pressure angle, and tooth width will have an impact on the spline stiffness. According to the calculation of the spline contact stiffness K, it can be determined that spline tooth width significantly impacts the spline contact stiffness. That was followed by pressure angle, number of teeth, and modulus. In addition, the internal friction damping coefficient of spline is inversely proportional

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Fig. 9. The self-excited vibration of spline changes with the spline width

to the radius of spline pitch circle, the pressure angle and the transverse displacement amplitude difference of tooth surface. During the calculation, one parameter changes, and the other three are fixed values. But the spline contact stiffness and the internal friction damping of the spline will change. The spline stiffness and damping change will change the self-excited vibration of the spline-rotor system. Therefore, the system response at 4600 r/min is also different. The variation of the self-excited vibration of splines is caused by the direct influence of spline structural parameters on spline stiffness and damping coefficient. The selfexcited vibration of splines is also affected by system vibration. Therefore, the spline structure parameters should be carefully calculated and then determined. In general, the spline tooth width has the most significant impact on the self-excited vibration of the spline, followed by the pressure angle. In contrast, the number of teeth and modulus have little impact on the self-excited vibration of the spline. Therefore, in terms of the sensitivity of spline self-excited vibration to each parameter, increasing the tooth width and pressure angle within a tolerable range can restrict the spline’s self-excited vibration. However, restricting the system’s self-excited vibration from the spline structure parameters only changes the spline’s production range or threshold of self-excited vibration. That does not completely eradicate the self-excited vibration of the spline. The

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internal friction damping of the spline is the main contributor to the self-excited vibration of splines. And it is possible to think about utilizing spline materials with lower friction coefficients in order to avoid the self-excited vibration of the spline-rotor system.

4 Conclusions In this paper, a constant spline stiffness and time-varying damping model is derived for the spline-rotor system of the helicopter tail transmission shaft. A multi-degree-offreedom model of the helicopter tail transmission shaft system is established by using the Timoshenko beam theory and lumped-mass method. The coupling dynamics equation of the system is solved by using the Newmark-β method, and the following conclusions are obtained: 1) The generation of spline self-excited vibration needs to cross the critical speed, and the spline self-excited vibration frequency is the first bending frequency of the short axis. 2) The spline tooth width has a significant influence on the self-excited vibration amplitude and speed range of the system, and the spline self-excited vibration is more sensitive to the tooth width. The spline modulus and tooth number have relatively little effect on the amplitude of self-excited vibration of the system, and the spline self-excited vibration is less sensitive to modulus and number of teeth. Spline pressure angle tremendously influences the speed range of self-excited vibration. 3) To reduce the amplitude of spline self-excited vibration, it is necessary to consider all structural parameters comprehensively. Acknowledgement. The authors gratefully acknowledge the support provided by the National Natural Science Foundation of China (No. 12172073) and National Science and Technology major projects (779608000000200007).

References 1. Fu, C.G., Zheng, D.P., Ou, Y.X., Zhou, S.J., Zhao, X.M.: Aeroengine Design Manual, vol. 19. Aviation Industry Press, Beijing, China (2001) 2. Zhao, G., Zhao, X.Y., Qian, L.T., Yuan, Y.B., Ma, S., Guo, M.: A review of aviation spline research. Lubricants 11(1), 1–20 (2023) 3. Wang, Y.L., Zhao, G., Sun, X.C., Li, S.X.: Review on research of aviation spline. Aeronaut. Manuf. Technol. 3, 91–100 (2017) 4. Walton, J., Artiles, A., Lund, J., Dill, J., Zorzi, E.: Internal rotor friction instability. NASA Report, MTI-88TR39 (1990) 5. Williams, R., Trent, R.: The effect of nonlinear asymmetric supports on turbine engine rotor stability. SAE Trans. 79, 1010–1020 (1970) 6. Marmol, R.A., Smalley, A.J., Tecza, J.A.: Spline coupling induced nonsynchronous rotor vibrations. J. Mech. Des. 102(1), 168–176 (1980) 7. Marmol, R.A.: Engine rotor dynamics, synchronous and nonsynchronous whirl control. Army Research and Technology Labs Report, USARTL-TR-79-2 (1979)

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8. Roger Ku, C.P., Walton, J.F., Lund, J.W.: Dynamic coefficients of axial spline couplings in high-speed rotating machinery. J. Vib. Acoust. 116(3), 250–256 (1994). https://doi.org/10. 1115/1.2930421 9. Park, S.K.: Determination of loose spline coupling coefficients of rotor bearing systems in turbomachinery. PhD Thesis, Texas A&M University, Texas, USA (1991) 10. Zhu, H., Chen, W., Zhu, R., Gao, J., Liao, M.: Modelling and dynamic analysis of splineconnected multi-span rotor system. Meccanica 55(6), 1413–1433 (2020). https://doi.org/10. 1007/s11012-020-01163-9 11. Brommundt, E., Kramer, E.: Instability and self-excitation caused by a gear coupling in a simple rotor system. Forsch Ingenieurwesen-Eng. Res. 70(1), 25–37 (2005) 12. GAO, T.: Research on the stability of spline connected rotor system. Master’s Thesis, Shanghai Jiao Tong University, Shanghai, China (2016) 13. Zhao, G., Liu, Z.S., Chen, F.: Influences of spline coupling on stability of rotor-bearing system. J. Vib. Eng. 22(3), 280–286 (2009) 14. Zhao, G., Long, X., Ge, C.F., Liu, Z.S.: Study on stability of rotor-bearing systems with different types of coupling. In: Proceedings of the 9th National Symposium on Rotor Dynamics ROTDYN’ 2010, Guiyang, China (2010) 15. Kang, L.X., Cao, Y.H., Mei, Q.: Dynamic instability of helicopter transmission rotating shafts with spline coupling. J. Beijing Univ. Aeronaut. Astronaut. 36(6), 645–649 (2010) 16. Li, Y.J., Zhao, G., Wu, X.S., Li, J., Yuan, W., Mei, Q.: Summary of research on self-excited vibration of aviation spline-rotor system. Acta Aeronautica et Astronautica Sinica 43(08), 21–35 (2022) 17. Wang, T., Wang, L., Liao, M.F.: Stability analysis of rotor with spline coupling. Aeroengine 47(03), 66–71 (2021) 18. Wang, T., Wang, Y.K., Liu, M.R., Zhong, Z.C.: Stability analysis of rotor with a spline coupling. In: 2022 International Symposium on Aerospace Engineering and Systems, China (2022) 19. Huang, J.B., Liao, M.F., Liu, Q.Y., Xue, Y.G., Wang, S.J., Zhou, X.: Experiment study on vibration stability of rotor system with spline connection structure. J. Propul. Technol. 43(02), 275–285 (2022) 20. Zhao, G.: Study on coupled dynamics of rotor-coupling-bearing-isolator system. PhD. Thesis, Harbin Institute of Technology, Harbin, China (2009) 21. Lin, H.H., Lundberg, G.: Prediction of gear dynamics using fast Fourier transform of static transmission error. Mech. Struct. Mach. 21(2), 237–260 (1993)

Parametric Optimization of BNES in Torsional Vibration Suppression of Rotor Systems Jinxin Dou1(B) , Rui Xue2 , Hongliang Yao1 , Hui Li1 , and Jianlei Li1 1 School of Mechanical Engineering and Automation, Northeastern University,

Shenyang 110819, China [email protected] 2 China North Vehicle Research Institute, Beijing 100072, China

Abstract. This investigation presents an optimal design of a bi-stable nonlinear energy sink (BNES), which is used to suppress the torsional vibration of the rotor system. The specific structure and working principles of the BNES are introduced, and the dynamic equations of the rotor system equipped with the BNES are established. In steady-state responses, a single objective optimization strategy is established to minimize the angular displacement amplitude of the vibration response; In transient responses, the objective function is selected to maximize the energy dissipation rate of the BNES. The parameters are optimized by the genetic algorithm (GA). The results show that the BNES with optimal parameters has better vibration suppression performance on the transient responses and steady-state responses of the rotor system. Keywords: Rotor system · Torsional vibration suppression · Nonlinear energy sink · Bistability · Optimization

1 Introduction Currently, numerous methods for suppressing vibration have been proposed by scholars, including dynamic balancing [1], support stiffness adjustment [2], and active control [3]. In addition, passive control techniques have gained widespread use in vibration mitigation due to their simple design and lack of need for sophisticated algorithms [4]. While attaching a linear dynamic absorber to the main structure is a frequently used passive control method, it only provides good vibration suppression effects near the anti-resonance point [5, 6]. As a result, vibration control techniques such as damping optimization [7], semi-active control [8], and nonlinearization of linear absorbers [9] are gradually gaining prominence. Nonlinear energy sinks (NESs) are a passive control method that primarily consists of nonlinear stiffness, damping elements, and NES mass [10]. Nonlinear stiffness provides a nonlinear restoring force [11], while NES enhances vibration suppression efficiency by transferring vibrational energy unidirectionally from the main system to the NES mass through the target energy transfer (TET) mechanism [12]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 133–146, 2024. https://doi.org/10.1007/978-3-031-40459-7_9

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Research studies have demonstrated that multi-stable NESs have a superior advantage over mono-stable ones due to their ability to consume more energy through jumps between multiple stable states. Therefore, researchers have proposed various multistable NESs, including the bi-stable NES introduced in reference [13], which has been effectively utilized to suppress bending vibration in rotor systems. In reference [14], a comparison was made between the vibration suppression performance of tri-stable and bi-stable absorbers. In reference [15], the dynamic response of a bi-stable absorber was studied from a nonlinear characteristics perspective. The aforementioned research has shown the effectiveness of using multi-stable NESs to suppress vibration. However, there has been limited investigation into the application of multi-stable NESs for torsional vibration suppression, with only one experimental study currently available [16]. Therefore, to broaden the scope of multi-stable NESs in torsional vibration suppression, this paper proposes a bi-stable NES for use in rotor systems. The structural principles, vibration characteristics, and vibration suppression effects of the proposed NES are examined through theoretical and experimental research.

2 BNES Structure and System Dynamic Equations 2.1 BNES Structure As shown in Fig. 1(a), the outer magnetic holder is fixed on the shaft, while the inner magnetic holder contains a built-in bearing, which is equivalent to the NES mass. The nonlinear stiffness is achieved through the positive and negative stiffness magnetic pair units (four in total). The specific structure of the positive and negative stiffness magnetic pair units is shown in Fig. 1(b). The magnetic force in the x-direction of magnetic pair N1 is used to provide negative stiffness, while the magnetic force in the x-direction of magnetic pairs P1 and P2 is used to provide positive stiffness. The permanent magnets in the structure have the same size and material, and their magnetization directions are along their axial directions. The magnets in each magnetic pair are configured in a mutually exclusive manner. Assuming that the NES mass only has displacement in the x-direction during torsional vibration. When the NES mass is at the initial position, it is in a critical (a)

Outer magnet holder

(b) Magnet pair P1

The initial position

Magnet pair P2

z Inner magnet holder

rm

Magnet

Magnet pair N1

Fig. 1. Structure of BNES

y θm

x

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equilibrium state. If the system generates torsional vibration that deviates from the static equilibrium position, the positive and negative stiffness magnetic pairs will jointly produce the required nonlinear restoring torque. r m is the rotation radius of the magnets. 2.2 Magnet Restoring Force The magnetic calculation model of the permanent magnet is shown in Fig. 2. The inner and outer surfaces of the left magnet are defined as face 1 and face 2, and the inner and outer surfaces of the right magnet are defined as face 3 and face 4, respectively. The radius and thickness of the magnet are Rm and cm , and the distance between the two magnets is hm . The displacement difference between the two magnets in the x-direction is em . 1

2 p2 (r2, α)

3

r23 o x' y'

Rm z'

em p3 (r3, β) hm

4

cm

Fig. 2. Magnetic calculation model

Based on the theory of static magnetism [17], the surface density of a magnet can be expressed as σ = μ0 · M = Br ,

(1)

where μ0 is the magnetic permeability of vacuum; M is the magnetization intensity; Br is the residual magnetic flux density of the magnet. Faces 2 and 3 have been chosen as the reference for calculating the force exerted between the permanent magnets. The magnetic charge density at any point p2 on face 2 is q2 = Br r2 dr2 dα,

(2)

where r 2 and α are the polar coordinates of any point on face 2. The magnetic field strength generated at any point p3 on face 3 by the magnetic charge at point p2 is H=

1 q2 Br r2 dr2 dα 1 r23 = r23 . 4πμ0 |r23 |3 4πμ0 |r23 |3

(3)

The magnetic charge at any point p3 on face 3 is q3 = Br r3 dr3 dβ,

(4)

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where r 3 and β are the polar coordinates of any point on face 3. According to the theory of equivalent magnetic charges, the magnetic field force experienced by point p3 can be expressed as dF23 = Hq3 =

Br2 r2 r3 dr2 dr3 dαdβ r23 . 4πμ0 |r23 |3

(5)

In Cartesian coordinate system, the coordinates of point p2 are (r 2 cosα, r 2 sinα, z2 ), and the coordinates of point p3 are (r 3 cosβ + em , r 3 sinβ, z3 ), then r23 = (r3 cos β − r2 cos α + em )i + (r3 sin β − r2 sin α)j + hm k,

(6)

where i, j, and k are unit vectors in the x-, y-, and z-direction respectively. On face 3, the magnetic force in the x-direction at point p3 is x dF23 = 

r3 cos β − r2 cos α + em (r3 cos β − r2 cos α + em )2 + (r3 sin β − r2 sin α)2 + (hm )2

3/2 r2 r3 dr2 dr3 dαdβ.

(7)

Similarly, on face 3, the magnetic force in the z-direction at point p3 is z dF23 = 

−hm (r3 cos β − r2 cos α + em ) + (r3 sin β − r2 sin α)2 + (hm )2 2

3/2 r2 r3 dr2 dr3 dαdβ.

(8)

By integrating Eq. (7) and Eq. (9), the force between the two faces along the x- and z-direction is x = F23

 2π  2π  Rm  Rm r3 cos β − r2 cos α + em Br2  3/2 r2 r3 dr2 dr3 dα dβ, 4πμ0 0 0 0 0 (r3 cos β − r2 cos α + em )2 + (r3 sin β − r2 sin α)2 + (hm )2

(9)

2  2π  2π  Rm  Rm −hm z = Br F23  3/2 r2 r3 dr2 dr3 dα dβ. 4πμ0 0 0 0 0 (r3 cos β − r2 cos α + em )2 + (r3 sin β − r2 sin α)2 + (hm )2

(10) The force in the x-direction between other faces is  2π  2π  Rm  Rm Aij ri rj dri drj dαdβ Br2 Fijx =  3/2 . 4πμ0 0 0 0 0 A2ij + Bij2 + Cij2

(11)

Similarly, the force along the z-direction between other faces is Fijz =

Br2 4πμ0



2π  2π  Rm 0

0

0

 0

Rm

−Cij ri rj dri drj dαdβ  3/2 . A2ij + Bij2 + Cij2

(12)

The expressions of Aij , Bij and C ij (i = 1,2; j = 3,4) can be written as ⎧ ⎨ A13 = r3 cos β − r1 cos α + em , B = (r3 sin β − r1 sin α)2 , ⎩ 13 C13 = cm + hm ;

(13)

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⎧ ⎨ A24 = r4 cos β − r2 cos α + em , B = (r4 sin β − r2 sin α)2 , ⎩ 24 C24 = cm +hm ; ⎧ ⎨ A14 = r4 cos β − r1 cos α + em , B = (r4 sin β − r1 sin α)2 , ⎩ 14 C14 = 2cm +hm .

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(14)

(15)

The force between two permanent magnets is the combined force of four magnetic charge surfaces, which can be expressed as x(z)

x(z)

x(z)

x(z)

x(z) Fsum = F23 + F14 − F13 − F24 .

(16)

Then, the torque produced by each magnetic pair can be expressed as x(z) Tm = Fsum rm .

(17)

It is assumed that the radius and thickness of the negative stiffness magnet pair are Rmn and cmn respectively, and the distance between the two magnets is hmn . The radius and thickness of the positive stiffness magnet pair are Rmp and cmp respectively, and the distance between the two magnets is hmp . The magnet material used is Neodymium Iron Boron, and the specific parameters are shown in Table 1. Table 1. BNES parameters Parameter

Value

Parameter

Value

Rmn /mm

5

hmp / mm

2.3

cmn / mm

4

r m /mm

70

Rmp /mm

2.5

μ0 /(T·m−1 ·A)

4π × 10–7

cmp / mm

5

Br / T

1.34

hmn / mm

3

Based on the parameters above, the relationship curve between magnetic torque T p and displacement of the positive stiffness magnetic pair P1 and P2 is shown in Fig. 3(a), and the relationship curve between magnetic torque T n and displacement of the negative stiffness magnetic pair N1 is shown in Fig. 3(b). Adding the magnetic torques acting on the NES mass, the relationship curve between the magnetic torque and displacement of the positive and negative stiffness magnetic pair structure can be obtained, as shown in Fig. 3(c). Fitting the precise values in Fig. 3(c), an approximate expression for the nonlinear torque T N can be written as   4 3 2 TN (θ ) = 4 × 2.89 × 108 rm θ − 396.94rm θ . (18)

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(a)

0.0

-0.4 -0.8 -3

0.0

3

0.0

A

B

C

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-0.2 0 θ (deg)

Accurate value Fitting value

(c)

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0.2 Tn (N·m)

Tp (N·m)

0.4

0.6

(b) TN (N·m)

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-0.4 -3

0 θ (deg)

-0.6 -3

3

0

θ (deg)

3

Fig. 3. Nonlinear force of the BNES

2.3 Dynamic Model of the Rotor-BNES System The application of BNES to torsional vibration suppression in a single-disc rotor system is shown in the simplified model of the coupled system in Fig. 4. The material for the shaft and the disc is steel, with a density of 7850 kg/m3 . Tr

K1

J1

C1

J2

TN

C2

J3

Fig. 4. Rotor-BNES system dynamics model

where J 1 is the equivalent moment of inertia of the motor rotor; J 2 is the equivalent moment of inertia of the disk; J 3 is the equivalent moment of inertia of BNES; K 1 represents√the torsion stiffness of the shaft section between the motor and the disk; C1 = 2ξ1 Jr K1 is the damping of the rotor system, where J r is the equivalent moment of inertia of the rotor system, ξ 1 is the damping ratio of the rotor system; C 2 is the damping of BNES; T r = T ra cos(ωt) is the external excitation torque, where ω is the rotational speed and T ra is the excitation amplitude; T N is the nonlinear restoring torque of BNES, and its expression is TN = kn1 (θ2 − θ3 ) + kn2 (θ2 − θ3 )3 ,

(19)

where k n1 and k n2 are obtained from Eq. (18) through curve fitting. According to the torsional vibration dynamic model shown in Fig. 4, the motion equation of the coupled system is

J1 θ1 + C1 θ1 − θ2 + K1 (θ1 − θ2 ) = 0

(20) J2 θ2 − C1 θ1 − θ2 − K1 (θ1 − θ2 ) + C2 θ2 − θ3 + TN = Tr .    J3 θ3 − C2 θ2 − θ3 − TN = 0

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By substituting ϕ i = θ i − θ i+1 (i = 1, 2) into Eq. (20), the rotation of the rigid body is eliminated, and Eq. (20) can be rewritten as         J1 +J2 J1 +J2 1 1 1 1 C − C K − K T + T 1 0  1 2 1 2 r N  J2 J2 ϕ + J1 J12 ϕ = J1 J2 +J3J2 . ϕ + J1 J12 n 3 − J2 J3 TN 01 − J2 C1 JJ22+J − J2 K1 JJ22+J J3 C2 Jn K2 (21)

3 Parameter Optimization and Result Analysis 3.1 Parameter Setting In the numerical simulation, BNES parameters are shown in Table 1, and some BNES parameters and rotor system parameters are shown in Table 2. The Runge-Kutta method is used to solve the vibration response of the coupled system. Table 2. Simulation parameters Parameter

Value

J 1 /(kg·m2 ) J 3 /(kg·m2 )

Parameter

Value

0.003

K 1 /(N·m/rad)

20.5

0.0015

C 2 /(N·m·s/rad)

0.01

d j2 /mm

120

ξ1

0.02

l j2 /mm

40

3.2 Parameter Optimization Under Transient Response The energy absorption efficiency of BNES under transient excitation can be quantitatively analyzed by the ratio between the dissipated energy of BNES and the input energy. Then the energy dissipation rate of BNES can be written as [18] t ηd (t) =

0

 2 C2 θ˙3 (t) − θ˙2 (t) d τ × 100%. 1 J2 θ˙2 (t0 )2

(22)

2

The sum of instantaneous kinetic energy and potential energy En can be expressed as [4] En =

1 1 J3 θ˙3 (t0 )2 + kn2 [θ3 (t) − θ2 (t)]4 . 2 4

(23)

Then, the energy percentage, which represents the proportion of the energy of the NES and the mechanical energy of the rotor system, can be expressed as ηm =

En En + 21 J2 θ˙2 (t0 )2 + 21 k1 [θ2 (t) − θ1 (t)]2

× 100%.

(24)

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The modifications in magnetic parameters are ultimately reflected in the changes of stiffness coefficient. Therefore, J3 , kn1 , kn3 , and c2 are directly selected for parameter optimization. To enhance the energy dissipation performance of BNES, the parameter optimization problem for BNES under transient response is solved by maximizing ηd as the objective. As a result, the objective function, optimization variables, and their corresponding constraint ranges can be expressed as max

J3 ,kn1 ,kn3 ,c2

ηd

⎧ 0.0005 kg · m2 ≤ J3 ≤ 0.0015 kg · m2 ⎪ ⎪ ⎨ . −7.6 N · m/rad ≤ kn1 ≤ −7.8 N · m/rad s.t. ⎪ 24000 N · m/rad3 ≤ kn3 ≤ 30000 N · m/rad3 ⎪ ⎩ 0.005 N · m · s/rad ≤ c2 ≤ 0.015 N · m · s/rad

(25)

The parameters of the BNES are optimized using a genetic algorithm, with a population size of 30 individuals and a maximum number of 30 iterations. The possibility of crossover is 0.8, while the probability of mutation is 0.01. 3.3 Vibration Reduction Analysis Under Transient Response The magnitude of the impulse is set as θ˙2 = 3 rad/s. By optimizing within the parameter space, convergence is achieved, and the optimal design parameters are J3 = 0.001 kg·m2 , kn1 = 7.7 N·m/rad, kn3 = 26000 N·m/rad3 , c2 = 0.015 N·m·s/rad. Figure 5(a) shows the transient response curves of the rotor system without BNES. The black dashed line represents 5% of the initial amplitude. It is seen that the vibration amplitude of the rotor system decays faster with the addition of BNES. This can be explained by analyzing Fig. 5(c), which shows that BNES undergoes inter-well motion in the time interval of 0–0.28 s, rapidly consuming the initial energy between its two equilibrium points. Finally, BNES stabilizes at equilibrium point C. Figure 6 shows the dynamic characteristics of the proposed BNES. According to Fig. 6(a), it is found that the largest energy dissipation rate ηd reaches 33.66%. Based on Fig. 6(b), it is seen that E n reaches the maximum value of 7.14 × 10−3 J in an instant of times in the snap-through motion region, indicating that the energy is transferred from the rotor system to the BNES. Inspecting Fig. 6(c), it is observed that the energy of the BNES occupies a large proportion in the snap-through motion region. The wavelet spectrum of the rotor system after adding BNES is shown in Fig. 7. It is seen that the 1:1 resonance occurred between the rotor system and BNES.

(a) 2

(b) 2

(c) 2

Amplitude (deg)

Amplitude (deg)

Amplitude (deg)

Parametric Optimization of BNES in Torsional Vibration Suppression

1 0 -1 -2 0

1 Time (s)

2

1 0 -1 -2 0

1 Time (s)

2

1

141

C

0 A

-1 -2 0

1 Time (s)

2

Fig. 5. Transient response curves of the rotor-BNES system under θ˙2 (t0 ) = 3 rad/s: (a) the rotor system without BNES, (b) the rotor system with BNES, and (c) the BNES displacement. (b) 8

ηd (%)

En (J)×10-3

(1.5, 33.66) 20

(c) 100 80

6 ηm (%)

(a) 40

4 2

0 0

1 Time (s)

2

60 40 20

0 0

1 Time (s)

2

0 0

1 Time (s)

2

Fig. 6. Dynamic characteristics of the BNES under θ˙2 (t0 ) = 3 rad/s: (a) the energy dissipation rate, (b) the energy of BNES, and (c) the energy percentage of BNES.

Fig. 7. Wavelet spectra of the rotor-BNES system under θ˙2 (t0 ) = 3 rad/s: (a) the rotor system without BNES, (b) the rotor system with BNES, and (c) the BNES displacement.

3.4 Parameter Optimization Under Steady-State Responses The vibration suppression rate is a metric used to evaluate the vibration suppression performance of BNES under steady-state response. It can be defined as p=

AL-NES −AA-NES , AL-NES

(26)

where AL-NES and AA-NES represent the resonance peak of the rotor system with BNES and active BNES respectively. The aim of parameter optimization is to minimize AA-NES . Therefore, a singleobjective optimization strategy subject to Eq. (21) is proposed, which entails an objective

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function and constraints that can be expressed as min

J3 ,kn1 ,kn3 ,c2

AA - NES

⎧ 0.0005 kg · m2 ≤ J3 ≤ 0.0015 kg · m2 ⎪ ⎪ ⎨ . −7.6 N · m/rad ≤ kn1 ≤ −7.8 N · m/rad s.t. ⎪ 24000 N · m/rad3 ≤ kn3 ≤ 30000 N · m/rad3 ⎪ ⎩ 0.005 N · m · s/rad ≤ c2 ≤ 0.015 N · m · s/rad

(27)

The parameter setting of GA is consistent with Sect. 3.2. 3.5 Vibration Reduction Analysis Under Steady-State Response In this section, T ra = 0.2 N·m is set. The optimal parameters corresponding to the convergence results are J3 = 0.0015 kg·m2 , kn1 = 7.71 N·m/rad, kn3 = 27850 N·m/rad3 , c2 = 0.01 N·m·s/rad. The amplitude-frequency response curve of the rotor system is shown in Fig. 8(a). The amplitude of the rotor system without BNES at 16 Hz is 4.36 deg. After adding BNES, the rotor system exhibits a strongly modulated response (SMR) phenomenon in the frequency range of 14–22 Hz, indicating that BNES effectively suppresses the vibration of the rotor system near the resonance frequency. The maximum amplitude of the rotor system is 2.05 deg, which is a 52.98% reduction compared to the system without BNES. Figure 9 shows the time-domain responses of the rotor system at 16.5 Hz. As depicted in Fig. 9(a), the rotor system without BNES performs periodic motion. As described in Fig. 9(b), the rotor system exhibits nonlinear beating vibration. Figure 9(c) shows that BNES performs chaotic motion and oscillates between its two equilibrium points. According to Fig. 10(b), the Poincaré diagram shows a distributed point set lacking annular features, while the phase diagram exhibits irregular reciprocating motion, indicating that the rotor system is in a chaotic state. Figure 10(c) illustrates the sweeping motion of BNES between its two equilibrium points and its chaotic behavior, providing additional confirmation to the findings in Fig. 8.

with locked BNES with active BNES

4

2

0 10

15 20 Frequence (Hz)

25

(b) 4

Amplitude (deg)

Amplitude (deg)

(a) 6

3 2 1 0 10

15 20 Frequence (Hz)

25

Fig. 8. Frequency response curves of the coupled system under T ra = 0.2 N·m: (a) the rotor system without and with BNES, and (b) the BNES.

(a) 3

(b) 3

(c) 3

Amplitude (deg)

Amplitude (deg)

Amplitude (deg)

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0

-3 2

3

Time (s)

4

5

0

-3 0

1

2 3 Time (s)

4

5

143

0

-3 0

1

2 3 Time (s)

4

5

Fig. 9. Time response curves of the coupled system under T ra = 0.2 N·m: (a) the rotor system without BNES, (b) the rotor system with BNES, and (c) the BNES. 2

(b) 3 ×10

(c) 3

2 dx1/dt (rad/s)

2 dx1/dt (rad/s)

2

1 0

1 0

-1

-1 -2

-2

-3 -3

-3 -3

-2

-1

0 1 x1 (mm)

2

3

×102

2 dx2/dt (rad/s)

(a) 3 ×10

1 0

-1 -2

-2

-1

0 1 x1 (mm)

2

3

-3 -3

-2

-1

0 1 x2 (mm)

2

3

Fig. 10. Phase portraits and Poincaré maps of the coupled system under T ra = 0.2 N·m: (a) the rotor system without BNES, (b) the rotor system with BNES, and (c) the BNES.

4 Experimental Research 4.1 Experimental Set-Up for BNES Effectiveness To verify the capability of BNES to suppress torsional vibrations, a rotor system test rig (shown in Fig. 11) is established. BNES is installed near the disk, and zebra tape is applied to the outer surface of both the disk and the PLA light disk. The rotating pulse is measured using a laser sensor, and the data is collected and analyzed using LMS.Testlab. The laser sensor is mounted on an adjustable support base, while corresponding zebra belts are installed on each end of the rotating shaft. By measuring the relative displacement of the two belts, the amount of torsional vibration in the disk can be determined. Disk

Motor

Zebra tape

Coupling

Remote optical laser sensor

NES

Fig. 11. Test device

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4.2 Experimental Results (1) Sweep frequency The motor speed increased steadily from 60 r/min to 1500 r/min, and the colormap of the rotor system is obtained, as shown in Fig. 12. As illustrated in Fig. 12(a), the rotor system without BNES exhibits multi-order torsional vibration excitation. However, the addition of BNES, as depicted in Fig. 12(b), resulted in a significant weakening of the intensity of harmonic components. Moreover, both figures indicate that the first-order resonance frequency of the rotor system is approximately 16.6 Hz, consistent with the simulation results.

Fig. 12. Colormaps of the rotor system: (a)with lock BNES, and (b)with active BNES.

Figure 13 displays the angular displacement fluctuation curves of the rotor system. The red dotted line depicts the vibration response of the rotor system without BNES, revealing a large torsional fluctuation amplitude. However, upon the addition of BNES, the response amplitude significantly decreases, as demonstrated by the blue solid line. The root mean square of the angular displacement fluctuation curve decreases from 1.73 deg to 0.41 deg, and the vibration suppression rate is 76.30%, indicating BNES possesses excellent vibration suppression capability.

Without BNES With BNES

0

-4

5

80 Time (s)

155

(b) 4 Amplitude (deg)

Amplitude (deg)

(a) 4

Without BNES With BNES

0

-4

90

90.25 Time (s)

90.5

Fig. 13. Time response curves of the rotor system: (a) comparison result with and without NES, and (b) partially enlarged view.

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5 Conclusions (1) By formulating the optimization strategies of the BNES under transient excitation, the optimal parameters of the BNES are obtained. (2) The dynamic response of the rotor system under different transient excitation shows that BNES can quickly dissipate the energy of the main structure. (3) The vibration suppression rate of BNES for steady-state response can reach 52.98% in simulations and 76.30% in experiments. Acknowledgments. The authors would like to gratefully acknowledge the National Natural Science Foundation of China (Grant No. 52075084), the Foundation of Equipment Pre-research Area (Grant No. 50910050302), and the Fundamental Research Funds for the Central Universities (Grant No. N2303005) for the financial support for this study.

References 1. Haidar, A.M., Palacios, J.L.: A general model for passive balancing of supercritical shafts with experimental validation of friction and collision effects. J. Sound Vib. 384, 273–293 (2016) 2. Song, Y.S., Sun, X.T.: Modeling and dynamics of a MDOF isolation system. Appl. Sci. 7, 393 (2017) 3. Poplawski, B., Mikułowski, G., Wiszowaty, R., et al.: Mitigation of forced vibrations by semi-active control of local transfer of moments. Mech. Syst. Signal Process. 157, 107733 (2021) 4. Zhang, Y.W., Lu, Y.N., Zhang, W., et al.: Nonlinear energy sink with inerter. Mech. Syst. Signal Process. 125, 52–64 (2019) 5. Taghipour, J., Dardel, M., Pashaei, M.H.: Vibration mitigation of a nonlinear rotor system with linear and nonlinear vibration absorbers. Mech. Mach. Theory 128, 586–615 (2018) 6. Lu, Z., Wang, Z.X., Zhou, Y., et al.: Nonlinear dissipative devices in structural vibration control: a review. J. Sound Vib. 423, 18–49 (2018) 7. Zhang, X.P., Kang, Z.: Vibration suppression using integrated topology optimization of host structures and damping layers. J. Vib. Control 22, 60–76 (2014) 8. Arrigan, J., Pakrashi, V., Basu, B., et al.: Control of flapwise vibrations in wind turbine blades using semi-active tuned mass dampers. Struct. Control. Health Monit. 18, 840–851 (2011) 9. Li, X., Zhang, Y.W., Ding, H., et al.: Dynamics and evaluation of a nonlinear energy sink integrated by a piezoelectric energy harvester under a harmonic excitation. J. Vib. Control 25, 851–867 (2018) 10. Gendelman, O.V.: Targeted energy transfer in systems with non-polynomial nonlinearity. J. Sound Vib. 315, 732–745 (2008) 11. Bab, S., Khadem, S.E., Shahgholi, M., et al.: Vibration attenuation of a continuous rotorblisk-journal bearing system employing smooth nonlinear energy sinks. Mech. Syst. Signal Process. 84, 128–157 (2017) 12. Dou, J.X., Li, Z.P., Cao, Y.B., et al.: Magnet based bi-stable nonlinear energy sink for torsional vibration suppression of rotor system. Mech. Syst. Signal Process. 186, 109859 (2023) 13. Yao, H.L., Wang, Y.W., Xie, L.Q., et al.: Bi-stable buckled beam nonlinear energy sink applied to rotor system. Mech. Syst. Signal Process. 138, 106546 (2020)

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14. Huang, X., Yang, B.: Investigation on the energy trapping and conversion performances of a multi-stable vibration absorber. Mech. Syst. Signal Process. 160, 107938 (2021) 15. Rezaei, M., Talebitooti, R., Liao, W.H.: Exploiting bi-stable magneto-piezoelastic absorber for simultaneous energy harvesting and vibration mitigation. Int. J. Mech. Sci. 207, 107938 (2021) 16. Haris, A., Motato, E., Mohammadpour, M., et al.: On the effect of multiple parallel nonlinear absorbers in palliation of torsional response of automotive drivetrain. Int. J. Non-Linear Mech. 96, 22–35 (2017) 17. 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) 18. Liu, C., Zhao, R., Yu, K., et al.: In-plane quasi-zero-stiffness vibration isolator using magnetic interaction and cables: theoretical and experimental study. Appl. Math. Model. 96, 497–522 (2021)

Aviation Spline Wear Test Bench and Fretting Wear Measurement Xiangyang Zhao1 , Guang Zhao1 , Yunbo Yuan2(B) , Fanrong Kuang1 , Mei Guo3 , and Haofan Li4 1 School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024,

China 2 School of Control Science and Engineering, Dalian University of Technology, Dalian 116024,

China [email protected] 3 AVIC Shenyang Engine Design Institute, Shenyang 110066, China 4 AVIC Shenyang Aircraft Design and Research Institute, Shenyang 110035, China

Abstract. Splines are widely used in aircraft power plants for their high load carrying capacity. Fretting wear and fatigue occur sometimes and have been a prominent problem for aviation splines. Currently, the wear mechanism of aviation splines is still not clear, which endangers and shortens their service life seriously. This paper establishes an aviation spline wear test bench, which is mainly composed of spline rotor system, lubrication system, control and test systems. The test bench can rotate continuously for several hours without any interruption and can realize accurate angular misalignment through adjusting the horizontal position of support bearings. Based on the proposed test bench, some fretting wear experiments are conducted using involute aviation splines under both aligned and misaligned conditions. Spline weight loss, wear debris weight and tooth wear depth are selected to quantify fretting wear behavior. Experimental results show that angular misaligned splines have obvious fretting wear, whereas the aligned ones almost have no wear. Keywords: Aviation splines · Fretting wear · Angular misalignment · Test bench · Experiment

1 Introduction Splines are widely used in power plants as an important part of mechanical power transmission systems because of its high bearing capacity [1]. The spline bears complex load and environmental conditions, and various factors lead to extensive and long-term failure. In many forms of spline failure, the wear caused by misalignment is particularly prominent [2]. Wear seriously affects the safety and reliability of aviation splines. Exploring the mechanism of spline wear and developing anti-wear design technology have become the focus of many researchers. The spline bears both periodic and impact loads, which produce alternating contact stress on the working tooth surface of the spline and will lead to slight vibration between © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 147–155, 2024. https://doi.org/10.1007/978-3-031-40459-7_10

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the contact tooth surfaces. Because the internal and external excitations that promote this vibration cannot be eliminated, the micro-amplitude vibration is inevitable, resulting in fretting wear of the spline. Fretting wear cannot be ignored in the dynamic model of the aviation transmission system. To ensure the accuracy of calculation and the reliability of the transmission system, the prediction of spline wear is important [3–5]. Many researchers have researched spline wear through experiments and finite element analysis. Ku et al. [6] conducted a spline wear test and believed that it is difficult to achieve a perfect centering state in practice. Their results indicate that misalignment has an important impact on the reliability and life of splines. In practical applications, tooth fatigue and fracture can be avoided by reducing the number of teeth. Cuffaro et al. [7] studied the behavior of early wear and fatigue failure caused by abnormal contact and vibration caused by geometric errors. Zhao G et al. [8] established an aviation spline vibration wear test bench and predicted the amount of spline wear. Cuffaro et al. [9, 10] developed a spline wear test bench, tested the influence of misalignment, surface roughness and lubrication on spline wear, and studied the experimental method of surface pressure measurement and monitoring of spline teeth. Xue et al. [11] studied the tooth surface wear of floating involute spline couplings and proposed a wear prediction model suitable for floating involute spline couplings. Zhao et al. [12] established a numerical model of spline wear with multiple errors using an improved Archard wear model. However, the above work rarely realizes the mechanism research on the coupling characteristics of spline misalignment, vibration and wear failure behavior, and most of them only focus on one. Previous studies lack effective experimental evidence to fully reveal this failure mechanism. Obtaining first-hand data through experiments to clarify the failure mechanism is of great significance for the management and maintenance of aviation splines. To this end, this paper develops an aviation spline wear performance test bench. The study of spline misalignment, vibration and wear coupling characteristics, and a speed and torque working condition loading function can be realized. In a long-term wear process, the vibration, torque and temperature monitoring and alarm are realized. At the same time, an accurate measurement method of fretting wear was developed, and the influence of angular misalignment spline on wear was discussed.

2 Spline Wear Test Bench According to the force form and friction and wear characteristics of an aviation involute spline pair in the actual working process, the quantitative experimental study of spline wear is carried out, and the aviation spline wears performance test bench is developed, as shown in Fig. 1. The test bench is mainly composed of a spline rotor system, lubrication system, control system and test system. The two-dimensional sketch model of the test bench is shown in Fig. 1a, and it has been assembled and debugged in the laboratory, as shown in Fig. 1b. The test platform adopts two 75 kW variable frequency motors, which are driven and loaded by two inverters. The maximum speed of the motor is 4500 rpm. The load generator current is fed back to the drive motor through the common DC bus to achieve energy savings. According to the spline torque and power of the central transmission rod, the rated torque of the test platform is selected at 213 Nm,

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and the rated speed is 3200 rpm. The torque sensor is connected to the control screen through a data line to facilitate the reading of the load and speed values. The maximum allowable inclination misalignment of the centering adjustment device is 0.5°. In order to facilitate the disassembly of the spline from the test bench, the inner and outer splines are rigidly connected to their respective flange shafts through a ring of bolts. The geometric parameters of the spline are shown in Table 1. The spline material and hardness are 16Cr3NiWMoVNbE @ HRC36–44, the spline tooth surface roughness is 1.6, and the spline tooth surface has no coating and hardening.

Fig. 1. Spline wear test bench: (a) two-dimensional sketch model, (b) test bench object.

Table 1. Parameters of spline. Types

Internal spline

External spline

Number of teeth

19

19

Module (mm)

1.5

1.5

Pressure angle (z)

30

30

Tooth width(mm)

32

32

Major diameter (mm)

30.75

30

Minor diameter (mm)

27.18

26.25

Pitch diameter (mm)

28.5

28.5

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Figure 2. Shows the test system of the test bench, which is developed to monitor the vibration signal of the test bench. The arrangement of acceleration sensor, displacement sensor, temperature sensor and torque sensor is shown in Fig. 1 and Fig. 2. The acceleration sensor is used to monitor the vibration signal of the spline. The sensitivity of the acceleration sensor at room temperature is about 100 mV/g. The displacement sensor is used to monitor the vibration displacement of the inner and outer splines in X and Y directions. The sensitivity of the displacement sensor at room temperature is about 8mV/mm; the temperature sensor is located near the inner bearing of the motor and is used to monitor the temperature of the motor bearing. The signals of the above sensors are collected and recorded in real-time by an 8-channel acquisition instrument. The data is processed and analyzed on the data analysis software by connecting the data transmission line to the PC. The collected signals are transmitted to the controller at the same time to realize the feedback control of torque and temperature and the alarm of torque, temperature and vibration.

Fig. 2. Configuration of test system for the test bench

3 Functional Analysis of the Test Bench 3.1 Implementation of Angular Misalignment Before setting the spline angular misalignment in the horizontal direction, the whole tester is fully aligned, including the radial and axial directions. The misalignment of each connection structure ( coupling, spline, bearing seat) should be guaranteed within 0.05 mm, each structure should be fixed, and the angular misalignment adjustment should be carried out after the complete alignment adjustment is completed. The adjustable bearing seat is fixed on the cast iron platform through the bottom plate, the guide rail and the guide key. The vertical plate is fixed on the bottom plate by bolts, the bolts are adjusted by the horizontal displacement, and the guide rail and the guide key are adjusted horizontally. The radial displacement of the horizontal bearing seat is changed by adjusting the horizontal displacement of the bolt to realize the angular misalignment control of the spline. After the misalignment angle is reached, the fixed bolt is fixed, and after adjustment, the accurate angular misalignment is measured by the laser alignment instrument. The test bench alignment spline adjustment method is shown in Fig. 3.

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Fig. 3. Test bench alignment spline adjustment method

3.2 Spline Torque Application and Feedback Measurement The drive frequency converter and the passive frequency converter are used to control the speed of their respective motors to achieve loading. During the long-term experiment, the shaft speed may fluctuate slightly, but it is generally stable at the set value. At present, the speed control method is adopted, and the torque is applied through the speed difference between the two. The controller is implemented by a touch screen combined with a controller. The torque of the actual spline is tested and calibrated by the passive motor side torque speed sensor. Through the actual measurement, the average torque displayed by the controller is the actual torque of the spline. The advantages of the spline wear test bench are: (1) Research on the coupling characteristics of spline misalignment, vibration and wear: Because misalignment will affect the vibration of the rotor, the misalignment and vibration adjustment of the vibration and wear performance tester of the aviation spline-rotor system is realized by the angular misalignment adjustment device and the unbalanced mass applied to the stepped flange shaft bolt. The contact stress and fretting wear between the external spline test piece and the internal spline test piece are affected by the two, so as to realize the research on the coupling characteristics of spline misalignment, vibration and wear. (2) Loading function of speed and torque conditions: Through the drive and loading system, the adjustment of the large cycle of the spline condition in the actual engine is realized.

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4 Wear Measurement Spline weight loss, wear debris weight and tooth wear depth are selected to quantify fretting wear behavior. The spline weight loss is the sum of internal spline weight loss and external spline weight loss. The wear debris weight is the common wear debris of the inner and outer splines. The tooth wear depth includes the tooth thickness at the internal spline pitch circle and the tooth thickness at the external spline pitch circle. Measurement method of spline weight loss: The spline needs to be cleaned, weighed, and data recorded before and after the experiment, and the weighing operation is repeated three times and averaged; the precision of the high-precision weighing instrument is 0.01 g. Measurement method of wear debris weight: At the end of the experiment, the wear debris left on the spline tooth surface was collected, weighed and data recorded, and the weighing operation was repeated three times, and the average value was taken. Before and after the experiment, the spline was scanned by a blue light scanning instrument, and then the tooth thickness at the pitch circle before and after the spline experiment was obtained by post-processing software. The wear amount was estimated according to the reduction of the tooth thickness at the pitch circle. The precision of the blue light scanning instrument is 0.01 mm. The tooth wear depth measurement process is shown in Fig. 4.

Fig. 4. Flow chart of spline tooth thickness measurement

The process is described in detail in Fig. 4: 1,Before the spline is scanned by blue light, the impurities on the surface of the spline are cleaned up. 2,The titanium powder solution is sprayed on the surface of the spline by using an air pump pen to form a uniform thin layer of titanium powder. 3, More than three reference points are calibrated on the spline end surface, which is used as a reference system for shape modeling to

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facilitate scanning. 4, Open the blue light scanning instrument, adjust the height and angle of the blue light scanning instrument, and ensure that all reference points are in the blue light irradiation area. Open the software set the relevant parameters and start scanning. 5, Using post-processing software, the tooth thickness at the spline pitch circle is measured. Clean the spline surface, oil the spline surface, seal and save.

5 Test Result Analysis Based on the test bench, the experimental temperature was 110 °C, the experimental speed was 3000 rpm, the experimental torque was 200 Nm, and the spline fretting wear tests with angular misalignment of 0.00° and 0.15° were carried out under dry friction. The total running time of each angular misalignment condition is 50h, and the spline weight loss measurement, wear debris weight collection and measurement are performed every 10h. After the operation, the statistical measurement of the tooth wear depth is performed. Two replicate experiments were conducted for each of the above misalignment conditions. The comparison of spline wear morphology after the operation of each working condition is shown in Fig. 5. It can be seen that when completely aligned, the internal and external splines have almost no wear, and the tooth surface is intact; when the angular misalignment is 0.15°, a layer of wear marks appears on the tooth surface of the internal and external splines.

Fig. 5. Comparison of the morphology of the spline after wear at each misalignment angle

The above-mentioned measured spline weight loss, wear debris weight, and tooth wear depth are the results of the average of two repeated experiments, as shown in Table 2.

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It can be seen that the wear quality can be neglected when the spline angular misalignment is 0.00°. With the increase of spline angular misalignment, the wear quality of the spline increases. When the angular misalignment is 0.15°, the average grinding tooth thickness at each pitch circle of the inner and outer spline is 0.167 mm and 0.176 mm, respectively. Table 2. Experimental results of spline wear at various misalignment angles Misalignment angle (z)

Spline weight loss (g)

Internal spline weight loss (g)

External spline weight loss (g)

Wear debris weight (g)

Internal spline wear depth (mm)

External spline wear depth (mm)

0.00

0.000

0.000

0.000

0.000

0.000

0.000

0.15

1.750

0.835

0.915

2.250

0.167

0.176

According to the experimental results, the curve of spline wear loss with wear operating time is drawn, as shown in Fig. 6. It can be seen that as the spline wear time continues, the weight loss of the inner and outer splines and the weight of the wear debris generally increase approximately linearly, and the wear rate decreases slightly. The wear debris weight is always greater than the spline weight loss, possibly due to the oxidation reaction on the spline tooth surface, and oxygen is stored in the wear debris.

Fig. 6. Change curve of spline wear with wear operation time

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6 Conclusion By introducing the self-made aviation spline wear test bench and fretting wear measurement method, the influence of angular misalignment on spline wear is analyzed. The main conclusions are: 1) The test bench has the function of large torque quantitative loading, misalignment, vibration and wears quantitative coupling loading. It provides technical support for the study of spline misalignment, vibration and wear coupling characteristics. 2) The experimental platform can realize the precise control of spline angular misalignment. 3) An accurate fretting wear measurement method was developed. 4) The misaligned spline has obvious fretting wear, and almost no wear on the aligned spline.

References 1. Zhao, G., Zhao, X.Y., Qian, L.T., Yuan, Y.B., Ma, S., Guo, M.: A review of aviation spline research. Lubricants 11(1), 1–20 (2023) 2. Chen, C.H.: Common Faults of Aeroengine Mechanical System, 1st edn. Aviation Industry Press, Beijing (2013) 3. Marmol, R.A., Smalley, A.J., Tecza, J.A.: Spline Coupling Induced Nonsynchronous Rotor Vibrations. J. Mech. Des. 102(1), 168–176 (1980) 4. Sang, K.P.: Determination of loose spline coupling coefficients of rotor bearing systems in turbomachinery. PhD. Thesis. Texas A&M Univ, College Station, USA (1991) 5. Al-Hussain, K.M.: Dynamic stability of two rigid rotors connected by a flexible coupling with angular misalignment. J. Sound Vib. 266(2), 217–234 (2003) 6. Ku, P.M., Valtierra, M.L.: Spline wear-effects of design and lubrication. J. Eng. Indu. 97(4), 1263–1265 (1975) 7. Cuffaro, V., Cura, F., Mura, A.: Experimental investigation about surface damage in straight and crowned misaligned splined couplings. Key Eng. Mater. 577, 353–356 (2014) 8. Zhao, G., Li, S.X., Guo, M., Sun, H., Sun, X.C., Han, Q.K.: Aerospace spline vibration wears prediction and experiment. Journal of Aeronautical Dynamics 33(12), 2958–2964 (2018) 9. Cuffaro, V., Cura, F., Mura, A.: Analysis of the pressure distribution in spline couplings. Mechan. Eng. Sci. 226(12), 2852–2859 (2012) 10. Cuffaro, V., Curà, F., Mura, A.: Surface characterization of spline coupling teeth subjected to fretting wear. Procedia Engineering 74, 135–142 (2014) 11. Xue, X., Liu, J., Jia, J., Yang, S., Li, Y.: Simulation and verification of involute spline tooth surface wear before and after carburizing based on energy dissipation method. Machines 11(1), 78 (2023) 12. Zhao, Q., Yu, T., Pang, T., Song, B.: Spline wear life prediction considering multiple errors. Eng. Fail. Anal. 131, 105804 (2022)

Dynamic Analysis of the Finger Seal-Rotor System Sai Zhang, Xiuli Hu, Renwei Che, and Yinghou Jiao(B) School of Mechatronics Engineering, Harbin Institute of Technology, No. 92 XiDaZhi Street, Harbin 150001, China [email protected]

Abstract. Finger seals are widely concerned because of their good sealing performance and low cost. Due to its unique structure and installation, there are complex interactions between the seal laminates, which undoubtedly affect the overall stiffness of the seal. Therefore, it is necessary to conduct an in-depth study on the sealing force model of finger seals. This paper divides the interaction seal force of the finger seal into two parts, the finger beam and finger boot, and establishes a nonlinear seal force model that considers the interaction between finger laminates. A nonlinear dynamics rotor-finger seal system is built, and its dynamic laws are studied by Poincaré mapping, axis trajectory, spectrum diagram and phase diagram. The research in this paper can be used to predict the dynamic behavior of finger seal-rotor systems and ensure the stable operation of rotor systems. Keywords: Finger Seal · Rotor Dynamic · Friction

1 Introduction Seals are important elements of gas turbines. The specific fuel consumption due to leakage can increase by as much as 0.58% for high-pressure turbines [1]. A good sealing system is key to improving the performance and reducing the fuel consumption of the engine. Finger seals have been widely considered in turbines due to their low leakage and low cost. The first finger seal was proposed by Johnson et al. [2], in which contact between finger feet and shaft existed, and such contact would reduce the sealing service life. To eliminate wear, a new type of noncontacting finger seal was proposed [3, 4]. The downstream fingers of the noncontacting finger seal have lift pads on their ends to provide the necessary film force, and then the fingers do not contact the shaft. To ensure the good performance of noncontacting finger seals, research has been carried out. The major topics to be investigated are lifting force and leakage performance. The mass-spring-damper models of noncontacting finger seals are widely used to investigate the leakage and hysteresis performance [5–7]. Braun et al. [8, 9] used the Navier–Stokes equation and obtained three-dimensional temperature and pressure results where the high-pressure side, the speed of rotation, and the heat transfer coefficient are the controlling parameters. Li et al. [10] investigated the effects of rotation speed, axial pressure differential, and lifting distance on the lifting force and leakage. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 156–167, 2024. https://doi.org/10.1007/978-3-031-40459-7_11

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Du et al. [11, 12] presented a model that combines seal dynamic performance and seal leakage to analyze the dynamic performance through leakage. The effects of the change in the clearance between the seal and rotor, the maximum amplitude of rotor motion, and the ratio of friction force to finger stiffness were investigated using the model. Zhao et al. [13] considered the effect of side leakage, which was transformed into seepage flow through a porous medium, to predict the leakage performance. Cutting grooved structures on the lift pad or the shaft surface is useful to improve the hydrodynamic film force. The effect on the lifting force and flow performance of the grooved structures under the lift pad was investigated by Jia et al. [14] and Zhang et al. [15, 16] using a two-way FSI method. Smith et al. [17] investigated the lifting properties and sealing capability of four finger seal embodiments by parametric experiments. The sealing performance of a noncontacting finger seal on a herringbone-grooved rotor was tested by Proctor et al. [18]. Based on the studies above, the seal performance of finger seals has been widely investigated. However, there is little research on the effects of the nonlinear finger seal force on the dynamic characteristics of the rotor system. It is well known that the seal force is one of the main causes of rotor instability [19–21]. Thus, further studies on the dynamic behavior of finger seal-rotor systems are needed. In this paper, a nonlinear finger seal force model considering the interaction between finger laminates is established. The model comprehensively considers the effects of key parameters such as the pressure difference, surface roughness and structure parameters of finger seals on the nonlinear sealing force. The dynamic characteristics of the rotor-finger seal system with and without considering the interaction between laminates are compared. The results of this paper can provide a theoretical prediction method for the dynamic behavior of finger seal-rotor systems.

2 Method 2.1 Geometric Relationship Based on the structural characteristics of finger seals, the following assumptions are made in the analysis process: the seal finger is simplified as a circular arc cantilever beam with equal section; it is assumed that the seal finger is always elastically deformed; and the friction between the back plate and finger laminate is not considered because it only acts on one laminate and is much smaller than the other forces. Figure 1 shows the geometric relationship between the rotor and fingers. The direction of the finger deformation is not along the rotor radius. From the geometric relationship, the equations between deformation and rotor displacement can be obtained:    2 er + 2 Rr · er · cos α + β + ϕ − θf = u2 + v2 + 2 Rr (u · sin m2 + v cos m2 ) u2 + v2 + CO22 + 2 CO2 (u sin m3 + v cos m3 ) − R2r = 0 (1)

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   where er is the rotor displacement; CO2 = er2 + R2r − 2er Rr cos ϕ − θf ; Rr is the rotor diameter; u and v are the radial and tangential deformation at the end of the finger,    R2r +R2s −R2cc e respectively; m2 = arccos 2Rr ·Rs , m3 = m2 + arcsin CO2 sin ϕ − θf .

Fig. 1. Geometric relationship between rotor and finger

2.2 Interaction Between Seal Laminates The finger seal is installed in a staggered manner that allows the interstices on the seal laminates to be covered. However, this type of mounting can affect the seal stiffness. Two cantilever beams installed in a staggered manner can be used as an example to illustrate this impact in a more visual way. As shown in Fig. 2a, a rigid plate moves upward, supporting two cantilever beams on it. In the contact area between the two beams, the frictional force acting on beam 2 is toward the lower right due to its greater deformation than beam 1. In comparison, the point of action of the frictional force on beam 2 is lower than that on beam 1; thus, its effect on downward deformation is greater. This in turn enhances the overall stiffness. Figure 2b shows the structure of the two laminates. It can be seen that, due to the staggered installation of the laminates, a pair of interaction friction forces produce different deformations on different fingers. The friction forces acting on the finger beam and finger foot are discussed separately. For the friction force acting on the finger foot, it can be seen that the arms of Ff 1 and Ff 2 are changed according to the finger deformation. Because the deformations of the finger in contact are very close, Ff 1 and Ff 2 can be considered the same (Fig. 3).

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Beam 1

159

Fixed

Force acting on beam 1 Beam 2

Force acting on beam 1 Rigid plate with upward translation a. Diagram of two staggered beams

b. Diagram of two seal laminates

Fig. 2. Staggered structure force analysis schematic

Fig. 3. Friction force acting on the finger foot

For the laminates at the edge, each finger contacts two fingers of the adjacent laminate. For the other laminates, each finger contacts four fingers of the adjacent laminate. Figure 4 shows the friction force acting on the finger beam. Due to the misalignment installation of the laminates, the ranges of the friction force are consistent. Over the entire angular, the magnitudes of Ffb1 and Ffb2 are:  α −0.5αr   i,j 1  i+1,j+1 (θ + 0.5α ) R d θ h − v P − I − v Ffb1 = 0 ul (θ ) s r s f 2   i,j 1  (2) αul i+1,j (θ ) R d θ h − v P − I − 0.5α − v Ffb2 = 0.5α (θ ) s r s f 2 r where hf is the section height, Is is the interstice width, and Rs is the finger arc radius.

Fig. 4. Friction force acting on the finger beam

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The deformations and rotation angle of the friction forces are calculated by the following equations: ⎧ 2 d v MR2s ⎪ ⎪ ⎨ 2 +v =− dθ EJz (3) ⎪ du NR M ⎪ ⎩ −v = − dθ EA EA where E is the elastic modulus, A is the section area, N and M are the axial force and bend moment, respectively, and J z is the section parameter. Mohr’s integral is also used:  αul N e Ni M e Mi Q Q ( (4) i = + + η e i )Rs d θ EA EI GA 0 where i represents the rotation angles, tangential deformations and normal deformations generated by forces. Then, the total tangential and normal deformations can be obtained: ⎧ ⎨ u = uN + uf + ufb (5) v = vN + vf + vfb ⎩ αAll = αN + αf + αfb where the lower corner markers N, f , fb represent normal force and friction force acted by the rotor, friction force acting on the foot, and friction force acting on the beam. Combining Eq. (1) and (5), the normal force acting on the finger can be obtained using the Newton–Raphson method. It is important to know that Ffb1 and Ffb2 in Eq. (2) depends on the final tangential deformation. Therefore, it is not possible to find the unknown force at once. First, it is necessary to assume the initial value of the deformation caused by Ffb1 and Ffb2 and use the overrelaxation method to obtain the final result. In rotor dynamic calculations, it is undoubtedly impractical to repeat the above calculation process for each solution. Therefore, the obtained seal force is fitted as a polynomial function of rotor displacement to greatly improve the solution speed. Since the mass of the fingers is much smaller than that of the rotor, according to the momentum theorem, when the rotor comes into contact with the seal, the change in seal velocity has little effect on the rotor velocity. In comparison, the seal stiffness and the interaction between laminates have a greater impact on rotor motion. In addition, the degree of freedom of the rotor-seal system will greatly increase if the rotor velocity is considered in the nonlinear seal force model because the dynamic equations of fingers should be induced. Thus, the nonlinear seal force in this paper does not consider the influence of the rotor velocity. 2.3 Finger Seal-Rotor System As shown in Fig. 5, to clearly investigate the effects of the nonlinear seal force, a finger seal-rotor system with rigid supports is established. The parameters of the rotor system are as follows (Table 1):

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Table 1. Parameters of the rotor system Parameters

Symbol

Value

Units

Disk radius

Rr

107.95

mm

Thickness of finger laminate

B

1.00

mm

Inner diameter of the seal

Ri

107.95

mm

Outer diameter of the seal

Ro

122.76

mm

Rotor diameter

Dsh

36.00

mm

Rotor length

L sh

1150.00

mm

Rotor damping

Dr

200

Ns/m

Unbalanced distance

e

0.10

mm

Dimensionless scale

Is

0.40

mm

Fig. 5. Finger seal-rotor system



The dynamic equation of the system is:               Fsx x¨ x˙ x 0 mr 0 meω2 cos ωt cr 0 kr 0 − + + + = Fsy 0 mr y¨ 0 cr y˙ 0 kr y meω2 sin ωt mr g (6) Dimensionless transformations are adopted:           X¨ D 0 X˙ K 0 X Gx + + = ¨ ˙ Gy Y 0D Y 0 K Y

y x Is ,Y = Is ,τ nl +meω2 sin τ −m g Fsy r . mr ω2 Is

where X =

= ωt,Gx =

nl +meω2 cos τ Fsx ,K mr ω2 Is

=

kr ,D mr ω2

=

(7) cr mr ω ,

and Gy =

3 Results and Discussion Predicting rotor system dynamic behaviors mainly focuses on the rotational speed. Figure 6a shows the bifurcation diagram of the rotor system for various rotational speeds considering the interaction between laminates. When the rotational speed is less than

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2200 rpm, the system maintains period-1 motion. In the range of ω ∈ [2200, 3200] rpm, the system exhibits quasiperiodic motion. Then, the system turns into a period one motion in the range of ω ∈ [3300, 7900] rpm. When the rotational speed is greater than 7900 rpm, the system exhibits quasiperiodic motion again. Figure 6b shows the bifurcation diagram of the rotor system for various rotational speeds without considering the interaction. There is a jump in the diagram at 1900 rpm. Figure 6c shows the frequency

Fig. 6. Dynamic characteristics with and without laminate interaction

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component at this rotational speed, and 2f r appears. Figure 6d shows the response time history during a virtual runup without sealing. The trend is basically consistent with the bifurcation diagram. Comparing Fig. 6a and 6b, it can be seen that when the interaction between laminates is considered, the system shows more nonlinear phenomena, and the amplitude of period one motions slightly decreases. Figure 7 shows the bifurcation diagrams with different rotor configurations. Compared with Fig. 7, it can be seen that the rotor structures with different length-to-diameter ratios all exhibit quasiperiodic motion at lower rotational speeds. Moreover, the longer the rotor is, the lower the speed at which the quasiperiodic phenomenon occurs. When the rotor length equals 1.3 m, the quasiperiodic motion occurs again at 4500 rpm. In contrast, when the rotor length is 1 m, the system maintains period-one motion at a higher rotational speed.

Fig. 7. Bifurcation diagrams with different rotor configurations

Figure 8 shows the dynamic response at 2700 rpm with and without considering the interaction between laminates. When the interaction is not considered, there is only one rotational frequency. The corresponding axis orbit and phase diagram are circles, and the Poincaré map is a single point. In contrast, when the interaction is considered, a new frequency component f w appears in Fig. 8a. The Poincaré map is a circle, as shown in Fig. 8f, and the system exhibits quasiperiodic motion. Figure 9 shows the dynamic response at 9550 rpm with and without considering the interaction between laminates. When the interaction is considered, the frequency component fw and combined frequency component appear. The Poincaré map is a circle, as shown in Fig. 9f. The axis orbit and phase diagram show a petal shape (Fig. 9d, 9 h). It should be noted that the f w appearing in Fig. 9b is the same as that appearing in Fig. 8b, which means that the two nonlinear phenomena are caused by the seal force. When the interaction is not considered, only the rotational frequency appears in Fig. 9a, and the system exhibits period-1 motion.

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Fig. 8. Dynamic response at 2700 rpm

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Fig. 9. Dynamic response at 9550 rpm

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4 Conclusion In this paper, a nonlinear finger seal force model was established. The model considered the interaction between finger laminates, which is determined by the key structure and working parameters. A finger seal-Jeffcott rotor system was established. The dynamic responses of the rotor system were obtained. The dynamic characteristics with and without considering the interaction were compared. The main conclusions are summarized as follows: 1. The interaction slightly decreases the vibration amplitude of the system when no nonlinear phenomenon appears. 2. The interaction between laminates makes the system show more nonlinear phenomena. When the interaction is considered, the system exhibits quasiperiodic motion in the range of ω ∈ [2200, 3200] and ω > 7900 rpm. 3. As the rotor length decreases, the nonlinear phenomena at high speeds weaken and eventually disappear. However, at lower rotational speeds, quasiperiodic motion always appears. Acknowledgment. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11972131 and 12072089).

References 1. Ludwig, L.P.: Self-acting shaft seals (1978) 2. Johnson, M.C., Medlin, E.G.: Laminated finger seal with logarithmic curvature. Patent No. 5108116. USA (1992) 3. Arora, G.K.: Noncontacting finger seal with hydrodynamic foot portion. Patent No. 5755445. USA (1998) 4. Proctor, M.P., Steinetz, B.M.: Noncontacting Finger Seal (2004) 5. Marie, H.: Dynamic simulation of finger seal-rotor interaction using variable dynamic coefficients. In: 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, p. 4931 (2006) 6. Temis, J.M., Selivanov, A.V., Dzeva, I.J.: Finger seal design based on fluid-solid interaction model. In: Turbo Expo: Power for Land, Sea, and Air, pp. V3A-V15A. American Society of Mechanical Engineers (2013) 7. Braun, M.J., Kudriavtsev, V.V., Steinetz, B.M., et al.: Two-and three-dimensional numerical experiments representing two limiting cases of an in-line pair of finger seal components. Int. J. Rotating Mach. 9, 171–179 (2003) 8. Braun, M.J., Pierson, H.M., Deng, D.: Thermofluids considerations and the dynamic behavior of a finger seal assembly. Tribol. T. 48, 531–547 (2005) 9. Braun, M.J., Pierson, H.M., Kudriavtsev, V.V.: Finger seal solid modeling design and some solid/fluid interaction considerations. Tribol. T. 46, 566–575 (2003) 10. Li, H., Braun, M.J.: The sealing behavior and force analysis of a double-laminate singlepadded finger seal. In: Turbo Expo: Power for Land, Sea, and Air, pp.1279–1290 (2007) 11. Du, K., Li, Y., Suo, S., et al.: Dynamic leakage analysis of noncontacting finger seals based on dynamic model. J. Eng. Gas Turbines and Power, 137 (2015)

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12. Du, K., Li, Y., Suo, S., et al.: Semi-analytical dynamic analysis of noncontacting finger seals. Int J. Struct Stab Dy 15, 1450060 (2015) 13. Zhang, Y., Yan, T., Liu, K., et al.: Analysis of performance for contact finger seal with variable stiffness. Material Engineering and Mechanical Engineering: Proceedings of Material Engineering and Mechanical Engineering (MEES2015). World Scientific, pp. 608–616 (2016) 14. Jia, X., Zheng, Q., Tian, Z., et al.: Numerical investigations on lifting and flow performance of finger seal with grooved pad. Aerosp. Sci. Technol. 81, 225–236 (2018) 15. Zhang, H., Chai, B., Jiang, B., et al.: Numerical analysis of finger seal with grooves on lifting pads. J. Propul. Power 31, 805–814 (2015) 16. Zhang, H., Zheng, Q., Yue, G., et al.: Unsteady numerical analysis of a whole ring of finger seal with grooves on finger pads. In: Turbo Expo: Power for Land, Sea, and Air, pp. V3A-V15A. American Society of Mechanical Engineers (2013) 17. Smith, I.M., Braun, M.J.: A parametric experimental investigation and performance comparison of four finger seal embodiments. In: International Joint Tribology Conference, pp. 349–351 (2007) 18. Proctor, M., Delgado, I.: Preliminary Test Results of Non-Contacting Finger Seal on Herringbone-Grooved Rotor. In: 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, p. 4506 (2008) 19. Zhang, E.J., Jiao, Y.H., Chen, Z.B.: Dynamic behavior analysis of a rotor system based on a nonlinear labyrinth-seal forces model. J. Comput. Nonlinear Dyn. 13(10), 101002 (2018) 20. Xu, Y.Y., Jiao, Y.H., Chen, Z.B.: On an independent subharmonic sequence for vibration isolation and suppression in a nonlinear rotor system. Mech. Syst. Signal Process. 178, 109259 (2022) 21. Wei, Y., Liu, S.: Numerical analysis of the dynamic behavior of a rotor-bearing-brush seal system with bristle interference. J. Mech. Sci. Technol. 33(8), 3895–3903 (2019)

Investigation on Information Assessment for Vibration Sensor Locations Installed in Aero-Engine Based on Unbalance Response Analysis Alexander A. Inozemtsev1 , Konstantin V. Shaposhnikov2(B) , Sergey A. Degtyarev2 , Mikhail K. Leontiev3 , and Ivan L. Gladkiy1 1 UEC-Aviadvigatel JSC, Perm, Russia 2 Engineering and Consulting Center On Rotordynamics Alfa-Tranzit Co., Ltd., Moscow, Russia

[email protected] 3 Moscow Aviation Institute, Moscow, Russia

Abstract. During design and development of every new aero-engine it is often necessary to verify its numerical model. Vibration readings obtained from permanent sensors often are not suitable for such verification or for engine detailed diagnostics. Therefore, it is frequently necessary to equip aero-engine with additional sensors and conduct special experiments. Proposed in the paper method allows to estimate preliminary information assessment for vibration sensor locations based on damped critical speed calculation and their mode shape analysis. Vibration sensor information assessment is further clarified based on unbalance response analysis. Investigation of various unbalance cases for engine’s rotors permit to obtain vibration sensors information assessment scatter as their maximum and minimum values. Approbation of the method was performed during simulation of high-bypass turbofan engine in software for rotordynamics simulation DYNAMICS R4. Performed analysis allowed to develop vibration sensors scheme installed on engine cases with high information assessment on aero-engine critical speeds and their mode shapes. Keywords: Aero-engine · vibration sensors · rotordynamics · scheme of vibration sensors · critical speed · unbalance response · rotor-case-suspension · modeling · DYNAMICS R4

1 Introduction During design and development of every new aero-engine it is often necessary to verify its numerical model. Such verification usually requires numerical reproduction of all critical speeds that are allowed in operating range of the engine. Permanent sensors are usually installed at engine suspension mounts and control vibration only in one or two directions (radial and axial, or just radial) [1–3], thus their readings may not be very suitable for numerical model verification, since aero-engine critical speeds due © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 168–187, 2024. https://doi.org/10.1007/978-3-031-40459-7_12

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to dynamic system orthotropy may have significant displacements at directions that are not controlled by permanent sensors. In addition, vibration amplitudes recorded by permanent sensors can be small and without obvious vibration peaks, increasing difficulty of clear identification of certain critical speeds. Conventional aircraft vibration monitoring systems are also limited in areas of vibration diagnostics [4]. Therefore, for numerical model verification and frequently for diagnostics it is necessary to equip aeroengine with additional sensors and conduct special experiments. Aero-engine OEM’s often do not openly disclose such experiments, and if some vibration sensors schemes are shown, there are usually no details on how sensor locations were chosen. However, in case of vibration diagnostics it is common in aviation industry to set vibration sensors at cases of the engine as close as possible to bearing supports [5]. Example for vibration sensors scheme developed for diagnostics of F100-PW-220 can be found in paper [6]. Vibration sensor selection methods for prognostics and health monitoring was proposed by Xu [7], but application requires large number of fault history information detected by each sensor, and hence proposed method is not applicable to newly designed engine or engine with small database vibration history. Hou and Cao proposed method for vibration sensor selection for aero-engine based on analysis of its variable vibration indexes and demonstrated that for some engine configurations vibration sensors installed in the middle section of the engine case may be more efficient for engine vibration monitoring, than sensors installed close to mounts [8]. Current paper is focused on problem of vibration sensors scheme development with high information assessment on aero-engine critical speeds based on their calculation and analysis for its numerical model.

2 Model Description High-bypass turbofan engine model built with usage of two-node finite elements – beams and shells in commercial software for rotordynamics simulation DYNAMICS R4 [9] is shown in Fig. 1a. Low pressure (LP) and high pressure (HP) rotors, cases and suspension mounts are main subsystems of the model, Fig. 1b. Principle of large structure representation in form of multiple subsystems is common in rotordynamics [10–12]. Each subsystem is represented by corresponding mass and inertia, stiffness and damping matrices. Auxiliary units of the engine were considered in the model by their mass and inertia properties attached to the engine cases in accordance with their real angular position. Case struts damping was taken into account using nondimensional relative damping coefficient ξ = 0.1 as recommended in [13], while damping coefficients for all dampers of the engine were assumed to be constant and equal to Cxx = Cyy = 105 N·s/m. The value of damping was set equal for each damper to simplify the simulation of the model and to keep vibration of the supports of the engine within allowable safe limits. Matrices of each subsystem are combined together at corresponding connection nodes, forming joint matrices of the entire dynamic system of aero-engine. In short form the general equation of motion for the aero-engine dynamic system can be written as: M q¨ + C q˙ + Kq = Q

(1)

M – mass and inertia matrix; C – damping and gyroscopic matrix; K – stiffness matrix; Q – generalized vector of external forces; q – displacement vector. Usage of lightweight

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Fig. 1. Aero-engine model in DYNAMICS R4: (a) 3-D representation (b) Subsystem model structure

cases in structure schemes of modern aero-engine may lead to appearance of their natural frequencies in the operating range of the engine. Such case structure natural frequencies can be lower than bending natural frequencies of the engine rotors. In such a way for accurate rotordynamic modeling aero-engine dynamic system should be considered as uniform rotor-case-suspension system. Aero-engine resonant modes which may appear in operating range of the engine can be generally classified into three main groups: rotor, case and coupled modes. Each group can be further classified as axial, torsional or lateral modes. Main types of aero-engine lateral resonant modes are shown in Fig. 2. Green color relates to forward precession motion of the dynamic system, blue – to backward. The term critical speed will be further used for description of engine model vibrations only in lateral direction.

Fig. 2. Aero-engine lateral resonant modes (critical speeds) classification: (a) Rotor mode; (b) Case mode; (c) Coupled mode (rotor-case-suspension)

For the rotor modes the largest deformations will be inherent to structure elements of the engine rotors, while for case modes – to structure elements of its cases. For the coupled modes significant deformations will be located both on rotor and case structural

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elements. Coupled rotor-case-suspension modes are more complex than the rest modes and more difficult to predict, since their critical speeds and mode shapes may significantly differ from the critical speeds and mode shapes of separate rotors or engine case modes. For the correct mode shape classification, it is necessary to analyze each mode shape taking into account mode shapes of main subsystems of the engine model, distinguishing subsystems with obvious significant deformations. It should be noted that observed aeroengine critical speed mode shapes classification is general, which assumes that even for pronounced rotor modes, small deformations will be still observed on its cases, while for the case modes small deformations will be observed on its rotors. When selecting places for vibration sensors in aero-engine for verification of its numerical model it is necessary to take into account the nature of the main type of mode shapes which are planned to be identified during experimental measurements. Each individual mode shape is characterized by position and number of its node and antinode points [14]. For efficient identification of each specific aero-engine resonant mode, vibration sensors should be installed in places distant from the node points of corresponding mode shape. Design features of engine mounts, asymmetric arrangement of massive auxiliary units on its cases, existence of clearances in its bearings and dampers lead to appearance of orthotropy for dynamic systems elastic properties in orthogonal direction, which in turn affects the orbits of engine elements transforming them to pronounced elliptical shape. Presence of orthotropy in the dynamic system of aero-engine also leads to necessity take into account in its critical speed analysis not only direct but also backward precession mode shapes. Mixed precession mode shapes, where one part of the rotor is subjected to direct precession, while the other – to backward are also inherent to appear in such complex dynamic systems as aero-engine. Existence of dynamic systems orthotropy may also significantly influence on vibration sensors information assessment, especially when they are installed in orthogonal directions. Critical speed mode shape analysis performed for aero-engine model helped to form statistics on potential resonant mode types excited by its LP and HP rotors in the range of rotor speeds 0%–130% of its maximum operating speed, Fig. 3.

Fig. 3. Statistics on aero-engine critical speed mode shapes excited by LP and HP rotor

It can be seen from the obtained results that 57%–69% of the aero-engine critical speeds obtained by simulation were related with coupled modes, while 29%–31% were

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associated with rotor modes. At the same time case modes were not detected in the operating range of the engine when excitation was considered by LP rotor, and 14% of resonant modes total number were identified as the case modes when HP rotor excitation was taken into account. High percentage of coupled modes detected in the operating range of the engine shows that vibration sensors location on cases of aero-engine will be efficient for its critical speeds control.

3 Method 3.1 Method for Vibration Sensors Preliminary Information Assessment Evaluation Vibration sensors preliminary information assessment can be evaluated based on analysis of damped critical speed mode shapes obtained by simulation for model of the aeroengine. Proposed method was first described in details in [15]. Main steps of the method are shown in Fig. 4.

Fig. 4. Method for vibration sensors location information assessment preliminary evaluation

Damped critical speeds are calculated from transformation of general equation of motion of the dynamic system to free vibration equation, assuming Q = 0 in Eq. (1). Then, the solution of the homogeneous equation can be obtained in exponential form as: q(t) = u · eλt

(2)

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where u – are eigenvectors of rotordynamic system; λ− are its eigenvalues. Both eigenvectors and eigenvalues are complex, where each eigenvalue has the form: λ = σ + iωd

(3)

where σ – system damping exponent, and ωd – is the damped natural frequency or the whirl speed. When dynamic system of aero-engine is subjected to harmonic excitation, coincidence of its natural frequency with excitation frequency brings to appearance of critical speed. As stated in the proposed method each critical speed mode shape is preliminary analyzed and classified based on mode shape type (rotor mode, case mode, coupled mode). Nodal and antinode points are defined for each mode shape. Vibration vectors projections (Utx , Uty ) of aero-engine model elements at places of virtual sensors locations for each of the selected N critical speed mode shapes are implemented as: Utx = Re(Utx ) + Im(Utx )i

(4)

    Uty = Re Uty + Im Uty i

(5)   Absolute values of vector projections for model elements |Utx |, Uty  for each of the selected critical speed mode shapes are obtained as:  |Utx | = (Re(Utx ))2 + (Im(Utx ))2 (6)     2   2 Uty  = Re Uty + Im Uty

(7)

In order to form the tables of information assessment, grades GxN , GyN are calculated for each sensor for each of the selected N critical speed mode shape. It should be noted that sensors are evaluated in groups by subsystems in which they are installed in model e.g. sensors of the outer case including permanent sensors of the front mount belong to one subsystem, while sensors of the inner case including permanent sensors of the rear mount belong to another subsystem. Thus, grade of each sensor represents the ratio of absolute vibration vector projection of model element in place where the sensor is installed to maximum value of vibration vector projections defined for a group of sensors which includes this sensor: |Utx | max|Utx |   Uty  N   Gy = maxUty  GxN =

(8) (9)

Assessment criteria: 1 – best value, 0 – worst value and all grades are set in range: 0 ≤ GxN ; GyN ≤ 1

(10)

Cells in all tables are also highlighted with colors set in accordance with proposed assessment criteria. Graphical interpretation for each table of information assessment

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for sensors installed in horizontal and vertical direction is represented with contour color plots with color distribution set in compliance with assessment criteria. Overall grade for each sensor is calculated in the table as a sum of its grades for each of the observed N critical speed mode shapes:  N 1 2 N = Gx,y + Gx,y + ... + Gx,y (11) Gx,y Each sensor information assessment is further determined in percentage as a ratio of overall sensor’s grade to number of considered N critical speed mode shapes:  N Gx,y Ix,y = · 100% (12) N 3.2 Clarification of Vibration Sensors Information Assessment Based on Unbalance Response Analysis Obtained sensors preliminary information assessment can be potentially overestimated since mode number N in its denominator includes excitation of all possible critical speeds for aero-engine dynamic system. Excitation of specific critical speed depends on a certain residual unbalance distribution, hence some critical speeds may not be excited simultaneously, while the other critical speeds may require «exotic» combination of residual unbalances located at specific rotor parts and may not be excited with typical unbalance distribution of rotor modules. Thus, vibration sensors information assessment can be further clarified based on unbalance response analysis. For method implementation different cases of residual unbalance distribution for LP and HP rotor of the engine can be modeled, while unbalance response readings can be recorded for all sensor locations. General description of main steps of the method is shown in Fig. 5. Tables of information assessment can be formed for each unbalance case, where the grades for each sensor represent the ratio of vibration amplitudes Ax , Ay recorded by the sensor for each N critical speed detected in unbalance response to maximum vibration amplitude defined for the group of sensors which includes this sensor: GxN =

Ax max(Ax )

(13)

GyN =

Ay   max Ay

(14)

Overall grade for each sensor is calculated following Eq. (11), while each sensor information assessment is determined in percentage as a ratio of overall sensor’s grade to number of critical speeds N detected in each unbalance case:  I x,y =

N Gx,y

N

· 100%

(15)

When information assessment tables and their color plots are obtained for variety of unbalance cases general data processing is performed. Tables processing allow to obtain

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Fig. 5. Method for vibration sensors information assessment clarification based on unbalance response analysis

arithmetic mean for sensors grades, based on which averaged versions of color plots can be built. Application of boxplot analysis for sensors information assessment can help to estimate the scatter of maximum and minimum values predicted by unbalance response analysis for each sensor and clarify preliminary vibration sensors scheme developed based on aero-engine damped critical speed analysis. Small scatter means that sensors information assessment is not sensitive to unbalance distribution on engine rotors and sensor will be efficient to control critical speeds based on his grades.

4 Simulation Results 4.1 Model Set Up For approbation of the proposed methods, model of high-bypass turbofan engine was prepared for further analysis and set on operating mode thermal state. At the flange joints of its outer and inner cases 14 sections were selected for evaluation of vibration sensor information assessment, Fig. 6. To simplify simulation in each section vibration sensors were strictly oriented in horizontal and vertical directions. Permanent vibration sensors were installed in front (Vfront ) and rear (Vaft ) suspension mounts of the engine.

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Fig. 6. Vibration sensor locations in aero-engine model: (a) Outer case sensors; (b) Inner case sensors

4.2 Vibration Sensors Preliminary Information Assessment Evaluation Preliminary information assessment for vibration sensor locations in model of aeroengine was evaluated based on the proposed method (see Fig. 4). Analysis was implemented for engine model critical speed mode shapes excited by LP and HP rotors in range of speeds 0%–130% of each rotor maximum operating speed. For LP rotor excitation 13 critical speeds were predicted in the operating range, while for HP rotor – 58 critical speeds. Obtained simulation results helped to form tables of information assessment for sensor locations in horizontal and vertical direction, Fig. 7. Resonant mode types in table were designated as: R – rotor mode, C – case mode, RCS – Coupled mode (rotor-case-suspension). Operating modes were highlighted with separate lines: ground idle mode (IDLE) and maximum operating mode (MAX). Graphical interpretation as contour color plots for each table are shown in Figs. 8 and 9. Grade scale for each plot was set based on assessment criteria: 1 – best value, 0 – worst value. Each color plot was subdivided in outer and inner case in accordance with sensor groups position in scheme of the engine. Y-axis in each subplot represents the sensor number, while X-axis of each plot corresponds to critical speed mode ordinal number. Color plots usage in conjunction with tables of information assessment gives quick visual representation on zones of critical speed with high and low grades, allowing to form vibration sensors scheme from sensors with high information assessment about engine critical speeds.

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Fig. 7. Preliminary tables of information assessment for vibration sensor locations to control critical speeds of aero-engine model: (a) excited by LP rotor (b) excited by HP rotor

Preliminary information assessment for all sensors installed in the model was calculated based on sensor overall grades obtained from tables of information assessment, Fig. 10. Simulation results demonstrated that none of the sensors can fully detect and control all of the critical speeds in the operating range. Thus, for efficient identification of the aero-engine resonant modes it is necessary to form the scheme of vibration sensors with high information assessment about its critical speeds excited both by LP and HP rotor. Hence based on the obtained results preliminary vibration sensors scheme can be formed from uniaxial vibration sensors for vibration control in X and Y direction installed at the brackets attached to the flange joints of the outer case of the engine in sections № 1, 4, 5. Simultaneously at the inner case uniaxial vibration sensors for vibration control in X and Y direction should be installed at the brackets attached to flange joints in sections № 6–12. From initially observed 14 sections for sensor location only 10 sections were used to form vibration sensors scheme for identification of the aero-engine critical speeds.

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Fig. 8. Contour color plots for preliminary evaluation of information assessment for vibration sensor locations to control critical speeds excited by LP rotor: (a) horizontal (b) vertical

For sensors type selection (general usage/high temperature) aero-engine cases thermal state analysis should be performed. Increase of temperature of engine cases may significantly influence on sensitivity change of installed on them accelerometer sensors. 4.3 Vibration Sensors Information Assessment Clarification based on Unbalance Response Analysis In order to clarify vibration sensors preliminary information assessment, model of aeroengine was prepared for further unbalance response analysis. Residual unbalance for LP and HP rotors of the engine was set corresponding to G 6.3 balance grade [16]. Due to volume restrictions of the current paper only three common cases of residual unbalance distribution were observed for each rotor: “in phase unbalance case” where residual unbalance was equally spaced among two lumped unbalances set with the same phase angle in c.g. of compressor and turbine modules of each rotor; “out of phase case” where residual unbalance was also equally spaced among two lumped unbalances with the 180 degree phase difference related to each other in c.g. of compressor and turbine modules of each rotor; “first bending mode unbalance case” where residual unbalance was equally spaced among three lumped unbalances set along rotor body in antinode places corresponding to first bending mode of each rotor. It should be noted that proposed method is not limited to observed three cases of residual unbalance distribution and number of cases can be increased as many as necessary if required for the analysis.

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Fig. 9. Contour color plots for preliminary evaluation of information assessment for vibration sensor locations to control critical speeds excited by HP rotor: (a) horizontal (b) vertical

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Fig. 10. Preliminary information assessment and overall grades for sensor locations to control critical speeds in aero-engine model: (a) excited by LP rotor; (b) excited by HP rotor

Obtained simulation results were summarized in form of multi-curve 3-D plots for each unbalance case shown in Figs. 11 and 12. Permanent vibration sensor readings were indicated with orange color. Operating modes of the engine were highlighted with separate lines. Speed ratio on the X-axis for each plot presents the ratio of exact rotor speed (ω) to rotor speed at MAX mode (). Sensor numbers were marked on the Y-axis, while Z-axis corresponds to vibrovelocity vibration amplitude. Based on proposed method (see Fig. 5) results processing was further performed for each unbalance response plot of each unbalance case of each rotor. As an outcome of the analysis vibration sensors information assessment tables, color plots and sensor information assessment diagrams for vibration control in vertical and horizontal directions were obtained for each unbalance case. Simulation results revealed that for LP rotor excitation in all unbalance cases only 9 critical speeds were detected in operating range of the rotor from 13 predicted by critical speed preliminary analysis. At the same time for HP rotor excitation only 47 critical speeds were detected in operating range of the rotor from 58 predicted by critical speed preliminary analysis. It can be seen from comparison of the obtained simulation results that amplitude frequency spectrum of the HP rotor is denser, than for LP rotor. Data processing for each unbalance case of the HP rotor requires significantly more time, since vibration peaks identification for HP rotor unbalance response is more difficult. It is worth noting that vibration amplitude curves obtained for permanent sensors by simulation is smoother than for some additional sensors installed on case of the engine model. Number of obvious vibration peaks corresponding to engine resonant modes is smaller, confirming that usage of permanent sensors readings for detailed verification of numerical model of the engine will be difficult.

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Fig. 11. Unbalance response simulation results – LP rotor excitation: (a) in phase unbalance case (b) out of phase unbalance case (c) first bending mode unbalance case

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Fig. 12. Unbalance response simulation results – HP rotor excitation: (a) in phase unbalance case (b) out of phase unbalance case (c) first bending mode unbalance case

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General result processing performed for all unbalance cases of each rotor allowed to obtain mean tables of information assessment for vibration sensor locations to control critical speeds excited by LP and HP rotors in vertical and horizontal directions, Fig. 13. Their graphical interpretation in form of contour color plots for evaluation of information assessment for vibration sensor locations to control critical speeds excited by engine LP and HP rotor is shown in Figs. 14 and 15.

Fig. 13. Mean tables of information assessment for vibration sensor locations to control critical speeds of aero-engine model: (a) excited by LP rotor (b) excited by HP rotor

Evaluation of overall grades for each sensor installed in the model to control critical speeds of the aero-engine for observed unbalance cases allowed to find information assessment maximum and minimum values. Further performed result processing permit to obtain scatter plots for all sensors to control critical speeds excited by LP and HP rotor of the aero-engine, Fig. 16. Based on calculation results established initial vibration sensors scheme formed on preliminary information assessment evaluation can be adjusted and number of recommended sensors can be reduced. At the outer case of the engine, it is recommended to use uniaxial vibration sensors for vibration control in X and Y directions installed at

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Fig. 14. Averaged contour color plots for evaluation of information assessment for vibration sensor locations to control critical speeds excited by LP rotor built based on unbalance response analysis: (a) horizontal (b) vertical

the brackets attached to the flange joints in sections № 1 and 5, since performed unbalance response analysis confirmed their high information assessment. Sensor № 4 was removed from the scheme, since unbalance response analysis revealed lower values of its information assessment. At the inner case of the engine, it is recommended to leave in initial scheme uniaxial vibration sensors in sections № 6, 8–12 to control vibration in X and Y directions, since unbalance response analysis confirmed their high information assessment, which is also higher than information assessment of permanent aft vibration sensor installed on the mount. Sensor № 7 was removed from the scheme because set of critical speeds with high grades controlled by this sensor is similar to № 6, but his overall grade was generally lower. Sensors № 8 – 12 showed similar overall grades to control critical speeds excited by HP rotor, but they have high grades for different sets of critical speeds, thus in order to control all of them, these sensors were left in the clarified aero-engine vibration sensor scheme. Vibration sensors scheme developed based on preliminary analysis for identification of aero-engine critical speeds which was consisting from 10 sections was finally reduced to 8 sections for sensor installation.

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Fig. 15. Averaged contour color plots for evaluation of information assessment for vibration sensor locations to control critical speeds excited by HP rotor built based on unbalance response analysis: (a) horizontal (b) vertical

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Fig. 16. Scatter of information assessment for vibration sensor locations clarified by unbalance response analysis to control aero-engine critical speeds: (a) excited by LP rotor (b) excited by HP rotor

5 Conclusion To sum up, proposed method allows to estimate preliminary information assessment for vibration sensor locations based on damped critical speed calculation and their mode shape analysis. Performed analysis allows to form vibration sensors scheme installed on cases of aero-engine with high information about its critical speeds. Vibration sensor information assessment can be further clarified based on unbalance response analysis. Investigation of various unbalance cases for engine’s rotors can help to estimate vibration sensors information assessment scatter as their maximum and minimum values and reduce the number of sensors necessary for efficient identification of engine critical speeds. Approbation of the method was performed on model of high-bypass turbofan engine, allowing to evaluate information assessment for vibration sensor locations and further form the scheme of sensors with high information assessment on aero-engine critical speeds and their mode shapes. Developed sensors scheme can be used in real aero-engine special tests performed for verification of its numerical model.

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Acknowledgement. The authors would like to thank UEC-Aviadvigatel JSC for their permission to publish this work.

References 1. Royce, R. The Jet Engine. John Wiley & Sons (1996) 2. Hünecke, K.: Jet engines: fundamentals of theory, design, and operation. Zenith Imprint (1997) 3. Linke-Diesinger, A.: Systems of commercial turbofan engines: an introduction to systems functions. Springer Science & Business Media (2008) 4. Simon, D.L., et al.: Sensor needs for control and health management of intelligent aircraft engines. Turbo Expo: Power for Land, Sea, and Air 41677, 873–882 (2004) 5. Keller, J.A., Grabill, P.: Inserted fault vibration monitoring tests for a CH-74D AFT swashplate bearing. In: Proceedings of 61st American Helicopter Society Annual Forum, pp. 151–160. Grapevine, TX, USA (2005) 6. Wroblewski, D., Grabill, P.: Analysis of gas turbine vibration signals for augmentor fault detection. In: 37th Joint Propulsion Conference and Exhibit, pp. 3767–3773 (2001) 7. Xu, J., Wang, Y., Xu, L.: PHM-oriented sensor optimization selection based on multiobjective model for aircraft engines. IEEE Sens. J. 15(9), 4836–4844 (2015) 8. Hou, L., Cao, S.: Evaluation method for vibration measurement on casing in aeroengine: theoretical analysis and experimental study. Shock Vib. 2019, 1–15 (2019) 9. DYNAMICS R4. Program system for analysis of rotor dynamics of turbomachines. User Guide. Alfa-Tranzit Co., Ltd. (2023). http://alfatran.com/dyn/userguide_eng.pdf 10. Chen, W.J.: A note on computational rotor dynamics. ASME. J. Vib. Acoust. 120(1), 228–233 (1998) 11. Leontiev, M., Zvonarev, S.: Truncation errors in the modal vibration analysis of the rotor systems. In: New Advances in Modal Synthesis of Large Structures, pp. 29–40 (1997) 12. Hong, J., Shaposhnikov, K., Zhang, D., Ma, Y.: Theoretical modeling for a rotor-bearingfoundation system and its dynamic characteristics analysis. In: Pennacchi, P. (ed.) Proceedings of the 9th IFToMM International Conference on Rotor Dynamics, pp. 2199–2214. Springer International Publishing, Cham (2015). https://doi.org/10.1007/978-3-319-06590-8_181 13. Nicholas, J.C., Whalen, J.K., Franklin, S.D.: Improving critical speed calculations using flexible bearing support FRF compliance data. In: Proceedings of the 15th Turbomachinery Symposium. Texas A&M University. Turbomachinery Laboratories (1986) 14. Bently, D.E., Hatch, C.T., Grissom, B.: Fundamentals of Rotating Machinery Diagnostics (2003) 15. Shaposhnikov, K.V., Davydov, A.V., Degtyarev, S.A., Leontev, M.K., Gladkii, I.L.: Method for equipping the aircraft gas turbine engine with vibration sensors by evaluating their information content based on mathematical modeling. Russ. Aeronaut. 65(4), 810–821 (2022). https://doi. org/10.3103/S1068799822040225 16. Standard I. S. O. 1940-1:2003: Mechanical Vibrations—Balance quality requirements for rotors in a constant (rigid) state — Part 1: specification and verification of balance tolerances (IDT) (2003)

Torsional Vibration Modelling of a Two-Stage Closed Differential Planetary Gear Train Guanghe Huo1,2 , Yinghou Jiao1(B) , Miguel Iglesias Santamaria2 , Xiang Zhang1 , Javier Sanchez-Espiga2 , Alfonso Fernandez-del-Rincon2 , and Fernando Viadero-Rueda2(B) 1 School of Mechatronics Engineering, Harbin Institute of Technology, Nangang Dist, 92 West

Dazhi Street, Harbin, People’s Republic of China [email protected] 2 Department of Structural and Mechanical Engineering, ETSIIT University of Cantabria, Avda. de los Castros S/N, 39005 Santander, Spain [email protected]

Abstract. In this paper, a torsional dynamic model of a two-stage double-helical planetary gear train is developed, and the vibration characteristics and coupling relationship are studied. Firstly, a purely torsional model was established based on structure diagram and different mesh phasing. Then, the runout error is taken into account. Finally, according to frequency spectrum analysis, the coupling relationship of two stages is studied. It is shown that the runout error of the planet in different stages has a different impact on the torsional vibration of transmission error. It also shows that the coupling relationship exists and affects each other between two stages, and the excitation frequency for the output element of two stages can be composed of a series of the mesh frequencies of each stage, the meshing frequency of high stage (1st stage) is dominated in two stages. Keywords: Vibration · Double-helical Planetary gear · Coupling

1 Introduction Single and multistage planetary gear train is widely used in helicopter, wind turbine and ship because of its large transmission ratio, high transmission efficiency, strong bearing capacity and small size. Especially for a large heavy-duty machine, multistage are commonly used based on a design given gear transmission ratio. Early model focused on linear time-invariant, Kahraman [1] studied and summarized the free vibration of different types compound planetary gear set and different modes were observed. Later, the nonlinear time-varying models [2, 3], includes time-varying meshing stiffness, backlash and friction, were developed. Dai et al. [4] studied theatrically and experimentally the meshing force of single planetary gear system. Therefore, it is clearly indicted from above literatures that time-varying meshing stiffness is basic parameter that affect dynamic characteristics. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 188–201, 2024. https://doi.org/10.1007/978-3-031-40459-7_13

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Moreover, in terms of meshing stiffness, mesh phasing is an important component. For the planet lord sharing characteristics, Sanchez et al. [5] studied the influence of different mesh phasing. Vangipuram et al. [6] investigated the influence of mesh phasing on parametric instabilities of a planetary gear with a flexible ring gear. Wang et al. [7] studied the suppression effects of planet mesh phasing on vibrations of planetary gears and proposed several rules for how mesh phasing generated. Peng et al. [8] used mesh phasing determine gear’s fault. Nomenclature S, P, C, R

Sun, planet, carrier and ring

TE

Transmission error

θ zs , θ zpi , θ zc , θ zr

Angular displacement of sun, planet, carrier and ring

λs , λpi , λr

The phase angle of run-out error of sun, planet and ring

Rbs , Rbpi , Rbc, Rbr

Base circle’s radius J zs , J zpi , J zr , J zce of sun, planet, carrier and ring

The moment of inertia of sun, planet, carrier and ring

β b , α sp , α rp

Base helix angle, meshing angle of sun-planet and ring-planet

k spi , cspi , k rpi , crpi

Meshing stiffness and damping of sun-planet i and ring-planet i

espi , erpi

Meshing error of sun-planet i and ring-planet i

f m, f e

Meshing frequency and error frequency

δ spi , δ rpi

Meshing displacement of sun-planet i and ring-planet i

ESSP, ESIP

Equally spaced sequential phase, equally spaced in phase

ωs , ωpi , ωc , ωr

Angular speed of the sun, planet, carrier and ring

ees , eepi , eer

The magnitude of run-out error of sun, planet and ring

First stage

2

Second stage

Superscripts 1

With the emergence of various machines, the single-stage planetary gear system can no longer meet the work requirements of high transmission ratio due to gear size limitations. So mechanical products of multi-stage planetary gear trains appeared. Li et al. [2] established a torsional model and studied the nonlinear characteristics, includes bifurcation and chao, and the impact of damping ratio, working speed and backlash on motion. It is revealed that Variation law of speed from single period motion to chaotic motion. Considering the elastic ring component, Wei et al. [9] analyzed theatrically and experimentally the modulation sideband phenomenon caused by unavoidable manufacturing errors. Zhang et al. [3] proposed a translation–torsion coupling model of a wind turbine, studied the influence of friction on nonlinear characteristics.

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For double-helical planetary gear, Lu et al. [10] developed a nonlinear model, includes sliding friction, time-varying meshing stiffness, backlash, axial stagger as well as gear mesh errors, and the influence of tooth friction on the periodic vibration and nonlinear vibration was investigated. Wang et al. [11–13] proposed step by step three nonlinear model such as torsional model and 3D model, comprehensively and deeply studied the nonlinear behaviour of double-helical gear train in GTF by taking spline stiffness, bearing stiffness and tooth friction into account. For the vibration analysis of multistage planetary gear system with a complex structure, the torsional vibration model is a basic model and the purpose of this paper is to preliminarily discuss the coupling characteristics of closed differential planetary gear train from the perspective of torsional dynamics model (1DOF). Therefore, this study establishes a purely torsional model of two stage planetary gear system, and this model considers time-varying meshing stiffness, connecting stiffness between two stages. Moreover, equations of motion in general coordinates are obtained by using the Lagrange equation. The vibration behaviour and coupling between two stages of whole system is analysed.

2 Torsional Model The schematic of double-helical two-stage planetary gear system is shown in Fig. 1, this system is comprised of two stages, epicyclic planetary gear system (1st stage) and star planetary gear system (2nd stage). The ring of stage 1 and sun of stage 2 are connected, so do carrier of stage 1 and ring of stage 2, i.e., the connected components rotate at the same speed. The model is obtained based on following several assumptions: 1). Each gear body is assumed to be rigid, and the flexibilities of the teeth of each gear are replaced by a spring-damper unit along the meshing line. 2). Each component is assumed to move in the torsional direction, i.e., they only have 1 degree of freedom. 3). Each planet gear is absolutely same and assigned around sun gear.

2.1 Relative Displacements Here, defining angular displacement θ (m) zi (i = s, r, c and pn (n = 1, 2,…, N), m = 1 and 2) as only motion of each component in each stage, which is obtained in a global coordinate system. It is noted that the upper right letter 1 is stage 1, number 2 is stage 2 and it is assumed that counter clockwise direction is positive. Therefore, for the mesh between sun or ring and each planet i in first stage, the equivalent meshing displacements in their contact direction are defined as shown in Eq. (1). (1)

(1)

(1) (1)

(1)

(1)

(1)

(1) ) cos(βb ) − espi (t) δspi = (Rbs θzs(1) + Rbpi θzpi − Rbc θzc (1) (1) (1) (1) (1) (1) (1) (1) δrpi = (R(1) br θzr − Rbpi θzpi − Rbc θzc ) cos(βb ) − erpi (t)

(1)

where, Rbi (i = s, r, c and pn (n = 1, 2,…, N)), β b and espi are the radius of base circle, helix angle of base circle and transmission error and, respectively.

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Fig. 1. The structure schematic of double-helical two stage planetary gear system

In same way, the equivalent meshing displacements in their contact direction of second stage are defined as shown in Eq. (2). (2)

(2)

(2) (2)

(2)

(2)

δspi = (−Rbs θzs(2) − Rbpi θzpi ) cos(βb ) − espi (t) (2) (2) (2) (2) (2) (2) δrpi = (−R(2) br θzr + Rbpi θzpi ) cos(βb ) − erpi (t)

(2)

Moreover, defining the relative angular displacements between connected members in two stages as δ (1,2) rs and δ (1,2) cr, which are shown in Eq. (3). It should be pointed out that a torsional spring is used to connect two members. (1,2) (1,2) (1) = θzr(1) − θzs(2) ,δcr = θzc − θzr(2) δrs

(3)

2.2 Meshing Error Excitation The meshing error must exist because of manufacture and assemble process, which can be composed of run-out error, planet position error and index error. The error value in this studied model is all 5Micron. Here, run-out error is taking an example, the equivalent meshing displacements projected onto the corresponding meshing line direction by the

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eccentricity error of each component in each stage are presented as following equations. (1)

(1)

(1) (1) sin[(ωs(1) − ωc(1) )t + λ(1) espi−s (t) = ees s + αsp − φpi ] (1) (1) (1) (1) espi−pi (t) = −eepi sin[(ωpi )t + λ(1) pi + αsp ] (1)

(1)

(1) (1) erpi−r (t) = eer sin[(ωr(1) − ωc(1) )t + λ(1) r − αrp − φpi ] (1) (1) (1) (1) erpi−pi (t) = −eepi sin[(ωpi )t + λ(1) pi − αrp ] (2)

(2)

(2) (2) espi−s (t) = −ees sin(ωs(2) t + λ(2) s − αsp − φpi )

(4)

(2) (2) (2) (2) espi−pi (t) = eepi sin(ωpi t + λ(2) pi − αsp ) (2)

(2)

(2) (2) erpi−r (t) = −eer sin(ωr(2) t + λ(2) r + αrp − φpi ) (2)

(2)

(2)

(2)

(2) erpi−pi (t) = eepi sin(ωpi t + λpi + αrp )

where, eejpi ( j = s or r), λj ( j = s, pi and r), α and φ pi are magnitude of error, initial assembly angle, pressure angle and orientation angle of planet #i, respectively. 2.3 Equations of Motion According to Lagrange equation [2, 3], the equations of motion of two stages can be derived, are shown in Eq. (5). Jzs(1) θzs(1) +

n 

(1) (1)

(1) (1)

(1)

(1)

(kspi δspi + cspi δ˙spi )Rbs cos(βb ) = Tin

i=1 (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) + (kspi δspi + cspi δ˙spi − krpi δrpi − crpi δ˙rpi )Rbpi cos(βb ) = 0 n  (1) (1) (1) (1) (1) (1) (1,2) (1,2) Jzr(1) θzr(1) + (krpi δrpi + crpi δ˙rpi )Rbr cos(βb ) + krs δrs = 0 i=1 n  (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) Jzce θzc − (kspi δspi + cspi δ˙spi + krpi δrpi + crpi δ˙rpi )Rbc cos(βb ) i=1 (1,2) (1,2) +kcr δcr = 0 n  (2) (2) (2) (2) (2) (2) (1,2) (1,2) (kspi δspi + cspi δ˙spi )Rbs cos(βb ) − krs δrs = 0 Jzs(2) θzs(2) − i=1 (2) (2) (2) (2) (2) ˙ (2) (2) (2) (2) ˙ (2) (2) δspi − krpi δrpi )Rbpi cos(βb(2) ) = 0 Jzpi θzpi − (kspi δspi + cspi δrpi − crpi n  (2) (2) (2) ˙ (2) (2) (2) (2) (1,2) (1,2) δrpi )Rbr cos(βb(2) ) − kcr (krpi δrpi + crpi δcr = −Tout Jzr θzr − i=1

(1) (1) Jzpi θzpi

(5)

In which, J i is moment of inertia of each part except the carrier in 1st stage, and J zce represents equivalent moment of inertia of the carrier in 1st stage, which is different with moment of inertia of the carrier, is shown as J zce = J c + n × mpi × Rc 2 . K jpi and cjpi are the meshing stiffness and corresponding meshing damping. It is assumed that each

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pair of gears (S-Pi and R-Pi) is meshing at the initial moment, but as the gear rotates, the mesh may separate, so the mesh displacement is a piecewise function as follows  δ ,δ >0 δjpi = jpi jpi , j = s or r (6) 0, else

3 Results and Discuss For the simulation study, the two-stage planetary gear train with the basic parameters is studied. Table 1 lists the basic parameters of whole system. Table 1. Parameters of the two-stage double-helical planetary gear train studied Stage

First Stage

Element

Sun

Planet

Ring

Sun

Second Stage Planet

Ring

Teeth number

38

76

190

80

55

190

Normal module (mm)

6

Normal pressure angle (°)

20

Helix angle (°)

25

Number of planets (N) Planets spacing angle (°) Mesh Phasing

3

5

120

72

ESSP

ESIP

The different stages have different mesh frequencies, but there is a relationship between two stages, which is f 1 m = ψ f2 m, and ψ = 3.375 depends on teeth number. 3.1 Stability Time-Domain Response The Newmark-beta method is adapted to solve the dynamic equations, shown in Eq. (5). In order to obtain a stable solution, the results of the about first 6.5 s are removed. In 1st stage, the time-domain stationary response curves of each element (sun, planet, ring) without any meshing error, are shown in Figs. 2, 3 and 4 under a constant 1200 rpm input speed and a constant 72400 Nm load torque. Here, the result of the angular displacement is a linear curve with respect to time the angular displacement, and the only difference is the slope and the positive and negative values. Therefore, taking the angular velocity (negative value) of planet as an example, i.e., its rotation direction (clock wise) is opposite to the definition of the positive direction. Moreover, the angular displacement of the part rotating in the positive direction increases linearly and vice versa.

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As shown in Figs. 2, 3, and 4, the angular speed of sun is more stable and less fluctuating, the ideal angular speed is 125.66 rad/s and angular acceleration is between −40 rad/s2 and 40 rad/s2 . Angular velocity of planet fluctuates between −52.2 rad/s and −26.38 rad/s, and the ideal angular speed is −52.28 rad/s. Angular velocity of ring fluctuates between −16.75 rad/s and −16.66 rad/s, and the ideal angular speed is − 16.39 rad/s, therefore, the angular speed agrees well with ideal value. Moreover, the acceleration fluctuation of the planet gear is more severe than that of the sun and the ring gear.

a) Angular velocity

b) Angular acceleration

Fig. 2. Stationary response of sun in 1st stage

a) Angular velocity

b) Angular acceleration

Fig. 3. Stationary response of planet in 1st stage

Similarly, in 2nd stage, the time-domain stationary response curves are shown in Figs. 5, 6 and 7. As shown in Figs. 5, 6 and 7, it can be known that the direction and of the angular displacement and magnitude of average angular velocity for each component correspond to the theoretical rotation direction and theoretical value. According to the curves of the angular displacement and the calculated velocity value, as shown in the above Figures, it is noted that the mean calculated velocity value agree well with ideal angular velocity value.

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a) Angular velocity

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b) Angular acceleration

Fig. 4. Stationary response of ring gear in 1st stage

a) Angular velocity

b) Angular acceleration

Fig. 5. Stationary response of sun in 2nd stage

a) Angular velocity

b) Angular acceleration

Fig. 6. Stationary response of planet in 2nd stage

3.2 Coupling Analysis It is critical to study coupling properties of two stage due to the structure diagram, shown in Fig. 1. In order to study the coupling characteristics, the acceleration of planet in each

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a) Angular velocity

b) Angular acceleration

Fig. 7. Stationary response of ring in 2nd stage

stage is chosen to study the coupling characteristics. The reason is that the planet meshes with both the sun and the ring simultaneously, which causes the excitation of planet is complicated and it could reflect the main excitation frequency of each stage. 3.2.1 Without error Fig. 7 is the acceleration in the frequency domain under input speed is 1200rpm. It is observed from Fig. 8a) that meshing frequency and its frequency multiplication, such as f 1 m and 2f 1 m, is the main excitation frequency. Moreover, the meshing frequency of 2nd stage also appear. Excepting that, the modulations of f 1 m and f2 m, also become the main excitation frequencies, particularly the peak at f 1 m + f2 m. As shown in Fig. 8b), there is some different situation. Although main excitation frequencies include it own meshing frequency and harmonic frequencies, it is clear that frequencies, are related to meshing frequency of 1st stage, not only appear, but also are the dominant frequency such as 2f 1 m, which is similar with Fig. 8a). Overall, influence of the other frequencies which is related to 2f 1 m, cannot be ignored.

a) 1st stage

b) 2nd stage

Fig. 8. Angular acceleration frequency spectrum of planet in each stage

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3.2.2 Run out Error Next, the influence of run out error on the coupling is studied. As shown in Fig. 9a), sideband effects appear on either side of the location corresponding to the meshing frequency in angular acceleration frequency spectrum of planet in 1st stage. Also, the same phenomenon occurs in the 2nd stage, shown in Fig. 9b), because the ring gear of the 1st stage is the input of the 2nd stage. The runout error of planet in 1st stage excites some frequencies of 2nd stage, although its magnitude is small.

a) 1st stage

b) 2nd stage Fig. 9. Angular acceleration frequency spectrum of planet #1 in 1st stage with runout error

When the runout error in planet of 2nd stage, the corresponding figure is given in Fig. 10. The exciting frequency almost do not change. Comparing with the results of Fig. 8, runout error of planet in 2nd stage do not change coupling relationship. 3.3 Torsional Vibration Analysis Here, the torsional vibration can also be called transmission error (TE) because of pure torsional dynamic model, which is the difference of angular displacement, is shown as

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a) 1st stage

b) 2nd stage

Fig. 10. Acceleration frequency spectrum of planet #1 in 2nd stage with runout error

follows: TE = θreal − θideal

(7)

Here, the global transmission error is defined as the transmission error of the output element and is chosen to study the influence of error on torsional vibration. 3.3.1 Without Error The steady time domain global transmission error result and the corresponding frequency domain result of output element without any meshing error are shown below. As shown in Fig. 11, the excitation frequency of output element is composed of a variety of frequencies, which are related to meshing frequency of each stage, specifically for 2nd stage.

a) Time domain

b) Frequency domain

Fig. 11. Torsional vibration response of output element without any error

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3.3.2 Run Out Error In this section, Figs. 12 and 13 show time domain global transmission error and the corresponding frequency domain result when considering the runout error of planet #1. It shows that the eccentricity error affects the vibration of the output shaft, but for the same magnitude of error value, the influence of the second stage is greater than that of the first stage. For torsional vibration of output element, the meshing frequency is the main frequency.

a) Time domain

b) Frequency domain

Fig. 12. Torsional vibration of output element with run out error of planet #1 in 1st stage

a) Time domain

b) Frequency domain

Fig. 13. Torsional vibration of output element with run out error of planet #1 in 2nd stage

4 Conclusions A torsional dynamic model of two-stage double-helical planetary gear train with different mesh phasing is established, and the vibration and coupling relationship are analysed. In order to verify the accuracy of the established model, the calculated angular displacement and angular velocity are compared with the theoretical values. The conclusions from this study are:

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1) The run-out error of planet has an important impact on excitation frequency of output element and the influence of 2nd stage is bigger than 1st stage. The runout error of planet in 1st stage affects the excitation frequency in 2nd stage, but the runout error of planet in 2nd stage almost do not affect the excitation frequency in 1st stage. 2) According to the coupling analysis, the coupling characteristics between the two stages exists and affects each other. Among the exciting frequencies, the meshing frequency and its harmonic frequency of each stage dominates. Moreover, some combination meshing frequencies, coupled to each other between two stages, are also not negligible. 3) This paper’s focus is one degree of freedom, i.e., the established model is pure torsional model. Moreover, the coupling results of this paper shows that exciting frequencies in each stage include meshing frequencies of two stages, which is a nice coupling result. The results of the coupling characteristics of the torsion model can be used as the research basis for the further research of the bending-torsion model. Acknowledgments. The research was supported by the National Natural Science Foundation of China (Grant No. 11972131) and (Grant No. 12072089). The first author Guanghe Huo also would like to acknowledges financing of his stay at the University of Cantabria by means of a CSC (China Scholarship Council) grant number 202106120133, which is financed by the government of the People’s Republic of China.

References 1. Kahraman, A.: Free torsional vibration characteristics of compound planetary gear sets. Mech. Mach. Theory. 36, 953–971 (2001) 2. Li, S., Wu, Q., Zhang, Z.: Bifurcation and chaos analysis of multistage planetary gear train. Nonlinear Dyn. 75, 217–233 (2013) 3. Zhang, Q., Wang, X., Wu, S., Cheng, S., Xie, F.: Nonlinear characteristics of a multi-degreeof-freedom wind turbine’s gear transmission system involving friction. Nonlinear Dyn. 107, 3313–3338 (2022) 4. Dai, H., Chen, F., Xun, C., Long, X.: Numerical calculation and experimental measurement for gear mesh force of planetary gear transmissions. Mech. Syst. Signal Process. 162, 108085 (2021) 5. Sanchez-Espiga, J., Fernandez-del-Rincon, A., Iglesias, M., Viadero, F.: Influence of errors in planetary transmissions load sharing under different mesh phasing. Mech. Mach. Theory. 153, 104012 (2020) 6. Canchi, S.V., Parker, R.G.: Effect of ring-planet mesh phasing and contact ratio on the parametric instabilities of a planetary gear ring. J. Mech. Des. Trans. ASME 130, 014501 (2008) 7. Wang, C., Dong, B., Parker, R.G.: Impact of planet mesh phasing on the vibration of threedimensional planetary/epicyclic gears. Mech. Mach. Theor. 164, 104422 (2021) 8. Peng, D., Smith, W.A., Randall, R.B., Peng, Z.: Use of mesh phasing to locate faulty planet gears. Mech. Syst. Signal Process. 116, 12–24 (2019) 9. Wei, L., Kai, S., Hailong, S., Wanyou, L.: Modulation sideband analysis of a two-stage planetary gear system with an elastic continuum ring gear. J. Sound Vib. 527, 116874 (2022)

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10. Lu, F., Zhu, R.. Wang, H., Bao, H., Li, M.: Dynamic characteristics of double helical planetary gear train with tooth friction. In: Proceedings of the ASME Design Engineering Technical Conference (2015) 11. Wang, S., Zhu, R.: Nonlinear torsional dynamics of star gearing transmission system of GTF gearbox. Shock Vib. 2020, 1–15 (2020) 12. Wang, S., Zhu, R.: Theoretical investigation of the improved nonlinear dynamic model for star gearing system in GTF gearbox based on dynamic meshing parameters. Mech. Mach. Theory. 156, 104108 (2021) 13. Wang, S., Zhu, R.: Nonlinear dynamic analysis of GTF gearbox under friction excitation with vibration characteristics recognition and control in frequency domain. Mech. Syst. Signal Process. 151, 107373 (2021)

Research on Robustness Analysis and Evaluation Method of Bearing-Support System Fangming Liu1 , Jie Hong2 , Yanhong Ma1 , and Xueqi Chen1(B) 1 Research Institute of Aero-Engine, Beihang University, Beijing 100191, China

[email protected] 2 School of Energy and Power Engineering, Beihang University, Beijing 100191, China

Abstract. Modern engines often use bearing outer ring with mounting edge in the front support structure of the rotor fan to reduce the number of parts and reduce the weight of the engine. During the working process, the front bearing of the fan is in a complex load environment. Among them, the bearing inner ring rotates at high speed with the rotor, which is subjected to significant centrifugal loads. Coupled with the periodic impact excitation caused by the non-synchronous procession, the lateral or angular impact load of the rotor, etc., the bearing and its supporting structure will produce a variety of damage accumulations, and even structural failures. To solve this problem, taking the structural system as the research object, considering the structural form and its rotor motion state under different load environments, and guided by the failure modes and their resulting mechanical processes, the quantitative evaluation parameters and calculation methods for two types of failure modes, interface contact failure and bearing failure, are proposed Among them, interface contact failure includes contact sliding, wear, and contact fatigue, and bearing failure includes tilt of inner and outer rings of bearings, rollers fall off, unsmooth movement when engine shutdown. On this basis, a method and process for evaluating the robustness of the bearing-support system have been developed. The results show that the proposed evaluation method can effectively characterize the damage of bearing-support system under different loads, and determine the main damage modes, which can provide evaluation method support for bearing-support system damage control and robust decision. Keywords: Bearing-support system · Interface contact state · Robust design · Failure mode

1 Introduction During the motion of the aero-engine, the rotor will inevitably excite the support structure, and the motion state of the rotor will change accordingly, resulting in damage accumulation and structural failure of the bearing and its support structure. The fan front bearing-support system also exposed some problems during the whole machine test, such as wear of the bearing inner and outer rings, unsmooth movement when engine shutdown, and excessive vibration. Therefore, it is urgent to form a complete robustness evaluation method and process to assess the damage of the bearing-support system. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 202–223, 2024. https://doi.org/10.1007/978-3-031-40459-7_14

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In terms of bearing damage and failure modes, scholars have carried out sufficient research. The commonly-seen failures of bearings mainly consist of plastic deformation, wear, corrosion, crack, and fracture [1–3]. The reasons for these failures include extreme operating conditions, inadequate lubrication, improper mounting, and deficient sealing [2]. El Laithy et al. [4] reviewed the research on rolling contact fatigue in bearings and the associated microstructural alterations. Darmo et al. [5] investigated the fatigue fracture on a double-row tapered roller bearing, and the thermal softening was the leading reason for the initial crack. Zhang et al. [6] investigated that the failure mechanism of the fatigue failure of the main bearing threaded hole in a high-strength diesel engine was the high cycle fatigue failure caused by the gas-dominated stress amplitude exceeding the bearing capacity of the material and structure. Xu et al. [7] studied the dynamic behaviors and contact characteristics of ball bearings under the effects of 3D clearance fit. The results indicate that the larger fit clearance of the unilateral bearing may lead to misalignment. However, there are few reports on the damage and failure of bearing-support system containing connect interfaces. Improving the robustness of the mechanical properties of structural systems has become the main goal of current engineering structure design [8]. The goal of robust design of structural systems is to reduce the sensitivity of structural mechanical properties to changes in working conditions and load environments by controlling damage, thereby improving the robustness of structural systems. Hong Jie et al. [9] analyzed the failure of the rotor connection interface. Based on the tolerance model, they carried out a robust design for the contact stress of the connection interface of the rotor structure system. Yue Wei et al. [11] established the stiffness damage model of the rabbet joint structure and proposed the robust design method of the rabbet joint structure. Jie Hong [12] proposed a mechanical model for evaluating sliding damage in bolted connections. Jie Hong [13] established a mechanical model of joint interface stiffness loss and proposed a joint structure stiffness loss suppression method based on strain energy distribution optimization for dynamic design of discontinuous rotor system. LI Junhui et al. [14] used the finite element method to study the main factors affecting the joint stiffness and contact stress of the gear structure, and proposed the corresponding structural design method of the gear structure. Bing-long Lei [15] established quantitative evaluation parameters based on interface deformation coordination, and optimized geometric characteristic parameters of rotor. Xueqi Chen [16] established an evaluation method for the influence of interface contact damage evolution on rotor dynamic characteristics. L. A. Sosnovskiy [17] proposed a prediction method of interface wear and fatigue caused by friction. To sum up, existing research on the damage and failure of bearing structures mostly focuses on the contact fatigue and wear damage of single components of bearings, with less consideration given to the bearing-support system. Especially for bearings and multiple connection structures under complex load environments, there is still a lack of systematic and effective methods for evaluating the robustness of interface contact damage and mechanical properties. For the typical fan front bearing and its support structure, the structural system is taken as the research object, comprehensively considering the changes in the mechanical properties of the bearing-support system under different load environments and rotor motion states, the damage evaluation parameters for the bearing and the two connection interfaces are proposed,

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2 Structure System and Load 2.1 Composition of Structural System As shown in Fig. 1, the typical fan front bearing-support system is composed of fan rotor, bearing, bearing seat, bearing frame, fan case and other structures. And the front bearing is the roller bearing, whose mounting edge of the outer ring is connected to the inner mounting edge of the bearing seat by bolts, forming a bearing-support system. It provides lateral and angular displacement constraints for the front end of the fan rotor and transmits part of the radial load to the bearing seat, the bearing frame and the main mounting.

Bearing frame Bearing seat Bolted joint The bearing outer ring

Fan rotor

Roller The bearing inner ring

Cage Front journal

Fig. 1. Typical bearing-support system in aero-engine

2.2 Interface Contact Characteristic A typical fan front bearing-support system consists of multiple components and forms a whole through the connection structure. This paper focuses on the two connections: the roller-bearing connection, the outer ring-bearing seat connection. As shown in the Fig. 2, the contact interface between the roller and bearing includes bearing outer ringbearing seat cylindrical face, bearing outer ring-bearing seat end-face, front and rear bolt end-face, which form a flange-bolt connection interface. The contact interface between the bearing outer ring and the bearing seat includes the outer ring-roller face and the inner ring-roller face. 2.3 Load Environment Assembly and Centrifugal Load. During the initial assembly, the bearing outer ring and bearing seat are connected by bolts, under the stress caused by the initial bolt pretension and the small interference fit of the cylindrical-face. Among them, the initial pretension

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Bearing outer ring-Bearing seat end-face p Bolt end-face Bearing outer ring-Bearing seat cylindrical face

f

p Bolt end-face

f f

Bearing outer ring-Roller face

Bearing inner ring-Roller face

Fig. 2. Contact interface and contact characteristic parameters

ensures the axial compression under complex load. And the initial interference fit ensures the connection structure always maintains centering, preventing the interface damage and increased vibration caused by looseness of cylindrical-face. Under the action of assembly load, due to different configuration and material parameters, the deformation ratio of bearing outer ring and bearing seat is different, which may lead to certain contact damage in the assembly state, as shown in Fig. 3. At the same time, the angular deformation will lead to the change of the contact position between the bearing outer ring and the roller, resulting in collision and contact damage.

l pre,0

Bolt Pretens ion

l pre

Bolt Pretens ion

Cylindrical interference fit

(a) Before assembly

(b) After assembly

Fig. 3. Deformation trend of structural system before and after assembly

The fan rotor is supported by the front and rear fulcrums. Due to the positive Poisson’s ratio of the material, the rotor contracts axially while expanding radially under centrifugal load. Moreover, since the rear fulcrum is positioned axially by ball bearing, the front journal moves backward during axial contraction, resulting in axial displacement of the bearing inner ring, as shown in Fig. 4b. If the axial deformation of the inner ring does not occur, the center surfaces of the bearing inner ring and the bearing inner ring coincide. The rotor and the bearing inner ring will transfer radial load to the bearing outer ring through the roller under centrifugal load. This radial load contributes to the radial compression at the cylindrical-face, as shown in

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Centrifug al load

Axial displacement of inner ring

Axial deformation direction

(a) Static state

ω

(b) Rotating state

Fig. 4. Deformation trend of fan rotor under centrifugal load

Fig. 5a. When the fan rotor deformation causes axial displacement of the bearing inner ring, the radial load caused by the centrifugal load will produce additional bending load on the bearing outer ring, and the outer ring will have angular deformation compared with the inner mounting edge of the bearing seat, and lead to bolt pretension loss, contact state change, constraint stiffness decrease, local separation of contact interface and even contact damage occur, as shown in Fig. 5b.

Radial load

The interface keep compression

(a)

Additional bending load

Interface separation on local area

Radial load

(b)

Fig. 5. Influence of axial displacement of bearing inner ring on deformation

Rotational Inertia Load. The rotating inertia load caused by the asymmetric mass of the rotor will generate the bearing dynamic load, as shown in Fig. 6.

Force transmission route of rotational inertia load

Fig. 6. Force transmission route of rotational inertia load on bearing-support system

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Looking along the face fore, the fan rotor rotates clockwise. At the same time, the rotational inertia load and the resulting bearing dynamic load are the rotational excitation that rotates clockwise with the fan rotor, and generate periodic excitation Fs,r for the bearing-support system which remains stationary, as shown in Fig. 7. Fs,r

L

(a) T=t0

Fs,r

L

(b) T=t1>t0

Fig. 7. Periodic excitation characteristics of bearing-support system under rotational inertia load

Generally, the rotor works above the multiple critical speeds. If the asymmetric distribution of the rotor mass distribution in the initial state only manifests as a center of mass center eccentricity, the self-centering effect in post critical range makes the lateral displacement of the rotor caused by the rotational inertia load and the amplitude of the periodic excitation Fs,r on the bearing-support system small, and the interface contact damage caused by the rotating inertia load is slight. However, when the rotor has a certain skewness in principal axis of inertia, a large rotational inertia moment will be generated in post critical range. And a large value of periodic excitation will be generated on the bearing-support system. Such periodic excitation will cause periodic changes in the contact state at the interface, which may lead to interface wear or fatigue damage. Stable Lateral (Angular) Inertia Load. When the engine is in an extreme working environment such as lateral overload or maneuvering flight, the bearing-support system will have obvious lateral or angular deformation because of lateral inertia load caused by overload or angular inertia load caused by maneuvering flight. For fan rotor, deformation can produce a large additional lateral or angular load Fs,in , as shown in Fig. 8. For bearing-support system, the direction and amplitude of lateral and angular inertial load is always the same, so the additional load is static load for bearing-support system. The static load has a constant influence on the contact state, which may cause some changes in the contact characteristics. When the support system is under stable lateral (angular) inertia load, asymmetry of load results in asymmetry of deformation, thus leading to different stiffness in each direction (nonlinearity of bearing stiffness), as shown in Fig. 9. Under lateral overload, the stiffness in vertical direction ks,V 1 = ks,V 2 is different, but the stiffness in horizontal direction ks,H 1 = ks,H 2 is the same. The vertical and horizontal stiffness ks,V 1 = ks,V 2 = ks,H 1 = ks,H 2 are different under maneuvering flight. The nonlinear stiffness of the support may cause the rotor to be in the state of non-synchronous procession, which may lead to the vibration exceeding the limit. At the same time, there may also be misalignment, and then produce frequency doubling, leading to vibration intensification.

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Before deformation

After deformation

After deformation

5g

Ω

ω

(a) Lateral overload

(b) Maneuver flight

Fig. 8. Fan rotor deformation under extreme working conditions

Before deformation After deformation

ks,V1

cs,V1

ks,H1

ks,H2 cs,H2

cs,H1 ks,V2

(a) Deformation diagram

cs,V2

(b) Mechanical model

Fig. 9. Bearing frame deformation under extreme working conditions

Impact Excitation. The generation of impact excitations mainly includes the following two types: first is the lateral impact excitation generating during aircraft hard landing, second is the rotational impact excitation that the rotor may produce on the bearingsupport system when the rotor is in non-synchronous procession. During aircraft hard landing, the rotor may have a large lateral deformation instantaneously, and collide with the bearing-support system, resulting in impact excitation. This kind of impact excitation shows the characteristics of extremely short time and high peak value in the time domain Although the load at this time is similar to that of the steady lateral (angular) inertial load, the interface is more prone to damage and failure because the excitation amplitude is much larger. The fan rotor is usually in non-synchronous procession state due to the influence of factors such as rotor-stator rubbing, asymmetric structural mechanical properties, interactive excitation between high and low rotors and so on. The rotor’s procession trajectory is not circular. When the procession radius is large, the rotor may produce periodic impact excitation on the bearing-support system. The amplitude of the periodic impact excitation is relatively small, and the impact on the contact state and the interface damage caused by the periodic impact excitation are similar to that of the steady rotational inertia load. In addition to directly causing the contact state change and contact damage of the interface, the periodic impact excitation will also arouse the modal vibration of the bearing-support system. If the excitation frequency is close to or even equal to the

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modal frequency of the bearing-support system itself, the modal vibration amplitude will increase greatly, and the bearing-support system will have modal resonance, leading to the response amplitude is increased. And the modal resonance will lead to contact damage or failure.

3 Damage and Failure Analysis of Bearing-Support System 3.1 Failure Mode According to the different failure locations, the failure modes of the bearing-support system can be divided into two types: One is the failure of the two contact interfaces in the structural system, and the other is the failure of the bearing function. Interface Contact Failure. For the interface, three types of interface contact damage, namely sliding damage, wear and contact fatigue, may occur in the working process, and thus result in failure. The magnitude of the quantitative evaluation parameter indicates the magnitude of the interface damage accumulation during working cycle. Interface Wear. Wear is the process of continuous damage to surface materials of objects in contact with each other during relative motion. It is the inevitable result of friction. When wear accumulates to a certain extent, the interface morphology changes, and the surface material peels off and loses, resulting in wear damage. In this paper, Archard friction work law to describe the amount of interface wear damage under external load changes Contact friction work is used to quantitatively evaluate the wear damage, as shown in “Eq. (1)”. Wwear = Ff dr

(1)

where, W wear is the contact friction work, F f is the friction force between the two members pressed together, and d r is the relative sliding distance. When the contact pair is sticking, the friction work generated by the position is ignored. When the contact pair is sliding, the friction work is calculated only with the interface contact characteristic parameters under the maximum load state in the loading cycle. Interfacial wear damage is usually caused by periodic loads. According to different load types, the possible wear and damage mechanism of the bearing-support system at the interface can be divided into two types: One is the interface wear caused by the periodic excitation caused by the rotating inertia load or the non-synchronous procession of the rotor. The other is the centrifugal load and the resulting interface sliding change with the rotating speed. In the process of multiple working cycles, the interface wear occurs. However, from the point of view of the degree of interface wear, the former is much higher than the latter. Because clearance fit is used at the cylindrical-face, the radial constraint stiffness is provided only by the bending stiffness of bolts and the tangential friction at the end-face. And the axial pretension of the bolt decreases after the end-face wears, tangential friction also decreases, resulting in stiffness loss and nonlinear change of bearing stiffness with load. The flatness of the end-face surface cannot be guaranteed after wear, and the outer ring may skewness, causing misalignment.

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Interface Sliding. The mechanism of sliding damage is that the friction between the interfaces cannot restrain the uncoordinated deformation under complex loads, which is mainly represented by the change of tangential relative position between the interfaces. The relative sliding distance d r is used to describe the interface sliding damage quantitatively. According to the above analysis of the load environment of the bearing-support system, it can be seen that the possible sliding damage mainly includes the following two forms: 1) Under centrifugal load, axial displacement occurs in the bearing inner ring. The radial load caused by centrifugal load in the rotor and bearing inner ring will generate additional bending load on the bearing outer ring through the roller. The mounting edge of the bearing outer ring will undergo angular deformation, which may cause relative sliding in end-face dr,1 and cylindrical-face dr,2 , as shown in Fig. 10a. 2) Under the lateral inertial load or impact load, the mounting edge of the bearing outer ring has a certain lateral displacement. Moreover, because the outer bearing ring and the bearing seat usually adopt clearance or transition fit at the cylindrical-face surface, the centering effect is not strong, resulting in the sliding of the end-face d r , as shown in Fig. 10b. Under the action of inertia load, the position relation of the bearing and support structure changes step by step. Under the impact load, the interface sliding shows a sudden change. Before deformation

After deformation

End-face sliding dr,1

Additional bend Centrifugal load

(a) under centrifugal load

Cylinderdr,2

Before deformmation

After deformation End-face sliding dr

Lateral inertial load

(b) under lateral

Fig. 10. Different interface sliding forms of bearing-support system

Interface sliding directly leads to the decrease of radial constraint stiffness of the bearing-support system (under the same lateral load, the greater the interface sliding, the greater the lateral displacement of the bearing-support system, and the lower the constraint stiffness). In addition, interface sliding will cause misalignment, and abnormal frequencies such as double frequency will appear in dynamic response of rotor. Interface Contact Fatigue. Fatigue damage refers to the accumulation of damage in the process of cyclic loading, which can be generally divided into high period fatigue and low period fatigue. It is mainly manifested as the initiation and expansion of cracks in interfacial materials under internal forces, which is generated by the accumulation of damage in cyclic loading. At present, the critical surface method is usually used to study the fatigue damage characteristics of the interface, and the SWT model is a more commonly used fatigue damage life model [18], as shown in “Eq. (2)”. Where, σmax is the maximum stress perpendicular to the plane, and ε is the maximum strain difference

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perpendicular to the plane. θ = σmax

ε 2

(2)

For a plane at angle θ , substitute τy,x = μp into “Eq. (2)”. The expression for θ can be written as shown in “Eq. (3)”. Where, pm is the maximum contact stress, and p is the represents the change in contact stress. The value of C f depends on the magnitude of the friction coefficient. Therefore, the multiaxial fatigue damage parameters θ mainly depend on the product pm p . Therefore, indicator parameter of fatigue strength Pfatigue = pm p were used to quantitatively evaluate the interface fatigue damage.  1 d εθ · p = Cf pm p θ ≈ σθ p=pm · · 2 dp

(3)

The possible contact fatigue damage of bearing-support system at the interface is mainly concentrated on the end-face of the bearing outer ring and the bearing seat, and may also be caused by periodic excitation caused by rotating inertia load or nonsynchronous procession. When the cracks in the material extend to a certain extent, fatigue fracture may occur, leading to the reduction of radial constraint stiffness and even constraint failure of the bearing-support system. Bearing Failure. For bearing, three types of damage, tilt of inner and outer rings of bearings, rollers fall off, unsmooth movement when engine shutdown may occur. Tilt of Inner and Outer Rings of Bearings. In the working process, the bearing inner and outer rings the may have angular displacement, so that the bearing roller is in abnormal motion, resulting in the change of the rolling position of the roller on the inner and outer rings. After angular displacement is generated in the inner and outer rings. On the one hand, the roller will produce a large rolling load (radial force F N and tangential drag force F T ). On the other hand, the contact position between the roller and the inner and outer rings will change (from line contact to point contact). The rolling radial force will cause the roller to generate traveling wave excitation load. Rolling tangential force will produce greater friction force, which may lead to obvious wear and fatigue damage in the conflict area. What’s more, the tilt of inner and outer rings can also lead to misalignment Therefore, the tilt angle θr is used to describe the relative tilt of inner and outer rings quantitatively. According to engineering experience, when the tilt angle is greater than 6’, the bearing will suffer greater damage (Fig. 11). Rollers Fall Off. Typical fan front bearing-support system uses inner ring positioning, roller and bearing outer ring can slide against each other. Under centrifugal load, the rear fulcrum of the fan rotor expands radially and contracts axially. The rear fulcrum of the fan rotor is ball bearing, which limits its axial displacement. Therefore, the inner ring of the front journal bearing of the fan will have axial displacement backward. The roller and cage will be axial displaced backward along with the inner ring of the bearing, resulting in the change of the contact position between the outer ring and the roller. As shown in Fig. 12. When the roller is detached from the bearing outer ring (when the axial

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FN,outer FT,outer

Conflict area

Conflict area

FT,inner

r H

Conflict area

FN,inner

Fig. 11. Bearing rolling load diagram in working progress

displacement of the roller exceeds the maximum allowable displacement l − l ≤ 0), the contact area between the roller and the bearing outer ring becomes smaller, resulting in a sharp increase in the contact stress at the contact position, and contact damage is likely to occur.

l

l

l+ l

l- l=0

Axial deformation (a) No axial displacement of inner ring

(b) Axial displacement of inner ring

Fig. 12. Axial deformation diagram of bearing inner ring in working process

Unsmooth Movement when Engine Shutdown. Under normal conditions, there is a certain radial clearance of the bearing. However, under asymmetric loads (such as lateral inertial loads), there will be a certain relative displacement between the inner and outer rings, resulting in reduced clearance at some positions of the bearing and increased clearance at some positions, as shown in position A in Fig. 13b. When the roller is moved to the position A, the contact deformation δin between the roller and the inner ring raceway and the contact deformation δout between the roller and the outer ring raceway increases, which leads to greater resistance when the roller reaches here, and further leads to the wear of the roller and the inner and outer rings of the bearing. Because there is a flange-bolt connection between the bearing outer ring and the bearing seat, certain sliding will occur on the end-face under asymmetric load. After engine shutdown, the elastic restoring force of the flange edge cannot overcome the friction force at the position of the end-face, resulting in a certain residual deformation

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Bearing clearance

(a) Before deformation

(b) After deformation

Fig. 13. Unsmooth movement due to asymmetric load

of the bearing outer ring, which cannot be restored. Thus, the distance between the inner and outer rings of the bearing decreases, resulting in unsmooth movement when engine shutdown (Fig. 14).

(a) Under asymmetric load

(b) Engine shutdown

Fig. 14. Diagram of deformation during under asymmetric load and shutdown

The core of unsmooth movement is that the restraint state of the bolted connection structure under external load cannot be restored. In this paper, considering the changes in surplus bolt pretension caused by local deformation of the interface under external loads, the surplus bolt pretension is used to reflect the change of local constraints of the connection structure. The greater the change in the surplus bolt pretension compared to the initial assembly pretension, the more likely it is to cause deformation that cannot be restored, and the more likely it is to cause unsmooth movement when engine shutdown. 3.2 Structural System Robustness Evaluation Method Various types of interface damage may occur simultaneously in bearing-support system, and the different damage will interact with each other. Therefore, it is necessary to comprehensively consider the randomness of fit characteristic parameters and load characteristic parameters to carry out a complete robustness evaluation of bearing-support system. Based on this, the flow chart is as shown as Fig. 15: 1) Calculate damage under normal working loads. Firstly, analyze the deformation, the interface contact characteristics of the bearing-support system from the assembly to the maximum operating speed state. Secondly, use the three evaluation parameters

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proposed in the previous section to evaluate whether the bearing is damaged. Then, use the three types of interface contact damage evaluation parameters and algorithms established to calculate the wear, sliding and fatigue damage. 2) Calculate the damage caused by periodic impact excitation due to the bearing-support system clearance. Periodic impact excitation is one of the loads that need to be considered in addition to the design conditions and may occur during the working process. By calculating the structural dynamic response, deformation and interface contact characteristics, the damage of the bearing-support system in a cycle is evaluated. 3) Calculate damage under lateral loads. Comparing the bearing dynamic load under stable lateral inertial load or hard landing, it is believed that when the bearing dynamic load is large, the deformation of the connecting structure is large and the contact state is poor. And evaluate the failure of bearing and contact damage of interface in this state. 4) Calculate damage under maneuvering flight. Analyze the deformation, the interface contact characteristics. Then, evaluate the failure of bearing and contact damage of interface in Maneuver flight.

Bearing-support system design requirements Preliminary engineering design

Damage assessment under assembly and centrifugal load

Damage assessment under periodic impact excitation Yes

Damage assessment under lateral load Amplitude of lateral impact excitation hard landing

Damage assessment under stable lateral load

Damage assessment under maneuvering flight No

Damage assessment under hard landing

No Whether it meets the damage control requirements of bearing Yes No Whether it meets the damage control requirements of the outer ring-bearing seat interface Yes No Whether it meets the damage control requirements of the roller-outer ring interface Yes Completion of interface damage and robustness assessment

Fig. 15. The evaluation process robustness of the bearing-support system

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4 Examples of Robustness Evaluation 4.1 Modeling of Bearing-Support System Based on typical fan front bearing-support system, solid 186 elements are used in bearing-support system. For the components far away from the connected structure, the rough mesh with the element size of 6 mm can be used. A grid with an element size of 0.3 mm is used for members close to the connected structure. Establish the three-dimensional solid finite element model as shown in Fig. 16. The interface of finite element model of bearing-support system is set as shown in Fig. 16, yellow represents bonded and red represents frictional.And the friction coefficient is 0.15 and the gap of the outer ring-bearing seat cylindrical-face is set as 0.01 mm in diameter. Considering the connection of the bearing-support system, fixed support are applied to the position of mounting joint (A in Fig. 16) and ball bearing (B in Fig. 16). Under the action of axial compression of the bolt head and nut, the mounting edge of bearing outer ring and the rear journal of bearing seat will be in a state of compression at the end-face and the tightening torque is set to 10 N·m (equivalent bolt pretension of 5674 N). The fan rotor working speed of is given to 10000 RPM. A

B

Y

X Z

Fig. 16. Finite element model, connects and boundary condition of bearing-support system

4.2 Damage Evaluation Under Assembly and Centrifugal Load In this section, ANSYS workbench was used for simulation calculation. The deformation distribution characteristics of the bearing-support system were calculated, as shown in Fig. 17. It can be seen that the inner ring of the bearing undergoes axial displacement relative to the outer ring. Bearing damage evaluation parameters from assembly to maximum working speed were calculated, as shown in Table 1. The relative tilt angle θr of inner and outer rings is 1.647’, which far less than 6’. The maximum axial displacement reaches l − l = 4.713 mm ≥ 0. Therefore, the roller will not be separated from the outer ring of the bearing under centrifugal load. Since the connection structure will loosen due to the positive Poisson’s ratio and the deformation of the compressed part, the bolt surplus

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-4.132×10-3mm

-1.514×10-2mm

9.142×10-3mm

1.010×10-2mm

-4.029×10-3mm

1.285mm

1.296mm

-4.524×10-3mm

(a) Radial deformation

(b) Axial deformation

Fig. 17. Deformation diagram under centrifugal load

Table 1. Bearing damage evaluation parameters. Evaluation parameters

Tilt angle θr

l − l

Surplus pretension

1.647’

4.713 mm

5665 N

pretension in maximum working speed state has a relatively small change compared to that in assembly state. Therefore, the bearing damage is relatively small in this state. The calculated interface damage is shown in Table 2. The contact friction work on the outer ring-bearing seat cylindrical-face, end-face and roller-outer ring face increased by multiple order of magnitude. Therefore, the probability of wear damage on the three interface increases in order. The maximum unrecoverable sliding distance on flangebolt connection interface is relatively large (greater than initial offset). This means that sliding damage is prone to occur at the interface. The fatigue indicator parameters on the end-face is small, while which is relatively large on the other two interfaces. That is, it is easy to produce fatigue damage on cylindrical-face and roller-outer ring face. Table 2. Interface damage evaluation parameters.

Cylindrical-face End-face Roller-outer ring face

Contact Friction Work

Unrecoverable Sliding Distance MAX

Fatigue indicator parameter

5.433 × 10−3 J

0.021 mm

12321 MPa2

0.035 mm

2885 MPa2

1.437 J 10.86 J

–

28157 MPa2

Figure 18 shows the damage morphology of the roller and bearing outer ring during the engine shutdown inspection after 318 h of operation. There is obvious wear on the roller and the bearing outer ring raceway, and there is a certain phenomenon of swapping block on the roller. The calculation results are consistent with the actual engine test results.

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(b) Roller-outer ring face

Fig. 18. Morphology of the damaged bearing outer ring

4.3 Damage Evaluation Under Periodic Impact Excitation This section considers the damage evaluation of periodic collision impacts between the rotor and stator when there is clearance in the bearing. Based on the joint simulation of ANSYS Workbench and LS-Dyna, on the basis of considering the centrifugal load, the unbalance of the fan rotor is set to 300 g mm, the bearing clearance is set to 0.01 mm, the low pressure rotor speed is set to 10000 RPM, the structural damping ratio is 0.003. The calculation results are shown in Fig. 19.

Fig. 19. Load bearing under periodic impact excitation

Compared with the evaluation results of bearing damage under normal working conditions (as shown in Table 1), The change in bearing damage evaluation parameters is relatively small, as shown in Table 3. Therefore, the bearing damage is less likely to occur in this state. Table 3. Bearing damage evaluation parameters. Evaluation parameters

Tilt angle θr max

l − l max

Surplus pretension

1.144’

4.715 mm

5652.2 N

The calculated interface damage is shown in Table 4. Compared to centrifugal load, contact friction work, maximum unrecoverable sliding distance slightly increases, but the

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overall change is not significant. The fatigue indicator parameters on end-face and rollerouter ring face is small. This means that fatigue damage may only occur on cylindricalface. Table 4. Interface damage evaluation parameters.

Cylindrical-face End-face Roller-outer ring face

Contact Friction Work

Unrecoverable Sliding Distance MAX

Fatigue indicator parameter

5.709 × 10−3 J

0.026 mm

13028 MPa2

0.046 mm

256 MPa2

1.783 J 11.74 J

2601 MPa2

–

4.4 Damage Evaluation Under Lateral Load This section uses ANSYS workbench for simulation calculation. Firstly, it is necessary to compare bearing forces under steady lateral loads and “hard landing” impact loads. Therefore, in this section, on the basis of considering centrifugal load, the steady lateral load is set as the lateral force that does not change with time of 5 g. The “hard landing” impact load is set as a lateral force with an action duration of 11 ms and a size of 5 g, and the damping ratio is set as 0.003. It is calculated that the force of the outer ring is 875.9 N under steady lateral load. The maximum force of the outer ring is 1265 N, which is much greater than the lateral load. Therefore, it is considered that the “hard landing” state is a state of poor contact, which needs to be evaluated for damage.

Radial deformation Min 0.00352mm

Min 0.00352mm

Y

z

x

Radial deformation Max 0.0156mm

Max 0.0156mm

Fig. 20. Radial deformation distribution of bearing outer ring under “hard landing” condition

The radial deformation of the outer ring under lateral load is shown in Fig. 20. It can be seen that, under the action of centrifugal load on the rotor, the bearing outer ring has umbrella-shaped and radial outward deformation, and the distribution of radial deformation along the circumference is not uniform due to the huge lateral load on the fan rotor. The radial deformation on the lower side is relatively large, resulting in the maximum angular deformation on the lower side (3.877’≤6’). The maximum axial

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displacement reaches l−l = 4.712 mm ≥ 0. Therefore, the roller will not be separated from the outer ring of the bearing under centrifugal load. 2.561870 N

21. 40MPa

x

87

5.

N 77 807. 6N

x

x

Radial load

Y

36. 84MPa

o z

275. 18N

x x

55. 92MPa

61. 98MPa

x

Y

o

x

(a)Normal contact stress

5665N

5673.4N

5655.6N

Surplus bolt pretension

25. 76MPa

x

Only bolt pretension No hard landing Hard landing

No overload Hard landing

x

x

Y

z

1265. 2N

(b)Radial load of roller

o

Z

678.8N

(c)Surplus bolt pretension

Fig. 21. Normal contact stress, load distribution and surplus bolt pretension diagram

Figure 21 show that lateral load will change the deformation and load distribution characteristics of bearings. In the load direction, the roller load increases and the radial deformation of the outer ring increases. In addition, the radial deformation of the outer ring will affect the contact state between the roller-bearing seat at the interface. Therefore, it is necessary to further analyze the contact damage of two interface. Due to the asymmetry of the lateral load, the surplus pretension on the lower side is relatively large, up to 5678.8 N. But compared to the assembly state, the change is relatively small. Therefore, it is not easy to cause unsmooth movement when engine shutdown (Table 5). Table 5. Bearing damage evaluation parameters. Parameters

Tilt angle θr max

l − l max

Surplus pretension max

3.877’

4.712 mm

5678.8 N

The calculated interface damage is shown in Table 6. Compared to centrifugal load, the change of contact friction work and maximum unrecoverable sliding distance is not relatively small. Fatigue indicator parameter increases slightly in cylindrical-face and roller-outer ring face, but increases exponentially (53 times) on the end-face. This means that fatigue damage is likely to occur on end-face during hard landing. 4.5 Damage Evaluation Under Maneuvering Flight This section uses ANSYS workbench for simulation calculation. On the basis of considering the centrifugal load, the maneuvering angular velocity of the given structural system is 3.5 rad/s. The deformation diagram under maneuvering flight is shown in Fig. 22. It can be seen that the inertia torque generated by the fan rotor will generate additional lateral

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Cylindrical-face

Contact Friction Work

Unrecoverable Sliding Distance MAX

1.447 × 10−3 J

0.025 mm

14436 MPa2

0.038 mm

156414 MPa2

End-face

1.593 J

Roller-outer ring face

11.94 J

30920 MPa2

–

Y

Y

x

z

Fatigue indicator parameter

z

x

Local bending

(a) Outer ring

(b) Inner ring

Fig. 22. Deformation diagram under maneuvering flight conditions

inertial load, and the axial displacement of the inner ring under centrifugal load will cause the lateral inertial load to generate bending load at the same time. As a result, obvious local bending deformation occurs in the outer ring. The maximum relative tilt angle of inner and outer rings is 37.651’.Excessive tilting of the inner and outer rings will cause significant bearing damage. Roller out of outer ring x 0.546mm x 0.63mm 0.359mm

1 5 .3 6 2 7 8 0 N

Only bolt pretension Hard landing Maneuvering flight

No maneuver flight Maneuver flight

x

807. 6N

0.133mm

5799N

x

Y

x

o z

Radial load

Y

x

5787N

o

5673.4N

z

5678N

13003N

x

1388M MPa

x x

13861N

22523N 27780N

x

(a)Normal contact stress

Y

o

Roller press the inner ring

(b)Radial load of roller

Z

6155N

(c)Surplus bolt pretension

Fig. 23. Normal contact stress, load distribution and surplus bolt pretension diagram

Under maneuvering flight conditions, the contact state and radial load distribution are shown in Fig. 23. It can be seen that the tilt of the fan rotor will generate additional bending moment on the bearing under maneuvering flight, leading to local bending deformation of the bearing outer ring and non-uniform compression load of the roller. Compared with the lateral load, the load of roller caused by maneuvering flight changes more violently, which is manifested as that the bearing capacity of only one side roller and the maximum load of roller significantly increase under maneuvering flight, which will have an adverse

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effect on the strength and life of the rollers. This also leads to a significant asymmetry in the surplus bolt pretension. The maximum surplus bolt pretension reaches 6155 N, which increases by 8% compared to the assembly state and the dispersion increases, which may cause certain unsmooth movement when engine shutdown (Table 7). Table 7. Bearing damage evaluation parameters. Parameters

Tilt angle θr max

l − l max

Surplus pretension max

37.65’

4.734 mm

6155 N

The calculated interface damage is shown in Table 8. The contact friction work, unrecoverable sliding distance and fatigue indicator parameter increase by multiple orders of magnitude compared to the centrifugal load. This means that the wear, sliding and fatigue damage are easy to occur. Table 8. Interface damage evaluation parameters.

Cylindrical-face End-face Roller-outer ring face

Contact friction work

Unrecoverable sliding distance MAX

Fatigue indicator parameter

3.123 × 10−3 J

0.176 mm

43068 MPa2

0.201 mm

28619 MPa2

3.227 J 49.99 J

1.928 × 106 MPa2

5 Conclusion In this paper, for a typical fan front bearing-support system, the possible failure modes are analyzed for different loads. And according to the failure modes, a robustness evaluation method and process for the bearing-support system is proposed. 1. By analyzing the structural characteristics, motion status, and complex load environment of a typical fan front bearing-support system, the interface contact damage characteristics and bearing damage characteristics were mastered. 2. There are three types of interface contact failure mode that may affect the mechanical properties of the bearing-support system: contact fatigue damage, relative slip damage, and wear damage. And three quantitative evaluation parameters are proposed: unrecoverable sliding distance, contact friction work, and fatigue indicator parameter, which are used to evaluate the interface contact damage accumulation of the bearing-support system under complex load environments.

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3. There are also three types of bearing failure mode that may affect the mechanical properties of the bearing-support system: tilt of inner and outer rings of bearings, rollers fall off, unsmooth movement when engine shutdown may occur. And three quantitative evaluation parameters are proposed: relative tilt angle of inner and outer rings, maximum allowable displacement, surplus bolt pretension, which are used to evaluate the bearing damage accumulation under complex load environments. 4. Taking a typical bearing-support system as an example, robustness evaluation method and process are introduced. The results show that the evaluation method proposed in this paper can effectively characterize the impact of different loads on the robustness. And by comparing the evaluation results under different working conditions, the main failure mode under each working condition are determined. The failure mode under centrifugal load mainly focuses on interface contact failure. Each interface may experience a certain degree of sliding damage. At the same time, the cylindricalface is prone to fatigue damage, and the roller-outer ring face is prone to wear and fatigue damage. The influence of periodic impact excitation and lateral impact load mainly focuses on fatigue damage compared to centrifugal load. The possibility of fatigue damage to cylindrical-face doesn’t vary significantly, while the end-face and roller-outer ring face have a reduced likelihood of fatigue damage under periodic impact excitation and increased likelihood under hard landing. The failure mode under maneuvering flight includes interface failure and bearing failure, in which the possibility of wear, sliding, and fatigue damage of each interface is significantly increased. It is also easy to cause excessive tilting of the inner and outer rings and unsmooth movement when engine shutdown. Acknowledgements. The authors would like to acknowledge the financial support from National Natural Science Foundation of China (Grant Nos. 52075018. 52205082), and National Science and Technology.

References 1. Harris, T A, Kotzalas M.N. Essential Concepts of Bearing Technology. CRC Press, 2006 2. Harris, T.A., Kotzalas, M.N.: Advanced concepts of bearing technology. CRC press (2006) 3. Liu, Z., Zhang, L.: A review of failure modes, condition monitoring and fault diagnosis methods for large-scale wind turbine bearings. Measurement 149, 107002 (2020) 4. El Laithy, M., Wang, L., Harvey, T.J., et al.: Further understanding of rolling contact fatigue in rolling element bearings-a review. Tribol. Int. 140, 105849 (2019) 5. Darmo, S., Bahiuddin, I., Handoko, P., et al.: Failure analysis of double-row tapered roller bearing outer ring used in Coal Wagon Wheelset. Eng. Fail. Anal. 135, 106153 (2022) 6. Zhang, X., Jing, G., Wang, G., et al.: Mechanism investigation on fatigue failure in threaded hole of the main bearing in high-strength diesel engine. Eng. Fail. Anal. 143, 106921 (2023) 7. Xu, H., Wang, P., Ma, H., et al.: Dynamic behaviors and contact characteristics of ball bearings in a multi-supported rotor system under the effects of 3D clearance fit. Mech. Syst. Signal Process. 196, 110334 (2023) 8. Chen, L.Z.: Robust Design, pp. 108–109. China Machine Press, Beijing (2000) 9. Jie, H.O.N.G., Yanhong, M.A.: Structure and Design of Aircraft Gas Turbine Engine. Science Press, Beijing (2021)

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10. Hong, J., Xu, X., Liang, T., et al.: Interface failure analysis and robust design method in rotor structural system. J. Aerospace Power 33(3), 649–656 (2018) 11. 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) 12. Hong, J., Ma, Y.B., Feng, S.B., et al.: The Mechanism and Quantitative Evaluation of Slip Damage of Bolted Joint with Spigot//Turbo Expo: Power for Land, Sea, and Air. American Society of Mechanical Engineers, 86076: V08BT27A025 (2022) 13. Hong, J., Xu, X., Su, Z., et al.: Joint stiffness loss and vibration characteristics of high-speed rotor. J. Beijing Univ. Aeronaut. Astronaut. 45(1), 21–28 (2019) 14. Li, J., Ma, Y., Hong, J.: Research on dynamic design method of rotor system gear structure. Aeroengine 35(4), 36–39 (2009) 15. Lei, B.-l., Li, C., Jie, H., Ma, Y.: Robust design of mechanical characteristics of rotor connection structure. Aeroengine 47(2), 3844 (2021) 16. Xueqi, C.: Evolution of joint-interface damage and its effects on rotor dynamics. BUAA, Beijing (2021) 17. Sosnovskiy, L.A.: Wear-Fatigue Damage and Its Prediction. China University of Mining and Technology Press (2013) 18. Socie, D.F.: Multiaxial fatigue damage models. J. Eng. Mater. Technol. 109(4), 293–298 (1987) 19. Hasenkamp, T., Arvidsson, M., Gremyr, I.: A review of practices for robust design methodology. J. Eng. Des. 20(6), 645–657 (2009) 20. Liu, S., Ma, Y., Zhang, D., et al.: Studies on dynamic characteristics of the joint in the aero-engine rotor system. Mech. Syst. Signal Process. 29(5), 120–136 (2012) 21. Fatemi, A., Socie, D.F.: A critical plane approach to multiaxial fatigue damage including out of phase loading. Fatigue Fract. Eng. Mater. Struct. 11(3), 149–165 (1988) 22. Gaul, L., Lenz, J.: Nonlinear dynamics of structures assembled by bolted joints. Acta Mech. 125, 169–181 (1997) 23. Iwan, W.D.: A distributed-element model for hysteresis and its steady-state dynamic response (1966). https://doi.org/10.1115/1.3625199 24. Iwan, W.D.: On a class of models for the yielding behavior of continuous and composite systems. J. Appl. Mech. 34(3), 612–617 (1967) 25. Zhao, B., Wu, F., Sun, K., et al.: Study on tangential stiffness nonlinear softening of bolted joint in friction-sliding process. Tribol. Intl. 156, 106856 (2021) 26. Li, L., Guo, C., Yu, P., et al.: Nonlinear stiffness mechanism analysis and numerical simulation of rabbet-bolted connection structure. J. Aerospace Power 36(2), 358–368 (2021)

Investigation on the Transient Lateral Vibration of a Flexible Rotor System with Substantial Unbalance Pingchao Yu1(B) , Zihan Jiang1 , Cun Wang2 , and Li Hou1 1 College of Civil Aviation, Nanjing University of Aeronautics and Astronautics,

Nanjing 211106, People’s Republic of China [email protected] 2 Beijing Institute of Power Machinery, Beijing 100074, People’s Republic of China

Abstract. Substantial unbalance is a possible condition usually induced by large blade off or bird strike during operation of aero-engine, and related dynamics are very crucial to the safety design of aero-engine. The main purpose of this paper is to get insight into the nonlinear transient lateral vibration of a flexible rotor with considering the nonlinearity of substantial unbalance. A new nonlinear rotor dynamic model with substantial unbalance is first built based on Lagrange method and finite element method. Then the effects of substantial unbalance on rotor’s modal and stability characteristics are analyzed. Finally, the transient vibration responses during rotor decelerating process are discussed. The results show that there exist multiple modal coupling regions. In those regions, some vibration modes are instable, which may lead to the divergence of steady-state response or lead to several new resonance peaks. The rotation speed variation has a significant effect on rotor’s transient response. Obvious amplitude oscillation phenomenon can be observed, and it becomes more intense with the increase of decelerating time. Improving system damping can effectively reduce the nonlinear effect caused by substantial unbalance. Keywords: Transient lateral vibration · flexible rotor · substantial unbalance · speed variation

1 Introduction In gas turbine engines, substantial unbalance is a possibility during the operation period, and is usually induced by fan blade out, bird strike, etc. [1, 2]. Substantial unbalance may lead to large unbalance force and other nonlinear effects on the spinning rotor. As a result, several secondary structural failures such as bearing damage or vibration instability occur. How to maintain the safety of the rotor system becomes a major concern in aero-engine design [3]. Therefore, more attention should be paid to the dynamics of the substantial unbalance rotor system.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 224–242, 2024. https://doi.org/10.1007/978-3-031-40459-7_15

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As one of the most common troubles in rotating machinery, unbalance problem has been studied since the birth of the rotordynamic [4]. Traditional analysis mainly investigated the behavior of lateral vibration of the unbalance rotor system. It is well known that unbalance will produce the synchronous excitation whose amplitude is proportional to the square of the rotational speed, thus it usually causes the synchronous lateral vibration [5]. In addition, the unbalance is the basic vibration source and can lead to many other secondary faults such as rub-impact, misalignment, and etc. The vibration characteristics of the rotor system under those coupling excitations have also been researched extensively. For example, Patel [6] built an unbalance-rubbing-crack coupling dynamic model of rotor system and studied the nonlinear dynamic behaviors. Yang investigated the dynamic characteristics of rotor system with unbalance, pedestal looseness and rub-impact [7]. Generally, only unbalance forces are considered in traditional dynamic analysis. Under this case, it is a kind of linear system. However, in the case of substantial unbalance, some nonlinear factors may be excited. Over the past several decades, the nonlinear lateral-torsional coupled dynamics of the unbalance rotor have attracted the attention of many scholars. Bernasconi [8] established the equations of motion of a continuous shaft and analysed the bisynchronous torsional vibrations of rotating shafts. Applying Lagrangian dynamics, Mohiuddin [9] and Al-Bedoor [10] developed models for the coupled torsional and lateral vibrations of an unbalanced rotor. Hong [11] studied the bending-torsional modal characteristics and found the instability phenomena. Recently, Li [12] analysed the bending-torsional vibration responses of a rotor-bearing-coupling system based on experiment and numerical simulation. In fact, when there is a substantial unbalance in the disk, the moment of inertia will change and inertia asymmetry appears [13]. Besides, the angular and translation motions of the unbalance disk can generate some additional dynamic terms [14]. However, those aspects are not included in the existing dynamic modeling. Furthermore, the rotation speed must be slow down in actual aero-engine for the purpose of flight safety once substantial unbalance suddenly happens. At that time, the variation of rotation speed may affect rotordynamics significantly, but most studies mainly focus on the condition of constant rotation speed. This paper aims to understand the nonlinear transient vibration of a flexible rotor system with substantial unbalance. The lateral vibration responses are the focus of the paper; thus, the torsional vibration is not contained in the analysis. First, the nonlinear dynamic model of the substantial unbalance model is derived based on Lagrange method. Based on that, the dynamic equations of the flexible rotor are established by introducing above nonlinear model into the rotor finite element model. Then the modal and instability characteristics of the unbalanced rotor are discussed. Finally, the transient lateral vibrations are investigated in detail.

2 Dynamic Modeling of a Flexible Rotor with Substantial Unbalance In this paper, a simplified flexible rotor system is built to study the dynamic characteristics under the substantial unbalance condition, as shown in Fig. 1. The flexible rotor is similar to the low-pressure rotor of aero-engine in structure and dynamics. It has three supports

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and two disks. The cantilever disk1 represents fan part while disk2 represents turbine part. The parameters of the flexible rotor system are presented in Table 1. It should be noted that parameters of disk1 listed below are given without considering the effect of substantial unbalance.

l l1

l2

l3

D 2 d2

D1 d1

l4

D3 d3

D4 d4

md2 Jp2 Jd2

E ρ μ

md1 Jp1 Jd1

K

(1) b

C

Support 1 Disk 1

(1) b

K

(2) b

C

(2) b

x

Support 2 ψ y

Disk2

φ

ω

K b(3)

Cb(3)

Support 3

z

Fig. 1. Schematic diagram of a flexible rotor system

As can be seen in Fig. 2(a), it is symmetric in orthogonal direction for the ideal disk without unbalance. The diameter moments of inertia in x direction and y direction are same, i.e. Jdx = Jdy = Jp /2, where Jp is polar moment of inertia. In some cases, such as blade out, a substantial unbalance is generated, then the diameter moments of inertia are not same, as seen in Eq. (1). Jdx = Jp /2, Jdy = Jp /2 − mb rb2

(1)

where mb and rb are the mass and mass center of blade respectively, as shown in Fig. 2(b). Obviously, an inertial asymmetry is induced for the substantial balance disk. In general, only unbalance force is considered in dynamic analysis of the unbalanced rotor. This simplification is reasonable in small unbalance condition. However, if the unbalance is large, then not only the unbalance force but also the change of inertia caused by mass eccentric should be taken account. In following parts, it is assumed that a substantial unbalance exists in disk1 and corresponding dynamic modeling will be performed. 2.1 Finite Element Modeling of Rotor System The dynamic model of above flexible rotor system is built in ANSYS software, as shown in Fig. 3. The shaft is modeled by beam element named BEAM188. This beam element is based on Timoshenko beam theory. Each beam element has 2 nodes and 8 degree of freedoms (Dofs). The mass matrix and stiffness matrix of Timoshenko beam element

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Table 1. Parameters of the flexible rotor system Component

Parameter

Symbol

Value

Disk1 without unbalance

Mass (kg)

m1

120

Polar moment of inertia (kg·m2 )

J p1

15

Diameter moment of inertia (kg·m2 )

J d1

7.5

Mass (kg)

m2

80

Polar moment of inertia (kg·m2 )

Jp2

6

Diameter moment of inertia (kg·m2 )

Jd2

3

Shaft (mm)

l, l 1 , l 2 , l 3 , l 4

2000, 300, 300, 1200, 200

Outer diameter (mm)

D1 , D2 , D3 , D4

160, 160, 80, 80

Inner diameter (mm)

d1, d2, d3, d4

140, 140, 70, 70

Disk2

Shaft

Material

Support

Density (kg/m3 )

ρ

7800

Young’s modulus (GPa)

E

210

Poisson’s ratio

µ

0.3

Support stiffness (N/m)

k b1 , k b2 , k b3

5e7, 5e7, 5e7

Support damping (N m/s)

cb1 , cb2 , cb3

0,0,0

have been described in literatures [15, 16]. The support is simulated by spring element named COMBIN14, and the corresponding stiffness and damping are set in real data of COMBIN14. The disk2 is modeled by mass element named MASS21, and the mass and moment of inertia are also set in real data. The detail information for spring element and mass element can refer to literatures [17, 18]. The disk1 is not modeled in the beam finite element model, because the traditional mass element cannot be applied for this substantial unbalance disk. The corresponding dynamic equation will be derived in next subsection. There are 24 nodes and 84 Dofs for the final dynamic model, including 20 beam elements, 1 mass element and 3 spring elements. After the finite element model has been built, the dynamic matrix can then be exported through HBMAT command, and the governing equations of the rotor system without disk1 can be obtained, as follows: Mr q¨ + (Cr + Gr )q˙ + Kr q = 0

(2)

where Mr , Cr , Gr , Kr and q are the mass matrix, damping matrix, gyroscopic matrix, stiffness matrix and displacement vector, respectively. Gr = ωGr0 , where ω is the rotation speed of rotor and Gr0 is the gyroscopic matrix under the unit rotation speed. Rayleigh damping Cr = αMr + βKr is applied to form corresponding damping matrix, where α and β are coefficients.

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M, Jp, Jdx, Jdy

(a)

y o

x

(b)

y

e o

x

rb mb Lost blade Fig. 2. The disk (a) without unbalance, and (b) with unbalance

Beam element

Mass element

Spring element

Fig. 3. Beam finite element model of flexible rotor

2.2 Model of Disk with Substantial Unbalance Figure 4 presents an unbalanced disk with 4 lateral Dofs denoted as x, y, θ x , θ y . θ is the rotation angle about axis z, thus θ˙ and θ¨ are the rotation speed and rotation acceleration,

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respectively. In this paper, the variation of rotation speed will be considered, i.e., θ¨ = 0. In addition, Jdx = Jdy caused by substantial will also be included. Above two points lead to the big difference between the dynamic equation of the unbalanced disk with that of the traditional rigid disk. Lagrange method will be used to derive its dynamic equation.

x x

Jdx

Mass center geometric center

O y

z

Mass center

O

Jdy

m

y

Fig. 4. The Dofs and structure parameters of unbalanced disk

(1) Rotational kinetic energy The second kind of Euler angle is used to represent the rotation of the unbalanced disk, as shown in Fig. 5. Coordinate system O’x0 y0 z0 , whose center is fixed with disk center O’, is parallel with global fixed coordinate system Oxyz (not marked in Fig. 5). O’x2 y2 z2 is a moving coordinate system fixed with disk, where O’x 2 is along the line between the mass centroid and the geometric center, and O’y2 is perpendicular to O’x 2 in the disk plane. The rotating motion of disk is equivalent to the moving process of O’x0 y0 z0 to O’x2 y2 z2 through Euler angle α, β and ϕ. Thus, the angular velocity components in O’x2 y2 z2 are as follows: ⎧ ˙ ⎪ ⎨ ωx2 = α˙ cos β sin ϕ + β cos ϕ (3) ωy2 = α˙ cos β cos ϕ − β˙ sin ϕ ⎪ ⎩ ωz2 = −α˙ sin β + ϕ˙ Then the rotational kinetic energy of the unbalanced disk is: Tr =

1 (Jdx ωx22 + Jdy ωy22 + Jp ωz22 ) 2

(4)

As mentioned above, Jdx = Jdy for the unbalanced disk. Then three parameters named average diameter moment of inertia J , inertia asymmetry J and inertial asymmetry coefficient  are introduced, and they are defined as: J =

  J 1 1 Jdx + Jdy , J = Jdx − Jdy ,  = 2 2 J

(5)

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x1

x0

x0

x1

x2 z2 z1

O

z2

ϕ α

x2

Mass center

z0

O

β

y0

y0

β α

z1 z0

ϕ

y1

y1

y2

y2 (a)

(b)

Fig. 5. (a) The second kind of Euler angle, (b) rotation of unbalanced disk

where  is a non-dimensional parameter and can reflect the asymmetry degree of different disks. Substituting Eqs. (5) and (3) into Eq. (4), one can obtain: ⎫ ⎧ ˙ 2 + α˙ 2 ) − 2J α˙ β˙ sin 2ϕ ⎬ ⎨ J ( β 1

Tr = (6) ˙ − αβ) ⎭ 2 ⎩ −J (β˙ 2 − α˙ 2 ) cos 2ϕ + 2J ϕ˙ 2 + ϕ( ˙ βα ˙ Due to the fact that lateral vibration of the disk is a small quantity, there exist following expressions: α˙ = θ˙y , β˙ ≈ θ˙ , ϕ˙ = θ˙ = ω

(7)

The rotational kinetic energy can be changed in following form by substituting Eq. (7) into Eq. (6): ⎧ ⎫ ˙ ˙ ˙2 ˙2 − J (θ˙x2 − θ˙y2 )cos2θ ⎬ 1 ⎨ J (θx + θy ) − 2J θx θy sin2θ

Tr = (8) ⎭ 2 ⎩ +2J θ˙ 2 + θ˙ (θ˙x θy − θ˙y θx ) (2) Translational kinetic energy Based on composition principle of velocities, the velocity of disk mass center V in the fixed coordinate system Oxyz can be expressed as: ˜ V = VO + V

(9)

where VO is the velocity of the disk geometric center, i.e. convected velocity. VO =  T x˙ y˙ 0 , among which x˙ and y˙ is the velocity components in x and y direction of the fixed coordinate system Oxyz. The velocity component in z direction is regarded

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˜ is the as zero because the disk axial vibration is not considered in this study. V velocity of the disk mass center in the moving coordinate system O’x2 y2 z2 , i.e. ˜ is generated by the disk rotational motion. relative velocity. The relative velocity V Its expression is shown as: ⎞ −θ˙ sin θ ˜ = e⎝ ⎠ V θ˙ cos θ θy θ˙ sin θ − θ˙y cos θ + θ˙x sin θ + θx θ˙ cos θ ⎛

(10)

where e is the distance between disk mass center and geometric center, i.e., eccentricity. In traditional analysis, the axial velocity component in Eq. (10) is not considered ˜ the translational kinetic because e is very small. Based on expressions of VO and V, energy of the unbalanced disk is obtained: Tt =

1 1 m(˙x − eθ˙ sin θ )2 + m(˙y + eθ˙ cos θ )2 2 2

1 ˙ y sin θ + θ˙ θx cos θ + θ˙x sin θ − θ˙y cos θ )e]2 + m[(θθ 2

(11)

Then Lagrange method is used to derive the dynamic equation, as shown in Eq. (12):    T d ∂Td ∂Td − = 0, qd = x, y, θx , θy (12) dt ∂ q˙ d ∂qd where qd is the generalized displacement and Td = Tt + Tr is the kinetic energy of the unbalanced disk. Substituting expressions of kinetic energy of the unbalanced disk into Eq. (12), the dynamic equation is derived as:     Md + Mdc cos 2θ + Mds sin 2θ q¨ d + Gd + Gdc cos 2θ + Gds sin 2θ q˙ d        + Kd θ˙ + Kd θ¨ + Kdcθ˙ + Kdcθ¨ cos 2θ + Kdsθ˙ + Kdsθ¨ sin 2θ qd = f(t) (13)   where Md , Gd and Kd θ˙ + Kd θ¨ are the constant parts of the mass matrix, gyro  scopic matrix and stiffness matrix, respectively. Mdc , Mds , Gdc , Gds , Kdcθ˙ + Kdcθ¨   and Kdsθ˙ + Kdsθ¨ are the coefficients of the time-varying parts of the mass matrix, gyroscopic matrix and stiffness matrix, respectively. f(t) is the excitation vector. Under constant rotation speed, θ = ωt. The expressions of above dynamic matrix

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are as: ⎡ ⎢ Md = ⎢ ⎣

⎤

m m 1 2 2 me

+J 1 2 2 me

⎡

Mds

0 ⎢ 0 =⎢ ⎣ ⎡

+J

⎢ 0 Gd = θ˙ ⎢ ⎣ ⎡

0

⎢ 0 Gdc = θ˙ ⎢ ⎣ ⎡

0

⎤

0

⎥ ⎢ ⎥, Mdc = ⎢ 0 ⎦ ⎣ − 21 me2 −J

0 −J − −J − 21 me2 0

0

⎡

⎤

⎥ ⎥, ⎦ 1 2 2 me

+ J

⎥ ⎥

1 2 ⎦, 2 me

⎤ ⎥ ⎥

, + me2 ⎦

 2J  0 0 − 2J + me2

⎤

⎥ ⎥  2 ⎦, 0 − 2J + me   0 − 2J + me2 ⎤

⎡

0

⎤

⎥ ⎢ ⎥ ⎢ 0 ⎥, K ˙ = θ˙ 2 ⎢ 0 ⎥, Gds = θ˙ ⎢ 1 dθ 2 ⎦ ⎣ ⎦ ⎣ − 0 me 2J + me2  2  1 2 2 − 2 me 0 − 2J + me ⎤ ⎡ ⎤ ⎡ 0 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎥, ˙2⎢ 0 1 Kd θ¨ = θ¨ ⎢ 1 2 + J ⎦, Kdcθ˙ = θ ⎣ 2 ⎦ ⎣ me me 0 2 2  1 2 1 2 0 − 2 me − 2 me + J ⎤ ⎤ ⎡ ⎡ 0 0 ⎥ ⎥ ⎢ ⎢ 0 ⎥, K ¨ = θ¨ ⎢ 0 ⎥, Kdsθ˙ = θ˙ 2 ⎢ 1 1 dc θ 2 2 ⎣ ⎣ me ⎦ 0 0 − me ⎦ ⎡

Kdsθ¨

1 2 2 me

0 ⎢ 0 = θ¨ ⎢ 1 2 ⎣ 2 me

2

0

⎤ ⎥ ⎥, ⎦

− 21 me2 ⎤ ⎡ ˙2 meθ cos θ ⎢ meθ˙ 2 sin θ ⎥ ⎥. f(t) = ⎢ ⎦ ⎣

2

0

− 21 me2

Different with the traditional disk, the inertial asymmetry due to the substantial unbalance is included. Meantime, an axial velocity component is also considered when the inclination motion occurs for the unbalanced disk. As a result, the additional terms −2J θ˙x θ˙y sin 2θ − J (θ˙x2 − θ˙y2 ) cos 2θ and ˙ y sin θ + θθ ˙ x cos θ + θ˙x sin θ − θ˙y cos θ )e]2 /2 appear in the rotational kinetic m[(θθ energy and the translational kinetic energy respectively. Those additional terms lead

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to the parametric terms in equation of motion Eq. (13). To summary, the mass eccentricity and the resulting inertial asymmetry are the physical reason of the parametric terms in the equation of motion. It should be noted that the time-varying terms related to Mdc , Mds , Gdc , Gds are similar to that of the asymmetric rotor, but the value in the matrix must be corrected by me2 . In addition, the unbalance leads to the additional stiffness terms. Among them, Kd θ˙ , Kdcθ˙ , and Kdsθ˙ are all related to rotation speed ¨ θ˙ , while Kd θ¨ , Kdcθ¨ and Kdsθ¨ are all related to angle acceleration θ. 2.3 Governing Equation of Whole Rotor System Based on the model established in Sects. 2.1 and 2.2, the dynamic governing equation for the whole system is given as follows: (Mr + Md + Mdc cos 2θ + Mds sin 2θ)q¨ + (Cr + Gr + Gd + Gdc cos 2θ + Gds sin 2θ)q˙         + Kr + Kd θ˙ + Kd θ¨ + Kdcθ˙ + Kdcθ¨ cos 2θ + Kdsθ˙ + Kdsθ¨ sin 2θ q = f(t)

(14)

The dynamic matrixes Mr , Cr , Gr and Kr correspond to the rotor system without substantial unbalanced disk1, and they have been described in Sect.    2.1. The dynamic matrixes Md , Mdc , Mds , Gd , Gdc , Gds , Kd θ˙ + Kd θ¨ , Kdcθ˙ + Kdcθ¨ and Kdsθ˙ + Kdsθ¨ correspond to the unbalanced disk1. Those dynamic matrixes are all expanded to 84 × 84 dimension in order to be added with dynamic matrixes Mr , Cr , Gr and Kr . Since the unbalanced disk is located at node 1, only the elements corresponding to node 1 are not zero, while other elements are all zero. The dynamic model built in this paper not only precisely considers the influence of substantial unbalance, but also includes the effect of the variation of rotation speed, which is a development on the existing model.

3 Mode and Stability Analysis of the Flexible Rotor System The substantial unbalance condition is generally caused by Fan blade out (FBO). For a typical high bypass ratio turbofan engine, the unbalance produced by the loss of the fan blade can reach 1–7 kg·m, and the corresponding asymmetry coefficient ranges from 0.04 to 0.3 [19]. In the subsequent analysis, we assumed that the diks1 has the substantial unbalance. The corresponding unbalance parameters are chosen referring to the fan structure with FBO in actual aero-engine, and their values are listed in Table 2. Table 2. The unbalance parameters of disk1 No

Diameter moment of inertia (kg·m2 ) Jdx1

Inertia asymmetry 

Eccentricity e (mm)

Jdy1

1

9

6

0.15

30

2

10.5

4.5

0.3

60

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Firstly, the modes and stability of substantial unbalance rotor are analyzed. In modal analysis, the rotor speed is constant. Therefore, θ = ωt, Kd θ¨ , Kdcθ¨ and Kdsθ¨ terms are all zero. In this case, Eq. (14) degenerates into a parametric vibration equation with time-varying frequency being 2ω, as shown below: (Mr + Md + Mdc cos 2ωt + Mds sin 2ωt)q¨ + (Cr + Gr + Gd + Gdc cos 2ωt + Gds sin 2ωt)q˙   (15) + Kr + Kd θ˙ + Kdcθ˙ cos 2ωt + Kdsθ˙ sin 2ωt q = f(t)

Without considering external excitation f(t) and damping Cr , the system mode can be obtained by the Hill method [20, 21]. Figure 6 shows the modal frequency and stability results when e = 60 mm. Because of the time-varying features of the dynamic matrix, there contain both the fundamental frequency and the harmonic modal frequency for any order modes. For example, ωn1+,0 is the fundamental frequency (subscript 0) of 1st order forward whirl mode (subscript n1+), and ωn2+,−1 is the negative 1st order harmonic frequency (subscript −1) of the 2nd order forward whirl mode (subscript n2+). The fundamental frequency corresponds to the modal frequency of the traditional linear rotor (without time-varying dynamic matrix). The harmonic frequency is determined by both the fundamental frequency and the time-varying frequency. Figure 6 (a) shows the modal coupling phenomenon between fundamental frequency and harmonic frequency occurring in the substantial unbalanced rotor. The modal coupling means that two modal frequency curves coincide with each other in a certain rotation speed region. One can find three regions. The first is [2910, 3210] rpm, where modal coupling occurs between ωn1+,0 and ωn1+,−1 . This region coincides with the 1st order critical speed of rotor system. The second is near 4320 rpm, where modal coupling occurs between ωn2+,0 and ωn2+,−1 . This region coincides with 2nd order critical speed of rotor system. The third region is [3690, 3780] rpm, where modal coupling occurs between ωn2+,0 and ωn1+,−1 as well as ωn1+,0 and ωn2+,−1 . Figure 6 (b) shows the real part of eigenvalue. In above three modal coupling regions, the real parts of the rotor eigenvalue are all greater than 0. As a result, the rotor modal response will diverge gradually over time, leading to the instability of the system dynamics. Therefore, we can infer that modal coupling will lead to system’s instability. In nature, the instable phenomenon of the substantial unbalanced rotor system is induced by the time-varying parametric terms in mass, gyroscopic, stiffness matrices. The modes and instability of the time-varying rotor system have been studied in many publications. More details can be found in those researches [22–24]. Furthermore, Newmark method is used to solve Eq. (15) to obtain the steady-state vibration of the rotor at constant rotation speed. In the calculation, the eccentricity e = 60 mm and the damping ratio is 0.01. The damping ratio is coveted into the Rayleigh damping factor based on the process described in literature [25], and then it is applied into dynamic equation. In order to intuitively compare vibration amplitudes under different eccentricities, normalized displacement x = x/e is introduced in the subsequent analysis, where x is the vibration displacement actually calculated. Figure 7 shows the rotor amplitude-speed curves, where “nonlinear” means that the time variation of each dynamic matrix due to substantial unbalance is considered; “linear” means that only the unbalance force f(t) is considered and it will be regarded as a traditional unbalanced rotor. The results show that the linear rotor has two peak

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ωn2+,0

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5 0 -5

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-20

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Fig. 6. (a) Modal frequency, (b) modal stability of the unbalanced rotor under constant rotation speed

rotation speeds of 3070 rpm and 4330 rpm, which correspond to the 1st and 2nd order critical speeds of the linear rotor. When considering the time-varying effect caused by substantial unbalance, it can be seen that the nonlinear rotor becomes unstable in the 1st order resonance region, and a new peak rotation speed appears at 3730 rpm. These two rotation speed regions just correspond to the instability region in Fig. 6. At 4330 rpm, the amplitude of the nonlinear rotor is slightly larger than the linear rotor, but the whole rotor is stable. In general, the modal coupling and instability will lead to more complicated response characteristics of rotor system: If the modal coupling is serious, the vibration response usually diverges; If the modal coupling is relatively weak, the vibration amplitude may increase which induces a new resonance peak point. Figure 8 shows the rotor amplitude-speed curves with different damping ratios. It can be seen that with the increase of system damping, the vibration response at the 1st order resonance speed tends to converge, while the resonance peak of 3730 rpm disappears. Therefore, we can conclude that the rotor’s stability can be effectively improved by increasing system’s damping.

4 Transient Responses of the Flexible Rotor System Generally, once FBO occurs, the substantial unbalanced rotor will quickly run down, and finally operate at the low-speed windmilling condition. Therefore, this section focuses on analyzing the vibration response of the substantial unbalanced rotor during the process of deceleration. By comparing Eqs. (14) and (15), one can find that when the rotation speed is changing, the dynamic equations are no longer a periodic time-varying system. Meantime, an additional dynamic matrix terms will be generated. Consequently, the vibration response will be different from that under the constant rotation speed. 4.1 Linear Case Figure 9 shows three kinds of decelerating curves. It assumed that the rotor evenly decelerates from 8000 rpm to 1000 rpm, and the deceleration time td is set as 2 s, 4 s and

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10

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4

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Rotatioin speed (rpm) Fig. 7. Steady-state vibration amplitude of the unbalanced rotor system 6

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Rotation speed (rpm) Fig. 8. Effect of rotor damping on the stability of the unbalanced rotor system

8 s respectively. First of all, without considering the time variation of dynamic matrix, the transient response of linear rotor system during decelerating process is calculated by Newmark method, as shown in Fig. 10. Figure 10(b) is the comparison of the envelope for the time responses in Fig. 10(a) under different td . For the purpose of comparison, the steady-state response results are also given in Fig. 10(b). It can be seen that under the decelerating condition, rotor resonance speed and the corresponding peak value are slightly lower than those in the steady-state condition. Amplitude oscillation occurs when rotor passes through the critical speed. The shorter the decelerating time is, the more obvious this phenomenon is.

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Rotation speed (rpm)

8000

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td=2s t =4s d

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td=8s Curve2

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Curve 3 6

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10

Time (s)

Decelerating

Normalized amplitude

(a)

8

6

td=2s

td=4s

td=8s

Rotation speed(rpm)

(b)

Normalized amplitude

Fig. 9. Three kind of decelerating curves used in calculation

Steady state td=2s

Decelerating

td=4s td=8s

4

2

0 8000

7000

6000

5000

4000

3000

Rotation speed (rpm)

2000

1000

Fig. 10. The vibration response in time-domain for the linear unbalanced rotor system

The time-frequency results of the time-domain responses for td = 2 and 8 s are obtained through the wavelet transform, as shown in Fig. 11. One can see that at the resonance speed, the modal frequency component is obvious. If the deceleration time is long enough (td = 8 s), the frequency component decays quickly, and the rotor response is mainly dominated by the rotation frequency. However, when the rotor decelerates rapidly (td = 2 s), the modal frequency component is not fully attenuated after the rotor passes through the resonance speed. At that time, both the modal frequency and rotation frequency exist in the rotor’s response. Therefore, the amplitude oscillation occurs when rotor just passes through the resonance speed is generated. Meanwhile, this is also the reason for the different transient responses of the rotor under different deceleration times.

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(a)

fn2+ Resonance

fn2+ fn1+

Rotation speed

Resonance

fn1+

Fig. 11. Time-frequency responses of the linear unbalanced rotor for (a) td = 2 s, (b) td = 8 s

4.2 Nonlinear Case Considering the time-varying of dynamic matrix brought by substantial unbalance, the nonlinear transient response during deceleration process is analyzed. Figures 12 and 13 show the time-domain vibration when e = 60 mm and e = 30 mm, and the damping ratio is 0.05 in the calculation. When e = 30 mm, the time-domain vibration response of the nonlinear rotor is similar to that of the linear rotor (presented in Sect. 4.1). The amplitudes and resonance speeds under different deceleration time are also close. One major difference between nonlinear case and linear case is that the responses are more complex before the rotor passes through the 2nd resonance speed. One can see that the nonlinear rotor presents an extremely significant amplitude oscillation phenomenon at that time. When e = 60 mm, the transient response and amplitude oscillation of the substantial unbalanced rotor become more complicated. When the rotor passes through two resonance regions, the rotor vibration level is very high, and no obvious resonance point can be observed. At the same time, the longer the deceleration time is, the higher the rotor amplitude is, and the rotor’s transient responses under different deceleration time have obvious difference. Moreover, the vibration amplitude of the substantial unbalance rotor when passing through the instable regions is not magnified infinitely. The reason may be that the vibration energy has not been fully accumulated during the deceleration condition. Figure 14 shows the time-frequency results obtained by the wavelet transform of time-domain responses. When e = 30 mm, the frequency-domain response of the rotor in the process of deceleration is still dominated by the rotation frequency, and its amplitude reaches the maximum at the 2nd order resonance point. However, it should be noted that when the rotor is close to the 2nd order resonance point, the non-negligible backward whirl modal frequency components fn1− and fn2− appear in the frequency-domain response. These two frequency components last for a certain time before they decay. Obviously, the appearance of these two frequency components is the main reason for the aforementioned rotor amplitude oscillation phenomenon. In addition, one can see that the amplitude of above modal frequency components is significantly lower than that of the rotation frequency, indicating that the transient vibration response of the rotor is still dominated by the unbalanced force at that time. Therefore, the nonlinear transient

Investigation on the Transient Lateral Vibration of a Flexible Rotor System Decelerating

Normalized amplitude

td=2s

td=4s

td=8s

td=2s

Decelerating

(b)

Normalized amplitude

(a)

239

td=4s

td=8s

Rotation speed(rpm)

Rotation speed(rpm)

Fig. 12. Time response of the nonlinear unbalanced rotor for (a) e = 60 mm, (b) e = 30 mm 300

Normalized amplitude

250 200 150 100

(b)

Instability region for steady-state case

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(a)

td=2s td=4s td=8s Steady state

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td=2s td=4s td=8s Steady state

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3000

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2000

1000

0 8000

7000

6000

5000

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3000

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Fig. 13. Envelope of the time-domain response in Fig. 12, (a) e = 60 mm, (b) e = 30mm

response for e = 30 mm is generally similar to the result of the linear rotor. Moreover, it can be found that compared with linear rotors, the backward whirl modal frequency components of substantial unbalanced rotors are more likely to be excited during the decelerating process. When e = 70 mm, before and after the rotor passes through the two-order resonance speeds, the frequency components f n1− and f n2− are more obvious. Meantime, the corresponding amplitude and the lasting region are longer. More importantly, the amplitude of those frequency components is basically equivalent to the rotation frequency amplitude, resulting in a significant increase of rotor amplitude and more complex vibration characteristics. By comparing the response results of td = 2 s and 8 s, one can find that the with the increase of deceleration time, the amplitude of the backward whirl modal frequency when passing through the resonance region becomes larger. At that time, the backward whirl modes have a significant effect on rotor response. Figure 15 shows the nonlinear transient responses under different damping ratios. In the calculation, e = 60 mm. It can be seen that when the damping ratio is 0.01, the rotor amplitude is higher than that when the damping ratio is 0.05. Obvious instability occurs when rotor passes through the critical speed. For example, the rotor amplitude

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(a)

(c)

(b)

fn2+:70Hz

fn2+ fn1+

fn1+ fn2-:39~44Hz

fn2-:37Hz

fn1-:24~29Hz

fn1-:24Hz

4200rpm fn1+ 3000rpm

fn2fn1-

Fig. 14. Time-frequency responses when (a) e = 30 mm, td = 2 s, (b) e = 60 mm, td = 2s, (c) e = 60 mm, td = 8 s

reaches 105 when td = 4 s. With the increase of rotor damping, rotor amplitude decreases and vibration response tends to be stable. When the damping ratio is 0.1, the response characteristics are close to those in Fig. 12(b), i.e., there are two obvious resonance points during decelerating process, and the resonance speed and resonance amplitude are close to linear rotor. Obviously, increasing the system’s damping is beneficial to improve the stability of transient response. td=2s

Decelerating

(b)

Decelerating

Normalized amplitude

Normalized amplitude

(a)

td=4s

td=8s

Rotation speed(rpm)

td=2s

td=4s

td=8s

Rotation speed(rpm)

Fig. 15. Influence of damping on response of nonlinear unbalanced rotor, (a) damping ratio 0.01, (b) damping ratio 0.1

5 Conclusions The main subject of this paper is to understand the effect of substantial unbalance on nonlinear lateral transient dynamics of a flexible rotor system. Based on Lagrange method and finite element method, a new nonlinear rotor dynamic model with the substantial unbalance is built. The model shows that the substantial unbalance not only generates traditional unbalance force, but also leads to the time-varying of dynamic matrix and introduces additional stiffness terms. As a result, some new nonlinear phenomena with regard to modes and vibration responses appear in the substantial unbalanced rotor. Through the modal analysis, it is found that substantial unbalance can cause the modal coupling and instability in some speed regions. Those instable regions mainly include

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two types: One is in the vicinity of resonance speeds just corresponding to the modal coupling of ωn1+,0 with ωn1+,−1 as well as ωn2+,0 with ωn2+,−1 ; Another corresponding to modal coupling regions of ωn2+,0 with ωn1+,−1 (or ωn1+,0 with ωn2+,−1 ). The former may lead to the divergence of steady-state response, while the latter generally causes new resonance peaks in steady-state response. The speed variation has a significant effect on nonlinear transient response. Obvious amplitude oscillation phenomenon can be observed before rotor passes through the 2nd resonance speeds during decelerating process. The longer the decelerating time is, the more intense the amplitude oscillation is. The main mechanism is that backward whirl modes are excited when the substantial unbalanced rotor decelerates. In addition, with the decrease of the eccentricity or the increase of damping, both the steady-state and transient responses tend to behave like the traditional linear unbalance rotor, in which the unbalance force dominates system’s response. Acknowledgements. The authors would like to acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 52005252, 52105130), Jiangsu Province Natural Science Foundation (Grant No. BK20211187), the Aeronautical Science Foundation of China (Grant No. 2020Z039052007) and the Postdoctoral Science Foundation Project (Grant No. 2022M711615).

References 1. Yu, P., Zhang, D., Ma, Y., et al.: Dynamic modeling and vibration characteristics analysis of the aero-engine dual-rotor system with fan blade out. Mech. Syst. Signal Process. 106, 158–175 (2018) 2. Sinha, S.K.: Rotordynamic analysis of asymmetric turbofan rotor due to fan blade-loss event with contact-impact rub loads. J. Sound Vib. 332(9), 2253–2283 (2013) 3. Von Groll, G., Ewins, D.J.: On the dynamics of windmilling in aero-engines. IMechE Conf. Trans. 6, 721–730 (2000) 4. Muszynska, A.: Rotordynamics. CRC press (2005) 5. Chen, X., Liao, M.: Transient characteristics of a dual-rotor system with intershaft bearing subjected to mass unbalance and base motions during start-up. In: Turbo Expo: Power for Land, Sea, and Air. American Society of Mechanical Engineers 51135, V07AT33A007 (2018) 6. Patel, T.H., Darpe, A.K.: Coupled bending-torsional vibration analysis of rotor with rub and crack. J. Sound Vib. 326(3–5), 740–752 (2009) 7. Yang, Y., Yang, Y., Cao, D., et al.: Response evaluation of imbalance-rub-pedestal looseness coupling fault on a geometrically nonlinear rotor system. Mech. Syst. Signal Process. 118, 423–442 (2019) 8. Bernasconi, O.: Bisynchronous torsional vibrations in rotating shafts. J. Appl. Mech. 54(4), 893–897 (1987) 9. Mohiuddin, M.A., Khulief, Y.A.: Coupled bending torsional vibration of rotors using finite element. J. Sound Vib. 223(2), 297–316 (1999) 10. Al-Bedoor, B.O.: Modeling the coupled torsional and lateral vibrations of unbalanced rotors. Comput. Methods Appl. Mech. Eng. 190(45), 5999–6008 (2001) 11. Jie, H., Pingchao, Y.U., Yanhong, M.A., et al.: Investigation on nonlinear lateral-torsional coupled vibration of a rotor system with substantial unbalance. Chin. J. Aeronaut. 33(6), 1642–1660 (2020)

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12. Li, Q., Chen, C., Gao, S., et al.: The coupled bending-torsional dynamic behavior in the rotating machinery: modeling, simulation and experiment validation. Mech. Syst. Signal Process. 178, 109306 (2022) 13. Ye, D., Xuan, H.J., Liu, L.L.: Research overview of full aero-engine dynamic response caused by blade-off. Appl. Mech. Mater. 423, 1552–1557 (2013) 14. Yuan, Z., Chu, F., Lin, Y.: External and internal coupling effects of rotor’s bending and torsional vibrations under unbalances. J. Sound Vib. 299(1–2), 339–347 (2007) 15. Editorial Board of Aero-engine Design Manual, Aero-engine Design Manual (19th Part): Rotor Dynamics and Whole Machine Vibration, pp. 208–226. Aerospace Industry Press, Beijing (2000) 16. Ehrich, F.F.: Handbook of Rotordynamics. McGraw-Hill, New York (1992) 17. Chen, G.: Vibration modelling and verifications for whole aero-engine. J. Sound Vib. 349, 163–176 (2015) 18. Ma, H., Yin, F., Wu, Z., et al.: Nonlinear vibration response analysis of a rotor-blade system with blade-tip rubbing. Nonlinear Dyn. 84, 1225–1258 (2016) 19. Hong, J., Ma, Y.: Structure and Design of Aircraft Gas Turbine Engine, Science Press (2021) 20. Jie, H., Pingchao, Y.U., Zhang, D., et al.: Modal characteristics analysis for a flexible rotor with non-smooth constraint due to intermittent rub-impact. Chin. J. Aeronaut. 31(3), 498–513 (2018) 21. Zheng, Z., Zhu, F., Zhang, D., et al.: A developed component mode synthesis for parametric response analysis of large-scale asymmetric rotor. J. Mech. Sci. Technol. 33, 995–1005 (2019) 22. Ishida, Y., Liu, J., Inoue, T., et al.: Vibrations of an asymmetrical shaft with gravity and nonlinear spring characteristics (isolated resonances and internal resonances). J. Vibr. Acoust. 130(4) (2008) 23. Filippi, M., Carrera, E.: Stability and transient analyses of asymmetric rotors on anisotropic supports. J. Sound Vib. 500, 116006 (2021) 24. Ishida, Y., Inoue, T.: Detection of a rotor crack using a harmonic excitation and nonlinear vibration analysis. J. Vib. Acoust. 128(6), 741–749 (2006) 25. Ma, H., Wu, Z., Tai, X., et al.: Dynamic characteristics analysis of a rotor system with two types of limiters. Int. J. Mech. Sci. 88, 192–201 (2014)

Dynamic Behaviors of a Bolted Joint Rotor System Considering the Contact State at Mating Interface Yuqi Li1,2,4(B) , Zhimin Zhu1 , Zhong Luo2 , Chuanmei Wen3 , Lei Li2 , and Long Jin1 1 School of Mechanical and Automotive Engineering, Guangxi University of Science and

Technology, Liuzhou 545006, People’s Republic of China [email protected] 2 School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, People’s Republic of China 3 School of Electronic Engineering, Guangxi University of Science and Technology, Liuzhou 545006, People’s Republic of China 4 Guangxi Earthmoving Machinery Collaborative Innovation Center, Guangxi University of Science and Technology, Liuzhou 545006, People’s Republic of China

Abstract. The bolted joint is widely used in the aero-engine rotor system, which introduced a larger number of contact interfaces. These bolted joints are subjected to complex vibrations during operation and cause changes in the contact state at the mating interface, leading to a time-varying mechanical characteristics. This study aims to investigate the effect of the contact state at the mating interface on the dynamic behavior of the rotor system. Details are as follows: (1) obtaining the hysteresis curve of the bolted joint using the finite element method to explore the contact behavior at the mating interface; (2) an Iwan-based analytical model of bolted joint considering the contact state is proposed; (3) an adaptive parameter identification method for the analytical model is developed by using the initial loading data; (4) the dynamic model of bolted joint rotor system considering the contact state at the mating interface is established by multiscale relevance between the local joint structure contact state and rotor system, then relevant numerical studies were carried out, and (5) the numerical simulation results are verified by established bolted rotor test rig. The studies would provide guidance for the design of the aero-engine rotor system. Keywords: Dynamic modeling · vibration analysis · bolted joint rotor · mating interface contact · hysteresis characteristic

1 Introduction In large rotating machinery, the bolted joint rotor systems are widely applied since different materials and assembly requirements of the various components, such as the aero-engine, turbomachinery, etc. [1–3]. As for the aero-engine shown in Fig. 1a, it would introduce several mating interfaces inevitably due to the bolted joint rotor system © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 243–256, 2024. https://doi.org/10.1007/978-3-031-40459-7_16

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used [4]. And it would cause the vibration response of joint rotor to exist distinguishable characteristics significantly with the integral rotor due to the frictional dissipation and contact state time-varying characteristic at the mating interface [5]. However, the existing studies mainly focused on the integral rotor vibration performance and bolted rotor system’s piecewise linear characteristics, the contact state at the mating interface is ignored, which leads to the real vibration characteristics of the bolted joint rotor fail to be revealed by numerical simulation results [6]. Based on the description of the above present situation, deriving a dynamical model of a bolted joint rotor system considering the contact state at the mating interface has become one of interest for investigations at present. In the past decades, abundant efforts have been attempted to inquire into the mechanical performance of the bolted jointed structure considering the contact state at mating interface from various viewpoints [7, 8]. Meanwhile, several nonlinear mechanics models are established, which be applied to analyze that mechanical performance of bolted joint structure, such as Iwan model, LuGre model, Valanis model and Bouc-Wen. For instance, Tan et al. [9] took a single bolt lap beam as an example, established a Jenkinsbased analytical model of bolted joint with threads, and then analyzed the effect of the non-parallel bearing surface and external force on connection performance. Liang et al. [10] proposed a modified analytical four-stage model used to explore the preload relaxation mechanism of the bolted joint structure under a static tension case. By using the virtual material model, Liu et al. [11] accurately simulated and analyzed the influence of the non-uniform distribution of pressure and variable stiffness of bolted joint on its vibration performance. Taking bolted flange connections in an aero-engine as the object, Mir-Haidari et al. [12] developed a lumped model of bolted flange joint, and parametric studies on the effect of different loads on its nonlinear dynamic behavior were carried out. Aiming at revealing the hysteretic behavior and contact stiffness of bolted joint mating interface under lateral vibration, Li et al. [13] established a multi-scale contact model of the bolted joint based on the Iwan model to explore the above characteristics in detail, and verified the model using the developed test rig which could measure the interface fretting response of bolted joints. In the past few years, research focused on the modeling method and dynamic analysis of bolted jointed rotor systems have in-depth even more [14–16]. To name a few examples, Yu et al. [17] presented a numerical model of bolted flange joint with a spigot adopting the Jenkins element considering the sticking and sliding statuses of spigot, and then the effect of damping introduced by bolted joint on the dynamic performance of joint rotor system by introducing the above numerical model into rotor system. Li et al. [18] combined the Lagrange modeling method to derive a numerical model of the rod-fastened rotor-bearing system considering the contact effect of the mating interface and investigated the bistable behaviour and hysteretic cycle of the rotor system, which the contact was be regarded as an addition damping. Wu et al. [19] established a rod-fastening rotor system considering the nonlinear contact stiffness between the disks in bending direction and further discussed the effect of several typical paraments on the dynamic response of rod-fastening rotor-bearing system, including the preload and friction coefficient of the mating interface. Li et al. [20], based on the mechanical relationship between the adjacent disks, developed a joint element with two nodes and

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explored the effect of dynamic joint parameters on the vibration performance of the rotor system. Based on the exploration of the above literature, despite the nonlinear mechanics model of bolted jointed structure has perfect enough, and the dynamic investigation of bolted joint rotor system has intensified enough, the effect of contact state at the mating interface for the dynamic characteristics of a bolted joint rotor system is rarely considered in the existing research. Thus, in the present work, a bolted joint structure is developed shown in Fig. 1b, and its nonlinear mechanical model considering the contact state at mating interface is derived according to its hysteresis characteristic. Furthermore, a parameter identification method is presented employing the initial loading data of the hysteresis curve, which is acquired by FE analysis. Based on that, a dynamic model of a bolted joint rotor system considering the contact state at the mating interface is established, then experimentally and numerically investigated the effect of preload on the dynamic performance of bolted rotor system. The observed results of the present work could further enhance the vibration performance prediction accuracy of the joint rotor. (a)

High pressure rotor

Low pressure rotor

(b)

Mating interface

Bearing

Bolted joint

Bearing

Fig. 1. Sketch of a certain type of aero-engine, (b) structure of the bolted joint established in this paper.

2 Nonlinear Mechanical Modeling of Bolted Joint Structure As shown in Fig. 2a, a FE model of the bolted joint structure is developed consisting of an upper disk, lower disk, and bolts, where all degree-of-free-doms (DOFs) of the lower disk are constrained, and the upper disk is subjected to lateral harmonic excitation. Additionally, its relative displacement and corresponding lateral harmonic excitation are acquired and plotted in one diagram, as shown in Fig. 2b. It can be observed that there exists significant hysteresis behavior in the force-displacement curve which is referred to hysteretic curve, and the area of hysteretic curve is equivalent the dissipation of frictional energy in one cycle of loading [21, 22]. That is owing to the variation of contact state at the mating interface further causes the stiffness softening behavior. According to the

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lateral force loading direction, the hysteretic curve is divided into three states, described as initial loading, reloading, and unloading [23]. Furthermore, as can be seen from Fig. 2c that the initial loading state consists of two linear parts and one nonlinear part, which is called stick state, micro slip state, and macro slip state, respectively, based on the contact state at the mating interface [24, 25]. Actually, the mating interface could be regarded as composed of infinite elastic contact points, as exhibited in Fig. 3a, and each contact point would generate a certain friction owing to the normal pressure provided by the preload. However, the maximum friction generated by each elastic contact point is different, attributed to the non-uniform distribution of normal pressure provided by preload. That causes the slip behavior of each elastic contact point of the mating interface does not occur synchronously, described as the contact behavior alternate changes process from stick state (i.e., no relative sliding occurred at all) to microslip state (i.e., part of contact positions occurred relative sliding) to macroslip state (i.e., the whole contact interface occurred relative sliding) under lateral harmonic excitation. According to the actual scenarios, a well-known Jenkins element is employed and form an Iwan model through parallel n Jenkins elements and a spring element as depicted in Fig. 3b, which could be used to describe that hysteresis characteristic. For a Jenkins element, it would enter into a slip state with its relative displacement ui exceed to the critical slip displacement u˜ i , , and its friction force would be a constant value (i.e., the critical slip force f˜i ). Therefore, a Jenkins element can be expressed as [26]  k˜ u , 0 < |ui | ≤ u˜ i Stick (1) fi (ui ) = i i ˜ Slip ±fi , u˜ i < |ui | where k˜i is the spring stiffness of Jenkins element, f i is the friction of the ith Jenkins element. (b) 10 Lateral force / kN

(c) 12

10 -2 Initial loading Reloading Unloading

5 0 -5

Lateral force / kN

(a)

10 -2 Stick Microslip Macroslip

8

4

-10 -3

-2 -1 0 1 2 Relative displacement / m

3 -6 10

0

0

0.5

1

1.5

2

Relative displacement / m

2.5 10 -6

Fig. 2. The FE model of the bolted joint, (b) force-displacement curve of the bolted joint under lateral harmonic excitation, (c) force-displacement curve of the initial loading state.

Dynamic Behaviors of a Bolted Joint Rotor System (a)

Jenkins element m

(b) F

247

k1

k2

F kn

ka

Fig. 3. (a) Sketch of the mating interface of the bolted joint under microcosmic view, (b) Sketch of the Iwan model of the bolted joint

Based on the above theoretical framework, the Iwan model can be described as follows by adding the friction of each element and the resilience of spring [7, 13]. ⎧ l N ⎪   ⎪ ⎪ ˜ ⎪ + f k˜i δ + ka δ Initial loading i ⎪ ⎪ ⎪ ⎪ i=1 i=l+1 ⎪ ⎪ ⎪ ⎪ l N  ⎨   k˜i ( − δ) + f˜i + ka δ Unloading f˜i + F(δ) = − (2) ⎪ ⎪ i=1 i=l+1 ⎪ ⎪ ⎪ ⎪ ⎪ l N  ⎪  ⎪ ⎪ ⎪ k˜i (δ + ) − f˜i + ka δ Reloading f˜i + ⎪ ⎩ i=1

i=l+1

where  is the maximum displacement at last loading state, k a is the stiffness of spring, δ is the relative displacement between upper and lower disk. In Eq. (2) the parameters associated with the Jenkins element, i.e. u˜ i , f˜i , k˜i and the stiffness of spring k a must be identified. Therefore, a discrete parameter identification method is proposed using the initial loading data of the hysteresis curve, and the detailed identification process is listed in Fig. 4. It should be mentioned that the contact behavior introduced by a bolted joint could be captured accurately using the Iwan model (i.e., Eq. (2)) and the above parameter identification method. Initial loading data A least square procedure Determine the stick and macroslip states

Calculate the slope of stick k1

Calculate the slope of macroslip kn+1

Discretize the microslip state into n-1 linear parts

Record the two endpoints abscissa of each part xi (i=1,2 ,n)

Calculate the slope of each part (k2-kn)

Fig. 4. Flowchart of the discrete parameter identification method.

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3 Dynamic Modelling of the Bolted Joint Rotor

Mating interface

A

B

Disk O1

Disk O2

Fig. 5. Sketch of the bolted joint rotor system.

In the present work, a dynamic model of bolted joint rotor considering the contact state combines the Lagrange modeling method and the above mechanical model of the bolted joint. Meanwhile, the schematic diagram of the bolted joint rotor is shown in Fig. 5, where the system is divided into four lumped points, described as rolling bearings mA , mB , and jointed disks mo1 , mo2 . Moreover, the shafts are defined as massless, and only the horizontal and vertical degrees of freedom are considered. To derive the dynamic model bolted joint rotor, the shaft stiffness is seated k, cA and cB are designed as the damping coefficient in the rolling bearings, cO1 and cO2 are defined as the damping coefficient introduced by the shafts, and the nonlinear force, in the x and y directions, introduced by the contact behavior at the mating interface of bolted joint is expressed as the F cx and F cy , the detailed expression is shown in Eq. (2). Furthermore, the kinetic, potential energy, and dissipation function of the rotor system can be written as follows:  1

 1

 1

 1

2 2 2 2 + mO2 x˙ O2 V = mA x˙ A2 + y˙ A2 + mB x˙ B2 + y˙ B2 + mO1 x˙ O1 + y˙ O1 + y˙ O2 2 2 2 2 (3) 1 1 1 1 k(xA − xO1 )2 + k(yA − yO1 )2 + k(xO2 − xB )2 + k(yO2 − yB )2 2 2 2 2 +Fcx (xO1 − xO2 ) + Fcy (yO1 − yO2 )

U =

Ug = mA gyA + mB gyB + mO1 gyO1 + mO2 gO2 D=

(4) (5)

 1

 1

 1

 1 2 2 2 2 2 cA x˙ A + y˙ A2 + cB x˙ B2 + y˙ B2 + cO1 x˙ O1 + c2 x˙ O2 (6) + y˙ O1 + y˙ O2 2 2 2 2

Based on the above formula, the dynamic governing equation of rotor system can be obtained employed Lagrange’s equations. According to the literature [27], the Lagrange’s equations can be expressed as follows:

∂L d ∂L ∂D − + (7) Fi = dt ∂ q˙ i ∂qi ∂ q˙ i

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where L = V −U , F i is the external force; qi is the generalized coordinate of the system. In this manner, the governing equation of the rotor system can be established by substituting Eqs. (3), (4), (5), (6) into Eq. (7), which can be represented as: ⎧ m x¨ + cA x˙ A + k(xA − xO1 ) = Fx ⎪ ⎪ A A ⎪ ⎪ ⎪ mA y¨ A + cA y˙ A + k(yA − yO1 ) + mA g = Fy ⎪ ⎪ ⎪ ⎪ ⎪ mO1 x¨ O1 + cO1 x˙ O1 − k(xA − xO1 ) + Fcx = Fux1 ⎪ ⎪ ⎪ ⎨ m y¨ + c y˙ − k(y − y ) + F + m g = F O1 O1 O1 O1 A O1 cx O1 uy1 (8) ⎪ m x ¨ + c x ˙ + k(x − x = F − F ) O2 O2 O2 B cy ux2 ⎪ ⎪ O2 O2 ⎪ ⎪ ⎪ mO2 y¨ O2 + cO2 y˙ 3 + k(yO2 − yB ) − Fcy + mO2 g = Fuy2 ⎪ ⎪ ⎪ ⎪ ⎪ mB x¨ B + cB x˙ B − k(xO2 − xB ) = Fx ⎪ ⎪ ⎩ mB y¨ B + cB y˙ B − k(yO2 − yB ) + mB g = Fy where F x and F y represent the nonlinear bearing force in x and y direction; F uyi (i = 1,2) is the unbalanced force introduced by disks O1 and O2 , respectively, and that can be expressed as follows: ⎧ ⎪ ⎪ Fux1 ⎨ Fuy1 ⎪ F ⎪ ⎩ ux2 Fuy2

= mO1 e1 ω2 cos(ωt) = mO1 e1 ω2 sin(ωt) = mO2 e2 ω2 cos(ωt + σ ) = mO2 e2 ω2 sin(ωt + σ )

(9)

where ω is the angular velocity of rotor system; σ is the eccentric phase difference between the two jointed disks. Based on the Hertz contact theory and combining the literature [28], the nonlinear bearing force can be written as: ⎧ Nb  ⎪ 2/3   ⎪ = −C δj H δj cos θj F ⎪ x b ⎨ j=1 (10) Nb  ⎪ 2/3   ⎪ ⎪ ⎩ Fy = −Cb δj H δj sin θj j=1

where the C b is the contact stiffness of the rolling bearing; θ j is the rotation location of jth ball of bearing at t time, expressed as θj = 2π(j − 1)/Nb + ωc t; the count of balls are be written as N b , and the ωc is rotating speed of the bearing cage, which is given by ωc = ω · r/(R + r); ω, r, R is the rotating speed of the shaft, radius of the outer race and inner race, respectively; the contact deformation δ j between the ball and race is obtained by δj = x cos θj + y sin θj − r0 ; r 0 is the radial clearance of the bearing; H is the Heaviside function used to neglect the case of δ j < 0;

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Physical parameter

Value

mA , mB , mO1 , mO2 (kg)

2, 2, 5, 5

cA , cB (N s/m)

1050, 1050

cO1 , cO2 (N s/m)

2100

k (N/m)

2 × 107

σ

π

g (m/s2 )

9.8

e1 , e2 (m)

2 × 10–3

Table 2. Structure parameter values of the ball bearing. Radius of outer Radius of race R (mm) inner race r (mm) 63.9

40.1

10

Radial Numbers of ball Contact stiffness clearance r 0 elements N b C b (N/m3/2 ) (µm) 8

13.31 × 109

BN

3.08

4 Numerical Simulation Results and Discussions It should be mentioned that the effect of the contact behavior of mating interface on the dynamic performance of bolted joint rotor, the evolution of the contact state, and the dissipation of frictional energy generated by slip behavior could be captured accurately using the above model. However, for the bolted joint structure in references 14 to 20, the contact behavior in the radial direction was regarded as linear, and the contact state and the dissipation of frictional energy were ignored in general, which would lead their model to fail to describe the real mechanical properties of the bolted joint structure. Comparing the Iwan-based analytical model and the adaptive parameter identification method presented in the present work, the model in references 14 to 20 was complicated and led to a certain limitation. Therefore, the research of the present work has a certain innovation, which improved the research gaps of the effect of contact behavior and the dissipation of frictional energy on the vibration characteristic of the bolted joint rotor, and further understanding of the contact mechanism profoundly.

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Based on the above theoretical framework and established model, this section focuses on the dynamic performance of the bolted joint rotor system considering the contact state at the mating interface, in which the effect of preload on its dynamic characteristics is emphasized. The physical parameters of the rotor system and the parameters of the ball bearing are listed in Table 1 and Table 2, respectively. Then, by using the Runge–Kutta– Fehlberg method, the vibration response, and spectra of bolted joint rotor system at rotating speed n = 1200 rpm are demonstrated to reveal the dynamic characteristics of the rotor system in different preload cases. And the preload cases include three kinds: low, middle, and high. In addition, at the beginning of solve, what needs to be emphasized is that the bolted joint should be regarded as a linear spring until the response of the system reaches a steady state, which could significantly reduce the effect of transient response on initial loading and the identification of maximum displacement . (a)

10-6

Low preload

Middle

High

2

x-Amplitude / m

2

10-6

1.5 1

1

0.5 0

0 -0.5

-1

-1 -2

-1.5

6.25 (b)

-5

10

6.3 6.35 6.4 6.45 Times / s Middle High Low preload

6.345

6.35

6.355

-5

-1.49

10

-1.5 -1.5 y-Amplitude / m

-1.52

-1.51

-1.54

-1.52

-1.56

-1.53 -1.54

-1.58 6.25

6.3

6.35 Times / s

6.4

6.45

6.255

6.26

6.265

6.27

6.275

Fig. 6. Vibration response of the bolted joint rotor system at different preload cases: (a) in the x direction, (b) in the y direction.

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Then, the vibration response of the system in horizontal (x) and vertical (y) directions are obtained, as shown in Fig. 6. It can be seen from these diagrams that show the higher amplitude of vibration response in the lower preload case, indicating that the higher preload would restrain the system vibration both horizontally and vertically. To further analyze the effect of preload on the vibrational frequency of the system, the spectra system in different preload cases are obtained, as depicted in Fig. 7. Unsurprisingly, the same tendency as their amplitude of vibration response can be observed in their first-and second-order rotating frequencies, whereas there exists no effect of different preload cases on their frequency component. This phenomenon is because the higher preload would extend the stick state of bolted joint structure, which be found in our previous research. That would cause a higher coupling stiffness and reduce frictional dissipation, leading to a lower vibration amplitude.

0.6 0.4 0.2

-8

0 0

0.4 0.2

2

Order

3

4

-8

4.35×10

0 0

5

1 10-7

(e) 3

2 3 Order

4

5.87×10-8 2.28×10

0.5

2

1

5.82×10-8

2.27×10-8

0.5 0

1

2 3 Order

4

5

0

0.4 0.2

4.33×10-8 0

1

2 3 Order

4

5

4

5

3

1.5

-8

0.6

-7 (f) 3.5 10

2.5

1

6.94×10-7

0.8

0

5

y-Amplitude / m

y-Amplitude / m

1 10-7

1.5

0

0.6

4.41×10

(d) 2

6.96×10-7

0.8

x-Amplitude / m

6.99×10-7

0.8

-6 (c) 1 10

10-6

(b) 1

x-Amplitude / m

10-6

1

y-Amplitude / m

x-Amplitude / m

(a)

2.5 2 1.5 1

5.80×10-8

2.26×10-8

0.5 0

1

2

Order

3

4

5

0 0

1

2

Order

3

Fig. 7. Frequency spectrum of the bolted joint rotor system at different preload cases: (a) in the x direction with low preload, (b) in the x direction with middle preload, (c) in the x direction with high preload, (d) in the y direction with low preload, (e) in the y direction with middle preload, (f) in the y direction with high preload.

5 Experimental Study In order to verify the above dynamic characteristic of the bolted joint rotor system considering the contact state at the mating interface demonstrated at different preload cases, a bolted joint rotor system test rig is set up for some validation, described as a PC, a data acquisition equipment, two displacement sensors, etc (Fig. 8). Considering the experimental result under the case of a large slip between the two disks are obviously, a rotating speed at n = 1500 rpm is selected, which approaches the rotating speed when impact between the screw hole and bolts occurs. Based on this point, the effect of the preload on rotor vibration performance is further explored and verified

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in detail by comparing its vibration response and frequency spectra under preload of 3 N and 6 N, as shown in Fig. 9. By comparing the amplitude of vibration response and frequency spectrum under different preload cases in Fig. 9, it can be observed that a higher amplitude of vibration response would be obtained for a lower preload at the same rotating speed. In addition, the evolving tendency of their frequency spectrums is consistent with the vibration response. That is, the higher preload would suppress the rotor dynamic, which demonstrates the obtained numerical simulation results exist a good agreement with the experimental result. In future work, the effect of some typical parameters on the motion state of bolted joint rotor system considering the contact state at the mating interface would be further explored, including preload, the area of mating interface, and the friction coefficient of the mating interface, etc.

Fig. 8. Test rig of bolted joint rotor system: (a) global view of experimental setup, (b) bolted joint structure, (c) vibration signal collection position.

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(a)

0.450 x-Amplitude (mm)

x-Amplitude (mm)

0.5

0

0.25 X Y

0.2 0.15 0.1

X Y

0.05 -0.5 2

-0.413 2.1

(c)

0

2.4

x-Amplitude (mm)

0.2 0 -0.2

-0.363

-0.4 2

2.1

2.2 2.3 Time (s)

0

20

(d)0.2

0.404

0.4

x-Amplitude (mm)

2.2 2.3 Time (s)

2.4

2.5

25 0.210

40 60 80 Frequency (Hz) X Y

0.15

50 0.028

25 0.187

0.1 X Y

0.05 0

0

20

50 0.026

40 60 80 Frequency (Hz)

100

Fig. 9. Vibration response and frequency spectrum at rotating speed n = 1500 rpm: (a) vibration response with preload of 3 N, (b) frequency spectrum with preload of 3 N, (c) vibration response with preload of 6 N, (d) frequency spectrum with preload of 6 N.

6 Conclusions In the present work, the nonlinear mechanical model of the bolted joint structure is established based on its hysteresis characteristic using the Iwan model. Then, according to the evolution process of the contact state, a discrete parameter identification method is proposed using the initial loading data of the hysteresis curve obtained from the FE result. By combining the above nonlinear mechanical model of bolted joint and the Lagrange modeling method, the dynamic model of bolted joint rotor system considering the contact state at the mating interface was established. Furthermore, taking three different preloads, for example, their dynamic response and frequency spectrum are numerically solved and discussed using the Runge–Kutta–Fehlberg method. Finally, the dynamic performance of the bolted joint rotor system considering the mating interface under different preload cases is verified by an experimentally investigation. Several representative conclusions of the present work are summarized as follows: 1) For the bolted joint structure, owing to the alternate changes of contact state at the mating interface under a lateral harmonic excitation, the hysteresis behavior would be found in its force-displacement curve, which would cause frictional dissipation. 2) The amplitude of response and frequency spectrum of bolted joint rotor system would increase in a lower preload, which is attributed to large coupling stiffness and lesser frictional dissipation generated by bolted joint in the higher preload case. And the

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same phenomena could be observed in the experimental result, which agree with the simulation results. 3) There exists no effect on the frequency component of the system under different preload cases. That is, the frequency components fail to be a standard to explore the dynamics performance of bolted joint rotor systems under different preload cases. Credit Authorship Contribution Statement Yuqi Li: Software, Writing - original draft, Funding acquisition. Zhimin Zhu: Formal analysis. Zhong Luo: Conceptualization, Funding acquisition, Project administration. Chuanmei Wen: Data curation, Writing - review & editing. Lei Li: Formal analysis, Investigation. Long Jin: Writing - review & editing. Acknowledgments. This research was supported by Guangxi Natural Science Foundation under Grant No. 2022GXNSFBA035488, the National Natural Science Foundation of China under Grant No. 11872148 and U190820023, Science and Technology Project of Guangxi under Grant No. GK AD22080042, and Doctoral foundation of Guangxi University of Science and Technology under Grant No. XKB 21Z64.

Conflict of Interest. The authors declare no conflict of interest, including specific financial interests and relationships relevant to the subject of this paper.

References 1. Qin, Z.Y., Han, Q.K., Chu, F.L.: Analytical model of bolted disk–drum joints and its application to dynamic analysis of jointed rotor. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 228(4), 646–663 (2014) 2. Yang, T., Ma, H., Qin, Z., Guan, H., Xiong, Q.: Coupling vibration characteristics of the shaft-disk-drum rotor system with bolted joints. Mech. Syst. Signal Pr. 169, 108747 (2022) 3. Yang, Y., Ouyang, H.J., Zeng, J., Ma, H., Yang, Y., Cao, D.: Investigation on dynamic characteristics of a rod fastening rotor-bearing coupling system with fixed-point rubbing. Appl. Math. Mech. 43(7), 1063–1080 (2022) 4. Qin, Z., Han, Q., Chu, F.: Bolt loosening at rotating joint interface and its influence on rotor dynamics. Eng. Fail Anal. 59, 456–466 (2016) 5. Hong, J., Chen, X., Wang, Y., Ma, Y.: Optimization of dynamics of non-continuous rotor based on model of rotor stiffness. Mech. Syst. Signal Pr. 131, 166–182 (2019) 6. Luo, Z., Li, Y., Li, L., Liu, Z.: Nonlinear dynamic properties of the rotor-bearing system involving bolted disk-disk joint. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 235(20), 4884– 4899 (2021) 7. Yang, D., Wang, L., Lu, Z.: Parameters identification of Iwan bolted joint models based on enhanced hysteretic force response sensitivity approach. Int. J. Nonlin. Mech. 143, 104022 (2022) 8. Li, C., Jiang, Y., Qiao, R., Miao, X.: Modeling and parameters identification of the connection interface of bolted joints based on an improved micro-slip model. Mech. Syst. Signal Pr. 153, 107514 (2021)

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9. Tan, L., Wang, C., Liu, Y., Sun, W., Zhang, W.: Study on hysteresis and threaded fitting behavior of bolted joint with non-parallel bearing surface. Mech. Syst. Signal Pr. 168, 108655 (2022) 10. Liang, Y., Xu, F., Zhang, X., Wang, A., Ma, C.: An analytically improved four-stage model for single-lap torqued bolted joints accounting for preload relaxation. Thin. Wall Struct. 182, 110252 (2023) 11. Liu, X., Sun, W., Liu, H., Du, D., Ma, H.: Semi-analytical modeling and analysis of nonlinear vibration of bolted thin plate based on virtual material method. Nonlinear Dyn. 108(2), 1247– 1268 (2022). https://doi.org/10.1007/s11071-022-07288-8 12. Mir-Haidari, S., Behdinan, K.: Nonlinear effects of bolted flange connections in aeroengine casing assemblies. Mech. Syst. Signal Pr. 166, 108433 (2022) 13. Li, D., Botto, D., Li, R., Xu, C., Zhang, W.: Experimental and theoretical studies on friction contact of bolted joint interfaces. Int. J. Mech. Sci. 107773 (2022) 14. Wang, L., Wang, A., Jin, M., Yin, Y., Heng, X., Ma, P.: Nonlinear dynamic response and stability of a rod fastening rotor with internal damping effect. Arch. Appl. Mech. 91(9), 3851–3867 (2021). https://doi.org/10.1007/s00419-021-01981-7 15. Li, J., Li, Y., Cui, L., Mo, G., Zhong, W., Feng, Y.: Nonlinear dynamic behavior of circumferential rod fastened rotor with initial thermal deflection. J. Vibr. Eng. Technol. 10, 767–780 (2021) 16. Li, Y., Wen, C., Luo, Z., Jin, L.: Vibration analysis of a multi-disk bolted joint rotor-bearing system subjected to fixed-point rubbing fault. Int. J. Nonlin. Mech. 146, 104165 (2022) 17. Yu, P., Li, L., Chen, G., Yang, M.: Dynamic modelling and vibration characteristics analysis for the bolted joint with spigot in the rotor system. Appl. Math. Model 94, 306–331 (2021) 18. Li, J., Yang, Z., Ren, Q., Mo, G., Zhong, W., Feng, Y., Li, G.: Study on the bistable vibration behaviour of a rod-fastened rotor-bearing system, Nonlinear Dynam. 109(2), 609–629 (2022) 19. 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 20. 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 Pr. 158, 107778 (2021) 21. Li, D., Xu, C., Liu, T., Gola, M.M., Wen, L.: A modified IWAN model for micro-slip in the context of dampers for turbine blade dynamics. Mech. Syst. Signal Pr. 121, 14–30 (2019) 22. Liu, J., et al.: Dynamic behaviour of a bolted joint subjected to torsional excitation. Tribol. Int. 140, 105877 (2019) 23. Li, Y., Hao, Z., Feng, J., Zhang, D.: Investigation into discretization methods of the sixparameter Iwan model. Mech. Syst. Signal Pr. 85, 98–110 (2017) 24. Jamia, N., Jalali, H., Taghipour, J., Friswell, M.I., Khodaparast, H.H.: An equivalent model of a nonlinear bolted flange joint. Mech. Syst. Signal Pr. 153 107507 (2021) 25. Li, D., Botto, D., Xu, C., Gola, M.: Fretting wear of bolted joint interfaces. Wear 458–459, 203411 (2020) 26. Li, D., Botto, D., Xu, C., Gola, M.: A new approach for the determination of the Iwan density function in modeling friction contact. Int. J. Mech. Sci. 180, 105671 (2020) 27. Li, Y., Luo, Z., Shi, B., Liu, J.: Influence of contact interface friction of bolted disk joint on motion stability of rotor-bearing system. Arch. Appl. Mech. 91(5), 2225–2236 (2021). https:// doi.org/10.1007/s00419-020-01879-w 28. Chen, G.: Study on nonlinear dynamic response of an unbalanced rotor supported on ball bearing. ASME. J. Vib. Acoust. 131(6), 1980–1998 (2009)

Identification of High-Speed Gear Traveling Wave Resonance Based on Phase Space Reconstruction Method Ziyang Xu1 , Jing Wei1,3(B) , Haibo Wei2 , Zhirou Liu1 , Yujie Zhang1 , and Hao Lin1 1 College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing 400044,

People’s Republic of China [email protected] 2 College of Aerospace Engineering, Chongqing University, Chongqing 400044, People’s Republic of China 3 State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, People’s Republic of China

Abstract. High-speed thin-walled gears are widely used in the aviation geared drivetrain, but they face the threat of traveling wave resonance (TWR). Therefore, it is necessary to identify them, and this paper proposes a method based on phase space reconstruction, called phase space characterization of resonance (PSCOR). High-speed gearing experiments are designed to verify this method. Besides, for the experimental gear, its instantaneous displacement and dynamic stress in the TWR state are analysed. The results show that the TWR of high-speed gears can be accurately identified by the PSCOR method. And when the TWR occurs, gear’s displacement and stress fields rapidly rotate, whose speeds depend on the resonant frequency. Furthermore, the petaloid distribution of high stress fits the mode shape with nodal diameters, and its fast movement leads to the alternating stress with high frequency. Keywords: traveling wave resonance · high-speed gears · phase space reconstruction

Nomenclatures Aw

Amplitude correction factor

ri

B(i)

i-th reduction matrix

Eigenvectors

ci

Center of the minimum sphere

V1 , V2 , V3 xi

f

Phase space mapping function

X (i)

i-th reconstruction matrix

Exciting force frequency

Y (i)

i-th inner product matrix

Natural frequency of k-th TWR

β (i)

i-th sub-time-series

fm

Meshing frequency

β (r)

Subseries of a resonance peak

fs

Shaft frequency

ζi

Phase point expansion rate

fe fk

Boundary radius (i)

(i)

(i)

i-th time response data

(continued) © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 257–267, 2024. https://doi.org/10.1007/978-3-031-40459-7_17

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(continued) (i)

Gk

Phase point space coordinates

 (t)

Time window

k

Order of the TWR

θ (t)

Response signal of β (r)

Nr

Number of sampling points

θ (t)

Windowed signal

1 Introduction High-speed gears are widely used in aviation transmission systems. Under the background of lightweight structure, it mostly adopts the thin-walled design. This also leads to the occurrence of traveling wave resonance (TWR) at high speed and heavy load. TWR refers to the resonance phenomenon where the nodal diameter (ND) modal shape is activated, which will greatly increase the risk of gear failure and even destroy the transmission system. Therefore, it is very important to accurately identify the traveling wave resonance. This is not only helpful to guide the design of aviation gears to avoid the influence of resonance, but also conducive to the maintenance of flight safety. The characteristic of TWR is hidden in the dynamic response information of gears. And phase space reconstruction is a method to mine the hidden nonlinear behavior in time series [1], which has been used in wind speed prediction [2], rotating machinery state characterization [3], gear fault diagnosis [4] and so on. The phase point distribution obtained from the reconstructed phase space is related to the state of the system [5–7]. Actually, resonance occurs referring to the system state changes, which can be characterized by reconstructing phase space in theory, but it is lack of related research support. Therefore, the phase space reconstruction method is used to deal with the dynamic response data of high-speed gears. The phase point distribution in the phase space is constructed to reflect the state of gear, and the index to describe the structural characteristics of the gear is proposed according to the characteristics of TWR. This method is called the phase space characterization of resonance (PSCOR). A high-speed gear transmission experiment is designed to verify the proposed method. At the same time, the instantaneous displacement and dynamic stress characteristics of the gear under TWR are analyzed.

2 Methodology of the PSCOR It is assumed that the unstable dynamic response of the transmission system working n times is expressed as β = (x1 , x2 , . . . , xi , . . . , xn ),

(1)

where x i is the i-th time response data. It is divided into p sub-time series which are approximately steady-state:   (2) X = X (1) , X (2) , . . . X (i) , . . . , X (p) .

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On the basis of the phase space mapping relationship f : β (i) → X (i) , any sub-timing β is mapped to a multi-dimensional space and reconstructed into a matrix X (i) . The key to the mapping process is to determine the required time delay τi and the embedding dimension mi . The former is calculated by the average mutual information method [8], while the latter is calculated by the false nearest neighbour method [9]. For all sub-time series, AMI method and FNN method are used to obtain the time delay and embedding dimensions that map them to phase space. Then the corresponding reconstruction matrix X (i) is obtained and placed in the cell array X of length p: (i)

X = [X (1) , X (2) , . . . X (i) , . . . , X (p) ],

(3)

X (i) = f (β (i) ), i = 1, 2, ... , p.

(4)

Principal component analysis (PCA) is used to reduce the phase point distribution from high-dimensional space to three-dimensional space to observe its structure [6]. For the reconstruction matrix X (i) , the inner product matrix is constructed: Y (i) = X (i) X (i) . T

(i)

(i)

(5)

(i)

The eigenvectors V 1 , V 2 , V 3 are called the main vectors, and they are the first three eigenvalues. X (i) is projected to these three directions to get the reduction matrix (i)

(i)

(i)

B(i) = X (i) [V 1 , V 2 , V 3 ] ⊂ RNi ×3 .

(6)

A visible phase trajectory is formed by the successive connection of the phase points. If its motion is always constrained in a certain volume of phase space [7], then the limiting surface is called the boundary of the spatial distribution of phase points. According to the minimum bounding sphere calculation method proposed by Welzl based on linear programming algorithm [10], the phase point of the reconstructed reduction matrix B(i) is enclosed in the minimum sphere with ci as the center and ri as the radius. The boundary radius ri and the phase point expansion rate ζi are used to quantify the phase point spatial distribution characteristics to reflect the sub-time series β (i) corresponding to the state of the system. The former characterizes the volume of the phase point distribution, while the latter is defined to measure the average level of the phase point deviating from the spherical center:    (i) Ni  − c G   i k 1 , (7) ζi = Ni ri k=1

where, the phase point Gk(i) deviates from the center of the sphere and approaches the  indicating that the higher its activity in the phase space, the closer  boundary,   (i) Gk − ci /ri to 1. Average the corresponding ratio of all phase points to reflect the overall level. Therefore, the higher the vibration frequency of the system is, the larger the proportion of phase points with high activity is, and the expansion rate of phase points increases.

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After the high frequency traveling wave resonance is excited, the structural vibration increases significantly, and the local peak value is formed in the distribution of ri and ζi . Because the TWR with the same order is divided into two types, namely the forward and backward ones, two adjacent peaks will overlap, so it is necessary to further judge the properties of peaks. Given that β (r) is the subseries of a resonance peak, it is mapped to the frequency domain for two purposes. The first one is to quantitatively obtain the amplitude at the resonance frequency, which could be further used to develop the traveling wave model. Another is to verify that the peak is indeed induced by the TWR. This is judged by whether its frequency matches the resonance excitation frequencies. In a rotating coordinate system, they can be expressed as [11]: f e = f m ∓ k · fs ,

(8)

where, fe is the exciting force frequency of TWR; fs and fm are the shaft frequency and meshing frequency of the k-order TWR, respectively, and the negative sign corresponds to the forward traveling wave resonance (FTWR), and the positive sign corresponds to the backward traveling wave resonance (BTWR). Furthermore, in order to reduce the leakage caused by signal truncation, the time window is applied: θ (t) = θ (t) ·  (t),

(9)

where θ (t) is the response signal corresponding to β (r) , with Nr sampling points;  (t) is the time window, such as Rectangular, Hanning, and Hamming window. For the windowed signal θ (t), Fourier transform is used to get its frequency spectrum. However, the calculated amplitude declines due to the effect of windows, and thus it needs to be restored by the amplitude correction factor Aw [12], which is defined as Nr Aw = N −1 r t=0

 (t)

,

(10)

where, Aw depends on the type of windows. By multiplying it and each spectral line, the spectrum is rescaled to obtain the final result of mapping β (r) to the frequency domain.

3 Experimental Design on High-Speed Gear TWR TWR can cause structural deformation, which can be reflected by dynamic strain. The strain testing method obtains the gear strain signal in the complex sound field environment formed by the high-speed test, and its testing accuracy is high. Furthermore, the continuous acceleration test is used to study the TWR characteristics of thin web gears under unsteady conditions. A test gearbox for bevel gear pair is designed, as shown in Fig. 1. Among them, the active bevel gear is used as the objective gear to analyze its TWR. The dynamic strain of the gear is collected by the strain gauge, and then the strain signal is converted into an electrical signal, which is transmitted to the data acquisition system outside the test gearbox through the slip ring.

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The test gearbox is placed on a high-speed experimental platform to carry out a uniform acceleration test with an input torque of 30N·m and an input speed of 4500– 8000 rpm. The speed-up time is 70s. The high-speed experimental platform also includes an input motor, a gear reducer and an output motor, as shown in Fig. 2. The input one controls the torque and the output one controls the speed.

Diving gear Input shaft

Slip ring 1

Driven gear

Slip ring 2

Output shaft

Fig. 1. The configuration of the experimental gearbox.

Fig. 2. Layout of high-speed experimental platform.

4 TWR Characteristics of the Experimental Gear 4.1 Instantaneous Displacement Characteristics Limited by the high-speed, airtight, oil-mist working environment of the gearbox, it is difficult to directly measure the instantaneous displacement and dynamic stress on the web of high-speed rotating gears. Thus, the time-varying displacement of the gear drivetrain in the experiment is simulated by the finite element method.

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Due to the low stiffness of thin-walled bevel gears in the axial direction, the TWR behaviour is most obvious in this direction. Therefore, the axial component is used to study the displacement response. Regardless of BTWR or FTWR, the instantaneous displacement field is similar to the 3-ND mode shape: the field is divided into six sectors of approximately equal size, including three positive sectors (distributing positive displacement) and another three negative parts, and these two types alternate over time, as shown in Fig. 3 and Fig. 4. There is a larger axial displacement near the outer edge of the gear, where a node is selected and taken as the starting point, i.e. θ = 0. Taking t 0 = 0.05 s, t = 2.5 × 10–5 s as the initial time and sampling period, axial displacements of the node set are captured. As shown in Fig. 3(b) and Fig. 4 (b), the axial displacement on the same circumference is distributed in the form of waves, which are called deformation waves. However, this wave is not a simple harmonic same as the standard mode shape. Instead, affected by the meshing force, additional deformations are superimposed on waves in the meshing area and its adjacencies.

Fig. 3. Movement of axial displacement during the 3-ND BTWR: (a) displacement field, (b) deformation wave movement over time on the selected circle.

Additionally, the BTWR and the FTWR have obvious phase differences: for the FTWR, its deformation wave moves in the same direction as the gear rotates, and the opposite is for the BTWR. The circle angle corresponding to the peak at t 0 will experience another peak after 4.4t, which is consistent with the resonance period 1/f k . f k is the natural frequency of k-th TWR, and it is equal to 9151.7 Hz (k = 3) from the modal analysis. In the same time period, the angle rotated by the deformation wave is nearly 30 times that of the gear. It suggests that even if the gear rotates at a high speed, the deformation wave caused by TWR still rotates considerably faster than it, resulting in the rapid time-space movement of the displacement field.

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Fig. 4. The time-space movement of displacement during the 3-ND FTWR: (a) displacement field, (b) deformation wave.

4.2 Dynamic Stress Characteristics The stress at the node is measured by the Mises stress, and another non-resonant working condition (only changing the input speed to 4500 rpm) is introduced to highlight dynamic stress characteristics by comparing them with the resonant ones. As shown in Fig. 5(a), the meshing influence area A generates excessive stress caused by the tooth surface contact and the tooth root bending, which is greater than the rest parts. Hence, for each working condition, the stress field is normalized by using 80 MPa as the maximum value, to avoid the stress on the web being diluted or even masked. Moreover, due to the input torque, the stress is also accumulated at the shaft area B. However, when the TWR is excited, as proposed in Fig. 5(b) and (c), the stress on the web gathers and forms a high-stress region C that fits the 3-ND mode shape. For both BTWR and FTWR, this area runs from the outer surface of the gear shaft to the outside of the rim, like a petal, so it is called the petal-type high-stress area. In addition, different from the areas A and B fixed in the stationary reference frame, the petal-type area C shows up movement characteristics similar to the displacement field: the field corresponding to the FTWR rotates in the same direction of the gear, while the BTWR moves in reverse. Since low-stress parts exist between adjacent petals in area C, for any point on the web, the suffered stress alternates between high and low levels as the gear rotates. On the other hand, the moving speed of this area is faster than that of the gear. A couple of these two phenomena result in the web being subjected to alternating stress with a high frequency. Taking the selected node in Fig. 3 as the observation, the dynamic stress is further analyzed: the responses under three working conditions are normalized based on the maximum stress and mapped to the frequency domain, as shown in Fig. 6. By contrast, for the TWR, the dynamic stress has obvious high-frequency harmonics that severely impact on the dynamic responses. Especially in the frequency domain, the dominant

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Fig. 5. The stress field of the targeted gear under different working conditions: (a) the non-resonant one; (b) the backward traveling wave resonance; (c) the forward one.

amplitudes are formed at the excitation frequencies f m + 3f s and f m -3f s for BTWR and FTWR, respectively.

5 Experimental Validation of the Proposed Method 5.1 Time-Frequency Response of the Captured Experimental Strains The experimental strain obtained is described by micro-strain με. In the process of increasing speed, the gear expands continuously under the increasing centrifugal force, so the strain increases with the increase of rotational speed, as shown in Fig. 7. There are local peaks in the regions I-IV. Among them, the region I is similar to II, which may be TWR. But it’s impossible to judge accurately. Therefore, the Short-Time Fourier transform (STFT) is used to process the dynamic strain signal to further quantify the resonance characteristics. In the spectrum as shown in Fig. 8, the excitation frequency is an oblique line, while the natural frequency of the gear is almost horizontal. They intersect and form a peak. Two neighbourly resonance peaks occur, by the action of excitation frequencies f m ± 3f s . This shows that the 3-ND mode (k = 3) is excited. The left peak is for the BTWR and the right peak is for the FTWR. 5.2 Identification of the TWR by PSCOR Because the boundary radius r can reflect the vibration amplitude, and the phase point expansion rate ζ is related to the main vibration frequency, both are used to quantify the state evolution of the vibration gear, as shown in Fig. 9. The peak overlaps I and II of these two indexes are in line with the judgment conditions of TWR: high frequency,

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Fig. 6. Stress response in time and frequency domains for the targeted gear under different working conditions: (a), (b) for the non-resonant one; (c), (b) for the backward traveling wave resonance; and (e), (f) for the forward one.

Fig. 7. Evolution of the acquired experimental strain with time.

high amplitude and adjacent peaks. The rotational speed of the left peak is relatively low, corresponding to BTWR and the right one for FTWR. For the other two r peak in Fig. 9, their phase point expansion rate ζ is low, indicating that the phase points near the center of the sphere, and the system is in low frequency vibration. In addition, comparing the speeds corresponding to peaks in Fig. 8 and Fig. 9,

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Fig. 8. Time-frequency analysis of the captured dynamic strains.

the recognition results of the two methods are basically the same for the TWR, which shows that the PSCOR method is accurate.

Fig. 9. Identification of the TWR by PSCOR where strains are used as response to construct time series.

6 Conclusion In this paper, the methodology of phase space characterization of resonance (PSCOR) is proposed and verified by high-speed gearing experiments. The results show that the traveling wave resonance (TWR) of high-speed gears can be accurately identified by the PSCOR method. For responses of TWR, the amplitudes, corresponding to excitation frequencies f m ± kf s , are dominant in the spectrum. On the other hand, displacement and stress fields rapidly rotate, and speeds of that depend on the resonant frequency. Particularly, the petaloid distribution of high stress fits the mode shape with nodal diameters, and its fast movement leads to the alternating stress with high frequency.

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Acknowledgements. This work was supported by the National Natural Science Foundation of China [grant numbers 52275048, 51775058]; the National Key R&D Program of China [grant number 2019YFE0121300]; and the Fundamental Research Funds for the Central Universities [grant number SKLMT-ZZKT- 2021Z02]. The authors would like to express their gratitude for the support of the funding authority.

References 1. Aydin, I., Karakose, M., Akin, E.: A new method for time series classification using multidimensional phase space and a statistical control chart. Neural Comput. Appl. 32, 7439–7453 (2020) 2. Fu, W.L., Wang, K., Tan, J.W., Zhang, K.: A composite framework coupling multiple feature selection, compound prediction models and novel hybrid swarm optimizer-based synchronization optimization strategy for multi-step ahead short-term wind speed forecasting. Energy Convers. Manage. 205, 112461 (2020) 3. Yan, R.Q., Liu, Y.B., Gao, R.X.: Permutation entropy: a nonlinear statistical measure for status characterization of rotary machines. Mech. Syst. Sig. Process. 29, 474–484 (2012) 4. Ma, Y.L., Cheng, J.S., Hu, N.Q., Cheng, Z., Yang, Y.: Symplectic quaternion singular mode decomposition with application in gear fault diagnosis. Mech. Mach. Theory 160, 104266 (2021) 5. Ding, C., Zhu, H., Jiang, Y., Sun, G., Wei, C.: Recursive characteristics of a running-in attractor in a ringon-disk tribosystem. J. Tribol. 141, 011604 (2019) 6. Lang, S.H., Zhu, H., Sun, G.D., Jiang, Y., Wei, C.L.: A study on methods for determining phase space reconstruction parameters. J. Comput. Nonlinear Dyn. 17, 011006 (2022) 7. Sun, G., Zhu, H., Ding, C., Jiang, Y., Wei, C.: On the boundedness of running-in attractors based on recurrence plot and recurrence qualification analysis. Friction 7(5), 432–443 (2018). https://doi.org/10.1007/s40544-018-0218-6 8. Wallot, S., Monster, D.: Calculation of Average Mutual Information (AMI) and False-Nearest Neighbors (FNN) for the estimation of embedding parameters of multidimensional time series in matlab. Front. Psychol. 9, 1679 (2018) 9. Chelidze, D.: Reliable estimation of minimum embedding dimension through statistical analysis of nearest neighbors. J. Comput. Nonlinear Dyn. 12, 051024 (2017) 10. Welzl, E.: Smallest enclosing disks (balls and ellipsoids). Lect. Notes Comput. Sci. 555, 359–370 (1991) 11. Drago, R.J., Brown, F.W.: The analytical and experimental evaluation of resonant response in high-speed, lightweight, highly loaded gearing J. Mech. Des.-Trans. Asme 103, 346–356 (1981) 12. Brandt, A.: Noise and Vibration Analysis, 1st edn. John Wiley and Sons, Chichester, West Sussex, United Kingdom (2011)

Rotordynamics of a Vibroflot Florian Tezenas du Montcel1,3(B) , Sébastien Baguet1 , Marie-Ange Andrianoely1 , Régis Dufour1 , Stéphane Grange2 , Laurent Briançon2 , and Piotr Kanty3 1 Univ Lyon, INSA Lyon, CNRS, LaMCoS, UMR5259, 69621 Villeurbanne, France

[email protected]

2 Univ Lyon, INSA Lyon, GEOMAS, EA7495, 69621 Villeurbanne, France 3 Menard, 22 rue Jean Rostand, 91400 Orsay, France

Abstract. This paper presents a modelling of a vibroflot dynamic behaviour. Vibroflots are used by ground improvement companies to perform vibro compaction. This technic consists in deeply densifying sandy soils by vibrations in order to make stable future infrastructures. A classical vibroflot is a slender structure composed of several rotors which are not rotating or driven by an asynchronous electrical motor that produces orbital vibration and therefore the soil compaction. The final objective of the investigation is to establish a global multi-physics model coupling electromagnetic motor, rotordynamics and soil models in order, for example, to adjust the speed of rotation that generates the soil resonance, making the soil compaction easier. The present paper presents only the electro-magnetic and mechanical models, in order to predict the vibroflot dynamic behaviour during the free hanging operation. Among the predicted results, the lateral displacement and motor amperage are of particular interest because they can be compared with the experimental measures collected on an in-situ full-scale instrumented vibroflot. Keywords: Rotordynamics · Asynchronous electric motor · Multi-physics models · Vibroflot · Vibro compaction

1 Introduction Vibro compaction is a ground improvement technique which aims to deeply densify sandy soil in order to make stable future infrastructure [1]. It is used to control and reduce settlement, mitigate liquefaction [2], stabilize or treat hydraulic fill and limit lateral earth pressure behind quay walls. Loose soil can be compacted through insertion of vibrating probes, called vibroflots, together with a large volume of water. This enables the sand particles to rearrange themselves in a denser formation and thus increases the overall density of the soil. Vibro compaction is executed using a rig – excavator, drilling rig or crane –, a vibroflot, and auxiliary equipment such as generator, compressor, and water pumps. Under the effect of its own weight, the machine’s pull-down force (if any), the effect of the jetted water and the sustained horizontal vibrations, the vibrating probe rapidly reaches the desired depth. The probe is then gradually lifted in successive passes, producing in this way a cylinder of compacted ground (see Fig. 1). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 268–275, 2024. https://doi.org/10.1007/978-3-031-40459-7_18

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Fig. 1. Vibro compaction process.

Since the development of vibro compaction during the 1930s vibroflots were progressively improved following operational experiences. The current design can be described as a multirotor system coupled with a motor (see Fig. 2). A non-rotating tube casing, in contact with the soil, is supporting an eccentric rotor by means of rolling bearings. This internal rotor, driven in rotation by an electrical motor, produces orbital vibrations. This paper focuses on a vibroflot set in motion by an asynchronous electrical motor but vibroflots working with hydraulic motors also exist. To prevent the vibrations to be transmitted to the extension tubes and therefore to the rig, two rubber coupling parts are used to support the external tube and isolate the vibrations from the vibroflot. It appears now necessary for soil improvement companies to accurately model this system to better know and improve their vibro compaction equipment and processes [3]. For instance, to be able to improve the lifetime of the equipment – bearings, rubber couplings, motor – and to improve vibro compaction process efficiency by working at resonance frequency [4]. This study was also motivated by the need to develop a new approach regarding vibro-compaction numerical models. Indeed, many studies, such as [5] or [6] are trying to simulate this particular soil improvement process but without any strong coupling between the soil and the vibroflot models – that is to say the amplitude and shape of vibrations are given to the soil model by the user. Others are seeing the importance to also model the vibroflot to better control vibro compaction process [7], but the model used is very simplified. The novelty presented here consists in model the vibroflot more accurately, including its electrical motor, which has not already been done. This paper presents only the vibroflot model in free hanging, which means without soil effects. At first the electrical model is described, then the multi-rotors model and the strong coupling made to solve both physics simultaneously are also presented. In a second part in-situ experimental investigations are described. Finally, numerical results are compared with in-situ measurements.

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Fig. 2. Vibroflot kinematic scheme.

2 Numerical Model 2.1 Electro-Magnetic Model The asynchronous electrical motor modelled here is made of a three-phase wound stator with p pairs of poles, and a squirrel cage rotor. The magnetic circuit is supposed to be not saturated and has a constant permeability; ferromagnetic loss is negligible; and the influences of skin effect and heating are not taken into account on motor parameters. Applying Lenz and Faraday electromagnetic laws to each stator and rotor phase the constitutive equations linking voltages and amperages are obtained. Then, after application of Park transformation [8], which consists in rewriting the tree-phase system − → → into a two-phase model including only direct d and quadratic − q axes, the set of four constitutive equation becomes: vsd = Rs isd +

d θs d ∅sd − ∅sq dt dt

(1)

vsq = Rs isq +

d ∅sq d θs + ∅sd dt dt

(2)

vrd = Rr ird +

d θr d ∅rd − ∅rq dt dt

(3)

vrq = Rr ird +

d ∅rq d θr + ∅rd dt dt

(4)

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with v, i, ∅, θ , R, stand respectively for the voltages, amperages, magnetic flux, angular positions, electrical resistance and with s, r, d , q subscripts refer respectively to the − → → stator, rotor, direct axis, and quadratic axis. In this case, ( d , − q ) axes are oriented so − → → q) that they are joined to the rotating field. Thus, the angular speed ωs = ddtθs of ( d , − − → − → d θr axes in the stator coordinates, and the angular speed ω = , of ( d , q ) axes in the r

dt

rotor coordinates are defined. Furthermore, because of the squirrel cage, the voltage is applied only on the electrical stator and thus vrd = vrq = 0. In these conditions the electromagnetic parameters are reduced to five: Ls the stator inductance, Lr the rotor inductance, M the mutual inductance between the stator and the 2 rotor, Tr = RLrr the rotor time constant, and σ = 1 − LMr Ls the total leakage factor. Finally, introducing the magnetic flux expression, Eqs. (1), (2), (3) and (4) are rewritten in a matrix form, such as: {v} = [B]

d {i} + [A]{i} dt

(5)

− → → with [A] and [B] matrices depending on stator and rotor parameters, as well as ( d , − q) axes angular speeds. To simplify equations, the following changes of variables are stated:  = Lr i and i  = Lr i . Solving Eq. (5) provides the four electrical intensities and ird rq M rd M rq therefore the electrical torque:     (6) Cm = p(1 − σ )Ls isq ird − isd irq that permits a strong coupling between electromagnetic and mechanical models. In order to access motor parameters easily, on a first estimation, nameplate information is used [9]. Indeed, these properties are not directly available for this kind of motor. It should be noted that the parameters are supposed constant, at least close to the nominal working state. More precise techniques could have been used but would have required electro-mechanical measurements or more complex identification methods. Only the stator resistance Rs is measured. Then, the other parameters are estimated as follows: σ =

1 − cos(ϕn ) 1 + cos(ϕn )

ωsln = 2π fsn − Nrn p

(7) 2π 60

1 σ ωsln √   1−σ σ Vsn 1+ Ls = Isn 2π fsn σ Tr = √

(8) (9) (10)

with ϕn the nominal power factor, fsn the nominal stator electric frequency, Nrn the nominal rotor speed, Vsn the nominal voltage, and Isn the nominal amperage.

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2.2 Multi-rotor Model Multi-rotor model permits simulating the transient dynamic behaviour of the three rotors composing the vibroflot. The rotordynamics theory is inspired from [10, 11] and [12]. Each rotor shaft is modelled by 1D Timoshenko finite elements taking into account, on each node, the six degrees of freedom devoted to bending, axial and torsion motions. Two-node bearing, and two-node nonlinear rubber coupling FE are also available. Furthermore, in this case-study, mass unbalance and gravity are considered. Finally, after the assembly, the application of the Lagrange equations yields the multi-rotor equation of motion:    ˙ S ] + [Cb ]){˙q} + ϕ[D ¨ S ] + [KS ] + KgS + [Kb ] {q} [MS ]{¨q}+ (ϕ[G

 (11) = ϕ{F ¨ kS } + {Fu } + FgS,t + {Fkb } + {Fkm } with [MS ] the shafts mass matrix, [GS ] the shaft gyroscopic matrix, [Cb ] the rolling bearings damping matrix, [DS ] the shafts kinetic stiffness matrix, [KS ] the shafts stiffness  {FkS } matrix, KgS the shafts gravity matrix, [Kb ] the rolling bearings stiffness

matrix,  the shaft torsion kinetic force vector, {Fu } the unbalance force vector, FgS,t the shafts gravity traction force vector, {Fkb } the bearings resistive torque vector, {Fkm } the motor torque vector, ϕ˙ the speed of rotation of the internal rotor, and {q} the vector containing all rotors degrees of freedom. 2.3 Global Model Assembly The global system consists of the electro-magnetic first order differential equations and of the mechanical second order differential equations. Coupling strongly the models requires to rewrite the system of equation as a Cauchy problem: y˙ (t) = f (t, y(t)) and to solve all equations together at each time step. With the following change of variables:

(12) Y T = {q} {˙q} {i} ϕ ϕ˙ the global system related to Eqs. (5), (6) and (11) becomes: ⎡ ⎤ {Y (2)} ⎢ [M ]−1 (−[C]{Y (2)} − [K]{Y (1)} + {F}) ⎥ ⎢ ⎥ ⎢ ⎥ Y˙ = ⎢ [B]−1 ({v} − [A]{Y (3)}) ⎥ ⎢ ⎥ ⎣ ⎦ {Y (4)} (Cm ({Y (3)}) − Cr ({Y (4)}))/IP

(13)

Then the time integration of the coupled models is carried out using Matlab ODE15s solver [13], which is a variable-step, variable-order (VSVO) solver based on the numerical differentiation formulas (NDFs) of orders 1 to 5. After defining all rotors and motor parameters the only input variable is the voltage applied between the motor phases. The predicted results presented in Sect. 4 are obtained with 6 degrees of freedom for the electro-magnetic model and with 345 degrees of freedom for the finite element model. Thus, the length of vector Y defined in Eq. (13) is 696.

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3 In-situ Experimental Investigation To validate the numerical model, an in-situ measurement campaign was conducted on a full-scale experiment V23 vibroflot. The whole process was instrumented but only the measurements related to the models discussed previously are presented in what follows. Sensors were fixed on the vibroflot, and on the electrical cabinet to monitor the dynamic behaviour of the vibroflot hung above the ground. Five triaxial accelerometers were used with a 2 kHz sampling frequency: three fixed on the external tube of the vibroflot and two fixed at the bottom of the extension tube (see Fig. 3). To access vibroflot’s displacements in 3D the accelerations recorded are integrated using a Newmark scheme with high-pass filters.

Fig. 3. Position of sensors fixed on the instrumented vibroflot.

To measure the rotation speed of the internal rotor an eddy current displacement sensor was fixed in front of a ten teeth wheel. Then the speed is calculated by detecting rising edges. In order to have accurate data of changing speeds the sampling frequency was 20 kHz. Motor amperage – effective value – and frequency were directly measured by the frequency inverter used to control the vibroflot and recorded by the same data acquisition system as the other sensors respectively at 200 Hz and 100 Hz.

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4 Results and Comparison with In-situ Experiments The predicted results are compared with the measurements carried out on site. Motor output state after start-up is summarized in Table 1. Simulated rotational speed and amperage values in steady state are satisfactory, especially since the motor parameters are roughly estimated. Table 1. Comparison of motor parameters at 60 Hz in steady state. Parameter

Measured value

Simulated value

Relative gap

Rotational speed

1795 rpm

1789 rpm

-0,3%

Amperage

76 A

75 A

-1,3%

Regarding the multi-rotors model, results are also close to the real vibroflot’s behaviour. Figure 4 clearly shows the double cone shape of vibration, with the highest amplitude at the tip of the vibroflot. Tip and top amplitudes are of the same order as the measured ones.

Fig. 4. Orbits and vibration envelope of the external tube of the vibroflot set in motion by its electrical motor operating at 60 Hz – numerical results.

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5 Conclusion and Perspectives The prediction of the transient dynamic behaviour of a vibroflot hung above the ground has been numerically simulated using a model coupling strongly the electro-magnetic model based on the Park transformation and the mechanical model based on 1Delements. These models take the gravity force into account and are able to reproduce the global shape of vibration of the vibroflot by applying only a voltage between the electrical stator model phases. This model can still be improved by implementing a more complex model for predicting bearing moment loss. Moreover, motor parameters were only estimated, on a first approach, by simplified formulations. To increase the global model accuracy at different frequencies motor parameters must be better defined. The next step is devoted to the prediction of the vibroflot behaviour during vibro compaction in a sandy soil, using the developed electro-mechanical finite element model. Acknowledgements. The authors would like to thank Menard company for its financial and technical support, Pr. Xuefang Lin-Shi for her advice.

References 1. Kirsch, K., Kirsch, F.: Ground Improvement by Deep Vibratory Methods, 2nd edn. CRC Press, Boca Raton (2017) 2. Debats, J.M., Mollereau, C.: Ground improvement and reinforcement in seismic zones by vibrocompaction and stone columns. Géo Montréal (2013) 3. Nagy P.: Deep vibrocompaction - dynamic compaction control based on the vibrator movement. Ph.D. thesis, Vienna University of Technology (2018) 4. Massarch, K.R, Zackrisson, P., Fellenius, B.H: Underwater resonance compaction of sand fill. In: 19th International Conference on Soil Mechanics and Geotechnical Engineering Proceedings, Seoul (2017) 5. Nagula, S.S., Grabe, J.: Coupled Eulerian Lagrangian based numerical modelling of vibrocompaction with model vibrator. Comput. Geotech. 123, 103545 (2020) 6. Triantafyllidis, T., Kimmig, I.: A simplified model for vibro compaction of granular soils. Soil Dyn. Earthq. Eng. 122, 261–276 (2019) 7. Nagy, P., Adam, D.: Quality control of deep vibrocompaction based on the vibrator movement. In: XVII ECSMGE-1019 Proceedings. Geotechnical Engineering Foundation of the Future (2019) 8. Kostenko, M., Piotrovski, L.: Machines électriques: Tome 2: Machines à courant alternatif. Edition Mir, Moscow (1969) 9. Boglietti, A., Ferraris, P., Pastorelli, M., Profumo, F., Zimagliac, C.: Induction motors field oriented controllers using data sheets motor parameters. In: International Conference on Electrical Machine (ICEM), Paris, vol. 2, pp. 656–660 (1994) 10. Nelson, H.D.: A finite rotating shaft element using timoshenko beam theory. ASME J. Mech. Des. (102), 793–803 (1980) 11. Lalanne, M., Ferraris, G.: Rotordynamics Prediction in Engineering, 2nd edn. Wiley, England (2011) 12. Nguyen, K.L., et al.: Nonlinear rotordynamics of a drillstringin curved wells: models and numerical techniques. Int. J. Mech. Sci. 166, 105225 (2020) 13. Shampine, L.F., Reichelt, M.W.: The MATLAB ODE suite. SIAM J. Sci. Comput. 18, 1–22 (1997)

Complex Stable and Unstable Subharmonic Vibrations of a Nonlinear Brush-Seal Rotor System Wenbo Ma1 , Yeyin Xu2(B) , Yinghou Jiao3 , and Zhaobo Chen3 1 School of Aerospace Engineering, Xi’an Jiaotong University, Xi’an 710049, People’s

Republic of China 2 State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong

University, Xi’an 710049, People’s Republic of China [email protected] 3 School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, People’s Republic of China

Abstract. This research investigates the complex subharmonic vibrations of the nonlinear brush-seal rotor system. The stable and unstable nonlinear vibrations are calculated and presented as discrete Poincare node results. For such study, we first derive an analytical seal force model from elasticity and bristle-rotor coupling effects. The analytical predictions in terms of the discrete eigenvalues for the stability and bifurcations of the independent and global nonlinear vibrations are presented. Independent vibrations are characterized with solitary rotating speed range and limited within two Saddle node bifurcations. Independent subharmonic1/3 and -1/7 vibrations are discovered in the research. A route from synchronous vibration to subharmonic-1/2 vibration is predicted also. The discrete eigenvalue analysis successfully predicts such route and the corresponding stability evolutions. The stable and unstable subharmonic-1/2 vibrations are obtained. Saddle node, Neimark and period-doubling bifurcations are obtained accurately. The jumping during synchronous vibration and subharmonic-1/2 vibration is obvious while unstable switching which starts from unstable subharmonic-1/2 to unstable subharmonic-1/2 is also successfully predicted. Complex subharmonic-1/2 vibration is presented with the discrete Poincare diagram. Subharmonic vibration illustration is provided with displacement orbits and velocity planes. Unstable synchronous vibration and stable subharmonic-1/2 vibration are illustrated. The provided method and corresponding results serve as guidelines for the theoretical research and practical applications. Keywords: Brush-seal · Subharmonic vibration · Stability · Bifurcations

1 Introduction Sealing machinery with a brush seal structure is an effective and efficient element in rotor system. Brush seals have significant effects on reducing the leakage compared to other types of seals. The brush seal does not appear to be centered between the rotor © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 276–288, 2024. https://doi.org/10.1007/978-3-031-40459-7_19

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and the stator despite the fact that the sealing capacity has not altered during operation. This not only increases the rotor’s thermal efficiency but also the machinery’s level of steadiness which is a significant benefit. As a result, it is one of the most sophisticated seals in rotating equipment and has seen application in a variety of turbines, including industrial turbines, gas turbines used in aviation, etc. [1–4]. The sealing force is primarily determined by a combination of experimental testing and theoretical analysis. For instance, Sun [5–7] et al. constructed a brush-seal rotor experimental bench in order to examine the features of brush filament deformation and vibration. They discovered that the brush filament vibration primarily occured in the area of the brush filament density relaxation. It increased when the pressure ratio was increased and the free end of the brush filament was oscillating in response to this movement. A 3D solid brush seal heat transfer dynamic model was developed in order to conduct an investigation of the thermodynamic properties of the flow and stability fields associated with the brush seal. It is difficult to produce an exact model of the sealing force that is used in the test since the brush filaments have a varying degree of elasticity [8]. A number of researchers suggested approximate theoretical models to determine the approximation of the sealing force by theoretical derivation [9–12]. Because the brush seal force has complicated nonlinear characteristics, its influence on the dynamic characteristics of the rotor system has to be completely studied. Fluent module was utilized by Chai [13] in order to perform a study on the brush seal leakage rate as well as to establish a theoretical model of brush seal rotor system to investigate the dynamics. Wei and Chen [14–16] et al. evaluated nonlinear sealing forces and short bearing oil layer. They employed the Runge-Kutta integration scheme to calculate the nonlinear response of the rotor. The adjustment of the parameters produced a rotor system that was more stable and had a less eccentric phase deviation as a consequence. Zhang et al. [17] used numerical simulation to study the effect of the installation position of the porous media brush filament model on the brush seal-rotor dynamics characteristics. They discovered that the direct stiffness, cross-coupling stiffness, and direct damping of the brush seal installed upstream of the surface were lower than those installed downstream. The direct damping of the brush seal installed upstream of the surface was lower than the direct damping of the brush seal installed downstream. Zhang [18] created a nonlinear rotorseal system that was based on a nonlinear Darcian porous media seal model. The rotor dynamics for various pressure ratios and inlet spin rates were evaluated. Seal part

K,C mg Seal part

Fig. 1. The schematic of the brush-seal rotor system.

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2 Governing Equations 2.1 Nonlinear Brush-Seal Rotor Models For analysis of the subharmonic vibrations of the brush-seal rotor system, we consider a Jeffcott rotor system subjected to brush-seal forces. The schematic of the brush-seal rotor system is presented Fig. 1. The governing equations can be given as ¨ + CX ˙ + KX = F∗ + f (ω) − G MX

(1)

where M is the mass matrix; C is the damping coefficient matrix; K is the stiffness matrix; F∗ is the brush seal force with F∗ = [Fx∗ , Fy∗ ]T ; f (ω) is the centrifugal force and G is the gravity force on the disk. The Fx∗ and Fy∗ have the expression as Fx∗ = A∗ σ2 [(μ1 + 2μ2 σ3 + 3μ3 σ32 + 4μ4 σ33 )e + (0.75μ3 σ22 + μ4 σ22 σ3 )e3 ] × (sin ϕ + cos ϕ)(sin γ + μ cos γ ), Fy∗ = A∗ σ2 [(μ1 + 2μ2 σ3 + 3μ3 σ32 + 4μ4 σ33 )e + (0.75μ3 σ22 + μ4 σ22 σ3 )e3 ] × (sin ϕ + cos ϕ)(cos γ − μ sin γ )

(2)

where μ0 = −2.11271 × 103 , μ1 = 9.08387 × 103 , μ2 = −1.46323 × 104 , μ3 = 1.04705 ×104 ,μ4 =2.80927×103 . A∗ and σi (i = 1 and 2) are the variables related to the brush-seal forces and can be referred to (19). The governing Eq. (1) is processed to be non-dimensional by   √ x = X/Ys , τ = t m k,  = ω M /k, G = (0, mg)T , M = diag(m,m), C = diag(c,c), K = diag(k,k), F∗ = (Fx , Fx )T , f (ω) = (meω2 cos ωt, meω2 sin ωt)T , X = [X , Y ]T , x = [x, y]T , [x, x˙ , y, y˙ ]T = [x1 , x2 , y1 , y2 ]T .

(3)

where x = x1 , x˙ = x2 and y = y1 , y˙ = y2 are the nondimensional displacement and velocity in in x- and y-directions. Ys is the static deflection of the rotor system in y-direction.

3 Results 3.1 Subharmonic Vibration Prediction In this section, the subharmonic vibrations are predicted by the discrete eigenvalue analysis. The prediction is illustrated by the eigenvalues varying with rotating frequency in Fig. 2. The abbreviation “SN”, “NB” and “PD” stand for Saddle node, Neimark and period-doubling bifurcations. “U” stand for unstable subharmonic vibrations. “P-m” (m = 1, 2, 3, · · · ) means subharmonic-1/m vibration.

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Fig. 2. Subharmonic vibration prediction of independent motions based on eigenvalues varying with rotating frequency.

The subharmonic vibration prediction of independent motions is presented in Fig. 2 via eigenvalues varying with rotating frequency. The nonlinear brush-seal rotor system contains four eigenvalues after discretization so that all these four eigenvalue curves are considered in the prediction. During scanning the rotating frequency, two ranges of solitary frequencies are discovered. The solitary frequencies go to two ends where one real eigenvalue crosses positive lines and the rests remain with magnitudes less than one. The subharmonic-1/3 vibration is predicted at  ∈ (4.452, 5.301) and subharmonic-1/7 vibration is predicted at  ∈ (5.503, 5.791). While one point we need to pay close attention is  = 5.791 which characterizes with one real eigenvalue equal to one, one pair of the complex eigenvalues with magnitudes greater than one and the rest with magnitude smaller than one. So an unstable Saddle node bifurcation happens at  = 5.791.

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Fig. 3. Subharmonic vibration prediction of global motions based on eigenvalues varying with rotating frequency.

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Fig. 3. (continued)

The subharmonic vibration prediction of global motions is presented in Fig. 3. The global subharmonic vibration becomes complex and the illustrations are presented in two

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rotating frequency ranges. The global subharmonic vibrations during  ∈ (5.0, 10.0) are presented in Figs. 3(a) and (b). The eigenvalues of the subharmonic vibrations from  ∈ (0.0, 5.0) are with magnitudes less than one so that the nonlinear vibration is stable. At  = 9.23, an unstable period-doubling bifurcation occurs and an unstable subharmonic-1/2 vibration is predicted. The unstable subharmonic-1/2 vibration is generated from unstable synchronous vibration. During such process, the nonlinear brush rotor system may behavior large displacement. At  = 7.85, a stable period-doubling bifurcation occurs which generates subharmonic-1/4 vibrations. The global subharmonic vibrations during  ∈ (10.1, 12.6) are presented in Figs. 3(c) and (d). During this process, two period-doubling bifurcations occur at  = 11.47 and  = 12.32 for generating subharmonic-1/2 and 1/4 vibrations, respectively. The real parts and magnitudes of the eigenvalues are presented together for better understanding the stability evolution and subharmonic vibration prediction.

Fig. 4. Stable and unstable subharmonic-1/3 and 1/7 vibrations in the brush-seal rotor system.

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3.2 Subharmonic-1/3 and 1/7 Vibration In Fig. 4, the stable and unstable independent subharmonic-1/3 and 1/7 vibrations are presented with discrete bifurcation diagram. The independent subharmonic-1/3 vibration is simple with two boundaries at  = 4.452 and 5.302 where two Saddle node bifurcations occur. The nonlinear brush-seal rotor system would experiences jumping between independent subharmonic-1/3 vibration and global vibrations. The subharmonic-1/7 vibration occurs between  ∈ (5.503, 5.791) where a stable Saddle node bifurcation occurs on the left and an unstable Saddle node bifurcation happens on the right. A Neimark bifurcation occurs at  = 5.76 where quasi subharmonic-1/7 vibration is generated. The brush-seal rotor system behaviors quasi periodic vibration and becomes unstable when increasing speed. In Fig. 5, the stable and unstable subharmonic-1/2 vibrations and the route from synchronous vibration into subharmonic-1/2 vibration are presented. The synchronous vibration is stable from rotating frequency  = 0.0. At  = 5.917 and  = 6.53, a Saddle node bifurcation and a Neimark bifurcation occur. The nonlinear brush-seal rotor system tends to experience chatting and quasi periodic vibrations. The nonlinear rotor system should increase speed faster to go through this part of range in practical. A subharmonic-1/2 vibration occurs at  = 9.23 where unstable nonlinear vibrations contain half frequency components. At  = 7.85, another period-doubling bifurcation occurs and subharmonic-1/4 vibration happens. The nonlinear brush-seal rotor system oscillates with quarter frequency component. The subharmonic-1/2 vibration is more complex with various bifurcations and complicate stability shift. Neimark bifurcations happen at  = 10.37 and  = 11.08 produce quasi periodic subharmonic-1/2 vibrations. Saddle node bifurcations happen at  = 10.197 and  = 14.307 generate jumping vibrations. Period-doubling bifurcation at  = 11.47 and  = 12.32 trigger subharmonic-1/4 and subharmonic-1/2 vibrations in the nonlinear brush-seal rotor system. Operation during these frequency ranges should be more care to avoid dangerous subharmonic vibrations.

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Fig. 5. Stable and unstable subharmonic-1/2 vibrations in the brush-seal rotor system.

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Fig. 6. The synchronous vibration illustration of the brush-seal rotor system.

3.3 Subharmonic Vibration Illustration The subharmonic vibration illustrations are presented in this section. The unstable synchronous vibration is presented first and then is the subharmonic-1/2 vibration illustration. In the synchronous vibration, the red circular symbols represent the unstable vibration solutions. Since it is an unstable vibration, the transient orbit starts from the unstable vibration solution and moves off after tens of periods approaching another stable vibration. The transient process can be observed in displacement orbit in Fig. 6(a) and velocity plane in Fig. 7(b).

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Fig. 7. The Subharmonic vibration illustration of the brush-seal rotor system.

The subharmonic-1/2 vibration is presented in Fig. 7. The circular symbols represent the initial conditions (“I.C”) and starting point of the second period (“1T”). It is a stable subharmonic-1/2 oscillation and the displacement orbit and velocity plane are clear presented. Half frequency components are contained in the vibrations.

4 Conclusion Subharmonic vibrations of a nonlinear brush-seal rotor system were investigated in this research. The stable and unstable subharmonic-1/3, 1/7 and 1/2 vibrations were obtained. The first two above subharmonic vibrations were independent and only occurred in solitary rotating frequency range. The route from synchronous vibration to subharmonic-1/2 vibration was also obtained. Such route contains Saddle node bifurcations, Neimark bifurcations and period-doubling bifurcations. Jumping phenomenon of the subharmonic vibrations was observed. The obtained data and applied methods provide a good understanding of the brush-seal rotor system.

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References 1. Holle, G.F., Krishnan, M.R.: Gas turbine engine brush seal applications. In: Proceedings of the 26st AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Orlando, FL, USA, AIAA 90-2142 (1990) 2. Menendez, R.P., Xia, J.: Recent developments in brush seals for large industrial gas turbines. In: Proceedings of the 36nd AIAA/ASME/SAE/ASEE/Joint Propulsion Conference and Exhibit, Huntsville, AL USA, AIAA 00-3374 (2000) 3. Aslan-Zada, F.E., Mammadov, V.A., Dohnal, F.: Brush seals and labyrinth seals in gas turbine applications. Proc. IMech. A. J. Pow. 227(2), 216–230 (2013) 4. Chupp, R.E., Hendricks, R.C., Lattime, S.B., Steinetz, B.M.: Sealing in turbomachinery. J. Propul. Power 22(2), 313–349 (2006) 5. Sun, D., Du, C.Y., Liu, Y.Q., et al.: Experiment on bristle deflection and oscillation characteristics of brush seals. Acta Aeronautica et Astronautica Sinica 41(10), 123364 (2020). (in Chinese). https://doi.org/10.7527/S1000-6893.2019.23364 6. Sun, D., Li, G., AI, Y., et al.: Numerical study on heat transfer mechanism of brush seal based on three-dimensional solid modeling. J. Aerospace Power 34(8), 1633–1643 (2019) 7. Sun, D., Ding, H., Li, G., et al.: Theory and experiment on the leakage characteristics of brush seals based on fluid-structure interaction. J. Aerospace Power 34(7), 1519–1529 (2019) 8. Lv, B., Li, W., Ouyang, H.: Moving force-induced vibration of a rotating beam with elastic boundary conditions. Int. J. Str. Stab. Dyn. 15(1), 1450035 (2015) 9. Sharatchandra, M.C., Rhode, D.L.: Computed effects of rotor-induced swirl on brush seal performance-part 2: bristle force analysis. J. Tribol. 118, 920–926 (1996) 10. Stango, R.J., Zhao, H., Shia, C.Y.: Analysis of contact mechanics for rotor-bristle interference of brush seal. J. Tribol. 125(2), 414–420 (2003) 11. Zhao, H., Stango, R.J.: Effect of flow-induced radial load on brush/rotor contact mechanics. J. Tribol. 126(1), 208–214 (2004) 12. Huang, S., Suo, S., Li, Y., Wang, Y.: Theoretical and experimental investigation on tip forces and temperature distributions of the brush seal coupled aerodynamic force. J. Eng. Gas Turbines Power 136, 052502 (2014) 13. Chai, B., Fu, X.: Numerical simulation on flow and temperature distributions of brush seals. Lubr. Eng. 41(2), 0254-0150 (2016) 14. Cavalca, K.L., Weber, H.I. (eds.) Proceedings of the 10th International Conference on Rotor Dynamics–IFToMM, vol. 2, no. 61. Springer, Cham (2018) 15. Wei, Y., Chen, Z., Jiao, Y., Liu, S.: Computational analysis of nonlinear dynamics of a multi-disk rotor-bearing-brush seal system. In: Cavalca, K.L., Weber, H.I. (eds.) IFToMM 2018. MMS, vol. 62, pp. 350–362. Springer, Cham (2019). https://doi.org/10.1007/978-3319-99270-9_25 16. Wei, Y., Chen, Z., Dowell, E.H.: Nonlinear characteristics analysis of a rotor-bearing-brush seal system. Int. J. Struct. Stab. Dyn. 18(05), 1850063 (2018) 17. Ha, Y., Ha, T., Byun, J., Lee, Y.: Leakage effects due to bristle deflection and wear in hybrid brush seal of high-pressure steam turbine. Tribol. Int. 106325 (2020) 18. Zhang, Y., Li, Z., Li, J., Yan, X.: Numerical investigation on the leakage flow and rotordynamic characteristics of brush-labyrinth seal. J. Xi’An Jiao Tong Univ. 53(3) (2019) 19. Xu, Y., Zhao, R., Jiao, Y., Chen, Z.: Stability and bifurcations of complex vibrations in a nonlinear brush-seal rotor system. Chaos 33, 033113 (2023)

Multi-objective Optimization of Active Dry Friction Damper-Rotor Systems Based on Predictive Control Minghong Jiang, Wengheng Li, Xianghong Gao, and Changsheng Zhu(B) College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China [email protected]

Abstract. Optimization on rotor-dynamics has increasingly attracted attention from numerous researchers, although concentrations on the optimization of timeinvariant system parameters result in relatively low versatilities. In this paper, a multi-objective optimization framework based on a novel predictive control method is proposed for rotor systems with active dry friction dampers. Instead of optimizing normal forces of friction dampers as most current researches do, we optimize the controller parameters instead to improve optimization efficiency. The predictive controller, incorporated with the augmented Kalman filter, is inserted into the non-dominated sorting genetic algorithm II to accomplish the multiobjective optimization. To validate the efficacy of the optimization procedure, numerical investigations are performed for a rotor system with two active friction dampers. Results demonstrate that the proposed control method can reduce the rotor vibration significantly after optimization with constraints on equivalent stiffnesses of the friction dampers, and it is superior to traditional direct optimization methods when the optimization on a specific system with different unbalance levels is considered. Keywords: Active Dry Friction Damper · Multi-objective Optimization · Rotor · Predictive Control · Augmented Kalman Filter

Rotating machinery play important roles in modern engineering industries. The safe operation of turbomachinery highly depends on the vibration level of rotor systems. This is especially true within the context of pursuit for higher power density in turbomachinery such as aero-engines [1, 2], where the high operating speeds and slender rotor structures result in several critical speeds to be passed before the operational speed range is reached. To ensure that critical speeds can be traversed safely, some early researches concentrated on the optimization of geometrical shapes [3, 4] or support parameters [5, 6] during the rotor design process. However, the performances of those optimization-based methods can not be further improved once the rotor is manufactured, which limits its applications to variant operating conditions. In the past few decades, the developments of the active dry friction dampers (ADFDs) provide the basis for active vibration control of rotors. The vibration attenuation capacity The present work is fully supported by National Science and Technology Major Project (J2019IV-0005-0073). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 289–303, 2024. https://doi.org/10.1007/978-3-031-40459-7_20

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of the ADFD can be easily improved by applying appropriate clamping forces acted on the friction interfaces. Meanwhile, they are also robust to temperature change and more convenient to maintain compared with other types of dynamically-similar active devices [7], e.g., magneto-rheological dampers [8]. Thus, their applications in active vibration control have attracted attention from numerous researchers. Earlier investigations on ADFD include parametric analysis and optimization on parameters. Sorge [9] investigated the unbalanced responses of a shaft with ADFDs attached to both ends, where the optimal clamping forces for the ADFDs are derived by searching for the local minimum of the analytical expression. In Refs [10] and [11], the equivalent damping techniques are used to analyze the vibration attenuation effect numerically such that optimum clamping force can be achieved. Similar to the research performed by Sorge, they presented the lumped parameter model as the numerical example. To extend the optimization procedure to more complex rotors, Cavalini et al. [12] further applied the genetic algorithm to improve the vibration control efficiency. The aforementioned researches, however, restrict themselves to optimizing constant normal forces for ADFDs, which has been proved as inefficient since the vibration attenuation capacity of the ADFD is related to the vibration level as well. Several control schemes have been proposed for online optimization of clamping forces according to the real-time rotor vibration levels. In 2019, Liao [13] and Wang et al. [14] proposed the proportional control to optimize the clamping force of the ADFD such that the rotor vibration amplitude can be sustained within a prescribed level. The efficacy of the proposed method is validated experimentally in their researches. Li et al. [15] proposed an open-loop polynomial interpolation method to generate the optimum normal force under every single rotational speed. Liu et al. [16] optimized the timevariant normal force based on the results of nonlinear complex mode analysis, and the modal damping ratio is maximized over a wide speed range. However, those methods could either suffer from the defects of time-consuming, or ambiguous knowledge to damping capacities. On the other hand, nearly all these existing optimizations performed for ADFDs-rotor systems were based on direct optimization (DO) of normal forces, which could change for different loads. In consequence, the efficiency of those methods should be further improved. To address those problems, a predictive-control-based optimization (PCO) framework is proposed in this paper for the multi-objective optimization of the ADFDs-rotor systems. Instead of directly tuning normal forces of friction dampers as most current researches do, inspired by [17], PCO optimizes the parameters embedded in the control algorithm instead, which shows the robustness to different unbalanced loads. To get the unmeasurable displacements required for controller implementation, we further integrate the augmented Kalman filter (AKF) with the PCO to reconstruct the rotor vibrations. The remaining of this work is organized in the following sequence. In Sect. 1.1, the configuration of the ADFD and corresponding dynamic modeling method is presented. The design of the PCO and the AKF are elaborated in Sects. 1.2 and 1.3. In Sect. 2.1 we discussed the accuracy of the AKF and the surrogate models and the optimization results are presented in Sect. 2.2. In Sect. 2.3, the optimization results of the PCO are further compared with the traditional DO methods. Finally, conclusions are drawn in Sect. 3.

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1 Problem Formulation 1.1 Dynamic Modeling of the ADFDs-rotor System The overall configuration of the investigated ADFDs-rotor system, which is a simulation test setup for a high pressure aeroengine rotor equipped with a novel electromagnetic ADFD, is displayed in Fig. 1(a) and the detailed structure of the ADFD is presented in Fig. 1(b). As shown in Fig. 1(b), The movable friction pad is attached to the elastic ring while the stationary pad is fixed on the iron core. The rotor shaft is supported by the rolling bearing placed on the elastic ring, thus the whirling motion of the rotor will result in the planar motion of moving friction pad. Then the relative motion between the moving friction pad and the stationary pad is induced and vibration energy is dissipated. By controlling the coil current during operation, the normal force acted on the contact interface is regulated online thus the damping capacity of the ADFD is adjusted.

Fig. 1. Schematics of the ADFDs-rotor system: (a) configuration of the rotor system equipped with the novel ADFD; (b) cross-section of the ADFD; (c) schematic diagram of the Coulombspring element

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In this paper, we particularly focus on the control of transient vibration when critical speeds are traversed. For the ADFDs-rotor system shown in Fig. 1(a), the transient dynamic equation can be obtained based on the finite element method as ¨ + (D + ωG)X ˙ + (K + αKs )X = fub + fdf MX

(1)

where M, D, G and K are, respectively, the mass, damping, gyroscopic and stiffness matrices; Ks is the skew symmetric matrix induced by speed variation [12, 18]; ω and α are the rotational speed and its change rate; fub and fdf are the unbalanced force and dry friction force vector. The Craig-Bampton method is utilized to perform coordinate transformation, where friction nodes are taken as interface degrees of freedom (DOFs) and other DOFs are treated as interior DOFs. The corresponding transformation matrix is given as   m  c (2) = 0 I where  is the modal transformation matrix; m is the normal modes obtained by solving the undamped and non-gyroscopic eigenvalue problem with only interior DOFs taken into consideration;  c is the constrained mode set by executing unit displacement on each master DOF. The details of the order reduction procedure can be found in Ref [19]. Further pre-multiplying Eq. (1) with the transpose of matrix  yields   ˜ q¨ + (D ˜ + ωG) ˜ q˙ + K ˜ + αK ˜ s q = f˜ub + f˜df (3) M ˜ = T M is the reduced mass matrix and the superscript T stands for the where M transpose of matrix; f˜ub = T fub is the reduced unbalanced force vector; other matrices and vectors take similar transformations and q is the modal displacement vector with constraint modes included. For relatively uniform surface-to-surface contact, as presented in Fig. 1 (c), the Coulomb-spring element is applied to model the friction force [7, 20]. Although this model ignores the influences of the Stribeck effect and frictional lag, it’s much simpler than other dynamic friction models. Thus, it has been widely used for the controller design of ADFDs and experimental investigations [7] also suggest the effectiveness of this model as far as the design of normal force controller is concerned. Generally, the friction force in the Coulomb-spring element can be expressed as   a b (4) − Zi,k Fi,k = −ki Zi,k where the subscript i stands for the ith ADFD; Fi,k = fxi,k + jfyi,k is the friction force at a = x a + jy a is the displacement of the time instant k; ki is the tangential stiffness; Zi,k i,k i,k b = x b + jy b is the ith node a, which is displacement of the movable friction pad; Zi,k i,k i,k displacement of the ith Coulomb element b.

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Generally, two contact states, including stiction and slippery, are considered in this friction model. The trajectory tracking method is utilized herein to distinguish them from each other [21]. It uses the following criteria to check whether stiction occurs for the ith ADFD    a  b ki Zi,k − Zi,k−1 (5)  ≤ μi Ni,k where μi and Ni,k are the dynamic friction coefficient and normal force of the ith ADFD. 1.2 Development of the Controller and Observer In 2004, Lu [17] proposed a predictive control method for ADFDs. However, his research is mainly restricted to one-dimensional cases and the excitations are known a priori, which are not satisfied for most ADFDs-rotor systems. In this section, we further extended the formalization of the predictive control method into two-dimensional case and an AKF is integrated with it to reconstruct the displacements with unknown system inputs. The basic principle of the predictive control method is that the ADFD should stay in slippery state such that vibratory energy can be dissipated. To achieve this goal, an extra controller parameter, βi , is introduced and the normal force is expressed as    a  b βi ki Zi,k − Zi,k−1  Ni,k = (6) μi where βi ∈ [0, 1] is the parameter controlling the extent of slippery. Comparing Eq. (4) and Eq. (5), the contact interface of the ADFD is separated if βi equals to 0 and will stay sticked when βi is taken as 1. In other words, the predictive control method guarantees that the friction interface keeps sliding if only βi < 1. By tuning the value of βi , the extent of sliding is also adjusted. Then, the optimization can be performed to search for the optimal parameter βi . However, the displacements of node a are hard to measure. Thus, the reconstruction of rotor vibration is necessary. To accomplish this task, expressing Eq. (2) in its state-space form, i.e.     O I O     ˙ Q= Q + ˜ −1 K ˜ + αK ˜ s −M ˜ D ˜ + ωG ˜ ˜ −1 T (fub + fdf ) (7) −M −M = AQ + B(fub + fdf ) It can be seen from Eq. (7) that the unbalanced disturbance emerges in the input vector, which will induce significant displacement reconstruction errors if they are not considered. To solve this problem, the AKF [22] is applied to estimate the displacements and forces simultaneously. For the accelerating rotor, the unbalanced forces can be expressed as [23] 

  cos(ϕ) − sin(ϕ) mi ei cos ϕeq,i 4 2 = A(ω, ϕ)faub,i (8) fub,i = ω + α sin(ϕ) cos(ϕ) mi ei sin ϕeq,i

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where fub,i is the unbalanced force of the ith disc; mi ei and ϕeq,i are, respectively, the equivalent unbalance magnitude and azimuth of the ith disc; ϕ is the rotation angle of the rotor. For the ADFDs-rotor system with fixed speed variation rate, these two terms, namely mi ei and ϕeq,i , are slowly time-varying parameters. They are considered as the additional constant state variable faub,i . Thus, the unbalanced disturbance is expressed as ⎤ ⎡ faub,1 ⎢ .. ⎥ ⎢ . ⎥ ⎥ ⎢ ⎥ fub = I ⊗ A(ω, ϕ)⎢ (9) ⎢ faub,i ⎥ == f aub ⎢ . ⎥ ⎣ .. ⎦ faub,n where the operator ⊗ stands for the Kronecker product; faub is the additional state variable vector. Under the assumption that the system is time-invariant during the sampling time interval, discretizing Eq. (7) and substituting Eq. (9) into Eq. (7) yields     Gk−1 Hk−1  a Hk−1 fdf, k−1 Qk−1 + Qak = O O I (10) a a a = Gk−1 Qk−1 + Hk−1 uk−1 T  T is the augmented state variables at time instant iT ; Gk−1 = where Qai = qiT q˙ iT faub,i eAk−1 T is the discretized state matrix; Hk−1 = A−1 k−1 (Gk−1 − I)Bk−1 is the discretized a a and Hk−1 are, respectively, the augmented state and input matrices; input matrix; Gk−1 uk−1 is the generalized input vector. In the AKF, the covariance matrix P k|k is used to describe the estimation error e k|k , i.e.    T   a ˆa ˆa = E e k|k eTk|k Q (11) − Q P k|k = E Qak − Q k|k k k|k where the operator E(·) stands for the expectation of contents; P k|k is the covariance ˆ a at time instant kT and e k|k is the estimation error. matrix of the optimal estimates Q k|k A prediction can be performed based on last optimal estimations, i.e. a a ˆa ˆa Q k|k−1 = Gk−1 Q k−1|k−1 + Hk−1 uk−1 a P aT P k|k−1 = Gk−1 k−1|k−1 Gk−1 + S

(12)

ˆa where S stands for the covariance matrix of state noises; Q k|k−1 is the prediction derived a ˆ from last optimal estimate Q k−1|k−1 ; P k|k−1 is the a priori covariance matrix. Then the Kalman gain matrix Kk , the optimal estimation of the augmented state Qak|k and the covariance matrix for the optimal estimates P k|k can be updated as Kk =

P k|k−1 CT CP k|k−1 CT + R

(13)

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  ˆa =Q ˆa ˆ ak|k−1 Y Q + K − C Q k k k|k k−1|k−1

(14)

P k|k = (I − Kk C)P k|k−1

(15)

where R stands for the covariance matrix for measurement noises; C is the output matrix; Yk is the displacements measured by the displacement sensors. Combining Eq. (2) and Eq. (14), the displacements of node a for the ith ADFD can be directly obtained as a ˆa ˆa Zi,k =Q mix ,k + j Qmix ,k

(16)

ˆa ˆa where Q mix ,k and Qmiy ,k are, respectively, the observed displacements of the ith ADFD at time instant kT, which are retained in the modal transformation procedure as interface DOFs, as indicated by Eq. (2). Similarly, the complex displacements of node b at time instant (k-1)T can be obtained based on Eq. (2), i.e.  b a Zi,k−1 = Zi,k−1 + Fi,k−1 k i (17) where Fi,k−1 can be directly measured by the force sensor. Substituting Eqs. (16) and (17) back into Eq. (6), the required normal force at instant k is obtained. 1.3 Optimization Method It has been mentioned above that controller parameter βi influence the contact states of the ADFDs, thus they should be carefully optimized. In our research, we take the unbalanced responses and the equivalent stiffness of the ADFD as the fitness functions to be minimized. This is motivated by the fact that excessive normal force will induce significant increase in equivalent stiffness of the ADFD, which is not expected in the system. A general formalization of the optimization problem is presented as   Min f 1 (β), f2 (β) (18) subject to: fc (β) ≤ 0 0 ≤ βi ≤ 1, i= 1, · · · ,nd

where f 1 (β) is the unbalanced response fitness function normalized with no-DFD cases; f2 (β) is the equivalent stiffness fitness function; f c (β) is the normalized constraint function; ndf is the number of ADFDs, in this paper, ndf equals to 2. More specifically, the fitness functions given in Eq. (18) are expressed as the weighted sums of corresponding components, i.e. 

f 1 (β) = A1 + A2 /2  (19) f2 (β) = Keq,1 + Keq,2 /2

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where A1 and A2 are, respectively, the maximum amplitudes of disk 1 and 2, which are normalized with their ADFD-free counterparts; Keq,i is the equivalent stiffness [21] for the ith ADFD. To limit the vibration amplitude of disk 4, f c (β) is expressed as  f c (β) = Ap A4 − 1 (20) where A4 is the resonant amplitude of disk 4; Ap is a prescribed amplitude. Due to the nonlinearity introduced by the ADFD, the functions in Eqs. (19) and (20) can not be solved analytically. To overcome the low efficiency of numerical integration, the RBFNNs are introduced to generate the surrogate models. The structure of the RBFNNs is shown in Fig. 2. Three RBFNNs are constructed for f 1 (β), f2 (β) and f c (β), where Gaussian kernels are applied as basis functions. And the NSGA-II algorithm will be applied to search for the Pareto front once the surrogate model is available. NN : f1 ( β ) ϕ1,1

M

S1

fˆ1

ϕ1,m

β jˈ1

NN : f 2 ( β ) ϕ2,1

M

M

S2

fˆ2

S3

fˆc

ϕ2,m

β j ,n

d

NN : f c ( β ) ϕ3,1

M ϕ3,m

Fig. 2. Radial basis neural networks with the Gaussian kernels

In NSGA-II, the boundary constraints for β are guaranteed in the individual initialization process and the generated population is fed into the RBFNNs to calculate fitness functions. For individuals satisfying f c (β) ≤ 0, the nondominated sorting procedure is directly utilized to get their nondominated ranks. For those unsatisfying the constraints, they are sorted by their violation level of the constraint and then added to the rank set subsequently. In each nondominated rank, the crowding distances are computed to spread the solution evenly on the Pareto front. The parents are picked by the classical binary tour selection procedure and SBX crossover and polynomial mutation are applied in the genetic operators. The optimization procedure presented above is finally summarized in Fig. 3.

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9ZGXZ

5VZOSGR2GZOTN_VKXI[HK YGSVROTM

8GTJUSVUV[RGZOUT OTOZOGRO`GZOUT

)UTYZX[IZOUTUL ZNKY[XXUMGZKSUJKR

)GRI[RGZKLOZTKYY L[TIZOUTY

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INORJ

4U

4K]VUV[RGZOUT

3G^OS[SX[T%

_KY KTJ

Fig. 3. Flow chart of the optimization procedure

2 Results and Discussions 2.1 The Accuracy of the AKF and Surrogate Models Numerical investigations are performed in this section to validate the efficacy of the optimization procedure. The finite element model of the high-pressure aeroengine rotor equipped with the novel ADFD is given in Fig. 4 and the geometrical parameters are summarized in Table 1. The shaft and the disks are made of steel and the Young’s modulus, Poisson’s ratio and density are, respectively, 2 × 105 MPa, 0.3 and 7800 kg/m3 . The mass and diametric moment of inertia of disks are lumped at the nodes of corresponding shaft elements. The first critical speed for the rotor is 3147 r/min and the corresponding vibration shape is translational mode.

Fig. 4. Finite element model of the ADFDs-rotor system

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M. Jiang et al. Table 1. Geometrical parameters of the ADFDs-rotor system.

Items

Unit

Value

Length of shaft segments (S1–S11)

mm

113.0/64.9/27.8/92.0/78.4/77.5/154.7/24.6/79.6/80.4/70.0

Outer diameters of shaft S1–S11

mm

36.0/40.0/64.0/105.0/97.0/70.6/76.0/53.3/45.0/39.0/60.0

Inner diameters of shaft S1–S11

mm

27.0/27.0/56.0/100.0/92.0/64.0/64.0/42.8/19.0/19.0/19.0

Diameters of disk 1–4 mm

209/220/232/370

Length of disk 1–4

20.8/15.2/28/40

mm

The total weight of the investigated rotor is 62.7 kg and the laboratory operating speed is set as 6000 r/min. According to the standard ISO1940-1 with grade G2.5, the allowable residual unbalance is 24.95 g·cm. Thus, in order to activate the translational mode, the unbalance of 10 g·cm is added at node 6 and 13 with the same phase respectively. The ADFDs are mounted at node 2 and 17. And the contact material of the friction pair is steel/powdered metal, whose dynamic friction coefficient is 0.15 and tangential stiffness is 2 × 107 N/m. The accuracy of the reconstructed displacements determines whether the PCO can work reasonably, as indicated by Eq. (6). Thus, to verify that the AKF can reconstruct the vibration accurately, the estimated displacements for ADFD 1 and 2 are compared with the accurate displacements, as shown in Fig. 5. It can be concluded that, the AKF can reconstruct the unmeasurable displacements accurately, even at the high-frequency region where traditional Luenberger observer fails.

Fig. 5. Comparisons on the estimated and accurate displacements of ADFDs 1 and 2: (a) response curve of the ADFD 1; (b) response curve of the ADFD 2.

Meanwhile, the accuracy of the three RBFNNs also matters. A total of 200 samples and 150 neurons are applied to achieve sufficient accuracy. The final R2 correlation

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coefficients of the three RBFNNs are 0.998, 0.998 and 1.000 respectively, which indicates excellent regression performances of those networks on the data set. 2.2 Results of Optimization In this section, the optimization results are presented based on the aforementioned optimization procedure. Figure 6 shows the Pareto front obtained from the multi-objective optimization procedure for the ADFDs-rotor system. The Pareto front can be obtained evenly based on the NSGA-II algorithm. It can be seen from Fig. 6 that the objective of minimizing equivalent stiffness and minimizing unbalanced responses are generally in conflict with each other. Generally, the final choice of optimal control parameters can be performed based on practical requirements. To explain the influences of different choices from the Pareto set, five solutions, marked by a - e in Fig. 6, are selected for comparison. The controller parameters for each solution and the corresponding computed equivalent stiffnesses are tabulated in Table 2. The larger the controller parameter is, the larger equivalent stiffness of the ADFD will become. This indicates that, for the ADFDs-rotor system, the excessive controller parameter β will cause stiffening effect of the elastic suspension. To further illustrate the influences of this stiffening effect, the response curves of disk 1 for cases a - e are compared in Fig. 7. The larger equivalent stiffness is the larger critical speed will become. In this case, excessive equivalent stiffness will make the critical speeds of the ADFDs-rotor system to significantly deviate from the original design, which should be avoided for the sake of secure operation.

Fig. 6. The Pareto front obtained the by proposed multi-objective optimization framework for the ADFDs-rotor system

2.3 Comparison with Classical Direct Optimization Methods To show the superiority of the proposed PCO method for the ADFDs, the results are further compared with the DO method. A total of 500 populations are included in each

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M. Jiang et al. Table 2. The controller parameters and the equivalent stiffnesses for selected solutions. Keq,1 /N·m−1

Keq,2 /N·m−1

1.59 × 106

2.13 × 106

0.918

9.40 × 105

9.40 × 105 1.17 × 105

Case

β1

β2

a

0.959

0.982

b

0.917

→

c

0.652

0.689

7.39 × 104

d

0.000

0.406

0.65

2.67 × 104

e

0.000

0.222

0.04

9.65 × 103

Fig. 7. The response curves of disk 1 with different solutions applied

generation and the maximum number iteration is set as 50. The pareto fronts for the two methods when the unbalanced mass is 5 g·cm are compared in Fig. 8. It can be seen that the number of Pareto-optimal solutions obtained based on the PCO is much larger than that of DO. Although some solutions for passive system dominate those of the active system, the discrepancies are much smaller when f 1 (β) < 0.03 and f2 (β) is around 1 × 106 N/m. Figure 9 further compares the pareto fronts obtained by both two methods for the same system with different load levels (case 1: unbalanced mass is 5 g·cm; case 2: unbalanced mass is 10 g·cm). The fact that the pareto fronts are nearly the same for both two methods indicates that the dynamical similarity can be utilized to enhance the efficiency of the optimization procedure. For traditional DO method, however, this process is not easy to implement. As presented in Table 3, when the same objective is set as the selection criteria, the normal forces obtained by the DO method is quite different. This makes multiple runs of optimization, incorporated with appropriate initial guess for the range of normal forces, must be conducted for different load cases unless a posterior interpolation being performed to improve the optimization efficiency. For the proposed PCO, the results are generally much closer to each other and the solutions can be directly applied to the same system with different load levels, which would be much more efficient.

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Fig. 8. Comparison on the pareto fronts obtained by PCO and DO

Fig. 9. Pareto front obtained by the two methods for the system with different load levels

Table 3. The optimization results of the two methods with the same criteria considered (f2 (β) ≤ 5.5 × 105 N · m−1 , f 1 (β) < 0.03). Item

Unit

PCO, case 1

PCO, case 2

DO, case 1

DO, case 2

β1

/

0.685

0.742

/

/

β2

/

0.694

0.730

/

/

N1

N

/

/

117.29

251.05

N2

N

/

/

111.27

207.93

f 1 (β)

/

0.0261

0.0259

0.0165

0.0164

f2 (β)

N·m−1

5.30 × 105

5.33 × 105

5.541 × 105

5.547 × 105

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M. Jiang et al.

3 Conclusions In this paper, we present an optimization framework based on predictive control (PCO) to perform multi-objective optimization for active dry friction dampers-rotor systems. The feasible range for design variables is fixed in the PCO framework, thus no initial guess is required to get the appropriate range of normal forces as traditional direct optimization method does. Numerical investigations are performed to show the efficacy and efficiency of the proposed method. Results reveal that by optimizing the controller parameter instead of normal forces, the PCO manifest itself as robust to the changes of unbalance levels, thus the optimization results for one particular unbalance level can be easily applied to the same systems with different load levels.

References 1. Jia, S., Zheng, L., Mei, Q.: Flexible rotor optimization design with considering the uncertainty of unbalance distribution. Int. J. Simul. Multi. Design Optim. 11(19), 1–8 (2020) 2. Wang, S., Liao, M.: Experimental investigation of an active elastic support/dry friction damper on vibration control of rotor systems. Int. J. Turbo Jet-Engines 31(1), 13–17 (2014) 3. Soorajkrishna, S., Sekhar, A., Shankar, K.: Multiobjective optimization of rotor-bearing systems with an investigation of goal programming approach. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 233(12), 4270–4287 (2018) 4. Huang, J., Zheng, L., Mei, Q.: Design and optimization method of a two-disk rotor system. Int. J. Turbo Jet-Engines 33(1), 1–8 (2016) 5. Liu, S., Yang, B.: Optimal placement of water-lubricated rubber bearings for vibration reduction of flexible multistage rotor systems. J. Sound Vib. 407, 332–349 (2017) 6. Tarlani, B., Bahrami, A.: Dynamic analysis of a high-speed rotor supported by optimized bearings at steady and transient operating conditions. J. Vib. Eng. Technol. 11(3), 1151–1161 (2022) 7. Lu, L., Lin, C., Lin, G., et al.: Experiment and analysis of a fuzzy-controlled piezoelectric seismic isolation system. J. Sound Vib. 329(11), 1992–2014 (2010) 8. Zhu, C.: A disk-type magneto-rheological fluid damper for rotor system vibration control. J. Sound Vib. 283(3–5), 1051–1069 (2005) 9. Sorge, F.: Rotor whirl damping by dry friction suspension systems. Meccanica 43(6), 577–589 (2008) 10. Ni, D., Zhu, R.: Influencing factors of vibration suppression performance for a smart spring device. J. Vibr. Shock 31(23), 87–91+98 (2012) 11. Ni, D., Zhu, R., Lu, F.: A parametric design method for a smart damped spring vibration reduction system. J. Vibr. Shock 33(23), 116–121+144 (2014) 12. Cavalini, A., Galavotti, T., Morais, T., et al.: Vibration attenuation in rotating machines using smart spring mechanism. Math. Probl. Eng. 2011, 1–14 (2011) 13. Liao, M., Li, Y., Song, M., et al.: Dynamics modeling and numerical analysis of rotor with elastic support/dry friction dampers. Trans. Nanjing Univ. Aeronaut. Astronaut. 35(1), 69–83 (2018) 14. Wang, S., Liao, M., Song, M., et al.: An active elastic support/dry friction damper: new modeling and analysis for vibration control of rotor systems. In: Proceedings of the 10th International Conference on Rotor Dynamics – IFToMM, pp. 19–33 (2019) 15. Li, M., Ma, L., Wu, C., et al.: Study on the vibration active control of three-support shafting with smart spring while accelerating over the critical speed. Appl. Sci. 10(17), 6100 (2020)

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16. Liu, D., Zhou, L., Zhang, D., et al.: A strategy of vibration control for rotors with dry friction dampers. J. Vibr. Control 29(13–14), 2907–2920 (2022) 17. Lu, L.Y.: Predictive control of seismic structures with semi-active friction dampers. Earthq. Eng. Struct. Dynam. 33(5), 647–668 (2004) 18. Simoes, R., Steffen, J., Der Hagopian, J., et al.: Modal active vibration control of a rotor using piezoelectric stack actuators. J. Vib. Control 13(1), 45–64 (2007) 19. Zheng, Z., Zhu, F., Zhang, D., Xie, Y.: A developed component mode synthesis for parametric response analysis of large-scale asymmetric rotor. J. Mech. Sci. Technol. 33(3), 995–1005 (2019). https://doi.org/10.1007/s12206-019-0201-9 20. Park, Y., Kim, K.: Semi-active vibration control of space truss structures by friction damper for maximization of modal damping ratio. J. Sound Vib. 332(20), 4817–4828 (2013) 21. Xiao, F., Li, L., Wu, Y., et al.: A linearization method based on 3D contact model for the steadystate analysis towards complex engineering structures containing friction. In: Proceedings of ASME Turbo Expo 2022, V08BT27A019 (2022) 22. Zou, D., Zhao, H., Liu, G., et al.: Application of augmented Kalman filter to identify unbalance load of rotor-bearing system: theory and experiment. J. Sound Vib. 463, 1–22 (2019) 23. Zhao, S., Ren, X., Deng, W., et al.: A novel transient balancing technology of the rotor system based on multi modal analysis and feature points selection. J. Sound Vib. 510, 116321 (2021)

Applying Central Manifold Theory in the Definition of Active Gas Foil Bearing Configurations for High-Speed Stability of Rotors Ioannis Gavalas , Emmanouil Dimou , and Athanasios Chasalevris(B) National Technical University of Athens, 15780 Athens, Greece [email protected] Abstract. This work explores the extension of threshold speed of instability in higher speeds, in rotors mounted in Active Gas Foil Bearings (AGFBs). An alternative configuration of an AGFB is presented including actuator elements which act independently under an optimization scheme, in order to establish such a shape of the top foil, that the effective damping property is achieved at a specific value. In this way, the bump foil structure existing in the conventional Gas Foil Bearings is not included. Operating speed higher than 100 kRPM is achieved in stable regimes for a D100 shaft. A simple rigid rotor mounted on two identical AGFBs is examined in the quality of instability, this described by the type of Hopf bifurcation (supercritical or subcritical) through the central manifold theory. The active foil shapes are elastic, and they are produced by eight piezoelectric actuators located at the bearing’s circumference, establishing lobe-type configuration. The elastoaerodynamic lubrication problem is modeled by the coupled state equations of the rotating journal displacement, the gas pressure distribution, and the elastic foil deformation, evaluated by finite difference method and finite element method respectively. The optimization pattern targets to set the damping ratio of the system in specific values. Several design scenarios for the AGFB are studied through dimensionless design parameters. The threshold speed of instability is located at ultrahigh speeds and the type of Hopf bifurcation can be controlled with respective configurations. Keywords: bifurcation theory · center manifold theory · nonlinear rotor dynamics · active gas foil bearings · high speed rotors · oil free

1 Introduction In the past years, Gas Foil bearings (GFBs) have contributed to the transition to oilfree rotating machines. Due to the use of ambient air as lubrication medium, GFBs can operate in high temperatures and produce no contaminants, with little to no maintenance required. However, GFBs tend to suffer from high amplitude sub-synchronous vibrations at high speeds, in a phenomenon called gas whirl (similarly to the oil bearings producing oil whirl/whip), and from severely reduced load carrying capacity compared to equivalent size oil or roller bearings. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 304–323, 2024. https://doi.org/10.1007/978-3-031-40459-7_21

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There is significant research that aims to modify the bearing shape and in particular the top foil, which acts as the lubricating surface, to increase the threshold speed of instability. In the work of Kim and San Andres [1], three metallic shims were added between the bump foil and the bearing outer shell, in order to study the effect of mechanical preloading to the stability properties of GFBs. The authors concluded that using the metallic shims led to a reduction between the cross-coupling coefficients and an increase of the onset speed of instability. Sim et al. [2] studied the optimum combination between bearing clearance and preload for stability and power loss. Schiffman and Spakovszky [3] found that the optimum preload pattern should depend on the design characteristics of the GFB, and Walter and Sinapius [4] showed that a preload value can be calculated that optimizes the trade-off between stability, load capacity and lift-off speed. More recently, there has been extensive work to address the inherent trade-off between load capacity and stability by utilizing controllable Gas Foil Bearings (c-GFBs). Sadri et al. [5] and Feng et al. [6] used patch-type PZT actuators to deform the outer bearing shell that varied depending on the operating speed and performance requirements. Park and Sim [7] included 9 PZT actuators in order to deform the bearing housing and control the amplitude of vibrations when passing through critical speeds. It was shown that bearing clearance control has a significant influence on the direct stiffness and damping coefficients while control of the mechanical preload has a larger effect on the cross-coupling coefficients. Park et al. [8] performed a theoretical and experimental study using parameter identification on a c-GFB with three PZT actuators and showed an increase of the onset speed of instability. Guan et al. [9] used the same bearing concept of Feng et al. [6] and studied the performance and bifurcations of the open loop model and performed an experimental study of a closed loop system using a PID controller. Furthermore, alternative concepts of c-GFBs include the work of Nielsen and Santos [10] where a theoretical and experimental of a piezoelectric air foil bearing (PAFB) which included a composite layer with piezoelectric fibers in the top foil, and von Osmanski and Santos [11] who studied the feasibility of a hybrid Air Foil Bearing with pressurized air injection. Both works showed that an increase of the onset speed of instability and a reduction of sub-synchronous vibrations is indeed possible. Further characteristics and utilities of active gas foil bearings have been extensively investigated in [12–16]. The nonlinear dynamics of rotor-bearing systems using rigorous methods from bifurcation theory is a relatively new subject. In [17–23] the Hopf bifurcation has been studied using analytical methods for self-excited vibrations in fluid film bearings, allowing the prediction of the quality and amplitude of the self-excited limit cycles beyond the onset speed of instability. Recently, Papafragkos et al. [24, 25] identified the correlation of dissipating energy in the gas film with the respective instabilities in conventional gas foil bearings, and Gavalas et al. [26] investigated active configurations in gas foil bearings to achieve Hopf bifurcations at very high speeds. Motivated by the latest works, the authors proceed in this paper adding the following novelty in literature: 1) The possibility to place Hopf bifurcation at selected speed and at selected type, supercritical or subcritical, utilizing the central manifold theory. In this way, the central manifold close to the Hopf bifurcation is controlled, and thus the journal motion at a Hopf bifurcation point follows stable limit cycles of small extend (supercritical

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bifurcation) or large extend (close to radial clearance, when the Hopf bifurcation is subcritical). 2) The conventional layout of a gas foil bearing including a passively deformed top foil mounted in a bump foil structure is not the case in this paper. Alternatively, the bump foil is replaced by a series of piezoelectric actuators and the top foil configuration is actively applied through an optimization procedure which renders specific damping factor in the system. 3) Ultra-high speeds of operation are achieved in the theoretical basis, where the circumferential speed of a D100 journal exceeds the speed of sound, while the system damping factor is retained at a desired value.

2 Analytical Model of the Rotor-AGFB System The analytical model of a rigid rotor mounted on two AGFBs is described in this section. The AGFBs consists of a compliant foil structure which is pin-supported along the inner circumference of the outer rigid shell, see Fig. 1, and fixed (clamped) along the axial direction of its one edge, see Fig. 2. In the bearing symmetry plane, a row of PZT actuators also act on the compliant foil with a normal force, see Fig. 2. Similar configurations can be found in [24, 25].

Fig. 1. Illustration of the analytical model including (a) the AGFB front view with all relevant design properties, and (b) gas pressure and resultant journal forces acting on the journal and the top foil

In the AGFB proposed, the PZT actuators replace the bump foil of a conventional GFB deforming the structure through open loop control, see Fig. 1. The equations of lateral motion for a rigid rotor where no angular misalignment (tilting of the rotor) is considered are defined in Eq. (1) where xj and yj are the journal displacement in

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the horizontal and vertical direction (identical to disc center displacement), md is the disc mass, mj = md /2 is the journal mass, Fb,x and Fb,y are the resulting bearing forces (gas film forces) in horizontal and vertical direction, Fu,x = u2 cos(t) and Fu,y = u2 sin(t) are the unbalance forces components at the two directions evaluated for constant rotating speed .

Fig. 2. Rigid rotor mounted on two AGFBs. The actuators contact the foil and rigid shell in a larger area along the axial (z) direction, than depicted in the figure.

Unbalance is defined as u = 10−3 G/(r md )[kg · m] with r [rad/s], being the maximum operating speed, and G is the ISO balance G-grade. Fg = mj g is the gravity force acting on each journal, where g = 9.81 m/s2 . The equations of motion are defined in Eq. (2) in dimensionless form where two dimensionless design parameters  and g are included, these defined in Eq. (3), where ψ = R/cr . In Eq. (2) the dimensionless parameters are defined as xj = xj /cr , and yj = yj /cr . Further definitions for the time derivatives are included as x¨ = ∂ 2 x/∂t 2 , x = ∂ 2 x/∂τ 2 where τ = t/ is the dimensionless time where  = 6μψ 2 /p0 . Bearing forces Fb,x and Fb,y are defined in Eqs. (6) and (7) in Sect. 2.1 where the elastoaerodynamic lubrication is discussed. x¨ j =

1 (Fb,x + Fu,x ), mj

xj = F b,x + F u,x ,

y¨ j =

1 (Fb,y + Fu,y − Fg ) mj

yj = F b,y + F u,y − g

(1) (2)

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=

36μ2 Dψ 5 36μ2 ψ 5 , g= g p0 mj p02 R

(3)

The number of actuators is chosen to be 8, mainly for the following reasons. In [26] the authors have investigated AGFB layouts with 4 and 8 actuators, concluding that 8 actuators allow for more possibilities to enhance the stability threshold. This is explained by the fact that 8 actuators can establish lobes which increase the effective eccentricity of the journal with respect to the gas film; this is positive for the increment of stability threshold speed. More actuators were also applied, but the benefit in stability threshold was not that important. In this paper, the operating principle of the actuators is implemented through closed loop control. The actuators perform the respective elongation after the control current is applied. The optimization problem renders the respective actuator displacement required for a specific foil shape. The actuators have the possibility to measure their displacements and therefore feedback is provided regarding the actual displacement. The control system sets the error between the desired and the actual displacement of the actuator close to zero. The detailed implementation of the control method is a future work by the authors. 2.1 Solution of the Elasto-Aerodynamic Model for the AGFB Lubrication The computational model of the top foil follows the formulation from previous work of the authors in [26, 27]. The top foil is assumed to behave as a flat thin plate, formulated using the MZC (Melosh-Zienkiewicz-Cheung) finite element [28] using analytical integration. This formulation for the top foil using thin plate theory by Kirchoff-Love [29] has been shown to have a reasonably good agreement with experimental data and the more complicated curved shell model. A small amount of stiffness proportional damping is considered in the formulation as numerical damping. The mass, damping and stiffness matrices are then reduced using the mode acceleration method, considering the first four modes of the undamped system [30]. The matrix V in Eqs. (5) and (6) contains the eigenvectors corresponding to the selected modes and the vector v is the modal response vector. The mass, damping and stiffness matrices and the force vector are reduced using the modal matrix as shown in Eq. (5). Then the reduced equations of motion are defined in Eq. (4) where fp,i = (pi − p0 )xz and fa,j = ca,j (˙qj − q˙ a,j ) + ka,j (qj − qa,j ) are the normal forces from the gas film pressure to the foil, and from the actuators to the foil, respectively. Finally, the complete physical response of the system is given by Eq. (5). It is noted that the method is only appropriate in systems without rigid body motion (due to the inversion of the stiffness matrix), but has the benefit of accurately predicting the physical response when few modes are retained while exactly preserving the static displacement for constant forcing (Fig. 3). Mf ,r v¨ f + Cf ,r v˙ f + Kf ,r vf = ff ,r

(4)

Mf ,r = VT Mf V, Cf ,r = VT Cf V, Kf ,r = VT Kf V, ff ,r = VT ff

(5)

qf = Kf−1 ff − Kf−1 Mf V¨vf − Kf−1 Cf V˙vf

(6)

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Fig. 3. Representative Finite Element mesh and support conditions of the foil structure

The reduced foil ODEs can be recast in state-space form in Eq. (7).        v˙ f O O I vf + = −1 Mf−1 −Mf−1 v˙ f v¨ f ,r Kf ,r −Mf ,r Cf ,r ,r ff ,r

(7)

The gas lubrication is governed by the compressible and isothermal Reynolds equation which is transformed into Eq. (8) in non-dimensional form by setting χ = χ /R, χ ∈ [0, 2π R], z = 2z/Lb , z ∈ [−Lb /2, Lb /2], p = p/p0 , h = h/cr ,  =  . Boundary conditions for the gas pressure are defined as p(χ = 0, z) = p(χ = 2π R, z) = p(χ , z = −Lb /2) = p0 , and ∂p/∂z|z=0 = 0 due to symmetry. The Reynolds equation is coupled to the foil and rotor ODEs through the film thickness function defined in Eq. (9a, b) in dimensioned and dimensionless form where q = q/cr .  

 2 2 2  ∂p ∂p 2 ∂p 1 2 R ∂p ∂h R ∂p ∂h 3 = h + + + ph ∂τ 2 ∂χ Lb ∂z 2 ∂χ ∂χ Lb ∂z ∂z    

 ∂p p ∂h p ∂h R 2 ∂ 2p 1 2 ∂ 2p + (8) + − + ph − 2 Lb ∂z 2 2 ∂χ ∂χ ∂χ 2 h h ∂τ h(θ, z) = cr − xj cos(θ ) − yj sin(θ ) + qf (θ, z)

(9a)

h(θ, z) = 1 − xj cos(θ ) − yj sin(θ ) + qf (θ, z)

(9b)

The resulting forces from the gas to the journal are evaluated in Eq. (10a, b) in dimensioned and dimensionless form, respectively. F b,x = −

1 2π

∫ p(θ, z) cos(θ )d xd z ≈ −

−1 0

Nθ Nz 4π pi,j cos(θi ) Nz Nθ j=1 i=1

(10a)

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F b,y = −

1 2π

∫ p(θ, z) sin(θ )d xd z ≈ −

−1 0

Nθ Nz 4π pi,j sin(θi ) Nz Nθ

(10b)

j=1 i=1

 T   The unknown state vector sn×1 = and the coupled xj yj pT vfT vTf xj yj ODE system define a non-autonomous dynamic system in Eq. (11a) when unbalance forces are included, or an autonomous dynamic system in Eq. (11b) for the balanced rotor. s = f(s, , τ ), s = f(s, )

(11a,b)

The dynamic systems defined in Eq. (11) can produce asymptotically stable solutions around a fixed point or limit cycle, among other solutions. Limit cycles can be periodic or quasi periodic. Chaotic motions can be also generated. The respective fixed points, limit cycles, and trajectories can be calculated through time integration of Eq. (11) applying specific algorithms for numerically stiff systems like the current one. Time integration is used only to establish an initial prediction for the respective stable fixed point or limit cycle. A collocation type method is used for the computation of periodic limit cycles produced by the ODE system in Eq. (11), at a constant rotating speed. Numerical continuation of periodic limit cycles (pseudo arc length continuation [31–35]) has been programmed by the authors, in correspondence to [31], to evaluate the limit cycles or the fixed point as the rotating speed changes. The reader may also find the methodology in the recent works of the authors in [22, 24]. Only periodic limit cycles are evaluated in this work. The evaluation of quasi periodic limit cycles and the respective continuation method is under preparation by the authors for future work. Further to the limit cycle response, the iterative solution of the collocation method renders as a product the monodromy matrix whose eigenvalues are directly related to the Floquet multipliers of the periodic motion. The convergence of the large ill conditioned algebraic systems is achieved by numerical techniques including tools like analytical derivatives, damped Newton method.

3 Center Manifold Theory at the Hopf Bifurcation of Fixed-Point Equilibria The center manifold theory is a method that allows the computation of the normal form of several types of bifurcations [36–38]. The normal form is a simplified low dimensional dynamical system that exhibits the same qualitative characteristics as the original system around the bifurcation point utilizing nonlinear transformations. The normal form of the Hopf bifurcation is defined in Eq. (12). A number can be defined through δ as a = Re(δ) and is called First Lyapunov Coefficient (FLC). dξ = (λ + i)ξ + δ|ξ |2 ξ, ξ ∈ C, δ ∈ C, λ ∈ R. dt

(12)

If a < 0 there exists a stable limit cycle for λ > 0, and the bifurcation is called supercritical, while for a > 0 there exists an unstable limit cycle for λ < 0, and then the

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bifurcation is called subcritical. The Hopf bifurcation can be subcritical or supercritical, depending on the value and sign of the FLC which can be computed at λ = −th → 0. The center manifold can be computed using the projection method which does not require the transformation of the system into its eigen-basis. The eigenvectors corresponding to the critical eigenvalues of the linearized system (around the fixed-point equilibrium) are used to calculate the projection of the system to the critical eigenspace. For more information on the derivation, the reader is referred to [38]. The application of the method results in a convenient method to calculate an invariant expression for the first Lyapunov coefficient. In general, the Taylor expansion of f(u, 0) at u = 0 is written in Eq. (13) where b(u1 , u2 ) and c(u1 , u2 , u3 ) are the multilinear functions (directional derivatives) with the components defined in Eq. (14). 1 1 f(u, 0) = Au + b(u, u) + c(u, u, u) + O(||u||4 ) 2 6   n

∂ 2 fj (w, 0)  u1k u2l , bj (u1 , u2 ) = ∂wk ∂wl  k,l=1 w=0   n 3

∂ fj (w, 0)  cj (u1 , u2 , u3 ) = u1k u2l u3m ∂wk ∂wl ∂wm  k,l,m=1

(13)

(14)

w=0

The multilinear functions b(u1 , u2 ) and c(u1 , u2 , u3 ) are approximated by matrixvector products. If we let q˜ ∈ Cn the complex eigenvector of A corresponding to the eigenvalue iω0 , and p˜ ∈ Cn the adjoint eigenvector corresponding to −iω0 , then the first ˜ Lyapunov coefficient is calculated in Eq. (15) where q˜ is the complex conjugate of q. a=

1 ˜ − 2p, ˜ b(q, ˜ A−1 b(q, ˜ c(q, ˜ q, ˜ q) ˜ Q Re[p, q)) 2ω0 ˜ (2iω0 I − A)−1 b(q, ˜ q))] ˜ ˜ b(q, + p,

(15)

The eigenvectors are normalized as q˜ T q˜ = 1, and the sign of a is independent of the normalization used. The procedure is presented in the flow chart in Fig. 4.

4 AGFB Configurations to Locate Hopf Bifurcations and Determine the Type of Instability A reference system is defined with two dimensionless design parameters, these being  and g, see Eq. (3). The quality of stability of the reference system is investigated for four pairs of the bearing design parameters, in Fig. 5, corresponding to a small and a large AGFB; note that the parameter g defines directly the diameter of the bearing, for a given ratio ψ = R/cr = 500. For instance, g = 0.1 and g = 0.01 correspond to bearing diameter D = 10 mm and D = 100 mm respectively. Further physical parameters needed to be defined are included in Table 1. In the reference system, all actuators are set in the zero position (no displacement), and therefore no active configuration takes place in the foil.

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Fig. 4. Flow chart for the calculation of the first Lyapunov coefficient

Table 1. Physical properties for ambient air and foil material Ambient air

Foil material (Inconel type)

Dynamic viscosity

μ = 1.83 · 10−5 Pa s

Poisson’s ratio

νf = 0.28

Ambient pressure

p0 = 101.3 kPa

Young’s modulus

Ef = 205 GPa

Density

ρf = 8200 kg/m3

The design parameter  is an indicator of the effective loading of the bearing. A bearing of  = 0.1 is more loaded than a bearing of  = 1. The first Lyapunov coefficient (FLC)α is evaluated at the Hopf bifurcation which takes place when rotating speed  equals the threshold speed of instability th for the various values of the actuator compliance α α and its effective loss factor ηα . Both parameters are significant for the dynamic behavior of the foil structure. The dimensionless foil thickness t f is defined at a fixed value of t f = tf /R = 0.08. The specific load of the bearing designs included in Fig. 5 does not exceed the value W = mj g/(Lb D) = 5 kPa. Perfectly balanced rotors are considered. In Fig. 5 one notices that in the wide design range of the AGFB design, and for the conventional design where no displacement of actuators applies, supercritical and subcritical Hopf bifurcations take place, and that the threshold speed of instability exhibits wide range of variation. An AGFB able to receive higher loads (as  increases) is keener to generate subcritical instability (positive FLC), than an AGFB of smaller diameter is (as  decreases). The potential for supercritical instability is increased for smaller AGFBs and especially for lower loads (lower g). Small AGFBs receiving high load will generate subcritical instability. There are combinations of design configuration and load which render subcritical or supercritical instability. The quality of Hopf bifurcation is similar in the two designs presented in Fig. 5, where the compliance of actuators is

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Fig. 5. First Lyapunov coefficient – FLC α and threshold speed of instability th (speed at Hopf bifurcation) as a function of  and g for actuator compliance a) α α = 0.01, and b) α α = 0.1. Positive and negative surface values are depicted in different shading.

increased by one order of magnitude in Fig. 5b. A more flexible support in the foil results in higher threshold speed of instability th . Typical solution branches for subcritical and supercritical instability are presented in Fig. 6. These have been evaluated by numerical continuation of fixed points and of limit cycles of a perfectly balanced rotor. At both cases the transient response of the system will always attracted by stable limit cycles after a Hopf bifurcation occurs. However, at the case of a subcritical Hopf bifurcation, the limit cycles are bounded by the physical constraint of radial clearance. At this case, and from the engineering point of view, the system will produce violent vibrations immediately after the Hopf bifurcation takes place, as the system increases its speed. At the case of a supercritical Hopf bifurcation, the self-excited vibration will be attracted from stable limit cycles which progressively extent to a larger area inside the radial clearance. From the engineering point of view, supercritical instability is preferred in rotating systems. 4.1 Configuration Patterns with One Control Input A specific foil configuration is investigated in continue where all actuators simultaneously exhibit equal absolute displacement qα at the range qα,i ∈ [−0.3, 0.3] in the pattern qα,1 = −qα,2 = qα,3 = −qα,4 = . . . = −qα,8 = qα . One control signal is sufficient to control the actuator displacements, as these are equal to the absolute value qα . The resulting foil configurations and the respective actuator positions are depicted in Fig. 7. The corresponding fixed-point equilibria of the journal (see Fig. 7a and b) and the respective stability threshold and sign of the first Lyapunov coefficient α (see Fig. 7c and d) are included at all seven scenarios qα = −0.3, ..., 0.3 including the conventional design where qα = 0. The stability threshold is approximately doubled comparing the conventional configuration of qα = 0 and the case of qα = 0.3, at the case of a large lightly loaded bearing (see Fig. 7a and c). For the case of a smaller and higher loaded bearing, the threshold speed of instability does not severely increase , see Fig. 7b and d. Stiffer actuators provide the potential for higher instability threshold

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Fig. 6. Solution branches of the system in a) supercritical, and b) subcritical Hopf bifurcation, for selected AGFB configurations.

at the first case, while more compliant actuators work towards stability at the second case. The dimensionless pressure, the foil configuration, and the film thickness are presented in Fig. 8 for the respective configurations of Fig. 7. In Fig. 7 it is shown that the foil structure follows the alternation in shape through the circumferential direction. The actuators’ displacement of a maximum absolute value qα = 0.3 produces a minimum gas film thickness lower than 20% of radial clearance. Such a case should be avoided in the potential applications. However, the design considers much higher rotating speeds in which the journal will operate in lower eccentricity and therefore the minimum gas film thickness will increase. The operability of AGFB is discussed further in Sect. 4.2. The maximum gas pressure generated in the lubrication area exceeds 3.5 times the ambient pressure of the gas. The respective foil deflection does not exceed the value set by the actuators in any case. The passive deformation of the foil is minimized by selecting a foil thickness as tf = 0.08R. The foil shape is presented in Fig. 9a in a 3D view where the respective actuator forces required to produce the shape are depicted too. A representative configuration is depicted in polar coordinates in Fig. 9b. 4.2 Configuration Patterns with Multiple Control Inputs In this Section the displacement of each actuator is independent from any other. Therefore, random configurations are established in the AGFB. An optimization procedure is engaged with eight inputs to correspond to the eight independent actuator displacements, and an objective function such that Hopf bifurcation to be located at selected speed, by setting the value of damping ratio ζ to be equal to the target ζ0 = 0. Damping ratio ζ is evaluated through the eigenvalues of the Jacobian matrix of the linearized system around the respective fixed-point equilibrium. By setting the speed of operation at the desired value H , the optimization algorithm searches for the configuration qα,1 , qα,2 , ..., qα,8

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Fig. 7. Active configurations, fixed point equilibria, and bifurcation type, for a)  = 1, g = 0.1 and b)  = 0.1, g = 0.01; FLC and threshold speed of instability for the respective cases in c) and d). ◯ subcritical instability, ● supercritical instability at all figures

which renders a fixed-point equilibrium where ζ ≈ ζ0 . During the iterations of the   optimization algorithm, several fixed points are evaluated, for the respective sets of qα,i . The objective function for locating the Hopf bifurcation at a selected speed is defined in Eq. (16).   OBF qα,1 , qα,2 , ..., qα,8 ,  = H = (ζ − ζ0 )2 (16) A second scenario with practical interest considers a target ζ0 for damping ratio at a selected positive value, e.g. ζ0 = 0.02 is selected. At this case, the balanced system executes a virtual run up from a low speed. At each speed, the fixed-point equilibrium and the respective damping ratio ζ is evaluated. While ζ > ζ0 theAGFB acts at each conventional version, meaning that no actuation takes place, qα,i = {0}. Increasing the rotating speed , damping ratio reduces and as soon as ζ < ζ0 the optimization scheme is engaged in the algorithm to render ζ ≈ ζ0 . The respective configuration of the foil acts as initial shape for the next optimization occurring at the next higher speed

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Fig. 8. Pressure distribution p(θ, 0), film thickness h(θ, 0), and foil displacement qf (θ, 0) for selected configurations of a) Case 1, and b) Case 2.

where ζ < ζ0 . In this way, the potential of the AGFB to establish stability at ultra-high speeds is depicted in Fig. 10 where the optimization procedure is repeated a total of 9 times (at 9 corresponding rotating speeds). It is found that the threshold speed of instability achieves higher values compared to those for one control input, see Sect. 4.1, corresponding to ultra-high speeds. The optimization goal is to determine the critical eigenvalue pair in the case of fixed-point equilibria. The optimization problem is solved using the MATLAB function patternsearch from the Global Optimization Toolbox [34] which is recommended for non-expensive objective functions. Because of the magnitude of the design space, it can become difficult for the fixed point to converge, when the initial guess is far from the solution. To overcome this obstacle, the optimization problem is allowed to run with the actuators constrained between 10% of the radial clearance in the inward and outward direction with respect to the last optimum configuration found. Once a solution has been achieved, the initial guess for the next equilibria is the solution found. The operational parameter DN will be used in this Section to express the peripheral speed of the journal. DN [mm/min] =D [mm] ·  [RPM] is used in literature to express the circumferential speed of a rotor inside a bearing (normally ball bearing). Aiming to AGFB configurations for ultra-high speeds, in this paper the dimensionless parameter DNM = DN/DNS is also defined, where DNS ≈ 6.5e6 is the value of DN for a journal with peripheral speed R = 343.2 m/s(= Mach 1); therefore, 0 2.5e6 (DNM > 0.38) is hardly observed in existing applications. In Fig. 10a, the respective foil configurations are presented at the respective speed for the design defined at the top of the figure, for a shaft diameter of D = 10 mm. The respective configurations render damping ratio ζ = ζ0 = 0.02. However, each configuration refers to different rotating speed as shown in the color bar. The quality of all configurations appears similar; the actuators establish three asymmetric lobes in the bearing.

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Fig. 9. Representative 3D view of a) foil displacement qf (θ, z) and actuator forces b) foil configuration in polar coordinates, and c) pressure distribution p(θ, z).

The conventional design becomes unstable at  ≈ 2.5 and the optimized AGFB reaches  ≈ 10.6. This is an extend of the threshold speed of instability by approximately 4 times. In Fig. 10b one may notice the respective stability threshold of the system for each of the 9 configurations, if each of them was fixed. A fixed-point continuation renders the fixed-point equilibrium and the respective damping ratio, the latter depicted by a solid line while ζ > 0 (stable fixed point), and a dashed line for ζ < 0 (unstable fixed point). The optimization algorithm fails to achieve the targeted damping ratio at the cases 8 and 9 depicted in Fig. 10b. This is attributed to the fact that the actuators are not allowed to establish configurations which would render the minimum gas film thickness lower than the prescribed threshold; this would compromise the integrity of the system. Transient response is evaluated for an unbalanced rotor during a run up where the foil configurations follow interpolated patterns of those configured above. The result is depicted in Fig. 10c where one may notice the stable response up to the speed of  ≈ 10.6; this is translated to DNM = 0.52 or  = 341 kRPM for the D10 shaft.

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Fig. 10. a) optimum configurations at various speeds to establish stable operation of ζ = ζ0 = 0.02 for the D10 bearing, b) progress of damping ratio during the virtual run up, and c) verified stable operation during a virtual run up of an unbalanced rotor.

The respective foil deformation is depicted in the same chart together with minimum film thickness. The main advantage is that relatively high gas film thickness is achieved for stable operation at ultra-high speeds. Further to that, it is interesting to notice the displacement of the top foil qf where the oscillating motion evolves. The oscillation frequency consists of more than one component, with the one of major amplitude to be synchronous to the excitation frequency (rotating speed ). This is attributed to the fact that the natural frequencies of the foil structure, under the specific mounting of 8 actuators, are all higher than the ending speed of the systems investigated. Therefore, the oscillating deformation of the top foil will be following the quality and frequency of the oscillating motion of the journal. The foil structure does include a low portion of damping (as numerical damping), but the actuator model does include viscous damping factor and influence the effective damping of the system. The foil structure itself cannot become unstable as there is not any mechanism of self-exciting vibration in its equations of motion, Eq. (4). The gas film thickness depicts oscillations too, as shown in Fig. 10c

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but these oscillations do not introduce any self-excited motion; the system operates at stable regime as the effective damping is retained positive.

Fig. 11. a) optimum configurations at various speeds to establish stable operation of ζ = ζ0 = 0.02 for the D100 bearing, b) progress of damping ratio during the virtual run up, and c) verified stable operation during a virtual run up of an unbalanced rotor.

In Fig. 11a the respective foil configurations are presented at the respective speed for the design defined at the top of the figure, for a shaft diameter of D = 100 mm. The respective configurations give a damping ratio ζ = ζ0 = 0.02. Each configuration of the AGFB refers to different rotating speed as shown in the color bar. The quality of all configurations appears similar, and the actuators establish three asymmetric lobes in the bearing as happened also for the D10 shaft. The conventional design becomes unstable at  ≈ 0.9 and the optimized AGFB reaches  ≈ 5.0. In Fig. 11b the respective damping ratio for the 21 configurations is depicted closely around the target value ζ0 = 0.02. Transient response is evaluated for the unbalanced rotor during a virtual run up where the foil configurations follow interpolated patterns of the 21 configured above. The result is depicted in Fig. 11c where

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one may notice the stable response up to the speed of  ≈ 5.0; this is translated to DNM ≈ 2.4 or  = 157 kRPM for the D100 shaft. The respective foil deformation is depicted in the same chart together with minimum film thickness. It is worth noticing that the minimum gas film thickness reduces at the ending speed; the integrity of the AGFB is questioned at this case. In continue, the possibility to reverse the type of Hopf bifurcation is investigated. The idea is that the Hopf bifurcation of the system is set at a specific rotating speed as described at the beginning of this Section. Then, at this specific speed, an optimization algorithm searches for one configuration which gives positive or negative First Lyapunov Coefficient FLC α. The objective function is defined at this case in Eq. (17).   (17) OBF qα,1 , qα,2 , ..., qα,8 ,  = H = (ζ − ζ0 )2 + (α − α0 )2 In Eq. (17), ζ0 is set at a low positive value, e.g. ζ0 = 1e − 5 so as a Hopf bifurcation to occur, and the target for FLC α0 is set at a positive or negative value at each case, at a realistic order considering the results of Fig. 5, e.g. α0 = ±0.01. At the first case, for α0 = +0.01, a supercritical Hopf bifurcation should occur when the optimization achieves both targets and OBF → 0. At the second case, for α0 = −0.01, a subcritical Hopf bifurcation should occur. The design applied in the results of Fig. 10 is selected and the Hopf bifurcation speed is set at H = 0.38. The corresponding configurations are depicted in Fig. 12. To this end, the advantage of AGFB to establish the preferred supercritical instability is clarified, at least for a selected design case.

Fig. 12. AGFB configurations for supercritical and subcritical Hopf bifurcation at speed H = 1.5.

5 Conclusions The paper introduces an Active Gas Foil Bearing with optimized configuration in real time operation. The optimization targets to set the damping ratio of the system in specific values by altering the foil configuration at each speed, during a virtual run-up of

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the system. Positive damping ratio guarantees stable fixed-point solution, while a zero damping ratio renders a Hopf bifurcation point. Specific AGFB configurations are found to render supercritical or subcritical instability at a selected Hopf bifurcation speed. The most important conclusions of the work are summarized as follows. 1) The AGFB configuration follows a progressively changed shape of the same quality, as the optimization algorithm targets to retain the damping ratio of the fixed-point solution at a specific value; this is valid at all speeds and designs checked. Stable operation is achieved up to ultra-high speeds. The peripheral speed of the journal is found at speeds higher than Mach 1 at selected design cases where stable operation is achieved. 2) It is possible to place Hopf bifurcation at selected speed and at selected type, supercritical or subcritical, utilizing the central manifold theory. 3) The bump foil of a conventional gas foil bearing can be replaced by a series of actuators and the top foil configuration to be actively applied, establishing stable operation at high speeds. 4) Ultra-high speeds of operation are achieved in the theoretical basis, where the circumferential speed of a D100 journal exceeds the speed of sound, while the system damping factor is retained at a desired value. There are limitations regarding the ending speed as the foil structure can perform displacements up to a certain extent. Acknowledgements. The research work was supported by two separate funding sources: a) the Hellenic Foundation for Research and Innovation (HFRI) Fellowship Number: 9575, and b) the Alexander von Humboldt Foundation, Germany, with a Research Group Linkage Program between National Technical University of Athens and Karlsruhe Institute for Technology.

References 1. Kim, T.H., Andrés, L.: Effects of a mechanical preload on the dynamic force response of gas foil bearings: measurements and model predictions. Tribol. Trans. 52, 569–580 (2009). https://doi.org/10.1080/10402000902825721 2. Sim, K., Lee, Y.B., Kim, T.H.: Effects of mechanical preload and bearing clearance on rotordynamic performance of lobed gas foil bearings for oil-free turbochargers. Tribol. Trans. 56, 224–235 (2013). https://doi.org/10.1080/10402004.2012.737502 3. Schiffmann, J., Spakovszky, Z.S.: Foil bearing design guidelines for improved stability. ASME J. Tribol. 135, 011103 (2013). https://doi.org/10.1115/1.4007759 4. Walter, F., Sianpius, M.: Influence of aerodynamic preloads and clearance on the dynamic performance and stability characteristic of the bump-type foil air bearing. Machines 9(8), 178 (2021). https://doi.org/10.3390/machines9080178 5. Sadri, H., Schlums, H., Sinapius, M.: Investigation of structural conformity in a three-pad adaptive air foil bearing with regard to active control of radial clearance. ASME J. Tribol. 141 (2019). https://doi.org/10.1115/1.4043780 6. Feng, K., Guan, H.Q., Zhao, Z.L., Liu, T.Y.: Active bump-type foil bearing with controllable mechanical preloads. Tribol. Int. 120, 187–202 (2018). https://doi.org/10.1016/j.triboint. 2017.12.029

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7. Park, J., Sim, K.: A Feasibility study of controllable gas foil bearings with piezoelectric materials via rotordynamic model predictions. ASME J. Eng. Gas Turbine Power 141 (2019). https://doi.org/10.1115/1.4041384 8. Park, J., Kim, D., Sim, K.: Rotordynamic analysis of piezoelectric gas foil bearings with a mechanical preload control based on structural parameter identifications. Appl. Sci. 11, 1–21 (2021). https://doi.org/10.3390/app11052330 9. Guan, H.Q., Feng, K., Cao, Y.L., Huang, M., Wu, Y.H., Guo, Z.Y.: Experimental and theoretical investigation of rotordynamic characteristics of a rigid rotor supported by an active bump-type foil bearing. J. Sound Vibr. 466, 115049 (2020). https://doi.org/10.1016/j.jsv.2019. 115049 10. Nielsen, B.B.: Combining gas bearing and smart material technologies for improved machine performance theory and experiment. Ph.D. Thesis, DTU (2016) 11. von Osmanski, S., Santos, I.F.: Gas foil bearings with radial injection: multi-domain stability analysis and unbalance response. J. Sound Vib. 508, 116177 (2021). https://doi.org/10.1016/ j.jsv.2021.116177 12. Sekunda, A., Niemann, H., Poulsen, N.K., Santos, I.: Parametric fault diagnosis of an active gas bearing. Int. J. Control Autom. Syst. 17(1), 69–84 (2019). https://doi.org/10.1007/s12 555-017-0738-2 13. Theisen, L.R.S., Niemann, H.H., Galeazzi, R., Santos, I.F.: Enhancing damping of gas bearings using linear parameter-varying control. J. Sound Vib. 395, 48–64 (2017). https://doi.org/ 10.1016/j.jsv.2017.02.021 14. von Osmanski, S., Larsen, J.S., Santos, I.F.: Multi-domain stability and modal analysis applied to gas foil bearings: three approaches. J. Sound Vib. 472, 115174 (2020). https://doi.org/10. 1016/j.jsv.2020.115174 15. Heinemann, S.T., Jensen, J.W., von Osmanski, S., Santos, I.F.: Numerical modelling of compliant foil structure in gas foil bearings: comparison of four top foil models with and without radial injection. J. Sound Vib. 547, 117513 (2023). https://doi.org/10.1016/j.jsv.2022.117513 16. Nielsen, B.B., Santos, I.F.: Transient and steady state behaviour of elasto-aerodynamic air foil bearings, considering bump foil compliance and top foil inertia and flexibility: a numerical investigation. Proc. Inst. Mech. Eng. Part J: J. Eng. Tribol. 231(10), 1235–1253 (2017). https:// doi.org/10.1177/1350650117689985 17. Wang, J.K., Khonsari, M.M.: On the hysteresis phenomenon associated with instability of rotor-bearing systems. ASME J. Tribol. 128, 188–196 (2006). https://doi.org/10.1115/1.212 5927 18. Miraskari, M., Hemmati, F., Gadala, M.S.: Nonlinear dynamics of flexible rotors supported on journal bearings – Part II: Numerical bearing model. ASME J. Tribol. 140 (2018). https:// doi.org/10.1115/1.4037731 19. Miraskari, M., Hemmati, F., Gadala, M.S.: Nonlinear dynamics of flexible rotors supported on journal bearings – Part I: Analytical bearing model. ASME J. Tribol. 140 (2018). https:// doi.org/10.1115/1.4037730 20. Wang, J.K., Khonsari, M.M.: Application of Hopf bifurcation theory to rotor-bearing systems with consideration of turbulent effects. Tribol. Int. 39, 701–714 (2006). https://doi.org/10. 1016/j.triboint.2005.07.031 21. Chasalevris, A.: Stability and Hopf bifurcations in rotor-bearing-foundation systems of turbines and generators. Tribol. Int. 145, 106154 (2020). https://doi.org/10.1016/j.triboint.2019. 106154 22. Gavalas, I., Chasalevris, A.: Nonlinear dynamics of turbine generator shaft trains: evaluation of bifurcations sets applying numerical continuation. J. Eng. Gas Turbine Power 145(1), 011003 (2022). https://doi.org/10.1115/1.4055533

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23. Anastasopoulos, L., Chasalevris, A.: Bifurcations of limit cycles in rotating shafts mounted on partial arc and lemon bore journal bearings in elastic pedestals. J. Comput. Nonlinear Dyn. 17(6), 061003 (2022). https://doi.org/10.1115/1.4053593 24. Papafragkos, P., Gavalas, I., Raptopoulos, I., Chasalevris, A.: Optimizing energy dissipation in gas foil bearings to eliminate bifurcations of limit cycles in unbalanced rotor systems. Nonlin. Dyn. 111, 67–95 (2022). https://doi.org/10.1007/s11071-022-07837-1 25. Papafragkos, P., Gavalas, I., Raptopoulos, I., Chasalevris, A.: Bifurcation elimination in rotor gas bearing systems applying numerical continuation with embedded design optimization scheme. In: 10th European Nonlinear Dynamics Conference – ENOC 2022, Lyon, France (2022) 26. Gavalas, I., Papadopoulos, A., Chasalevris, A.: Investigation of active configuration in gas foil bearings for optimum load capacity and stability of rotating systems. In: 1st Workshop on Active Bearings in Rotating Machinery – ABROM 2022, Athens, Greece (2022) 27. Papadopoulos, A., Gavalas, I., Chasalevris, A.: Controlling Bifurcations in high-speed rotors utilizing active gas foil bearings. In: 15th European Conference on Rotordynamics – SIRM 2023, Darmstadt, Germany (2023) 28. Zienkiewicz, O.C.: The Finite Element Method in Engineering Science. McGraw-Hill, New York (1971) 29. Love, A.E.H.: On the small free vibrations and deformations of elastic shells. Philos. Trans. R. Soc. Lond. 17, 491–549 (1888) 30. Cornwell, R.E., Craig, R.R., Jr., Johnson, C.P.: On the application of the mode-acceleration method to structural engineering problems. Earthq. Eng. Struct. Dyn. 11, 679–688 (1983). https://doi.org/10.1002/eqe.4290110507 31. Doedel, E.J.: Lecture Notes on Numerical Analysis of Nonlinear Equations. Department of Computer Science, Concordia University, Montreal, QC, Canada 32. Doedel, E.J., Keller, H.B., Kernevez, J.P.: Numerical analysis and control of bifurcation problems (II) bifurcation in infinite dimensions. Int. J. Bifurcat. Chaos 1(3), 745–772 (1991). https://doi.org/10.1142/S0218127491000555 33. Nayfeh, A.H., Balachandran, B.L.: Applied Nonlinear Dynamics. John Wiley & Sons, Hoboken, NJ (1995) 34. Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. Society for Industrial and Applied Mathematics, Philadelphia, PA (2003). https://doi.org/10.1137/1.978 0898719154 35. Natsiavas, S.: Vibrations of Dynamic Systems with Nonlinear Characteristics. Ziti Publications, Thessaloniki (2000). (in Greek) 36. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York, NY (2004) 37. Kuznetsov, Y.A., Govaerts, W., Doedel, E.J., Dhooge, A.: Numerical periodic normalization for codim 1 bifurcations of limit cycles. SIAM J. Numer. Anal. 43, 1407–1435 (2005). https:// doi.org/10.1137/040611306 38. Carr, J.: Applications of Centre Manifold Theory. Springer-Verlag (1981) 39. MATLAB 2022a: Global Optimization Toolbox. The MathWorks Inc., Natick (2022)

Locating Period Doubling and Neimark-Sacker Bifurcations in Parametrically Excited Rotors on Active Gas Foil Bearings Emmanouil Dimou1 , Ioannis Gavalas1 , Fadi Dohnal2 and Athanasios Chasalevris1(B)

,

1 National Technical University of Athens, 15780 Athens, Greece

[email protected] 2 Vorarlberg University of Applied Sciences, 6850 Dornbirn, Austria

Abstract. In this work, parametric excitation is introduced in a fully balanced flexible rotor mounted on two identical active gas foil bearings. The active gas foil bearings change the top foil shape harmonically with a specific amplitude and frequency. The deformable foil shape is approximated by an analytical function, while the gas pressure distribution is evaluated by the numerical solution of the Reynolds equation for compressible flow. The harmonic variation of the foil shape generates a respective variation in the bearings’ stiffness and damping properties and the system experiences parametric resonances and antiresonances in specific excitation frequencies. The nonlinear gas bearing forces generate bifurcations in the solutions of the system at certain rotating speeds and excitation frequencies; period doubling and Neimark-Sacker bifurcations are noticed in the examined system, and their progress is evaluated as the two bifurcation parameters (rotating speed and parametric excitation frequency) are changed, though a codimension-2 numerical continuation of limit cycles. It is found that at specific range of excitation frequency there are parametric anti-resonances and the bifurcations collide and vanish. Therefore, a bifurcation-free operating range is established and the system can operate stable at a wide speed range. Keywords: parametric excitation · nonlinear rotor dynamics · gas foil bearings · bifurcations · numerical continuation

1 Introduction Systems with multiple degrees of freedom (MDoF) and periodically changing physical properties (parametrically excited systems) have gathered both the mathematical and engineering interest of the last few decades [1–3]. If the parameters of the excitation strategy are carefully chosen, the existing damping properties of the system will be more efficiently used [4, 5]. Therefore, the stability of an initially unstable system will potentially be retained. The aforementioned phenomenon is called parametric antiresonance and can be interpreted as beneficial modal interaction. In current work, parametric excitation is introduced in a realistic model of a high speed, turbopump rotor, mounted on two identical gas foil bearings. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 324–341, 2024. https://doi.org/10.1007/978-3-031-40459-7_22

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One of the first attempts to implement parametric excitation in realistic rotor models has been done in [6], where the potential to stabilize an equilibrium position was investigated. The stabilization of limit cycles was investigated in [7], where a turbine rotor, modeled with finite element method (FEM), was mounted on adjustable oil film bearings. The works hereby referred, do not consider complex rotor models coupled to active gas foil bearings (AGFBs) and do not examine the type of the occurring bifurcations. Additionally, numerical continuation methods for limit cycles and their bifurcations have been recently applied in simplistic nonlinear rotor bearing systems. In [8–10], simplified models of high-speed rotors were coupled to floating ring bearings, while in [11–14] Jeffcott rotor models on simple oil film bearings were investigated. Recent studies, focusing mainly on the bearing models, studied the bifurcation sets of simplistic rotor models on adjustable oil bearings [15] and on gas foil bearings [16] without implementing parametric excitation. In current work, a lot of emphasis was given on the programming of a robust and time efficient continuation method, applicable to parametrically excited, complex rotor bearing systems with multiple degrees of freedom. A nonlinear approach of the elastoaerodynamic problem is straightly adopted. Common assumptions about the gas lubrication problem are introduced and the Reynolds equation for the compressible gas flow is solved using a Finite Difference Method [FDM]. The Simple Elastic Foundation Model (SEFM) is adopted for the representation of the bump foil behavior. The structure consists of linear elements of stiffness and damping in the radial direction while the top foil is considered massless. Parametric excitation is introduced by a sinusoidal displacement of the outer, deformableringwith predefined amplitude and frequency, and a harmonic variation in bearing’s stiffness and damping properties is generated. This can practically be achieved using piezo-actuators [17]. In general, there are various experimental and theoretical investigations which show that increased damping and stabilization is possible using closed loop control techniques such as hydraulic servo systems [18]. In current work, an open loop, periodic excitation strategy is proposed, the frequency of which should be close to the lower critical speeds. The periodic solutions of the parametrically excited and perfectly balanced rotorbearing systems are considered as solutions of nonlinear Boundary Value Problems (BVPs) and are evaluated using the explicit Runge-Kutta scheme [19], as it is found to be more robust method than the widely known collocation method. The corresponding solution branches are evaluated using the most reputable continuation method, the pseudo-arc length continuation method [20–23]. This method has the primary advance to study MDoF systems where the nonlinear equations of motion can be many [24] and the occurring bifurcations of various types. Similar continuation methods are applied in order to accurately predict period doubling (PD) and Neimark-Sacker (NS) bifurcations as two bifurcation parameters, the rotating speed and the excitation frequency are changed. Finally, the type of the occurring Neimark-Sacker bifurcations is investigated [25]. All the aforementioned methods are programmed by the authors directly from the notes [20, 23, 26]. The motion of unbalanced rotors under the effect of parametric excitation has quasi periodic characteristics resulted by the simultaneous excitation and synchronous frequency and should be studied using the theory of nonlinear normal

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modes. Nevertheless, using time integration algorithms, the authors verified that balance quality grades for turbopump rotors do not dramatically affect the phenomenon of parametric antiresonance.

2 Analytical Model of the Parametrically Excited Rotor with AGFBs 2.1 Elastoaerodynamic Lubrication and Resulting Gas Forces A gas foil bearing with active configuration is presented in Fig. 1 in a schematic representation of the working principles. Under the assumptions of a) isothermal gas film, b) laminar flow, c) no slip boundary conditions, d) continuum flow, e) negligible fluid inertia, f) ideal isothermal gas law (p/ρ = ct), g) negligible entrance and exit effects and negligible curvature of the gas film, the compressible gas flow is described by the Reynolds equation, given in Eq. (1). This equation is written in dimensionless form and it is an implicit function of dimensionless time and journal and foil kinematics.     ∂ ∂   ∂   3 ∂p 3 ∂p 2 ∂ +κ = ph ph ph + 2 ph (1) ∂x ∂x ∂z ∂z ∂x ∂τ Since analytical solution for Eq. (1) cannot be defined, the Finite Difference Method (FDM) is used to approximate the gas pressure distribution. At first, the Reynolds equation is rewritten, defining the first time derivative of the pressure distribution, in Eq. (2).     ˙   ˙p = 1 ∂ ph3 ∂p + κ ∂ ph3 ∂p −  ∂ ph − ph ∂x ∂z 2h ∂x 2h ∂z 2h ∂x h

(2)

The pressure domain is converted into a grid of i = 1, . . . , Nx +1 and j = 1, . . . , Nz + 1 mesh points (i and j are the indices in the circumferential and axial direction, see Fig. 1), upon which, the first order partial derivatives of Eq. (2) are expressed with backward differences and the second order partial derivatives are expressed by central differences. It should be noted that the elastoaerodynamic lubrication problem of Eq. (2) includes the dimensionless parameters of gas pressure p, gas film thickness h, spatial coordinates in the circumferential and axial direction x = θ, z respectively, dimensionless time τ , dimensionless rotating speed , the ratio κ = R/Lb . The gas film thickness is defined in Eq. (3) for both the continuous and discrete spatial coordinates, where q = q(θ ) or qi = q(θi ) is the dimensionless foil deformation in radial direction, see Fig. 1. h = 1 − xj cos θ − yj sin θ + q, hi = 1 − xj cos θi − yj sin θi + qi

(3)

The symmetry of the gas lubrication problem in the axial direction is taken into account with the boundary conditions described in Eq. (4). These conditions are also expressed in the continuous and the discrete domain. p(τ, θ0 , z) = p(τ, θ0 + 2π, z) = 1, p1,j = pNx+1,j = 1 ∂p ∂z |z=1/2

= 0,

pi,Nz/2 −pi,Nz/2−1 z

=0

(4)

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Fig. 1. (a) Representation and key design properties of the gas foil bearing under the effect of parametric excitation force acting on the outer ring, (b) modeling of the bump foil and the respective forces acting on the components of the gas foil bearing.

It is of high importance to note that, when integrating the pressure distribution over the bearing’s surface in order to compute the impedance gas forces, sub ambient pressure values are neglected. The Gümbel boundary condition is imposed and in terms of numerical calculations, if the dimensionless fluid pressure is lower than 1, then it is replaced by 1; in this way the pressure in the cavitated areas is neglected. The schematic representation of the widely known Simple Elastic Foundation (SEF) model for the bump foil structure is also depicted at Fig. 1. According to the aforementioned model, the structure consists of equally valued linear elements of dimensionless stiffness k f (with the corresponding compliance af = 1/k f ) and damping cf in the radial direction, while the top foil is considered massless, see Fig. 1. Its stripes along the axial direction are assumed to remain parallel to the bearing surface during their motion. Therefore, no axial direction is needed for the description of the top foil motion. Instead, only the mean axial gas pressure pm is necessary. This pressure, is given in Eq. (5), in the continuous and the discrete domain, in the dimensional and the dimensionless form. pm (θ ) =

1 Lb



Lb 0

p(θ )dz, pm,i =

NZ  1  pi,j z , Lb j=2

pm,i =

NZ   1  pi,j NZ

(5)

j=2

Given the fact that the top foil’s motion is synchronous to the pressure excitation, the structural damping coefficient can be expressed as cf = η · k f , where η denotes the loss factor. Generally, the dimensionless foil stiffness coefficient k f is related to some specific physical properties of the bump. According to [27], the dimensional foil compliance af can be analytically approximated by the following formula:    2p0 Sbf lbf  2 af = 1 − vbf (6) cr Ebf tbf

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where Sbf is the pitch of bump foil, lbf is half bump foil’s length, tbf is bump foil’s thickness and Ebf , vbf are Young’s modulus and Poison’s ratio of the bump foil respectively. Therefore, the dimensionless foil stiffness coefficient can be defined as: kf = kb =

cr af p0

(7)

As it is clearly stated in the Introduction, the parametric excitation

is implemented  by a predefined harmonic variation of the bearing’s outer ring qr = qr,i . Therefore, the radial displacement qi of the ith top foil’s stripe under the effect of the mean axial gas pressure and the parametric excitation is defined in Eq. (8), see also Appendix. q˙ i = q˙ r,i +

[pm,i − k f (qi − qr,i )] , i = 2, 3, ..., Nx cf

(8)

Finally, it is denoted x = 2π/Nx , z = 1/Nz and the nonlinear gas forces can be evaluated according to Eq. (9). B

2π 1

Fx = −

B

(p − 1) cos θ d θ d z = −

(9) (p − 1) sin θ d θ d z = −

0

(pi,j − 1) cos θi x z

i=2 j=2

0 0 2π 1

Fy = −

Nx  Nz 

0

Nx  Nz 

(pi,j − 1) sin θi x z

i=2 j=2

2.2 Condensed Rotor Model A representative turbopump rotor, mounted on two identical AGFBs and designed to operate above 20 kRPM is implemented in the current work,see Fig. 2. The rotor has complex geometry with different material properties, directly related to the temperature distribution among its length, and additional masses in various locations. Thus the rotor is discretized with cylindrical finite elements, each one having two nodes and a total of eight degrees of freedom xi , two transverse displacements and two tilting angles per node. The individual beam element matrices of inertia, stiffness and gyroscopy are properly summated and finally construct the corresponding global matrices. The global damping matrix follows the classical Rayleigh formula and the equations of motion for the whole rotor system in form are derived in Eq. (10). On the

dimensionless B right-hand side, gas bearing forces F i are evaluated according to the aforementioned elastoaerodynamic approach and they are the only source of nonlinearity in the rotorG bearing system. Additionally, gravity forces F i are composed supposing that the mass of each element is equally divided to the two nodes of the element.           B G (10) M x¨ i + C + G x˙ i + K {xi } = F i + F i

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Fig. 2. Schematic representation of a slender high-speed rotor supported on two identical GFBs. Finite element discretization, bearing span Ls , and master and slave nodes are also depicted.

The rotor model described in Eq. (10) is then reduced using the Guyan (static) reduction method. The selection of master (retained) nodes has been performed in order to match the dynamic response of the full system to this of the reduced one in terms of unbalance response and modal properties. It should be noted herethat the harmonic variation of qi seems to be efficient if its frequency is around specific damped natural frequencies of the linearized rotor-bearing model. Therefore, it is of great importance the reduction method to be held carefully. In current work, the number of total master nodes is 7, including both the overhang nodes and 5 almost equally distributed rotor nodes. By definition, only transverse displacements at each node are retained and the equations of motion for the reduced rotor system in dimensionless form are derived in Eq. (11).     

  

 B G  

(11) Mr x¨ m,i + Cr + Gr x˙ m,i + Kr xm,i = F r,i + F r,i The reduced rotor model equations of motion can now be converted to the  following

 

x˙ m,i set of first order ordinary differential equations (ODEs), where ym = ym,i =

 xm,i  B G

and F r,i = F r,i + F r,i .

y˙ m,i



 28×1

=



014×14 −1

−Mr Kr

14×14

  014×1 I14×14 

 ym,i 28×1 + −1

 −1 −Mr Gr + Cr 14×14 M F r,i r



(12) 14×1

2.3 Composition of the Parametrically Excited Rotor-Bearing System The aerodynamic lubrication problemin Eq. (2) renders Nx − 1 first order ODEs, with respect to the time derivative of the dimensionless nodal pressures, in Eq. (13).  (13) p˙ = fB p, q, qr , ym In turn, the structural problem renders another Nx − 1 first order ODEs. Equation (8) can alternatively be written in the following form. 

 (14) q˙ = q˙ i = fF p, q, qr

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The reduced rotor model equations of motion, see Eq. (12), can be written in the form:  (15) y˙ m = fR p, ym The nonlinear rotor-gas bearing system is defined by the following first order ODEs,

T where s = p q ym . It should be noted that due to the periodic variation of the bearing’s outer ring dimensionless time still appears explicitly in Eq. (16).  s˙ = f s, , ex , τ (16) In order for the limit cycle solutions to be efficiently evaluated by the explicit RungeKutta method, the aforementioned system should be converted to autonomous and this can be achieved by augmenting an oscillator with two degrees of freedom whose unique solution is a harmonic function of frequency ex , see Eq. (17).   s˙ N +1 = fN +1 = sN +1 + ex sN +2 − sN +1 s2N +1 + s2N +2   (17) s˙ N +2 = fN +2 = −ex sN +1 + sN +2 − sN +2 s2N +1 + s2N +2 Finally, the autonomous system of first order ODEs is defined in Eq. (16), where  



˜s = sT sN +1 sN +2 T and f˜ = f T fN +1 fN +2 T .   ˜  s˙˜ = f˜ s,

(18)

3 Quality of Bifurcations of the Dynamic System 3.1 Location and Continuation of Limit Cycles Away to find isolated periodic solutions (limit cycles) of the Dynamic System defined in Eq. (18) should be established. If the system poses a stable limit cycle, then it is reasonable to approximate it by numerical integration with an initial condition which belongs to the basin of attraction of the cycle. Given an initial guess for the limit cycle s˜0 and an initial guess for the cycle period T0 = 2π/ex it is possible to formulate a periodic Boundary Value Problem (BVP), see Eq. (19) on a fixed time interval [0, 1] [0, 1]. ⎧   ⎪ d s˜ ⎪ ˜  =0 ˜ s, ⎪ − T f ⎪ ⎪ d τ1 ⎪ ⎪ ⎪ ⎨˜ ˜ s(1) − s(0) =0 (19) ⎪ 1 ⎪    ⎪ ⎪ ⎪ ˜ s˜0 d τ1 = 0 ⎪ s, ⎪ ⎪ ⎩ 0

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The first two conditions define a periodic solution to the partially defined BVP but not uniquely, since any time shift of such a solution, is another solution. Therefore, an extra condition has to be appended, known as phase condition, in order to select one and only periodic solution among all those corresponding to the cycle. The phase condition appended in current work is called integral phase condition and it is a necessary condition ˜ s˜0 with respect to any time shifts. The for a local minimum of the distance between s, aforementioned problem can be reduced to finite dimensional problem using the explicit ∗ Runge-Kutta discretization scheme and solved for the unknown periodic solution s˜ and ∗ the unknown period T . By definition, the periodic solution of the problem defined in Eq. (19) depends ∗ on the dimensionless rotating speed . The problem of computing the curve s˜  belongs to the general case of finite dimensional continuation problems. The numeri∗ ∗ ∗ cal solution of the continuation problem means computing a sequence of s˜1 , s˜2 , s˜3 , ... ∗ Approximating the curve s˜ (). This sequence is generated by an initial point s˜0 which is sufficiently close to the curve. In current work, the continuation algorithm implements a predictor-corrector method called pseudo-arc length continuation method. For more detailed information, the reader may refer to [22]. 3.2 Location and Continuation of Codim1 Bifurcations of Limit Cycles The problem of locating Codim 1 bifurcations of limit cycles is a more delicate problem and, in this case, should be approached again as BVP, since there are periodic solutions whose multipliers have magnitude much smaller than 1. In the case ofFlip (perioddoubling) bifurcation, a vector-valued function v(τ1 ) is introduced and a non-periodic BVP is considered on the fixed time interval in Eq. (18). The first three conditions specify the periodic BVP defined in Eq. (19), the fourth condition is the linearization ˜ the fifth condition corresponds to the flip of Eq. (18) around the periodic solution s, bifurcation condition and the last one provides a normalization to v(τ1 ). This problem can be reduced to its finite dimensional form using the explicit Runge-Kutta discretization ∗ scheme and solved for the unknowns s˜ , T ∗ , v, . .In the case of Secondary Hopf (Neimark Sacker) bifurcation, a complex eigen-function w(τ1 ) and the scalar variable θm (which parameterizes the critical multipliers) are introduced and the non-periodic

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BVP is considered on the fixed time interval in Eq. (20). ⎧   d s˜ ⎪ ⎪ ˜  =0 − Tf˜ s, ⎪ ⎪ ⎪ d τ1 ⎪ ⎪ ⎪ ⎪ ˜ ˜ ⎪ s(1) − s(0) =0 ⎪ ⎪ ⎪ ⎪ 1 ⎪   ⎪  ⎪ ⎪ ⎪ ˜ s˜0 d τ1 = 0 ⎪ s, ⎪ ⎪ ⎨ 0

⎪ dv ∂ f˜ ˜  ⎪ ⎪ ⎪ −T s,  v = 0 ⎪ ⎪ d τ1 ⎪ ∂ s˜ ⎪ ⎪ ⎪ ⎪ v(1) + v(0) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ v, v d τ1 = 0 ⎪ ⎩

(20)

0

⎧   ⎪ d s˜ ⎪ ˜  =0 ˜ s, ⎪ − T f ⎪ ⎪ ⎪ ⎪ d τ1 ⎪ ⎪ ⎪ ˜ ˜ − s(0) ⎪ s(1) =0 ⎪ ⎪ ⎪ ⎪ 1 ⎪   ⎪  ⎪ ⎪ ⎪ ˜ s˜0 d τ1 = 0 s, ⎪ ⎪ ⎪ ⎨ 0

⎪ dw ∂ f˜ ˜  ⎪ ⎪ ⎪ −T s,  w = 0 ⎪ ⎪ d τ1 ⎪ ∂ s˜ ⎪ ⎪ ⎪ w(1) − eiθm w(0) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ w, w d τ1 = 0 ⎪ ⎪ ⎩

(21)

0

The meaning of the augmented conditions in Eq. (21) is similar with the meaning of the augmented conditions in Eq. (20). It is suggested this problem to be written in its real form.Then it should be discretized using the Runge-Kutta scheme and finally ∗ solved for the unknowns s˜ , T ∗ , w, θm , . The presented BV problems can also be used to continue generic Flip (PD) and Secondary Hopf bifurcations (NS) of limit cycles. They are called fully extended augmented BVPs since the augmented conditions for the location of Codim 1 bifurcation can be replaced by one and only equation using bordering techniques analytically presented in [22]. It is important to note that Eq. (20) and Eq. (21) do not consider the degeneracy conditions of the corresponding bifurcations. Generally, the finite dimensional problem, arising after proper discretization of Eq. (19) is solved using the damped Newton method, analytically presented in [19]. Based on the Jacobian matrix of the aforementioned system of nonlinear equations, one can approximate the monodromy matrix of the isolated

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periodic solution, using a method similar to this presented in [26]. All the criteria to determine local flip (period doubling) and secondary Hopf (Neimark – Sacker) bifurcations arise from the eigenvalues of the monodromy matrix, called Floquet multipliers. More specifically, supposing that all the non-degeneracy conditions hold, if there is one pair of complex eigenvalues on the unlit circle, λ1,2 () = r()eiθ() , r = 1, 0 < θ < π, then Neimark-Sacker bifurcation occurs. The non-degeneracy conditions indicate that eikθ = 1, k = 1, 2, 3, ... (absence of strong resonances), ddr = 0 and d = 0, where d stands for a coefficient involved in the normal form of Neimark – Sacker bifurcation, see [25]. The case of Flip bifurcation is simpler. Supposing again that all the non-degeneracy conditions hold, if there is one real eigenvalue on the unit circle λ3  = −1, then period doubling bifurcation occurs. The non-degeneracy conditions now indicate that d λ3 = 0 and c = 0, where c stands for a coefficient in the normal form of period d doubling bifurcation, see [25].

4 Results In Fig. 3 full bifurcation sets for four different values of the dimensionless foil stiffness coefficient k b and for three different values of the maximum dimensionless vertical displacement of the outer ring δ are depicted. According to the literature, in theoretical investigations k b varies from 0.1 to 100. In our specific case, k b belongs to the aforementioned range and enhances the phenomenon of parametric antiresonance as much as possible. The evaluation of generic Neimark – Sacker bifurcations for bigger than presented values of k b was numerically difficult, thus omitted. The minimum value of δ is selected so as not to affect the threshold speed of instability of the reference rotor – bearing system. The maximum value of δ generally depends on the outer ring’s physical properties, the power supply availability and the excitation frequency. In our case, the maximum value is selected in order to avoid numerical difficulties in the continuation of Neimark Sacker bifurcations. Currently, alternative methods of continuation of Neimark-Sacker bifurcations are studied in order to overcome the aforementioned numerical difficulties. The occurring Neimark-Sacker bifurcations as the two bifurcation parameters , ex change are depicted for eachvalue of the dimensionless stiffness coefficient. The progress of period doubling bifurcations is evaluated by solving Eq. (20) in the context of a sequential continuation method and the progress of Neimark-Sacker bifurcations is evaluated by solving Eq. (19) in the context of the same continuation method, for simplicity reasons. It can be safely concluded that excitation frequencies around which parametric resonances and antiresonances occur can be approximately predicted by Eq. (22), where j,k denote the dimensionless critical speeds of the rotor-bearing system.   j ± k  2j 1 2 , ex,int  , j, k, n = 1, 2, 3, ... (22) ex,int  n n The denominator n denotes the order of the parametric resonance or antiresonance. In these results, only first and second order resonances and antiresonances are found.

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Fig. 3. Full bifurcation set for a) k b = 3, , b) k b = 10, c) k b = 20 and k b = 50.

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Fig. 3. (continued)

As it is expected, the strength of such phenomena is enhanced as the dimensionless parameter δ is increased. It is additionally observed that this strength is enhanced as the dimensionless stiffness coefficient is increased. Both the dimensionless parameters mentioned above are related to the variation of the bearing clearance. Therefore, it can be concluded that the greater the variation in clearance, the greater the difference in the threshold speed of instability is. Around some of the excitation frequencies of interest given by Eq. (22) and under specific circumstances, period doubling bifurcations occur too. For instance, it is observed that for the lowest value of the dimensionless parameter δ no period doubling bifurcation occurs. As this parameter increases, further bifurcations appear. In contrast, as the dimensionless stiffness coefficient increases, only Neimark-Sacker bifurcations appear. Finally, it is of high importance to note that in all examined cases there are zones of excitation frequencies at which the stability threshold of the rotor-bearing system is enhanced, and no other type of bifurcation occurs. The transient response of the rotor system is depicted in Fig. 4 for some operating conditions of interest. For each of the different foil stiffness values k b = 3, 10, 20, 50, the excitation frequencies where antiresonance occurs are selected from the stability maps in Fig. 3. These are found to be ex = 0.63, 0.70, 0.72, 0.74 respectively. The transient response of the system is evaluated under the parametric excitation of ex as before, and for the amplitude of excitation force to render δ = 0.2. The transient response is evaluated with time integration of the system in Eq. (16) and the envelop of response is depicted in each of Figs. 4a, 4b,4c,4d in the lower chart, together with the response envelop evaluated by sequential continuation of the limit cycles. The Floquet multipliers of each limit cycle motion depict the quality of bifurcations when these occur. At all cases depicted in the aforementioned charts, the system experiences a Neimark-Sacker bifurcation at the ending speed of c.a.  = 1.2 and this is considered as the threshold speed of instability. However, at the case of Fig. 4a, the system experiences period doubling bifurcation in much lower rotating speed, c.a.  = 0.4 and stable limit cycles

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Fig. 4. Response envelope for a) k b = 3, b) k b = 10, c) k b = 20, and d) k b = 50.

are generated; in Fig. 4a, and at the lower chart one may notice the unstable limit cycles and the stable limit cycles where the system oscillates after the PD bifurcation. In the

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Fig. 4. (continued)

upper charts of Figs. 4b,4c,4d one may notice the lower threshold speed of instability at c.a.  = 0.95 when parametric excitation is of low frequency, e.g. ex = 0.2. In Fig. 4a, and in the upper chart, the threshold speed of instability appears at  = 1.1 and this is due to the fact that the foil is compliant enough (k b = 3) and dissipation of energy takes place due to the higher motion of the foil. Considering the above, parametric excitation provides increase of the threshold speed of instability up to 30% at the specific application. More design sets are currently investigated by the authors. For clear observation of bifurcation trees of period doubling bifurcations occurring, bifurcation diagrams are depicted in Fig. 5.

Fig. 5. Bifurcation diagrams during and after a flip (period doubling) bifurcation

As one may notice in Fig. 5 (top), around the excitation frequency of interest the unstable limit cycles where the system oscillates after PD are distributed in a wide range of rotating speeds.

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The transient response of the unbalanced rotor system under the effect of parametric excitation is compared with the response of the balanced one in Fig. 6, for some operating conditions of interest. Generally, for turbopump rotors G = 6.3, 2.5 balance quality grades are considered. For each of the dimensionless foil stiffness values k b = 3, k b = 50, the excitation frequencies at which parametric antiresonance occurs are selected, and two balance quality grades are applied (G = 6.3, G = 1). As one may notice in Figs. 6a and 6b the phenomenon of parametric antiresonance is not affected by any level of unbalance.

Fig. 6. Evaluation of periodic response applying numerical continuation of limit cycles, and quasiperiodic response applying time integration for the respective design and operating parameters as depicted.

Time integration is the only tool in this paper to evaluate the quasi-periodic response under the simultaneous parametric and unbalance excitation; the time response proves that the stability threshold (NS bifurcation) is very similar in both periodic and quasiperiodic solutions, in the status of parametric antiresonance. In the case of compliant bump foil depicted in Figs. 6c and 6d where higher unbalance is applied, still the above

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comment holds. However, as the unbalance becomes higher, the quality of response includes further characteristics, which should be studied with the appropriate tools for quasi periodic solution evaluation, in future work.

5 Conclusions This work proves that parametric antiresonance is feasible in slender, high-speed rotors mounted on active gas foil bearings. Slender rotors retrieve stability in high rotating speeds under the effect of periodic load acting on the deformable ring of the gas foil bearings. A nonlinear approach for the elastoaerodynamic problem is adopted, according to which, the compressible gas flow is described by the Reynolds Equation and the bump foil’s behavior is represented by the simple elastic foundation model. Based on the following conclusions, this paper aims to raise further concerns on parametrically excited rotating systems. The investigation of full bifurcation sets at a wide range of rotating speed and excitation frequency and amplitude indicates that the zone of excitation frequencies at which para-metric resonances or antiresonances occur can be approximated using Eq. (20), existing in literature since long. The strength of both parametric resonance and antiresonance depends on the variation of the bearing clearance (amplitude of exciting force). The greater the variation of the clearance, the greater the difference in the threshold speed of instability is. Based on the literature, see [28] and on personal experience, authors firmly believe that all the aforementioned conclusions regarding the correlation between the threshold speed of instability and the variation in foil stiffness coefficient and clearance are valid for a wide range of slender rotors mounted on AGFBs. Parametric antiresonance and modal interaction are two simultaneous phenomena, and it is of quite interest to study the energy flow between the interacting modes. This can be achieved by comparing the unbalance response of a parametrically excited rotor with the unbalance response of the same rotor mounted on conventional gas foil bearings (without parametric excitation). It should be noted that in the former case, the rotorbearing system has quasi periodic characteristics due to the simultaneous synchronous and parametric excitation. Harmonic balance is currently under investigation in order to be embedded in the corresponding continuation scheme. It is furthermore of quite interest to evaluate the type of the occurring Neimark-Sacker bifurcations (subcritical/supercritical). This can be straight forward achieved by approximating the normal form of Neimark Sacker bifurcations. The validity of the coefficients involved in this normal form is currently under investigation.

Appendix: Implementation of Parametric Excitation The deformation of a ring with the physical and geometrical properties like Poisson’s ratio vr , Young’s modulus of elasticity, inner/outer radius Ri,r , Ro,r and polar moment of inertia I , is evaluated with approximate analytical formulas  obtained by the strength of materials. The effect of a periodic vertical load F0 1 + sin ex τ is the deformation

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in the horizontal (dh) and vertical (dv) direction of the ring, see also Fig. 1, as defined in Eq. (21). The constants κ1 and κ2 included are defined in Eq. (22).   R3o,r 2 2 κ1  F0 κ 1 + sin(ex τ ) − κ + 2 2 11 2 4.2 · 10 Ir π 2   3 Ro,r 2κ22  π −F0 κ1 − dv = qr (θ = π/2, 3π/2, τ ) = 1 + sin(ex τ ) 11 2 4.2 · 10 Ir 4 π (23) dh = qr (θ = 0, π, τ ) =

˙ d v˙ , Given the corresponding derivatives with respect to the dimensionless time d h, the deformation of the outer ring and its rate of change in the circumferential direction are evaluated in Eq. (23), where qr = qr /cr and q˙ r = q˙ r /cr .     R4o,r − R4i,r R4o,r − R4i,r 1.33(1 + 2vr )Ro,r  +  , κ2 = 1 −    κ1 = 1 − 2R2i,r R2o,r − R2i,r π R2o,r − R2i,r 2R2i,r R2o,r − R2i,r  2  2  Ri,r + dh cos θ + Ri,r + dv sin θ − Ri,r qr = qr (θ, τ ) =      Ri,r + dh cos θ d h˙ cos θ + Ri,r + dv sin θ [d v˙ sin θ ] q˙ r = q˙ r (θ, τ ) = qr + Ri,r

(24)

(25)

References 1. Bolotin, V.: The Dynamic Stability of Elastic Systems. Holden-Day, Australia (1964) 2. Seyramian, A.P., Mailybaev, A.A.: Multiparameter Stability Theory with Mechanical Applications, vol. 13. World Scientific Pub. Co., Singapore (2003) 3. Schmidt, G.: Parametererregte Schwingungen (In German, Translated Title ‘Parametrically Excited Oscillations’). Deutcher Verlag der Wissenschafte (1975) 4. Tondl, A.: On the interaction between self excited and parametric vibrations. Monogr. Memoranda, Natl Res. Inst. Mach. Des. 25 (1978) 5. Tondl, A.: To the problem of quenching self-excited vibrations. ACTA Technol. 43, 109–116 (1998) 6. Breunung, Thomas, Dohnal, Fadi, Pfau, Bastian: An approach to account for interfering parametric resonances and anti-resonances applied to examples from rotor dynamics. Nonlinear Dyn. 97(3), 1837–1851 (2019). https://doi.org/10.1007/s11071-019-04761-9 7. Dohnal, F., Chasalevris, A.: Improving stability and operation of turbine rotors using adjustable journal bearings. Tribol. Int. 104, 369–382 (2016) 8. Boyaci, A., Hetzler, H., Seemann, W., Proppe, C., Wauer, J.: Analytical bifurcation analysis of a rotor supported by floating ring bearings. Nonlinear Dyn. 57, 497–507 (2009) 9. Boyaci, A., Lu, D., Schweitzer, B.: Stability and bifurcation phenomena of Laval/Jeffcott rotors in semi-floating ring bearings. Nonlinear Dyn. 79, 1535–1561 (2015) 10. Van Breemen, F.C.: Stability analysis of a Laval rotor on hydrodynamic bearings by numerical continuation: Investigating the influence of rotor flexibility, rotor damping and external oil pressure on the rotor dynamicbehavior, M.Sc. thesis, Delft University of Technology (2016)

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11. Rubel, J.: Vibrations in nonlinear Rotordynamics Dissertation, PhD thesis, Ruprecht-KarlsUniversität Heidelberg (2009) 12. Amamou, A., Chouchane, M.: Bifurcation of limit cycles in fluid film bearings. Int. J. NonLinear Mech. 46, 1258–1264 (2011) 13. Sghir, R., Chouchane, M.: Prediction of the nonlinear hysteresis loop for fluid-film bearings by numerical continuation. Proc. IMechE Part C: J. Mech. Eng. Sci. 229(4), 651–662 (2015) 14. Sghir, R., Chouchane, M.: Nonlinear stability analysis of a flexible rotor-bearing system by numerical continuation. J. Vibr. Control 22(13), 3079–3089 (2016) 15. Becker, K.: Dynamisches Verhalten hydrodynamisch gelagerter Rotoren unter berücksichtigung veranderlicher Lagergeometrienm Ph.D. Thesis, Karlsruhe Institute of Technology, Germany (2019) 16. Leister, T.: Dynamics of rotors on refrigerant lubricated gas foil bearings, Ph.D. Thesis, Karlsruhe Institute of Technology, Germany (2021) 17. Dohnal, F.: Optimal dynamic stabilisation of a linear system by periodic stiffness excitation. J. Sound Vib. 320, 777–792 (2009) 18. Santos, I.F.: Design and evaluation of two types of active tilting pad journal bearings. IUTAM Symposium on Active Control of Vibration, Bath, England, pp. 79–87 19. Ascher, U.M., Mattheij, R.M.M., Russell, R.D.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, 1st edn. Society for Industrial and Applied Mathematics, Philadelphia (1995) 20. Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. Society for Industrial and Applied Mathematics, Philadelphia (2003) 21. Meijer, H., Dercole, F., Olderman, B.: Numerical bifurcation analysis. In: Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science, pp. 6329–6352. Springer, New York (2009). https://doi.org/10.1007/978-0-387-30440-3_373 22. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Applied Mathematical Sciences, Springer, New York (1998) 23. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics, Wiley Series in Nonlinear Science. J. Wiley & Sons, Hoboken (1995) 24. Doedel, E.J., Keller, H.B., Kernevez, J.P.: Numerical analysis and control of bifurcation problems (II) Bifurcation in infinite dimensions. Int. J. Bifurcat. Chaos 1(3), 745–772 (1991) 25. Kuznetsov, Y.A., Govaerts, W., Doedel E.J., Dhooge A.: Numerical periodic normalization for Codim 1 Bifurcations of limit cycles. J. Numer. Anal. 43, 1407–1435. Society for Industrial and Applied Mathematics, Philadelphia (2015) 26. Doedel, E.J.: Lecture notes on numerical analysis of nonlinear equations. In: Krauskopf, B., Osinga, H.M., Galán-Vioque, J. (eds.) Numerical Continuation Methods for Dynamical Systems. UCS, pp. 1–49. Springer, Dordrecht (2007). https://doi.org/10.1007/978-1-40206356-5_1 27. Heshmat, H., Walowit, J.A., Pinkus, O.: Analysis of gas-lubricated foil journal bearings. J. Lubricat. Technol. (1983) 28. Dohnal, F.: A contribution to the Mitigation of Transient Vibrations Parametric AntiResonance: Theory, Experiment and Interpretation. Technischen Universität Darmstadt, Habilitationsschrift (2012)

Dynamic Design of the High-Speed Rotor System Considering the Distribution of Strain Energy Cong Liu1 , Yongfeng Wang1(B) , Ruiqi Jia2 , and Jie Hong1 1 School of Energy and Power Engineering, Beihang University, Beijing 100191, People’s

Republic of China [email protected] 2 AVIC Shenyang Engine Research Institute, Shenyang 110066, People’s Republic of China

Abstract. In order to optimize the dynamic properties of typical high-speed rotor system, a mechanical model of the high-speed rotor system considering the key structural features was established. Intrinsic relationship between the first bending critical speed, the reaction force at the bearing and the strain energy distribution of the rotor was investigated based on Rayleigh method and Euler beam theory. The results show that the first bending modal strain energy distribution corresponds to the first bending critical speed of rotor and the reaction force at the bearing. The bending critical speed and the proportion of strain energy of drum shaft are positively correlated with local angular stiffness in front of the turbine, the reaction force at the rear bearing and the proportion of strain energy of rear neck are positively correlated with local angular stiffness behind the turbine. The feasibility of the correlations on practical rotor was proved through a typical high-speed rotor. Keywords: Design of dynamic properties · High-speed rotor · Strain energy distribution · Critical speeds · Reaction force at bearing

1 Introduction Generally, high-pressure rotors of modern aero-engine operate over the first two critical speeds, and under the third critical speed, which are the typical high-speed rotors. Highspeed rotors are tended to bear greater loads and cause less weight, which leads to higher operating speed and lower bending critical speed. This will cause insufficient safety margin between the operating speed and bending critical speed, resulting in huge bending deformation of the rotor [1]. The skew of principal axis of inertia of the large mass component caused by the bending deformation will produce huge rotatry inertia load at high rotational speed, and finally be balanced by the bearings and cause huge reaction force [2, 3]. To reduce the damage caused by the excessive reaction force, it is necessary to increase the bending critical speed and supress the reaction force at the bearing at the same time. This is a multi-objective optimization design, in which multiple structural parameters need to be adjusted and the dynamic properties of the rotor are solved cyclically to ensure the effectiveness of parameter adjustment. If the traditional methods such as cyclic modal analysis (Campbell diagram) and harmonic response © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 342–363, 2024. https://doi.org/10.1007/978-3-031-40459-7_23

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analysis are used [4–6], it will take a lot of time and computing power when the number of iterations is large. Therefore, in order to improve the efficiency of iterative design, it is necessary to find an equivalent evaluation parameter that is simple to calculate and can accurately measure the dynamic properties of the rotor. According to the definition of strain energy, the strain energy distribution of the rotor corresponds to its deformation, which can indirectly characterize the gyroscopic moment and structural internal force of the rotor, and then reflect multiple dynamic properties such as critical speed and reaction force at the bearing. If the response of the rotor in the working state is dominated by a certain mode shape, the dynamic properties of the rotor can be measured by the strain energy distribution of this mode shape. In fact, the strain energy analysis has already been applied in rotordynamics design by some scholars. Conry [7] studied the optimal design of rotor unbalance distribution, and proposed a rotor balancing method based on strain energy of rotor and bearing. Chen [8] explored the correlation between the strain energy distribution and dynamic properties of the rotor, and pointed out that the critical speed distribution of the rotor can be effectively adjusted by changing the stiffness of the high strain energy density shaft and bearing. Srinivas [9] evaluated the quality of different design schemes quantitatively by the strain energy proportion of shaft in the structural layout design of a medium thrust turbofan engine. Hong, Song, et al. [10] proposed a robust design method based on rotor strain energy distribution considering the bending stiffness loss at the connection structure. Wang [11] used the strain energy proportion of the rotor and the bearing structure to quantitatively evaluate the amount of bending deformation of the rotor and analyze the mode shape of the rotor system. Zheng [12] carried out strain energy analysis on a certain type of counter-rotating aero-engine and proposed a strain energy evaluation criterion. Sun [13] established a rough relationship between the natural frequency of the rotor system and the strain energy of the rotor based on the Rayleigh method. Hong, Xu, Yang et al. [14, 15] proposed to reduce the interface damage by controlling the strain energy level of the connection structure, improving the robustness of the rotor dynamic properties of high-speed rotor systems. The above studies have initially established the evaluation and design method of rotor dynamic properties based on strain energy. However, in the current design of rotor dynamic properties, strain energy is mainly used to evaluate the natural properties of the rotor and the robustness of the connection structure, and there is little relevant research on its application to the evaluation of rotor dynamic response. Moreover, the understanding of the correlation between strain energy and dynamic properties is mostly based on qualitative analysis and lacks theoretical support. And current research objects are mainly rotors that operates far below the bending critical speed. Therefore, this paper takes the high-speed rotor system that operates near the bending critical speed as the research object, extracts its key structural features and establishes an equivalent mechanical model. Based on the model, the dynamic design method based on rotor strain energy distribution is explored.

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2 Modeling of High-Speed Rotor 2.1 Strain Energy Distribution of High-Speed Rotor To analyse the deformation and strain energy distribution characteristics of high-speed rotors, the dynamic response of a typical high-speed rotor under working condition is calculated as an example. The engine is a long-span and thin-walled rotor supported by two bearings, whose maximum operating speed is up to 16000 r/min. According to reference [2] and [3], the skew of the principal axis of inertia will cause significant impact on the dynamic response of high-speed rotors. Therefore, in order to accurately describe the influence of the mass distribution characteristics, the skew angle of the principal axis of inertia should be considered in addition to the mass eccentricity. An unbalance of 1000 g·mm and a skew angle of the principal axis of inertia of 3 × 10–4 rad are applied to the turbine of the typical high-speed rotor, and then the vibration response of the rotor at 16000 r/min is calculated, as shown in Fig. 1, and the corresponding strain energy distribution is shown in Fig. 2. It can be seen from Fig. 1 that large relative angular deformation occurrs between the compressor and turbine, indicating that the rotor experiences significant bending deformation. Figure 2 indicates that most of the strain energy concentrates on the rear neck of the compressor, the drum shaft and the rear neck of the turbine, which means these structures experience huge flexible deformation. Moreover, the flexible deformation of the compressor and turbine is extremely small, they almost remain rigid in rotation. Therefore, in the modeling of high-speed rotor, the compressor and turbine can be treated as rigid disks that mainly consider the mass distribution, while the necks and drum shaft can be treated as flexible shafts that mainly consider the flexible deformation.

Fig. 1. Deformation of the typical high-speed rotor under working condition

2.2 Dynamical Model Based on the results above, a theoretical rotor model is established considering the key structural characteristics of aero-engine high-pressure rotors, as shown in Fig. 3. The compressor and turbine are treated as rigid disks, and the drum, the front and rear necks are treated as flexible shafts. The joints around the turbine disk are studied, which is

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modeled by two local springs. Disk 1 is fixed on the shaft, and disk 2 is connected to the shaft through two local springs. The axial thicknesses of the disks are not considered, but changes in shaft material and section size are considered. The shaft is divided into two parts at the thin disk (the left part is the front shaft and the right part is the rear shaft). X

Disk1

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Fig. 3. Theoretical model of typical high-speed rotor systems

As shown in Fig. 3, an absolute coordinate system O-XYZ is established at the center point of the front bearing, where the Z axis is along the rotation center line. The dynamic response of the rotor is described by the following displacement vectors: ⎧ r1 (z, t) = x1 (z, t) + iy1 (z, t) ⎪ ⎪ ⎪ ⎨ r (z, t) = x (z, t) + iy (z, t) 2 2 2 (1) ⎪ r (t) = x (t) + iy (t) d2 d2 d2 ⎪ ⎪ ⎩ θ2 (t) = −θ2x (t) + iθ2y (t)

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where r1 (z, t) is the lateral displacement of the front shaft, r2 (z, t) is the the lateral displacement of the rear shaft, rd 2 (t) and θd 2 (t) is the lateral and angular displacement of disk 2. The stiffness matrix of the local spring in front of disk 2 is: ⎡

0 kr1 0 ⎢ 0 kr1 −kθr1 K1 = ⎢ ⎣ 0 −kθr1 kθ1 0 kθr1 0

⎤ kθr1 0 ⎥ ⎥ 0 ⎦ kθ1

(2)

where kr1 is the local lateral stiffness between disk 2 and front shaft, kθ1 is the local angular stiffness, and kθr1 is the coupling stiffness of the lateral and angular degrees of freedom. Similarly, the stiffness matrix of the local spring behind disk 2 is: ⎡

0 kr2 0 ⎢ 0 kr2 −kθr2 K2 = ⎢ ⎣ 0 −kθr2 kθ2 0 kθr2 0

⎤ kθr2 0 ⎥ ⎥ 0 ⎦ kθ2

(3)

The energy of the rotor shown in Fig. 3 includes the kinetic energy of each disk, the kinetic energy and bending potential energy of the shaft, and the local spring potential energy [16]. The potential energy energy of the front and rear shafts of the rotor system are: ⎧ z2 ⎪

 2 ⎪ 1 ⎪

⎪ ⎪ ⎪ VShaft,1 = 2 E1 (z)I1 (z) r1 dz ⎪ ⎨ 0 (4) z3 ⎪ ⎪

⎪ 1 2 ⎪ ⎪ VShaft,2 = E2 I2 (z) r2 dz ⎪ ⎪ 2 ⎩ z2

where E1 (z)I1 (z) and E2 I2 (z) are the flexural stiffness of the front shaft and rear shaft, z2 , z3 are the axial coordinate of disk 2 and rear bearing, |•| denotes norm of vector, (•) denotes the second derivative with respect to the axial coordinate. The potential energy of local angular springs around disk 2 are: ⎧ 1 ⎪ ⎨ VSpring,1 = r1 · K1 · r1T 2 (5) 1 ⎪ T ⎩V r = · K · r Spring,2 2 2 2 2 where r1 is the relative deformation matrix between disk 2 and the front shaft, r2 is the relative deformation matrix between disk 2 and the rear shaft, which contains four degrees of freedom in lateral and angular directions.

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The kinetic energy of the shafts are: ⎧ z2 ⎪ ⎪ ⎪ ⎪ TShaft,1 = ρ1 (z)A1 (z)|˙r1 |2 dz ⎪ ⎪ ⎪ ⎨ 0

z3 ⎪ ⎪ ⎪ ⎪ ⎪ TShaft,2 = ρ2 A2 (z)|˙r2 |2 dz ⎪ ⎪ ⎩

(6)

z2

where ρ1 (z), ρ2 are the densities of the front shaft and rear shaft, A1 (z), A2 (z) are the cross-section areas of the front shaft and rear shaft. The kinetic energy of disk 1 and disk 2 are: ⎧

2

2 1 1 1 ⎪ ⎨ TDisk,1 = md 1 r˙de 1 + Jp1 ω2 (1 − |θd 1 |2 ) + Jd 1 ( θ˙d 1 + ω2 |θd 1 |2 ) 2 2 2 (7)

2

e 2 1 1 1 ⎪ 2 2 2 2 ⎩T



˙ | |θ | |θ θ r ˙ m J J = + ω (1 − ) + ( + ω ) p2 Disk,2 d2 d2 d2 d2 d2 d2 2 2 2 where md 1 , Jp1 , Jd 1 are the mass, polar moment of inertia and diameter moment of inertia of disk 1, md 2 , Jp2 , Jd 2 are the mass, polar moment of inertia and diameter moment of inertia of disk 2, ω is the rotational speed of the rotor, (•) denotes the derivative with respect to the axial coordinate, (˙•) is the derivative with respect to time. The expression of each displacement vector in Eq. (7) is: ⎧ ⎨ r e = r (z ) + e · eiωt+βe , θ = ∂r1 (z1 ) + τ ei(ωt+β1 ) 1 1 1 1 d1 d1 ∂z (8) ⎩ e rd 2 = rd 2 + e2 · eiωt , θd 2 = θ2 + τ2 ei(ωt+β2 ) where e1 is the mass eccentricity of disk 1, τ1 is the skew angle of the principal axis of inertia of disk 1, e2 is the mass eccentricity of disk 2, is the skew angle of the principal axis of inertia of disk 2, βe is the phase difference between the unbalance of disk 1 and the unbalance of disk 2 in the XOY plane, β1 , β2 are the initial phases of the principal axis of inertia of disk 1 and disk 2 in the XOY plane, z1 is the axial coordinate of disk 1. So, the Lagrangian of the rotor is: L=

2    TDisk,i + TShaft,i − VShaft,i − VSpring,i

(9)

i=1

Apply the Hamilton principle: t2 δ

Ldt = 0 t1

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Substitute Eq. (9) into the variation, then the differential equation of rotor motion can be obtained as follows:    ⎧ ∂ 2 r1 ∂2 ∂ 2 r1 ∂ 2 r1 ⎪ ⎪ ρ (z)A (z) + (z)I (z) E + md 1 2 − md 1 e1 ω2 ei(ωt+βe ) ⎪ 1 1 1 1 ⎪ 2 2 2 ⎪ ∂t ∂z ∂z ∂t ⎪ ⎪ ⎪   ⎪ 3r 2r ⎪   ∂ ∂ ∂ 1 1 ⎪ 2 i(ωt+β1 ) ⎪ − − iJ ω − J ω e τ δ(z − z1 ) J − J ⎪ p1 p1 1 d 1 d 1 ⎪ ⎪ ∂z ∂z∂t 2 ∂z∂t ⎪ ⎪ ⎪   ⎪ ⎪ ∂ 2 r1 ⎪ ⎪ − kr1 (rd 2 − r1 ) + kθ1 2 + kθr1 θ2 δ(z − z2 ) = 0 ⎪ ⎪ ⎪ ∂z ⎪ ⎨     ∂ 2 r2 ∂2 ∂ 2 r2 ∂ 2 r2 ⎪ ρ2 A2 (z) 2 + 2 E2 I2 (z) 2 − kr2 (rd 2 − r2 ) + kθ2 2 + kθr2 θ2 · δ(z − z2 ) = 0 ⎪ ⎪ ∂t ∂z ∂z ∂z ⎪ ⎪ ⎪     ⎪ 2r ⎪ ∂r1 ∂r2 ∂ ⎪ d2 ⎪ + k = md 2 e2 ω2 eiωt θ θ + k − r − r − − + k + k [r [r (z )] (z )] ⎪ md 2 r1 1 2 r2 2 2 θr1 2 θr2 2 d 2 d 2 ⎪ ⎪ ∂t 2 ∂z ∂z ⎪ ⎪ ⎪     ⎪ ⎪ ∂θ2 ∂r1 (z2 ) ∂r2 (z2 ) ∂ 2 θ2 ⎪ ⎪ ⎪ + kθ2 θ2 − + kθr1 (rd 2 − r1 ) Jd 2 2 − iJp2 ω + kθ1 θ2 − ⎪ ⎪ ∂t ∂t ∂z ∂z ⎪ ⎪ ⎪   ⎩  +kθr2 (r − r2 ) = Jp2 − J τ2 ω2 ei(ωt+β2 ) d2

d2

(11) where β1 equals to β1 − π2 and β2 equals to β2 − π2 . It can be seen from the above equation that the latter two equations describe the lateral and angular motion of disk 2, and the first two equations describe the lateral motion of the continuous shaft under the influence of the two disks. The rotary inertial loads of the disk change the dynamic characteristics of the rotor system by affecting the lateral and angular displacements of the connection points between the disk and shaft. Both ends of the rotor are elastically supported, and the supporting stiffnesses are kb1 and kb2 , so the boundary conditions of the rotor are: ⎧ ∂ 2 r(0, t) ∂ 3 r(0, t) ⎪ ⎪ = 0, EI = −kb1 · r(0, t) ⎨ EI ∂z 2 ∂z 3 (12) 2 3 ⎪ ⎪ ⎩ EI ∂ r(l, t) = 0, EI ∂ r(l, t) = kb2 · r(l, t) ∂z 2 ∂z 3 With the obtained equations, the dynamic response of the rotor could be solved numerically, the method used in this paper is the Laplace transformation, after which the equations are transformed into several algebraic equations.

3 Design on the Critical Speed In this section, the correlation between drum strain energy and bending critical speed is explored based on the theoretical model as shown in Fig. 3. The parameters of the theoretical model used in the following calculation are shown in Table 1, which are all based on a practical aero-engine high-pressure rotor. The first bending critical speed of a rotor is determined by the bending modal frequency and the gyroscopic effect. The bending modal frequency descripes the modal frequency at 0 r/min, which determines the starting point of the frequency curve in the Campbell diagram. While the gyroscopic effect descripes the effect of the gyroscopic moment on the rise of modal frequency, which determines the way how the frequency curve changes.

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Table 1. The parameters of the rotor system Parameters

Value

Total length of the rotor[mm]

870

Density of shaft ρ [kg/m3 ]

8240

Elastic modulus of shaft E [GPa]

197

Poisson’s ratio of shaft

0.32

Mass of disk 1 md 1 [kg]

53

Polar moment of inertia of disk 1 Jp1

975.2

[t·mm2 ] Diameter moment of inertia of disk 1 Jd 1 [t·mm2 ]

927.7

Mass of disk 2 md 2 [kg]

43

Polar moment of inertia of disk 2 Jp2

736.8

[t·mm2 ] Diameter moment of inertia of disk 2 Jd 2 [t·mm2 ]

389.8

Local lateral stiffness kr1 [N/m]

3 × 109

Local angular stiffness kθ1 [N·m/rad]

1 × 107

Local coupling stiffness kθr1 [N/rad]

1 × 106

Local lateral stiffness kr2 [N/m]

3 × 108

Local angular stiffness kθ2 [N·m/rad]

1 × 106

Local coupling stiffness kθr2 [N/rad]

1 × 105

Front and Rear bearing stiffness kb1 , kb2 [N/m]

1 × 107

3.1 Correlation Between Drum Strain Energy and Bending Modal Frequency For the rotor shown in Fig. 3, according to the Rayleigh method, the main vibration corresponding to the first bending mode can be assumed as: r(z, t) = R(z)eiωn t

(13)

where z represents the axial coordination, ωn represents the hypothetical frequency of the first bending mode, and R(z) represents the mode shape of the first bending mode. Under this assumption, the amplitude of the kinetic energy and potential energy of the rotor-bearing system are: Tmax =

1 M ωn2 , Vmax = V 2

(14)

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According to the energy balance of the system, Tmax equals to Vmax , so the hypothetical frequency of the first bending mode of the rotor is:  2V (15) ωn = M where M is the mass of the rotor in the modal coordinate system, which can be expressed as:      ∂R(z1 ) 2   ∂R(z2 ) 2  2 M = R (z)dm + Jp1 − Jd 1 + Jp2 − Jd 2 (16) ∂z ∂z Typical mode shape of the first bending mode of high-speed rotors is shown in Fig. 4, which is similar to the sine curve. Therefore, R(z) can be assumed as: R(z) = sin(λz + φ)

(17)

The centroid of the compressor and turbine

Fig. 4. Sketch of mode shape of the first bending mode of high-speed rotors

As shown in Fig. 4, the centroid of the compressor and turbine are approximately the vibration nodes. Therefore, the following assumptions can be made:  R(z1 ) = 0 (18) R(z2 ) = 0 Substitute Eqs. (17) and (18) into Eq. (16), and ignore the mass of the shaft, then it can be calculated that M ≈ Jp1 − Jd 1 + Jp2 − Jd 2 . Considering that most of the bending deformation of the high-speed rotor-bearing system under the first bending mode is concentrated on the drum shaft, the potential energy of the system is mainly the strain energy of the drum. Therefore, according to the Rayleigh method, the first bending modal frequency of the rotor is:   2V 2V Drum = f (V Drum ) (19) ≈ ωn = Jp1 − Jd 1 + Jp2 − Jd 2 M It can be seen from Eq. (19) that the first bending modal frequency of a high-speed rotor is approximately the function of the strain energy of the drum shaft. As the mode shape can be scaled, the strain energy under a particular mode has no practical significance. However, the relative magnitude of the strain energy of each elastic component ( the front and rear neck and drum shaft) can reveal the deformation characteristics of

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the rotor. In this paper, the relative magnitude of the strain energy is measured by the proportion of the strain energy of each elastic component to the total strain energy of the rotor, which is called the strain energy distribution of the rotor in the followings. Equation (11) indicates that the turbine disk influences the bending deformation of the drum shaft through the local spring in front of the disk. Therefore, the angular stiffness of the front local spring may be a key influencing factor on the strain energy distribution of the rotor. Based on the rotor model shown in Table 1, the first bending mode frequencies of the rotor under different angular stiffness of the front local spring are calculated, as shown in Table 2. And the relationship curve between the first bending modal frequency and the strain energy proportion of the drum shaft is shown in Fig. 5. Table 2. Strain energy distribution and first bending modal frequency of the rotor under different angular stiffness of the front local spring kθ1 /(N·m/rad)

kθ2 /(N·m/rad)

V Front /%

V Drum /%

V Spring /%

V Rear /%

ωn /Hz

1 × 107

1 × 106

5.80

79.33

14.38

0.49

230.54

2 × 107

1 × 106

5.74

85.84

8.03

0.39

235.45

5 × 107

1 × 106

5.67

89.38

4.61

0.34

238.31

1 × 108

1 × 106

5.64

90.49

3.54

0.33

239.25

2 × 108

1 × 106

5.63

91.04

3.02

0.31

239.71

5 × 108

1 × 106

5.62

91.36

2.71

0.31

239.99

Frequency of the first bending mode /Hz

where V Front , V Drum , V Spring , and V Rear represent the strain energy proportion of the front neck, the drum shaft, the local springs and the rear neck.

n-VDrum

240

curve

238

R = 0.9998 236

234

232

230 80

82

84

86

88

90

92

Strain energy proportion of the drum shaft /%

Fig. 5. Relationship curve between the first bending modal frequency and the strain energy proportion of the drum shaft

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It can be concluded from Table 2 that the angular stiffness of the front local spring has a significant influence on the strain energy proportion of the drum shaft, as well as the frequency of the first bending mode. As the angular stiffness of the front local spring increases, the strain energy moves from the local springs to the drum shaft and the strain energy proportions of the front and rear neck almost remain unchanged. According to Fig. 5, the correlation coefficient between the frequency of the first bending mode and the strain energy proportion of the drum shaft is up to 0.9998, which is almost linear. Therefore, the frequency of the first bending mode can be equivalently evaluated by the strain energy proportion of the drum shaft under the first bending mode, and the strain energy proportion of the drum shaft can be effectively changed by adjusting the angular stiffness of the local spring in front of the turbine disk. In practical design, the angular stiffness of the local spring in front of the turbine disk can be changed by adjusting the diameter, thickness or material of the drum shaft, as shown in Fig. 6.

D

D' : Diameter of the drum High angular stiffness

Low angular stiffness

D

D'

Fig. 6. Practical design of the drum shaft with different front local angular stiffness

3.2 The Influence of Strain Energy Distribution on Gyroscopic Effect The first bending modal frequency curves of the rotor under different angular stiffness of the front local spring are shown in Fig. 7. It can be seen from the figure that the first bending modal frequency increases more rapidly with higher front angular stiffness, which means stronger gyroscopic effect and results in higher critical speed. The gyroscopic moment of disk 2 can be expressed as:   (20) Mg = − Jp2 − Jd 2 θ ω2 where θ is the angular displacement of the principal axis of inertia of disk 2, which is related to the bending deformation of the shaft. As mentioned above, the bending deformation of the rotor under the first bending mode can be evaluated by the strain energy proportion of the drum shaft.

Dynamic Design of the High-Speed Rotor System 1×107 2×107 5×107 1×108 2×108 5×108 Excitation frequency curve

500

Modal frequency/Hz

353

400

300

200

100

0 0

5000

10000

15000

20000

25000

30000

Rotational speed/(r/min) Fig. 7. First bending modal frequency curves of the rotor under different front local angular stiffness

In order to quantitatively evaluate the effect of gyroscopic moment on the rise of the modal frquency, the rising ratio of the modal frequency ωn between 0 r/min and 18000 r/min as well as the bending critical speed ωcr are clalculated, as shown in Table 3. The relationship curve between the rising ratio of the modal frequency and the strain energy proportion of the drum shaft under the first bending mode is shown in Fig. 8. Table 3. The bending deformation and gyroscopic effect of the rotor under different front angular stiffness kθ1 /( N·m/rad)

kθ2 /( N·m/rad)

V Drum /%

ωn /(Hz/1000 rpm)

ωcr /(r/min)

1 × 107

1 × 106

79.33

6.48

21752.27

2 × 107

1 × 106

85.84

7.17

23584.13

5 × 107

1 × 106

89.38

7.55

24865.96

1 × 108

1 × 106

90.49

7.66

25317.05

2 × 108

1 × 106

91.04

7.71

25547.36

5 × 108

1 × 106

91.36

7.74

25688.05

Accoring to Table 3, higher front angular stiffness results in greater bending deformation of the drum shaft, causes stronger gyroscopic effect and higher bending critical speed. Accoring to Fig. 8, the correlation coefficient between the rising ratio of bending modal frquency and the strain energy proportion of the drum shaft is up to 0.9999, which is almost linear. As the bending modal frequency and gyroscopic effect both have a strong linear relationship with the strain energy proportion of the drum shaft, the bending critical speed may also be linear to the strain energy proportion of the drum shaft.

C. Liu et al.

Rising ratio of modal frequency /(Hz/1000rpm)

354

7.8

7.5

R2 = 0.9999 7.2

6.9

6.6

80

82

84

86

88

90

92

Strain energy proportion of the drum shaft /% Fig. 8. The relationship curve between the gyroscopic effect and the strain energy proportion of the drum shaft

As shown in Fig. 9, the correlation coefficient between the bending critical speed and the strain energy proportion of the drum shaft is up to 0.9971, which also shows strong linear relationship. It can be concluded from above results that the bending critical speed of high-speed rotors can be equivalently evaluated by the strain energy proportion of the drum shaft under the first bending mode.

First bendind critical speed /(r/min)

26000

The relationship curve The fitting curve 25000

R = 0.9971 24000

23000

22000 80

82

84

86

88

90

92

Strain energy proportion of the drum shaft /%

Fig. 9. The relationship curve between the bending critical speed and the strain energy proportion of the drum shaft

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4 Optimization of the Reaction Force at Bearing 4.1 Generating Mechanism of Reaction Force at Bearing According to the generation reason of rotary inertial loads, they can be divided into rotary inertial force and rotary inertial moment, the former is due to the mass eccentricity of the rotor, which is known as unbalance force, and the latter is due to the skew angle of the principal axis of inertia [2]. Generally, the magnitude of the rotary inertial load is equal to the change rate of its momentum and angular momentum, and the direction is opposite to the variation, as shown in Eq. (21). Fint = −

dP dL , M int = − dt dt

(21)

where Fint is the rotary inertial force, P is the momentum of the rotor, M int is the rotary inertial moment and L is the angular momentum. The momentum of the rotor is determined by the mass and the velocity of the center of mass. The angular momentum is determined by the angular velocity of the principal axis of inertia and the moment of inertia. In order to explore the generating mechanism of reaction force at the bearings, the influences of the rotary inertial force and the rotary inertial moment on the dynamic response are analyzed. Firstly, the influence of rotary inertial force on the dynamic response of the rotor is analyzed. Only the mass asymmetry of disk 2 is considered, and there is only mass eccentricity and no skew of the principal axis of inertia. Set the mass eccentricity of disk 2 to e2 = 0.0025mm and the initial skew angle of the principal axis of inertia to τ2 = 0 in Eq. (11). Then solve the differential equation of rotor motion. The dynamic response of the rotor system, specifically the displacement amplitude of the rear bearing is shown in Fig. 10. It can be seen from Fig. 10 that the dynamic response of the high-speed rotor is the same as that of the Jeffcott rotor under the mass eccentricity excitation. In the postcritical range ω > ωn1 , the turning of the center of mass occurs, and the center of mass gradually approaches the rotational centerline, namely self-centering [15]. At the same time, the vibration amplitude decreases and approaches a constant value. The above results reveal that the mass eccentricity will not cause a continuous increase in the dynamic response in the postcritical range. Set the mass eccentricity of disk 2 to e2 = 0 and the initial skew angle of the principal axis of inertia to τ2 = 3×10−4 rad in Eq. (11). The dynamic response of the rotor system under skew excitation is shown in Fig. 11. It can be seen from Fig. 11 that different from the mass eccentricity excitation, under the skew excitation of the principal axis of inertia, the vibration of the rotor does not decrease after passing the first two critical speeds. On the contrary, in the postcritical range, the dynamic response of the rotor increases with rotational speed. The internal mechanism of the above phenomenon is the same as the turning of the center of mass. When disk 2 rotates at a high speed, the skewed principal axis of inertia has a tendency to coincide with the rotational centerline, namely the self-centering of the principal axis, which is determined by its own rotary inertia [15]. The principal axis of inertia of the disk approximately coincides with the rotational centerline at high rotational speed,

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Dynamic response/(×10-3mm)

n1

1.8

1.2 n2

0.6

n3

0.0 0

5000

10000

15000

20000

25000

30000

Rotational speed/(r/min) Fig. 10. Dynamic response of the high-speed rotor system under the rotary inertial force 12

Dynamic response/(×10-3mm)

n2

10 n3

8

6

4 n1

2

0

0

5000

10000

15000

20000

25000

30000

Rotational speed/(r/min) Fig. 11. Dynamic response of the high-speed rotor system under the rotary inertial moment

which will lead to a gradual increase in the bending deformation of the shaft, causing the continuous increase in the reaction force at the bearings. It can be concluded from the above results that the reaction forces at the bearings of the high-speed rotor in the postcritical range are mainly affected by the rotary inertial moment caused by the skew of the principal axis of inertia. What’s more, self-centering of the principal axis of inertia at high speed will lead to the continuous increase of the

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357

rotor response. Therefore, special attention should be paid to the influence of rotary inertial moment during the dynamic design of high-speed rotors. 4.2 Correlation Between Strain Energy and Reaction Force at Bearing For the mechanical model of high-speed rotor system shown in Fig. 3, the local deformation and force of the rear neck in the rotating coordinate system are shown in Fig. 12, in which C represents the shape center of disk 2, G represents the mass center of disk 2, FI and MI are the rotary inertial force and rotary inertial moment of disk 2, Fb is the reaction force at the rear bearing. Principal axis of inertia

G

FI MI

Rear neck

Fb Rear bearing

Drum shaft

Rear local spring Front local spring

Disk 2

Fig. 12. Local deformation and force of the rear neck

The internal forces of the drum shaft and the rear neck at the end face near the disk 2 are balanced with the rotary inertial force and moment of the disk, as shown in Fig. 13.

Fb Rear neck Fs

Fb Rear bearing

Drum shaft

M1

M2

Fig. 13. The internal forces near disk 2

When neglecting the axial thickness of the turbine, the relationships between the internal force near the turbine the rotary inertial loads are:  FI = Fs + Fb (22) MI = M1 + M2 The axial length of the rear neck is assumed as L. When neglecting the mass of the rear neck, its shearing force is constant to Fb , and the bending moment can be expressed as: M (z  ) = Fb z  − M2

(23)

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where z  is the distance between the cross section and disk 2. The strain energy of the rear neck is: L Vε =

F 2 L3 M 2 (z  )  dz = b 2EA 6EA

(24)

0

Then the reaction force of the rear bearing can be expressed as:  Fb (ω) = k Vε (ω)

(25)

 where k = 6EA/L3 is a constant value determined by the shape and material of the rear neck. It can be seen from Eq. (25) that the reaction force at the rear bearing is proportional to the quadratic root of the strain energy of the rear neck at each speed. Considering the similarity of structure and force, the reaction force at the front bearing is also proportional to the quadratic root of the strain energy of the front neck. This is because the amount of the strain energy indicates the bending deformation of the neck. And greater bending deformation means greater constraint force on the neck. High-speed rotor operates near the bending critical speed, the dynamic response of the rotor is dominated by the first bending mode. Therefore, the bending deformation of the rear neck can be evaluated by the strain energy of the rear neck under the first bending mode. As the strain energy under a particular mode has no practical significance, the strain energy distribution of the rotor is used alternatively. In the postcritical range, the reaction forces at the bearings are mainly affected by the rotary inertial moment of the turbine, and the rotary inertial moment affects the bending deformation of the shaft through the local spring. Therefore, when considering load of the rear bearing, the angular stiffness of the rear local spring may have a significant influence on the reaction force of the rear bearing. The strain energy distribution under the first bending mode and the reaction force at operating speed (20000 r/min) under different angular stiffness of the rear local spring are shown in Table 4. Table 4. Strain energy distribution and reaction force at operating speed under different rear angular stiffness kθ1 /(N·m/rad)

kθ2 /(N·m/rad)

V Front /%

V Drum /%

V Spring /%

V Rear /%

1 × 107

1 × 106

Fb /N

5.80

79.33

14.38

0.49

85.31

1 × 107

2 × 106

5.74

80.66

12.78

0.82

112.55

1 × 107

5 × 106

5.66

82.25

10.93

1.16

140.03

1 × 107

1 × 107

5.60

82.99

10.10

1.31

151.93

1 × 107

2 × 107

5.57

83.40

9.64

1.39

159.22

1 × 107

5 × 107

5.55

83.65

9.36

1.44

163.94

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It can be concluded from Table 4 that the angular stiffness of the rear local spring is a key influencing factor on the strain energy proportion of the rear neck. As the rear local angular stiffness increases in the range between 1 × 106 and 1 × 107 N·m/rad, the strain energy proportion of the rear neck increases rapidly, while the strain energy proportions of the drum shaft and the front neck change slightly. As the strain energy proportion of the rear neck increases, the reaction force at the rear bearing increases as well. The relationship curve between them is shown in Fig. 14. It can be seen that the correlation coefficient between the reaction force at the rear bearing and the strain energy proportion of the rear neck under the first bending mode is up to 0.9999, which is almost linear. Therefore, the reaction force at the rear bearing can be equivalently evaluated by the strain energy proportion of the rear neck.

Reaction force at rear bearing /N

160

140

R = 0.9999

120

100

80 0.4

0.6

0.8

1.0

1.2

1.4

Strain energy proportion of rear neck /%

Fig. 14. Relationship curve between the reaction force at rear bearing and strain energy proportion of the rear neck

The relationship between the reaction force at the rear bearing and the strain energy proportion of the rear neck is different from Eq. (25). This is beacause the total strain energy of the rotor increases more rapidly than the strain energy of the rear neck, the slope of the curve increases when using the proportion of strain energy as abscissa. In practical design, the angular stiffness of the local spring behind the turbine disk can be changed by adjusting the diameter, thickness or material of the conical shell of the rear neck, as shown in Fig. 15. 4.3 Design on Practical Rotor Based on Strain Energy Distribution To prove the feasibility of the correlations above on practical high-speed rotors, the bending critical speed and reaction force at the rear bearing under operating speed of a typical high-speed rotor ( as shown in Fig. 16) under different strain energy distribution are calculated. The bearing stiffnesses at the front and rear bearing are all 2.5 × 107 N/m,

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D

D' : Diameter of the shaft

High angular stiffness

D

Low angular stiffness D'

Fig. 15. Practical design of the rear neck with different rear local angular stiffness

the unbalance and skew angle of the principal axis of inertia at the turbine disk are 1000 g·mm and 3 × 10–4 rad. And the local angular stiffness of the connection structure is adjusted by changing the diameter, as shown in Fig. 6 and Fig. 15. The results are shown in Table 5. 4

2

6

7

3

5

1

1-The front neck 2-The compressor 3-The drum shaft 4 6-The connection structures between the turbine and shaft 5-The turbine 7-The rear neck Fig. 16. Component division of the typical high-speed rotor

Table 5. Strain energy distribution and dynamic properties of the practical rotor at operating speed under different rear angular stiffness kθ 1 /(N·m/rad)

kθ 2 /(N·m/rad)

V Front /%

V Comp /%

V Drum /%

V Turbo /%

V Rear /%

ωcr /(r/min)

Fb /N

3.6 × 107

1.2 × 109

0.44

9.80

78.83

0.72

10.16

0.03

88157.2

524.4

3.6 × 107

1.2 × 108

0.11

4.52

63.95

24.58

6.81

0.03

76829.2

519.8

7.2 ×

107

1.2 × 108

0.19

6.04

66.21

18.90

8.63

0.02

78001.2

454.9

1.8 × 108

1.2 × 108

0.31

7.86

71.61

10.24

9.97

0.02

81203.4

423.2

3.6 × 108

1.2 × 108

0.37

8.77

74.78

5.83

10.25

0.01

83708.3

420.7

V Spring /%

where V Comp and V Turbo are the strain energy proportions of the compressor and turbine.

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Bending critical speed /(×103 r/min)

90

The relationship curve The fitting curve

88 86

R = 0.9883 84 82 80 78 76

62

64

66

68

70

72

74

76

78

Strain energy proportion of the drum shaft /%

80

Fig. 17. The relationship curve between the bending critical speed and the strain energy proportion of the drum shaft of the typical high-speed rotor

Reaction force at rear bearing /N

540 520

The relationship curve The fitting curve

500

R = 0.9977 480 460 440 420 400 0.005

0.010

0.015

0.020

0.025

0.030

0.035

Strain energy proportion of rear neck /% Fig. 18. The relationship curve between the reaction force at rear bearing and the strain energy proportion of the rear neck of the typical high-speed rotor

It can be seen from Table 5 that the strain energy distribution of the rotor is effectively changed when adjusting the angular stiffness of the connection structures between the turbine and the shaft, the bending critical speed and the reaction force at the rear bearing are greatly changed at the same time. And according to Fig. 17 and Fig. 18, the strain energy distribution and the rotor dynamic characteristics are still highly correlated for

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practical high-speed rotors, there are one-to-one correspondences between them. For the first and last case in Table 5, the dynamic response at the rear bearing is reduced by up to 19.77%, which proves the feasibility of the load suppression method through reduction of the strain energy of the rear neck. Therefore, the dynamic design of practical high-speed rotors can be carried out based on the strain energy proportions of the elastic component (namely the strain energy distribution) under the first bending mode. The relationship curves between the strain energy distribution and the dynamic characteristics can be determined by several trials before the design, during which the local angular stiffnesses of the connection structures between the turbine disk and the shaft are adjusted to the maximum extent. Then in the following design, the dynamic characteristics can be determined by the strain energy distribution and the relationship curves. As the modal anlysis of a rotor and corresponding statistics of strain energy distribution are much more easier than the plot of Campbell diagram and harmanic anlysis, dynamic design based on strain energy distribution will consume much less computing power and time.

5 Conclusion Aiming at the dynamic design of high-speed rotors, this paper establishes a dynamic model based on the key structural features of aero-engine high-pressure rotors, the correlations between the strain energy and the dynamic properties of the rotor are studied through numerical studies, and the following conclusions are stated: (1) There are strong linear relationship between the bending critical speed and the strain energy proportion of the drum shaft under the first bending mode. For the equivalent mechanical model established in this paper, the correlation coefficient between the bending critical speed and the strain energy proportion of the drum shaft is up to 0.9971. For the typical high-speed rotor, the same correlation coefficient is up to 0.9883. Therefore, the bending critical speed of high-speed rotors can be equivalently evaluated by the strain energy proportion of the drum shaft under the first bending mode. (2) There are strong linear relationship between the reaction force at the rear bearing and the strain energy proportion of the rear neck under the first bending mode. For the equivalent mechanical model established in this paper, the correlation coefficient between the reaction force at the rear bearing and the strain energy proportion of the rear neck under the first bending mode is up to 0.9999. For the typical high-speed rotor, the same correlation coefficient is up to 0.9977. Therefore, the reaction force at the rear bearing of high-speed rotors can be equivalently evaluated by the strain energy proportion of the rear neck under the first bending mode. (3) The strain energy distribution of high-speed rotors under the first bending mode can be effectively changed through the adjustment of the local angular stiffnesses of the connection structures between the turbine disk and the shaft. The strain energy proportion of the drum shaft is positively correlated with local angular stiffness in front of the turbine, and the strain energy proportion of the rear neck is positively correlated with local angular stiffness behind the turbine.

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Acknowledgments. The authors would like to acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 52205082), and the Science Center for Gas Turbine Project(P2021-A-I-002–002).

References 1. Yu, H., Ma, Y.H., Xiao, S., et al.: Mechanical and dynamic properties of bearing with looseness on high-speed flexible rotor. J. Beijing Univ. Aeronaut. Astronaut. 43(08), 1677–1683 (2017) 2. Hong, J., Yang, Z.F., Sun, B., et al.: Influence of local rotary inertia on the dynamic properties of rotor systems. J. Aerosp. Power 37(04), 673–683 (2022) 3. Hong, J., Yan, Q., Feng, S.B., et al.: Rotational inertia model and dynamic response characteristics of multi-disk rotor system with interface. J. Aerosp. Power 37(05), 897–908 (2022) 4. Yuan, S., Deng, W.Q., Xu, Y.L., et al.: Dynamic properties analysis of a cantilever flexible rotor with large length-to-diameter ratio. Aeronaut. Sci. Technol. 28(11), 62–68 (2017) 5. Wang, R., Liao, M.F., Cheng, R.H., et al.: Modal characteristics and their expression method for aeroengine dual-rotor system. Journal of Vibration and Shock 41(21), 209–215+278 (2022) 6. Murgayya, S.B., Suresh, H.N., Madhusudhan, N., et al: Effective rotordynamics analysis of high speed machine tool spindle–bearing system. In: Saravana, B. D., Keshavamurthy, R., Praveennath, G. K. (eds.) Materials Today: Proceedings, vol. 46, pp. 8905–8909 (2021) 7. Conry, T.F., Goglia, P.R., Cusano, C.: A minimum strain energy approach for obtaining optimal unbalance distribution in flexible rotors. J. Mech. Des. 104(4), 875–880 (1982) 8. Chen, W.J.: Energy analysis to the design of rotor-bearing systems. J. Eng. Gas Turbines Power 119(2), 411–417 (1997) 9. Srinivas, R.S., Mythu, S.E., Degaonkar, G.K.: Rotordynamic design studies of medium thrust class twin spool engine. In: Fakher, C., Francesco, G., Vitalii, I. (eds.) Lecture Notes in Mechanical Engineering, pp. 531–541. Springer, Singapore (2021). https://doi.org/10.1007/ 978-981-15-5701-9_43 10. Hong, J., Song, Z.H., Ma, Y.H., et al.: Robust design method for dynamics of high-speed rotor system with interface. In: Fakher, C., Francesco, G., Vitalii, I. (eds.) Lecture Notes in Mechanical Engineering, vol. 58, pp. 629–645. Springer, Singapore (2021). https://doi.org/ 10.1007/978-981-15-8049-9_39 11. Wang, M.L., Wen, B.G., Han, Q.K., et al.: Dynamic properties of a misaligned rigid rotor system with flexible supports. Shock Vibr. 2021, 1–16 (2021) 12. Zheng, X.D., Zhang, L.X., Liu, T.Y.: Calculation and analysis of vibration characteristics and strain energy of aeroengine. Aeroengine 2000(02), 42–46 (2000) 13. Sun, L.Q.: Application of strain energy method in rotor dynamic calculation of a gas turbine rotor. Mech. Eng. 2017(12), 105–106+108 (2017) 14. Hong, J., Xu, X.R., Su, Z.M., et al.: Joint stiffness loss and vibration characteristics of highspeed rotor. J. Beijing Univ. Aeronaut. Astronaut. 45(01), 18–25 (2019) 15. Hong, J., Yang, Z.F., Lyu, C.G., et al.: Robust design method for dynamic properties of high-speed flexible rotor systems. J. Beijing Univ. Aeronaut. Astronaut. 45(05), 855–862 (2019) 16. Ishida, Y., Yamamoto, T.: Linear and Nonlinear Rotordynamics: A Modern Treatment with Applications, 2nd edn. John Wiley & Sons, NY (2013)

Optimization of Journal Bearings Considering Their Adjustable Design and Rotor Dynamics Denis Shutin(B) , Alexander Fetisov, and Leonid Savin Orel State University, Orel 302026, Russian Federation [email protected]

Abstract. The procedures of optimal design of journal bearings usually come down to finding the extremums of one or more key bearing characteristics. The load capacity, the minimum film thickness and the friction torque are most often considered as the optimization criteria. The parameters of dynamic rotor behavior require a significant amount of calculations to be estimated and are much rarely considered as such. Variable design of fluid film bearings brings additional challenges to their optimal design. In particular, they should provide the maximum efficiency of the control system impact on the rotor. This work presents an approach to the optimal designing of journal fluid film bearings, which takes into account the factors of the dynamic rotor behavior and the variability of the controlled parameters of adjustable bearings. The approach is implemented in solving the problem of parametric synthesis of an actively lubricated journal bearing. The abovementioned factors are taken into account in the objective functions. The optimization problem was solved using a genetic algorithm. The resulting set of Pareto optimal solutions can be used for choosing a balance between friction, stability and controllability of the bearing. However, the issue of developing the optimal control is offered to be considered as a separate stage in the active bearing synthesis procedure and is taken out of the scope of this study. Finally, the conclusions are drawn about the directions for further improvement of methods and tools for the optimal design of conventional and adjustable fluid film bearings. Keywords: Bearings Design Optimization · Fluid Film Bearings · Adjustable Bearings · Design Criteria · Rotor Dynamics · Stability

Nomenclature x, y, z h0 , h O, O1 V U T fr μ L p , Wp Dp

Cartesian coordinates initial and local radial gap; center of bearing and shaft radial velocity; circumferential velocity; friction torque in bearing; dynamic viscosity of lubricant; oil pocket length and width oil pocket depth;

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 364–376, 2024. https://doi.org/10.1007/978-3-031-40459-7_24

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dh lh α S r R, D L B Fb mg Q d p n, ω G Rmax T tr

365

restrictor diameter; restrictor length; angular coordinate; bearing surface area; shaft radius; bearing radius and diameter bearing length; rotor length; bearing force; rotor weight; lubricant mass flow; rotor imbalance; pressure; rotation speed and angle speed design variables vector; maximum control force; transient time.

1 Introduction In many cases, fluid film bearings are designed according to simplified methods [1]. They usually do not require large amounts of calculations, but give only averaged solutions. The use of optimal synthesis procedures makes it possible to obtain more accurate solutions that are more in line with the requirements put forward [2–4]. These procedures utilize mathematical models of rotor-bearing systems, while also simplified analytical models and fairly simple objective functions are often used. The simplification of mathematical models is achieved through adoption of a number of assumptions, such as the assumption of small bearing length, absence of misalignments, omission of cavitation and turbulence effects, etc. [5, 6]. This approach is able to provide more accurate solutions compared to the basic methods. However, a number of aspects of operation of rotor-bearing systems, such as dynamic and nonlinear phenomena, are mainly also omitted. The aspects of the dynamic behavior of the rotor are almost not considered in scientific works as objective functions. Although they often become the limiting factors in achieving high performance of rotary machines. These problems are of special importance for rotor-bearing systems with flexible rotors [7, 8]. In most cases nonlinear rotor dynamics factors require huge amounts of calculations, and this is a significant reason for not considering it while solving optimization problems for fluid film bearings. In this regard, the work [9] can be noted, where the decrement of damping of the rotor oscillations is considered as an optimization criterion. However, a simplified analytical model of the fluid film bearing was also used by the authors. Adjustable design of fluid film bearings further complicates the design process. Despite the variety of approaches to implementation of active control in them [10–12], the design principles of conventional bearings are usually used when choosing their parameters. However, it is advisable to choose the parameters that would provide the

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most efficient conversion of the control signals into the control actions to maximize the efficiency of the control system. Synthesis of optimal controllers for active bearings should also be an essential element of their design [13], however, this work focuses only on the mechanical design issues. It should be noted that the progress in computing facilities and new approaches to the simplification of rotor systems [14, 15] actualizes the problem of development of methods for the optimal design of modern fluid film bearings again. This work demonstrates an approach that allows taking into account the complex of these factors in the procedure of parametric synthesis of a fluid film bearing design. The presented solution of the multicriteria optimization problem allows finding a balanced combination of tribological, dynamic and integral bearing characteristics for implementation of the active lubrication principle [16, 17]. The results obtained demonstrate both achievements and challenges related to the considered problems of optimal design of rotor-bearing systems.

2 Models and Methods 2.1 Mathematical Model of the Rotor-Bearing System The paper considers the problem of optimal design of a rotor-bearing system on fluid film bearings with active lubrication. Their behavior is studied in more detail in [16], the schematic is shown in Fig. 1. The bearing’s sleeve includes 4 lubrication channels in the center line with rectangular hydrostatic pockets. The lubricant pressure is controlled by separate servo valves in each channel. The adjustable supply pressure results in a control force impact on the rotor. Thus, the load capacity in this hybrid bearing is created by a combination of hydrodynamic and hydrostatic effects. The control force is created only by the hydrostatic effect.

Fig. 1. Actively lubricated journal hybrid bearing

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The mathematical model of the hybrid bearing is based on the main provisions of the hydrodynamic lubrication theory [18, 19]. The pressure distribution is calculated in accordance with the modified Reynolds equation:     ∂ h3 ∂p ∂ h3 ∂p ∂ + · = 6 (Uh) + 12V . (1) ∂x μ ∂x ∂z μ ∂z ∂x Equation (1) was numerically solved using the finite differences method [20, 21]. It was solved together with the flow balance equation considering the lubricant flow through the supply jets [22]: Q = QX + QZ + QY .

(2)

The fluid film forces are determined by integration of the obtained pressure distribution, also taking into account the pressure in the hydrostatic supply pockets: 2π RL fx =

p sin(α)dxdz, 0

0

(3)

2π RL fy =

p cos(α)dxdz. 0

0

The friction torque in the fluid film occurs due to the viscous forces is:  ¨  h ∂p U μ D Tfr = + dS. 2 2 ∂x h

(4)

S

The motion of the rotor in bearings considering the action of a combination of forces is determined by solving the Lagrange equations, as in [16]. The diagram of the rotorbearing system is shown in Fig. 2. Since the rotor is considered symmetrical, and the rotor system operates at a subcritical frequency without significant misalignments, only one bearing is considered in the calculations in this work. The control system of the considered actively lubricated bearing is based on a P-controller with feedback on the rotor position. A more detailed description of the controller is presented in [19]. 2.2 Description of the Optimization Problem The problem of parametric synthesis of bearings is traditionally reduced to minimizing an objective function of a set of functions. In this case, three different parameters are considered as the objective parameters at once. 1. The viscous friction torque created in the fluid film is one of the key energy parameters describing the efficiency of the rotor system. It also characterizes the intensity of heat generation in the friction zone, as well as the changes in the lubricant rheological properties. The averaged value of the friction torque Tfr during the steady operation of the rotor-bearing system was estimated according to Eq. (4).

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Fig. 2. Schematic of the symmetric rigid rotor on bearings

2. The transient time of a response to an external force disturbance is considered as the basic criterion for assessing the rotor stability in the bearing. This parameter is calculated during simulation tests carried out for each bearing configuration. At the beginning of a simulation the rotor freely travels to the equilibrium position from the bearing center. The corresponding time interval is called Transient time 1 in Fig. 3. After that a force pulse is applied to the rotor with a value of 3 times its weight and a duration of 0.2 ms. Then, the transient period 2 along the x and y coordinates is estimated using the sliding window method. A steady oscillation range of less than 3% of the bearing radial clearance was considered as the criterion for the end of the transient process. 3. The maximum control action defines the ultimate ability of controlling the rotor position in an actively lubricated bearing. It largely depends on the configuration of the lubricant supply channels, including the parameters of hydrostatic pockets. The throttling effect can significantly influence the resulting hydrostatic forces in a hybrid bearing. Therefore, the maximum control action was calculated for the centered rotor position in the bearing providing the equal hydraulic resistance in all lubricant supply channels. The maximum control force Rmax was calculated using Eq. (3) at the maximum control signal value umax set for both control channels, i.e., axes x and y. According to the control scheme used, the pressure in two adjacent supply channels is set maximal, while in the other two opposite channels it is set minimal [20]. It should be noted that among the considered objective parameters, at least the friction torque contradicts the other two, since they require higher value of the stabilizing and/or control forces. According to Eq. (4), reduction in the pressure variance in the bearing is the criterion for reducing the friction torque. Thus, the solution of the optimization problem in this case can be represented as a three-dimensional Pareto front. The optimization problem can be presented as follows:   minf (G) = Ttr ,Tfr ,1/Rmax   0, 5 < L/D < 1, 5 (5) subject to see Table 1 where G = [L, h0 , dh , lh , Wp , Lp , Dp ].

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Fig. 3. Estimation of the transient time in the rotor system

A number of constraints were applied to the design variables to ensure the manufacturability of the resulting solution and provide the hybrid mode of creating the load capacity. The list of variables and the corresponding constraints is given in Table 1. Table 1. Design variables and constraints №

Variable

Lower limit

1

Bearing length L, mm

40

80

2

Radial clearance h0 , µm

40

80

3

Lubrication channel diameter d h , mm

0.5

4

4

Lubrication channel length l h , mm

5

12

5

Hydrostatic pocket width W p , % of bearing length

5

60

6

Hydrostatic pocket length L p , degrees

5

40

7

Hydrostatic pocket depth Dp , % of radial clearance

200

Upper limit

1000

The list of other system parameters remained unchanged during solving the optimization problem is given in Table 2. These parameters characterize the conditions and the operation mode of the bearing.

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№

Variable

1

Lubricant

Water

2

Dynamic viscosity μ, mPa·s

1.14

3

Lubricant density ρ, kg/m3

1000

4

Bearing temperature T, °C

30

5

Rotation speed n, rpm

3000

6

Lubricant supply operating pressure Pop , MPa

0.2

7

Maximum lubricant supply pressure Ps , MPa

1.0

8

Rotor mass m, kg

4.5

9

Servo valve time constant T SV , sec

0.002

Value

3 Results and Discussion 3.1 Solution of the Optimization Problem The presented optimization problem (5) was solved using a genetic algorithm implemented in MATLAB software package. The population size was 200 elements. The number of generations was 20. The relative tolerance was 10–5 . The calculation was completed at the 14th generation, and Pareto curves for two pairs of objective parameters Tfr –Ttr and Tfr –Rmax were obtained, see Fig. 4.

Fig. 4. Pareto curves for objective parameters “Friction torque - Transient time” and “Friction torque - Maximum control action”

A decrease in the friction torque entails a decrease in the maximum control force (Fig. 4, a). The minimal friction torque is possible only with a significant decrease in the stability of the rotor system (Fig. 4, b). This confirms the previously noted conflict between the indicated pairs of objective functions. Thus, the choice of an appropriate

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solution should be based on a balance between the objective parameters. Also, a threedimensional Pareto front was calculated for the joint analysis of all three objective functions, as shown in Fig. 5.

Fig. 5. Three-dimensional Pareto front

The resulting Pareto front includes 200 points. Most of them are located closer to higher values of the control force Rmax . However, there are solutions that are characterized by a significantly lower value of Rmax . The coordinate axes in Fig. 5 are supplemented with additional designations for easy prioritizing when looking for the required solution according to the operating conditions of the rotary machine. Thus, the parameter T tr is related to the stability of the basic rotor system with passive bearings. The Rmax parameter reflects the controllability of an actively lubricated bearing, based on an initial passive bearing. Performance of the rotor-bearing system for several ultimate solutions (green dots in Fig. 5) were then tested using a simulation model of the rotary system in order to validate them. The results of the numerical tests are shown in the next subsection. 3.2 Solution Analysis Three solutions with the values of the objective parameters close to the extreme cases were selected from the resulting set of Pareto-optimal solutions. The corresponding values of the design variables for the solutions are shown in Table 3. The simulation tests for analysis of the performance of the chosen solutions were performed numerically for the rotary system operating at a rotation speed of 3000 rpm. During the tests the rotor was subjected to various loads with and without the position

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Parameter

Case #1

Case #2

Friction

Low

High

Low

Stability

Low

High

Low

Controllability

Medum

High

Low

Friction torque Tfr , N·m

0.0109

0.0271

0.0102

Transient time T tr , sec

7.75

1.1

7.2

Maximum control force Rmax , N

303

571.7

132.5

Bearing length L, mm

40

76

40

Radial clearance h0 , µm

78

53

80

1.01

0.57

Lubrication channel diameter d h , mm 1.86

Case #3

Lubrication channel length l h , mm

8.59

13.35

11.95

Hydrostatic pocket width W p , % of L

33.35

54.14

55.39

Hydrostatic pocket length L p , degrees 18.67

25.5

18.82

Hydrostatic pocket depth Dp , % of h0 690

690

595

control. The last case assumes a constant and equal lubricant supply pressure to all supply chambers, i.e., passive hybrid bearing. The simulation tests scenario included the following steps: 1) control is off; a perfectly balanced rotor moves freely from the bearing center to the equilibrium position; the transient time is estimated; 2) after the system stabilization a force impulse of 2 rotor weights with a duration of 3 ms is applied to the rotor; the displacement amplitude and the transient time are estimated; 3) an unbalanced mass is added to the rotor resulting in imbalance value of d = 1·10–4 m; the steady-state oscillation amplitude is estimated; 4) P-control is switched on with the setpoint at the equilibrium position of a perfectly balanced rotor (approximately the center of the rotor orbit); the amplitude reduction ratio relative to the amplitude at step 3 is estimated; 5) the rotor imbalance is removed; the controller remains on; 6) the force test from step 2 is repeated for the controlled system; the displacement amplitude and the transient time are estimated in comparison to the results of step 2; 7) the rotor setpoint is changed to another at a certain distance from the previous one; the transient time is estimated. Figure 6 shows the rotor motion along the coordinate axes x and y during the test for the chosen cases of bearing parameters. Figure 7 demonstrates the corresponding variation in viscous friction in the bearing. In general, the obtained results confirm the presence of the tested configurations of the features specified when choosing the appropriate solutions to the optimization problem.

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Configuration #3 is characterized by the smallest rotor stability margin and minimal friction torque in the system. The oscillation amplitudes of the unbalanced rotor are close to the largest, and the rotor displacements under the action of applied force impulses are also the highest among the tested cases. In this case, the smallest control force leads to the least significant reduction of the rotor displacements in the controlled system. Configuration #2 shows the fastest transient time and the highest friction torque as well. The degree of the rotor displacement reduction in the controlled system is the highest. At the same time, the transient time for configuration #1 is only slightly longer, but the friction torque is close to the minimum. In addition, the general system behavior in case #1 is less oscillatory than in case #2, where the vibrations of the unbalanced rotor in the passive system are the highest, and the transient process also includes more oscillations. Thus, configuration #1 may be considered as the preferred combination of characteristics among the tested cases. It should be noted that the difference in the transient time between configurations #1 and #2, is not as pronounced as the difference in the transient behavior, although these solutions are significantly far from each other in terms of the T tr value according to Fig. 5. This indicates that the criteria describing the rotor stability needs to be improved in order to describe the transient system behavior more fully. It should take into account not only the time, but also the type of the transient processes, including the oscillatory behavior, as well as the tendency of the system to the increased vibrations under the action of unbalanced masses, which is observed for case #2 in Fig. 5. At the same time, the parameter Rmax quite adequately reflects the ability of the adjustable bearing to minimize the rotor displacements from the setpoint both under impulse and harmonic force impacts. This parameter can be used to determine the bearing configuration that most effectively implements the control actions in the rotor-bearing system. In addition, we note that the use of the control system makes it possible to reduce the vibrations compared to the passive bearings for all the presented configurations, even for the least stable ones. Thus, the presented approach to the optimal design of fluid film bearings can be assessed as generally consistent, but requiring some improvements and additions. It allows obtaining solutions ready for the effective implementation of control systems, e.g., active lubrication. At the same time, the considered criteria for the dynamic behavior of the rotor must be improved. In addition, it looks rational to introduce some additional objective parameters, such as lubricant consumption, for better representation of the rotor-bearing system’s operational properties. Finally, it is also reasonable to further improve the calculation speed, especially when considering models of flexible rotors operating in supercritical regions, which may require more computations to evaluate the nonlinear vibrations.

4 Conclusions The work proposes an approach to the procedures of optimal design of fluid film bearings, taking into account both their conventional passive and adjustable design, as well as the dynamic behavior of the rotor-bearing system. A multicriteria optimization problem was formulated and solved for an actively lubricated journal bearing using a genetic

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Fig. 6. Rotor displacements during the simulation tests for the three bearing configurations

Fig. 7. Friction torque during the simulation tests for the three bearing configurations

algorithm. A set of Pareto-optimal solutions was found for three objective functions describing the viscous friction in the bearing, the rotor motion stability, and the bearing efficiency in implementing control actions. A three-dimensional representation of the Pareto front allows choosing solutions with specific features that best meet the requirements of the system being designed. The results of comparative simulation tests mostly confirmed the presence of such properties in the obtained bearing configurations. The proposed criterions for the friction and the maximum control efficiency quite adequately represent their behavior. However, the transient time does not reflect sufficiently the

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system stability and the rotor transient behavior. Improving this criterion by considering additionally the oscillatory rotor behavior is the way to make the results of the optimization procedures more predictable. Acknowledgements. The study was supported by the Russian Science Foundation grant No. 22–19-00789, https://rscf.ru/en/project/22-19-00789/.

References 1. Logan, E., Jr.: Handbook of Turbomachinery, p. 472. Marcel Dekker Inc., New York (1995) 2. Adams, M.L.: Rotating Machinery Vibration: From Analysis to Troubleshooting, p. 354. Marcel Dekker Inc., New York (2001) 3. Yamamoto, T., Ishida, Y.: Linear and Nonlinear Rotordynamics. A Modern Treatment with Applications, p. 326. John Willey & Sons, New York (2001) 4. Chen, W.J., Gunter, E.J.: Introduction to Dynamics of Rotor-Bearing Systems, p. 482. Manchester, Trafford Pub (2005) 5. Singh, R., Chaudhary, H., Singh, A.: Defect-free optimal synthesis of crank-rocker linkage using nature-inspired optimization algorithms. Mech. Mach. Theory 116, 105–122 (2017) 6. Panda, S., Nanda, P., Mishra, D.: Comparative study on optimum design of rolling element bearing. Tribol. Int. 92, 595–604 (2015) 7. Lu, K., et al.: The applications of POD method in dual rotor-bearing systems with coupling misalignment. Mech. Syst. Sig. Proc. 150, 107236 (2021) 8. Onunka, C., Grobler, H., Bright, G.: A stability optimization model for shaft rotor-bearing systems. Afr. J. Sci., Technol., Innov. Dev. 8, 1–12 (2016) 9. Saruhan, H.: Optimum design of rotor-bearing system stability performance comparing an evolutionary algorithm versus a conventional method. Int. J. Mech. Sci. 48, 1341–1351 (2006) 10. Haugaard, M.A., Santos, I.F.: Elastohydrodynamics applied to active tilting-pad journal bearings. ASME J. Tribol. 132, 10 (2010) 11. San Andres, L., Childs, D.: Angled injection–hydrostatic bearings analysis and comparison to test results. ASME J. Tribol. 119, 179–187 (1997) 12. Laukiavich, C.A.: A comparison between the performance of ferro– and magnetorheological fluids in a hydrodynamic bearing. In: Laukiavich, C.A., Braun, M.J., Chandy, A.J. (eds.) Proceedings of the Institution of Mechanical Engineers, Part J Journal of Engineering Tribology. Conference 2014, vol. 228, pp. 649–666 (2014) 13. Santos, I.: Controllable sliding bearings and controllable lubrication principles—an overview. Lubricants. 6, 16 (2018) 14. Jin, Y., Lu, K., Huang, C., Hou, L., Chen, Y.: Nonlinear dynamic analysis of a complex dual rotor-bearing system based on a novel model reduction method. Appl. Math. Model. 75, 553–571 (2019) 15. Lu, K., et al.: Review for order reduction based on proper orthogonal decomposition and outlooks of applications in mechanical systems. Mech. Syst. Sig. Process. 123, 264–297 (2018) 16. Li, S., et al.: Active Hybrid Journal Bearings with Lubrication Control: Towards Machine Learning. Tribology International 175, 107805 (2022) 17. Jensen, K., Santos, I.: Design of actively-controlled oil lubrication to reduce rotor-bearingfoundation coupled vibrations - theory & experiment. Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol. 236, 1493–1510 (2022) 18. Hori, Y.: Hydrodynamic lubrication. Hydrodyn. Lubr., 1–231 (2006)

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19. Mattox, D., Wang, Q., Chung, Y.: Encyclopedia of Tribology. Encycl. Tribol., 2717–2726 (2013) 20. Savin, L., Polyakov, R., Shutin, D., Babin, A.: Peculiarities of reactions control for rotor positioning in an active journal hybrid bearing. Int. J. Mech. 10, 62–67 (2016) 21. Dmochowski, W.M., Dadouche, A., Fillon, M.: Finite difference method for fluid-film bearings. Encycl. Tribol., 1137–1143 (2013) 22. San Andres, L.: Notes 14. Experimental identification of bearing force coefficients (2009). https://oaktrust.library.tamu.edu/handle/1969.1/93254. Accessed 09 Mar 2023 23. Shutin, D., Polyakov, R.: Adaptive nonlinear controller of rotor position in active hybrid bearings, PP. 1–6 (2016)

Stability Margin Optimization for Unsymmetrical Rotor/Stator Dynamic System Yaqun Jiang(B) Analytical Dynamics Inc., Michigan, USA [email protected]

Abstract. This paper introduces a computer simulation tool that optimizes the damping forces acting on a stationary body to maximize the stability margin of a rotating body. The proposed method is highly efficient and can be used to investigate the dynamic stability of any general rotating system, even those that are non-axisymmetric. To achieve this, the authors combine Floquet theory and Hill’s method to create an analytical tool that operates in the frequency domain and can compute the stability of unsymmetrical flexible rotor and stator systems in a shorter computation time. Using the simulation tool, the authors explore various damping design configurations for an I4 internal combustion engine to identify the optimal damping values that can maximize dynamic stability. The simulation results demonstrate how the damping of both the stator (engine block) and rotor can be optimized to achieve maximum dynamic stability. Overall, this paper presents a practical and effective approach to designing damping systems that can improve the stability of rotating machinery. Keywords: Rotor Stability · 3D Finite Element Method · Margin Optimization

1 Introduction The stability margin is a critical parameter in system design, as it provides insight into how robust the system is and how much margin there is for variations in the system’s parameters or external disturbances. Stability is a widely researched topic in rotating systems with asymmetry in shaft stiffness and inertia. However, the stability issue is often not considered during the design of internal combustion engines. A crankshaft has unsymmetrical inertia and stiffness about its rotation axis, as is common in a typical I4 engine which has five main bearings and eight unsymmetrical count weights. Under normal operating conditions, all the bearings are heavily loaded, and the crankshaft’s vibration is confined by the engine block. Its instability threshold caused by the unsymmetrical design is significantly higher than the operating speed. However, during no-load or light-load conditions, such as when a vehicle coasts down a hill with the engine brake engaged and the cylinders not firing, the crankshaft can run at extremely high RPMs with relatively small restriction from the main bearings, causing the instability threshold © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 377–389, 2024. https://doi.org/10.1007/978-3-031-40459-7_25

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to drop within the operating range and potentially leading to catastrophic engine failure. Thus, it is crucial to conduct a comprehensive investigation of instability during engine design for identifying the optimal design parameters that minimize the risk of failure. Over the last two decades, various FEA 3D modeling approaches have been developed to analyze general rotor and stator systems such as crankshafts and engine blocks. Based on axisymmetric hypotheses, Geradin and Kill [1] utilized a fully 3D FEA approach based on axisymmetric hypotheses for stability analysis, while Stephenson et al. [2] employed a similar approach to study the coupling effects of rotor-bearing systems. To model general unsymmetrical rotating machines, Lazarus et al. [3] developed a 3D FE approach using the component mode synthesis (CMS) technique. The fully 3D finite element model included the stator and rotor coupling, with the connection between them modeled by preventing the rotor rigid body motions. A similar approach was adopted by Meng et al. [4], who proposed two techniques to reduce the computational effort required by 3D FE models for stability and frequency analyses. Specifically, they established the equations in the rotating reference frame and utilized Floquet theory and Hill’s method to solve the time-variant systems. This study utilizes a Finite Element Analysis (FEA) method to simulate both the rotor and stator. The study investigates the influence of the crank damper ring inertia and evaluates system instability with different ring damping levels. The study also employs Floquet theory and Hill’s method to optimize the mount damping for achieving the most possible system stability.

2 Mathematical Model The mathematical model used for this investigation is based on Lagrange’s equations, which describe the dynamics of a system in terms of its energy. The methodology and simulation process have been presented and validated in reference [5]. In brief, the equations of motion are derived using a co-rotating coordinate system that accounts for the motion of the system and its components. To improve simulation performance, several simplifying assumptions have been made while still capturing the essential features of the system. These assumptions enable the equations to be formulated in a more manageable form. 1. The rotor nodal mass has either two translational degrees of freedom (u, v) or four degrees of freedom (u, v, ϕu , ϕv ), where ϕ represents the rotational degree of freedom. 2. The bearings in the model are assumed to have only linear stiffness and viscous damping. 3. Furthermore, the analysis is focused on the stability in the x and y axis, the lateral crankshaft vibration. The effects of axial and torsional vibrations on the stability analysis are considered negligible.

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Under the above assumptions, the equations of motion of the rotor system will have four degree of freedom for each of the node (u, v, ϕu , ϕv ). The homogeneous equation of the rotating system is established by the combined use of the conservation of energy and the Lagrangian approach [6]. [M ]{¨q} − i[G]{˙q} + ([Cn ] + [Cr ]){˙q}   + [K] + 2 ([G] − [M ]) + i[Cn ] {q} + [Ks ]{q} = {0}

(1)

where • [K] is the rotor stiffness matrix • [Ks ] is the stator stiffness matrix which is time dependent in co-rotating coordinate. • {q} = (u, v, ϕu , ϕv ) is nodal displacement vector defined in co-rotating coordinate. The rotating coordinate system (u, v) is defined with respect to the fixed reference frame (x, y, z), as shown in Fig. 1, where z-axis of the rotating frame is the same as the z-axis of the fixed reference. • Subscripts r and n refer to the rotating and the non-rotating parts of the system. • [M] is the mass matrix, describes the inertia effects of both the rotor (2–4 degree of freedom) and stator (2 degree of freedom) nodal mass. • [G], is the gyroscopic matrix, which describes the mass effect of rotating components. • [Cn ] and [Cr ] are matrices that represent the damping forces applied to the stator and rotor, respectively. Utilizing the Floquet theory for the equation, the displacement vector and its derivatives could be expressed as {q} = {}eλt   λt ˙ e + λ{}eλt {˙q} =    λt   λt ¨ e + 2λ  ˙ e + λ2 {}eλt {¨q} = 

(2)

The unknown mode vector {} is time dependent periodic function, which can be written in terms of a Fourier series as {} =

+∞    ji2t lj e j=−∞

+∞      ˙ =  lj ji2eji2t j=−∞ +∞      ¨ =−  lj (j2)2 eji2t

(3)

j=−∞

The final matrix representing the homogeneous Eq. (1) of the rotating system is as follows [4]:   +∞  −j 2 (2)2 + 2λji(2) + λ2 lj eji2t [M ] j=−∞

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+ [D]

+∞ j=−∞

  (ji(2) + λ) lj eji2t

+ ([Kr ] + [Ko ] + [Ka ]e−i2t + [Kb ]ei2t )

+∞ j=−∞

  ji2t lj e = {0}

(4)

where • [D] = −i[G] + ([Cn ]+[Cr ]) • Kr is the rotor stiffness matrix • (Ko + Ka e−i2t + Kb ei2t ) is the stator stiffness matrix, which is time-dependent in the co-rotating coordinate.

Fig. 1. (a) Diagram of the I4-engine crankshaft and block system. (b) Rotating coordinate (u, v) setup with respect to fixed coordinate (x, y, z)

3 Numerical Implementation Hill Method. An infinite set of algebraic equations is readily obtained by separately equating to zero of the various terms of Eq. (4). When j tends to infinity, the frequency spectrum width of {} is not independent. The infinite solution set is redundant and one needs only n independent eigenvectors to characterize the periodically time-varying system of Eq. (4) in the frequency domain. According to Hill determinant convergence, the number of harmonics significantly contributing to the quasi modes {} is limited. Thus, a finite harmonic truncation order jmax = 3 is chosen in this work to accurately determine their frequency spectrum [6]. By setting each harmonics individually equal to zero in Eq. (4), we obtain the truncated set of algebraic equations. ⎫ ⎤⎧ ⎡ l−2 ⎪ A−2 B−1 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ C A B 0 0 ⎥⎪ ⎨ l−1 ⎪ ⎬ ⎥⎪ ⎢ −2 −1 0 ⎥ ⎢ = {0} (5) ⎢ 0 C−1 A0 B+1 0 ⎥ l0 ⎥⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎣ 0 0 C0 A+1 B+2 ⎦⎪ l +1 ⎪ ⎪ ⎪ ⎪ 0 0 0 C+1 A+2 ⎩ l+2 ⎭

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The matrix [Aj ] holds the average properties of the system, while [Cj ] and [Bj ] pertain to the asymmetrical attributes of the rotor and stator [6]. Equation (5) can also be expressed as Hill’s eigenvalue problem.     ˆ + λ Cˆ + [K]){} = 0 (λ2 M (6) 

Finite Element Model Setup. Figure 1 illustrates the rotor/stator dynamic system for an I4-engine with linear spring as the main bearing connection. The crankshaft represents a typical I4 engine, with a uniform distributed counterweight of total length 500 mm and a nominal main bearing diameter of 45 mm. The rigid flywheel inertia is assumed to have equal values of 70 t.mm^2 in the x and y directions, and 140 t.mm^2 in the polar direction. The rotating system is also assumed to be perfectly balanced, with no shaft misalignment considered in the analysis. These assumptions simplify the model and allow for easier analysis. Using a co-rotating coordinate system (fixed to the crankshaft) for this analysis means that all degrees of freedom of the engine block (stator) will be present in all three stiffness matrices (Ko, Ka, and Kb in Eq. 4), which significantly increases the computational complexity. Therefore, a reduced number of nodes were selected for the simulations, following the model reduction approach outlined in [7]. Specifically, one node was selected for each of the 5 main bearings and each mount, allowing for improved computational efficiency. Engine mounts are designed to isolate engine vibrations from the rest of the vehicle, and they typically exhibit nonlinear behavior and frequency-dependent mechanical properties. Their stiffness and damping characteristics can change with the frequency of the engine’s vibration, making the behavior of the mount difficult to predict and model accurately. To simply this simulation, we assume the engine mount only have small displacement in its normal operations, and it oscillates around its balanced position with linear spring characteristics. In this work, the stiffness is choose to be 1000 N/mm. Figure 2. Illustrates the location of three mounts. All the mount stiffness will be assumed as constant, and their damping ratio will be used as optimization variables.

Fig. 2. Diagram of the I4-engine mount configuration (Top view)

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4 Simulation and Optimization Zero Damping Response. The initial simulation of Eigen problem Eq. 6 did not account for damping. The rotor/stator dynamics model used a simplified main bearing model with a constant stiffness of 4e4 N/mm. In Fig. 3, the decay rate (i.e., the real part of the eigenvalue) in the frequency range of 600–780 rad/s is plotted along with the Campbell diagram. These plots illustrate the instability behavior of the system, which is caused by the unsymmetrical crank rotating system. The dynamic coupling between the crank and block gives rise to multiple unstable regions at rotating speeds above 600 rad/s. The data presented in Fig. 3(a) and (b) indicate that there is no instability observed below 600 rad/s. The simulation employed co-rotating coordinates, so the zero frequency in the Campbell diagram corresponds to synchronous oscillation with respect to crank rotation.

Fig. 3. Campbell diagram with harmonic truncation order jmax = 2. (a) Campbell diagram, imaginary part of the eigenvalue (0–780 rad/s). (b) Decay rate, real part of the eigenvalue (0– 780 rad/s). (c) Campbell diagram, imaginary part of the eigenvalue (600–780 rad/s). (d) Decay rate, real part of the eigenvalue (600–780 rad/s)

In the Campbell diagram of Fig. 3, only positive frequencies are represented against the rotational speed. In fact, each parametric quasi-mode {} can beseen as the sum of forward and backward waves associated with the frequency spectra ωk ± j2 , where ωk is the fundamental frequencies of the k th quasi-mode [4]. To investigate the unstable region, simulation of the model was conducted using a smaller frequency step in three frequency ranges: 610–630 rad/s, 702–712 rad/s, and 750–775 rad/s. The results are presented in Fig. 4, which shows refined plots of the unstable regions. The interaction related to the crank and block dynamics is responsible for the instability observed in these regions. Specifically, the first instability at around frequency of 613 rad/s is the primary instability caused by the crank un-symmetricity. All the other positive decay rate regions are due to the dynamic interaction with the engine block.

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Optimization Setup. As discussed in the previous section, most of the instability was caused by harmonic coupling between crank and block interaction. These instability is easily be eliminated by applying a light damping on the non-rotating part (engine block).

Fig. 5. Decay rate after 5% engine mount damping applied

Figure 5 displays the decay rate following the application of a 5% damping force to the engine mount. The results indicate that the damping on the engine mount can effectively mitigate the instability caused by the interaction between the crank and block. However, the primary instability at approximately 613 rad/s remains, as evidenced by the positive decay rate. This instability is difficult to eliminate, and in the next section, we will utilize the decay rate of the primary instability as the optimization target to identify the optimal damping parameter and achieve the most stable dynamic system. Stability Margin Optimization. The system being analyzed consists of a 2 kg ring mass, with zero internal damping in the crank, and a base frequency of 150 rad/s for the

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engine mount. The only variable parameter in the design is the mount damping ratio, which can range from 10% to 80%. Figure 6 illustrate the decay rate plot of the primary instability for each damping ratio. In all cases, the system exhibits a negative decay rate, which indicates that it is dynamically stable. As the damping ratio increases in 5% increments, the decay rate decreases. However, when the damping ratio exceeds 45%, the decay rate begins to increase. The system’s most stable configuration is achieved at a damping ratio of 45%, which yields a maximum negative value of 2.7 for the decay rate. This value is referred to as the maximum stability margin in this study, and it quantifies the system’s robustness by measuring how much the damping ratio can be reduced before the system becomes unstable.

Fig. 6. Decay rate for a 2 kg ring mass with mount damping ranging from 0.1 to 0.8.

Influence of Rotational Damper - Mass and Internal Damping. A torsional damper is mounted on the front end of the crankshaft and consists of a ring mass that is free to rotate around the crankshaft axis, and a rubber or elastomeric material that connects the mass to the crankshaft (Fig. 7). The mass of the damper rotates out of phase with the crankshaft, creating an opposing force that helps to cancel out the torsional vibrations. The polar inertia of the ring and torsional stiffness of the rubber in the damper are carefully tuned to a specific frequency in order to effectively isolate torsional vibrations and reduce the amount of energy transferred to other engine components. However, the lateral and vertical vibrations of the ring mass can interact with the bending vibration of the crankshaft, which can have a detrimental effect on the stability of the system. Additionally, since the torsion damper rotates with the crankshaft, the damping effect is generated internally and can have a negative influence on the system stability when the crankshaft rotates above its threshold speed. According to our initial assumption, the displacement in the torsional direction is independent of other types of motion. Each node on the crank has a maximum of four degrees of freedom, including two for translation and two for bending. We have neglected any influence from axial and torsional motion in this work. Therefore, the ring damping between the ring and crank is referred to the damping in the lateral and vertical directions of the crankshaft (not torsional damping).

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Fig. 7. Crankshaft and torsion damper

Figure 6 shows a decay rate plot that illustrates the effect of a 2 kg torsional damper on the system’s decay rate. To further investigate this effect, we simulated the system with two additional ring masses of 1 kg and 3 kg. Figure 8 presents the decay rate plots for all three ring masses, enabling a clear comparison of their respective decay rates.

Fig. 8. Decay rate comparison for ring masses of 1, 2, and 3 kg with mount damping ranging from 0.1 to 0.8

In order to provide a clear visualization, only the oval shapes around the maximum decay rate values are plotted in Fig. 8. Each oval shape represents a decay rate for a specific combination of engine mount damping ratio (ranging from 0.1 to 0.8) and ring mass value (ranging from 1 to 3 kg), and corresponds to a potential instability field of the system when damping is reduced. In Fig. 8, the maximum point of each case is highlighted with a circle. These points represent the maximum decay rate value for each combination of ring mass and engine mount damping ratio. To facilitate comparison between the different cases, the decay rate values of these points are plotted against the mount damping ratio in Fig. 9.

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Figure 9 clearly shows that, for all three ring masses, the decay rates reach their maximum negative values at an engine mount damping ratio of 0.45. This suggests that a mount damping ratio of 0.45 provides the most effective damping for the system. Additionally, we observed that using a heavier damper ring results in a more stable dynamic system. The 3 kg ring provides nearly three times the stability margin as compared to a 1 kg ring with the same 0.45 mount damping ratio.

Fig. 9. Decay rate plots with respect to mount damping ratio with 0% ring damping

The above analysis is based on the assumption of zero lateral/vertical damping for the torsion damper (ring damping). As we mentioned at the beginning of this section, the ring damping is the internal damping of the rotating system and may have a negative impact on the system’s dynamic stability. To demonstrate this effect, additional optimization simulations are performed with 2% and 5% lateral ring damping. In the following analyses, the lateral/vertical frequency of the ring is fixed at 625 rad/s, and the corresponding results are displayed in Fig. 10.

Fig. 10. Decay rate plots with respect to mount damping ratio (a) 2% ring damping. (b) 5% ring damping

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Figure 10(a) shows the decay rate plots of three ring masses with 2% ring damping. Compared to the results obtained with 0% ring damping in Fig. 9, the stability margin almost doubled, indicating improved system stability. However, in Fig. 10(b), further increasing ring damping to 5% resulted in a decrease in stability margin. Therefore, to ensure optimal system stability, it is preferable to employ light ring damping.

Fig. 11. Decay rate plots for the ring damping 0%, 2%, and 5%

In practical application, ring damping is often greater than 2%. Ring damping exhibits a negative speed-dependent behavior, whereby the decay rate increases with the rotating speed. If the mount damping is insufficient, the system may become unstable in the supercritical range. Figure 11 illustrates the decay rate plots of three ring damping configurations with a 2 kg ring mass. Notably, the system becomes unstable once the speed exceeds 750 rad/s for the configuration with 5% ring damping. The ring damping, particularly when high, primarily influences the stability margin. Therefore, to guarantee optimal system stability, the damper design must be modified by elevating the lateral frequency and ring mass to shift the instability threshold further beyond the maximum operating speed (700 rad/s). Figure 12 depicts the decay rate for a damper with 650 rad/s lateral frequency, a 3 kg ring mass and 5% ring damping, where the instability threshold occurs at approximately 820 rad/s. Consequently, the aforementioned approach can still be employed for optimization, resulting in a highly dependable dynamic system.

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Fig. 12. Decay rate plots for the 5% ring damping and 3 kg ring mass

5 Conclusion This paper presents an efficient method for optimizing the stability of flexible rotor and stator engine systems. The study investigates the dynamic behaviors and vibration characteristics of an unsymmetrical rotor-stator system with internal and external damping. The negative effects of ring damping are incorporated into the dynamic model using Lagrange’s equation, and the unstable dynamic behaviors of the system are demonstrated graphically. Based on the findings, the following conclusions can be drawn: 1. Selecting the appropriate mount damping ratio can significantly enhance stability. For a typical I4 engine damper with a ring mass of around 2 kg and zero internal damping, using an optimized damping ratio of 40% to 50% can lead to a doubling of stability (decay rate) in comparison to levels of 10% to 20% damping ratio. However, if the damping ratio surpasses 60%, stability will begin to deteriorate. 2. Engine mount damping can effectively mitigate the instability caused by the interaction between the crank and block. 3. High ring damping (internal damping) negatively affects system stability. To ensure optimal system stability, it is recommended to elevate the lateral frequency and ring mass to shift the instability threshold further beyond the maximum operating speed 4. Increasing ring mass enhances the efficiency of mount damping in controlling system stability.

References 1. Geradin, M., Kill, N.: A new approach to finite element modelling of flexible rotors. Eng. Comput. 1(1), 52–64 (1984) 2. Stephenson, R.W., Rouch, K.E.: Modeling rotating shafts using axisymmetric solid finite elements with matrix reduction. J. Vib. Accoust. 115(4), 484–489 (1993) 3. Lazarus, A., Prabel, B., Combescure, D.: A 3D finite element model for the vibration analysis of asymmetric rotating machines. J. Sound Vib. 329(18), 3780–3797 (2010)

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4. Meng, M.W.: Frequency and stability analysis method of asymmetric anisotropic rotor bearing system based on three-dimensional solid finite element method. Trans. ASME 137, 102502 (2015) 5. Jiang, Y.: Harmonic balance method for unsymmetrical rotor dynamic analysis. In: SAE Noise and Vibration conference and Exhibition, September 2021 6. Genta, G.: Whirling of unsymmetrical rotors: a finite element approach based on complex co-ordinates. J. Vib. Accoust. 124, 27–53 (1988) 7. AVL EXCITE theory manual version 2010.1

Remaining Useful Life Prediction for Anti-friction Bearings Based on Envelope Spectrum and Extended Kalman Filter Haobin Wen1(B) , Long Zhang2 , and Jyoti K. Sinha1 1 Dynamics Laboratory, The Department of Mechanical Aerospace and Civil Engineering, The

University of Manchester, Manchester M13 9PL, UK [email protected] 2 The Department of Electrical and Electronic Engineering, The University of Manchester, Manchester M13 9PL, UK

Abstract. Anti-friction bearings (AFB) are essential parts of many rotating machines. It is also well known that the fault in the bearings keeps developing during machine operation. This bearing fault can propagate further and trigger other faults within the machine, and eventually lead to the machine failures and shutdown. Hence, the early prediction of remaining useful life (RUL) plays a significant role to optimize maintenance schedule for overhauls and avoiding failures. Recent research studies in the literature have used data-driven models using vibration-based indicators (HI) to estimate the RUL. In the current study, an envelope analysis-based indicator is used that truly reflects bearing conditions only. Then only a few initial vibration measurements are used once the bearing defect is identified to estimate RUL using the extended Kalman filter (EKF). The RUL prediction method is applied to the experimental vibration data on a rig. Keywords: Remaining Useful Life · Bearings · Predictive Maintenance · Extended Kalman Filter · Nonlinear State Estimation · Bayesian Statistics

1 Introduction Asset health management and smart maintenance are critical links in intelligent manufacturing to ensure safe, stable, and efficient industrial production. As fundamental components in rotating machineries, anti-friction bearings (AFB) are widely applied in various electromechanical systems, such as electric motors, combustion engines, gas turbines, industrial robots, etc. Given the integration of mechanical equipment, bearing faults are a major concern as they could cause secondary structural failures and systematic shutdown. Therefore, the early prediction of bearing remaining useful life (RUL) plays an important role in preventing unscheduled downtime and optimize maintenance planning. In recent research, various data-driven models for bearing RUL prediction have been developed based on machine learning theories and probabilistic approaches [1]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 390–397, 2024. https://doi.org/10.1007/978-3-031-40459-7_26

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The machine learning approaches can establish the nonlinear relationship between the adaptively extracted features and the RUL, such as random forest, support vector regression [2], artificial neural networks [3], and deep neural networks. While the probabilistic approaches rely on sophisticated statistical models and hand-crafted features to quantify the uncertainty in bearing RUL estimation, such as stochastic processes [4] and Bayesian filtering [5, 6]. These recent advancements have been demonstrated on realworld datasets collected from bearing life tests [7]. Among them, the recursive Bayesian filters, e.g., Kalman Filter [8] and Extended Kalman Filter (EKF) [6], have attracted lot of attention for their efficiency in the integration with bearing degradation models based on selected health indicators (HI), not only adapting to nonlinear state estimation but also avoiding the need of massive training data. However, the conventional bearing HIs based on signal statistics might be redundant due to the impacts from the coupled structure, e.g., involving shaft defect components. In the current study, the bearing life data from [7] is analyzed to develop RUL prediction techniques. First, the envelope analysis-based bearing HI is computed using the amplitudes of the bearing characteristic frequencies in the envelope spectrum. It is thus purely related to the bearing conditions. The use of envelope spectrum also helps detect bearing defects for determining first inspection time (FIT) and the end of life (EoL) bounds. After the bearing fault is detected, the extended Kalman filter (EKF) is applied to the HI history for predicting the bearing RUL since the FIT. Only few initial HI observations are used in the state-space model to extrapolate the bearing RUL. The method is then applied to the real-world bearing degradation data, the result of which shows its effectiveness in RUL prediction.

2 Experimental Settings [7] The development of RUL prediction methods relies on full bearing degradation data that are expected to cover all bearing failure modes and different operating conditions. In this work, the bearing vibration signals are analyzed using the XJTU-SY bearing dataset available online [7]. Figure 1 provides an overview of the experimental rig where the test bearings are hydraulically loaded from the horizontal direction. Two-channel vibration signals are available from the measurements of the horizontal and vertical accelerometers placed on the bearing housing. In this study, the vibration acceleration data from group ‘Bearing 1_3’ is analyzed. The test bearing in this case is said to run under the shaft speed of 35 Hz with a hydraulic loading force of 12 kN from the normal state until failure. The sampling frequency is 25.6 kHz and the sampling length is 1.28 s per minute. The specification of the bearing is listed in Table 1. Further details of the bearing test rig and the dataset are available in [7].

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Fig. 1. Experimental Rig [7] Table 1. The specifications of the test bearing Number of Balls

8

Fundamental Train Frequency

13.49 Hz

Inner Diameter

29.30 mm

Ball Pass Frequency Outer (BPFO)

107.91 Hz

Outer Diameter

39.80 mm

Ball Pass Frequency Inner (BPFI)

172.09 Hz

Contact Angle

0°

Ball Spin Frequency (BSF)

72.33 Hz

3 Methodology 3.1 Fault Detection Based on Envelope Spectrum In this study, ‘Bearing 1_3’ is run from the normal state until failure after a total of 158 min. The test bearing is found to have an outer-race fault by disassembling the rig after the end of the life test. To reveal the underlying bearing conditions during the test, envelope analysis is performed on the vibration signals of the horizontal accelerometer. Based on the envelope spectrum inspection, the test bearing is detected with outerrace defect at the 59th minute, which is verified the outer-race ball pass frequency (as marked in green dash line) clearly observed in the envelope spectrum, as shown in Fig. 2. Therefore, the first inspection time (FIT) for this bearing is set at the 59th minute. 3.2 Bearing Health Indicator To remove the interferences of the defect sources other than bearing faults, e.g., the shaft unbalance, the envelope spectral HI is crafted using the amplitudes of the bearing fault characteristic frequencies from the first to the third harmonics, which can be given by HI = {A(fBSFi× ) + A(fFTFi× ) + A(fBPFOi× ) + A(fBPFIi× )}/A, i = {1, 2, 3}

(1)

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Fig. 2. Envelope spectrum of the horizontal vibration signal at the 59th minutes.

where A( f ) is the amplitude of a frequency component in the envelope spectrum and the four characteristic fault frequencies are respectively BSF, FTF, BPFO, and BPFI. The envelope spectral HI for ‘Bearing 1_3’ is shown in Fig. 3 (marked by grey solid line) with the smoothed indicator (marked in blue) attained via 6-point moving average.

Fig. 3. The health indicator based on envelope analysis for the analyzed case. Original indicator vs. smoothed indicator.

In this work, the time when the HI reaches the maximum is considered the end-oflife (EoL), as the maximum HI indicates that the bearing fault frequencies have also gathered their highest energy. In practice, the EoL is not priorly known whereas the envelope spectral HI offers a proxy for generating the baseline RUL.

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3.3 RUL Prediction at FIT State-space models with recursive parameter identification is well suited for RUL estimation with limited observations. For linear state estimation, the well-known Kalman filtering (KF) [9] method provides optimal estimation under the squared error objective. Kalman Filtering The Kalman filter is a recursive state estimator that seeks for the optimal posterior estimate of the state vector x, from noisy observations z. The state vector is assumed to follow a linear discrete-time stochastic process, xk+1 = f (xk , vk ) where f is the linear state process model with Gaussian noise, vk [9]. The distorted sensor measurements are modelled by zk+1 = h(xk , wk ), where wk represents model inaccuracies and sensor noise [9]. Extended Kalman Filtering Method For nonlinear system dynamics, the EKF introduces the Taylor expansion to linearize the non-linear process model f and measurement model h by dropping the higher order term [10]. xk+1 ≈ x˜ k+1 + Fk+1 (xk − x˜ k ) + Gk+1 vk

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zk+1 ≈ z˜ k+1 + Hk+1 (xk+1 − x˜ k+1 ) + Wk+1 wk

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where xk+1 is the state vector at time step k + 1, x˜ k+1 is the prior estimate of state, zk+1 denotes the measurement, and z˜ k+1 its prior estimation. F, G, H, and W are the partial derivative terms, ∇f x , ∇f v , ∇hx , and ∇hv . EKF RUL Prediction To apply EKF for predicting bearing RUL, it is assumed that the highly nonlinear HI encompasses a latent state that correctly model the bearing degradation process. Thus, given the incremental trend of the HI since FIT, an exponential process model can be established for the actual state related to the bearing health. The prior prediction of state, the HI itself, is given by  b  e k xk , (4) x˜ k+1 = bk with the prediction of observations of the spectral HI, z˜ k+1 = h(xk+1 ) = xk+1 (1). The linearization of model f and h can be given by  b b  e k e k xk , Fk+1 = 0 1 Hk+1 = ( 1 0 ),

(5)

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After a selection of initial conditions for the state vector x˜ 0 = x0 and the state error covariance matrix P0 , the nonlinear state estimation of EKF involves the recursive iteration of the following steps: 1) Prediction of the a priori state and a priori covariance. x˜ k+1 = f (˜xk , 0),

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2) Take in the smoothed spectral HI as the actual observations, zk . 3) Correction of the prediction to give posteriori estimations. T T (Hk+1 P˜ k+1 Hk+1 + Rk+1 )−1 , Kk+1 = P˜ k+1 Hk+1

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x˜ k+1 ← x˜ k+1 + Kk+1 (zk − Hk+1 x˜ k+1 ),

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where Qk and Rk are the covariance matrices of the process noise vk and the observation noise wk . Finally, the RUL at inspection time k is extrapolated out where the state variable reaches the end-of-life (EoL) thresholds, γ lower and γ upper . Here, γ is computed empirically as an interval summarizing from all available bearing data. For example, the lower bound of RUL is given by RULlower = inf{RUL : f (˜xk+RUL−1 ) ≥ γlower }.

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4 RUL Prediction Results Table 2 summarizes the FIT and the baseline EoL of the test bearing determined by envelope spectrum with the RUL estimation results. After the outer race fault is detected at the 59th minutes, 20 data points of spectral HI are applied in EKF for bearing RUL prediction. From Table 2, it is seen that for ‘Bearing 1_3’, the EKF method offers correct prediction of the RUL at the lower end-of-life bound, i.e., 53 min from the inspection time at the 79th minute.

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Fig. 4. EKF-based RUL Prediction for ‘Bearing 1_3’.

Specifically, the estimated mean RUL is the period from the inspection time to the point where the extrapolated HI surpasses the two empirical EoL bounds. Also, the bearing RUL is predicted with uncertainty estimation and clamped by two predictive intervals (PI) with 95% confidence, as shown in Fig. 4. The EKF posterior state estimation of the bearing HI (blue solid line) is extrapolated out to the future time and touches the lower EoL bound exactly when the original HI (marked by grey solid line) reaches the maximum. At this time, the test bearing is considered to have a fatal failure and should be replaced. On the above basis, the bearing spectral HI has a clear indication of bearing fault severity and is combined with the EKF for estimating the nonlinear trend of bearing degradation. The application of the prognostic method to experimental bearing data has demonstrated its effectiveness in bearing RUL prediction.

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5 Conclusions The work presents the envelope-spectrum-based indicator for bearing health assessment and the EKF-based RUL prediction method with model simplicity. The bearing HI is shown to have a few advantages including a clear indication of the bearing condition. Finally, the applications of the RUL prediction method to the actual vibration data demonstrated its usefulness for bearing predictive maintenance. The prediction performance will be further supported by more examples using bearing data under different operating conditions.

References 1. Si, X.S., Wang, W., Hu, C.H., Zhou, D.H.: Remaining useful life estimation – a review on the statistical data driven approaches. Eur. J. Oper. Res. 213(1), 1–14 (2011) 2. Loutas, T.H., Roulias, D., Georgoulas, G.: Remaining useful life estimation in rolling bearings utilizing data-driven probabilistic E-support vectors regression. IEEE Trans. Reliab. 62(4), 821–832 (2013) 3. Heimes, F.O.: Recurrent neural networks for remaining useful life estimation. In: 2008 Int. Conf. Progn. Heal. Manag. PHM 2008 (2008) 4. Park, C., Padgett, W.J.: Accelerated degradation models for failure based on geometric Brownian motion and gamma processes. Lifetime Data Anal. 11(4), 511–527 (2005) 5. An, D., Choi, J.H., Kim, N.H.: Prognostics 101: a tutorial for particle filter-based prognostics algorithm using Matlab. Reliab. Eng. Syst. Saf. 115, 161–169 (2013) 6. Singleton, R.K., Strangas, E.G., Aviyente, S.: Extended Kalman filtering for remaining-usefullife estimation of bearings. IEEE Trans. Ind. Electron. 62(3), 1781–1790 (2015) 7. Wang, B., Lei, Y., Li, N., Li, N.: A hybrid prognostics approach for estimating remaining useful life of rolling element bearings. IEEE Trans. Reliab. 69(1), 401–412 (2018) 8. Wang, Y., Peng, Y., Zi, Y., Jin, X., Tsui, K.L.: A two-stage data-driven-based prognostic approach for bearing degradation problem. IEEE Trans. Ind. Inform. 12(3), 924–932 (2016) 9. Haykin, S.: Kalman Filtering and Neural Networks, vol. 5 (2001) 10. de Freitas, M., Coelho, K.B., Ahmed, K.: An improved extended Kalman filter for radar tracking of satellite trajectories. Designs 5(3), 54 (2021). https://doi.org/10.3390/designs50 30054

New Comprehensive Approach for Torsional Analyses of Industrial Powertrains Timo P. Holopainen(B) and Tommi Ryyppö ABB Large Motors and Generators, 00381 Helsinki, Finland [email protected]

Abstract. Calculation procedures are well-established for torsional powertrain analyses of industrial applications. The mechanical system is described by a onedimensional model. The solutions are obtained from a symmetric system of second order linear differential equations. The electromechanical and process interactions are usually neglected, and these effects are included to the solution as external loads. Generally, the model is tuned to the operating point to justify the linearized modeling approach. However, this approach neglects the interactions and non-linear effects of torque and speed variation. In addition, the modeling of powertrain components is usually conducted by the system integrator starting from the drawings and other design documents. This is a manual and laborious procedure and susceptible for mistakes. The main aim of this paper is to introduce a novel approach for industrial powertrain analyses. A distinctive feature of this approach is to rely only on the timedomain simulations and on the first-order differential equations. All powertrain components are modeled by using power parameters in the interfaces. This enables systematic inclusion of non-linear effects and electromechanical interaction. The proposed approach is compared to the traditional one. The pros and cons of these approaches are reviewed. A calculation example of a reciprocating motorcompressor is used to demonstrate the differences. Finally, the paper outlines the future research needs. Keywords: Torsional Rotordynamics · Electrical Machines · Powertrain · Electromechanical Interaction · Reciprocating Compressors

1 Introduction Transmission of torque with a specified speed is the main function of mechanical powertrains. This transmission must be achieved without excessive torsional vibrations. Thus, torsional analyses are routinely conducted for rotating machine systems. An overview of torsional vibrations as part of industrial practice is described in [1–3]. Recent research activities have focused on torsional issues of powertrain components like gears, couplings, electric machines, and reciprocating compressors. However, the fundamentals of powertrain analysis have remained mainly unchanged. A typical torsional analysis includes following items: natural frequencies and mode shapes, steady-state response, start-up transient, and electrical faults. The calculation © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 398–408, 2024. https://doi.org/10.1007/978-3-031-40459-7_27

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model is commonly one-dimensional, and the analysis types are eigenvalue, harmonic and transient. The problem is formulated as a system of second order linear differential equations. Torque loads of electric machines and pistons/impellers are calculated separately and are given as input to the torsional analysis. The parameters of the model, like coupling stiffness, are typically adjusted for the rated operating condition. The formulation of torsional vibrations as second order differential equations is somehow exceptional compared to the surrounding systems like electric drives, electric network, and fluid-flow piping systems. These systems are commonly formulated as firstorder differential equations. Further, the traditional linearization of model parameters to the operating point is usually a laborious process if required. The main aim of this paper is to outline a comprehensive approach for torsional analyses of powertrains. The actual rotation speed and transmitted torque are included to the model together with the non-linear characteristics of powertrain components. The problem is formulated with a system of first-order differential equations. Finally, all the analyses are conducted in time-domain and required results are obtained by post-processing. This paper starts by describing the proposed approach on a general level. Essentially, this new approach is based on a novel combination of established methods and techniques. Figure 1 shows an example of a motor-driven compressor system. In this paper the focus is on the mechanical section of the powertrain system. A digital twin of electric motors for torsional powertrain analyses has been presented in [4–6].

Fig. 1. Digital twin of a motor-driven variable-speed compressor system.

After description of the approach several calculation examples of a motor-driven reciprocating-compressor system are presented. Finally, the traditional and this novel approach are compared, and research needs outlined.

2 New Comprehensive Approach 2.1 Main Distinction Traditionally, industrial powertrain torsional analyses are conducted by using linear models. The basic formulation is a system of second order differential equations. The main tasks are the identification of the lowest natural frequencies (eigenvalue analysis), steady-state response (harmonic analysis) and transient response analyses of fault conditions and starting (time-stepping analysis). Usually, the linearization of the system components, like couplings, is explicitly made for the rated operating condition. The size of the problems is numerically small in industrial analyses.

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A driving force to search an alternative approach for torsional analyses has been the electromechanical interaction in electric machine. This interaction cannot be modelled easily as a part of mechanical model. At least, auxiliary degrees-of-freedom are needed, if more than one oscillation frequency is under consideration. This is the case with reciprocating compressors possessing substantial number of excitation orders. It is even more difficult to model the transient loading of a premature breaker reclosure with a simple and linear model [7]. The main idea of this novel approach is to calculate all required analyses in timedomain. The problem is formulated with a system of first-order differential equations with speed and torque as variables. 2.2 Modelling Mechanical drivetrains are traditionally modelled by a system of second order differential equations Mθ¨ + Cθ˙ + Kθ = T

(1)

where M, C and K are the inertia, damping and stiffness matrices, respectively, and θ and T are the vectors of angular displacements and loading torques, respectively. The size of the matrices is n × n and length of vectors n. Two-node linear elements are used with torsional stiffness k ij for element i, and the damping of the same element proportional to the stiffness, i.e., cij = αi k ij . The angle of twist and the torque can be written as ϕij = θj − θi τij = kij ϕij

(2)

Further, the symbol of angular velocity can be changed i = θ˙i , and the relation of twist and angular velocity written in the form ϕ˙ij = j − i

(3)

These operations yield the system of first order differential equations for the mechanical drive train         I 0 τ˙ 0 B τ 0 = + (4) ˙ 0M  A −C  T where I is the identity matrix of size n × n, and matrices A and B can be obtained from the original matrix K using Eqs. (2) and (3). These matrices are ⎤ ⎡ −1 0 0 0 0 ⎢ 1 −1 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 0 1 −1 0 0 ⎥ ⎥ ⎢ (5a) A(n×n−1) = ⎢ . . ⎥ ⎢ 0 0 .. .. 0 ⎥ ⎥ ⎢ ⎣ 0 0 0 1 −1 ⎦ 0 0 0 0 1

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⎡

B(n−1×n)

−k12 k12 ⎢ 0 −k23 ⎢ ⎢ =⎢ 0 ⎢ 0 ⎢ ⎣ 0 0 0 0

⎤ 0 0 0 0 k23 0 0 0 ⎥ ⎥ ⎥ .. .. . . 0 0 ⎥ ⎥ ⎥ .. 0 . kn−2,n−1 0 ⎦ 0 0 −kn−1,n kn−1,n

401

(5b)

In general, the symmetric structure of original equations of motions has been lost, but the system of first order differential equations achieved. The difference between the proposed and traditional approaches can be described also by their elements and degrees-of-freedom. Figure 2 shows two-node prismatic elements for these approaches. The derivation of 1st and 2nd order differential equations is based on the dynamic equilibrium of torque in nodal points.

Fig. 2. Traditional two-node prismatic elements used in 1st and 2nd order formulations together with the new first order formulation with an internal torque variable τ .

It can be added that the rotating rotors do not have boundary conditions defined as angular displacements. This enables the rejection of angular displacements as variables. Thus, the torsional rotordynamics seems to be a clearly abnormal area of modelling of mechanical vibrations but consistent with modelling of electrical and fluid systems. Here, the derivation of a new linear prismatic element was presented and the assembly of elements to a calculation model was shown. This derivation started from the traditional formulation and resulted in a new formulation. This was a heuristic approach. Most probably, it is possible to derive higher order elements and prove the convergence characteristics with mathematical rigor. Similarly, it is assumed to be possible to derive elements for conical and other element geometries. 2.3 Damping Damping of torsional mechanical systems is customarily modelled by damping ratio of natural modes. This approach requires the use of natural modes and modal superposition. In general, the damping of torsional mechanical systems is small and weakly known. A non-linear approach allows more advanced damping models for critical components like flexible coupling. The relative damping of a coupling is the ratio of damping energy, converted into heat during a vibration cycle, to the flexible strain energy [8]. For oscillations at one frequency this can be modelled by a viscous damping coefficient between velocities of element ends [7]. For general case, a mode advanced damping model is required.

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It can be mentioned that the angular velocity cannot be directly used for the modelling of damping, because the angular velocity is dominated by rotor speed. 2.4 Modularity An essential part of the developed approach is to allow and support modular preparation of calculation models of components. Thus, the aim is to define standardized interfaces for all types of components (electrical, mechanical, hydraulic, …). This reminds the approach under the name bond graphs, see e.g. [9]. The question arises how to define the component interfaces and how far the preprocessing of the component models is reasonable to process before combination of sub-models. This preprocessing and interfaces are also related to the general need to protect the intellectual property of separate manufacturers. 2.5 Connection of Component Models The mechanical components of powertrains are connected by couplings or shaft-end flanges. There are standardized approaches for modelling of these components and their connections [10] that can be automatically included to the assembly of sub-models. In general, the mechanical powertrain components are connected to each other from one degree-of-freedom, i.e., angular velocity of connection node. Similarly, equilibrium between components is governed by the torque in connection node. 2.6 Boundary Conditions The boundary conditions must be specified for the solution. The extension of calculation model and the selection of adequate boundary conditions is not straightforward for a power flow system. Usually, if an electric machine is included to the model, the upstream boundary condition is defined by the supply voltage and frequency. In downstream, the boundary condition is defined by the loading torque that is usually a function of rotating speed and sometimes a function of angular displacement like with reciprocating compressors. It might sound unusual, but the boundary conditions, at least in steady state, must be defined so that there will be a balance between the supply and load power. 2.7 Initial Values Time-domain simulation starts from the initial values. Thus, the computational efficiency depends on the quality of initial values. In some cases, like in start-up analysis, the initial values are readily available. However, often the aim of the simulation is related to an operating point. In these cases, the guess of initial values is crucially important. One tempting solution to find initial values is to conduct harmonic analysis in steady state before the time-domain simulation. Usually, this combination of analysis reduces the required length of simulation significantly.

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2.8 Solution The problem is formulated as a system of first-order differential equations. The component mass matrices are symmetric or diagonal, and combined stiffness/damping matrices asymmetrical (4). The component models are connected to each other by one degreeof-freedom, i.e., by angular velocity. The time-domain solution of this kind of a system consisting separate blocks can be conducted by various algorithms. The authors are not familiar with simulation algorithms for first order systems of component blocks. 2.9 Natural Frequencies and Damping The most important analysis type for torsional powertrains is the analysis of natural frequencies. Fundamentally, the eigenvalues can be determined only for a linear or linearized system. This seems to set an obstacle for the proposed approach enabling non-linearity and relying only on the time-domain simulation. However, a method based on the impulse response analysis can be applied. This method reminds the experimental modal testing.

3 Calculation Example 3.1 Example Powertrain A direct-on-line motor-compressor powertrain was used in calculation examples. Table 1 shows the main parameters of this powertrain. Table 1. Rated parameters of the example powertrain. Parameter

Value

Unit

Power

460

kW

Frequency

60

Hz

Speed

713

rpm

Number of poles

10

–

Voltage

4160

V

Rated torque

6.16

kNm

The example motor drives a compressor. The drive train consists of the following components with inertia in parenthesis: direct-on-line motor (52.2 kg m2 ), flexible coupling (0.9 kg m2 ), flywheel (129.4 kg m2 ) and reciprocating two-cylinder compressor (3.9 kg m2 ). The stiffness of the coupling is 46.9 kNm/rad and the relative damping 0.791.

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3.2 Calculation Model Figure 3 shows the geometry of the modelled mechanical system. The calculation model was generated between the airgap and load torques on another. The stiffness between these two locations was included to the calculation model. The outboard inertia on the left of airgap torque node was added to the node inertia. The same approach was applied for the inertia on the right of the load torque node. This mechanical powertrain model was fully coupled to the electromagnetic motor model in the airgap torque node [7].

Fig. 3. Powertrain model from the left: motor, coupling (yellow), flywheel and two-cylinder compressor (Color figure online).

3.3 Steady-State Response The initial values for steady-state simulation were calculated by harmonic analysis. Figure 4 shows the steady-state response in the rated condition directly after harmonic analysis.

Fig. 4. Steady state response: a) torque, and b) speed of the components.

Missing of initial transient on the left-hand side of Fig. 4 indicates that the harmonic analysis gives adequate initial values to the start of the simulation. 3.4 Natural Frequencies and Damping The natural frequencies and damping ratios were obtained by the time-stepping method with impulse response approach. Two time-stepping simulations in steady state shown in Fig. 4 were calculated. In the first one, the rotor was rotated by the rated voltage

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and frequency against the cyclic compressor load. In the second one, a torque pulse (0…360°) in the load position was added to the load torque. The parameters of the pulse, yielding continuous excitation up to 50 Hz, were: amplitude 0.1 MNm, length 0.0364 s. A constant and equal time-step length was applied for both simulations. The results of these two simulations, particularly the load torque and the angular speed of the rotor core, were subtracted from each other and used for the post-processing. In this case, Frequency Response Function (FRF) between the angular speed of the rotor core and the load torque was derived (Fig. 5). The damping ratio of natural modes was derived using half-power point method [11] or an advanced generalization of that method when the peak did not have the half-power points [12]. When the applied angle of half-power points is 90°, the angle of 45° was used for the generalized identification. The lowest natural frequency without electromagnetic effects is 5.48 Hz [7].

Fig. 5. Frequency response plot between motor angular speed and load torque oscillation. The identified modal parameters: f1 = 2.55 Hz, ζ1 = 18.9%, f2 = 8.98 Hz, ζ2 = 20.0%.

3.5 Start-Up Figure 6 shows a direct-on-line start-up of the example powertrain. The starting is made without compressor load cycle using a constant load. A long starting time is due to the large inertia flywheel. The initial air-gap torque transient consists mainly of 60 Hz oscillations. The largest coupling torque is reached in the breakdown point. The coupling torque oscillates during the initial transient with frequency ~5.5 Hz and after overshooting the rated speed with frequency ~2.5 Hz. 3.6 Two-Phase Short-Circuit Figure 7 shows the two-phase short-circuit loading of the example powertrain. The initial values of the simulation are obtained by the harmonic analysis as with the steady-state case. After short (0.021 s) steady-state simulation the short-circuit occurs. The coupling torque starts to oscillate with frequency ~5.5 Hz. The speed of the rotor starts to slow down.

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Fig. 6. Starting against constant load: a) torque, and b) speed of components.

Fig. 7. Two-phase short-circuit loading: a) torque, and b) speed of components.

3.7 Premature Breaker Reclosing Figure 8 shows a transient loading due to a premature breaker reclosing. As in the precious cases, the initial values of simulation are obtained by harmonic analysis. Then, after short (0.017 s) steady-state simulation a power cut-off occurs. The air-gap torque drops immediately to zero and the coupling torque starts to oscillate with frequency ~5.5 Hz. This means that the rotor is vibrating with its first undamped natural frequency together with decreasing rotational speed. Simultaneously, the magnetic field of the active parts starts to decay due to the resistive losses in the copper cage of the rotor and the iron-losses of the stator and rotor. After power interruption of 0.2 s, the supply voltage is reconnected with the resulting air-gap and coupling transient torques that die away quickly. A more detailed analysis can be found in [7]. 3.8 Comparison The proposed method is fully coupled electro-magneto-mechanical approach. Traditional industrial approaches are based on separately calculated electromagnetic air-gap torque loads and purely mechanical simulation models. Table 2 shows a comparison of predicted maximum coupling torques of these two methods. Here, the traditional uncoupled method consists of following features for short-circuit and reconnection cases: the initial speed and torque is zero, and the simulation starts from the instant of a shortcircuit or reconnection. As Table 2 shows the uncoupled analysis does not include the steady-state torque (6.16 kNm) affecting, at least, on the maximum short-circuit torques.

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Fig. 8. Breaker reclosing after 0.2 s power interruption: a) torque, and b) speed of components.

Table 2. Comparison of predicted torque values of uncoupled and coupled analyses. Load case

Maximum coupling torque, kNm Coupled

Difference, %

Uncoupled

Start-up, initial transient

8.57

7.20

−16.0

2-phase short-circuit

7.21

3.51

−51.4

3.12

−61.1

3-phase short-circuit Reconnection 0.2 s

8.02 23.1

29.5

27.6

4 Discussion The calculation examples show that all the industrial analyses can be conducted by the new proposed approach. It is significant, that the natural frequencies and damping ratios can be identified from simulation results for non-linear systems without any linearization of system equations. This means that the same model with the same boundary conditions can be used in all analysis types. The initial values of steady-state operating points can be obtained effectively by harmonic analysis using, naturally, the same model. The comparison of uncoupled and coupled analyses shows significant differences of maximum coupling torques (Table 2). A large part of the differences in short-circuit cases results from the initial condition of the uncoupled analyses. If the initial torque of the rotor (6.16 kNm) is included to the analysis, the difference will be most probably reduced. The most challenging loading case is the breaker reclosing analysis. In this analysis the electromagnetic system interacts strongly with the mechanical system. The supply voltage drops from rated value to zero and back again. In the example of Fig. 8, the speed of the motor drops from 713 rpm to 515 rpm and returns to rated speed. These phenomena would be exceedingly difficult to model with an uncoupled analysis. Figure 8 shows an example for 0.2 s power interruption. Actually, there is a need to find a representative or worst interruption time [7].

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5 Conclusions A novel comprehensive approach for powertrain torsional analyses was introduced. A distinctive feature of this approach is to rely only on the time-domain simulations and on the first-order differential equations. All powertrain components are modeled by using power parameters, i.e., angular speed and torque in mechanical interfaces. This enables systematic inclusion of non-linear effects and electromechanical interaction. The calculation example of a motor-driven reciprocating-compressor indicates strongly that this approach improves the accuracy particularly in premature breaker reclosing analysis.

References 1. Walker, D.: Torsional Vibration of Turbomachinery. McGraw-Hill, New York, USA (2003) 2. Anon.: Rotordynamic Tutorial: Lateral Critical Speeds, Unbalance Response, Stability, Train Torsionals, and Rotor Balancing. API Technical Report 684. American Petroleum Institute, Washington DC, USA (2019) 3. Corbo, M., Malanoski, S.: Practical design against torsional vibration. In: Proceedings of the 25th Turbomachinery Symposium, pp. 189–222. Turbomachinery Laboratory, Texas A&M University, College Station, Texas, USA (1996) 4. Holopainen, T., Roivainen, J., Ryyppö, T.: Digital twin of induction motors for torsional vibration analysis of electrical drive trains. In: Proceedings of the 12th International Conference on Vibrations in Rotating Machinery, pp 564–574. CRC Press, Balkema, The Netherlands (2020) 5. Holopainen, T.: Digital twin of induction motors for torsional analysis of powertrains. In: Proceedings of SIRM 2021 – 14th International Conference on Dynamics of Rotating Machines, 9 p., Gdansk, Poland (2021) 6. Holopainen, T., Ryyppö, T.: Digital twin of induction motors for response analysis of electric drive trains. In: Proceedings of Torsional Vibration Symposium 2022, 10 p., Salzburg, Austria (2022) 7. Holopainen, T., Ryyppö, T., Järvinen, J.: Maximum torques due to electrical reclosures for drivetrain components of motor driven reciprocating compressors. In: Proceedings of SIRM 2023 – 15th International Conference on Dynamics of Rotating Machines, 9 p., Darmstadt, Germany (2023) (Submitted for publication) 8. Anon.: Technical data: highly flexible couplings for industrial applications. Vulkan Drive Tech. (2022) www.vulkan.com 9. Karnopp, D., Margolis, D., Rosenberg, R.: System Dynamics: A Unified Approach, 2nd edn. John Wiley & Sons, New York, USA (1990) 10. Anon.: Flexible Couplings – Mass Elastic Properties and Other Characteristics. American National Standard, ANSI/AGMA 9004-B08 (2008) 11. Friswell, M., Penny, J., Garvey, S., Lees, A.: Dynamics of Rotating Machines. Cambridge University Press, New York, USA (2010) 12. Ewins, D.: Modal Testing: Theory, Practice and Application, 2nd edn. Research Studies Press, Baldock, Hertfordshire, England (2000)

Modeling of the Divergently Worn Annular Seal for the Two-Way Coupled Fluid–Structure Interaction Analysis of Shaft Vibration and Clearance Flow Shogo Kimura1(B) , Tsuyoshi Inoue1 , Hiroo Taura2 , and Akira Heya1 1 Nagoya University, Aichi, Japan [email protected], [email protected], [email protected] 2 Kindai University, Osaka, Japan [email protected]

Abstract. One of the causes of shaft vibration in turbomachinery is the rotordynamic (RD) fluid force. RD fluid force has a significant impact on the stability of the system, so it is necessary to carry out shaft vibration analysis considering the influence of RD fluid force at the design stage. The common methods for considering the effect of RD fluid force in shaft vibration analysis are linear RD coefficients and two-way coupled analysis. In the previous research, two-way coupled analysis was validated by experiment. However, Onset Speed of Instability (OSI) changed as experiment was repeated due to wear of seal stator. In order to explain this change, two-way coupled analysis of the clearance flow and shaft vibration of the divergent tapered annular seal was performed. In particular, the modeling of the clearance flow of the divergent tapered annular seal was investigated by detailed comparison with CFD analysis of the pre-swirl ratio and loss coefficient at the inlet. Two-way coupled analysis of the clearance flow and shaft vibration of the divergent tapered annular seal incorporating this model was conducted to determine the OSI. The results were compared with experimental results and discussed. Keywords: FSI analysis · Divergent tapered seal · Shaft vibration · Clearance flow

1 Introduction Rotordynamic (RD) fluid force generated at the seals is one of the causes contributing to shaft vibration in turbomachinery [1]. RD fluid force is generated by the interaction between shaft vibration and fluid and has a significant influence on the stability of the system, so shaft vibration analysis considering the influence of RD fluid force is necessary at the design stage. There are two types of shaft vibration analyses which consider the influence of RD fluid force: the method using linear RD coefficients and Fluid-Structure Interaction (FSI) © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 409–418, 2024. https://doi.org/10.1007/978-3-031-40459-7_28

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analysis. Bulk-Flow analysis [2] and CFD analysis [3] are two methods for obtaining linear RD coefficients. In these methods, the RD coefficient is obtained by linearising the RD fluid force around the centre of the seal or static eccentricity. However, the prediction of linear RD coefficients is based on the assumption that the whirling amplitude is small and cannot consider the non-linearity of the RD fluid force. On the other hand, FSI analysis is a method of shaft vibration analysis that can be applied when the whirling amplitude is large and the non-linearity of the RD fluid force is pronounced. Miyake et al. [4] and Kunori et al. [5] used this method to determine the Onset Speed of Instability (OSI) of a vertical shaft system supported by a plain annular seal and compared the OSI with experimental results. The frequency responses obtained from the experiments are shown in Fig. 1.

Fig. 1. Comparison of frequency responses obtained from experiments [4, 5]

Comparing the result of the experiment conducted by Miyake et al. [4] with one conducted by Kunori et al. [5], the OSI dropped from 3357 rpm to 2828 rpm as shown in Fig. 1, a difference of 529 rpm, even though the same experimental apparatus was used. This is thought to be due to the stator being worn as the experiment was repeated, as the rotor contacted the stator after the rotational speed exceeded the OSI in the experiment. The radial clearance was 200 µm when the seal was made, but as the experiment was repeated, the radial clearance increased due to wear, and it was found that the seal went from a plain annular seal to a divergent tapered annular seal. The measurement positions and the clearance are shown in Fig. 2. Since it has been shown that the direct damping and direct stiffness coefficients among the RD coefficients are reduced in divergent tapered annular seal [6], resulting in reduced stability, the FSI analysis of a vertical shaft supported by a divergent tapered annular seal is carried out by expanding the conventional FSI analysis for a plain annular seal [4, 5]. The change in stator geometry due to the taper causes change in the flow at the seal inlet and outlet compared to the plain annular seal case, and the boundary conditions such as pre-swirl ratio and loss coefficients are considered to change. These coefficients have a significant influence on the dynamic characteristics of the seal and

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Fig. 2. Measurement position and seal clearance

need to be estimated accurately. The pre-swirl ratio and loss coefficient at the seal inlet are obtained by CFD analysis and applied as boundary conditions in the clearance flow analysis of the FSI analysis to accurately model the divergent tapered annular seal. The results are compared with experimental results.

2 CFD analysis for boundary condition parameter estimation The analysis domain used in the CFD analysis is shown in Fig. 3. In addition to the seal section, a preliminary region is set at both the seal inlet and outlet sides as the analysis domain. The length of the preliminary region is 1/4 of the length of the seal section [7].

Fig. 3. Analysis domain of CFD analysis

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2.1 Estimation of pre-swirl ratio The average circumferential velocity inside the seal is calculated from the circumferential velocity distribution by CFD analysis and the circumferential velocity vin at the seal inlet is calculated by extrapolation to obtain the pre-swirl ratio α using Eq. (1). α=

vin Rω

(1)

where R is the rotor radius and ω is the rotor rotational speed. The area-weighted average is used to calculate the average value of the circumferential velocity. If the total area of the meshes at the seal inlet is A, the area of each mesh is Ai and the circumferential flow velocity at each mesh is vi , the circumferential flow velocity vin at the seal inlet is expressed by Eq. (2) [8]. vin =

1 vi Ai A

(2)

i

2.2 Estimation of loss coefficient From the axial velocity distribution, the average axial velocity inside the seal is determined and extrapolated to calculate the axial velocity at the seal inlet and outlet. The pressure at the seal inlet and outlet is calculated by extrapolation from the pressure distribution inside the seal. These are substituted into Eq. (3) (Bernoulli’s equation) to calculate the loss coefficients at the inlet and outlet. The area-weighted average is used to calculate the average axial velocity. 1 2 Ps = pin + ρ(1 + ξs )win 2 1 2 Pe = pout + ρ(1 − ξe )wout 2

(3)

3 FSI analysis for divergent tapered annular seal 3.1 Rotor system model As an analytical model, the two-disk vertical elastic shaft model used in the literature [4, 5] is used: two disks are mounted on a shaft, with simple support at the upper end and seal support at the lower end; the RD fluid force acts on the disk at the seal. The equation of motion for this rotor system is expressed by Eq. (4). ˙ + Kq = Funb + FRD M¨ q + Cq

(4)

where M is mass matrix, C is damping matrix, K is stiffness matrix, q is displacement vector, Funb is the unbalance force and FRD is the RD fluid force.

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3.2 Clearance flow analysis While the previous research [4, 5] focused on plain annular seal, the present study extends the scope to divergent tapered annular seal. The clearance h between the rotor and stator in divergent tapered annular seal is expressed by Eq. (5). h(θ, z) = Cr − qsx cos θ − qsy sin θ + ϕz

(5)

where C r is the seal radial clearance, qsx and qsy are the rotor displacements, ϕ is the taper angle, θ is the circumferential coordinate and z is the axial coordinate. The governing equations for the clearance flow analysis are Eqs. (6), (7) and (8).   ∂h ∂(hv) ∂(hw) + + =0 (6) ρ ∂t ∂y ∂z     ∂(hv) ∂ hv2 ∂(hwv) ∂p ρ + + + τry + τsy = 0 (7) +h ∂t ∂y ∂z ∂y    ∂(hw) ∂(hvw) ∂ hw2 ∂p + + + τrz + τsz = 0 ρ (8) +h ∂t ∂y ∂z ∂z where y (= Rθ ) is the circumferential coordinate, v is the circumferential flow velocity, w is the axial flow velocity, p is pressure, ρ is density of fluid and τ is sheer stress. The pressure distribution in the seal is obtained by solving Eq. (9) derived from these governing equations.   ∂ 2 h ∂ 2 (hv2 ) ∂ ∂(hv) ∂(hw) ∂h ∂ 2 (hvw) 2 + + − 2 + + 2 −(∇h · ∇p + h∇ p) = ρ ∂t ∂y ∂z ∂t ∂t ∂y2 ∂y∂z

2 2 ∂ (hw ) ∂ ∂ + + (τsy + τry ) + (τsz + τrz ) (9) ∂z 2 ∂y ∂z Equation (3) is used as a boundary condition when solving the Eq. (9). The obtained pressure is integrated over the rotor surface as shown in Eq. (10) to obtain the RD fluid force FRD acting on the rotor. ¨ ⎧ ⎪ ⎪ p cos θ d θ dz ⎨ FRDx = − ¨ (10) ⎪ ⎪ ⎩ FRDy = − p sin θ d θ dz

4 Results and discussion 4.1 Pre-swirl Ratio Figure 4(a) shows the distribution of circumferential flow velocities in all meshes of the plain annular seal. The average value of the circumferential velocity is shown as the red circle in Fig. 4(b). The circumferential flow velocity vin at the seal inlet was obtained by extrapolation from the average value of the circumferential flow velocity inside the seal.

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Fig. 4. Circumferential velocity distribution in all meshes (a) and average circumferential velocity (red circle) and approximating curve (blue line) (b) (Color figure online)

Fig. 5. Comparison of pre-swirl ratio in plain annular seal, divergent tapered annular seal and previous studies [4, 5]

The pre-swirl ratio obtained for the plain annular seal and the divergent tapered annular seal are shown in Fig. 5.

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In previous studies [4, 5], the pre-swirl ratio for plain annular seal was assumed to be constant at 0.2 irrespective of the rotational speed, as shown by the black line. The pre-swirl ratio increases with increasing rotational speed for both plain annular seal and divergent tapered annular seal. The pre-swirl ratio for the plain annular seal is higher than that for divergent tapered annular seal. This can be attributed to the fact that the circumferential shear force is reduced due to the increased average clearance caused by the taper, which also reduces the circumferential flow velocity. 4.2 Inlet Loss Coefficient The pressure and axial velocity distributions were calculated for the plain annular seal and the divergent tapered annular seal, respectively, and the axial velocity and pressure at the seal inlet and outlet were determined as shown in Figs. 6 and 7.

Fig. 6. Axial velocity distribution in all meshes (a) and average axial velocity (red circle) and approximating line (blue line) (b) (Color figure online)

The inlet loss coefficients obtained for the plain annular seal and the divergent tapered annular seal are shown in Fig. 8. In previous studies [4, 5], the inlet loss coefficient for plain annular seal was assumed to be constant at 0.5 irrespective of the rotational speed, as shown by the black line. For plain annular seal, the inlet loss coefficient increases with increasing rotational speed. On the other hand, the value is almost constant for the divergent tapered annular seal, irrespective of the rotational speed. The inlet loss coefficient for the divergent tapered annular seal is higher. This is due to the fact that, as shown in Fig. 9, the vortex is generated over a wider area at the inlet of the divergent tapered annular seal due to the longer distance between the flow separation and reattachment to the stator wall.

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Fig. 7. Pressure distribution in all meshes (red circle) and approximation line of pressure inside of the seal (blue line) (Color figure online)

Fig. 8. Comparison of inlet loss coefficient in plain annular seal, divergent tapered annular seal and previous studies [4, 5]

4.3 Onset Speed of Instability In the case of the divergent tapered annular seal, the pre-swirl ratio and inlet loss coefficient obtained from the CFD analysis were used in the FSI analysis to obtain the OSI. The RD coefficients were derived using the pre-swirl ratio and inlet loss coefficients obtained from the CFD analysis and the OSI was obtained by eigenvalue analysis. The OSI obtained from the FSI and eigenvalue analysis and the OSI obtained from the experiments [4, 5] are shown in Fig. 10, for the same pre-swirl ratio and inlet loss coefficient as in the previous study and for the pre-swirl ratio and inlet loss coefficient obtained from the CFD analysis, respectively. In the experiments, the OSI decreased with the development of the taper, while the FSI and eigenvalue analyses showed the opposite trend, with the OSI increasing with the taper. The following can be considered as possible causes: in the CFD analysis, the taper angle was assumed to be constant from the seal inlet to the seal outlet, but in reality the

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(a) Plain annular seal

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(b) Divergent tapered seal

Fig. 9. Contour of axial velocity for comparison of distance to reattachment

Fig. 10. Comparison of OSI obtained from FSI analysis, Eigenvalue analysis and experiment

magnitude of the taper angle varied depending on the axial position, as shown in Fig. 2. This may disturb the flow in the gap and cause a pressure loss. The pressure loss may have resulted in a weakening of the fluid film reaction force, which may have affected the OSI. In the FSI analysis, the pre-swirl ratio and loss coefficient were assumed to be constant in the circumferential direction, but in reality, the gap between the rotor and stator at the seal inlet is not constant due to rotor whirl, so these coefficients are not constant in the circumferential direction.

5 Conclusion In this paper, the estimation of the pre-swirl and loss coefficients by CFD analysis is first presented. Furthermore, the FSI analysis for divergent tapered annular seal was extended from the conventional FSI analysis for the plain annular seal. The OSI obtained from the FSI analysis was compared with the OSI obtained from the experiments carried out in previous research and the following conclusions were drawn.

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(1) The pre-swirl ratio increases with increasing rotational speed. The pre-swirl ratio for the plain annular seal is greater than that for the divergent tapered seal. (2) The inlet loss coefficient of the plain annular seal increases with rotational speed, whereas the inlet loss coefficient of the divergent tapered annular seal is almost constant irrespective of the rotational speed. (3) The FSI analysis of the divergent tapered annular taper seal with boundary condition parameters estimated by CFD analysis did not give the same OSI trend for the taper angle as in the experiment. It is necessary to conduct another experiment using a divergent tapered annular taper seal in the future.

References 1. Black, H.F.: Effects of hydraulic forces in annular pressure seals on the vibrations of centrifugal pump rotors. J. Mech. Eng. Sci. 11(2), 206–213 (1969) 2. Nelson, C.C.: Analysis for leakage and rotor dynamic coefficients of surface roughened tapered annular gas seals. ASME J. Eng. Gas Turbines Power 106(4), 927–934 (1984) 3. Ha, T.W., Choe, B.K.: Numerical simulation of rotordynamic coefficients for eccentric annulartype-plain-pump seal using CFD analysis. KSME J. Mech. Sci. Technol. 26(8), 1043–1048 (2012) 4. Miyake, K., Inoue, T., Watanabe, Y.: Two-way coupling fluid–structure interaction analysis and tests of shaft vibration and clearance flow across plain annular seal. ASME J. Appl. Mech. 86, 101002 (2019) 5. Kunori, Y., Inoue, T., Miyake, K.: Two-Way coupled shooting analysis of fluid force in the annular plain seal and vibration of the rotor system. ASME J. Vibr. Acoust. 143, 051006 (2021) 6. Todd Lindsey, W., Childs, D.W.: The effects of converging and diverging axial taper on the rotordynamic coefficients of liquid annular pressure seals: theory versus experiment. ASME J. Vibr. Acoust. 122, 126–131 (2000) 7. Yang, J., Andres, L.S.: On the influence of the entrance section on the rotordynamic performance of a pump seal with uniform clearance a sharp edge versus a round inlet. ASME J. Eng. Gas Turbines Power 141, 031029 (2019) 8. ANSYS Fluent 2019 R3 User’s Guide

Vibration Control of Rotor Bearing Systems Using Electro and Magneto Rheological Elastomers Mnaouar Chouchane(B) and Faiza Sakly Laboratory of Mechanical Engineering, National Engineering School of Monastir, University of Monastir, Monastir, Tunisia [email protected]

Abstract. Electro and magneto rheological elastomers exhibit dynamic stiffness and damping characteristics varying with the applied fields, loading frequencies and strain amplitudes. This paper proposes the use of a smart elastomer ring between the outer shell of the rolling bearing and the bearing housing so that the damping and the stiffness of the bearing can be modified by varying the electric or magnetic field intensity. A semi active control strategy is thus used to set the electric field intensity so that lower steady state vibration responses of the rotor bearing system are obtained. To illustrate the feasibility of the proposed vibration control strategy, a dynamic finite element model is derived for a rotor bearing system by taking into account the gyroscopic effect of the rotor and the internal damping of the shaft. The smart elastomer ring is dynamically modeled based on shear stress strain testing data. The effect of the electric field intensity on the steady state vibration responses due to unbalance excitations is investigated. It has been shown that a reduction of vibration responses can be achieved by varying the intensity of the applied electric field for different rotating speed ranges. Keywords: Vibration control · Rotor bearing system · Electro-rheological elastomers (ERE) · Magneto-rheological elastomers (MRE) · semi-active control

1 Introduction Rotating machines subjected to large excitation forces, near resonance conditions or instability may exhibit undesirable high vibration levels. Different techniques are applied to reduce and control machine vibrations. One of these techniques consists of enhancing damping at the bearings using squeeze film dampers or polymeric elastomer supports. Elastomer supports can be in the form of rings or sectors inserted between the outer shell of the bearing and the bearing housing or in the form of a pad inserted underneath the bearing housing. Several studies showed the efficiency of elastomer supports for vibration reduction when used as a passive mean of vibration control [1–4].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 419–428, 2024. https://doi.org/10.1007/978-3-031-40459-7_29

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Recent advances in active materials have extended the rheological effect observed in rheological fluids (RF) to rheological elastomers (RE). Similarly to rheological fluids, rheological elastomers are composed of polarized particles suspended in a polymeric matrix. The application of an electric or a magnetic field enhances the alignment of the particles in the direction of the field and thus increases the material modulus. The rheological effect varies with the field intensity and is reversible as the material returns to its initial state when the field effect is removed. The field dependent material properties provide a gradual and reversible means to modify the stiffness and damping of a rotor bearing which can be used as a semi active means to control the vibration of rotating machines. The field may be optimally selected for different operating conditions such as different rotating speeds, run up and run down and near stability conditions. In the absence or following the interruption of the field, the elastomer material becomes a passive means for vibration control and should provide adequate vibration control. Design considerations and field varying range of the elastomer properties should guide for the appropriate selection of a magnetic or an electric field and field intensity. Several publications described the fabrication of rheological elastomer material using different types of particles, matrix materials, different particle to matrix weight fractions, and using different fabrication techniques [5, 6]. Samples of the rheological elastomers are often tested under varying magnetic or electric field for different levels of harmonic shear strains and different frequencies to determine its storage and loss modulus [5]. Depending on the shape and geometry of the polymeric support, its stiffness and damping under dynamic load can be estimated and used in models developed for rotor-bearing analysis [1]. Rotor vibration response for different operating conditions under variable field intensities can thus be estimated [7–10]. It has been found that vibration reduction in steady state conditions is possible for some rotating speed ranges. Continuous progress in the fabrication and testing of the rheological elastomers provides research opportunities in the design of elastomer supports, modeling of rotor bearings using elastomer material and for the prediction of rotor-bearing system response at different operating conditions and under different field intensities. Optimal design and field settings can thus be determined for different operating conditions. This paper is organized as follows. Section 2 describes the dynamic modeling of a rotor bearing including a smart elastomer ring and the finite element modeling of the complete rotor-bearing systems. Section 3 presents typical steady state responses of a rotor using electrorheological (ER) and magnetorheological (MR) elastomer rings in the presence and in the absence of an electric or a magnetic field. The paper is concluded in Sect. 4.

2 Dynamic Modeling of a Rotor Bearing System with a Smart Elastomer Ring Insert 2.1 Static and Dynamic Modeling of a Passive Bearing Elastomer Ring Insert The elastomer ring used for vibration damping is typically inserted between the outer shell of the rolling element bearing and the bearing housing as shown in Fig. 1. The elastomer ring has a width b, an inner diameter Di , an outer diameter Do and a thickness i h = D0 −D 2 .

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Fig. 1. A rolling bearing with an elastomer damping ring

The model for the dynamic stiffness and damping of the elastomer ring will be derived based on the following stiffness of a rubber ring subjected to a static radial deformation [1, 11]: πb   (Ec + G)

ks = ln

Do Di

(1)

where Ec is the compression modulus and G is the shear modulus. The compression modulus can be expressed in terms of the elastomer Young’s modulus E and the geometric properties of the ring as follows [11]:   2  b 4 (2) Ec = E 1 + 3 2h Using an approximation for the logarithmic function in Eq. (1), the static stiffness of the ring can be expressed as follows:  β 2 (3) ks = π Dm E (5 + β ) 6 o is the average diameter of the elastomer ring and β is the width where Dm = Di +D 2 to thickness ratio β = bh . It should be noted that the term between braces in Eq. (3) depends only on the nondimensional ratio of geometrical dimensions β. For a passive elastomer under cyclic loading with angular frequency ω, the complex modulus of the elastomer E ∗ can be expressed as follows:

E ∗ (ω) = E  (ω) + j E  (ω) = E  (ω)(1 + j η(ω))

(4)

where E  (ω) is the storage modulus which represents the elastic stiffness of the material, E  (ω) is the loss modulus which represents the viscous damping of the material, and  (ω) η(ω) = EE  (ω) is the loss factor. For a viscoelastic material, the storage and loss modulus depend also on the strain amplitude and temperature in addition to the loading frequency. Under a dynamics loading of frequency ω, the elastomer ring can be modeled by a spring of stiffness kd (ω) and a viscous damper with damping coefficient cd (ω) [1]. The

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dynamic stiffness kd (ω) is found by replacing the modulus E in Eq. (3) by the storage modulus E  (ω). Thus,  β  2 (5) kd (ω) = π Dm E (ω) (5 + β ) 6 Similarly, the dynamic damping under cyclic loading cd (ω) is found by replacing E  in Eq. (3) by E ω(ω) . Thus, cd (ω) = π Dm

E  (ω) ω



β η(ω) (5 + β 2 ) = kd (ω) 6 ω

(6)

2.2 Dynamic Modeling of the Rheological Elastomer Ring Equations (5) and (6) require the knowledge of the complex modulus E ∗ (ω) = E  (ω) + j E  (ω). However, testing of the material under a cyclic load and a controlled magnetic or electric field is often carried out in the shear mode. Thus, the complex modulus E ∗ (ω) needs to be estimated from the shear modulus G ∗ (ω). For a viscoelastic material, the complex shear stress τ ∗ is related to the complex shear strain γ ∗ , for a given temperature and strain amplitude, as follows: τ ∗ (ω) = (G  (ω) + j G  (ω)) γ ∗

(7)

where G  and G  are respectively the shear storage and the shear loss modulus which depend on temperature T , the strain amplitude and loading frequency ω. Using a Poisson ratio ν ≈ 0.5 for the elastomeric material and using the isotropic relationship between the shear and elasticity modulus, the following approximation is used to estimate the storage and loss modulus required to estimate the dynamic stiffness and damping in Eq. (5) and (6): E ∗ (ω) = E  (ω) + jE  (ω) = 3G ∗ (ω) = 3G  (ω) + j3G  (ω)

(8)

Equations (5) and (6) can be used to form a complex stiffness for the elastomer ring in terms of G ∗ (ω) as follows:  β (5 + β 2 ) (9) kd∗ (ω) = 3π Dm G ∗ (ω) 6

2.3 Modeling of a Bearing with an ERE/MRE Ring Insert When the method of finite elements is applied to model a rotor bearing system with a bearing containing smart elastomer rings, an additional degree of freedom is required for each bearing which may complicate the modeling. To avoid the addition of a degree of freedom, Ribeiro et al. [4] suggested the use of a complex equivalent bearing stiffness which accounts for the dynamic stiffness and damping of the rolling bearing and the

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elastomer ring as well as the shell mass. This complex stiffness can be used for steady state response to harmonic excitation forces. For a bearing composed of a rolling bearing with an outer shell of mass mb and of negligible damping and a radial stiffness k, the equivalent complex dynamic stiffness of the rolling bearing including the ER/MR elastomer ring can be expressed as follows [4]: ∗ =k− keq

k

+ kd∗

k2 − mb ω2

(10)

The dynamic characteristics of the ER/MR elastomer depend on the angular excitation frequency ω which is assumed to be equal to the rotating speed of the rotor bearing system [6, 7]. Furthermore, The ER and MR elastomers used for rotor vibration control are characterized by a linear viscoelastic behavior, and are assumed to be approximately constant ambient temperature [7]. 2.4 Modeling of a Rotor Bearing System Using Electrorheological (ER) and Mangetorheological (MR) Elastomer Ring Supports Figure 2 shows a schematic view of a rotor system comprising a flexible shaft supported by two rolling bearings including ER of MR elastomer rings. A set of unbalanced disks are mounted on the shaft at selected locations.

Fig. 2. Schematic view of a rotor bearing system using ER/MR elastomer rings

Based on the finite element method, the rotor shaft is discretized into (n-1) finite beam elements and n nodes with four degrees of freedom per node: two lateral displacements yi and zi and two rotations θ i and ψi . Figure 3 shows a beam element located between the node i and the node i + 1 and the degrees of freedom at each node. The displacement vector (the vector of degrees of freedom) can thus be formed as follows: T

{u}= y1 z1 θ1 ψ1 ... yi zi θi ψi ... yn zn θn ψn (11) The equations of motion of the rotor bearing system at a rotating speed ω can be written in the following matrix form: [M ]{¨u} + ([C0 ] + [Cb ] + ω[G]){˙u} + ([K0 ] + [Kb ] + ηv ω[Kc ]){u} = {F(t)}

(12)

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Fig. 3. Degrees of freedom for a beam element.

where [M ] is the mass matrix including contributions from the shaft elements and the disk elements, [G] is the gyroscopic matrix of the shaft and the disks; [Cb ] and [Kb ] are respectively the damping and the stiffness matrices of the bearings; [Kc ] is the global circulation matrix; [K0 ] is the stiffness matrix of the shaft and [C0 ] is the damping matrix of the shaft which is assumed to be proportional to [K0 ]. Thus, [C0 ] = ηv [K0 ]

(13)

where ηv is the viscous damping coefficient. All the square matrices presented in Eq. (12) are of size 4n × 4n. The vector {F(t)} of size 4n × 1, represents the unbalance forces vector of the rotor system with nd disks which is expressed as follows: {F(t)} =

nd

{ek } mdk εk ω2 ej(ωt+αk )

(14)

k=1

where mdk , εk and αk are respectively the mass, the eccentricity and the phase angle of the disk located at node k. Vector {ek } is a (4n × 1) vector indicating the degree of freedom where the unbalance forces acting in the y and z directions. For an unbalance force applied on a disk dk fixed at the node i, vectors {ek } has only two non-zero components equal, respectively, to 1 and -j for the 4(i - 1) + 1) and (4(i - 1) + 2) components. Due to the incorporation of the ER/MR elastomer ring in the bearing, the damping matrix of the bearing [Cb ] is set to zero and the stiffness matrix of the bearing [Kb ] in Eq. (12) is replaced by the complex stiffness matrix of the bearings using an ER/MR elastomer ring. The four degrees of freedom associated with a bearing node are set to the following matrix: ⎡ ⎤ ∗ 0 00 keq ∗ 0 0⎥

∗ ⎢ ⎢ 0 keq ⎥ (15) Kb = ⎢ ⎥ ⎣ 0 0 0 0⎦ 0 0 00 ∗ is defined in Eq. 9. where keq

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3 Effect of the ER/MR Field on the Rotor-Bearing System Response The effect of using ER elastomer ring inserted between the outer shell of the ball bearing and the housing on the steady state response of the rotor under varying electrical field has been investigated for a simple Jeffcott rotor in references [7, 8] and for a bi-disk rotor in reference [9] using simulations based on a finite element model for the rotor-bearing system. The Jeffcott rotor is a rotor with a flexible shaft and a single disk located at the middle of the shaft whereas a bi-disc rotor contains two symmetrical disks fixed at two different locations of the shaft. In the research reported in references [7–9], it has been found that the use of an elastomer ring in passive mode reduces significantly the vibration amplitude at resonance speeds due to the added damping and, in addition, it reduces the resonance frequencies due to the added flexibility. The gradual increase of the electric field gradually increases the resonance frequencies and the resonance amplitudes because of the increase of the overall stiffness and decrease of damping of the bearing. However, lower vibration amplitudes are observed away from resonances under a supplied electric field compared to the case of absence of the field as shown in Fig. 4 for a bi-disk rotor. This makes the application of an electric field beneficial for steady operation at rotor speeds away from resonances.

Fig. 4. Vibration amplitude of a rotor system using an ER elastomer for different levels of the electric field [9]

The use of a rheological elastomer ring bearing insert subjected to a magnetic field has also been investigated by the authors [10] for the steady state response of a single disk rotor for an elastomer fabricated with added silicon oil plasticizer as described in reference [6]. The plasticizer is used to enhance particle alignment during the application of the magnetic field and thus generate a more pronounced effect of the magnetic field on the material properties. In the passive mode, the elastomer ring with the added silicone oil plasticizer provides significant reduction of the resonance amplitude and a reduction of the resonance frequency due to the higher damping and lower stiffness as shown in Fig. 5.

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Under a magnetic field the resonance amplitude and the resonance frequency increase but lower vibration amplitudes are observed at frequencies away from the resonance frequency. The addition of the oil plasticizer enhances the field effect on the dynamic characteristics of the elastomer ring and thus on rotor response. To benefit from lower vibration amplitude for different frequency ranges when the magnetic field is applied, an On-Off control strategy can be applied as shown in Fig. 6. For the simulated case in Fig. 5, the elastomer ring is subjected to a magnetic field of intensity 0.6 T to reduce the steady state vibration response for operating speeds lower than approximately 2800 rpm. For higher rotor speeds, the magnetic field is removed making the elastomer ring a passive control device. Significant reduction of rotor vibration can thus be achieved for an extended rotor speed range.

Fig. 5. Steady-state response at the disk position for the uncontrolled and controlled rotor system [10]

Fig. 6. Effect of the on–off control strategy on the disk vibration amplitude

In this paper, simulation results are presented only for the steady state rotor response as illustrated in Figs. 4, 5, and 6. Further research should consider the transient response during speed up and speed down and stability analysis. Experimental investigations are

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also needed to confirm the theoretical findings and to account for the variety of rheological elastomer fabrication and the design of damping devices. This paper discussed only the use of a single elastomer ring inserted in the bearing, elastomer pads can also be used for vibration control at the bearings as well as the use of a combination of rings and pads and the use of multiple concentric elastomer rings. Furthermore, future researches should consider the design constraints for using an elastomer ring subjected to an electrical or a magnetic field. Dividing the ring into sectors may be a more feasible design in this case. In addition, the use of multi-sectors can provide more control flexibility as the sectors to be subjected to the electric or a magnetic field can be selected depending on the desired stiffness and damping to be provided by the elastomer sectors. The sectors can also be structured into two or more concentric rings separated by metal shells, thus providing more control of the bearing dynamic properties. The use of semi active control strategy relies on the supply of electric power for the electric field. For the magnetic field, it can be generated by permanent magnets or electromagnets. When electric power is used, it is desirable to make the bearing independent of the external electric supply by using batteries or possibly power harvested from the rotor or bearing vibrations. The use of autonomous electric power supply to the bearing can only be used for lower power devices.

4 Conclusion Finite element modelling has been used to predict the steady state response of rotors supported by bearings containing an electro or a magneto-rheological elastomer ring inserted between the outer shell of the ball bearing and the bearing housing. The steady state response of the rotor is analysed under gradual application of an electric or a magnetic field. It has been shown that the presence of the elastomer ring in passive mode reduces significantly the vibration amplitude at the resonance frequencies and also reduces the resonance frequencies because of the added damping and flexibility to the bearing. The gradual increase of the electric or magnetic field increases the vibration amplitude at the resonance frequencies and increases the resonance frequencies but lower vibration amplitudes occur away from resonance frequencies. In this case, the operation in the presence of the field is beneficial at operating speeds far from resonances. An On-Off control strategy has been shown to be worth to be considered as lower vibration amplitudes can be obtained in the presence or absence of the electric or magnetic field depending on the rotor operating speed range.

References 1. Liebich, R., Scholz, A, Wieschalla, A.: Rotors supported by elastomer-ring-dampers – experimental and numerical investigations. In: 10th International Conference on Vibrations in Rotating Machinery, pp. 443–453. Woodhead Publishing, London (2012) 2. Dutt, J.K., Toi, T.: Rotor vibration reduction with polymeric sectors. J. Sound Vib. 262(2003), 769–793 (2003)

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3. Bavastri, C.A., Ferreira, E.M.D.S., De Espíndola, J.J., Lopes, E.M.D.O.: Modeling of dynamic rotors with flexible bearings due to the use of viscoelastic materials. J. Braz. Soc. Mech. Sci. Eng. 30(1), 22–29 (2008). https://doi.org/10.1590/S1678-58782008000100004 4. Ribeiro, E.A., Pereira, J.T., Alberto Bavastri, C.: Passive vibration control in rotor dynamics: Optimization of composed support using viscoelastic materials. J. Sound Vibr. 351, 43–56 (2015). https://doi.org/10.1016/j.jsv.2015.04.007 5. Li, W.H., Zhou, Y., Tian, T.F.: Viscoelastic properties of MR elastomers under harmonic loading. Rheol. Acta 49(7), 733–740 (2010). https://doi.org/10.1007/s00397-010-0446-9 6. Khairi, M.H.A., et al.: Enhancement of particle alignment using silicone oil plasticizer and its effects on the field-dependent properties of magnetorheological elastomers. Int. J. Mol. Sci. 20(17), 4085 (2019). https://doi.org/10.3390/ijms20174085 7. AL Rkabi, M., Moeenfard, H., Rezaeepazhand, J.: Vibration attenuation of rotor-bearing systems using smart electro-rheological elastomer supports. J. Braz. Soc. Mech. Sci. Eng. 41(6), 1–17 (2019). https://doi.org/10.1007/s40430-019-1748-1 8. Sakly, F., Chouchane, M.: Vibration control of a rotor using smart bearings with magnetorheological elastomer supports. In: Bouraoui, T., et al. (eds.) CoTuMe 2021. LNME, pp. 376– 382. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-86446-0_50 9. Sakly, F., Chouchane, M.: Vibration control of a bi-disk rotor using electro-rheological elastomers. Smart Mater. Struct. 31(6), 065009 (2022). https://doi.org/10.1088/1361-665X/ ac691a 10. Sakly, F., Chouchane, M.: Vibration analysis and control of a rotor-bearing system using a magneto-rheological elastomer containing silicone oil plasticizer. In: Walha, L., et al. (eds.) Design and Modeling of Mechanical Systems – V: Proceedings of the 9th Conference on Design and Modeling of Mechanical Systems, CMSM 2021. Lecture Notes in Mechanical Engineering, pp. 269–275. Springer International Publishing, Cham (2023). https://doi.org/ 10.1007/978-3-031-14615-2_31 11. Freakly, P.K., Payne, A.R.: Theory and Practice of Engineering with Rubber. Applied Science publication, London (1978)

A New Type of Inerter Nonlinear Energy Sink Using Chiral Metamaterials Hui Li, Hongliang Yao(B) , and Yangjun Wu Northeastern University, Shenyang 110819, People’s Republic of China [email protected]

Abstract. To reduce size and weight of the inertial mass required in traditional nonlinear energy sink (NES), a chiral metamaterials inerter nonlinear energy sink (CINES) for suppressing the torsional vibration of the rotor system is developed in this paper. The INES is a combination of a NES with piecewise linear stiffness and an inerter utilizing the compressive-torsional coupling effect of chiral metamaterials. The structure of the CINES is introduced, the inerter mechanism is analyzed and the dynamic model of the CINES-rotor system is built. Vibration attenuation performance of CINES on the rotor system is evaluated by the transient torsional vibration and steady-state torsional vibration. By using the inerter, a significant damping effect also appears. Keywords: Chiral inerter nonlinear energy sink · piecewise linear stiffness · torsional vibration suppression · genetic algorithm

1 Introduction A common method to dampen harmful vibrations in rotating machinery is to add dynamic vibration absorbers (DVAs), which are also known as tuned mass dampers (TMDs) [1– 3]. A common feature of DVAs is that they rely on linear resonance, which narrows their effective frequency band, so they need to be manipulated in real time through complex structures [4]. In order to obtain better damping effect, widen effective bandwidth, and improve robustness of DVA or TMD, the commonly used methods are to conduct optimization design studies or to use multiple damping devices simultaneously [5]. Likewise, nonlinear vibration reduction methods are often used to replace traditional vibration dampers, where nonlinear energy sinks (NESs) have attracted particular interest from researchers [6]. NESs usually are the small mass attachments with nonlinear parameters [7]. The small mass can lead a unidirectional energy flow from the primary system towards the NES that is called targeted energy transfer (TET) or energy pumping [8], in this process the vibration energy of the main system will be captured, modulated, absorbed or dissipated. TET and strongly modulated response (SMR) behaviors are the mechanisms by which NES suppresses transient and steady-state vibrations. When TET or SMR occurs, the NES can adjust its characteristics according to the main system response. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 429–438, 2024. https://doi.org/10.1007/978-3-031-40459-7_30

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Bergeot et al. [9]performed a steady-state response study by using the numerical analysis about the capacity of a NES to control a helicopter ground resonance. The inerter can provide inertial parameters much larger than its own mass to reduce the mass required for the system. The inerter is widely used in vibration absorption in engineering fields, such as in transportation [10] construction and bridges, etc. Inspired by the effectiveness of the inerter on TMDs, researchers combined inerter with NES to provide the large inertial parameters required, reduce the mass needed and enhance the vibration suppression. Zhang et al. [11]demonstrated the NES-inerter compared to the traditional NES is more effective in vibration suppression. Ref. [12]designed an inerter NES which uses an inerter to replace the NES mass and gave a series of effectiveness and superiority analyses. Javidialesaadi et al. [13] that the use of inerter can be an effective way to improve the control performance of NES passive structures. NESinerter reduces RMS response better than TMDI. Compared with the original spring and damping parts, the inerter usually requires more complex transmission mechanisms and cannot be applied to vibration reduction occasions with high size requirements such as precision instruments and aerospace. Recently, the concept of chirality was introduced into the design of mechanical metamaterial [14–16]. The combination of compressiontorsion coupling effect of chiral metamaterials and inerter structure has not been applied [17].

2 Design of CINES 2.1 Chiral Metamaterial Inerter Mechanism The structure of chiral metamaterial is depicted in Fig. 1(a). Figure 1(b) shows the simplified theoretical model of the structure. The relation between torsional motion φ and displacement x is expressed as x =

2π rnφ tanθ

Fig. 1. Chiral metamaterial: (a) structure, (b) geometrical relationship.

(1)

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The relationships between the θ n and the x n , as well as the θ i and the x n , are expressed as follows 2π r1 n1 θn tan θc1 2π r2 n2 xn = θi tan θc2 xn =

(2)

In order to achieve the amplification mechanism, the input and output of the chiral materials must be restricted to only one direction. As illustrated in Fig. 2, a compliant mechanism is constructed that includes rotational and translatory restraints. By constraining the compliant mechanism, the kinematic condition for the inerter amplification mechanism is obtained. The upper vibrator plate of CM1 is internally connected with a moving restraint that prevents the transverse movement of the connected disk. The spiral connecting rod allows it to rotate only around the z-axis. The lower vibrator plate is internally connected with a moving restraint that generates translational motion along the z-axis. The inner ring of the compliant mechanism is fixed.

Fig. 2. Compliant mechanism: (a) rotational restraints, (b) translatory restraints, (c) kinematic condition.

2.2 Piecewise Linear NES In Fig. 3(a), the piecewise linear element of the NES consists of one connection beam d 0 and three piecewise linear stiffness beams d i (i = 1, 2, 3) with increasing diameters. And the piecewise stiffness beams have different angular clearances ei (i = 1, 2, 3) rad with the NES mass. As the amplitude increases, the piecewise linear stiffness beam contacts the NES mass successively, the stiffness increases piecewise linearly, and elastic recovery force presents piecewise nonlinear characteristics. As shown in Fig. 3(b), the cubic stiffness can be fitted after appropriate parameters are selected to effectively suppress the vibration of the rotor system. 2.3 Structure of CINES In Fig. 4, the CINES is designed which consists of a piecewise linear NES part and a chiral metamaterial-inerter part. The NES part includes a NES mass, a bearing, elastic beams, and a support. The NES mass is connected to the support through the connecting

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Fig. 3. Schematic of piecewise linear stiffness: (a) angular clearances, (b) stiffness fitting.

beam. The inerter part consists of a NES mass, an inerter mass, two chiral metamaterials and the compliant joints. As the present study mainly focuses on the central substructure of the CINES, the compliant joints are neglected in the subsequent analysis.

Fig. 4. Schematic diagram of CINES.

3 Dynamic Model of the Rotor-CINES System 3.1 Model Analysis The rotor system is presented in Fig. 5. The torsional stiffnesses of the couplings are kc1 and kc2 , respectively. The motor at the left end provides rotational speed. NES’s support is fixed near the disk so that θd can be replaced by the torsional motion angle of the NES’s support. The piecewise linear torque of the NES is ⎧ ⎪ ⎪ ⎨

kn1 |θ | (e0 < |θ | ≤ e1 ) kn1 e1 + kn2 (|θ | − e1 ) (e1 < |θ | ≤ e2 ) Tn (θ ) = ⎪ k e + kn2 (e2 − e1 ) + kn3 (|θ | − e2 ) (e2 < |θ | ≤ e3 ) n1 1 ⎪ ⎩ kn1 e1 + kn2 (e2 − e1 ) + kn3 (e3 − e2 ) + kn4 (|θ | − e3 ) (|θ | > e3 )

(3)

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Fig. 5. Rotor system model.

where, θ =θd − θn , kni is the equivalent torsional stiffness of the ith piecewise linear beam. Now, the piecewise linear torsional stiffness of the NES can be obtained as ⎧ ⎪ ⎪ ⎨

kn1 (e0 < |θ | ≤ e1 ) kn1 + kn2 (e1 < |θ | ≤ e2 ) kn = ⎪ k + kn2 + kn3 (e2 < |θ | ≤ e3 ) n1 ⎪ ⎩ kn1 + kn2 + kn3 + kn4 (|θ | > e3 )

(4)

3.2 Dynamic Model Establishment The designed CINES structure is applied to the rotor system for torsional vibration suppression. As illustrated in Fig. 6, the dynamic model of the rotor-CINES system is established. According to Newton’s second law, the governing equations of the rotorCINES system are    Jd θ¨d + cd θ˙d + kd θd +cn θ˙d −θ˙n + Tn (θ ) = T (5) (Jn + Ji b)θ¨n − cn θ˙d − θ˙n − Tn (θ ) = 0

Fig. 6. Dynamic models of the rotor-CINES system.

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4 Simulations and Discussions 4.1 Parameters Definitions The rotor system under consideration consists of a disk with a diameter and thickness of 120 mm and 40 mm, respectively, and a shaft with a diameter of 10 mm. The shaft is divided into three segments by couplings with lengths of 100 mm, 300 mm, and 250 mm, respectively. The stiffness parameters of the couplings are listed in Table 1, and the material used in the rotor system is steel. The inertial mass rotational inertia is consistent with the NES mass. The material of the elastic beams is steel. The material of the NES mass and the inerter mass is polylactic acid (PLA). The parameters of the CINES will be determined in Sects. 4.2 and 4.3. Table 1. Couplings parameters. Coupling i

1

2

k ci (N.m/rad)

350

400

4.2 Transient Torsional Vibration Under the initial velocity, θd with the locked CINES displays damped vibration, with a slight decrease in amplitude due to the presence of damping, as well as a slight decrease in initial energy, as shown in Fig. 7(a). In Fig. 7(b), the addition of active CINES reduced the time needed for the rotor system’s amplitude to decay to 10% of initial value from 3 s to 0.95 s, demonstrating the system’s capacity to quickly dissipate the initial input energy. Figure 8 are the corresponding WT spectra of the rotor-CINES system.

Fig. 7. Time domain response: (a) with locked CINES, (b) with active CINES.

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Fig. 8. WT spectra: (a) with locked CINES, (b) with active CINES.

4.3 Steady-State Torsional Vibration As shown in Fig. 9. The resonance frequency of 1st-order torsional vibration critical speed of the rotor system with the locked INES is about 16.35 Hz, and the peak amplitudes of θd are about 2.28°. After adding the active CINES, θd in the resonance area are greatly suppressed where the maximum θd is about 0.27°. In the resonance range, the responses of θd − θn (|θ |) and θn indicates that CINES only has the 1st linear torsional stiffness kn1 and 2nd linear torsional stiffness (kn1 + kn2 ) participate in vibration reduction.

Fig. 9. Frequency domain responses of rotor-CINES system: (a) θd , (b) θn , (c) θd − θn .

5 Experimental Validation and Discussion 5.1 Experimental Setup An experimental apparatus is constructed as illustrated in Fig. 10 in order to confirm the capacity of the suggested CINES to suppress the torsional vibration of the rotor system. This experiment involves a speed tracking measurement and a constant speed measurement. For the first one, the motor speed is increased from 60 rpm to 1500 rpm. For the latter one, the torsional vibration of 16.5 Hz (990 rpm) is recorded that is in the first-order torsional critical speed resonance band. In speed tracking, Fig. 11 are the instantaneous angular displacement curves of θd with the locked CINES, when attaching the locked CINES, θd reaches the resonance frequency of the 1st-order torsional vibration critical speed with the approximately 2.2°

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Fig. 10. Test platform and structure of CINES.

Fig. 11. Instantaneous angular displacement curves of θ d at speed tracking.

resonance peak. CINES smoothly suppresses θd and produces obvious beating in which the maximum amplitude is about 0.78°. Figure 12 are the 3D spectrograms of θ d in the process of speed tracking. In Fig. 12(a), there is a narrow bright band starts from about 16 Hz on the frequency axis which denotes the resonance frequency of the 1st-order torsional vibration critical speed. After adding CINES, there is no obvious vertical bright band at 16 Hz, which indicates the resonance peak is suppressed and the resonance region is broadened. In Fig. 12(b), the responses of Order 1 can be suppressed by the INES.

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Fig. 12. Color maps of θ d at speed tracking. (a) with locked CINES, (b) with active CINES (Color figure online).

6 Conclusions In this paper, a CINES is proposed for suppressing the torsional vibration of the rotor system. The inerter uses CTC effect of the chiral metamaterial to amplify the rotary inertia of the NES in an adjustable way. Through the analyses of this paper, the following conclusions can be obtained. (1) The CINES developed which implements the inerter mechanism using the principle of CTC effect of the chiral metamaterial can well suppress torsional vibrations of rotor systems, both transient and steady-state. (2) When inhibiting the transient torsional vibration of the rotor system, CINES reduced the time needed for the rotor system’s amplitude to decay to 10% of initial value from 3 s to 0.95°. The steady-state vibration elimination of CINES is 88.5% in simulation and is 61.3% in test. This also illustrates that suppression abilities of torsional vibrations. (3) By introducing the inerter into the NES, it can be found that only a small amount of inertial mass can obtain a high level of vibration damping for the torsional vibration of the rotor system. In other words, this feature of the NES with inerter may make it an attractive potential alternative to the traditional NES. 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. Yang, Y., Dai, W., Liu, Q.: Design and implementation of two-degree-of-freedom tuned mass damper in milling vibration mitigation. J. Sound Vib. 335, 78–88 (2015). https://doi.org/10. 1016/j.jsv.2014.09.032

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2. Yao, H., Wang, T., Wen, B., Qiu, B.: A tunable dynamic vibration absorber for unbalanced rotor system. J. Mech. Sci. Technol. 32(4), 1519–1528 (2018). https://doi.org/10.1007/s12 206-018-0305-7 3. Yang, F., Sedaghati, R., Esmailzadeh, E.: Vibration suppression of structures using tuned mass damper technology: a state-of-the-art review. J. Vib. Control 28(7–8), 812–836 (2022). https://doi.org/10.1177/1077546320984305 4. Wang, Q., et al.: Dual-function quasi-zero-stiffness dynamic vibration absorber: lowfrequency vibration mitigation and energy harvesting. Appl. Math. Model. 116, 636–654 (2023). https://doi.org/10.1016/j.apm.2022.12.007 5. Yang, Y., Chen, G., Ouyang, H., Yang, Y., Cao, D.: Nonlinear vibration mitigation of a rotorcasing system subjected to imbalance–looseness–rub coupled fault. Int. J. Non-Linear Mech. 122, 103467 (2020). https://doi.org/10.1016/j.ijnonlinmec.2020.103467 6. Yao, H., Cao, Y., Ding, Z., Wen, B.: Using grounded nonlinear energy sinks to suppress lateral vibration in rotor systems. Mech. Syst. Signal Process. 124, 237–253 (2019). https://doi.org/ 10.1016/j.ymssp.2019.01.054 7. Dou, J., Li, Z., Cao, Y., Yao, H., Bai, R.: Magnet based bi-stable nonlinear energy sink for torsional vibration suppression of rotor system. Mech. Syst. Signal Process. 186, 109859 (2023). https://doi.org/10.1016/j.ymssp.2022.109859 8. McFarland, D., Bergman, L., Vakakis, A.: Experimental study of non-linear energy pumping occurring at a single fast frequency. Int. J. Non-Linear Mech. 40, 891–899 (2005). https://doi. org/10.1016/j.ijnonlinmec.2004.11.001 9. Bergeot, B., Bellizzi, S., Cochelin, B.: Analysis of steady-state response regimes of a helicopter ground resonance model including a non-linear energy sink attachment. Int. J. Non-Linear Mech. 78, 72–89 (2016). https://doi.org/10.1016/j.ijnonlinmec.2015.10.006 10. Shen, Y., Chen, L., Yang, X., Shi, D., Yang, J.: Improved design of dynamic vibration absorber by using the inerter and its application in vehicle suspension. J. Sound Vib. 361, 148–158 (2016). https://doi.org/10.1016/j.jsv.2015.06.045 11. Chen, Y., Tai, Y., Xu, J., Xu, X., Chen, N.: Vibration analysis of a 1-DOF system coupled with a nonlinear energy sink with a fractional order inerter. Sensors 22(17), 6408 (2022). https:// doi.org/10.3390/s22176408 12. Chen, H.Y., Mao, X.Y., Ding, H., Chen, L.Q.: Elimination of multimode resonances of composite plate by inertial nonlinear energy sinks. Mech. Syst. Signal Process. 135, 106383 (2020). https://doi.org/10.1016/j.ymssp.2019.106383 13. Javidialesaadi, A., Wierschem, N.E.: An inerter-enhanced nonlinear energy sink. Mech. Syst. Signal Process. 129, 449–454 (2019). https://doi.org/10.1016/j.ymssp.2019.04.047 14. Bergamini, A., et al.: Tacticity in chiral phononic crystals. Nat. Commun. 10(1), 4525 (2019). https://doi.org/10.1038/s41467-019-12587-7 15. Fernandez-Corbaton, I., et al.: New twists of 3D chiral metamaterials. Adv. Mater. 7 (2019) 16. Feng, X., Jing, X., Xu, Z., Guo, Y.: Bio-inspired anti-vibration with nonlinear inertia coupling. Mech. Syst. Signal Process. 124, 562–595 (2019). https://doi.org/10.1016/j.ymssp. 2019.02.001 17. Zheng, B.B., Zhong, R.C., Chen, X., Fu, M.H., Hu, L.L.: A novel metamaterial with tensiontorsion coupling effect. Mater. Des. 171, 107700 (2019). https://doi.org/10.1016/j.matdes. 2019.107700

Vibration Characteristic Analysis and Optimization of the Propulsion Shaft in the Underwater Vehicle Yuchen An1,2 , Jing Liu1,2(B) , Chiye Yang1,2 , and Guang Pan1,2 1 School of Marine Science and Technology, Northwestern Polytechnical University,

Xi’an 710072, People’s Republic of China [email protected], [email protected] 2 Laboratory for Unmanned Underwater Vehicle, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China

Abstract. The propulsion shaft is the main part of the underwater vehicle propulsion system, the vibration of the propulsion shaft has the great influence on the noise, stealth and performance of the underwater vehicle. It is necessary to analyze and optimize the vibration characteristic of the propulsion shaft in the underwater vehicle. This issue establishes the finite element model of the propulsion shaft system include the motor, shaft and the support. The vibration mode of the shaft is analyzed. To reduce the vibration, an optimization method to the propulsion shaft system is presented by using the rubber isolations. The parameters of the rubber isolations are selected in this paper. The harmonic response analysis of the propulsion shaft with the rubber isolations is conducted, and the characteristic frequencies of the rolling bearings of the propulsion shaft system are also considered. By using the rubber isolation on the shaft system, the vibration amplitudes reduced in the range of 75.4% to 89.7% at the different parts. This work can provide some guidance to the design of the propulsion shaft of the underwater vehicle. Keywords: Vibration characteristic · propulsion shaft · rubber isolation · harmonic response analysis

1 Introduction The propulsion shaft system is important to the power transmission of the underwater vehicle (UV). It has significant effects on the noise, vibration and stealth performance of the UV. It is necessary to conduct a study on the vibration characteristics and the optimization method of the propulsion shaft system. Many reports have been studied the vibrations and isolation methods for the propulsion shaft systems in the ships and underwater vehicles. Zou et al. [1, 2] conducted series studies on the coupled longitudinal-transverse vibration of the marine propulsion shaft system. Zhang et al. [3] studied the influence of the support structures on the vertical and longitudinal forces of the propeller, as well as the bending vibration of the propulsion shafting. Kim et al. [4] studied the whirling vibration and shaft alignment © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 439–451, 2024. https://doi.org/10.1007/978-3-031-40459-7_31

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of the propulsion shaft system with single and double stern bearings. The effects of the hull deformation on the shaft system were analyzed. Hu et al. [5] changed the vibration transmission path by installing a resonance charger in the thrust bearing of a marine propulsion shafting, the isolation performance was improved by change the resonance charger parameters. Xu et al. [6] established a vibration analysis model of a ship propulsion shaft system. The model considered the influence of the thrust force. The vibration energy transmission behavior was studied by using the power flow analysis. Liu et al. [7] presented an optimization algorithm for the rubber isolators of the propulsion system. Jee et al. [8] presented a viscous-spring damper to control the torsional vibration of the propulsion shaft system. The optimum stiffness and damping coefficients of the damper was obtained. Zhang et al. [9] studied the effect of thrust bearing location on vibration characteristics by using the transfer matrix method. Zhang et al. [10] conducted the numerical and finite elements analysis on vibration characteristic of the propulsion shaft system. The hull deformation excitations was considered. Zambon et al. [11] conducted the numerical and experiment study for the marine shaftline vibrations. The numerical model considered the diesel engine. Liu et al. [12] presented a semi-active dynamic vibration absorber for the propulsion shafting vibration. Huang et al. [13] conducted the critical speed analysis, harmonic analysis and transient analysis of the coupled transverse and longitudinal vibration of the propulsion shaft system by using the numerical simulations. The results were verified by using the finite element analysis and experiments. Above works focused on the propulsion shaft systems in ships. However, the propulsion shaft systems of the UVs are different from that of the marines. The propulsion shaft systems of the UVs have the smaller sizes, higher speeds, lighter loads and different support structures. Therefore, it is necessary to conduct a study for the propulsion shafting in the underwater vehicles. This paper conducts a vibration analysis about the propulsion shaft system of the UVs by using the finite element model. The model considers the installation lugs, motor, propulsion shaft, bearings, bearing support and a simplified propeller. The vibration modal analysis of the propulsion shaft is studied. To reduce the vibration, the rubber isolators are used in the paper. The sizes and material hardness of the rubber isolators are optimized through the harmonic analysis. Compared with the dynamic responses of the propulsion shaft system without the isolations, the vibrations are effectively reduced by using the presented method.

2 Model Description 2.1 Scientific Problem The schematic of the power and propulsion section of the UV is given in Fig. 1. The main vibration source is the propulsion shaft system. The radiated noise of the UV is mainly generated by the vibrations transmitted from the propulsion shaft system to the shell structure. The structures of the propulsion shaft, supports, and electric motor are fixed. The vibrations transmitted to the shell can only be reduced through the transmission path by using the rubber isolators. The vibration transmission path is given in Fig. 1. The propeller vibrations transmitted to the shell through the thrust bearing, propulsion shaft- support, and propulsion shaft- motor-isolators. The propulsion shaft vibrations

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Coupling Motor

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Thrust bearing Propulsion shaft

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Shaft vibration transmission path Motor vibration transmission path Propeller vibration transmission path Fig. 1. Schematic of the structure and vibration transmission path of the power and propulsion section in an UV.

transmitted to the shell through the support, thrust bearing and motor-isolators. The motor vibrations transmitted to the shell through the motor isolators, shaft-support and shaft thrust bearing. Due to the limitation of the shafting alignment, the rubber isolators are installed between the motor and shell. However, the frequency responses of the propulsion shaft system will be changed by the isolators. It is necessary to conduct the vibration characteristic analysis for the propulsion shaft system. 2.2 Analyze Model of the Propulsion Shaft System The geometry of the propulsion shaft system is shown in Fig. 2(a). The motor is mounted at the shell through the installation lugs. The rubber isolators are installed between the lugs and the motor. The motor shaft is connected with the propulsion shaft through the coupling, which can be modelled as a rigid connection in the model. The propeller is simplified as an equivalent disk. The propeller is fixed at the end of the propulsion shaft. The propulsion shaft is supported by the ball bearings and support structure. The schematic of the studied system is given in Fig. 2(b), the dynamic model of the propulsion shaft system is (Ms + Mm + Mp )q¨ + (Cs + Ciso + Cb )q˙ + (Ks + Kiso )q = Fb + F

(1)

where, Ms , Mm and Mp are the mass matrix of the flexible shaft, motor and propeller, Cs , Ciso and Cb are the damping matrix of the shaft, rubber isolator and bearings. Ks and Kiso are the stiffness matrix of the shaft and isolators. q = [x, y, z, θ x , θ y , θ z ]’ is the generalized displacement vector of the system, q˙ and q¨ are the velocity and acceleration vector of the system, which can be expressed as the first and second derivatives of q. Directions x, y and z indicates the axial, vertical and horizontal directions. The detail

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formulations of Eq. (1) can be derived according to the Ref. [14, 15], F is the excitation force vector, Fb is the bearing forces, which can be given by ⎧ N  ⎪ ⎪ ⎪ F = K βi δin cos θi ⎪ bx T ⎪ ⎨ i=1 (2) N ⎪  ⎪ ⎪ ⎪ βi δin sin θi ⎪ ⎩ Fby = KT i=1

where, KT is the contact stiffness of between the roller and raceway, which can be calculated through the Hertizan contact theory. N is the roller number of the bearing. δ i is the total contact deformation between the inner raceway and outer raceway, which can be expressed by the displacement of the shaft at the corresponding location. Loaddeformation exponent n = 1.5 for the ball bearing. β i is the exponent that determines whether the contact deformation occurs of the i-th ball:  0 δi < 0 βi = (3) 1 δi < 0

(a) Motor Installation lug

Shell

(b)

Shaft

Support and bearing

Deformed shaft

Bearing

Motor

Isolator

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Propeller Undeformed shaft

Support

Fig. 2. A (a) geometry model, (b) schematic of the propulsion shaft system.

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(a)

(b)

Fixed

Rotate Fixed Fig. 3. A (a) Finite element model and (b) boundry conditions of the propulsion shaft system.

To obtain the vibration characteristic of the propulsion shaft system numerically, the Finite Element Model is established as shown in Fig. 3. The max element size is 2 mm, the numbers of the elements and nodes are 6083514 and 9087782, respectively. 2.3 Rubber Isolator Design To reduce the vibration from the propulsion shaft system, the rubber isolators are designed, which mounted between the installation lungs and the motor. The installation and the structure of the vibration isolators is shown in Fig. 4. The thickness of the isolator T = 7 mm, the length L = 27 mm; The thickness of the installation lug T i = 15 mm; The diameter of the screw D = 10 mm; The width and thickness of the isolator end lug T r = 5 mm, W r = 5 mm; The Shore hardness of the isolator material HS = 80; The density of the isolator material ρ = 1100 kg/m3 ; The Poisson’s ratio of the isolator material υ = 0.49. The elastic modulus E of the isolator material can be calculated through the hardness, which is given by E=

15.75 + 2.15HS 100 − HS

(4)

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Installation lug

L Ti

Motor lug

T

D

Bolt Isolator

Isolator

Wr Tr

Fig. 4. The design of the rubber vibration isolator. (a) the installation of the isolator. (b) the structrue of the isolator.

3 Vibration Characteristic Analysis To obtain the characteristic frequencies of the parts in the shaft system, the modal analysis is conducted by using the FEA and theoretical method. The vibration responses in frequency domain at different locations of the propulsion shaft system are obtained by using the harmonic analysis. The responses of the system with the different sizes’ isolators are compared. Then, the optimized parameters of the isolator are obtained. 3.1 Characteristic Frequencies Analysis The characteristic frequencies and vibration mode shapes of the shaft are calculated by using the FEA. The first five natural frequencies of the shaft are given in Table 1. The corresponding mode shaped are given in Fig. 5. Table 1. The characteristic frequencies of the propulsion shaft Order

Frequency (Hz)

Mode shape

1

183

1st bending

2

505

2nd bending

3

982

3rd bending

4

1606

4th bending

5

2367

5th bending

The characteristic frequencies of the bearing are theoretically calculated through the rolling bearing kinematics, which are given by fc =

1 d d 1 fi (1 − cos α) + fo (1 + cos α) 2 D 2 D

(5)

1 d (fo − fi )(1 + cos α) 2 D

(6)

fri =

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Fig. 5. Vibration modal of the propulsion shaft. (a) First order, (b) second order, (c) third order, (d) fourth order and (e) fifth order.

fbpfi =

Z d (fo − fi )(1 + cos α) 2 D

(7)

fbpoi =

Z d (fo − fi )(1 − cos α) 2 D

(8)

D d (fo − fi )(1 − ( cos α)2 ) 2d D

(9)

fbsf =

where f c is the cage characteristic frequency; f ri is the relative frequency between the cage and inner race; f bpfi and f bpfo are the ball passing frequency about the inner and outer ring; f bsf is the ball spin frequency; f i and f o are the rotating frequencies of the bearing inner and outer ring; d is the ball diameter; D is the bearing pitch diameter; α is contact angle; Z is the ball number. The speed of the propulsion shaft system is 1553 rpm. The characteristic frequencies of the propulsion shaft system are given in Table 2. The frequency range of the harmonic analysis is 0–500 Hz. The range contains the characteristic frequencies and multiples of the motor, shaft and bearings.

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Frequency (Hz)

Motor

25.88

Shaft

183

bearing

fc

15.52

f ri

15.52

f bpfi

186.26

f bpfo

124.34

f bsf

57.17

3.2 Harmonic Responses of the Propulsion Shaft System The vibration responses in frequency domain of the propulsion shaft system are obtained by using the harmonic analysis. The harmonic response at the support structure, shaft, installation lug and motor are given in Fig. 6. The isolators have a significant effective on vibration reduction. When the excitation frequency in the region of 0–350 Hz, the vibration responses amplitudes are reduced by the isolators. The corresponding frequencies are also decrease. However, when the excitation frequency in the region of 350–500 Hz, the response amplitudes increases. 3.3 Vibration Isolation Analysis To reach the best vibration reduction effective, harmonic response analysis are conducted for the propulsion shaft system with different isolators. The parameters of the isolators are thickness T, length L and hardness HS. The first two peak amplitudes with T, L and HS at support structure, shaft, installation lug and motor are given in Figs. 7, 8 and 9. The vibrations of the system decreased with the thickness T. The best value of the thickness T is 7 mm. The vibrations of the system vary with the length L. When L = 15 mm, the first peak values reach to the minimum at the support structure, installation lug and motor, the second peak values reach to the minimum at the support structure and shaft. Which will minimize the vibrations of the system. The vibrations decrease with the hardness HS. The best value of HS is 80. Then, the optimized parameters of the isolators are obtained.

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Fig. 6. The harmonic responses of the propulsion system. (a) support. (b) shaft. (c) installation lug. (d) motor.

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Fig. 7. The effects of the T on the first two peaks at (a) support. (b) shaft. (c) installation lug. (d) motor.

Fig. 8. The effects of the L on the first two peaks at (a) support. (b) shaft. (c) installation lug. (d) motor.

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Fig. 9. The effects of the HS on the first two peaks at (a) support. (b) shaft. (c) installation lug. (d) motor.

The peak values and corresponding frequencies with the optimized isolators are given in Table 3. The vibration reduction effect is in the region of 75.4% to 89.7% at different parts of the propulsion shaft. The vibrations at the support structure is reduced by 89.7%. The vibrations of the shaft and motor are reduced by 88.4% and 82.9%, respectively. The vibrations of the installation lug is reduced by 75.4%. Table 3. Harmonic responses peak values and frequencies Position Support

Without isolators

With isolators

Peak (Hz)

Frequency (m/s2 )

Peak (Hz)

Frequency (m/s2 )

1272

298

131.1

286

Effect (%) 89.7

Shaft

20077

298

2329

286

88.4

Installation lug

357.7

298

87.86

286

75.4

Motor

1667.7

298

410.7

286

82.9

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4 Conclusions This study conducted a vibration characteristic analysis for a propulsion shaft system in an UUV. The vibration modes and characteristic frequencies of the propulsion shaft system was obtained by using the FEA and theoretical method. The rubber isolators are used to reduce the vibrations. The vibration responses in frequency domain of the system with the isolators are obtained by using the harmonic response analysis. Then, the geometric parameters and hardness of the isolators are optimized. The vibration peak values of the system are reduced by 75.4% to 89.7% with the optimized isolators. The presented method is effective on the vibration reduction of the propulsion system. However, the coupling effects of the isolators and the propulsion shaft system are not considered in the paper, which will be researched in the future works. Funding. Support provided by the National Natural Science Foundation of China under Contract No. 52175120 and 51975068; and the Fundamental Research Funds for the Central Universities (No. 3102020HHZY030001).

Conflict of Interest. The authors declared that they have no conflicts of interest.

References 1. Zou, D., Liu, L., Rao, Z., et al.: Coupled longitudinal–transverse dynamics of a marine propulsion shafting under primary and internal resonances. J. Sound Vib. 372, 299–316 (2016) 2. Zou, D., Rao, Z., Ta, N.: Coupled longitudinal-transverse dynamics of a marine propulsion shafting under superharmonic resonances. J. Sound Vib. 346, 248–264 (2015) 3. Zhang, Y., Xu, W., Li, Z., et al.: Dynamic characteristics analysis of marine propulsion shafting using multi-DOF vibration coupling model. Shock and Vibration, 2019 (2019) 4. Kim, Y.-G., Kim, U.-K.: Design and analysis of the propulsion shafting system in a ship with single stern tube bearing. J. Mar. Sci. Technol. 25(2), 536–548 (2019) 5. Zechao, H., Lin, H., Wei, X., et al.: Optimization design of resonance changer for marine propulsion shafting in longitudinal vibration. Chin. J. Ship Res. 14(1) (2019) 6. Xu, D., Du, J., Tian, C.: Vibration characteristics and power flow analyses of a ship propulsion shafting system with general support and thrust loading. Shock and Vibration (2020) 7. Liu, W., Zhou, Q., Li, H.: Research on optimal design of rubber isolator used in propulsion system. In: 2017 4th International Conference on Information Science and Control Engineering (ICISCE). IEEE, pp. 1171–1176 (2017) 8. Jee, J., Kim, C., Kim, Y.: Design improvement of a viscous-spring damper for controlling torsional vibration in a propulsion shafting system with an engine acceleration problem. J. Marine Sci. Eng. 8(6), 428 (2020) 9. Zhang, G., Zhao, Y., Li, T., et al.: Propeller excitation of longitudinal vibration characteristics of marine propulsion shafting system. Shock and Vibration (2014) 10. Zhang, C., Tian, Z., Yan, X.: Analytical analysis of the vibration of propulsion shaft under hull deformation excitations. J. Vibroengineering 18(1), 44–55 (2016) 11. Zambon, A., Moro, L.: Torsional vibration analysis of diesel driven propulsion systems: the case of a polar-class vessel. Ocean Eng. 245, 110330 (2022) 12. Liu, G., Lu, K., Zou, D., et al.: Development of a semi-active dynamic vibration absorber for longitudinal vibration of propulsion shaft system based on magnetorheological elastomer. Smart Mater. Struct. 26(7), 075009 (2017)

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13. Huang, Q., Yan, X., Wang, Y., Zhang, C., Jin, Y.: Numerical and experimental analysis of coupled transverse and longitudinal vibration of a marine propulsion shaft. J. Mech. Sci. Technol. 30(12), 5405–5412 (2016) 14. Yan, D., Wang, W., Chen, Q.: Fractional-order modeling and nonlinear dynamic analyses of the rotor-bearing-seal system. Chaos, Solitons Fractals 133, 109640 (2020) 15. Branagan, M.: Rotordynamic analyses using finite element method. Master Thesis, School of Engineering and Applied Science, University of Virginia, 66 (2014)

Computation of Components System Stiffness for Variable Stator Vane Mechanism Jing Chang

and Zhong Luo(B)

Northeastern University, Shenyang 110819, Liaoning, China [email protected]

Abstract. The linkage ring and rocker arm in the variable stator vane (VSV) mechanism have been proven to deform during movement, seriously affecting the motion accuracy of the mechanism. The most concerned and urgent issue is to discover the influence of the stiffness and deformation of the components on the output accuracy of the mechanism. For a given designed structure, the stiffness varies with the configuration of the mechanism. In this paper, the kinematics model of the system composed of components on the kinematic chain formed from the linkage ring to the blades is discussed, and then the stiffness model of the system is obtained. The stiffness obtained by theoretical calculation is compared with that obtained by the Finite Element Analysis model, and the correctness of the formulation is verified. Finally, a study is conducted on whether gravity is needed to be considered, and the results demonstrate its necessity for analysis of stiffness and motion accuracy. Keywords: VSV mechanism · Kinematics analysis · Stiffness model · Deviation

1 Introduction The VSV mechanism adjusts the angle of the stator blades to ensure that the aerodynamic performance of the engine meets the design conditions. In general, the blades are distributed throughout the entire circumference. In order to make the entire circumference of the blades adjustable at the same time, the designers adopted a structure of a linkage ring and a complete circle of rocker arms, so that the angle of the blades connected to each rocker arm can be adjusted. At this time, the focus of the adjusting mechanism design shifted to designing a mechanism that can drive the linkage ring. As a result, there were many types of adjusting mechanisms. They are all mechanically complex, nonlinear, and time-varying dynamic systems. Although existing differences in overall structure, they all have the same combination structure of linkage ring and rocker arms. Figure 1 is a commonly used type of adjustment mechanism [1], it is considered that the components will not deform, and the degrees of freedom of all components is uniquely determined based on the motion relationship at the time of design. But the fact often differs from the initial design thought. During flight or testing, the linkage ring and rocker arm often deform, and the mechanism’s motion accuracy cannot achieve the expected effect, and even the mechanism’s blocking © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 452–463, 2024. https://doi.org/10.1007/978-3-031-40459-7_32

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force increases. Given the existence of this phenomenon, some scholars are attempting to identify the cause. Zhang et al. [2] built a multi-level joint tuning rigid-flexible coupling dynamic model that can consider these above factors and analyzed the attribution of motion accuracy and retardation force from the perspectives of simulation and bench tests. In addition, these studies [3, 4] took into account the flexibility and deformation of components. Some of them used self-developed programs to study the rigid-flexible coupling of multi-level adjusting mechanisms [3], while others conducted an analysis of the impact of aerodynamic forces on the stiffness and strength of the IGV mechanism [4]. Rocker Spherical Pin hinge Linkage Rod Linkage Blade arm bar ring

Aero-engine case Fig. 1. A type of design for variable stator vane system construction.

Most of the above studies have been conducted from the perspective of dynamic simulation, and there is a lack of in-depth theoretical explanation for the phenomenon. The following research [5, 6] is based on full rigid body dynamics modeling of the single-stage adjustable mechanism. Although these studies do not consider component flexibility, they have a certain significance for the theoretical analysis of rigid flexible coupling modeling of mechanisms. When conducting stiffness analysis on the mechanisms in the field of robotics [7–11], researchers used various stiffness evaluation methods, such as the trace of the stiffness matrix [12], eigenvalues of stiffness matrix [13], determinant of stiffness matrix [14], and evaluating the position error of the mechanism under external forces [15]. The method used in this article is to evaluate the stiffness by analyzing the rotational deviation of the components system in contact under payload. Due to the fact that component flexibility has become a factor that cannot be ignored in regulating the motion accuracy of mechanisms, there is a lack of a deeper theoretical explanation of the impact of flexibility (stiffness) on accuracy in current research. Therefore, this paper first extracts the components that need to be carefully considered

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for deformation based on the actual situation to form a components system and then conducts stiffness research on the components system. The stiffness model of the components system is validated by the finite element method. In addition, it is also analyzed whether the influence of gravity was considered in the stiffness analysis.

2 Kinematic and Stiffness Models 2.1 Research Object Extraction Figure 2 shows the extraction process from the original single-stage VSV mechanism to the three-dimensional model of the linkage ring-blade components system, and then to the kinematic model of the components system. This paper is exactly based on the stiffness research of the components system kinematics model in Fig. 2.

Fig. 2. Schematic diagram of research object extraction.

2.2 Kinematic Model Figure 3 is a schematic diagram of the motion chain of the linkage ring-blade components system, which is different from the configuration in Fig. 2 (the blades are in the initial position). This diagram represents the other state within the range of motion of the mechanism except for the initial position. The thick yellow line, thick purple line, and thick black line represent the linkage ring, rocker arm, and aero-engine case, respectively. The origin of the coordinate system O-XYZ of the components system is located at the intersection point between the centroid of the linkage ring and the engine axis when

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the component is in its initial position. Assume that the assembly contains a total of i blades. For any combination of rocker arms and blades, the point on the linkage ring where the rocker arms are hinged is marked as point C i , the other end of the rocker arm where the blades are hinged is marked as point Bi, and the intersection point with the engine on the rotating axis of the blades is point A. The coordinate system {B} is the body coordinate system of the rocker arm, with the origin located at the hinge point B. The positive direction of the X-axis points towards the linkage ring along the symmetric axis of the rocker arm, the positive direction of the Y-axis is the direction of the blade’s rotation axis (pointing from point A to the origin of the body coordinate system), and the positive direction of the Z-axis is determined according to the right-hand rule.

YBi

Mi Bi ZB

l Ci XB

A Z

RR

Y O

X

Fig. 3. Kinematic and static model of components system.

In this paper, since the rocker arm will not reach the singular position, the Euler angle can be used to represent the orientation of the coordinate system {B}. By establishing a closed-loop constraint equation, the motion model of the components system can be established, and the constraint equation is as follows: Equations for constraint the motions of spherical plain bearing and rocker arm: ro + [0; RR × sin(θ );RR × cos(θ )] + A[h; 0;0] = bbb + A[0; 0;l]

(1)

where, ro and A are the coordinates of the point A under the global coordinate system and the rotation matrix of the body coordinate system {B} relative to the global coordinate system, respectively. If the Z-X-Z rotation way is adopted, the rotation matrix is expressed

456

as:

J. Chang and Z. Luo

⎡

⎤ cos ψ cos φ − sin ψ cos θ sin φ − cos ψ sin φ − sin ψ cos θ cos φ sin ψ sin θ A = ⎣ sin ψ cos φ + cos ψ cos θ sin φ − sin ψ sin φ + cos ψ cos θ cos φ − cos ψ sin θ ⎦ sin θ sin φ sin θ cos φ cos θ where, ψ, θ , φ represent the Z-axis, the X-axis after rotation and the Z-axis after secondary rotation respectively. For rotation matrix A, it can also be written as: ⎡ ⎤ cos α 0 − sin α A = A(ψ1 , θ1 , φ1 ) · ⎣ 0 1 0 ⎦ (2) sin α 0 cos α where, ψ1 , θ1 , φ1 respectively represent the Euler angles of the coordinate system {B} relative to the global coordinate system when the blade is in its initial position. α is the rotation angle of the blade around its axis of rotation (Y-axis of coordinate system {B}). 2.3 Stiffness Model Figure 4 presents the static sketch of the components system, where the coordinate system {O} and body coordinate system {B} still follow the definition method shown in Fig. 2. After applying inertia load to the blade end, equal force F1 and equivalent torque M 1 will be generated in the coordinate system {O}, as well as a force will be generated on the rocker arms to balance external forces. Therefore, the statics equation of the system can be written as follows: ⎧ i=n

⎪ ⎪ ⎪ f i + F1 = 0 ⎪ ⎪ ⎨ i=1 (7) i=n ⎪

⎪ ⎪ ⎪ ei × f i + M 1 = 0 ⎪ ⎩ i=1

where, f i is the internal force on any rocker arm F1 is the external force vector acting on the coordinate system {O} M 1 is the external moment vector acting on the coordinate system {O} ei is the position vector connecting the linkage ring and the hinge point Ci of the rocker arm with the origin O of the coordinate system {O} Convert Eq. (7) into a form containing a Jacobian matrix, i.e. JTf = W

(8) T  where, W is the torque acting on point O, being W= −F1 −M 1 . f is the force acting on the rocker arm. J is the Jacobian matrix of the components system, and it can be expressed as: T

... ni ... nm n1 (9) J= (eT1 × nT1 )T ... (eTi × nTi )T ... (eTm × nTm )T

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Fig. 4. Schematic of the static configuration of combined components.

where, ni is the unit vector in the direction of the symmetry axis of any rocker arm, and ei is the position vector connecting the origin O of the coordinate system {O} with the point C i on the linkage ring. Thus, the stiffness of the system can be obtained by taking the partial derivative of the torque W relative to the motion vector, K=

∂ T ∂W ∂f = JT + (J )f = K 1 + K 2 ∂S ∂S ∂S

(10)

T  where, S is the motion vector of the system, including the motion vector d T θ1 of the linkage ring, the motion scalars h1 ,h2 ,…,hm of all pins, and the motion scalars θ21 , θ22 , ..., θ2m of all rocker arms. d is the displacement vector of the linkage ring. The stiffness matrix K 1 is caused by the deformation of the rocker arm, K 2 is caused by the application of external forces to the system and is related to the structural stiffness of the system. For the stiffness matrix caused by rocker arm deformation, it can be written as: ∂f ∂f ∂L = JT = J T diag(k1 , k2 , .., km )J (11) ∂S ∂L ∂S where, L is the vector composed of all rocker arms in the direction of the symmetry axis. k indicates the stiffness of the rocker arm in the direction of the symmetry axis. For the structural stiffness K 2 of the system, it is related to the geometric configuration. Simplifying it can obtain:

 ∂ T ∂ ... ni ... nm n1 K2 = (J )f = f (12) ∂S ∂S (eT1 × nT1 )T ... (eTi × nTi )T ... (eTm × nTm )T K1 = JT

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In Eq. (12), the following expression can be extended to: 

∂ ... ni ... nm n1 ∂S (eT1 × nT1 )T ... (eTi × nTi )T ... (eTm × nTm )T 

  ∂ni ∂ ni ∂S = = (i = 1, 2, ..., m) ∂eT ∂nT ∂S (eTi × nTi )T ( ∂Si × nTi + eTi × ∂Si )T 

(13)

If the system only has one set of rocker arms, Eq. (13) can be written as:   ∂n ∂n1 ∂S

∂n1 ∂d T ∂e ∂nT T T T 1 × n1 + e1 × ( ∂d × n1 + eT1 × ∂d1 )T  ∂n1 ∂n1 ∂h ∂θ2 ∂eT ∂nT ∂eT ∂nT ( ∂h1 × nT1 + eT1 × ∂h1 )T ( ∂θ12 × nT1 + eT1 × ∂θ21 )T ∂eT ( ∂S1

=

∂nT1 T ∂S )

1

∂eT ( ∂θ11

∂θ1

× nT1 + eT1 ×

∂nT1 T ∂θ1 )

(14) Due to the limited freedom of translation and rotation of the linkage ring, d is a zero matrix and its value of θ1 is also zero. In addition, since the displacement h of the pin is a scalar, it can be concluded that if the partial derivative is taken, its value is zero, Eq. (13) can be further simplified as:   

∂ni ∂ni 03×4 ∂ ni ∂hi ∂θ2i = ∂eT ∂nT ∂eT ∂nT ∂S (eTi × nTi )T 03×4 ( ∂hii × nTi + eTi × ∂hii )T ( ∂θ2ii × nTi + eTi × ∂θ2ii )T   ∂ni 03×5 ∂θ2i = (i = 1, 2, ..., m) ∂eT ∂nT 03×5 ( ∂θ2ii × nTi + eTi × ∂θ2ii )T (15) When the linkage rotates around the engine axis with increment θ1 , the displacement of the linkage ring caused by this rotational motion can be represented as ∂ei = 2ei sin θ21 + ∂d = 2ei sin θ21 + l · [ cos θ2i cos θ1 sin θ2i sin θ1 sin θ2i ]T , then ∂eTi ∂θ2i

× nTi =

θ 2eTi sin 21 +l·[ cos θ2i

cos θ1 sin θ2i sin θ1 sin θ2i ]T ∂θ2i

× nTi

(16)

= l · [ − sin θ2i cos θ1 cos θ2i sin θ1 cos θ2i ]T × nTi and, eT i ×

∂nT i

∂θ2i

 = eT i ×

∂nT i ∂d

·

∂d ∂θ2i



 = l · eT i ×

 ∂nT i · − sin θ2i cos θ1 cos θ2i ∂d

sin θ1 cos θ2i

  T

=0

(17)

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∂ni According to Eq. (17), it is concluded that ∂θ should be parallel to the vector ei , so 2i the structural stiffness of the system is ultimately expressed as,   03×5 03×1 K2 =  ∂eTi T T 03×5 i=m i=1 fi ( ∂θ2i × ni ) (18)

 03×5 03×1  = T T T 03×5 i=m i=1 fi l([ − sin θ2i cos θ1 cos θ2i sin θ1 cos θ2i ] × ni )

Therefore, the stiffness of the components should be expressed as: K = K1 + K2

(19)

where, K 1 and K 2 are the stiffness related to the rocker arm and the structural stiffness of the components system. K 1 and K 2 are obtained by taking the derivative of the motion vector for torque W. Based on Hooke’s law, the value of Eq. (19) can be obtained. By deriving the above formula, it is not difficult to determine the combined stiffness of the assembly in the workspace. In the following section, the finite element method will be used to validate the proposed stiffness matrix model.

3 Simulation and Validation 3.1 Components System Stiffness Model Validation In this section, the calculation derivation of the stiffness matrix of the components system will be verified through the finite element model of the system. In order to verify, a finite element model case that is consistent with the theoretical model is proposed, as shown in the Fig. 5. The finite element model is designed to ensure the same force situation as the real mechanism, and the boundary condition is set to fix and constrain the connection surface between the linkage ring and the rod (Fig. 1). The assumed inertia torque applied in the theoretical calculation section corresponds to the torque applied to the blade in the FEA model, where the torque value is 500 Nmm. In order to verify the stiffness model of the components, 11 positions within the movement range of the blade and a complete quantity of components are selected to calculate the deviation, and then the simulation results are compared with the theoretical calculation results. The eleven selection points for the range of motion are -20°, -15°, -10°, -5°, 0°, 5°, 10°, 15°, 20°, 25°, and 30°, respectively. When the blade angle is 0°, it is considered the initial position of the mechanism’s motion, and the length direction of the rocker arm is parallel to the engine axis (X-axis in the Fig. 5). Hereinafter, when we describe the components located in the same motion branch chain as the loaded blades, we use the word “first” to describe them uniformly, such as the first rocker arm or the first pin. The position of the linkage ring connected to the first rocker arm is called the position of the first hole of the linkage ring. Figure 6 shows the angular displacement change of the system after applying a torque of 500 Nmm to the first blade when it’s in its original position. Figure 6(a) shows the angular displacement of the overall assembly, while Fig. 6(b) shows the rotational

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Fig. 5. FEA model case of components system.

angular displacement deviation of the maximum angular displacement (loaded blade) around its own rotation axis in the assembly. It is obvious that the angular displacement deviation of the first blade around its axis after applying moment is the largest, which coincides with our common sense. According to the simulation value of the first blade angle deviation in Fig. 6, it can be seen that when all the blades are in action, the angle deviation value of the first blade is relatively small. Therefore, in order to clearly distinguish the difference between theoretical calculation and simulation values, special numerical processing (taking negative logarithm) is performed on both results to amplify the deviation value. Figure 7 shows the angular displacement deviation of the enlarged first blade at different positions within the range of motion, which is used as a criterion for evaluating the stiffness of the assembly. It can be seen that the angle deviation value of the first blade is basically on the same order of magnitude as the simulation value, but the trend of the simulation value is relatively gentle. Two reasons can explain the difference between simulation and theoretical values: i When conducting simulation analysis, the contact between all components is taken into account, but the theoretical model ignores the stiffness between joints, which will lead to a calculation gap between the two; ii. When the blade angle is 0 degrees, the reason for the biggest difference between simulation and theoretical calculation values is that the theoretical model takes into account the contact between all components when the angle is zero degrees, The stiffness value K 2 contained within Eq. (19) of the component system is zero, but when the angle of rotation is not zero, the structural stiffness value is not zero, and the portion that accounts for a large proportion of the stiffness value of the component system plays a role. 3.2 Influence of Gravity on Stiffness The theoretical and simulation results mentioned above are based on the assumption that the gravity of all components is not taken into account, but all components of the real engine are in a gravity environment. Therefore, the following study begins to investigate the variation law of the first blade’s angle deviation within the range of blade motion under the action of all stage blades, as well as the influence law of the number of contact

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(a)

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(b)

Fig. 6. Statics simulation results of components system. (a) Angular displacement amplitude of the system (b) Maximum angular displacement of the first blade

Fig. 7. Deviation of the rotational angle of the first blade within the range of motion.

blades on the first blade’s angle at the initial position. It should be noted that when considering the number of contact blades, all blade positions are symmetrically set. All the calculations take into account the influence of gravity. Figure 8 shows the influence of whether to consider the gravity on the first blade angle deviation which can further illustrate the influence of gravity on the stiffness of the system. As before, all first blade angle deviations have been numerically processed. From the figures (a) and (b) in Fig. 8, it can be seen that the trend of considering the influence of gravity on the first blade angle deviation remains basically unchanged, only the deviation value changes. Moreover, considering gravity, the angle deviation value is smaller than not considering gravity, which can also be said to have a positive effect on the improvement of the stiffness of the components system. The reason why the consideration of gravity can cause changes in the motion deviation of the assembly is still due to the changes in the equal force and equivalent torque generated at the center

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of the linkage ring when considering gravity, which leads to changes in the theoretical solution results. From Fig. 8 (b), it can also be seen that when the number of contact blades is 1, the motion deviation is relatively large between considering gravity and not considering gravity. The reason is that during calculation, all blade positions are symmetrically arranged, and when the number of blades is 1, there are no other blades to balance the force with the first blade. Although the stiffness calculation conducted in this section is only for individual cases, combined with the previous research, it is not difficult to reach the same conclusion for the remaining cases that have not been studied.

Fig. 8. The effect of gravity on the angle deviation of the first blade. (a) Derivations of first blade angle within the range of motion under the combined motion of full blades (b) The influence of different blade numbers on the deviation of the first blade angle when the blade is in its original position

4 Conclusion This paper focuses on the stiffness evaluation method of a single stage VSV mechanism under a certain configuration. The obtained stiffness model is validated using the finite element method, and gravity impact analysis proved that considering the influence of gravity on the stiffness of the composite component is indispensable. Under the influence of gravity, the motion deviation of the component will increase.

References 1. Tang, Y., Guo, W.: Global dimensional optimization for the design of adjusting mechanism of variable stator vanes. J. Mechanical Eng. 11(56), 26–35 (2020) 2. Wang, H., Sun, H., Sun, H., Liu, D.: Attribution analysis of blocking force and adjustment accuracy of adjusting mechanism of variable stator vane. Acta Aeronautica et Astronautica Sinica 41(12), 1–11 (2020) 3. Sun, K., Lin, Q.S., Zhang, Y.S.: Kinematic optimization of compressor VSV system based on ADAMS and ISIGHT. In: 7th China Aviation Society Youth Science and Technology, pp. 364–368. Chinese society of aeronautics and astronautics, Zhongshan (2016)

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4. Hensges, M.: Simulation and optimization of an adjustable inlet guide vane for industrial turbo compressors. In: 53rd ASME turbo expo, pp. 11–20. ASME, Berlin (2008) 5. Cheng, S., et al.: Rough surface damping contact model and its space mechanism application Int. J. Mech. Sci. 214(106899), 1–20 (2022) 6. Chang, J., Luo, Z., Wei, K., Han, Q., Han, F.: Investigation of spatial plane joint characteristic for dynamic analysis of VSV mechanism based on similarity scaling technique Int. J. NonLinear Mech. 148(104300), 1–13 (2023) 7. Duan, X., Yang, Y., Cheng, B.: Modeling and analysis of a 2-dof spherical parallel manipulator. Sensors 16(1485), 1–15 (2016) 8. Behzadipour, S., Khajepour, A.: Stiffness of cable-based parallel manipulators with application to stability analysis. ASME J. Mech. Des. 128(309), 303–310 (2006) 9. Dong, X., Raffles, M., Guzman, S.C., Axinte, D., Kell, J.: Design and analysis of a family of snake arm robots connected by compliant joints. Mech. Mach. Theory. 77, 73–91 (2014) 10. Bolboli, J., Khosravi, M.A., Abdollahi, F.: Stiffness feasible workspace of cable-driven parallel robots with application to optimal design of a planar cable robot. Rob. Auton. Syst. 114, 19–28 (2019) 11. Li, C., Rahn, C.D.: Design of continuous backbone, cable-driven robots. J. Mech. Des. 124(2), 265–271 (2002) 12. Zhang, D.: Global stiffness modeling and optimization of a 5-dof parallel mechanism. In: Proceedings of the 2009 International Conference on IEEE Mechatronics and Automation (ICMA), pp. 3551–3556 (2009) 13. El-Khasawneh, B.S., Ferreira, P.M.: Computation of stiffness and stiffness bounds for parallel link manipulators. Int. J. Mach. Tools Manuf 39, 321–342 (1999) 14. Lim, W., Raja, V., Thing, V.: Generalized and lightweight algorithms for automated web forum content extraction. In: Proceedings of the 2013 IEEE International Conference on Computational Intelligence and Computing Research (ICCIC), pp. 1–8. IEEE society, Enathi (2013) 15. Ma, N., Yu, J., Dong, X., Axinte, D.: Design and stiffness analysis of a class of 2-DoF tendon driven parallel kinematics mechanism. Mech. Mach. Theory. 129, 202–217 (2018)

An Unbalance Identification Method of a Whole Aero-Engine Based on the Casing Vibrations Weimin Wang1,2(B) , Jiale Wang1 , and Qihang Li1,2 1 Beijing Key Laboratory of Health Monitoring and Self-Recovery for High-End Mechanical

Equipment, Beijing University of Chemical Technology, Beijing 100029, China [email protected] 2 State Key Laboratory of High-End Compressor and System Technology, Beijing, China

Abstract. Squeeze film dampers (SFDs) have been frequently employed in aeroengines due to their excellent vibration reduction performance. However, their nonlinear characteristics make it difficult for traditional balancing methods to achieve efficient vibration suppression. In addition, because of the harsh condition and restricted space inside the casing, sensors are only able to be installed on the surface of the casing to evaluate the operational state of an aero-engine. Therefore, it is vital to propose an unbalance identification method of a whole aero-engine with SFDs. In this paper, the equivalent unbalance calculation equation, which takes the vibration transfer characteristics of the casing, nonlinear oil film forces, the flexible support and gyroscopic moments into account, is derived by decoupling the differential equation of rotor motion in modal coordinates. The equivalent unbalance distribution of the rotor is generated by solving very ill-conditioned linear equations produced by modal parameters. Finally, a nonlinear unbalance identification method based on casing vibrations is developed. By establishing a whole aero-engine dynamics model, simulation is used to verify the accuracy of this unbalance identification method. The results show that, when the High Pressure rotor has unbalance alone, the vibration reduction rates for the aeroengine rotor system are no less than 93%. For the condition that HP and LP rotors are unbalanced at the same time, using the excitation source separation method proposed in this paper, the balancing effects are also very significant. Keywords: whole aero-engine dynamics model · squeeze film damper · flexible support · nonlinear unbalance identification

1 Introduction Establishing an accurate dynamics model is the most important for balancing the whole aero-engine. Yang et al. [1] ignored the casing deformation and replaced the casing with a concentrated mass on the rotor, while Wang et al. [2] established the casing by beam elements. Using finite element software, Hai and Bonello [3, 4] conducted a modal analysis for a rotor-casing coupling system and decreased the degrees of freedom in modal coordinates. Finally, using the impulsive receptance technique, the dynamic response of a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Chu and Z. Qin (Eds.): IFToMM 2023, MMS 140, pp. 464–481, 2024. https://doi.org/10.1007/978-3-031-40459-7_33

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whole aero-engine model with nonlinear bearings was calculated. Choi et al. [5] demonstrated the effectiveness of this method by combining the frequency response function (FRFs) of the flexible support with the rotor model. Similarly, Hu et al. [6] compared a rotor-support finite element model and a rotor finite element model combined with FRFs. Dewi et al. [7] also successfully identified the critical speed and damping coefficient of a rotor-support system. Traditional dynamic balancing techniques like the influence coefficient balancing method [8] and the modal balancing method [9] can resolve most unbalance problems. The advantages of these two approaches were combined by Darlow to present the Unified Balancing Approach [10]. Although some academics have recently conducted extensive studies on balancing methods and presented new approaches [11–13], the majority of them only apply to linear systems. For nonlinear rotor systems, Krodkiewski et al. [14] and Ding et al. [15] employed the finite difference method and Fourier coefficient expansions to obtain the nonlinear oil film forces. The unbalance was identified through the first-order vibrations of the rotor bearing. Sinha et al. [16] determined the equivalent stiffness and damping of a rotor at a fixed speed and balanced the rotor by assuming that the bearing forces are linear. For the non-invasive inverse problem, Cedillo et al. has made outstanding contributions. In Cedillo’s research [17], a finite element software is used to calculate the eigenforms and eigenfrequencies of rotor-casing systems. Through the first-order differential equation of motion, Cedillo eliminated the nonlinear oil film forces of the SFDs by employing the relationship between the vibrations of the casing and the equivalent unbalance. The results indicate that this method is effective for the unbalance identification of a rotor-casing system. This paper establishes a dynamic model of a whole aero-engine and proposes a method to calculate the equivalent unbalance of the nonlinear system, while taking nonlinear oil film forces, flexible casing, and gyroscopic moments into account. The balance effects under different conditions are researched.

2 Theoretical Methods 2.1 Rotor-Casing Modeling

Fig. 1. Structure of a rotor-casing system: (a) sketch of a rotor-casing system; (b) calculation process of the equivalent support stiffness and displacement.

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The rotor and the casing are no longer independent of each other once the flexible support of the casing is considered. The interaction between the rotor and the casing is depicted in Fig. 1 (a), where f by1 , f bx1 , f by2 , f bx2 are the bearing forces and f uy , f ux are the unbalanced forces of the rotor. As illustrated in Fig. 1 (b), Fu , Fb are the unbalanced and bearing forces on a rotor. K b is the bearing stiffness. Gr , Gc are the flexibility of the rotor and the casing, respectively. X b , X are the vibrations of the rotor at the bearings and other positions, while X cb , X c are the vibrations of the casing at the bearing housings and other positions. Gc consists bc bb of the flexibility Gbb c and Gc . Gc is the flexibility matrix from one bearing housing of the casing to all bearing housings, while Gbc c is the flexibility matrix from one bearing housing of the casing to the casing vibration monitoring points. The rotor and the casing are coupled to each other through the Gc . The presence of casing displacements X cb change the bearing forces Fb , which affect the rotor displacements X b , X and the casing displacements X cb , X c in turn. Taking the bearing force as an external force, the equation of motion of a rotor system is M · X¨ + C · X˙ + K · X = Su · Fu + Sb · Fb

(1)

  T T where Fu = fux , fuy and Fb = fbx1 , fby1 , fbx2 , fby2 , · · · , fbxn , fbyn are the unbalanced and the bearing forces matrices. The function of the selection matrices Su and Sb are to set the elements of Fu and Fb to the corresponding position of the finite element model. M, C and K are the mass, damping, and stiffness matrices of the rotor finite element model. Since rolling bearings are commonly used in aero-engine rotor systems, the damping of bearings can be ignored, and the bearing forces matrix is Fb = −K b (X b − X cb )

(2)

where K b is the stiffness matrix of bearings. The relationship between the bearing forces and the casing displacements satisfies X cb = −Gbb c · Fb

(3)

The flexibility matrix of the casing Gbb c represents the flexibility relationship among the bearing housings of the casing, which can be obtained by experiment or simulation. ⎡

Gbb c

g11 ⎢ g12 ⎢ =⎢ . ⎣ .. g1n

g21 · · · gn1 g22 · · · gn2 .. . . .. . . . g2n · · · gnn

⎤ ⎥ ⎥ ⎥ ⎦

(4)

According to Eq. (2) and Eq. (3), the vibrations of the casing at the bearing housings X cb is eliminated. Thus Fb can be written as −1 Fb = −[I + K b · Gbb c ] Kb · Xb

(5)

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The relationship between the bearing forces and the displacements of a rotor at the bearings is determined as Fb = −Kˆ · X b

(6a)

−1

Kˆ = I + K b · Gbb Kb c

(6b)

where Kˆ is the equivalent support stiffness of a rotor with the influence of the flexible casing. Furthermore, the displacements of the monitoring points of the casing can be estimated using the flexibility matrix Gbc c . X c = −Gbc c · Fb

(7)

2.2 Rotor-SFD-Casing Modeling

Fig. 2. Structure diagram of the support.

Figure 2 depicts the relative position of the rotor, support, SFD, and casing in an aero-engine. The bearing is attached to the casing through a squirrel-cage, and the SFD is formed by filling the gap between the outer diameter of the squirrel-cage and the casing with an oil film. The distance between the oil film force of the SFD and the elastic force of squirrel-cage acting on the casing is especially short compared to the length of the casing. As a result, it is assumed that F 2 and F 1 act at the same place and are both in the position of F 1 when calculating the dynamics of the whole aero-engine. Under the influence of the oil film force and the elastic force of squirrel-cage, Eq. (1) can be rewritten as: M · X¨ + C · X˙ + K˜ · X = Su · Fu + Ss · Fs

(8)

The equivalent stiffness of the rotor K˜ includes the rotor stiffness K and the equivalent support stiffness Kˆ induced by flexible casing. Fs and Ss are the nonlinear oil film forces matrix and its selection matrix, respectively. The vibrations at the monitoring points of the casing are X c = −Gbc c · (Fb +F s )

(9)

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2.3 Unbalance Identification Method of a Nonlinear System In order to evaluate the operational state of an aero-engine, sensors are only able to be installed on the surface of the casing due to the harsh condition and restricted space inside the casing. As a result, it is essential to identify the rotor unbalance through the casing vibrations. Based on the vibrations of the casing, the forces of the casing at the bearing housings Fc can be obtained using Eq. (10).

−1 Xc (10) Fc = Gbc c where the flexibility matrix Gbc c can be acquired through simulation or experiment. The forces applied to the rotor support positions are equal to Fc in magnitude with opposing directions. These forces, named Fr , include elastic support forces and nonlinear oil film forces. Fr = −Kˆ · X b + Fs

(11)

For a π film approximation based on short bearing theory, the oil film force can be expressed as μRL3 2ε2 π ε˙ (1 + 2ε2 ) [ + ] C 2 (1 − ε2 )2 2 (1 − ε2 )5/2 μRL3 π ε 2˙εε Ft = [ + ] C 2 2(1 − ε2 )3/2 (1 − ε2 )2

Fr =

(12)

where R, c are the radius and radial clearance of the SFD, μ is the dynamic viscosity of lubricant, and ε = e/c is the eccentricity ratio. Given that the frequency of vibration caused by unbalance is consistent with the rotational frequency, an effective method for accurately identifying the unbalance of a rotor is to obtain the first-order differential equation of motion through Fourier series expansion to eliminate the influence of forces at other frequencies on the rotor vibration. Thus, Fr can be written as Fr = f 0 +

k

 f ic cos(iωt) + f is sin(iωt)

(13)

i=1

The first-order Fourier coefficients f c and f s are  2 T f c = f 1c = Fr cos(iωt)dt T 0  2 T f s = f 1s = Fr sin(iωt)dt T 0

(14)

Similarly, after evaluating the Fourier coefficients of X b and Fs , Eq. (11) can be rewritten as ˆ b +Fs F = − KX

(15)

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T  where F = f c f s , X b = [xbc xbs ]T , Fs = [f sc f ss ]T . Through the Newton-Raphson iterative algorithm, X b and Fs can be calculated after substituting Eq. (6b) and Eq. (12) into Eq. (15). At this step, the data used to identify the equivalent unbalance has been converted from the vibrations of the casing to the vibrations of the rotor at bearings. When determining the oil film forces Fs , it should be noted that the e in Eq. (12) are the displacements of the rotor relative to the casing. The displacements of the casing at the bearing housings are X cb = Gbb c · Fc

(16)

The equation of motion of a rotor-damper-casing system can be written as M · X¨ + K · X = Su · Fu + Ss · Fs + Fc + Sb · Fb

(17)

Fc includes the damping forces and gyroscopic moments of the rotor system. And Fc = −C · X˙

(18a)

Fb = −Kˆ · X b

(18b)

M and K in Eq. (17) are symmetric matrices. Thus this system can be completely decoupled in modal coordinates. Equation (17) can be expressed in a modal coordinate as [18] q¨ +q = ET · Su · Fu + ET · Ss · Fs + ET · Fc + ET · Sb · Fb

(19)

where E is the mass-normalized eigenvectors. q is the displacement in the modal coordinate and X = Eq.  is the diagonal matrix of ω12 、ω22 ,…,ωR2 , with ωi ( i = 1, 2, . . . , R) are the eigenfrequencies of this system. Set T

(20) Eq = ET(b1) ET(b2) · · · ET(bn) with E(bi) is the vector of E in row bi (i = 1,2,…,n), defining bi are the nodes number of the rotor finite element model where the degrees of freedom of bearings are located. Left multiplying Eq. (19) by Eq and simplifying this equation, gives X b = H u Fu + H s Fs + H c Fc + H b Fb with H u = R Eq ·ET ·Sb i=1 ω2 −2 i

R

Eq ·ET ·Su i=1 ω2 −2 , i

Hs =

R

Eq ·ET ·Ss i=1 ω2 −2 , i

Hc =

(21) R

Eq ·ET i=1 ω2 −2 , i

Hb =

, where  is the rotor speed.

For fundamental frequency component, Eq. (21) can be described as               Hu f uc Hs f sc Hc f cc Hb f bc xbc = + + + xbs H u f us H s f ss H c f cs H b f bs (22)

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where



f cc f cs



 =

·C − · C



xc xs

 (23)

with f uc , f us , f sc , f ss , f cc , f cs , f bc , f bs , xc , xs are the Fourier coefficients of Fu , Fs , Fc , Fb , and X. For all degrees of freedom of the rotor system, Eq. (22) can be expanded as    a    a    a    a   xc Hu f uc Hs f sc Hc f cc Hb f bc = + + + xs H au f us H as f ss H ac f cs H ab f bs (24) with H au =

R

E·ET ·Su a i=1 ω2 −2 ,H s i

=

R

E·ET ·Ss a i=1 ω2 −2 ,H c i

=

R

E·ET a i=1 ω2 −2 ,H b i

=

R

E·ET ·Sb i=1 ω2 −2 . i

Substituting xc , xs back into Eq. (23) and eliminating f cc , f cs , Eq. (22) can be expressed as         xbc f f f = A bc + B uc + C sc (25) xbs f bs f us f ss where



H c CDH ab −H b − H c CDPH ab a −H c CDH b −H b − H c CDPH ab   H u +H c CDPH au −H c CDH au B= H c CDH au H u +H c CDPH au   H s +H c CDPH as −H c CDH as C= H c CDH as H s +H c CDPH as

A=

 (26a) (26b) (26c)

with P = −H ac ·  · C

(27a)

−1 D = I + P2

(27b)

Then rearranging Eq. (25) get         x f f f B uc = bc − A bc − C sc f us xbs f bs f ss Set

(28)



     xbc f f − A bc − C sc xbs f bs f ss     Uc f = B2 VU c = QU c B uc = B2 f us Us Z=

(29a) (29b)

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Equation (28) can be rewritten as QU c = Z

(30)

According to the solution of Eq. (30), the equivalent unbalance matrix U c of the rotor system can be acquired. T

U c = Ux1 Uy1 Ux2 Uy2 · · · UxJ UyJ

(31)

with J is the number of disks to applied balancing weights. The value U j and phase φj of the equivalent unbalance can be calculated by  Uj =

2 2 j j Ux + Uy

(32a)

⎧ ⎨



j j j arctan Uy /Ux , Ux > 0

φj = ⎩ arctan Uyj /Uxj + π, Uxj < 0

(32b)

3 Simulation Verification 3.1 Rotors-SFDs-Casing System In the rotors-SFDs-casing system calculated in this Section, the rotors are installed to the casing through bearings, and there is an intershaft bearing between the High Pressure (HP) and Low Pressure (LP) rotors. The finite element model of the dual-rotor system is built using the Timoshenko beam element, as shown in Fig. 3. The LP rotor consists of 42 elements with a 1–2-1 support type equipped with three fans and one turbine. The HP rotor consists of 48 elements with a 1–0-0 support type, equipped with six compressors and one turbine. The speed ratio between the LP and HP rotors is 1:1.2.

Fig. 3. Finite element model of the dual-rotor system.

In the support positions, elastic components like squirrel-cages are frequently used on aero-engine rotors. As the influence of the elastic support is taken into account, the stiffness of the bearing is adjusted to 107 N/m. Thus, the elastic support structures should be disregarded when building the casing model. As displayed in Fig. 4, a computation

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FH

Z Displacement MH

FV

B1

MV

B2

B3

BH

BV

B4

B5

Fig. 4. Model of the casing.

software was used to create the finite element model of the casing, while the material is titanium alloy. Three restrictions are used in the casing model. The full degrees of freedom of the Fixed Support regions shown in Fig. 4 are constrained, while the Z directional degree of freedom of the Z Displacement region is constrained. B1B5 in Fig. 4 illustrate the position of the rotor supports on the casing. In fact, the support structures are installed on the inner surface of the casing, and these symbols are merely intended to indicate the relative position of them. The FRFs of each support position, named FRFs-SS (support position to support positions), are calculated by applying a series of sweeping forces from 0 Hz to 260 Hz in the horizontal (Y) and vertical (Z) directions at B1B5 (the direction is consistent with that shown in Fig. 4). Taking the condition that the sweeping force is applied in B1 as an example, the FRFs-SS are shown in Fig. 5. When the sweeping force is applied in the Y direction, it can be seen that the FRFs-SS in the Y direction are significantly larger than those in the Z direction. Similarly, if the sweeping forces are applied in the vertical direction, the FRFs-SS in the vertical direction are more significant than those in the horizontal direction. In the casing’s front, middle, and back sections, the points at the horizontal and vertical of each section are chosen as vibration monitoring points. These six points are designated as FH, FV, MH, MV, BH, and BV. Their positions are illustrated in Fig. 8. Same as FRFs-SS, the FRFs of vibration monitoring points are obtained by applying a series of sweeping forces at B1B5, named FRFs-SM (support position to monitoring points). Figure 6 shows the results that the sweeping force is applied in B1. bc The flexibility matrix Gbb c and Gc can be obtained according to FRFs-SS and FRFsbc SM, respectively. What’s more, the flexibility matrix Gbb c and Gc can also be experimentally acquired through a real engine casing. However, in the experiment, accelerations rather than displacements of the casing are being evaluated for vibrations. In order to acquire the flexibility matrix, it is necessary to integrate the accelerations in the frequency domain. The procedure is not repeated here. 3.2 Unbalance Exists in the HP Rotor Squeeze film dampers are installed in the dual-rotor system displayed in Fig. 3. The locations and structures of SFDs are listed in Table 1.

An Unbalance Identification Method of a Whole Aero-Engine (a)

(b)

(c)

(d)

473

Fig. 5. FRFs-SS with sweeping force is applied in B1: (a) FRFs-SS of the Y direction with the Y direction force is applied; (b) FRFs-SS of the Y direction with the Y direction force is applied; (c) FRFs-SS of the Z direction with the Y direction force is applied; (d) FRFs-SS of the Z direction with the Z direction force is applied.

Table 2 lists the unbalance distribution of the HP rotor. According to Eq. (17) and Eq. (7), the vibrations of the monitoring points of the casing and the turbine on the HP rotor with the influence of flexible support and SFDs are obtained through the FRFs-SS and FRFs-SS. Figure 7 exhibits the vibrations of the monitoring points of the casing. Affected by the flexible casing and nonlinear oil film forces of SFDs, the vibrations are very complex with multiple peaks and messy phases. There are two reasons for this phenomenon. Initially, the resonances appear near not only at the natural frequencies of the casing but the natural frequencies of the rotor-casing coupled system because of the flexible casing. Furthermore, the HP and LP rotors are connected by the casing and intershaft bearing, leading the forces of bearings will cause the LP rotor to vibrate, although the unbalance only emerges in the HP rotor. In Fig. 7, the maximum vibration of the casing monitoring points appears at 6000 rpm when the unbalance is as shown in Table 2. In addition, it can be found from Fig. 5 (d) and Fig. 6 (d) that the FRFs have peaks around 105–110 Hz, which indicates that a natural frequency of the casing exists between 105 and 110 Hz. After coupling with the rotor, the natural frequency changes, resulting the significant vibration at 6000 rpm. Thus, 6000 rpm has been picked as the balancing speed, and the disks at the primary compressor, 2nd stage compressor, 5th stage compressor, 6th stage compressor and

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(a)

(b)

(c)

(d)

Fig. 6. FRFs-SM with sweeping force is applied in B1: (a) FRFs-SM of the Y direction with the Y direction force is applied; (b) FRFs-SM of the Y direction with the Z direction force is applied; (c) FRFs-SM of the Z direction with the Y direction force is applied; (d) FRFs-SM of the Z direction with the Z direction force is applied.

Table 1. Locations and structures of SFDs Location

Oil

Width (mm)

Diameter (mm)

Clearance (mm)

Bearing 1

VG32

18

180

0.13

Bearing 4

VG32

20

200

0.14

Bearing 5

VG32

22

220

0.16

turbine of the HP rotor are chosen as the balancing disks to add balancing weights. Using the method mentioned in Sect. 2.3, the equivalent unbalance of the HP rotor has been determined according to the vibrations of the casing monitoring points. The results are displayed in Table 3. It should be noticed that there is a 180° difference between the phase of balancing weights and equivalent unbalance. Comparing Fig. 7 and Fig. 8 reveals that the vibrations of the casing monitoring points are reduced after adding balancing weights to the balancing disks, especially at the balancing speed (6000 rpm). Table 4 lists the vibrations of the casing monitoring

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Table 2. Initial unbalance distribution of the HP rotor Location

Unbalance (g.mm) Phase (°)

Primary compressor

300

-60

3rd stage compressor 120

60

4th stage compressor 180

50

5th stage compressor 130

70

Turbine

-80

200

(b)

(a)

Fig. 7. Vibrations of the monitoring points of casing at initial unbalance (unbalance exists in the HP rotor): (a) Y direction; (b) Z direction. Table 3. Identification of equivalent unbalance of the HP rotor Location

Unbalance (g.mm)

Phase (°)

Primary compressor

100

43

2nd stage compressor

111

37

5th stage compressor

116

26

6th stage compressor

103

19

Turbine

76

−100

points before and after balancing at 6000 rpm, with the vibration reduction rate Au is    A2 − A1   × 100%  (33) Au =  A1 

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(a) 10-5

1.5 1 0.5 0

10-5

2.5 FH FV MH MV BH BV

Amplitude m

Amplitude m

2

FH FV MH MV BH BV

2 1.5 1 0.5

0 0

6000

12000

18000

0

6000

4

2

2

Phase rad

Phase rad

4

0

18000

12000

18000

0

-2

-2 -4

12000

Speed rpm

Speed rpm

0

6000

12000

18000

-4

0

6000

Speed rpm

Speed rpm

Fig. 8. Vibrations of the monitoring points of casing after balancing (balancing at 6000 rpm): (a) Y direction; (b) Z direction. Table 4. Vibrations comparison of the casing monitoring points (balancing at 6000 rpm) Location

Vibrations in the Y direction

Vibrations in the Z direction

Before (μm)

After (μm)

Au (%)

After (μm)

Au (%)

FH

36.74

0.14

99.62

58.89

0.24

99.59

FV

33.04

0.08

99.76

101.02

0.4

99.6

MH

25.49

0.07

99.73

11.42

0.04

99.65

MV

8.37

0.03

99.64

39.02

0.16

99.59

BH

2.83

0.04

98.59

16.5

0.07

99.58

BV

36.53

0.17

99.53

51.18

0.2

99.61

Before (μm)

In conclusion, the Aus of all monitoring points of the casing are above 99.5% after the rotor system being balanced at 6000 rpm. The results demonstrate that the equivalent unbalance identification method can successfully identify the unbalance for the HP rotor at a single balancing speed with an outstanding performance. 3.3 Unbalance Exists in the LP and HP Rotors When the HP and LP rotors of the engine exist unbalance simultaneously, the vibrations of the casing are the superposition of vibrations aroused by the HP rotor and the LP rotor. In order to determine the unbalance of the whole aero-engine, it is necessary to separate the vibrations caused by the LP rotor and the HP rotor, and identify the equivalent unbalance of the LP and HP rotors, respectively. The Fourier series expansion of the casing vibrations X c in the time domain is Xc = X0 +

k  i=1

X ic cos(ωi t) + X is sin(ωi t)

(34)

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with ωi are the frequencies of the casing vibrations, such as the rotational frequencies of the HP and LP rotors, and the frequencies generated by other forces. The amplitude and phase of the vibration for ωi are   i 2  i 2 X ia = Xc + Xs (35a) X ip =

⎧ i ⎨ arctan − Xis , X ic > 0 X c

⎩ π + arctan − X s , X i < 0 c Xi i

(35b)

c

with

 2 T X c cos(ωi t)dt T 0  2 T i Xs = X c sin(ωi t)dt T 0 X ic =

(36)

Given that the speed ratio between the LP and HP rotors is 1:1.2, the vibrations excited by the unbalanced forces of the LP rotor and the HP rotor can be obtained by substituting ωL and ωH into Eq. (34) and Eq. (35), with ωL and ωH are the rotation frequencies of the LP and HP rotors. Same as 6000 rpm in Sect. 3.2, 10500 rpm is selected as the balancing speed, as the casing vibrates significant at this speed. What’s more the FRFs in Fig. 5 (a) and Fig. 6 (a) have peeks near 175 Hz. The vibrations of the casing monitoring points in the time domain between the Keyphasor marks of the LP or HP rotor are picked up to evaluate the Fourier series. If the Keyphasor is installed on the LP rotor, the data within n × 5T L should be selected for analysis, as the speed ratio between the LP and HP rotors is 1:1.2. Where T L is the rotation period of the LP rotor and n is an integer. This process is depicted in Fig. 9.

Fig. 9. Process of calculating the Fourier series of the vibrations of casing monitoring points.

The amplitudes and phases of the vibrations of each casing monitoring point induced by LP and HP rotors are estimated and displayed in Table 6, with the unbalance distribution is the same as depicted in Table 5.

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It is feasible to determine the unbalance of the whole aero-engine based on the vibrations created by the LP and HP rotors after separating the excitation sources of the vibrations. Table 7 displays the equivalent unbalance of the LP and HP rotors when the vibrations are as indicated in Table 6. Table 5. Initial unbalance distribution of the LP and HP rotors Location

Unbalance (g.mm)

Phase (°)

3th stage fan

70

60

LP turbine

150

0

2nd compressor

180

110

5th compressor

130

60

HP turbine

170

-80

Table 6. Vibrations separation of the casing monitoring points (10500 rpm) Point

Vibrations caused by the LP rotor

Vibrations caused by the HP rotor

Y direction

Y direction

Amplitude (μm)

Z direction Phase (rad)

Amplitude (μm)

Z direction

Phase (rad)

Amplitude (μm)

Phase (rad)

Amplitude (μm)

Phase (rad)

FH

95.12

−1.52

24.52

−1.97

15.11

−2.21

13.72

0.05

FV

112.31

−1.58

15.24

−2.1

15.17

−2.33

14.48

0

MH

51.81

−1.67

39.9

1.35

8077

1.66

8.75

1.02

MV

31.44

−1.59

50.34

1.36

11.75

1.56

12.34

0.93

BH

20.04

1.64

3.59

1.15

4.06

−1.58

8.41

−2.08

BV

6.16

−1.75

2.89

-0.53

4.58

−1.86

−3.02

20.08

Figure 10 displays the vibration orbits of each casing monitoring point before and after balancing. Due to the different frequencies of unbalanced forces excited by LP and HP rotors and the effect of the nonlinear oil film forces, the vibration orbit is no longer a circle or ellipse. Figure 10 shows that the vibrations of the casing are significantly decreased after balancing through the method proposed in this Section.

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Table 7. Identification of equivalent unbalance (10500 rpm) The equivalent unbalance of LP rotor

The equivalent unbalance of HP rotor

Location

Unbalance (g.mm)

Phase (°)

Location

Unbalance (g.mm)

Phase (°)

Primary fan

4.15

18

Primary compressor

48.8

128 104

66.27

65

2nd compressor

64.7

3th stage fan

5th compressor

71.9

76

6th compressor

48.3

56

HP turbine

203

83

LP turbine

149

-1

FH

FV

MH

BH

BV

Before balanceing After balancing

MV

Fig. 10. Comparison of casing vibration orbits before and after balancing (10500 rpm).

4 Conclusion In this paper, a dynamic model of a whole aero-engine considering the flexible casing is established to determine the unbalanced response of the rotor and casing under the influence of the SFDs. According to the model, a nonlinear unbalance identification method is proposed based on the vibrations of casing monitoring points. This method accounts for the impacts of the flexible casing, the nonlinear oil film forces, and the gyroscopic moments of rotor. It has been demonstrated that this technology is suitable for non-invasive unbalance identification of an engine and other rotating equipment for

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its excellent ability to reduce the vibrations of the rotor. Through the simulation of a dual-rotor system, the following conclusions are obtained. (1) When the unbalance exists in the HP rotor, the vibration reduction rate Aus are higher than 99.5% after balancing. (2) In the case that the LP and HP rotor exists unbalance simultaneously, the method described in this research can effectively separate the vibrations induced by LP and HP rotors according to the vibrations of casing monitoring points. By balancing the LP and HP rotors separately, the vibration of the casing and rotor can be greatly decreased.

References 1. Yang, Y., Ouyang, H., Yang, Y., et al.: Vibration analysis of a dual-rotor-bearing-double casing system with pedestal looseness and multi-stage turbine blade-casing rub. Mechanical Systems and Signal Process. 143 (2020) 2. Wang, N., Liu, C., Jiang, D., et al.: Casing vibration response prediction of dual-rotor-bladecasing system with blade-casing rubbing. Mech. Syst. Signal Process. 118, 61–77 (2019) 3. Hai, P.M., Bonello, P.: An impulsive receptance technique for the time domain computation of the vibration of a whole aero-engine model with nonlinear bearings. J. Sound Vib. 318, 592–605 (2008) 4. Bonello, P., Minh Hai, P.: A receptance harmonic balance technique for the computation of the vibration of a whole aero-engine model with nonlinear bearings. J. Sound and Vibration 324, 221–242 (2009) 5. Choi, B.L., Park, J.M.: An improved rotor model with equivalent dynamic effects of the support structure. J. Sound Vib. 244, 569–581 (2001) 6. Hu, L., Palazzolo, A.: Solid element rotordynamic modeling of a rotor on a flexible support structure utilizing multiple-input and multiple-output support transfer functions. J. Eng. Gas Turbines & Power 139 (2017) 7. Dewi, D.K., Abidin, Z., Budiwantoro, B., Malta, J.: Dimensional analysis of a rotor system through FRF using transfer function and finite element methods. J. Mech. Sci. Technol. 34(5), 1863–1870 (2020) 8. Goodman, T.P.: A least-squares method for computing balance corrections. J. Manuf. Sci. Eng. 86, 273–277 (1964) 9. Bishop, R., Gladwell, G.: The vibration and balancing of an unbalanced flexible rotor. Archive J. Mechanical Eng. Sci. 1959–1982, 1(1), 66–77 (1959) 10. Darlow, M.S.: Balancing of high-speed machinery: theory, methods and experimental results. Mech. Syst. Signal Process. 1, 105–134 (1987) 11. Chatzisavvas, I., Dohnal, F.: Unbalance identification using the least angle regression technique. Mech. Syst. Signal Process. 50, 706–717 (2015) 12. 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) 13. Yao, J., Yang, F., Su, Y., et al.: Balancing optimization of a multiple speeds flexible rotor. J. Sound Vib. 480, 115405 (2020) 14. Krodkiewski, J.M., Ding, J., Zhang, N.: Identification of unbalance change using a non-linear mathematical model for multi-bearing rotor systems. J. Sound Vib. 169, 685–698 (1994) 15. Ding, J., Al-Jumaily, A.: A linear regression model for the identification of unbalance changes in rotating machines. J. Sound Vib. 231, 125–144 (2000)

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16. Sinha, J.K., Lees, A.W., Friswell, M.I.: Estimating unbalance and misalignment of a flexible rotating machine from a single run-down. J. Sound Vib. 272, 967–989 (2004) 17. Torres Cedillo, S.G., Bonello, P.: An equivalent unbalance identification method for the balancing of nonlinear squeeze-film damped rotordynamic systems. J. Sound Vibration 360, 53–73 (2016) 18. Cedillo, T., Guillermo, S.: The Identification of Unbalance in a Nonlinear Squeeze-Film Damped System using an Inverse method - a Computational and Experimental study (2015)

Author Index

A Ai, Yan-ting 76 An, Yuchen 439 Andrianoely, Marie-Ange

H Heya, Akira 409 Holopainen, Timo P. 398 Hong, Jie 34, 202, 342 Hou, Li 11, 224 Hu, Xiuli 156 Huo, Guanghe 188

268

B Baguet, Sébastien 268 Briançon, Laurent 268 C Chang, Jing 452 Chasalevris, Athanasios 304, 324 Che, Renwei 156 Chen, Ren-zhen 76 Chen, Xueqi 202 Chen, Zhaobo 1, 91, 276 Chen, Zhoudian 101 Chouchane, Mnaouar 419

G Gao, Xianghong 289 Gavalas, Ioannis 304, 324 Gladkiy, Ivan L. 168 Grange, Stéphane 268 Guo, Mei 147

J Jia, Ruiqi 342 Jiang, Minghong 289 Jiang, Yaqun 377 Jiang, Zihan 224 Jiao, Yinghou 1, 91, 156, 188, 276 Jin, Long 243 K Kanty, Piotr 268 Kimura, Shogo 409 Kuang, Fanrong 147

D Dai, Huwei 56 Degtyarev, Sergey A. 168 Dimou, Emmanouil 304, 324 Dohnal, Fadi 324 Dou, Jinxin 133 Dufour, Régis 268 F Fernandez-del-Rincon, Alfonso Fetisov, Alexander 364 Fu, Jie 34

I Inoue, Tsuyoshi 409 Inozemtsev, Alexander A. 168

188

L Leontiev, Mikhail K. Li, Chao 34 Li, Haofan 147 Li, Hui 133, 429 Li, Jian 117 Li, Jianlei 133 Li, Lei 243 Li, Qihang 464 Li, Wengheng 289 Li, Yingjie 117 Li, Yuqi 243 Li, Zhitong 91 Lin, Dafang 101 Lin, Hao 257

© 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 140, pp. 483–484, 2024. https://doi.org/10.1007/978-3-031-40459-7

168

484

Author Index

Lin, Jiewei 56 Liu, Cong 342 Liu, Fangming 202 Liu, Jing 439 Liu, Yuan 101 Liu, Zhirou 257 Lu, Xin 56 Luo, Zhong 243, 452

Wang, Jiale 464 Wang, Siji 101 Wang, Weimin 464 Wang, Yongfeng 342 Wang, Yongquan 117 Wang, Zhi 76 Wei, Haibo 257 Wei, Jing 257 Wei, Yuan 1 Wen, Chuanmei 243 Wen, Haobin 390 Wu, Bin 56 Wu, Yangjun 429

M Ma, Wenbo 276 Ma, Yanhong 34, 202 P Pan, Guang

439

R Ran, Xuhe 1 Ryyppö, Tommi

X Xu, Jian 56 Xu, Yeyin 91, 276 Xu, Ziyang 257 Xue, Rui 133

398

S Sakly, Faiza 419 Sanchez-Espiga, Javier 188 Santamaria, Miguel Iglesias 188 Savin, Leonid 364 Shaposhnikov, Konstantin V. 168 Shutin, Denis 364 Sinha, Jyoti K. 390 T Taura, Hiroo 409 Tezenas du Montcel, Florian Tian, Jing 76 V Viadero-Rueda, Fernando W Wang, Cai 76 Wang, Chengyang 101 Wang, Cun 11, 224

268

188

Y Yang, Chiye 439 Yao, Hongliang 133, 429 Yu, Pingchao 11, 224 Yuan, Yunbo 117, 147 Z Zhang, Feng-ling 76 Zhang, Jinqi 101 Zhang, Junhong 56 Zhang, Long 390 Zhang, Sai 156 Zhang, Xiang 188 Zhang, Yujie 257 Zhang, Zexin 117 Zhao, Guang 117, 147 Zhao, Runchao 91 Zhao, Xiangyang 147 Zhu, Changsheng 289 Zhu, Zhimin 243