Vibration Engineering and Technology of Machinery: Proceedings of VETOMAC XV 2019 3030606937, 9783030606930

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Mechanisms and Machine Science

José Manoel Balthazar   Editor

Vibration Engineering and Technology of Machinery Proceedings of VETOMAC XV 2019

Mechanisms and Machine Science Volume 95

Series Editor Marco Ceccarelli , Department of Industrial Engineering, University of Rome Tor Vergata, Roma, Italy Editorial Board Alfonso Hernandez, Mechanical Engineering, University of the Basque Country, Bilbao, Vizcaya, Spain Tian Huang, Department of Mechatronical Engineering, Tianjin University, Tianjin, China Yukio Takeda, Mechanical Engineering, Tokyo Institute of Technology, Tokyo, Japan Burkhard Corves, Institute of Mechanism Theory, Machine Dynamics and Robotics, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany Sunil Agrawal, Department of Mechanical Engineering, Columbia University, New York, NY, USA

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.

More information about this series at http://www.springer.com/series/8779

José Manoel Balthazar Editor

Vibration Engineering and Technology of Machinery Proceedings of VETOMAC XV 2019

Editor José Manoel Balthazar São Paulo State University Rio Claro, São Paulo, Brazil

ISSN 2211-0984 ISSN 2211-0992 (electronic) Mechanisms and Machine Science ISBN 978-3-030-60693-0 ISBN 978-3-030-60694-7 (eBook) https://doi.org/10.1007/978-3-030-60694-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 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

Machines of all kinds are used in nearly every aspect of our daily lives from the vacuum cleaner and washing machine we use at home to the industrial machinery used to manufacture nearly every product we use on a daily basis. On other hand, the research development on nonlinear dynamics, nowadays, continuously reveals that nonlinear phenomena can bring many amazing and advantageous effects in every practical machinery engineering problem, such as vibration control, energy harvesting, structure health monitoring, micro/nano-electro-mechanical systems, and so on. Recent trends in machinery vibration and technology including nonlinear systems and phenomena (cases studies) are the main strength of this book of interest to both researchers and practicing engineers. Following the scientific tradition of the conference VETOMAC, its 15th edition is internationally recognized as a central forum for discussing scientific achievements and is intended to provide a widely selected forum among scientists and engineers to exchange methods, techniques, and ideas related to Vibration Engineering and Technology of Machinery problems So, this book presents the most significant contributions to the VETOMAC 2019 Conference held in Hotel Nacional Inn, Curitiba, Paraná, Brazil from November 10 to 15, 2019, covering a range of Vibration Engineering and Technology of Machinery problems to provide insights into recent trends and advances in a broad variety of fields in dynamics and control. VETOMAC 2019 was promoted by The Vibration Institute of India through its International Steering of JVET (Journal of Vibration Engineering & Technologies) and Local Committees at Curitiba, Brazil. All the papers gathered here will be of interest to all researchers, graduate students, and engineering professionals working in the fields of Vibration Engineering and Technology of Machinery problems and related areas around the globe. It should be emphasized that all the chapters have been reviewed by two independent referrers and authors are responsible for their opinions expressed in their work. This book comprises 29 contributions from different countries. The main keywords are vibration engineering, technology of machinery problems, nonlinear phenomena, and control design. v

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The chapters will be subdivided into four main areas: • • • •

Concepts and methods in dynamics, containing seven sub-chapters, Dynamics of mechanical and structural system, containing five sub-chapters, Dynamics and control containing seven sub-chapters, and Recent and emergent trends in dynamics and control containing ten sub-chapters.

I would like to thank the authors, presenters, and session chairs for their participation. Special gratitude must be extended to several individuals whose invaluable help enabled the organization of the VETOMAC XV 2019 Conference in Hotel Nacional Inn, Curitiba, Paraná, Brazil. Sincere gratitude is also expressed to the Steering Committee: Prof. Jammi Srinivasa Rao—India (in memorium), Prof. Jose Manoel Balthazar— Brazil, Prof. Chee Wah Lim—China, Prof. Jyoti K. Sinha—UK, Prof. C Nataraj— USA, Prof. Romuald Rz˛adkowski—Poland, Prof. Ronald L. Eshleman—USA; to the Scientific Committee: Prof. Mohamed Belhaq—Morroco, Prof. Giuseppe Rega—Italy, Prof. Livija Cveti´canin—Serbia, Prof. José A. Tenreiro Machado—Portugal, Prof. Ferdinand Verhulst—The Netherlands, Prof. Oded Gottlieb—Israel, Prof. Marcelo José Santos de Lemos—Brazil, Prof. Dumitru I. Caruntu—USA, Prof. Dr. Mariano Febbo— Argentina, Prof. Walter Lacarbonara—Italy, Prof. El˙zbieta Jarz˛ebowska—Poland; and to the Local Committee: Prof. Dr. Jose Manoel Balthazar—Chair, Prof. Dr. Angelo Marcelo Tusset—Vice Chair, Prof. Dr. Giane Gonçalves Lenzi, Prof. Dr. Mauricio Aparecido Ribeiro, Prof. Dr. Dailhiane Grabowski Bassinello, Engg. Wagner Barth Lenz, Prof. Dr. Edson Hideki Koroishi, Prof. Dr. Atila Madureira Bueno, Prof. Dr. Suelia De Siqueira Rodrigues Fleury Rosa, Prof. Dr. Américo Barbosa Cunha Junior, Prof. Dr. Airton Nabarrete, Prof. Dr. Silvio Luiz Thomaz de Souza, Eng. Civil. Giovanna Gonçalves, Prof. Dr. Eduardo Marcio de Oliveira Lopes, Prof. Dr. Carlos Alberto Bavastri, and all reviewers. I would like to give a special thanks to Ms. Nathalie Jacobs for her help and encouragements to the publication of this volume. Finally, the history of VETOMAC Series of Conferences (in collaboration with the Vibration Institute of India) can be summarized as follows: • • • •

VETOMAC I, 2000: Indian Institute of Science, Bangalore, India; VETOMAC II, 2002: Bhabha Atomic Research Centre, Mumbai, India; VETOMAC III, 2004: Indian Institute of Technology, Kanpur, India; VETOMAC IV, 2007: University College of Engineering, Osmania University, Hyderabad, India, Bharat Heavy Electricals Limited, Hyderabad, India; • VETOMAC V, 2009: Huazhong University of Science and Technology, Wuhan, China and City University of Hong Kong, China;

Preface

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• VETOMAC VI, 2010: Indian Institute of Technology, New Delhi, India, VETOMAC VII, 2011: Shanghai Jiao Tong University and City University of Hong Kong, China, VETOMAC VIII, 2012: Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdansk, Poland; • VETOMAC IX, 2013: Nanjing University of Aeronautics and Astronautics, Nanjing, China; • VETOMAC X, 2014. University of Manchester, UK, VETOMAC XI. 2015, National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan; • VETOMAC XII, 2016, Organized by Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdansk and Air Force Institute of Technology, Warsaw, Poland; • VETOMAC XIII, 2017, Organization 12th World Congress on Engineering ASSET Management and 13th International Conference on Vibration Engineering and Technology of Machinery Brisbane, Queensland, Australia; • VETOMAC XIV, 2018, organized by Faculdade de Ciências e Tecnologia of Universidade Nova de Lisboa (DEC/FCT/UNL) and IDMEC—Institute of Engineering Mechanics of Instituto Superior Técnico of University of Lisbon (IDMEC/IST/UL); and • VETOMAC XV, 2019, organized by Universidade Tecnológica Federal do Paraná, Câmpus Ponta Grossa, Paraná/Brazil.

Curitiba, PR, Brazil

Prof. José Manoel Balthazar UNESP-Universidade Estadual Paulista Bauru, SP, Brazil UTFPR-Universidade Tecnológica Federal do Paraná Ponta Grossa, PR, Brazil

Contents

Concepts and Methods in Dynamics Stabilization of Chaos Via Strong Nonlinearities: The Lorenz-Malkus Wheel Under Coulomb and Hystersis Frictions . . . . . . . . Mikhail E. Semenov, Evgeny A. Karpov, Sergey G. Tikhomirov, Peter A. Meleshenko, and Margarita Teplyakova Drive Dynamics of Vibratory Machines with Inertia Excitation . . . . . . . . Nikolay Yaroshevich, Olha Yaroshevych, and Viktor Lyshuk

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Quasiperiodic Stability Diagram in a Nonlinear Delayed Self-Excited Oscillator Under Parametric Coupling . . . . . . . . . . . . . . . . . . . Ilham Kirrou and Mohamed Belhaq

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Harmonic Balance of Bouc-Wen Model to Identify Hysteresis Effects in Bolted Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luccas Pereira Miguel, Rafael de Oliveira Teloli, and Samuel da Silva

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Dynamic Friction Model Study Applied to a Servomechanism at Low Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rudnei Barbosa, Átila Madureira Bueno, José Manoel Balthazar, Paulo José Amaral Serni, and Daniel Celso Daltin Signal Analysis Through the Ensemble Empirical Mode Decomposition and Hilbert-Huang Transform-Application to Vortex Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ana Paula Ost, Alexandre Vagtinski de Paula, and Sergio Viçosa Möller

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Numerical Assessment of the Pressure Recovery of the Turbulent Flow in a Venturi-Type Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Naítha Mallmann Caetano and Luiz Eduardo Melo Lima

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Dynamics of Mechanical and Structural System Delamination Fault Compensation in Composite Structures . . . . . . . . . . . 141 Luke Megonigal, Foad Nazari, Amirhassan Abbasi, T. Haj Mohamad, and C. Nataraj Space Robotics and Associated Space Applications . . . . . . . . . . . . . . . . . . . 151 Ijar M. da Fonseca Stick-Slip Phenomenon: Experimental and Numerical Studies . . . . . . . . . 171 Ingrid Pires and Hans Ingo Weber Determination of Wheel-Rail Interaction Forces of Railway Vehicles for Evaluation of Safety Against Derailment at Running on Twisted Tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Ion Manea, Marius Ene, Ion Girnita, Gabi Prenta, and Radu Zglimbea A Small-Scale Dynamometer Roller Analysis by Laval Rotor Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Maria Augusta M. Lourenço, Fabricio L. Silva, Ludmila C. A. Silva, Jony J. Eckert, and Fernanda C. Corrêa Dynamics and Control Optimal Control for Path Planning on a 2 DOF Robotic Arm with Prismatic and Revolute Elastic Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Jose A. G. Luz Junior, Angelo M. Tusset, Mauricio A. Ribeiro, and Jose M. Balthazar Numerical and Experimental Analysis of a Hybrid (Passive-Adaptive) Vibration Control System in a Cantilever Beam Under Broadband Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Maurizio Radloff Barghouthi, Eduardo Luiz Ortiz Batista, and Eduardo Márcio de Oliveira Lopes Retroactive Control Applied to a BLDC Motor . . . . . . . . . . . . . . . . . . . . . . . 233 Carlos da Conceição Castilho Neto, Lenon Diniz Seixas, and Fernanda Cristina Corrêa Modeling, Construction and Control of Quadrotors . . . . . . . . . . . . . . . . . . . 245 Fernando M. B. Lima, Átila M. Bueno, and Paulo S. Silva Stabilization of a Flexible Inverted Pendulum via Hysteresis Control: The Bouc-Wen Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Mikhail E. Semenov, Andrey M. Solovyov, Peter A. Meleshenko, and Olesya I. Kanishcheva

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State Observer Applied to Position and Vibration Control Using Flexible Link Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Daniel Celso Daltin, Átila Madureira Bueno, José Manoel Balthazar, Paulo José Amaral Serni, and Rudnei Barbosa Time-Delayed Feedback Control Applied in a Circuit with a (PbTiO3 ) Ferroelectric Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Thiago G. do Prado, Vinícius Piccirillo, Angelo Marcelo Tusset, Frederic Conrad Janzen, and Jose Manoel Balthazar Recent and Emergent Trends in Dynamics and Control Modeling, Identification and Controller Design for an Electrostatic Microgripper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Andrei A. Felix, Diego Colón, Bruno M. Verona, Luciana W. S. L. Ramos, Houari Cobas-Gomez, and Mario R. Gongora-Rubio Using Different Approximations of Averaging Method in Theory of Micro Electromechanical Systems (MEMS) . . . . . . . . . . . . . . . . . . . . . . . . 333 G. A. Kurina, J. M. Balthazar, and A. M. Tusset Mathematical Analysis of Electroencephalography Applied to Control Brain Machine Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Cristhiane Gonçalves and Sergio Okida Remarks on a PVDF Piezo-Wind Generator . . . . . . . . . . . . . . . . . . . . . . . . . 357 Itamar Iliuk, Felipe A. Nazario, Jose M. Balthazar, Angelo M. Tusset, and Jose R. C. Piqueira A Bond Graph Approach to Modelling of the Human Skin . . . . . . . . . . . . 369 Marcos Augusto Moutinho Fonseca, Rebeca Hannah de Melo Oliveira, Ludmila Evangelista dos Santos, Luciana Alves Fernandes, Murilo Venturin, and Suélia de Siqueira Rodrigues Fleury Rosa Dynamics and Control of Energy Harvesting from a Non-ideally Excited Portal Frame System with Fractional Damping . . . . . . . . . . . . . . . 383 Angelo M. Tusset, Rodrigo T. Rocha, Itamar Iliuk, Jose M. Balthazar, and Grzegorz Litak On the Classical and Fractional Control of a Nonlinear Inverted Cart-Pendulum System: A Comparative Analysis . . . . . . . . . . . . . . . . . . . . 397 José Geraldo Telles Ribeiro, Julio Cesar de Castro Basilio, Americo Cunha Jr, and Tiago Roux Oliveira

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Dynamics of the Pressure Fluctuation in the Riser of a Small Scale Circulating Fluidized Bed: Effect of the Solids Inventory and Fluidization Velocity Under the Absolute Mean Deviation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Flavia Tramontin Silveira Schaffka, Jhon Jairo Ramírez Behainne, Maria Regina Parise, and Guilherme José De Castilho Attenuation of the Vibration in a Non-ideal Excited Flexible Electromechanical System Using a Shape Memory Alloy Actuator . . . . . 431 Adriano Kossoski, Angelo M. Tusset, Frederic C. Janzen, Mauricio A. Ribeiro, and Jose M. Balthazar Dynamics of Coupled Nonlinear Oscillators with Mistuning . . . . . . . . . . . 445 Grzegorz Litak, Grzegorz Kudra, and Jan Awrejcewicz Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

Concepts and Methods in Dynamics

Stabilization of Chaos Via Strong Nonlinearities: The Lorenz-Malkus Wheel Under Coulomb and Hystersis Frictions Mikhail E. Semenov, Evgeny A. Karpov, Sergey G. Tikhomirov, Peter A. Meleshenko, and Margarita Teplyakova Abstract In this chapter we consider the modified Lorenz-Malkus water wheel model. Within a novel approach we take into account friction features on the rim of the water wheel formalized by strong nonlinearities. Namely, the dry friction (within the Coulomb model) and hysteresis friction (within the Bouc-Wen model and the Dahl model) are considered. The dynamic characteristics such as fixed points, Lyapunov characteristic exponents, bifurcation diagrams, are presented and discussed. Detailed analysis of a 2-dimensional Lorenz-Malkus system (where the third coordinate is supposed to be constant) is also presented and discussed. Namely we show the bifurcation process where two saddles and stable node birth from a saddle. It is shown that the static friction (formalized within the Coulomb model) leads to stabilization of the system at the origin independent on the value of the friction coefThis chapter is an extension of the work “Chaos vs Hysteresis: water wheel under dry friction” presented on VETOMAC-2019, Curitiba, Brazil. M. E. Semenov (B) Geophysical Survey of Russia Academy of Sciences, Lenina av. 189, 249035 Obninsk, Russia e-mail: [email protected] Meteorology Department, Zhukovsky–Gagarin Air Force Academy, Starykh Bolshevikov st. 54 “A”, 394064 Voronezh, Russia Mathematics Department, Voronezh State Technical University, XX-letiya Oktyabrya st. 84, 394006 Voronezh, Russia M. E. Semenov · E. A. Karpov · P. A. Meleshenko · M. Teplyakova Digital Technologies Department, Voronezh State University, Universitetskaya sq. 1, 394006 Voronezh, Russia e-mail: [email protected] S. G. Tikhomirov Voronezh State University of Engineering Technologies, Revolutsii Prospect 19, 394036 Voronezh, Russia P. A. Meleshenko Target Search Lab of Groundbreaking Radio Communication Technologies of Advanced Research Foundation, Plekhanovskaya st. 14, 394018 Voronezh, Russia

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_1

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ficient. At the same time we show that using certain parameters within the Bouc-Wen and Dahl models, chaotic behaviour can be controlled. A way to use the modified Lorenz-Malkus system with certain parameters as a “natural” pseudo-random numbers generator is also discussed. Keywords Chaos · Lorenz system · Coulomb friction · Hysteresis · Bouc-wen model · Dahl model · Lyapunov characteristic exponents · Bifurcation diagrams

1 Introduction 1.1 The Lorenz System In 1963 American researcher Edward Lorenz engaged with weather forecasting problems and published famous paper 3 [1] “Deterministic Nonperiodic Flow” in the Journal of the Atmospheric Sciences where he showed that relatively simple system of three ordinary differential equations (which is well-known now as the “Lorenz system”), which was obtained during the analysis of the convection process in a fluid layer, demonstrated unexpected behaviour. This research was a starting point for such a modern field as the chaos theory. The Lorenz system ⎧ ⎨ x˙ = σ (y − x), y˙ = x(r − z) − y, (1) ⎩ z˙ = x y − bz, includes variables x, y and z that have the following sense (in terms of the convection problem): • x is proportional to the rate of convection; • y is proportional to the horizontal temperature variation; • z is proportional to the vertical temperature variation. Parameters σ , r and b are proportional to the Prandtl number, the Rayleigh number, and the coefficient corresponding to the geometry of the convective region, respectively. During numerical simulations, Lorenz obtained some unexpected feature of system (1). Such a feature plays a special role in the following investigation of the system and reflects the high sensitivity of the solution to small deviations of initial conditions. Using the standard iterative procedure it was shown that the sensitivity to small deviations of initial conditions leads to the divergence of the initially close solutions. In our days this feature1 is called the dependence of the solution on the initial conditions (Fig. 1). 1 Note,

that it can be served as a main sign of the chaotic behaviour.

Stabilization of Chaos Via Strong Nonlinearities …

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Fig. 1 A solution to system (1). Dependence of variable x on the time for two systems with close initial conditions: initial condition for red curve is (2.01, −2, 2) and for blue curve is (2, −2, 2)

For many years researchers detected chaotic behavior in different physical models such as lasers [2], dissipative oscillator with inertial excitation [3], chemical reactions [4], and others. Models of these systems can be described in terms of the Lorenz system (1). Recent works [5, 6] are dedicated to modifications of the Lorenz system where ordinary derivatives were replaced by fractional derivatives (the so-called fractional Lorenz system). Particularly, in [5] athors reported a new algorithm of calculus for fractional derivatives, moreover they obtained that chaos in the fractional-order system exists with order lower than 2.97. Novel results in chaos control and system synchronization together with the corresponding analytical results were presented and disccused in [6], where the Malkus water wheel model (which is also described by the Lorenz system) was considered in terms of the fractional calculus. Attracting regions of the Lorenz system are also important charcteristics of chaotic systems and have to be examined in details as was done in [9]. Particularly, authors investigated the form of attractors and their behavior by expanding the space R into space C. Results of [9] are proved that the obtained attractors are not chaotic. Another important chracteristics within the nonlinear dynamics are the so-called homoclinic orbits. In [7, 8] the main objects for analysis are homoclinic orbits of the Lorenz system as well as their construction. One of the major questions in the chaos theory (from both fundamental and applied points of view) is chaos control. Nowadays, a wide range of techniques to reduce chaos has invented. Firstly, let us note a well-known method to control chaos which is based on feedback principles. For example, in [13] authors investigated changes in dynamics of the Lorentz system under feedback control. Particularly, it was shown that the trajectory of the Lorenz system is located at one stable point. In [11], the global attracting set of the simplified Lorenz model was investigated within the Lyapunov stability theory. It was shown that the method of constructing the Lyapunov function applied to classical systems with chaos, does not work for a simplified model of the Lorenz system. In [12], the dynamics of unidirectionally connected chaotic Lorenz systems were investigated. It was shown that chaos is observed in the system independently on the generalized synchronization. To demonstrate the lack of synchronization, the Lyapunov exponents were used. In some cases, this result

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can be applied in the field of the weather forecasting. In [10] the Lorenz system was considered as a model of atmospheric disturbances and as a part of the so-called LDWNN (Lorenz Distribution Wavelet Neural Network) model. It was established that the Lorenz system gives more accurate results of the wind speed prediction comparing to the WNN (Wavelet Neural Network) model. In recent years an interest to the Lorenz system is growing up in the field of cryptography (signal generated by the Lorenz system can be considered as a pseudorandom sequence). For instance, in [14], a method for generating sequences of pseudo-random numbers based on the generalized Lorenz system is proposed (in this paper the Lorenz system used to generate a binary sequence). It was shown that this method shows an effective crypto-robustness. Also, a comparison between the presented method and other methods for generating of pseudo-random sequences is considered. In [15] authors propose a novel asymmetric watermarking mechanism using the Lorenz system. This feature added nonlinear properties to this mechanism, as well as expanded the space of generated keys. Thus, the Lorenz system remains relevant even after more than fifty years of its discovering and gives promising and rich ideas to researchers in various fields of modern science. In this chapter we take a fresh look to the Lorenz system under various strongly nonlinear control (Coulomb friction and hysteresis friction ) and discuss some features occurring in this case.

1.2 Chaos: Analysis and Control Analysis of chaotic dynamics in nonlinear systems usually reduces to two separate problems: diagnosis of the system’s behaviour (analysis) and control of the dynamics (usually, by means of an appropriate excitation). At the same time, two main methods usually implement for chaos control: program control (by means of an initially defined time function) and feedback control principles. Diagnosis allows to investigate behaviour of the system at different values of the system parameters or, for example, depending on the initial conditions, etc. From both fundamental and applied points of view it is very important to analyze dynamics of the system depending on its parameters. An excellent example is a changing of

Fig. 2 Phase portraits of system (1) with parameters σ = 10, b = 8/3 and varying parameter r : left panel—r = 0.5 , middle panel—r = 5, right panel—r = 28

Stabilization of Chaos Via Strong Nonlinearities …

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Fig. 3 Left panel—Phase portrait of system (1) with the corresponding parameters: σ = 10, r = 300, b = 8/3. Right panel—the x-coordinate versus time

the Rayleigh number in the Lorenz system (see Fig. 2). Varying this parameter a critical value when system (1) loss stability was found. Determining the dependence of solutions on the initial conditions is also an important task in the analysis of chaotic systems. Particularly, this task makes it possible to determine the attraction regions, as well as to identify the type of behaviour as it was done by Lorenz [1]. One of the well-known approaches to chaos control is based on the parameters varying. Indeed, by increasing parameter r in the classical Lorenz system (1), we can achieve completely regular dynamics (for r = 300 an absolutely stable limit cycle is observed, see Fig. 3). Another implementation of chaos control is based on using the control function in one (or more) equation of the Lorenz system (modified Lorenz system): ⎧ ⎪ ⎨ x˙ = σ (y − x) y˙ = x(r − z) − y + u , ⎪ ⎩ z˙ = x y − bz

(2)

where u is the control input. Additional variable introduced in order to control dynamics of the system. Depending on how it acts, dynamics of the system will also change.

2 Modified Lorenz System: Water Wheel Under Coulomb Friction 2.1 Water Wheel Model One of the real physical model described by Lorenz equations is a water wheel model. This model has been constructed by Willem Malkus and his colleagues in1970s. A water wheel includes some leaking cups on the rim of the wheel, liquid flows from the

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Fig. 4 A water wheel model with different rates of water flow. Left panel—low rate and the water wheel remains motionless; middle panel—moderate rate and the water wheel moves with a steady speed; right panel—high rate and the water wheel moves chaotically

top and each cup leaks from the bottom. Under different input flow rates, the wheel will fixed, spin one way, or spin chaotically (direction is changing unpredictable) (see Fig. 4).

2.2 Main Equations Usually, authors consider an ideal model [16–19] and do not take into account the Coulomb friction on the rim [20] which, obviously, exists in a real physical model. In this chapter we pay attention to this fact and modify the classical Lorenz-Malkus water wheel model. Following the derivation of equations describing the LorenzMalkus model [16], ω is an angular velocity of the water wheel. It is clear, that the dry friction (the Coulomb model) affects only the speed of the wheel: Nd f = m sign(ω),

(3)

where m is the coefficient of the dry friction, sign(·) is a standard signum function. Following the second Newton’s law (as it was done in [17] with one remark: in our consideration the water wheel is placed in the vertical plane relative to the Earth surface), the rate of change of the angular momentum equals to the torque which is the sum of N g (gravity component), Nd f (a component describing by equation (3)), Nν (viscous friction component), Nw (a component corresponding to bringing the incoming water flow up to the speed of the cup into which it falls): I

dω = N g + Nw + Nμ + Nd f , dt

(4)

Stabilization of Chaos Via Strong Nonlinearities …

9

and rewriting each of these components in an explicit form, equation (4) take the form: Mg α Iw m dω = y − ω + λ ω − sign(ω), (5) dt I I I I where g is the garvitational acceleration, α is a coefficient of the viscous friction, m is a coefficient of the dry friction, Iw is the moment of inertia of the water, λ is the cup leakage parameter, M is the total mass of the water. Equations for y and z can be represented as (following to [16]): dy = ωz − λy, dt

(6)

dz = −ωy + λ(R − z), dt

(7)

where R is a radius of the water wheel. Finally, bringing equations and parameters in (5–7) to the dimensionless form (in this case ω turns to x), the modified LorenzMalkus water wheel model reads: ⎧ ⎪ ⎨ x˙ = σ (y − x) − Msign(x), y˙ = x(r − z) − y, (8) ⎪ ⎩ z˙ = x y − bz, where Md f is a coefficient of the dry friction moment. In this model variables have the following meaning: • x is an angular velocity of the water wheel; • y and z are coordinates of center of mass of the water.

2.3 Stationary Points It is known that investigation of any nonlinear system described by differential equations starts from analysis of stationary points. These points play a special role in system behaviour. Therefore, we start from identification of stationary points for the modified Lorenz-Malkus system (8). Following the standard approach, x, ˙ y˙ and z˙ are supposed to be zero, then system (8) reads: ⎧ ⎪ ⎨ σ (y − x) − Md f sign(x) = 0 x(r − z) − y = 0 . ⎪ ⎩ x y − bz = 0

(9)

A solution to algebraic equations (9) determines a set of stationary points. Additional term Md f sign(x) complicates the process of finding stationary points unlike for the

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classical Lorenz system and an explicit form of the obtained points has a quite complicated for analysis form (see Appendix). However, we show numerical values of these points for the system with parameters σ = 10, r = 28, b = 8/3, Md f = 3: P0 = (0, 0, 0) P1 = (8.642059263, 8.34205926, 27.03471396) P2 = (−8.330947618, −8.630947618, 26.9639896) . P3 = (−0.011111644, −0.311111644, 0.001296360) P4 = (0.011111644, 0.311111644, 0.001296360)

(10)

A set of stationary points is denoted by Pi , i = 0 . . . 4. As can be seen from Eq. (10) we obtain two more stationary points (points P3 and P4 ) comparing to the classical system, however other three points have similar values. An analysis of the system behaviour in the neighborhood of these points was carried out using the first Lyapunov method. An analysis of the trajectories behaviour near point P0 starts from the identification of type of this point. For that reason, the standard linearization procedure is implemented. A solution to system (8) can be written in the form: A, A = A0 + 

(11)

where A = (x(t), y(t), z(t))T is a column vector of the time-dependent variables, A0 = (x0 , y0 , z 0 )T is a column vector of coordinates of stationary point (in our case this vector is (0, 0, 0)), and  A = ( x (t),  y(t), z(t))T is a column vector of small additives. Further, keeping only the main terms in right-hand sides of the linearised system, we obtain: ⎧ x ), y − x ) − Md f sign( ⎪ ⎨ x˙ = σ ( ⎪ ⎩

y˙ = r x − y − x0 z − x z0 , z˙ = −b z + x 0 y + x y0 ,

(12)

where terms of zero-order have removed, because (x0 , y0 , z 0 ) is a stationary point, x + x0 ) is replaced by2 : Md f (sign( x ) + sign(x0 )). and term Md f sign( Note, that linearization of the right-hand side of the first equation in (12) is impossible due to the non-smoothness of the dry friction term. However, assuming that time dependence of the perturbation of small additives is exponential and keeping x ) in the first equation, we obtain the eigenvalue problem. The only term −Md f sign( results of numerical simulation show that all eigenvalues are real and negative independent on the Md f value (in our case only positive values of this parameter has an

the following obvious relation: sign(x) + sign(y) − 1 ≤ sign(x + y) ≤ sign(x) + sign(y) + 1.

2 Using

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Fig. 5 Trajectories behaviour in a small neighborhood of stationary points for system (8)

interest), therefore point P0 is a stable node. An illustration of the behaviour of phase trajectories near P0 and other four points3 are presented in Fig. 5.

2.4 Dynamical Features This section discusses dynamical features of the modified Lorenz-Malkus system. First of all, it seems interesting to consider the system dynamics depending on the parameter corresponding to the moment of dry friction Md f , and obviously this new parameter must change dynamics. Firstly, note that dependence of the time during which the system exhibit chaotic dynamics on the dry friction parameter is essentially nonlinear. Qualitatively this fact is shown in Fig. 6. For example, at Md f = 3 the system is staying in chaotic regime during a longer period of time as compared to the case when Md f = 2. Systems were solved with same initial conditions, however, this property holds for any initial conditions. At the same time, a period during which the system is in chaotic regime for various values of Md f depends on the initial conditions. This feature is due to the fact that additional term −Md f sign(x) leads to an additional parametric dependence described by a non-smooth function, which, in turn, complicates the structure of the phase space. Also, the time when the system trajectory comes to a stable zero point directly depends on how soon it enters the attraction basin. An additional contribution to understanding the system dynamics gives an evolution of the x-coordinate depending on the parameter Md f . As can be seen in Fig. 7 for time instant t = 10 the behaviour of function x(Md f ) for small values of the dry friction parameter has a completely deterministic structure. However, starting from a certain value of Md f , x(Md f ) exhibits chaotic behaviour and tends to zero while Md f increases (this is clearly seen from the right panel in Fig. 7). That means that the solution trajectory entered in an attraction basin and rapidly came to stable point P0 . Moreover, we also note (as follows from the results presented in the left panel of Fig. 7) that for any fixed time instant with a non-zero value of Md f the system inevitably comes to a small neighborhood of the origin. The modelling results 3 During

numerical simulations it was established that P1 and P2 are saddle-node points, and P3 and P4 are unstable focuses.

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were obtained for a fixed time instant (200 units of model time), initial conditions are (1, 1, 1), and parameters of the system are σ = 10, r = 28, b = 8/3.

2.5 Lyapunov Characteristic Exponents The presence or absence of chaotic behaviour is closely related to instability. One of the most important methods that allow to identify the stability of the system is the Lyapunov method. This method characterizes the behaviour of a given trajectory depending on the behaviour of trajectories located in its small neighborhood. An informative characteristic of this behaviour is the spectrum of Lyapunov characteristic exponents.4 Chaotic behavior is fully determined by the largest Lyapunov exponent. Its value uniquely determines the type of dynamical regime. However, a well-known fact that following the fundamental Kolmogorov-Arnold-Moser the-

Fig. 6 Phase portraits of the modified Lorenz-Malkus system (8) with parameters σ = 10, r = 28, b = 8/3 and varying parameter Md f : left panel – Md f = 0.5, middle panel – Md f = 2, right panel – Md f = 3

Fig. 7 Left panel – coordinate x versus Md f and time. Right panel – the x-coordinate versus Md f at different time instants: t = 3 – orange curve, t = 5 – blue curve, t = 10 – red curve

4 Recall, that number of Lyapunov exponents is equal to the dimension of the system’s phase space.

Stabilization of Chaos Via Strong Nonlinearities …

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Fig. 8 Left panel – The Lyapunov spectrum versus time (the largest Lyapunov exponent is indicated by blue curve); Right panel – phase portrait of the corresponding modified Lorenz-Malkus system. Modelling parameter are: σ = 10, r = 28, b = 8/3, Md f = 3. Initial conditions are (1, 1, 1)

orem [3], to obtain a spectrum of Lyapunov characteristic exponents, it is enough to consider an evolution of the perturbation only for the individual solution of the Lorenz system (Fig. 8). In this work the spectrum of Lyapunov exponents were calculated using the standard Gram-Schmidt orthogonalization procedure together with the Wolf algorithm (for details, see [21]). An analysis of the spectrum of Lyapunov exponents for system (8) with the corresponding parameters σ = 10, r = 28, b = 8/3, Md f = 3 showed that on the initial time period (around 150 units of model time) the largest Lyapunov exponent is positive, therefore the solution is chaotic. However, after that, the largest Lyapunov exponent monotonically decreases down to the region of negative values, which indicates the stabilization of the system in a stable fixed point.5 Assumption 1 Stabilization of the modified Lorenz-Malkus system occurs at any non-zero value of the dry friction parameter Md f . In this work we also analyze the dependence of the dry friction parameter Md f on parameter r in a situation when the largest Lyapunov exponent is equal to zero. Note, that dependence Md f (r ), obtained during numerical simulations, has a very irregular structure as can be seen in Fig. 9. Based on the features presented above, it can be concluded that there is an attractive manifold, in which the solution trajectory inevitably ends up in an arbitrarily small neighborhood of a stable stationary point P0 .

5 This

point is P0 with coordinates (0, 0, 0).

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Fig. 9 The dependence of parameter Md f on r , corresponding to zero value of the largest Lyapunov exponent. Numerical simulations were carried out for initial conditions (10, 10, 10) and calculated during 500 units of model time

2.6 Stochastic Features This section discusses stochastic properties of the modified Lorenz-Malkus system. Particularly, we investigate the time during which the system is in chaotic regime depending on the initial conditions. Figure 10 shows histograms of the distribution of time during which the system is in chaotic regime for different values of the dry friction moment. To simulate the corresponding histograms in a random way,6 initial conditions were selected from the cube with a side of 20 units centered at the origin (ten thousand initial values were generated). Next, we considered a quantity corresponding to the time during which the solution trajectory was not in a neighborhood of the origin. It is important to note, that obtained histograms are stable enough for various sets of initial conditions. The shape of histograms obviously depends on the parameter Md f , however, in this work, we do not identify the universal distribution law which describes histograms for various values of the dry friction parameter. At the same time, it was found that for some values of Md f (particularly, for Md f = 10), the obtained histograms can be approximated with a good accuracy by the Gamma distribution which has the form: f (α, β) =

1 (α)β

 α−1 x e−x/β , β

(13)

where α and β are parameters of the Gamma distribution, and (·) is a standard Gamma function. In the right panel of Fig. 10 we show approximation of the histogram by the Gamma distribution (13) with parameters α = 4.9, β = 1.98. 6 We

supposed the uniform distribution of initial conditions.

Stabilization of Chaos Via Strong Nonlinearities …

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Fig. 10 Histograms describing how much time needs to solution to achieve the origin. Numerical simulations was carried out with following parameters: σ = 10,r = 28, b = 8/3 and Md f = 3 (left panel), Md f = 10 (right panel). Red curve on the right panel corresponds to the Gamma distribution with parameters α = 4.9, β = 1.98

The obtained result can be used in Bayesian statistics, biology, medicine and many other fields where the Gamma distribution is traditionally used. In other words, for some values of Md f , the modified Lorenz system may serves as a “natural” pseudo-random number generator due to stochastic properties satisfying the Gamma distribution law.

2.7 Bifurcation Analysis It is well-known, that a bifurcation analysis and the corresponding bifurcation diagrams are powerful tools in analysis of systems with chaotic regimes. Most of these systems have internal parameters which determine in general the behaviour of systems. During a monotonic slight change of these parameters, a transition from one mode to another can occur. A set of events that happen during these changes are called bifurcations. In this work for the modified Lorentz-Malkus system we construct the bifurcation diagram in not a common sense, but merging a set of bifurcation diagrams because, during the numerical simulation, it has to be taken into account that when the trajectory hit the attracting region of the origin, a solution to the system is localized in the zero-dynamics region.7 Therefore, in presented figures there are many points in the domain of zero values. As can be seen from the left top and bottom panels in Fig. 11 for the modified system (8), transition to chaos becomes a bit later comparing to the classical Lorenz system. For large values of the Rayleigh parameter, transition to chaos occurs through a cascade of period-doubling, which can be seen from the right top and bottom panels in Fig. 11.

7 Note,

that zero is absolutely stable regardless of the value of parameter r .

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Fig. 11 Bifurcation diagrams depending on the parameter r . Top panels correspond to the classical Lorenz system; Bottom panels correspond to the modified Lorenz-Malkus system (8) with parameter Md f = 3

2.8 Analysis of a “flat” System Analysis of trajectories near stationary points is important for studying the dynamic features of the system, and also gives some ideas about the structure of the attracting manifold. To simplify an analysis of the modified Lorenz-Malkus system (8), we consider a situation when z = const. Further, consider a two-dimensional system: ⎧ ⎨ x˙ = σ (y − x) − Md f sign(x) = (x, y), ⎩ y˙ = x(r − 1 x y) − y = (x, y). b

(14)

In this section, we will consider vector fields of the “flat” system. This will help us to understand the trajectories behaviour, especially near stationary points. One of the most important characteristics in the analysis of vector fields is an index of the stationary point, which, first of all, gives an opportunity to establish the type of the singular point, namely, whether it belongs to the saddle type or not. The calculation of the index of the stationary point is carried out by finding a sign of the determinant which includes partial derivatives of the right-hand sides of equations (14): (x, y)x (x, y) y , sign

(x, y)x (x, y) y

(15)

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Fig. 12 Attraction manifold for trajectories of the “flat” modified Lorenz-Malkus system

where lower indexes denote partial derivatives. If the obtained value equals to −1, then the investigated point is a saddle-type point, otherwise when the value is 1, then this point is either center, or node, or focus. For system (14) determinant (15) can be written as: −σ − 2Md f δ(0) σ . r −1

(16)

Obviously, the determinant (16) is σ (1 − r ) + 2Md f δ(0) (here δ(·) is the Dirac deltafunction), and it is clear that 2Md f δ(0)  σ (1 − r ). Based on the obtained results the following conclusion can be formulated: zero in the modified Lorenz system is not a saddle regardless of the value r , which is clearly seen from the resulting expression. Let us analyze the behaviour of trajectories near stationary point P0 . The vector field of the modified flat system sufficiently differs from the vector field of the classical system, particularly, because P0 is a stable node. Points P1 and P2 also bring changes to the vector space near zero since they are saddle points. As can be seen from the right panel in Fig. 12 a set of all paths is separated into three subsets that correspond to three attractive points (two focuses and one stable node). Note, that dry friction parameter Md f is a bifurcation parameter. When parameter Md f becomes positive, then from the saddle point, two symmetric saddle points and a stable node at the origin are born (see Fig. 13). It is clear that the size of the region depends on the value of parameter Md f and is determined by positions of two saddle points P1 and P2 . The larger attraction area corresponds to the larger value of parameter Md f . Moreover, as can be seen from

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Fig. 13 Bifurcation process from a saddle-node to two saddle-nodes and a stable node during varying parameter Md f from zero to positive value Fig. 14 Separatrices of two saddle-nodes. Blue curve corresponds to eigenvectors of the linearized matrix of system (14)

1.0

y

0.5

0.0

- 0.5

- 1.0 - 1.0

- 0.5

0.0

0.5

1.0

x

Fig. 14 the structure of the attracting region has a complex structure even in a twodimensional case. An additional fact: the tangent to the boundary of the attracting set in a small neighborhood of saddle points obviously coincides with eigenvectors of the matrix of the linearized system in the neighborhood of the corresponding points (see the Fig. 14). Separatrices are shown in Fig. 14 and were constructed by solving the system in the reverse time.

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3 Modified Lorenz-Malkus System Under Hysteresis Friction: The Bouc-Wen Model 3.1 Some Preliminaries to the Bouc-Wen Model One of the famous approaches that allow describing many real-life physical systems and processes with memory is based on the model of hysteresis phenomena. It is caused, on the one hand, by design features of such systems, and, on the other hand, by features of external conditions that directly effect the system. Hysteresis phenomena are quite spread within physical, chemical, biological, and other processes in which states of the object are ambiguously dependent on external conditions. One of the main examples of a hysteresis relation is the dependence of the magnetization on the external field strength. Note that this dependence is caused not only by current state of the object (system) but also depends on its history. Historically, one of the first references to hysteresis dependencies was presented in the work of J. Ewing [22]. From a mathematical point of view, a hysteresis operator can be considered as a “black box”, which respond to an external action (input) with a certain reaction (output). At the same time, the output value depends on the state of the object at the initial time instant. In this work, the carrier of hysteresis properties is understood as operator W (depending on its initial state as a parameter), which corresponds to continuous output u(t) and receive continuous input signal x(t) (see Fig. 15). The parametric relationship between input and output of such an operator is called as a hysteresis loop. Note that hysteresis curves are usually identical for periodic input signals with different frequencies, in other words, properties of operator W do not depend on the time scale. In this case such an operator is called static. The shape (in particular, the area of the loop) significantly effects the dynamics of systems containing hysteresis term. This feature makes it possible to introduce hysteresis operators in the control tasks. In recent years many hysteresis models have been obtained to solve various problems in a wide field of engineering tasks. First of all, let us note work [25], which has made a huge contribution to hysteresis analysis. Particularly, this work is a base to the so-called design approach to hysteresis phenomena. One of the most frequently used phenomenological models of hysteresis is the Bouc-Wen model. This model was first proposed by Bouc [23] in 1971 and generalized by Wen in 1976 [24]. The Bouc-Wen model is represented as the first-order nonlinear differential equation of the following form:

Fig. 15 The “black-box” approach. Here x(t) is an input signal, u(t) is an output signal, W is a hysteresis operator

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M. E. Semenov et al. n−1 ˙ , F˙ = δ x˙ − β x|F| ˙ n − γ F|x||F|

(17)

where x is an input, F is an output, δ, β, γ are dimensionless parameters which correspond to the loop shape, and parameter n corresponds to smooth transition from elastic to plastic output. The main advantage of this model is flexibility, thus, the Bouc-Wen model can be used to model a wide class of systems that have hysteresis properties, such as piezoelectric elements, magneto-rheological dampers, and many others. Today the Bouc-Wen model is frequently used by researchers to model hysteresis phenomena in various fields [26–31]. Particularly, in [26] the authors consider the problem of analytical description of limit cycles within the Bouc-Wen model. It was shown that a periodic T -input leads to a periodic T -output. An explicit analytical description of the limit cycle is also provided. In [27] the authors considered dynamics of a mechanical system under external load when the damping part is of hysteresis nature. A comparison between the hysteresis damper and nonlinear viscous damper was made. Let us note also works [29, 30], which deal with algorithms of parameters quantification within the Bouc-Wen model. A detailed analysis of the speed and accuracy together with comparison with other well-known algorithms was made. Additionally, the Bouc-Wen model can be used to model dynamics of constructions under external load as it was done in [31]. This paper is studied isolated bridge, which is described by the Bouc-Wen model with two degrees of freedom, simulated the internal stochastic nature of seismic events via the random oscillation method. It was shown that the high-security of the construction achieved by reducing the input energy by using certain parameters of the Bouc-Wen model. The obtained result can be useful during planning a building and may help to choose the more correct material for the required conditions. In this section, the Bouc-Wen model (17) is used to describe the friction phenomena within the hysteresis approach (the Coulomb term in equations of motion is replaced by a hysteresis one). Moreover, to the classical Lorenz system (1) is added fourth equation (17) corresponding to the Bouc-Wen model. An expression representing friction feedback −τ F is added to the first equation of (1). Finally, we obtain the modified Lorenz-Malkus system with hysteresis friction: ⎧ x˙ = σ (y − x) − τ F ⎪ ⎪ ⎪ ⎨ y˙ = x(r − z) − y ⎪ z˙ = x y − bz ⎪ ⎪ ⎩ n−1 F˙ = δ x˙ − β x|F| ˙ n − γ F|x||F| ˙ where τ is a parameter characterizing the strength of hysteresis friction.

(18)

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Fig. 16 Left panel – stationary points positions (magenta, red and cyan curves represent first, second and third point, respectively) depending on F. Right panel – phase portrait of the system (18) and stationary points curves

3.2 Analysis of Stationary Points Analysis of the modified Lorenz-Malkus system with hysteresis term we start from identification of stationary points. Note, that in the fourth equation of (18), derivative x˙ takes place in each term and we do not take it into account considering stationary points. Thus, we have the system of algebraic equations: ⎧ ⎪ ⎨ σ (y − x) ± τ F = 0 x(r − z) − y = 0 ⎪ ⎩ x y − bz = 0

(19)

where ± is used because τ F might have any sign. As can be seen from system (19), the number of unknowns is greater than the number of equations. Hence, this system has infinitely many solutions according to the Rouché-Capelli theorem. Therefore, let us present the solution to system (19) in the form of a parametrically defined curve8 depending on F as a parameter. Equations for the obtained curves are presented in Appendix (Fig. 16). Next we study the stability of the obtained stationary points. The stability analysis was carried out within the first Lyapunov method. We have:

8 In

this case we have dynamically changing stationary points the position of which relies on hysteresis output.

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⎧ ⎪ ⎪ ⎪ ⎪ ⎨

 x˙ = σ ( y − x) − τ ω ˙ y = r x − y − x0 z − x z0

⎪  z˙ = −b z + x 0 y + x y0 ⎪ ⎪ ⎪

 ⎩  F˙ = δ x˙ = δ σ ( y − x) − τ F

(20)

To obtain eigenvalues of the 4 × 4 matrix composed from coefficients of equation (20), we use the fact that the determinant of this matrix must be equal to zero (to find a non-trivial solution). Obviously, the first and fourth equations are proportional to coefficient δ, therefore the first eigenvalue will always be zero. Thus, the asymptotic stability does not make sense in this case. Theorem 1 The asymptotic stability of any stationary points in the hysteresis Lorenz-Malkus system does not exist. Equating the determinant to zero, we obtain a cubic equation, the solutions of which define eigenvalues: (21) λ3 + a1 λ2 + a2 λ + a3 = 0, where a1 = b + δτ + σ + 1, a2 = b(δτ + σ ) − σ (r + z 0 ) + x02 + δτ, a3 = b(δτ + σ (1 − r + z 0 )) + σ (x02 + x0 y0 ) + δτ x02 . This equation can be solved in an explicit analytical form using Cardano formulas, for instance. Obviously, that solutions to equation (21) are quite complicated (see Appendix) in explicit form. Next we consider the case when parameters σ , b, τ and δ are known and can be substituted to expressions for a1 , a2 and a3 . Following this assumption, we get eigenvalues as functions depending on F and r : λi (F, r )|(x0 ,y0 ,z0 ) ,

(22)

where i = 1..3 and (x0 , y0 , z 0 ) is the investigated stationary point. Let parameters are: σ = 10, b = 8/3, τ = 1 and δ = 1. The eigenvalue defines the type of stationary point and, at the same time, determines the stability of this point. The real and imaginary parts of each eigenvalue are shown in Fig. 17. As can be seen, eigenvalues which correspond to the first two stationary points have non-zero imaginary part. This means that they have a focus-like type. Whereas, eigenvalues which correspond to the third point have an insignificant deviation of imaginary part from zero in the range of parameter r from 0 to 5. Beyond this range it is equal to zero. Hence, within the interval, the third point has a saddle-like type, and beyond it—a node-like type (all eigenvalues are real). Stability analysis showed that first two points are stable (all real parts are negative) independent of r and F values, whereas the third point is unstable (see Fig. 17). Therefore, we conclude that two focus-like

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Fig. 17 Each panel represents a certain eigenvalue λ (obtained by solving equation (22) with fixed parameters σ = 10, b = 8/3, τ = 1, δ = 1). The eigenvalue number corresponds to the number of column (e.g.. top-left panel corresponds to the first eigenvalue with the first stationary point). Top, middle and bottom panels correspond to eigenvalues with the first, second and third stationary points, respectively. Red surface refers to the imaginary part of eigenvalue, blue surface refers to the real part of eigenvalue

points in the hysteresis Lorenz-Malkus system are stable, unlike the classical Lorenz system, while the third saddle point is still unstable.

3.3 Dynamical Features In this section we analyze the dynamics of the modified Lorenz-Malkus system with a hysteresis term represented by the Bouc-Wen model. Particularly, we investigate the dependence of dynamical characteristics of system (18) on the shape of the hysteresis

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Fig. 18 Hysteresis loops obtained by solving system (18) with parameters τ = 1, δ = 1, β = 1, n = 3: left panel – γ = 0.1, right panel – γ = 0.6. Initial conditions are (1, 1, 1, 1)

Fig. 19 Phase portraits of system (18) with parameters τ = 1, δ = 1, β = 1, n = 3: left panel – γ = 0.1, right panel – γ = 0.6. Initial conditions are (1, 1, 1, 1)

loop. Figure 18 shows the parametric dependence of the input represented as the xvalue on the hysteresis output. As was noted above, changing the Bouc-Wen model parameters we change the shape of the loop. Using numerical simulations it was found that system behaviour certainly depends on the specified parameters. Namely, we changed parameter γ which corresponds to the skew and area of the hysteresis loop (see Fig. 18 where the loop area in the left panel is larger than in the right one). During numerical simulations it was found that system behaviour certainly depends on the shape of the loop, which is clearly seen in Figs. 18 and 19. As can be seen from presented results, decreasing the value of parameter τ (from 0.6 to 0.15) makes it possible to change chaotic behaviour to regular. Hence, we can conclude that changing parameters of the Bouc-Wen model allows to control dynamical regimes in system (18).

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Fig. 20 Left panel—a set of randomly generated points; Right panel—a bar chart representing the number of points which are came in a certain focus. Colours on the left and right panels are the same

3.4 Stochastic Features As was noted above, in the modified Lorenz-Malkus system (18), two stable manifolds corresponding to two points of the focus type are observed. Figure 20b shows the number of points falling into first or second focuses. The trajectory of the solution more often comes into a stable focus which is placed in the first octant than in the other. To illustrate this statement, 500 of initial conditions were randomly generated (the normal distribution is used). After that, for a short time (10 model time units), system (18) had been solved and the final position of the trajectory was analyzed. As a result, two sets were constructed: the region of attraction of the left and right focuses. We found that attracting regions are located asymmetrically with respect to focuses.

3.5 Bifurcation Diagrams An additional information about dynamical features of the Lorenz-Malkus system with hysteresis friction gives bifurcation diagrams. Numerical modelling of bifurcation diagrams was carried out depending on parameter τ , since it defines how much energy will be “taken” by the hysteresis. As can be seen in Fig. 21, the hysteresis friction makes it possible to control the dynamics of system (18), namely, to reduce chaos. Based on the obtained results we can conclude that the hysteresis friction (described within the Bouc-Wen model) makes it possible to control dynamical regimes of the system, namely, to realize transition from chaos to regular dynamics. Using these results it is possible to construct the switch source (at least on the software level) for changing dynamical regimes in the system from chaos to regular motion when required.

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Fig. 21 Bifurcation diagram as function of parameter τ for the modified Lorenz-Malkus system under hysteresis friction (18) with the following parameters: σ = 10, r = 28, b = 8/3, δ = 1, β = 1, γ = 0.6, n = 3. Initial conditions are (10, 10, 10, 1)

4 Modified Lorenz-Malkus System Under Hysteresis Friction: The Dahl Model 4.1 Some Preliminaries to the Dahl Model Here we start from the statement that friction effects have a great importance in models of real-life physical systems. From one hand, friction can decrease the system performance, but from other hand, friction can be used to control systems and and can be considered as a useful phenomenon. Each separate task has a certain friction type, and a number of approaches to modelling the friction phenomena were introduced. All models describing friction process can be separated into two parts: static models and hysteresis models. The static friction usually arises in linear tasks, while the hysteresis friction arises in non-linear systems, when the friction process can not be described explicitly. Together with the Bouc-Wen model (recall that this model describe a wide class of hysteresis phenomena) describing the friction in the previous section, it seems useful to consider anoter models, which were introduced in the friction modelling only. One of the most famous hysteresis models in this case is the Dahl model (due to its simplicity):   F ˙ |x| ˙ F = κ x˙ − FC

(23)

where x is the velocity of the considering system, F is the control friction force (in the dimensionless form), FC is the Coulomb friction force, κ is the stiffness coefficient. Following Dahl’s proposal [32, 33], an object subjected to small displacements returned to its original position, what from some point of view, related to the bonding forces occurring between two surfaces. In recent years, a large number of research works are dedicated to the Dahl model. Let us note last works dedicated to the Dahl model. In [32] authors studied loss energy through analysis of minor loops and with

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Fig. 22 Phase portraits of system (24) with parameters τ = 1, κ = 3: left panel – FC = 2, right panel – FC = 6

corresponding analytical results. A helpful work is [35] where authors investigate the Dahl model parameter identification in a control system. The method proposed in this article based on frictional resonances and proofs of the developed method are presented. In this section we investigate the Lorenz-Malkus water wheel under control in the form of the hysteresis friction formalized by the Dahl model9 : ⎧ x˙ = σ (y − x) − τ F ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ y˙ = x(r − z) − y z˙ = x y − bz ⎪   ⎪ ⎪ ⎪ F ⎪ ⎩ F˙ = κ x˙ − |x| ˙ FC

(24)

The corresponding phase portraits are shown in Fig. 22.

4.2 Dynamical Features In this section we investigate dynamical features of system (24) depending on parameters of the Dahl model. Note, that parameter τ corresponds to the amount of energy taken by hysteresis, and parameters κ and FC correspond to the smoothness and area of the hysteresis loop.

9 Let

us note, that analysis of stationary points is similar to that in the case of the Bouc-Wen model with replacement δ → κ.

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Fig. 23 Hysteresis output depending on the input signal (represented as the x-coordinate) for system (24) with parameters τ = 1, κ = 3: left panel – FC = 2, right panel – FC = 6

Numerical results show that the system behaviour depends on the loop area as can be seen in Figs. 22 and 23: changing the parameter responsible for stiffness, we can effect the system dynamics. During numerical simulations we established that larger area of the hysteresis loop corresponds to higher energy dissipation in the friction element (described within the hysteresis model). Note, that the LorenzMalkus system under hysteresis friction formalized by the Dahl model has the same number and the same coordinates of stationary points as in the case of the Bouc-Wen model. Thus, the hysteresis friction can be considered as a “natural” method for chaos control. Moreover, it is worth nothing to construct a real-life tool which can change friction parameters (e.g.., a tool for changing the contact area, the material type on the supporting construction, etc.).

4.3 Bifurcation Diagrams It seems interesting to investigate the dynamics of system (24) depending on the Dahl model parameters, namely, to identify critical values of these parameters. We propose to use bifurcation diagrams for this purpose because the explicit form of the obtained results regarding equilibrium analysis is really hard to analyze. In this section we investigate the system dynamics depending on two parameters FC and κ. As can be seen from the left panel in Fig. 24 in the interval from 0 to 4.1 system (24) exhibits chaotic dynamics. However, when the value of parameter FC crosses 4.1 dynamics qualitatively change: it is clearly seen a sharp transition from chaotic to regular dynamics. According to the right panel in Fig. 24, where bifurcation diagram was calculated depending on parameter κ, we have an absolutely same situation. Namely, when the value of parameter κ crosses 2.3 a sharp transition from chaotic to regular dynamics takes place.

Stabilization of Chaos Via Strong Nonlinearities …

29

Fig. 24 Left panel – the bifurcation diagram depending on FC ; Right panel – the bifurcation diagram depending on κ

5 Conclusion In this work we studied the dynamics of a water wheel model described by the modified Lorentz system (the Lorenz-Malkus model) taking into account various types of friction on the rim. In the case of the dry friction (modelled within the Coulomb approach) it was found that the origin is a stable stationary point, moreover, any trajectory (regardless of the initial conditions) in a finite time comes to an arbitrarily small neighborhood of the origin. At the same time, during initial period of time, the system exhibits chaotic regime (the largest Lyapunov characteristic exponent is positive), and this time interval (for some values of parameters) satisfies the stochastic distribution law. We presented histograms of the distribution of time of chaotic behaviour with a random choice of initial conditions. It is established that, at certain values of the dry friction parameter, this distribution can be rather correctly approximated by the gamma distribution. Also, the stationary points of the modified Lorentz system are investigated and classified, bifurcation diagrams are constructed, the general properties and fundamental differences of the modified and classical Lorentz systems are established. The dependence of the largest Lyapunov exponent on the dry friction parameter was studied and it was established that this dependence (under fixed initial conditions and parameters of the system) is essentially non-monotonic. Additionally, a two-dimensional system obtained by considering the modified system under the fixed value of variable z is investigated. It is established that the dry friction parameter is a bifurcation parameter: at zero value of this parameter two saddle-shaped equilibrium points and a stable node were born from the saddle point. Also in this work, we studied a model of the water wheel under hysteresis friction formalized by means the phenomenological Bouk-Wen and Dahl models. Stability analysis and analysis of dynamical features are presented and discussed. It was shown that unstable within the classical Lorentz system foci become stable. It is shown that dynamics of the system depends on the shape and area of the hysteresis loop. The regions (basins of attraction) from which the trajectories in a short period of time come in a small neighborhood of one of the foci are studied. Histograms of the distribution of trajectories hitting one of the foci, as well as the structure of the

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attraction domains for each of the foci are presented. It was found that these domains are placed asymmetrically relative to the focus in which they come. Acknowledgements This work was supported by the RFBR (Grants 18-08-00053-a and 19-0800158). The contributions by M.E. Semenov and P.A. Meleshenko (“Modified Lorenz-Malkus system under hysteresis friction: the Bouc-Wen model” and “Modified Lorenz-Malkus system under hysteresis friction: the Dahl model”) were supported by the RSF grant No. 19-11-00197.

Appendix 5.1 Stationary Points Within the Coulomb Friction Model 2 Let − 18bMσ 2 − 2M 3 , a2 = 3σ (bσ − br σ ) − √M 2√, and γ = a1 + a1 = −9bMr σ √ √ √ 1+i 3 )a2 − (1−i√ 3) 3γ − M , ν = ( 3γ 32a M 4a23 + a12 , β = − 3√3γ σ2 + 3√32σ − 3σ , α = 3 22/3 √3γ 3σ σ 6 32σ √ √ √ (1−i √3)a2 − (1+i√ 3) 3γ − M . Further a (F), a (F), a (F), γ (F) are denoted as 3 22/3 3γ σ

6 32σ

3σ

1

2

3

a1 , a2 , a3 and γ , respectively.

5.1.1

First Point x =β

(25)

 √ √ √ 22/3 βγ 2/3 + 6 32bβ(r − 1)σ 2 + 2 32β M 2 − 2 3 β 2 + β 3γ M y= √ 6β 3γ σ (26) √ √ √ √ M 32bσ 3γ 32br σ 32M 2 M z= √ + r (27) + − √ − √ − √ + 3βσ β 3γ 3 32βσ β 3γ 3β 3γ σ β2σ 5.1.2

Second Point x =α 



√  √ √ √ i 32α 32 3 + i γ 2/3 − 6 3 − i b(r − 1)σ 2 y= + √ 12α 3γ σ 

 √ √ √ √ 3 − i α M 2 − 4 3 α2 + α 3γ M −2i 32 + √ 12α 3γ σ

(28)

(29)

Stabilization of Chaos Via Strong Nonlinearities …

31

√ √ √ √ 3γ i 3γ i 3bσ bσ i 3br σ z=− √ √ + √ − − + √ √ √ + 2 32 3ασ 6 32ασ 22/3 α 3γ 22/3 α 3γ 22/3 α 3γ M br σ i M2 M2 M +r + + + + +√ √ √ √ √ 3ασ 22/3 α 3γ 22/3 3α 3γ σ 3 22/3 α 3γ σ α2 σ (30)

5.1.3

Third Point x =ν

 √ √ √  6i 32 3 + i bν(r − 1)σ 2 + 22/3 −1 − i 3 γ 2/3 ν y= + √ 12 3γ νσ



 √ √ √ √ 3 + i ν M 2 − 4 3γ 3 ν 2 + ν M 2i 32 + √ 12 3γ νσ

√ √ √ i 3bσ bσ i 3br σ br σ i 3γ z= − − + + √ + √ √ √ √ √ 22/3 3γ ν 22/3 3γ ν 22/3 3γ ν 22/3 3γ ν 2 3 2 3νσ √ 3 γ M M i M2 M2 +√ +r + + √ − + √ √ √ 3 2/3 3 γ νσ 2/3 2 3 3 2 3νσ 6 2νσ 2 3 γ νσ ν σ 5.1.4

(31)

(32)

(33)

Fourth Point x = −β

(34)



 √ √ √ √ − 3 2β 6b(r − 1)σ 2 + 3 2γ 2/3 − 2 3 2β M 2 + 2 3 β 2 + β 3 γ M y= (35) √ 6β 3 γ σ z=

5.1.5

√ √ √ √ 3 3 3 3 γ M M 2bσ 2br σ 2M 2 − + +r − + − √ √ √ √ 3 2 β3γ β3γ 3β 3 γ σ 3βσ 3 2βσ β σ

(36)

Fifth Point x = −α

(37)

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√  √  √  √ √ 22/3 1 − i 3 αγ 2/3 + 6 3 2 1 + i 3 αb(r − 1)σ 2 + 2 3 2 1 + i 3 α M 2 + y= √ 12α 3 γ σ

√ √ 4 3 α2 + α 3 γ M + √ 12α 3 γ σ

(38)

√ √ √ 3 γ i3γ i 3bσ bσ z=− √ + − − √ √ √ √ + 2 3 2 3ασ 6 32ασ 22/3 α 3γ 22/3 α 3γ √ M br σ i M2 M2 M i 3br σ +r + + + + +√ + √ √ √ √ √ 2/3 2/3 2/3 2/3 3ασ 2 α 3γ 2 α 3γ 2 3α 3γ σ 3 2 α 3γ σ α2 σ

(39)

5.1.6

Sixth Point x = −ν

(40)

√ √  √  √ √  6 3 1 − i 3 bν(r − 1)σ 2 + 22/3 1 + i 3 γ 2/3 ν + 2 32 1 − i 3 ν M 2 y= + √ 12 3γ νσ  √ √ 4 3γ 3 ν 2 + ν M + √ 12 3γ

(41) √ √ √ i 3bσ bσ i 3br σ br σ i 3γ z= − − + + √ √ + √ √ √ √ 22/3 3γ ν 22/3 3γ ν 22/3 3γ ν 22/3 3γ ν 2 32 3νσ √ 3γ M i M2 M2 M + r (42) + + √ − + +√ √ √ √ 3νσ 6 32νσ 22/3 3 3γ νσ 3 22/3 3γ νσ ν2σ

5.2 Stationary Points Within the Bouc-Wen and Dahl Models a2 (F) = −9bFr σ 2 − 18bFσ 2 − 2 F 3 , − bσ ) − F 2 , and γ (F) = a2 (F) + 4a33 (F) + a22 (F).

Let

a1 (F) =

F 3σ

,

a3 (F) = −3σ (br σ

Stabilization of Chaos Via Strong Nonlinearities …

5.2.1

33

First Point x(F) = ±

√ √ 6a1 3γ σ + 2 32a3 − 22/3 γ 2/3 √ 6 3γ σ

√ √ √ √ −6 3br σ 2 + 6 32bσ 2 − 22/3 γ 2/3 − 2 3 2F 2 − 4 3 γ F y(F) = ± √ 6 3 γσ

z(F) = ±

5.2.2

√ √ 9 22/3 a1 γ σ + 18 22/3 b2 (r − 1)2 σ 4 + 12 22/3 bF 2 (r − 1)σ 2 + 6 3 2b 3 γ F(r − 1)σ 2 ± 18bγ 2/3 σ 2 √ √ √ 12bγ 2/3 (r − 1)σ 2 + 3 2γ 4/3 + 2 22/3 F 4 + 2 3 2 3 γ F 3 − 2 22/3 γ F ± 2/3 2 18bγ σ

(43)

(44)

(45)

Second Point ⎛ x(F) = ± ⎝a1 −

√ 2 3√ −2a3 3 γ

√ ⎞ + (−2)2/3 3 γ ⎠ 6σ

√ √ √ 6 3 −2b(r − 1)σ 2 − (−2)2/3 γ 2/3 + 2 3 −2F 2 − 4 3 γ F y(F) = ± √ 6 3 γσ √ 3

z(F) = ±

−2

√ 3

√ 

 −2F− 3γ γ +2 F 3 bσ 2

 √ √ + 6(r − 1) 2γ 2/3 + 2(−2)2/3 F 2 − 3−2 3γ F 18γ 2/3 ±

(46)

(47)

±

18(−2)2/3 b(r − 1)2 σ 2 18γ 2/3

(48)

5.2.3

Third Point √ √ √ √ √ 12a1 3γ σ + 2i 3 2 3a3 − 2 3 2a3 + i22/3 3γ 2/3 + 22/3 γ 2/3 x(F) = ± √ 12 3 γ σ

(49)

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y(F) = ±

√ √ √ √ √ √ √ −6i 3 2 3br σ 2 + 6 3 2br σ 2 + 6i 3 2 3bσ 2 − 6 3 2bσ 2 + i22/3 3γ 2/3 ± √ 12 3 γ σ √ √ √ √ 22/3 γ 2/3 − 2i 3 2 3F 2 + 2 3 2F 2 − 8 3 γ F ± √ 12 3 γ σ

(50) z(F) = ±

√  √

√

√  √ √ 1 3 3 3 + i γ 4/3 + 2i 2 3 + i 3 γ F 3 + 22/3 −1 − i 3 γ F+ {i 2 36bγ 2/3 σ 2

√

√  √  √ 3 3 + i b 3 γ F(r − 1)σ 2 − 12i22/3 3 − i bF 2 (r − 1)σ 2 + +6i 2

√ 

√  3 − i b2 (r − 1)2 σ 4 + 24bγ 2/3 (r − 1)σ 2 − 2i22/3 3 − i F 4} + − 18i22/3

(51)

5.3 Stability Analysis To proof that three obtained in Appendix 5.2 points are correct, let F equals to zero and other parameters (it is no needed to specify parameter τ because this parameter multiplied to F in each equation) are σ = 10, b = 8/3. Hence, we have a parametrically defined curve depending on r . As can be seen in Fig. 25 the obtained

Fig. 25 Stationary points coordinates depending on r (for stationary points from Appendix 5.2 when parameters σ = 10 and b = 8 are fixed, and parameter F is equal to zero). Here the red curve is dashed and placed over the blue curve

Stabilization of Chaos Via Strong Nonlinearities …

35

curve absolutely the same as the position of stationary points in the classical Lorenz system (1).10

References 1. Lorenz E (1963) Deterministic non-periodic flows. Journal of Atmospheric Sciences 20:130– 141 2. Haken H (1985) Laser Light Dyhamics. North Holland 3. Sagdeev RZ, Usikov DA, Zaslavski GM (1988) Nonlinear physics: from the pendulum to turbulence and chaos, vol 4. Harwood Academic Publishers 4. Field RJ, Burger M (1985) Oscillations and traveling waves in chemical systems. WileyInterscience 5. Xiang-Jun W, Shi-Lei S (2009) Chaos in the fractional-order Lorenz system. International Journal of Computer Mathematics 86(7):1274–1282 6. Mehmet Ali Akinlar, Fairouz Tchier, Mustafa Inc. Chaos control and solutions of fractionalorder Malkus waterwheel model. Chaos, Solitons & Fractals, 2020, v. 135, Pp. 109746 7. Song Juan, Niu Yanmin, Li Xiong (2019) The existence of homoclinic orbits in the Lorenz system via the undetermined coefficient method. Applied Mathematics and Computation 335:497– 515 8. Leonov GA, Kuznetsov NV, Mokaev TN (2015) Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity. Applied Mathematics and Computation 28:166–174 9. Xavier Gomez-Mont, Jose-Job Flores-Godoy, Guillermo Fernandez-Anaya. The Attractors in the Complex Lorenz Model. 3rd IFAC Conference on Analysis and Control of Chaotic Systems, 2012, v. 45, Issue 12, Pp. 87–92 10. Zhang Yagang, Yang Jingyun, Wang Kangcheng, Wang Zengping, Wang Yinding (2015) Improved wind prediction based on the Lorenz system. Renewable Energy 81:219–226 11. Zhang F, Shu Y (2015) Global dynamics for the simplified Lorenz system model. Applied Mathematics and Computation 259:53–60 12. Fen Mehmet Onur (2017) Persistence of chaos in coupled Lorenz systems. Chaos, Solitons & Fractals 95:200–206 13. Chaohai T, Chunde Y (2008) Three control strategies for the Lorenz chaotic system. Chaos, Solitons & Fractals 35:1009–1014 14. V. Lynnyk, N. Sakamoto, S. Celikovsky. Pseudo random number generator based on the generalized Lorenz chaotic system. 4th IFAC Conference on Analysis and Control of Chaotic Systems, 2015, v. 48, Issue 18, Pp. 257–261 15. Rakheja Pankaj, Vig Rekha, Singh Phool (2019) An asymmetric watermarking scheme based on random decomposition in hybrid multi-resolution wavelet domain using 3D Lorenz chaotic system. Optik 198:163289 16. Matson LE (2007) The Malkus-Lorenz water wheel revisited. American Journal of Physics 75:1114–1122 10 Stationary

points for system (1) are: x = 0, y = 0, z = 0   x = b(r − 1), y = b(r − 1), z = r − 1   x = − b(r − 1), y = − b(r − 1), z = r − 1

.

(52) (53) (54)

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17. Illing L, Fordyce RF, Saunders AM, Ormond R (2012) Experiments with a Malkus-Lorenz water wheel: Chaos and synchronization. American Journal Of Physics 80:192–202 18. Mishra AA, Sanghi S (2006) A study of the asymmetric Malkus waterwheel: The biased Lorenz equations. Chaos 16:14 19. Tongen A, Thelwell RJ, Becerra-Alonso D (2013) Reinventing the wheel: The chaotic sandwheel. American Journal of Physics 81(2):127–133 20. Karpov EA, Semenov ME, Meleshenko PA, Klinskikh AF, Pervezentcev RE, Glazkova MYu (2019) Modified Lorenz-Malkus water wheel model: dry friction versus chaos. Journal of Physics: Conference Series 1368:042044 21. Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time-series. Physica D: Nonlinear Phenomena 16(3):285–317 22. Ewing JW (1885) Experimental researches in magnetism. Transactions of the Royal Society of London 176:523–640 23. Bouc R (1967) Forced vibrations of mechanical systems with hysteresis. Proceedings of the Fourth Conference on Nonlinear Oscillations 2:156 24. Wen YK (1976) Method for random vibration of hysteretic systems. Journal of the Engineering Mechanics Division 102(2):249–263 25. Krasnoselskii MA, Pokrovskii AV (1983) Systems with hysteresis. Springer 26. Ikhouane F, Rodellar J (2005) On the Hysteretic Bouc-Wen Model. Nonlinear Dynamics 42:63– 78 27. Solovyov AM, Semenov ME, Meleshenko PA, Barsukov AI (2017) Bouc-Wen model of hysteretic damping. Procedia Engineering 201:549–555 28. Semenov M.E., Reshetova O.O., Tolkachev A.V., Solovyov A.M. and Meleshenko P.A. Oscillations Under Hysteretic Conditions: From Simple Oscillator to Discrete Sine-Gordon Model. In: Belhaq M. (eds) Topics in Nonlinear Mechanics and Physics. Springer Proceedings in Physics, 2019, vol 228, Pp. 229–253 29. Zaman MA (2015) Bouc-Wen hysteresis model identification using Modified Firefly Algorithm. Journal of Magnetism and Magnetic Materials 395:229–233 30. Charalampakis AE, Dimou CK (2010) Identification of Bouc-Wen hysteretic systems using particle swarm optimization. Computers and Structures 88(21–22):1197–1205 31. Marano GC, Sgobba S (2007) Stochastic energy analysis of seismic isolated bridges. Soil Dynamics and Earthquake Engineering 27(8):759–773 32. Dahl PR (1968) A solid friction model. The Aerospace Corporation, Technical report 33. Dahl PR (1977) Measurement of solid friction parameters of ball bearings. The Aerospace Corporation, Technical report 34. Ikhouane Faycal, Manosa Victor, Pujol Gisela (2020) Minor loops of the Dahl and LuGre models. Applied Mathematical Modelling 77:1679–1690 35. Young JY, Trumper DL (2014) Friction modeling, identification, and compensation based on friction hysteresis and Dahl resonance. Mechatronics 24(6):734–741

Drive Dynamics of Vibratory Machines with Inertia Excitation Nikolay Yaroshevich, Olha Yaroshevych, and Viktor Lyshuk

Abstract Dynamics of the vibration machine taking into account the elasticity of coupling of the unbalance vibration exciter with a limited- power electric motor is considered. A solution describing the relative torsional oscillations of the vibration exciter and motor rotors in steady (near-steady) motion modes and the formula for vibrational torque was obtained. The above mentioned dependencies are also valid for the case of motor speed “sticking” in the natural frequency region of the vibration machine. The correlation between the vibrations of the unbalance drive and the vibration machine bearing body is established. Recommendations for avoiding resonant vibrations of the vibration machine drive are formulated. Keywords Vibration machine · Unbalanced drive · Elastic coupling · Somerfeld effect

1 Introduction Unbalance driven vibration machines are widely used in various industries. Typically, inertial exciter vibration machines are above resonance. When a vibration machine with limited-power passes the natural frequency region its resonant oscillations can be excited. Such oscillations are accompanied by a significant increase in dynamic loads in the structural elements of the machine. It should be noted that if the motor power of the vibration machine is not greatly overmotored, then the probability of the manifestation of the Sommerfeld effect at its start-up (run-up) is quite real. The practice of operation of vibratory sites for the manufacture of concrete and reinforced concrete products proves the frequent failure of the drive parts (such as N. Yaroshevich (B) · O. Yaroshevych · V. Lyshuk Lutsk National Technical University, St. Lvivska 75, Lutsk 43018, Ukraine e-mail: [email protected] O. Yaroshevych e-mail: [email protected] V. Lyshuk e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_2

37

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cardan shafts, elastic couplings, bearings of vibration exciters) due to their destructive vibrations [1]. Therefore, the study of the dynamics of the vibration machine taking into account the elasticity of coupling of the unbalance vibration exciter with an electric motor is a relevant scientific and applied problem.

2 Analysis of Researches on Dynamics of Unbalance Driven Vibration Machines The investigation of dynamic processes in vibration machines with unbalance drive, the passage of the inertial vibration exciter through the region of the machine natural frequency and the emergence of the Sommerfeld effect in particular are described in many papers, an overview of them can be found for example in [2, 3]. The theory of so-called limited-excitation systems has been comprehensively developed in the works of Kononenko [4] as well as in numerous papers of his students and followers. Significant theoretical results of the study of resonant interaction of a limited power source with an elastic subsystem (the Sommerfeld effect) were substantiated [5–7]. In particular in [5] it is analyzed the dynamical coupling between energy sources and structural response that must not be ignored in real engineering problems, since real motors have limited output power. Among the recent works, which consider these problems, the papers [8, 9] should be noted. A lot of studies are devoted to the analysis of dynamic processes in the drive of machine assemblies with elastic units. In particular, in [10], it is established that the dependence of the moment of resistance forces in the machine assembly on the angle of rotation, the elasticity of the drive units, as well as the limited power of the motor at its start-up can lead to a phenomenon similar to the Sommerfeld effect, and in steady motion it leads to a noticeable decrease in rotation frequency. However, the existing results are generally obtained for the most common machine assemblies and can be quite limitedly applied for vibration machines with unbalance drive because they are characterized by a number of specific features. In [3, 11], the advantages of the vibration mechanics approach for solving the applied tasks of vibrotechnics are demonstrated. In [11], using the method of direct separation of motions for the simplest system, the oscillating part of which is linear and has a single degree of freedom, the equation of slow motions of the rotor of the vibration exciter in the near-steady rotation modes is obtained; it is shown the graphical interpretation of stationary modes occurrence and their stability. This equation looks like the equation of a machine assembly, but an additional term, namely, the vibration torque (given to the motor rotor of the average torque of the forces of resistance caused by dissipative forces during oscillations) is added. The main dependences obtained for a vibration machine with straight-line oscillations of the bearing body, in [12] are generalized for the practically interesting case for machines with flat oscillations.

Drive Dynamics of Vibratory Machines with Inertia Excitation

39

Among the papers, which analyze the influence of the unbalance drive elasticity on the course of dynamic processes in vibration machine, it is worth mentioning papers [13, 14]. Chosen works show that the elastic coupling of the electric motor rotors and the vibration exciter can cause the resonant vibrations of the drive. These features should be taken into account when choosing the parameters for vibration machines with the unbalance drive while designing them. In this article, the results obtained in [13, 14] are revised and improved.

3 Description and Equations of Turning System Motion The bearing solid body is elastically mounted on a fixed base and can perform rectilinear oscillations along the x axis (Fig. 1). The vibration exciter, which is rotated by the motor, is fixed on the supporting body. The rotors both of the motor and vibration exciter are connected with the help of an elastic-damping element, let’s call it a clutch. It should be noted that in general, the rotors can be connected by any elastic element (such as a belt drive or a universal joint shaft). The elastic-damping element is considered inertial; its stiffness and resistance coefficients are denoted by cc and βc , respectively. The motion equations of such a system are as follows: I1 ϕ¨1 + βc (ϕ˙1 − ϕ˙ 2 ) + cc (ϕ1 − ϕ2 ) = L(ϕ˙1 ), I2 ϕ¨2 − βc (ϕ˙1 − ϕ˙2 ) − cc (ϕ1 − ϕ2 ) = −R(ϕ˙ 2 ) + mε(x¨ sin ϕ2 + g cos ϕ2 ), M x¨ + βx x˙ + cx x = mε(ϕ¨2 sin ϕ2 + ϕ˙22 cos ϕ2 ),

(1) (2)

where x, ϕ1 , ϕ2 —are the generalized coordinates of the bearing body, the motor rotor and the vibration exciter, respectively; I1 , I2 —are moments of inertia of the electric motor rotor and the unbalance vibration exciter; m, ε—are the mass of the vibration exciter and its eccentricity; M—is the total mass of the bearing body of the vibration machine and the vibration exciter; cx , βx —are coefficients of rigidity and Fig. 1 Schematic representation of vibration machine with rectilinear oscillations of bearing body

40

N. Yaroshevich et al.

viscous friction of the elastic elements of the suspension bracket of the bearing body; L(ϕ˙1 ), R(ϕ˙2 )—is the torque of the electric motor and the moment of resistance forces of rotation of the vibration exciter; g—is gravity acceleration.

4 Understanding Motion Equations We use the method of direct separation of motions [3, 11] for our study. We will look for solutions of equation systems (1), (2) in the following form: ϕ˙i = ω(t) + ψ˙ i (t, ωt), ϕi = ω(t)t + αi (t) + ψi (t, ωt), i = 1, 2, x = x(t, ωt),

(3)

 ˙ slow, and ψi and x—are fast time functions t, where ω and αi (t) = ω(t)dt—are whilst the latter 2π - being periodic for τ = ωt and their average values during this period are equal to zero; we also assume that ψ˙ i 0

(16)

B = (S1 S2 + S4 S5 ) > 0

(17)

Furthermore, Eq. (14) has only one positive root if the following condition is satisfied C = S12 + S42 − S32 < 0

(18)

In what follows, the parameters are fixed as: α1 = 0.05, α2 = 0.05, β1 = 0.4, β2 = 0.4, γ1 = 0.2, γ2 = 0.2, δ1 = 1, δ2 = 1, μ = 0.2, M = 0.5, δ12 = 0.3, λ12 = 4.754, λ22 = −0.421, χ = 0.192, ψ1 = 0.919, ψ2 = −0.0813, η1 = 1.112, η2 = −0.112, p1 = 0.766 and p2 = 1.168.

3 Quasiperiodic Solutions and Stability Diagram In this Section, QP responses of Eqs. (3), (4) corresponding to periodic solutions of the slow flow (13) are determined near the first naturel frequency p1 applying the second-step multiple scales method on the slow flow [1]. This method has previously been used to approximate QP solutions and frequency-response curves in various nonlinear oscillators; see for instance [4, 32–34]. To approximate periodic solutions of the slow flow (13), it is convenient to transform the modulation equations from the polar form (13) to the following Cartesian system using the variable change u = a1 cos φ1 , v = −a1 sin φ1 ⎧ du ⎪ = (S4 − S5 )v + μ{S1 u + (S2 u + S5 v)(u 2 + v2 )} ⎨ dt ⎪ ⎩ dv = −(S + S )u + μ{S + (S − S u)(u 2 + v2 )} 4 3 1 2 5 dt

(19)

where μ is a new bookkeeping parameter introduced to implement a second-step perturbation procedure. A periodic solution of the slow flow (19) can be sought in the form

Quasiperiodic Stability Diagram in a Nonlinear Delayed Self-Excited …

55

u(t) = u 0 (T0 , T1 ) + μu 1 (T0 , T1 ) + O(ε2 ) v(t) = v0 (T0 , T1 ) + μv1 (T0 , T1 ) + O(ε2 )

(20)

where T0 = t and T1 = μt. Introducing Di = ∂∂Ti yields dtd = D0 + μD1 + O(μ2 ), substituting (20) into (19) and collecting terms, one obtains at different order of μ: D02 u 0 + ν 2 u 0 = 0 (S4 − S5 )v0 = D0 u 0

(21)

D02 u 1 + ν 2 u 1 = −D0 D1 u 0 + (S4 − S5 )[−D1 v0 + S1 v0 + (S2 v0 − S5 u 0 )(u 20 + v02 )] + D0 [S1 u 0 + (S2 u 0 + S5 v0 )(u 20 + v02 )] (22)

(S4 − S5 )v1 = D0 u 1 + D1 u 0 − [S1 u 0 + (S2 u 0 + S5 v0 )(u 20 + v02 )]

(23)

√ where ν = (S4 + S3 )(S4 − S5 ) is the frequency of the QP modulation. Up to the first order, the solution reads u 0 (T0 , T1 ) = R(T1 ) cos(νT0 + ψ(T1 )), −ν R(T1 ) sin(νT0 + ψ(T0 )) v0 (T0 , T1 ) = (S4 − S5 )

(24)

where R and ψ are, respectively, the amplitude and the phase of the QP solution. Substituting (24) into (22), (23) and removing secular terms, we obtain the following modulation equations of the slow flow (slow-slow flow) ⎧

S2  3 dR ν 2 S2 ⎪ ⎪ ⎪ ⎨ dt = 2 + 2(S − S )2 R + S1 R 4 3 3 ⎪

 3ν 3 S ν S5 dψ ⎪ 5 ⎪ ⎩R = + S5 (S4 − S3 ) R 3 + 3 dt 8(S4 − S3 ) 4(S4 − S3 ) 8ν

(25)

Equilibria of this slow-slow flow corresponding to periodic solutions of the slow flow (19) determine QP solutions of the original Eqs. (6), (7). In addition to the trivial equilibrium, R = 0, the nontrivial equilibrium obtained by setting ddtR = 0 is given by R=

S1 (S3 − S4 ) S2 S4

(26)

Thus, the approximate periodic solution of the slow flow (19) is given by u(t) = R cos(ϕt) νR v(t) = − sin(ϕt) (S4 − S5 )

(27)

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and the amplitude a(t) of the QP oscillations reads a(t) =

 R 2 cos2 (ϕt) + −

ν R 2 2 2ν R 2 sin(2ϕt) sin (ϕt) ± (S4 − S5 ) (S4 − S5 )

(28)

Consequently, the modulation envelope of the QP response is delimited by amin and amax given by amin = min





 R 2 cos2 (ϕt) + −



 ν R 2 2 2ν R 2 sin(2ϕt) sin (ϕt) ± (S4 − S5 ) (S4 − S5 ) (29)

 ν R 2 2 2ν R 2 sin(2ϕt) sin (ϕt) ± (S4 − S5 ) (S4 − S5 ) (30) Note that in the case of parametric resonance with respect to the second frequency of free vibrations ω2 = p22 + εσ2 where σ2 is a detuning parameter the slow flow modulation equation of amplitude a2 and phase φ2 can be obtained as before and it is given in Appendix. The same analysis is carried out near the second naturel frequency p2 and the results are also reported in this paper. Figure 2 illustrates the effect of delay amplitudes on the frequency-response near the first naturel frequency p1 for time delay T = 3. The effect of the delay amplitude in the velocity gv and in position g p is shown in Fig. 2a, b, respectively. Black curves correspond to the undelayed case (solid for stable and dashed for unstable), while blue lines correspond to the delayed case. Circles are obtained by numerical simulations carried out using dde23 algorithm [35]. It can be seen from these plots amax = max

4

 R 2 cos2 (ϕt) + −

gv=0.2

(a)

4

3.5

3.5

3

3

2.5

a1



v

a

1

2 1.5

2 1.5

1

1

0.5

0.5

0

gp=0

2.5

g =0

(b)

gp=0.2

0.72

0.74

0.76

0.78

ω

0.8

0.82

0

0.72

0.74

0.76

ω

0.78

0.8

0.82

Fig. 2 Frequency-response curves near the naturel frequencies p1 for T = 3; a g p = 0, b gv = 0, Color online

Quasiperiodic Stability Diagram in a Nonlinear Delayed Self-Excited … 2

ω=0.75

g =0 p

g =0.2

4

(a)

v

1

2

0.5

1

1

x (t)

1

gp=0

g =0.2

0

0 −1

−0.5 −1

−2

−1.5

−3 −4

−2 500

1000

1500

2000

2500

500

3000

1000

1500

Time, t

2

ω=0.77

gv=0

gp=0.2

(c)

2500

3000

4

ω=0.74

g =0 v

g =0.2

(d)

p

3

1

2

0.5

1

x1(t)

1

2000

Time, t

1.5

x (t)

(b)

v

3

1.5

x (t)

ω=0.8

57

0 −0.5

0 −1

−1 −2

−1.5 −3

−2 500

1000

1500

2000

2500

3000

−4

Time, t

500

1000

1500

2000

2500

3000

Time, t

Fig. 3 Time histories for values of ω picked from Fig. 2

that the delay decreases the pick of the periodic amplitude and increases substantially the QP envelop in both cases of delay when present in position (Fig. 2a) or in velocity (Fig. 2b). Time histories presented in Fig. 3a, b, c, d are drawn for ω = 0.75, ω = 0.8, ω = 0.77 and ω = 0.74, respectively. These correspond to values of circles picked from Fig. 2a, b. In Fig. 4 is shown the same results as in Fig. 2 but near the second naturel frequency p2 and for time delay T = 1. It is observed from the figures that the amplitude of periodic and QP responses increases in both cases of delay except that the magnitude of amplitudes is smaller comparing to that near p1 (Fig. 2). In the previous case (Fig. 2) only the amplitude of the QP response increases while that of the periodic solution decreases. It is interesting to examine the variation of the amplitude of periodic and QP responses versus time delay T and locate the intervals of time delay for which the magnitude of amplitudes are maximum or minimum. This variation is presented in Fig. 5 near the first naturel frequency p1 for various values of delay amplitudes g p and gv . Figures 5a, b show, respectively, the variation when the delay is present only in the position, while in Fig. 5c, d the delay is present only in the velocity. In each situation, the result are compared to the case where the delay is absent (dashed line). These plots indicate the intervals of time delay in which the values of amplitudes are

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(b)

(a) gv=0.2

gp=0.2

1

0.8

0.8 0.6

a

a2

2 0.6

0.4 0.4

gv=0

g =0 p

0.2

0.2

1.05

1.1

1.15

ω

1.2

1.25

1.3

1.05

1.1

1.15

1.2

ω

1.25

1.3

Fig. 4 Frequency-response curves near the naturel frequencies for p2 for T = 1; a gv = 0, b g p = 0. Color online 4.5

4.5

ω=0.74 ω=0.76

4

3.5

3.5 3

a1p

3

g =0.2

g =0.2 p

p

2.5

g =0

a

p

1.5

1.5

1

1

0.5

0.5

2

p

1QP 2

0

g =0

2.5

2

0

(b)

4

(a)

6

4

8

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12

10

0

2

4

Time delay, T 4.5

4.5

ω=0.76

4

10

12

(d)

4 3.5

g =0.2 v

3

a1QP

v

2

1.5

1.5

1

1

0.5

0.5

2

4

6

Time delay, T

8

v

10

12

g =0

2.5

2

0

g =0.2

3

g =0

2.5

0

8

ω=0.74

(c)

3.5

a1p

6

Time delay, T

0

v

0

2

4

6

8

10

12

Time delay, T

Fig. 5 Periodic and QP responses versus time delay T near the naturel frequency p1 , a, b gv = 0, c, d g p = 0

Quasiperiodic Stability Diagram in a Nonlinear Delayed Self-Excited … 1.2

ω=1.17

(a)

ω=1.1

1.2

1

59 (b)

1

g =0.2 p

g =0.2 p

gp=0

0.8

0.8

g =0.1

a

a

p

2p

2QP

0.6

0.4

0.4

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0.2

0

gp=0

0.6

0

2

4

6

8

10

0

12

0

2

4

Time delay, T 1.2

6

8

10

12

Time delay, T

(c)

ω=1.17

ω=1.1

1.2

(d)

1 1

g =0.2

g =0.2

v

v

0.8

0.8

g =0.1

a2p

a2QP

v

0.6

g =0 v

0.6

0.4

0.4

g =0 v

0.2

0

0.2

0

2

4

6

Time delay, T

8

10

12

0

0

2

4

6

8

10

12

Time delay, T

Fig. 6 Periodic and QP responses versus time delay T near the naturel frequency p2 a, b gv = 0, c, d g p = 0

optimum. The same results are given in Fig. 6 but near the second naturel frequency p2 and for different values of the delay amplitudes g p and gv . Figure 7a shows, in the absence of time delay in the position, g p = 0, the stability chart in the parameter plane (gv , T ) near p1 for ω = 0.79. Three regions can be distinguished. Namely, the white region where stable limit cycle (SLC) exists, the lime region in which stable QP (SQP) solution occurs and the aqua region over which SQP and SLC responses are both present. This bistability of SQP and SLC solutions can be clearly seen in the two time histories (Fig. 7b, bottom) corresponding both to cross 2 picked from Fig. 7a for two different initial conditions. The bistability zone is also depicted in Fig. 7c showing the effect of delay in the position g p on the stability diagram in the case where both delays are present. In Fig. 8a, b is presented the stability chart in the parameter plane (g p , T ) near the natural frequency p1 in the absence and presence of time delay in velocity, respectively. The bistability zone can also be depicted.

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Fig. 7 Stability chart in the plane (gv , T) near p1 ; (SLC) stable LC, (SQP) stable QP, a g p = 0, c g p = 0.15, ω = 0.79. b time histories. Color online

Fig. 8 Stability chart in the plane (g p , T) near p1 ; (SLC) stable LC, (SQP) stable QP, a gv = 0, b gv = 0.15, ω = 0.79. Color online

Quasiperiodic Stability Diagram in a Nonlinear Delayed Self-Excited …

61

Fig. 9 Stability chart in the plane (gv , T) near p2 ; (SLC) stable LC, (SQP) stable QP, a g p = 0, c g p = 0.15, ω = 1.2. b time histories. Color online

Finally, Fig. 9a, c show, respectively, the stability chart in the parameter plane (gv , T ) in the absence and presence of time delay in the position. The results are given near p2 for ω = 1.2. As before, the small domain of bistability can be observed. The two time histories (Fig. 9b, bottom) corresponding to cross 2 picked from Fig. 9a clearly confirm the coexistence of the SQP and SLC responses for the frequency ω = 1.2. The stability chart in the parameter plane (g p , T ) is presented near p2 in Fig. 10a in the absence of time delay in the velocity and in Fig. 10b in its presence.

4 Conclusions We have studied QP solution and frequency-response curves in a two degree-offreedom nonlinear delayed coupled oscillators. The coupling is introduced through a linear parametrically modulated stiffness and the time delay is considered in the position and velocity. The delay is acting as feedback control on one component of the coupled system. Analytical approximation of QP solution was obtained applying

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Fig. 10 Stability chart in the plane (g p , T) near p1 ; (SLC) stable LC, (SQP) stable QP, a gv = 0, b gv = 0.15, ω = 0.79. Color online

the second-step perturbation method on the slow flow of the coupled oscillators near the first and the second naturel frequencies. A stability analysis was performed near the two naturel frequencies and the stability charts were obtained. Results shown that the delay increases substantially the amplitude of the QP response away from the resonance. This can be of interest in some engineering applications where largeamplitude response is desirable. In this case, the delay parameters can be tuned optimally to ensure a best performance of the system. It was also proved that for appropriate values of time delay parameters and frequency of the modulation, there exists small regions in the stability chart where two stable solutions may coexist. This bistability phenomenon, involving stable LC and QP solutions, can be suppressed by a small increase of delay amplitude in position or in velocity. Acknowledgements Part of the results of this chapter was presented by The first author I.K. at the 15th International Conference on Vibration Engineering and Technology of Machinery (VETOMAC 2019), Curitiba, Brazil, November 10-15, 2019. Prof. Jose´ Manoel Balthazar is gratefully acknowledged for his support.

Appendix The slow flow modulation equation of amplitude a2 and phase φ2 near the second natural frequency p2 is given by ⎧ da2    ⎪ ⎪ = S1 a2 + S2 a23 + S3 a2 sin(2φ2 ) ⎪ ⎨ dt ⎪ ⎪ dφ ⎪ ⎩ a2 2 = S4 a2 + S5 a23 + S3 a2 cos(2φ2 ) dt

(31)

Quasiperiodic Stability Diagram in a Nonlinear Delayed Self-Excited …  gp χ sin(ν12 ωT ) − g2v χ cos(ν12 ωT ), S2 2ω 2   ν σ S3 = − 4νη112μω + 4νη112μω Mλ22 , S4 = − 2ν2112 ω2 − 38 β1 (ν12 ω)2 ψ13 − 18 β2 χ 3 Mλ22 ,  gp 12 γ1 χ cos(ν12 ωT ) − g2v χ sin(ν12 ωT ) and S5 = 3ν ψ 3 − 8ν3γ122ω χ 3 Mλ22 . 2ν12 ω 8ν12 ω 1

where



S1 = 21 α1 ψ1 + 21 α2 χ Mλ22 +

63

= +

References 1. Belhaq M, Houssni M (1999) Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations. Nonlinear Dyn 18:1–24 2. Belhaq M, Guennoun K, Houssni M (2002) Asymptotic solutions for a nonlinear quasi-periodic Mathieu equation. Int J Non-Linear Mech 37:45–460 3. Guennoun K, Houssni M, Belhaq M (2002) Quasi-periodic solutions and stability in a weakly damped nonlinear quasi-periodic Mathieu equation. Nonlinear Dyn 27:211–236 4. Rand R, Guennoun K, Belhaq M (2003) (2003), 2:2:1 Resonance in the quasi-periodic Mathieu equation. Nonlinear Dyn 31:367–374 5. Abouhazim N, Rand R, Belhaq M (2006) The damped nonlinear quasiperiodic Mathieu equation near 2:2:1 resonance. Nonlinear Dyn 45, 237-247 6. Lakrad F, Abouhazim N, Azouani A, Belhaq M (2005) Busters and quasi-periodic solutions of a self-excited quasi-periodic Mathieu oscillator. Chaos, Solitons Fractals 24:813–824 7. Kirrou I, Mokni L, Belhaq M (2013) On the quasiperiodic galloping of a wind-excited tower. J Sound Vib 332:4059–4066 8. Belhaq M, Kirrou I, Mokni L (2013) Periodic and quasiperiodic galloping of a wind-excited tower under external excitation. Nonlinear Dyn 74:849–867 9. Hamdi M, Belhaq M (2013) Quasi-periodic oscillation envelopes and frequency locking in excited nonlinear systems with time delay. Nonlinear Dyn 73:1–15 10. Hamdi M, Belhaq M (2015) Quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay amplitude. J Vib Control 24:5726–5734 11. Lakrad F, Belhaq M (2015) Quasi-periodically actuated capacitive MEMS. In: Belhaq M (ed) Structural nonlinear dynamics and diagnosis, vol 168. Springer, Switzerland, pp 183–200 ISBN 978-3-319-19850-7 12. Belhaq M, Hamdi M (2016) Energy harvesting from quasi-periodic vibrations. Nonlinear Dyn. 86:2193–2205 13. Ghouli Z, Hamdi M, Belhaq M (2017) Energy harvesting from quasi-periodic vibrations using electromagnetic coupling with delay. Nonlinear Dyn 89:271–285 14. Belhaq M, Ghouli Z, Hamdi M (2018) Energy harvesting in a Mathieu-van der Pol-Duffing MEMS device using time delay. Nonlinear Dyn 94:2537–2546 15. Warminski J, Litak G, Szabelski (2000) Synchronisation and Chaos in a Parametrically and self-excited system with two degrees of freedom. Nonlinear Dyn 22:135–153 16. Warminski J (2012) Regular and chaotic vibrations of van der Pol and Rayleigh oscillators driven by parametric excitation. Procedia IUTAM 78–87 17. Belykh VN, Pankratova EV, Pogromsky AY, Nijmeijer H (2008) Two van der Pol-Duffing oscillators with Huygens coupling. In: ENOC-2008. Saint Petersburg, Russia 18. Camacho E, Rand R, Howland H (2004) Dynamics of two van der Pol oscillators coupled via a bath. Int J Solids Struct 41:2133–2143 19. Kozlowski J, Parlitz U, Lauterborn W (1994) Bifurcation analysis of two coupled periodically driven Duffing oscillators. Phys Rev E 51(3) 20. Kenfack A (2003) Bifurcation structure of two coupled periodically driven double-well Duffing oscillators. Chaos Solitons Fractals 15:205–218 21. Cai J, Shen J (2008) Hopf bifurcation analysis and approximation of limit cycle in coupled van der Pol and Duffing oscillators. Open Acoust J 1:19–23

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Harmonic Balance of Bouc-Wen Model to Identify Hysteresis Effects in Bolted Joints Luccas Pereira Miguel, Rafael de Oliveira Teloli, and Samuel da Silva

Abstract This chapter presents a new methodology through a harmonic balance method to identify the parameters to describe the hysteresis effect in bolted joints using the Bouc-Wen model. The main idea is to adopt a priori smoothing procedure in the restoring force that divides it into smooth polynomial intervals that describe, separately, the loading and unloading hysteresis cycles. This previous step allows us to obtain a closed-form analytical solution through the harmonic balance method to a nonlinear system with a weak hysteretic restoring force based on its parameters. The methodology is illustrated on the BERT benchmark, that consists of a cantilever aluminum beam with a bolted joint connection at its middle. The experimental displacement amplitudes are used as input of an inverse problem in which the output is the parameters obtained through an optimization procedure. The advantage of this approach is the direct use of the harmonic balance equations on the objective function. The results show that the identified Bouc-Wen model can adequately describe the hysteresis effect of the bolted joint. Keywords Hysteresis · Bolted joints · Harmonic balance method

1 Introduction Bolted joints play a vital role in structural engineering and are also a significant source of local nonlinearity in assembled structures. The nonlinear behavior occurs due to the friction on the contact surface between the joined components, which L. Pereira Miguel (B) · R. de Oliveira Teloli · S. da Silva UNESP - Universidade Estadual Paulista, Departamento de Engenharia Mecânica, Ilha Solteira, SP, Brazil e-mail: [email protected] R. de Oliveira Teloli e-mail: [email protected] S. da Silva e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_4

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induces hysteresis effects [1]. In a general way, hysteresis is the interaction between three variables: an input, an output, and an evolutionary variable which induces delay and memory effects on both others [2]. In the case of mechanical systems, the input is equivalent to an excitation force, the output a displacement, and the evolutionary variable a nonlinear restoring force which generates changes on the stiffness and mainly, for bolted joints, damping, reaching up to 90 % of the global damping of the structure [3]. Due to the mathematical and phenomenological complexity, which includes features like nonsmoothness, multiple solutions, and memory effect, dealing with hysteresis carries some challenging issues. To deal with it, several models had been proposed to describe the phenomena, such as the LuGre model [4, 5] and the Dahl model [6]. Among them, stands out also the Bouc-Wen model [7], as one of the most versatile models. As it is not a physical but a mathematical model, it is included the possibility of applications even for other knowledge areas [2]. However, some practical questions remain underexplored in the specific literature, such as the obtaining of closed-form solutions and the adjustment of a parameter set that fits the proposed models to real structures. Therefore, in the present chapter, a new identification method is presented to the Bouc-Wen model. The methodology is based on an analytical solution proposed by Miguel, Teloli and da Silva [8] through a piecewise harmonic balance approach, led by a priori smoothing procedure. This procedure is based on dividing the hysteresis loop into smooth intervals, delimited mainly by the transition between the loading and unloading cycles. Hereupon arises another challenging issue about how to describe the smooth paths, which involves the particularities of each hysteresis model and the chosen approach to represent it. Ikhouane and Rodellar [2] proposed to divide the hysteresis loop of the LuGre model in two paths, considering the loading and unloading cycles with the concept of consistency and strong-consistency. Considering the Bouc-Wen model, Teloli and da Silva [9] divided both cycles into two paths, each one yielding in the four intervals approach applied to this chapter. The complete method is illustrated using experimental data acquired on a test bench composed of two aluminum beams with a bolted joint between them in cantilever boundary. A reduced model approaches the system with a single degree of freedom is considered on the free end. The outline of this chapter is organized as follows. First, Sect. 2 describes the Bouc-Wen model and the piecewise smoothing procedure. Then, the harmonic balance methodology developed both for general nonlinear systems and for nonsmooth systems is presented in Sect. 3. Next, Sect. 4 provides the formulation of the optimization problem solved on the identification procedure. Further, Sect. 5 presents the experimental test bench identified to illustrate the methodology. Section 6 presents the results comparing the Bouc-Wen identified model to the experimental response. Finally, some remarks and future steps are highlighted on Sect. 7.

Harmonic Balance of Bouc-Wen Model to Identify Hysteresis Effects in Bolted Joints

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2 The Bouc-Wen Model The Bouc-Wen model to describe mechanical hysteretic systems is given by [7]: m y¨ (t) + c y˙ (t) + ky (t) + Z (y, y˙ ) = u (t)

(1)

where m [kg] is the mass, c [Ns/m] the linear damping and k [N/m] the linear stiffness. Additionally, y(t) [m], y˙ (t) [m/s] and y¨ (t) [m/s2 ] are displacement, velocity and acceleration for an excitation force u(t) [N] in an instant of time t [s]. Lastly, Z (y, y˙ ) [N] is the nonlinear restoring force, which follows the nonlinear differential equation:   ˙ (y, y˙ ) = α y˙ (t) − γ | y˙ (t)| |Z (y, y˙ )|ν−1 Z (y, y˙ ) + δ y˙ (t)|Z (y, y˙ )|ν Z

(2)

where α [N/m], γ [m−1 ], δ [m−1 ] and ν are the Bouc-Wen parameters. Also, it is important to notice that in this paper, the specific case in which ν = 1 is considered. Due to absolute value functions that introduce the nonsmooth feature of the phenomena, this equation does not provide an explicit expansion in terms of displacement y(t) [10]. Hence, it is not possible to apply the harmonic balance or other whitebox methods directly. To deal with it, Teloli and da Silva [9] proposed a piecewise smoothing procedure in terms of the Bouc-Wen parameters α, δ, and γ. For this, some mathematical manipulations must be done, starting with a division of the time derivative of nonlinear restoring force by the velocity term, yielding:

⇔

dZ dy

dZ dy dZ ⇔ dy

⇔



 γ | y˙ | |Z|ν−1 Z ν + δ |Z| ⇔ y˙   γ | y˙ | |Z|ν−1 Z ν = α − |Z| +δ ⇔ y˙ |Z|ν   γ | y˙ | Z ν = α − |Z| +δ ⇔ y˙ |Z|   γ | y˙ Z| ν = α − |Z| +δ , for y˙ Z = 0 y˙ Z

dZ =α− dy

(3)

which is a differential equation of the displacement y. The definite integral of the Eq. 3, generates four smooth intervals based on the displacement and velocity directions. These intervals are described as: – path (i): y˙  0, Z  0 Z1 =

 α  1 − e−(δ−γ)(y−y0 ) (δ − γ)

(4)

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(a) Hysteresis loop.

(b) Exemplifying hysteretic output.

Fig. 1 Illustrative example of the hysteresis loop of a Bouc-Wen model. ˜ 2 , represents Z ˜ 3 and is Z ˜ 4 [8] ˜ 1 , is for Z described by Z

represents the path

• path (ii): y˙  0, Z  0  α  1 − e(δ+γ)(y−y0 ) (δ + γ)

(5)

 α  1 − e(δ−γ)(y+y0 ) (δ − γ)

(6)

 α  1 − e−(δ+γ)(y+y0 ) (δ + γ)

(7)

Z2 = − • path (iii): y˙  0, Z  0 Z3 = − • path (iv): y˙  0, Z  0 Z4 =

Figure 1 illustrates the four smooth paths on the y versus Z plan, where the hysteresis loop is also observed. They are limited by the maximum or minimum displacement, and the displacements y0 on the points B and C, for which the nonlinear restoring force is null. Thus, the intervals AC, C D, D B and B A correspond to the paths (i), (ii), (iii), (iv) respectively. Further, the path D B A composes the so-called loading cycle, whereas AC D the unloading cycle, characterized by the velocity direction. Since it is a limited problem along the hysteresis loop in a steady-state response, it is possible to expand on Taylor series:

Harmonic Balance of Bouc-Wen Model to Identify Hysteresis Effects in Bolted Joints

  ∞ [−(δ − γ)]n (y − y0 )n α Z1 ≈ 1− n! (δ − γ) n=0

  ∞ [(δ + γ)]n (y − y0 )n α Z2 ≈ − 1− n! (δ + γ) n=0

  ∞ [(δ − γ)]n (y + y0 )n α Z3 ≈ − 1− n! (δ − γ) n=0

  ∞ [−(δ + γ)]n (y + y0 )n α Z4 ≈ 1− n! (δ + γ) n=0

69

(8)

(9)

(10)

(11)

3 The Harmonic Balance Method Among the different methods to approximate the response of nonlinear systems, such as harmonic probing [9], the harmonic balance method stands out with a simple formulation and a clear physical meaning. On the one hand, when a single-harmonic excitation is applied on a linear system, the output expected is also a single-harmonic displacement with the same frequency of the input. On the other hand, for nonlinear systems, this assumption is not valid. Instead, there is harmonic distortion on the output signal due to the presence of high-order harmonic terms, multiples of the input frequency. Based on this assumption, the harmonic balance method proposes an analytical closed-form solution as a sum of sine and cosine terms as the Fourier series [11]. This section details the application of the method to obtain the Fourier coefficients on general nonlinear systems and then deals with the specific case of hysteretic systems, for which the methodology of this chapter is proposed [8]. Lastly, the formulation for the Bouc-Wen oscillator is presented.

3.1 General Harmonic Balance Assuming a motion equation such as the Eq. 1, a general form to represent the response of the system is: y(t) = a0 +

κ

[an cos(nωt) + bn sin(nωt)]

(12)

n=1

where an and bn are the nth Fourier coefficients, in a sum of κ considered terms. The accuracy of the method increases with the value of κ, but also the computational cost. So, the number of terms chosen must be evaluated according to the necessary precision. Once the displacement y(t) has been substituted by the Eq. 12 on the

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motion equation, the method also proposes the expansion of the nonlinear restoring force as Fourier series: κ

A0 Z(t) = + [An cos(nωt) + Bn sin(nωt)] 2 n=1

(13)

where An and Bn are also the nth Fourier coefficients but for the Z(t) force. For a general formulation, it is assumed that the nonlinear term Z(t) assumes the form of ]. In this a smooth and continuous function during all the period of oscillation [0, 2π ω case, the amplitudes An and Bn can be obtained by the classical Fourier analyses: 2π

ω An = π

ω

Z(t) cos(nωt)dt

(14)

Z(t) sin(nωt)dt

(15)

0 2π

ω Bn = π

ω 0

So, substituting the Eqs. 12–15 on the motion equation, yields: A sin (ωt) = −mω 2 + cω

κ

n 2 [an cos(nωt) + bn sin(nωt)]

n=1 κ

n [−an sin(nωt) + bn cos(nωt)]

n=1

+ k a0 + +

κ

n=1 κ

[an cos(nωt) + bn sin(nωt)]

A0 + [An cos(nωt) + Bn sin(nωt)] 2 n=1

(16)

To obtain the Fourier coefficients based on the system parameters, or even the inverse problem, a system of equations must be extracted from Eq. 16. For this, the proposal is the balance of the series that multiply the same harmonic term, resulting in the following 2κ + 1 equations:

Harmonic Balance of Bouc-Wen Model to Identify Hysteresis Effects in Bolted Joints

⎧ κ  ⎪  ⎪ −m (ωn)2 an + c (nω) bn + kan + An = 0 ⎪ ⎪ ⎪ ⎪ n=1 ⎪ ⎪    ⎪ ⎪ ⎪ cosine terms ⎪ ⎪ 2 ⎪ ⎪ + b1 k + B1 = A ⎨ −m (ω) b1 − cωa  1  first-order sine terms ⎪ κ ⎪   ⎪ ⎪ ⎪ −m (ωn)2 bn − c (nω) an + kbn + Bn = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n=2   ⎪ ⎪ ⎪ sine terms ⎪ ⎩ under braceka0 + A20 remaining terms = 0

71

(17)

Thus, the Fourier coefficients are obtained, solving the system of equations (17). The numerical method adopted here is the classical Newton-Raphson root-finding algorithm.

3.2 Piecewise Harmonic Balance for Hysteretic Systems As highlighted in Sect. 3.1, the simplicity of applying the harmonic balance is presented on the assumption of continuity and smoothness. Dealing with nonsmooth systems, in special the hysteretic ones, demands a more careful approach. It is because the nonlinear restoring force is not described as a single and smooth equation due to the transition between different regimes of motion. In the hysteresis case, it is necessary to divide the loop on smooth paths delimited mainly by the transitions between loading and unloading cycles. It is because smooth functions can not approximate these region. So, the idea of a piecewise harmonic balance approach for hysteresis consists of taking into account the individual contribution of each smooth path separately along with the limited range in which it is valid, that also delimit the integration intervals of the Fourier series of the Eqs. 14 and 15 [8]. Thus, the sum of the contributions of the integrals around the hysteresis loop approximates the Fourier coefficients of an average nonlinear restoring force. As discussed before and applied at this work, Teloli and da Silva [9] further divided both cycles, loading and unloading, into other two paths each one, resulting in the four intervals Z1 , Z2 , Z3 and Z4 , described by the Eqs. 8–10. It is important to highlight that it is also assumed for the harmonic balance a scenario in which there is a weak hysteresis that is necessary to admit an available approximation by Fourier series. So, an average hysteretic force of the Bouc-Wen model is given by the Fourier coefficients [8]:

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⎞ ⎛ π π 2ω ω ⎜ Z cos(nωt)dt + Z cos(nωt)dt+⎟ 1 2 ⎟ ⎜ ⎟ ⎜ π ⎟ ⎜0 2ω ⎟ ω⎜ ⎟ ⎜ An = ⎜ ⎟ 3π 2π π ⎜ 2ω ⎟ ω ⎟ ⎜ ⎟ ⎜ Z3 cos(nωt)dt + Z4 cos(nωt)dt ⎠ ⎝ π ω

(18)

3π 2ω

⎞ ⎛ π π 2ω ω ⎜ Z sin(nωt)dt + Z sin(nωt)dt+⎟ 1 2 ⎟ ⎜ ⎟ ⎜ π ⎟ ⎜0 2ω ⎟ ω⎜ ⎟ ⎜ Bn = ⎜ ⎟ 3π 2π π ⎜ 2ω ⎟ ω ⎟ ⎜ ⎟ ⎜ Z3 sin(nωt)dt + Z4 sin(nωt)dt ⎠ ⎝ π ω

(19)

3π 2ω

Moreover, the output amplitudes an and bn can be obtained, solving the system of equations of the Eq. 17. It is worth mentioning that in this work we consider the case in which the hysteresis loop of the Bouc-Wen model is symmetrical. Consequently, the Fourier term in which n = 0 that is related to asymmetric motion is null.

4 Nonlinear Parameter Estimation Thus, the harmonic balance equations are employed on a parameter estimation procedure. Firstly, for controlled tests on the structure of interest, data acquisition followed by signal processing of experimental measurements is done. Next, formulating the identification procedure of a model that predicts the experimental behavior of testing structures, it falls into an optimization problem. In this context, to overcome convergence problems during the estimation of parameters, we propose a two-step procedure: it starts extracting linear modal parameters through the adjustment of a FRF when the structure behaves linearly, which is given by: H( jΩ) =

1 X( jΩ) = 2 U( jΩ) ( jΩ) + 2ξωn ( jΩ) + ωn2

(20)

where the resonance frequency Ωn [rad/s] and damping ratio ξ can be adjusted by classical modal analysis and curve fitting employing line-fit methods [12]. Further, the second step of the estimation procedure is based on the harmonic balance equations for nonlinear identification of the Bouc-Wen parameters. It is

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adopted a mass normalization of the parameters, that turn to adopt the notation ∼ at each one. The optimization problem is now reduced in a direct comparison between the experimental Fourier amplitudes and the analytical ones for a set of Bouc-Wen parameters. Once we are dealing with a system with a weak hysteretic force, the first Fourier coefficients are enough to describe properly the frequency response curve of the assembled structure. The experimental displacement coefficients are extract from the experimental frequency curve when the structure is subjected to a stepped-sine around the first bending mode. These coefficients are given in terms of amplitude Xex p and phase φex p [13]:  (21) Xex p = a12ex p + b12ex p φex p and consequently



a1ex p = tan 1 b1ex p −



        X ex p a1  =    ex p    tan21φ + 1 

(22)

(23)

ex p

   b1  = X2 − a 2 ex p ex p 1ex p

(24)

Also assuming that the nonlinear restoring force has lowdelay concerning the sinusoidal excitation, one can conclude that the term y0 = a1ex p , that is a fixed term on the harmonic balance equations for a frequency value. Then, the optimization procedure is performed according to the objective function that follows:                 a1ex p  − a1hbm (λ) b1ex p  − b1hbm (λ) 2 2 + y(t)ex p − y(t)int (λ) 2     Fob (λ) = +     y(t)ex p 2 a1ex p  b1ex p  2 2

(25)

! " ˜ γ˜ is the vector of Bouc-Wen parameters, a1hbm and b1hbm are the where λ = α, ˜ δ, displacement Fourier amplitudes of the Eq. 12, obtained by the harmonic balance, and a1ex p and b1ex p are the experimental ones. Additionally, the terms y(t)ex p and y(t)int are displacement vectors obtained respectively experimentally and by numerical integration for a swept-sine input through the Runge Kutta 4t h order method. These terms are considered to take into account transient response, since the harmonic balance only considers steady-state response. The cross-entropy method is selected to perform the optimization procedure. The method consists of picking samples of parameter sets from truncated distributions, one for each parameter, and sorting the best ones. Then, the selected sets are so used to update the distributions, repositioning the means, and reducing the standard deviations. So the algorithm is repeated until the distributions converges to the best set of

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parameters to describe the experimental case [14]. The convergence criterion adopted in this work is the value of standard deviation of the distributions. Optimization ends when the greatest one reaches less than 1 × 10−5 .

5 Experimental Setup To illustrate the methodology, a bolted joint structure is assembled based on some literature examples [15–19]. The idea is to generate hysteretic behavior due to the stick-slip motion between the jointed parts and to model it by a Bouc-Wen model. Figure 2 shows the BERT1 beam, that is composed of two aluminum beams connected by a bolted joint in a clamped-free configuration. As presented in detail in the schematic representation of Fig. 2b, the assembled structure is instrumented by four accelerometers along its length, and a laser vibrometer at the free end. As this work focuses on the first bending mode, a reduced model is considered taking the laser vibrometer signal into account, once it provides better observability around the movement at the tip of the beam. The tests include whitenoise, swept-sine, and stepped-sine excitations, and the application of each one will be presented in the development of the following results.

Shaker Measurement Points

Bolted Joint

Laser Vibrometer

Top view of the bolted joint:

(a) General view

(b) Schematic representation

Fig. 2 Setup of the BERT beam

1 https://github.com/shm-unesp/DATASET_BOLTEDBEAM.

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Receptance [m/N]

10−2

10−3

10−4 17

18

19 20 Frequency [Hz]

(a) Swept-sine test.

21

(b) Stepped-sine test.

Fig. 3 Indicatives of nonlinear behavior on frequency domain. medium and high 0.25 V

is low input level

6 Results Figure 3 presents initial evidences of nonlinear behavior in the structure response, which are related to the presence of hysteric behavior. Figure 3a presents frequency response curves for a swept-sine considering several excitation amplitudes. One can see that the peak amplitude decreases when the excitation level increases, which indicates the nonlinear damping behavior. Additionally, stepped-sine tests shown in the Fig. 3b ranging from 13 up to 23 Hz, with increment step of 0.10 Hz, indicate a frequency shift in the resonance when the excitation level increases. This occurs due to the softening effect present in bolted structures upon nonlinear interactions at the joint interface. Thus, due to the known tendency of bolted joints to present hysteretic behavior [1] and to the observed nonlinear physical phenomena, it is assumed that the developed identification procedure is suitable for application on the assembled test bench. The procedure starts obtaining the model parameters by adjusting the experimental FRF with the line-fitting strategy, as discussed in Sect. 4. For this, it is used a whitenoise excitation as input with low amplitude level supplied in the shaker amplifier (0.05 V). Besides that, all the tests are performed with a sampling rate of 1024 Hz. The white-noise input is used here due to it is less energetic, which is useful to obtain the linear behavior. Figure 4 shows the experimental FRF compared to the line-fit result using Eq. 20. The identified resonance frequency is ωn = 18.8 Hz and damping ratio ξ = 0.44 %. For the nonlinear identification procedure, a stepped-sine test is performed with 32 seconds in each frequency to ensure a steady-state response. Figure 5 shows the computed amplitude and phase, which are used to extract the fundamental harmonic terms with Eqs. 23 and 24. Then the obtained terms are compared to the harmonic balance amplitudes in function of a set of Bouc-Wen parameters as in Eq. 25. In addition, it is also performed a swept-sine test to consider the transient behavior.

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Fig. 4 Comparison of the receptance FRF between

experimental and

(a) Amplitude stepped-sine.

linear FRF

(b) Phase stepped-sine.

Fig. 5 Amplitude and phase of the stepped sine test Table 1 Bouc-Wen parameters of the model identified k˜ (rad2 /s2 ) ωn (Hz) ξ (%) α˜ (rad2 /s2 ) 18.8

0.44

457.84

13495.41

δ˜

γ˜

534.52

37.27

Hereupon, employing the cross-entropy optimization method the results presented on Table 1 are obtained. Figure 6 correlates an experimental receptance FRF and the Bouc-Wen model identified in the frequency domain for low (0.05 V) and medium (0.15 V) excitation level to evaluate the results obtained, and both conditions attest a good agreement of the model. On the time domain, a comparison between the experimental response and the identified model is also presented.

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10

Displacement [mm]

Displacement [mm]

Fig. 6 Comparison of the receptance FRF between experimental (markers) and Bouc-Wen model (lines). and to is to

5 0 -5 -10 0

2

4

Time [s]

6

8

(a) Swept-sine time response. Fig. 7 Comparison of the time response between

10 5 0 -5 -10 3.5

4

4.5

Time [s]

5

5.5

(b) Zoom in the resonance frequency. experimental and

identified model

As the swept-sine test is primarily a transient input, Fig. 7 ensures that the nonlinear model identified is able to represent the transient effects on the time domain accurately. Lastly, as the main result, Fig. 8 compares the hysteresis loop for the Bouc-Wen model to the experimental data for three excitations level that are high, medium, and low. Figure 8 allows noticing that the nonlinear restoring force modeled by the BoucWen model identified is able to predict the hysteretic behavior. Another important feature that can be observed is the energy dissipated by the hysteresis per oscillation cycle, which is proportional to the internal area of the loop. So, as the form and scale of the predicted loop fits the experimental, the energy dissipated has a good agreement too. Furthermore, it increased with the amplitude is verified on the hysteresis loops, as also pointed out on the nonlinear FRF in Fig. 3a.

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Nonlinear Force [N]

Nonlinear Force [N]

1 0.5 0 -0.5 -1 -5

0

0 -0.5 -1 -20

5

Displacement [mm]

0.5

(a) Low excitation level.

-10

0

10

Displacement [mm]

20

(b) Medium excitation level.

Nonlinear Force [N]

1 0.5 0 -0.5 -1 -40

-20

0

20

Displacement [mm]

40

(c) High excitation level. Fig. 8 Comparison of the time response between

experimental and

identified model

7 Conclusion A smoothing method was developed to admit to implement the harmonic balance to identify Bouc-Wen parameters for describing of a bolted joint. The comparison between the experimental and identified model permits us to conclude that the identified model could adequately describe the experimental hysteresis in the bolted joint studied. On this structure, the main hysteresis feature was observed, which is the hysteresis loop due to the memory effect. For the loops obtained the identified model could accurately predict the energy dissipation. It also allows concluding that the piecewise approach of the hysteresis is beneficial to deal with the nonsmooth feature of the phenomena, making possible obtaining analytical response not only through the harmonic balance but other methods that find on the nonsmoothness a challenging issue, such as harmonic probing. Future steps on this scope include the identification not only of a deterministic model but also of a stochastic model that takes into account the uncertainty quantification through a bayesian approach.

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Acknowledgements The authors would like to thank the financial support provided by FAPESP grant numbers 2016/21973–5, 2017/15512–8, 2018/14528–0, and 2019/19684–3, and CNPq grant number 307520/2016–1 and 306526/2019–0.

References 1. Brake MRW (2017) The mechanics of jointed structures: Recent research and open challenges for developing predictive models for structural dynamics. Springer, Berlin 2. Ikhouane F, Rodellar J (2007) Systems with hysteresis: analysis, identification and control using the Bouc-Wen model. Wiley, New York 3. Beards CF (1983) The damping of structural vibration by controlled interfacial slip in joints. J Vib Acoust Stress Reliab Des 105(3):369–373 4. De Wit CC, Olsson H, Åström KJ, Lischinsky P (1995) A new model for control of systems with friction. IEEE Trans Autom Control 40(3):419–425 5. Åström KJ, De-Wit CC (2008) Revisiting the lugre friction model. IEEE Control Syst 28(6):101–114 6. Dahl PR (1968) A solid friction model. Technical report 7. Wen YK (1976) Method for random vibration of hysteretic systems. J Eng Mech Div 102(2):249–263 8. Miguel LP, Teloli RO, da Silva S (2020) Some practical regards on the application of the harmonic balance method for hysteresis models. Mech Syst Signal Process 143:106842 9. Teloli RO, da Silva S (2019) A new way for harmonic probing of hysteretic systems through nonlinear smooth operators. Mech Syst Signal Process 121:856–875 10. Jalali H (2014) An alternative linearization approach applicable to hysteretic systems. Commun Nonlinear Sci Numer Simul 19(1):245–257 11. Worden K, Tomlinson GR (2001) Nonlinearity in structural dynamics. Institute of Physics Publishing 12. Maia NMM, Silva JMM (1997) Theoretical and experimental modal analysis. Research Studies Press 13. Rao SS (1986) Mechanical vibrations. Addison-Wesley Longman, Incorporated 14. Rubinstein RY, Kroese DP (2013) The cross-entropy method: a unified approach to combinatorial optimization, Monte-Carlo simulation and machine learning. Springer Science and Business Media 15. Song Y, Hartwigsen CJ, McFarland DM, Vakakis AF, Bergman LA (2004) Simulation of dynamics of beam structures with bolted joints using adjusted Iwan beam elements. J Sound Vib 273(1):249–276 16. Ahmadian H, Jalali H (2007) Generic element formulation for modelling bolted lap joints. Mech Syst Signal Process 21(5):2318–2334 17. Jaumouillé V, Sinou J-J, Petitjean B (2010) An adaptive harmonic balance method for predicting the nonlinear dynamic responses of mechanical systems-application to bolted structures. J Sound Vib 329(19):4048–4067 18. Bograd S, Reuss P, Schmidt A, Gaul L, Mayer M (2011) Modeling the dynamics of mechanical joints. Mech Syst Signal Process 25(8):2801–2826 19. Süß D, Willner K (2015) Investigation of a jointed friction oscillator using the multiharmonic balance method. Mech Syst Signal Process 52–53:73–87

Dynamic Friction Model Study Applied to a Servomechanism at Low Velocities Rudnei Barbosa, Átila Madureira Bueno, José Manoel Balthazar, Paulo José Amaral Serni, and Daniel Celso Daltin

Abstract The following work contains an experimental method for identifying dynamic friction in a DC servomotor, whose identification and subsequent compensation is highly important. Friction has characteristics that are difficult to equate and for this reason it becomes very important and necessary to identify nonlinearities and subsequently compensate them. In this paper, based on the DC motor equation using Newton’s law of motion, feasible models are presented and friction is measured and analyzed through Matlab and a Quanser data acquisition card. Nonlinear dynamic friction torque is used and compared with the experimental results of a real servomotor. Keywords Nonlinear friction model · Nonlinear control systems · Mathematical modeling

R. Barbosa · Á. M. Bueno (B) · P. J. A. Serni Institute of Science and Technology at Sorocaba (ICTS), São Paulo State University (UNESP), Sorocaba, SP, Brazil e-mail: [email protected] R. Barbosa e-mail: [email protected] P. J. A. Serni e-mail: [email protected] J. M. Balthazar · D. C. Daltin School of Engineering (FEB), São Paulo State University (UNESP), Bauru, SP, Brazil e-mail: [email protected] D. C. Daltin e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_5

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1 Introduction Most physical phenomena have a nonlinear behavior and some of these phenomena have an approximate linear dynamic model, but many are extremely nonlinear as for example the relative motion between masses in the presence of friction as the actuation of a robotic joint by a DC servomotor [3, 5, 7, 11, 17]. Friction is the main source of nonlinearity that hinders the response time of a system that depends on a desired reference amplitude input value. Most published studies that study the phenomenon of friction under control systems consider the dynamics of each mechanism as linear and a linear model is obtained, not taking into account the nonlinearities present in the system [3–7, 11, 13]. The paper “Dynamic model study applied to a servo motor at low speed” was presented on 15th International conference on Vibration and Engineering and Tecnhology of Machinery in Brazil was held at Curitiba, that happenned on 10–15 November 2019 [1]. The conference focused on the four main areas from Concepts and methods in dynamics, Dynamics of mechanical and structural system, Dynamics and Control , Recent and emergent trends in dynamics and Control view [1]. In order to linearize the dynamics of a harmonic drive servomotor used in robotic joints and dynamic automation systems, an initial model based on open-loop experimental results is proposed, in search of a functional friction model through the measurement of angular velocities [1–5, 11, 13]. The objective of testing a dynamic model is always present during the development of this work, and for this reason, all the results of the experiments are compared to the results obtained in real time environment simulations using the Quanser data acquisition modules, preserving the essence of capturing analog signals from DC servomotor armature current transducers and Quanser SRV02 module speed and position transducers that will both be taken into consideration, which is a major factor in obtaining a position and velocity dependent dynamic friction model with the torque component present [1, 2, 11, 14, 15].

2 Types of Friction Models Lately there has been a significant interest in dynamic friction models, and the desire for knowledge plus the demand for precision servos and hardware advances makes it possible to implement friction compensators [4, 6, 8]. Several examples are available in the literature that generally correspond to the behavior of the effects of friction on the systems (Fig. 1a , b, c) [6]. The choice of the friction model used in this work has the Stribeck phenomenon [1] and was initially proposed by Gomes and Rosa [8]. The Stribeck curve is a more advanced velocity friction model even though it is still valid only at steady state, includes the Coulomb model and viscous friction as embedded elements represented in Fig. 1d [6].

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Fig. 1 Friction models. (H. Olsson, K. J. Åström, C. Canudas de Wit, M. Gäfvert, P. Lischinsky, 1998)

The Stribeck effect is a type of friction that occurs when a solid liquid or lubricant is used for contact surfaces of mechanical parts that do not move together. The phenomenon occurs with the reduction of the coefficient of friction F with the increase of velocity v until the zone called velocity versus Stribeck. When this speed starts at the threshold of the phenomenon (stick-slip) it affects especially the viscous friction [1, 3–7]. Several authors who focus on the nonlinear frictional behavior approach use the Coulomb friction depicted in Figure 1a, b, c. The Stribeck curve is a more advanced velocity friction model although only valid at a steady state, it includes the Coulomb model and viscous friction as embedded elements represented in Fig. 1d [5–7].

3 Gomes and Rosa’s Friction Model The friction model used in this study is based on Stribeck’s phenomenon [1] and was first proposed by Gomes and Rosa [8]. According to Gomes and Rosa [8] the dynamic model of the system is described by the following equations:   I r θ + k1 θr − ηθ¨ s = τ m − τ at

(1)

 θr ¨  − θ s = −τ at I r θ + k1 η

(2)

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Where Tat is the nonlinear friction torque [1, 5, 8, 10] so we will consider a rigid approach [8, 14], where the model is replaced by one degree of freedom and so we make the Eqs. (1) and (2) reduce to Eq. (3): I s θ¨ s + f v∗ θ˙ s = T m − T l

(3)

4 Calculating the Parameters of the Moment of Inertia in an Electromechanical System The viscous damping coefficient Beq was experimentally set by Quanser [14] Be q = 0.015 Nm/rad in the larger gear setting. Another parameter to be calculated is the motor’s moment of inertia. It is the sum of the rotor’s moments of inertia plus the tachometer speed sensor’s moment of inertia [10, 14, 17]. J m = J tach + J m r otor

(4)

Using the values specified in the table, we can calculate the total inertia moment of the electromechanical system [14, 17]. The load consisting of a steel cylinder coupled to the gear set with one of the gears having 24 teeth, the other gear having 70 teeth and the final one having 120 teeth coupled to the DC motor shaft. The formula according to Beer [17] for calculating the moment of inertia of the steel cylinder is: J cil =

2 mrcil 2

(5)

Where m is the mass and r is the radius of the steel cylinder. Assuming the gears use the specifications in the appendix, the moment of inertia of the 24-tooth, 72-tooth and 120-tooth gears. We have coupled to the shaft of the DC servomotor a gear set according to Table 2 [12]. Therefore the moment of inertia of the gears is: J g = J24 + J72 + J120

(6)

As the gears are coupled to the steel cylinder and the cylinder has a diameter of 50 mm and a mass of 504 g, we get the total moment of inertia of the servomechanism (J cil1) According to the mass of the steel cylinder (Table 3). Jl = Jg + Jcil(...)

(7)

Dynamic Friction Model Study Applied to a Servomechanism at Low Velocities Table 1 Physical parameters of the system Quanser 2011 Parameter Description Value Beq Viscous damping coefficient 0.004 Jeq Jl Kg ηm ηg km kt Vnom Rm Lm

Moment of inertia (motor shaft) Moment of inertia 1 (center of mass) Gear ratio Motor efficiency Gearbox efficiency Counter electromotive force coefficient Motor torque coefficient Nominal input voltage in the motor Armature resistance Armature inductance

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Unit Nm/(rad/s)

2.08 x 10−3 0.003372

kgm2 kgm2

70 0.69 0.90 7.68 × 10−3

V/(rad/s)

7.68 × 10−3 6.0 2.6 0.18

Nm/A V  mH

Table 2 Gear ratio of the servomechanism coupled to the DC motor Symbol Description Value Kgi Kge, high

Internal reduction rate of the servomotor Reduction rate of the largest gear

Table 3 Gears’ moments of inertia Inertia Value J24 J72 J120

1.03.10–7 5.44.10–6 4.18.10–5

14 5

Unit Nm/V Nm/V Nm/V

Therefore the final value of the moment of inertia is: Jl(l) = 3.37.10− 3Nm/V (massa = 504 g) Using the Eq. 8, Jeq = ηg K g2 Jm + Jl and the gain value found: Am = 0.129 Nm/V, results in the total inertia moment over the DC motor shaft: J e(q)1 = 2.08.10− 3Kgm2 (504 g) This value is represented at Table 1.

(8)

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5 Searching for a Dynamic Friction Model The motor torque applied to the inertial is τm and rotor I r and I s are current parameter besides applied load inertia represented by a steel cylinder coupled to the reducer output shaft, is the reduction ratio and θr and θs represent the angular positions of the rotor and associated load, respectively. In this type of dynamics in question, nonlinear friction is present and the motor torque is not at its fullness at the gear unit output [5]. According to Gomes and Rosa [8] the dynamic model of this system is described by the equations:   I r θ + k1 θr − ηθ¨ s = τ m − τ at (9)   θr ¨ − θ s = −τ at I r θ + k1 η

(10)

Where Tat is the nonlinear friction torque . We will consider a rigid approach, where the model is replaced by one degree of freedom. So we make the Eqs. (9) and (10) reduce to Eq. (11): I s θ¨ s + f v∗ θ˙ s = T m − T l

(11)

In search of an improvement regarding the nonlinearities present in the physical system, such as stick-slip modes and the Stribeck friction effect we chose the experimental method described below [5, 8, 9]. At each motor torque level the speed was measured with a speed sensor (tachometer) and the motor torque was obtained by editing the armature current multiplied by the motor constant km (see at Table 2). As with each engine torque, speed is measured so that the friction torque is equal to the applied motor torque for each speed in steady state, due to the absence of other external torques [5, 8]. Given this, we are able to obtain a set of rotor speed and friction torque points. For each direction of rotation an adjustment curve is plotted against these points. The polynomial equation that represents this adjustment has the following form: τt = f i + f i v + (θ + ci)(θ¨ )2

(12)

According Gomes and Rosa [5, 8] where the index i is equal to p at positive rotor speeds and n at negative rotor speed. Friction torque is written as an internal dynamic effect in the form: τt = f v (θ˙ )

(13)

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Fig. 2 Servomechanism. (Quanser 2011)

6 Implementation and Results to Obtain the Dynamic Friction Model For this purpose we will use the MATLAB / Simulink–Real Time toolbox software and a servomechanism. Figure 2 presents the servomechanism in question. A Simulink diagram was created as showed in Fig. 3. A servomotor is a key element in Quanser’s control experiments consisting of a reducing servo mechanism system, a direct current (DC) motor with a gear set. The servomechanism is also equipped with an encoder and a potentiometer for measuring output position and a tachometer for measuring motor speed [12]. The values found are represented in graphs with the mass of 504 g coupled to the servomechanism. Nine experiment points with 504 g mass, friction torque, and negative angular velocity of the DC motor and on that we can get a set of rotor speed and friction torque points that will be represented in the graphs. For each direction of rotation an adjustment curve is plotted against these points. In the previously developed study the polynomial curve had two partials, a positive velocity side polynomial and a negative velocity side polynomial. By using poly f it function of Matlab it is possible to perform a polynomial approximation of points by a polinomial of any order and thus bring the set closer together [3, 10, 16]. As an example the third degree polynomial curve was verified independently of the graph in Fig. 4 [3]. The syntax is : >> coef = polyfit(x, y, n) ; Where n is the order of the polynomial. Note that the function is valid for nonlinear approximations, being shown the cases n = 3, n = 5, n = 7 and n = 9.

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Fig. 3 Simulink experiment friction torque. (Quanser 2011)

In the graphs presented the polynomial curve of order: n = 3, n = 5, n = 7 and n = 9. In the first graph presented the polynomial curve of order: n = 3 with coupled load of 504 g. In the second graph presented the polynomial curve of order: n = 5 with coupled load of 504 g. In the third graph is presented the polynomial curve of order: n = 7 with coupled load of 504 g. And finally we will see the ninth order polynomial approximation curve. In this curve we note that the values of the extremes tend to not converge but as we increase the order of the polynomial equation the curve gets closer and closer to converging with the points, but in some high order polynomials above the ninth order it starts to get difficult for the curve to convergence with the points at the extremes [3, 10, 16]. Figures 4, 5, 6, and 7 represent the dynamic friction model generating a polynomial of the 3rd to 9th order extracted from the 18 points initially sampled.

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Fig. 4 Friction curve of the third order versus angular velocity

Fig. 5 Friction curve of the fifth order versus angular velocity

Notice that there was a need to increase the sampled points and that through numerical interpolation we could have more intermediate points to improve the approximation in search of the polynomials that represent the nonlinear behavior of Stribeck friction [3–8] (Fig. 8).

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Fig. 6 Seventh order curve of friction torque versus angular velocity

Fig. 7 Ninth order curve of friction torque versus angular velocity

7 Validating the Dynamic Friction Model Experimental identification of friction in a DC motor was performed on a servomechanism with a Faulhaber Coreless DC Motor type 2338S006, with maximum rated torque of 7.682 * 10–3 Nm/amp, stall torque of 17.0 kgm2 and constant current of 0.92 A [3, 12]. To allow the collection of velocity data on the motor, an analog tachometer was connected to the shaft through gears 1 and 2.

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Fig. 8 From second to ninth order curve of friction torque versus angular velocity

Gears 1 and 2 give out a reduction ratio of 1:70 to increase the sensitivity of the tachometer, whose resolution is 1.5 mV per revolution [12]. To allow the collection of current data on the motor, a current sensor with instrumentation amplifier was connected to the DC motor [12]. These signals were applied through a acquisition card to the servo amplifier. The real current input to the motor, which is assumed to correspond to the torque was measured and recorded using the same acquisition unit [3, 12]. By means of a nonlinear regression process, the optimized sets of parameters in each model are obtained (Figs. 9 and 10). A diagram model was develop at MATLAB/Simulink–Real Time toolbox software and a servomechanism to validate the friction model [3, 15]. According to Fig. 10, there are three blocks with polynomial functions that represent the friction models of the third, sixth and ninth order. These models were inserted in their respective blocks, and they all influence the system’s velocity, current and torque. The values of polynomial funcions are coefficients that were inserted into the polynomial function blocks on Simulink/MATLAB. In the Fig. 11, for example, ninth friction block were built and we demonstrated internal blocks that represents a dynamic servomechanism system to measure speed at experimental test [13]. For example we demonstrate the coefficients of third polynomial function which represents the velocity curve with friction torque included [10]. >> coef = polyfit(x, y, n) ;. >> coef. 3rd order = (−5.151618890e–006, 9.764693666e–007, +6.409008976e–004, −3.846869453e–005); The results were found and represented on graphs validated with the mass of 504 g coupled to the servomechanism.

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Fig. 9 Friction model validate diagram third order polynomial function. (Quanser 2011)

Fig. 10 Validate diagram—velocity with friction model third order polynomial function

In the simulation, five velocity signals are represented on graph 12, which represents the velocity real plant and we validates a friction model, while the friction model is approaching the real model as we increse the order of the polynomial function. The outcomes were presented on the graphs in Fig. 12, with the blocks n = 3, n = 6 and n = 9 adding friction and getting the theoretical model closer to the real model [10, 16].

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In the friction model that was previously developed, a friction model could be validated in which polynomial functions were represented and results were obtained [10, 16].

8 Conclusion The results obtained through simulations and the choice of a conventional actuator or a specific and precise actuator (with speed reducer) represented the search for a dynamic friction model that would help to understand and compensate for the nonlinear behavior present in this type of electric actuator and the load applied to it. From compliance modeling results, it has been illustrated that viscous dry friction and damping together will solve the reported difficulties in determining the compliance parameters. Furthermore, it has been shown that a non linear model better captures the behavior of system when combined with appropriate parameters for capturing nonlinearities. Friction losses of the harmonic drive are modeled at both low velocity and high velocity operating regimes. Finally, the model performance is confirmed by a simulation that verifies the friction model found and real model plant, presenting velocity measurements of both on graph 12. The DC servo actuator, despite being very precise, has some highly nonlinear internal friction that has been taken into account. A dynamic model has been studied which at very low angular velocities approaches a significant torque dead zone that, if not compensated, degrades control performance (by 10 to 20% of their maximum torque). Additionally, several experimental results were obtained during the present work by always using the same servomechanism for this purpose with satisfactory results, some of which focus on the identification of parameters and subsequent validation of the dynamic model, since obtaining a realistic dynamic model is an important basis for efficient control design.

References 1. Balthazar JM, Tusset AM, Ribeiro MA, Lenz WB (2019) Booklet of abstracts. In: 15th international conference on vibration engineering and tecnhology of machinery. Vectomac XV 2. Canudas de Wit C, Astrom KJ, Lischinsky P (1995) A new model for control of systems with friction. IEEE Trans Autom Contro 40–43 3. Daltin DC (2017) Modelagem e Controle de Sistemas Dinâmicos ModCon 4. Ferdinand B, Russell PE (1980) Mecânica vetorial para engenheiros. McGraw-Hill do Brasil, Johnston Jr 5. Garcia C (2005) Modelagem e simulação de processos industriais e de sistemas eletromecânicos, EDUSP

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6. Gomes SCP, Da Rosa VS (2003) A new approach to compensate friction in robotic actuators. In: 2003 IEEE international conference on robotics and automation, (Cat. No. 03CH37422), Vol 1. IEEE, pp 622–627 7. Gomes SCP, Rosa VS (2003) A new approach to compensate friction in robotic actuators. In: Proceedings of IEEE international conference on robotics and automation 8. Grimble MJ (2006) Robust industrial control systems: optimal design approach for polynomial systems. Wiley, New York 9. Hagedorn P, Stadler W (1998) Non-linear oscillations. Oxford engineering science series. Clarendon Press 10. Lynch S (2004) Dynamical systems with applications using MATLAB, 1st edn, Birkhauser, p 459 11. Machado CC (2007) PhD Thesis in Applied Mathematics UFRGS 2007, Mathematical modelling and active control of a manipulator with a flexible link, São Paulo State University Rio Grande (FURG), Brazil 12. MATHWORKS Inc. SimMechanics User’s Guide, November 2002 13. Ogata K (2003) Engenharia de controle moderno, Ogata, Katsuhiko, São Paulo: 4 edição 14. Olsson H, Åström KJ, Canudas de Wit C, Gäfvert M, Lischinsky P (1998) Friction models and friction compensation. Eur J Control (4):176–195 15. Quanser (2011) SRV02 Rotary Servo Unit User Manual 16. Ramasubramanian D (2016) Identification and control of DC motors, MS thesis. Universitat Politècnica de Catalunya 17. Taghirad HD, Belanger PR, Helmy A (1996) An experimental study on harmonic drives. Technical Report submitted 18. Virgala I, Kelemen M (2013) Experimental friction identification of a DC motor. Int J Mech Appl 3. https://doi.org/10.5923/j.mechanics.20130301.04 19. Wang X, Wang S-P (2012) High performance adaptive control of mechanical servo system with LuGre friction model: identification and compensation. J Dyn Syst Meas Control 134

Signal Analysis Through the Ensemble Empirical Mode Decomposition and Hilbert-Huang Transform-Application to Vortex Shedding Ana Paula Ost, Alexandre Vagtinski de Paula, and Sergio Viçosa Möller Abstract This paper presents the application of the Hilbert-Huang transform as a tool for signal analysis. The Hilbert-Huang transform (HHT) enables the analysis of nonstationary and nonlinear data being a combination of the Empirical Mode Decomposition (EMD) and the Hilbert Spectral Analysis (HSA). A noise assisted data analysis method, called Ensemble Empirical Mode Decomposition (EEMD), based on the statistical properties of white noise, was used to overcome EMD problems like mode mixing, and successfully deal with the scale separation. The EEMD was applied to each velocity signal to obtain a collection of Intrinsic Mode Functions (IMF). Vortex shedding in turbulent flows past circular cylinders represents a canonical problem suitable for validating new approaches in experimental analysis, as it has well known features. Experimental data in form of time series from hot wire signals of the turbulent flow past circular cylinders in an aerodynamic channel are employed for the analysis. Cylinder had a diameter of 32 mm. The Reynolds numbers varied from 1.05 × 104 to 3.28 × 104 . EEMD provided the localization of particular events in time-frequency space, allowing a complete physical interpretation of the dynamic processes at the corresponding flow scales. The Hilbert Spectral analysis provided an accurate representation of the amplitude-frequency-time distribution of the flow. The most energetic IMFs and the corresponding flow scales were identified from the Mean Square Energy of the IMF components. The application of HHT to the analysis of the hot wire signals obtained in the turbulent wake behind cylinders showed that it is a useful tool for the analysis of turbulent data, allowing the identification of turbulent structures and flow scales.

A. P. Ost · A. V. de Paula · S. V. Möller (B) Mechanical Engineering, Federal University of Rio Grande do Sul – UFRGS, Rua Sarmento Leite, 425, 90.050-170, Porto Alegre, RS, Brazil e-mail: [email protected] A. P. Ost e-mail: [email protected] A. V. de Paula e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_6

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Keywords Signal analysis · Hilbert-Huang transform · Ensemble empirical mode decomposition · Single cylinder · Turbulent flow · Hot wires

Nomenclature a(t) aj (t) B cj (t) cji (t) D E(t) f H(ω,t) k k* L MSE j N P Re  r ki (t) St T t* t, t’ U U* u(t) u x X(t) X H (t) X i (t) w wi (t) Y (t) Y j (t) Z(t) θ (t) ν

Instantaneous amplitude [m/s] Instantaneous amplitude of the jth IMF component [m/s] Aerodynamic channel width [m] Ensemble mean of the jth IMF The jth IMF component of the ith realization of noise-added data Diameter [m] Spline envelope Vortex shedding frequency [Hz] Hilbert energy spectrum Wavenumber [1/m] Dimensionless wavenumber Aerodynamic channel high [m] Mean Square Energy of the jth IMF component Length of time series or number of trials in EEMD Cauchy principal value Reynolds number Re = U D/ν Real part of the IMF Residue after extracting k IMFs in the ith realization of the noise added data Strouhal number Str = f D/U Sampling time [s] Dimensionless time Time [s] Reference velocity [m/s] Dimensionless instantaneous velocity Instantaneous velocity values [m/s] Mean velocity [m/s] Longitudinal position [m] Time series IMF data after Hilbert-transform The ith observation of the noise added time series Transversal position [m] White noise series Hilbert transform Normalized IMF, where j is the IMF mode number Complex analytical signal Phase function Kinematic viscosity [m2 /s]

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(k) *(k*) ω(t) ωj (t)

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Power spectral density [m2 /s] Dimensionless power spectral density Instantaneous frequency [Hz] Instantaneous frequency of the jth IMF component [Hz]

1 Introduction Like many other areas of the knowledge, Engineering requires data analysis as a mean to determine the parameters necessary for models construction, and to confirm that those models represent the phenomenon studied, being most of this data nonlinear and/or nonstationary. In this case, the application of Fourier analysis [6] is limited since it cannot deal with series where the mean values vary with time. In this case, literature suggests the use of wavelets [14]. Fourier spectral analysis has provided a general method for examining the global energy-frequency distributions. The method has dominated the data analysis efforts since soon after its introduction, because of its accomplishments and because of its simplicity, and has been applied to all kinds of data. The Fourier transform analyzes a signal over its whole temporal duration; hence, it yields information over the complete spectrum of the signal, but it is only accurate when applied to linear and stationary data [1]. Wavelet analysis arises from the idea of stretching and compressing the window of the windowed Fourier transform, according to the frequency to be localized, thus allowing the definition of all scales of interest in time and frequency domain [18, 35]. It is a suitable method for linear and non-stationary data. Indeed it is very useful in analyzing data with gradual frequency changes; however the method has limitations, and a non-adaptive nature, giving only a physically meaningful interpretation for linear phenomena [29]. Those methods have found many successful applications, however they have considerable limitations when applied to analyze nonlinear, nonstationary data. To overcome those limitations the empirical mode decomposition (EMD) and its subsequent modification, the ensemble empirical mode decomposition (EEMD), were developed [27]. The Hilbert-Huang transform proved to be a promising tool for non-linear and non-stationary data, being applied in many areas of research, for example, the analysis of financial time series [32], molecular dynamics [49], structural damage detection [60], vibration and modal analysis [12, 19], biological data analysis for cardiac and neurological problems detection [55], seismic and geophysics [3], and nuclear power plants [23] among others. There are many examples of HHT application in turbulence studies: in atmospheric boundary layer [25], nonlinear ocean waves [56], Lagrangian velocity analysis [26], grid generated turbulence [21], wind turbines [38], etc., but not so many in wakes behind cylinders.

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Single cylinders and cylinder arrangements submitted to a transversal flow are used to simulate a wide range of practical situations, such as shell and tube heat exchangers, chimney groups, risers and offshore platforms. Cylinders are an example of bluff bodies with a strong adverse pressure gradient and a wake formed on the opposite side impinged by the flow. For relatively small Reynolds numbers, the wake is characterized by the Kármán vortex street with a defined frequency which value varies with the flow velocity. The turbulent structures of this flow consist of a wide range of scales, including large-scale spanwise structures and relatively small-scale structures such as the secondary vortices, Kelvin–Helmholtz vortices and longitudinal rib-like structures [50], which makes this canonical flow suitable for studying turbulent structures in terms of their organized aspects and transport process; and for comparisons and validation of experimental and numerical methods. Since the works of Strouhal [53] and Bénard [8, 9], the flow on a single cylinder has been subject of many experimental and numerical studies concerning the flow features like drag and lift coefficients [57], Fage and Falkner [17], Achenbach [2]; velocity fields and Reynolds stresses [16, 46]; shedding frequency (Strouhal number) [51] and pressure distribution [11]. Many reviews on the flow features of single cylinders can also be found concerning the many aspects of experimental results, such as the influences of smooth or rough surfaces on the laminar and turbulent flows on circular cylinders by [61, 62], vortex-induced vibrations [4], wake configurations an flow visualizations [13]. A review of experiments on flow past circular cylinders was presented by Niemann and Hölscher [45], and a review on the wake formation of bluff bodies was presented by Derakhshandeh and Alam [15]. More recently the studies combined experimental and numerical simulations e.g. Parnadeau et al. [48], Capone et al. [10] and the application alternative tools of analyses, like wavelet transforms e.g. Indrusiak and Möller [34], and Hilbert-Huang transform e.g. Silveira and Möller [52], Franzini et al. [22]. Miau et al. [40] used Hilbert and Wavelet transforms to analyze the signals of MEMS sensors situated spanwisely on a circular cylinder for examining the unsteady, three dimensional behavior of vortex shedding. Analysis allowed the identification of the strong three-dimensional vortex shedding process due to the occurrences of vortex dislocation. This was also demonstrated in a further work, where hot wires and MEMS sensors were applied [41]. A comparison between Fourier Spectral Analysis, Wavelets and Hilbert-Huang Transform for the analysis of the shedding process of a single cylinder in turbulent flow was presented by Silveira and Möller [52]. The results were still preliminary, but showed that the Hilbert-Huang Transform could complement Wavelets, once it is directly decomposed from the original data, and the physical characteristics are shown clearly in the Intrinsic Mode Functions [27]. EMD was also used by Marzellier et al. [37] to analyze the time series obtained with Laser Doppler Velocimetry on the near wake of a square cylinder. Authors show the ability of EMD to identify coherent and turbulent fluctuations from the time series, concluding that the turbulence contribution is much smaller than the coherent part of the signal.

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More recently, Mironov et al. [42] compared HHT with Wavelet transform in the analysis of hot wire signals in the wake of a cylinder with instantaneous change of frequency. Results showed higher temporal resolution of HHT, although with some difficulties due to separation of fluctuations of interest into different IMF (Intrinsic Mode Functions). Analysis of the literature shows that the use of modern mathematical tools, like wavelets and HHT Transform, has increased the knowledge in many fields of science and engineering due to their ability of capturing non stationary features of the phenomena analyzed. According to Catallano et al. [11], the flow around a circular cylinder, with its complex features, represents a canonical problem for validating new approaches in computational fluid dynamics. This statement can be extended to new approaches in experimental analysis. This paper presents the analysis of hot wire signals from vortex shedding measurements in turbulent cross flow past a circular cylinder by means of Hilbert-Huang transform as tool for nonstationary and nonlinear turbulent data. Thus, time series of flow velocity are obtained in laboratory from hot wire measurements at several different Reynolds numbers by varying the air velocity and the cylinder diameter [47]. Since the flow around cylinders has well know features, the present study aims to verify the effectiveness of the Hilbert-Huang in characterizing properly a turbulent flow by means of the signal analysis from hot wire measurements and properly identifying turbulent features, such as coherent structures and flow scales. EEMD results are supported by Fourier and Wavelet analysis.

2 Hilbert Huang Transform The combination of the empirical mode decomposition (EMD) and the Hilbert spectral analysis is known as the Hilbert-Huang transform (HHT), which enables the analysis of non-linear and non-stationary data. Data-analysis methods are traditionally based on linear and stationary assumptions. Methods as Fourier analysis and Wavelet transform may not provide complete and physically meaningful results for turbulent data [28].

2.1 Hilbert Transform As Huang et al. [29] pointed out; one of the typical characteristics of a nonlinear process is their intra-wave frequency modulation, which indicates the instantaneous frequency changes within one oscillation cycle. One way to describe the system is in terms of the instantaneous frequency, which will reveal the intra-wave frequency modulations, and the easiest method to compute the instantaneous frequency is by using the Hilbert transform, which for an arbitrary time series X(t), is given by

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  1 ∞ X t  Y(t) = P ∫ dt π −∞ t − t

(1)

where P indicates the Cauchy principal value, which is a standard method applied in mathematical applications by which an improper integral is measured in a balanced way around singularities or at infinity [36]. By this definition HHT is, therefore, a convolution of X(t) with 1/t; hence, the transform emphasizes the local properties of X(t). X(t) and Y(t) form a complex conjugate pair by definition, so it is possible to have a complex analytical signal, Z(t) in the form Z(t) = X(t) + iY(t) = a(t)eiθ(t)

(2)

1/2  a(t) = X2 (t) + Y2 (t)

(3)

 Y(t) . θ (t) = arctan X(t)

(4)

where

and 

In Eqs. (3) and (4), a(t) is the instantaneous amplitude, and θ (t) is the phase function. Based on the Hilbert transform, the instantaneous frequency can be defined as ω(t) =

dθ (t) . dt

(5)

At any given time, it is possible that the signal may involve more than one oscillation mode, and consequently the signal has more than one local instantaneous frequency [27]. A complete description of the Hilbert transform with the emphasis on its mathematical formalities can be found in Hahn [24], and in Bendat and Piersol [6].

2.2 The Ensemble Empirical Mode Decomposition The concept of the Ensemble Empirical Mode Decomposition (EEMD) is based in the following observations [59]: (1) A collection of white noise cancels each other out in time-space ensemble mean, surviving and persisting, only the final noise-added signal ensemble mean; (2) Finite amplitude white noise is necessary to force the ensemble to exhaust all possible solutions; the finite magnitude noise makes the different scale signals reside in the corresponding IMF, dictated by the dyadic filter banks;

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(3) The true physically meaningful result to the EMD is designated to be the ensemble mean of the noise-added signal, after a large number of trials. From the EEMD, a collection of IMFs is generated. An IMF, Intrinsic Mode Function, represents the oscillation mode imbedded in the data, being based on the direct extraction of the energy associated with the time scales. Expressed in terms of IMFs, data will have well-behaved Hilbert Transform allowing the determination of the instantaneous frequencies [29]. The approach is based on the studies of the statistical properties of the white noise, which showed that the EMD is effectively an adaptive dyadic filter bank when applied to white noise. The method was proposed to successfully deal with the scale separation problem, and avoid mixing mode caused by intermittencies in the flow [58]. The EEMD algorithm is developed as follows: (a) Add a white noise series to the data; (b) Decompose the noise-added data into IMFs; (c) Repeat steps 1 and 2 N number of trials, with different white noise series each time; (d) Obtain the ensemble mean of corresponding IMFs of the decompositions. The noise-added signal is treated as the possible random noise that would be encountered in the measured data. In those conditions, the observed data with the added noise is Xi (t) = X(t) + wi (t)

(6)

where X i (t) is the ith observation of noise added data, X(t) is the observed data set, wi (t) is the ith realization of the white noise series. As the ensemble number approaches infinity, the final IMF, cj can be defined as c j = lim

N →∞

N  1  c ji (t) + rki (t) N i=1

(7)

in which cji (t) is the jth IMF component of the ith realization of noise-added data, and r ki (t) is the residue after k number of IMFs are extracted in each sifting process. The ensemble number of trials, N, has to be large, in order to the ensemble mean to cancel out the added noise. A flowchart of the EEMD steps is presented in Fig. 1. The EEMD principle is simple: white noise of finite amplitude is added to the signal, populating the whole time-frequency space uniformly. This white noise background causes the different scales in the data to automatically project themselves onto their proper scales of reference, eliminating mode mixing. Although each individual trial may produce results with a large amount of noise, the noise in each trial is expected to be canceled out in the ensemble mean after enough trials, and this ensemble mean is treated as the final physical result [58]. The effect of the added white noise in EEMD is to provide a dyadic filtering reference frame in the time-frequency space. Therefore, the added noise collects and

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Fig. 1 Flowchart of EEMD steps

combines the portion of the signal of comparable scale in one IMF, significantly reducing the chance of mode mixing [27].

2.3 Hilbert Spectral Analysis After obtaining the intrinsic mode functions, one can apply the Hilbert transform to each IMF component, and compute the instantaneous frequency using Eqs. (1–5). The IMF data, after the Hilbert transform can be expressed as the real part  of the IMF in the form

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⎧ ⎫ n ⎨  ⎬ X H (t) =  a j (t) exp i ∫ ω j (t)dt ⎩ ⎭

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(8)

j=1

Although the Hilbert transform can treat the monotonic trend as part of a longer oscillation, the energy involved in the residual trend representing a mean offset could be overpowering [27]. With the IMF expansion, the amplitude and the frequency modulations are also clearly separated. This frequency-time distribution of the amplitude is designated as the Hilbert spectrum H (ω,t).

2.4 Normalized Hilbert Transform The normalized Hilbert transform was proposed by Huang [31], and detailed by Huang et al. [30], to satisfy the conditions imposed by the Bedrosian theorem on the Hilbert transform (further information on the Bedrosian theorem can be found in Bedrosian [5] and Hahn [24]. The normalization of the IMFs consists of the following steps: starting from the IMFs. (1) Find all the maxima of the IMFs; (2) define the envelope by a spline through all the maxima; and designate the envelope as E(t); (3) normalize the IMF by dividing it by the envelope E(t). Y j (t) =

c j (t) E(t)

(9)

with Y j (t) as the normalized data. Thus the normalized function will have all local maxima equal to unity. With the normalized IMF, Huang [31], proposed to compute a variable error as the difference between unity and the squared amplitude of the normalized IMF. If the Hilbert transform is exactly the quadrature, the difference between the squared amplitude of the normalized IMF and the unitary value will be zero. If the Hilbert transform is not the quadrature than the error will be obtained through the difference between the normalized square amplitude and unity.

3 Experimental Procedure The aerodynamic channel used in the experiments, Fig. 2a, is made of acrylic, with a rectangular test section of 0.146 m height, 0.193 m width and 1.020 m length. The air flow is driven by a centrifugal blower of 0.75 kW, and passes through a diffuser, two honeycombs and two screens, that homogenize the flow and reduce the turbulence intensity to about 0.6% in the test section. Before the test section, a Pitot tube is placed at a fixed position to measure the steady reference velocity of the experiments which

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Fig. 2 Schematic view of the aerodynamic channel (a) and probe position (b)

Table 1 Geometric parameters D (mm)

Aspect ratio [L/D]

Blockage [D/B] (%)

x/D

w/D

32.0

4.56

16.58

1.64

2.65

is used for the definition of the Reynolds number. A frequency inverter controlled the blower speed, so that the flow velocity in the aerodynamic channel could be varied from 0 to 16 m/s. The velocity and its fluctuations are measured by means of a DANTEC StreamLine constant hot-wire anemometry system, with a single hot wire probe (type DANTEC 55P11), with the wire perpendicular to the main flow. Data acquisition was performed with a 16-bit A/D board (National Instruments 9215-A) with a USB interface, with a sampling frequency of 1000 Hz and a low pass filter at 300 Hz to avoid aliasing. These frequencies were chosen to obtain a good resolution (low bandwidth) of the spectra in the frequencies corresponding to the large scales in the flow. Each data series had 217 elements, being 131.072 s long. The cylinder diameter used was 32 mm, made of commercial PVC. The corresponding blockage ratio (D/B, were D is the cylinder diameter and B is the channel width) was 16.58%. Considerations on the effects of the blockage ratio on the Strouhal number are given by Möller et al. [44], showing that the increase in the blockage ratio leads to higher Strouhal numbers. Velocity measurements were performed at constant free stream velocities of 5.12, 8.24, 10.5 and 16 m/s for all the cases. Finally a ramp was applied, following a similar procedure of [34].

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Table 2 Vortex shedding frequency and Strouhal number for various Reynolds numbers and corresponding velocities D (mm) 32.0

U [m/s]

Re

f [Hz]

St

5.08

1.05 ×

104

37.11

0.232

8.24

1.71 × 104

56.64

0.218

10.5

2.18 × 104

74.22

0.226

16.0

3.28 ×

109.40

0.219

104

Geometric parameters are summarized in Table 1. The non-dimensional parameters of measurement, like aspect ratio (L/D, where L is the channel high) and blockage ratio in percentage; and non-dimensional positioning of the probe, where x/D is the longitudinal position, w/D is the transversal position; those parameters were chosen so that the probe was located out of the recirculation zone in the wake behind the cylinder. Figure 2b shows a scheme of the cylinder and the probe position parameters. The Reynolds number calculated based on the cylinder diameter and the free stream velocity are given in Table 2. Uncertainties associated with the mean velocity values are about ±3% and for the instantaneous velocity values, about ±5%. Uncertainty analysis is based on value of the primary uncertainties in the experiment and the uncertainty of the considered variable, as presented by Moffat [43]. The uncertainty in the evaluation of the velocity depends on the conversion of the voltage output of the hot wire bridge in velocity values through a calibration procedure. All time-velocity series from the measurements presented in this paper are available at Ost et al. [47].

4 Results and Discussion 4.1 Steady State Flow Table 2 summarizes the vortex shedding frequencies and the corresponding Strouhal number for all Reynolds numbers for the diameter investigated in this research work, in agreement with previous literature, [7, 44, 62]. The first harmonics, not given in Table 2, were also observed. The nonlinear hydrodynamic stability theory and experimental findings says that higher harmonics are generated when the amplitude of the fundamental components exceeds about 4% of the mean flow [64]. The Ensemble Empirical Mode Decomposition was performed on the velocity signals for four different incident velocities. In each case the EEMD resulted in 16 Intrinsic Mode Functions (IMF) and a residue, which is a monotonic function from which no more IMF can be extracted. Each oscillatory mode present in the flow is represented by an IMF; small order IMF components represent the high frequency terms.

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Figure 3 shows the velocity signal and the resulting IMF components for a Reynolds number Re = 2.18 × 104 as an example, since all IMF components for the remaining diameters and Reynolds numbers, were very similar to the shown case. In Fig. 3 it is possible to see that the total amplitude of the IMFs reduces as the frequency reduces. The oscillatory behavior that can be associated to the vortex

U(t)

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C1

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formations and flow scales can be well identified from IMF 1 to 8 (C1 to C8). Looking in detail to IMF 3 and 4 (C3 and C4) it is possible to see an oscillatory behavior resembling a beat, which corresponds to the vortex shedding caused by the cylinder in the flow. This condition is also seen in smaller amplitudes in IMF 6 and 7 (C6 and C7). High order IMFs (IMF C8 to C16) present an almost monotonic behavior, and frequencies very close to zero. Those IMFs do not have any physical significance in the flow and can be considered as pseudo frequencies originated from the decomposition method, and will be, therefore, disregarded in this analysis. According to Huang et al. [26] the physical meaning of IMFs is the decomposition of the original signal in quasieigenmodes with locally homogeneous oscillating properties. The intrinsic mode functions represent, therefore, the flow wave modulations in their characteristic time scale. For a better interpretation the spectra were converted in dimensionless wavenumber, defined as k∗ =

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and the dimensionless energy spectrum function can be described as Φ ∗ (k∗) = Φ(k)

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Since each IMF represents locally homogeneous oscillating properties, it is possible to associate each IMF to a corresponding dimensionless wavenumber k*, and consequently to its flow scale. Figure 4 shows the resulting dimensionless wavenumber spectra of IMF components and velocity signals for D = 32 mm. The five first IMF indicated by arrows (C1 to C5), are the ones containing information about the main scales in the flow, like the vortex shedding frequencies. Third to fifth IMFs (C3 to C5) present the contents corresponding to the energy of vortex shedding. IMFs C6 to C8 correspond to the wave modulations of large scales in the flow and contribute to the fluid motion energy, they also are related to vortices generated by the channel itself. Higher orders IMFs do not have a significant energetic contribution to the fluid motion and therefore will be disregarded. For several cylinder diameters and Reynolds numbers, the IMF components behave essentially the same for all, demonstrating the hypothesis of similarity and self-preservation of the flow by associating the IMF corresponding scale to a wavenumber in spectra. Those hypothesis states that at all high Reynolds numbers the processes that determine the main structure of turbulent motion are independent of the fluid viscosity and that the structure at all instant of time is similar [54]. As one can see the self-preservation does not depend on the Reynolds number. As the Reynolds number increases the order of IMFs containing frequencies corresponding to the vortex shedding diminishes. Looking at Re = 1.05 × 104 the vortex shedding peaks of frequency are still visible in C6, while at Re = 3.28 × 104 , those frequencies are only identified until C3.

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From the spectra of the IMF components, it is observed that the IMFs tend to organize themselves like a filter bank structure. The filter associated with the first mode (C1) is essentially a high pass filter, and the modes of higher order are characterized by a set of overlapping bandpass filters. Furthermore, each mode of index (i + 1), i ≥ 2 occupies a frequency domain which is approximately the upper half-band of that of the previous index [20]. After performing the EEMD on the velocity signal, the Hilbert transform is applied on each IMF component. The Hilbert spectral analysis shows each instantaneous frequency at its corresponding time and energy amplitude. It can be interpreted as a weighted non-normalized joint amplitude-frequency-time distribution, the local

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amplitude being the weight assigned to each time-frequency cell, giving the exact occurrence time of the oscillations [29]. Figure 5 shows the Normalized Hilbert Energy Spectra (NHES) for the various Reynolds numbers. The vertical axis on the left side of the graphic represent frequency in Hz, the vertical axis on the right side is the corresponding Strouhal number. The lower horizontal axis represents time in seconds, and the upper horizontal axis is the dimensionless time defined as t∗ =

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It is observed that the higher concentrations of amplitude corresponding to the most energetic IMFs, are in the band range that corresponds to the vortex shedding frequencies in Hz, and corresponding Strouhal number, whose values are summarized in Table 2. It is also visible that the shedding frequency is not constant; rather it oscillates about a constant value, corresponding to a St ≈ 0.21 along time. The energy contents of the first harmonics are also present in the NHES but in a larger frequency band, giving to them a more disperse appearance and very poor resolution,

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with difficult identification. The pseudo frequencies created by the decomposition method appear concentrated in the frequencies close to zero and have no physical meaning. For quantitative analysis of the energy of each IMF, the dimensionless Mean Square Energy (MSE), as an estimation measure, was defined as follows [39] 1 MSEi = [Ci ]2 N t=0 T

(13)

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energy values, most likely because they correspond to pseudo frequencies generated by the decomposition method, and do not have physical significance. Figure 7 presents the joint Probability Density Function (PDF) of the normalized amplitude and frequency of the first five IMF. The lower horizontal axis represents frequency in Hz, and the upper horizontal axis represents the dimensionless wave number (k*). It is possible to observe that the first IMF contains higher frequencies, and its amplitude is lower, as expected, with the concentrations distributed in a wide range of frequency values, from around 200 Hz to 500 Hz. As the IMF order increases, the distributions concentrate in smaller ranges of frequencies, since the increasing in the IMF order means larger flow scales and a more monotonic IMF component. Also the overlapping of IMF components is observed in the range of the vortex shedding frequencies; this behaviour is probably due to more than one IMF containing those frequencies. By definition, a coherent structure is a large-scale turbulent mass of fluid with phase-correlated vorticity over its spatial extent [33], distinguished in a turbulent flow from how much energy it contains. The EEMD method also recognizes the coherent structure in the sense of the mean energy. A mode can be considered a coherent structure, if its mean energy exceeds 10% of the whole fluctuation energy [63]. Results from Fig. 7 show the IMFs which are the carriers of the features of the signals corresponding to the vortex shedding and the first harmonic, characterized by the peaks in turbulence spectra and of the corresponding IMF. Subsequent IMFs have an almost uniform energy for a gradual increase of the scales, characterized by the reduction of the dimensionless wave numbers. In Fig. 5, it must be considered that wave numbers with lower values than the inverse of the channel width (0.14 m) have no meaning in the physical space. Therefore, the IMFs in the values of k* below 0.2286 cannot be considered as turbulence and therefore should not be considered in this analysis. By observing Fig. 7, the IMF with higher energy for Re = 1.71 × 104 is C3. That IMF has two important features, compared to the others: it is concentrated on

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a region about the shedding frequency, showing that the shedding process does not occur on a sharp fixed frequency, rather it wanders about that value, and there is a high probability of these frequencies present high amplitudes combined with high energy values. IMF C3 corresponds to the shedding vortices characterizing, according to Hussain [33], the so called preferred mode, the characteristic geometric configuration of the structure in the physical space. The wandering shedding frequency may be a consequence of the three dimensionality of the wake flow resulting from vortex dislocation along the cylinder and time [40]. Figure 7 can be considered as a second representation of the IMF spectra (Fig. 5), complementing the spectra by showing the location of the IMFs in a PDF representation with the normalized amplitude and showing also the location of the structures with high energy and amplitude, with a clear separation of the scales. This allows the association of the scale range with the energy content. Therefore, EEMD is an efficient process of eduction of coherent structures.

4.2 Accelerating Flow Vortex shedding frequency is 109.4 Hz and the first harmonic appears at 216.8 Hz. Figure 8 represents the accelerating flow signal for D = 32 mm, and Reynolds number Re = 3.23 × 104 and the first seven IMF components resulting from the EEMD. The EEMD performed in the transient resulted 15 IMF and a residue. Each oscillatory mode of the flow is represented by an IMF. It is possible to observe that the first IMF (C1) is a monotonic component with a very wide range of frequencies, this IMF contains in this case all the noise introduced by the method to work as a filter, and does not present physical significance. From IMF C2 to C4 is possible to identify regions where the instantaneous amplitude of the IMF increased. In C2 this region begins around the time that would correspond to the stabilization of the ramp in its maximum velocity value. However, C3 presents three amplitudes zone, the small amplitude until around 10 s, an increase in amplitude from 13 to 27 s and an intermediate amplitude from 27 s that persists in the rest of the IMF. The same behavior can be observed in C4, where the highest amplitude is located from 7 to 15 s. IMFs from C5 to C6 also present this characteristic but less evident, due to the difference in the maximum values being smaller. This behavior in the IMF components can be associated to the accelerating characteristic of the ramp. The dimensionless wavenumber spectra of the accelerating flow and the eight first IMF components is represented in Fig. 9. It is possible to observe that when compared to Fig. 5, the main difference is in the first IMF, C1, which in the accelerating flow concentrates all the noise and does not fit in the original velocity power spectrum. Concerning the other IMFs, their behavior is very similar to the stationary case presenting the vortex shedding peaks in C2 and C3 and the filter bank characteristic of the method.

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After the EEMD performed, the Hilbert transform is applied in each IMF component resulting in the Hilbert Spectrum. Figure 10 shows the Hilbert Spectrum of the accelerating flow IMFs. For a better representation, only the accelerating part of the signal is considered in the spectra. The first IMF, C1, was omitted from Fig. 10, once it contained the noise part of the accelerating signal and no physical meaning. Higher order IMFs were also omitted considering that they are pseudo frequencies generated by the method and their frequency contents are close to zero. Figure 10 shows that there is a growth in the concentration of amplitudes corresponding to the accelerating part, in the frequencies around 40 Hz to 150 Hz. It is possible to observe that the IMF component with frequency around 40 Hz persists under the ramp, and that the component around 150 Hz, decreases around 20 s, being absorbed by the accelerating part. Looking at the Mean Square Energy, in Fig. 11, is possible to see that IMF C1 seems to be the most energetic, that effect is caused by the noise contents of this IMF, therefore it will be disregarded from the analysis. Another interesting characteristic is that the energy in the accelerating flow decreases, and does not present peaks in intermediary modes, like the stationary cases. IMFs higher than 8 do not present significant energy contents.

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5 Concluding Remarks In this paper the Hilbert-Huang transform is used for signal analysis. Velocity and velocity fluctuations of the vortex shedding behind circular cylinders were measured by means of hot wire anemometry. The cylinder was at first submitted to the constant velocity flow, and secondly to an accelerating flow in order to analyze the influence of Reynolds number in IMF (Intrinsic Mode Functions) components resulting from EEMD (Ensemble Empirical Mode Decomposition) and the ability of the method dealing with non-stationary phenomena. The dimensionless results made possible to identify that the IMF components behave essentially the same, demonstrating the self-preservation and the similarity hypothesis of the flow. Those hypotheses do not depend on the Reynolds number. It was also possible to associate each IMF to its corresponding scale through wavenumber spectra. The vortex shedding frequencies can be identified from IMF C3 to C5. The large vortices responsible for the flow motion are represented in IMFs C6 to C8. IMFs with order higher than C8 do not represent significant energetic contribution to the turbulent motion. The Normalized Hilbert spectral analysis makes possible to comprehend the energy distribution on the studied turbulent flow, the amplitudes being more concentrated on the main frequencies of the flow. The vortex shedding frequencies and the first harmonics were represented, as well as low amplitudes corresponding to the large scales. Those characteristics were not distinguished by the other methods such as wavelet transform. By the accounting of the dimensionless mean square energy, it was possible to identify the most energetic IMF components, which varied from IMF 4 in the lowest Reynolds number to IMF 2 in the highest Reynolds number. Those IMFs are located in the energetic region of the spectra and correspond to coherent structures in the flow. This result complemented the spectral analysis, confirming that IMFs higher than C8 do not contribute with the energy of the turbulent motion, consisting of pseudo frequencies originated by the decomposition method. The joint PDF analysis of the amplitude-frequency distribution complemented the previous wavenumber and Hilbert spectral analyses, showing clearly that the higher concentrations of occurrences are in the main vortex shedding frequency, the so called preferred mode, the characteristic geometric configuration of the structure in the physical space. The joint PDF analysis also represents the dispersion in frequency present in the first IMF component, consequence of the small scales decomposition. The characterization of the coherent part of the motion, and, as a consequence, of the uncoherent motion is an important result of the joint PDF analysis of the amplitudefrequency distribution, showing also that EEMD is an efficient process of eduction of coherent structures. Concerning the analysis of the accelerating flow by means of HHT, the decomposition method showed that the accelerating characteristic of the flow was distributed along many IMFs, or flow scales. By the PSD of those IMFs, the same selfpreservation hypothesis could be made. The Hilbert Spectral Analysis of the transient

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part of the signal identified the accelerating characteristics of the flow. It was not possible to identify the presence of coherent structures by the mean square energy of the accelerating part of the signal. This study highlights the potential of the Hilbert-Huang transform algorithm in the study of turbulent flows, taking the flow past circular cylinders as a base study. The method which is suitable for non-linear and non-stationary data, due to its adaptive basis, provides a description of the turbulent system in terms of the instantaneous frequency, which characterizes the turbulence intra-wave frequency modulations. Also it allows the identification of the many scales in the flow, including coherent structures; and the recognition of properties that other tools, such as Fourier and wavelet transform, may overpass or make unclear. The methodology presented in this paper can be applied in the analysis of a variety of flows, especially in the identification of coherent motion like the one found in vortex shedding process. Acknowledgements Authors are grateful to the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil, for the partial financial support (BPP-Program). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.

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Numerical Assessment of the Pressure Recovery of the Turbulent Flow in a Venturi-Type Device Naítha Mallmann Caetano and Luiz Eduardo Melo Lima

Abstract Venturi-type devices can vary from minor adjustments in their geometry to more complex engineering adaptations, depending on their industrial application: automotive, food, agricultural, oil, among others. Thus, their study still becomes necessary nowadays. And the computational modeling aggregates as an important additional tool for investigation and determination of flow variables as the pressure recovery, that affect stability and performance in the processes that apply the Venturi principle. Therefore, this numerical study aims to analyze the flow pressure recovery in Venturi-type devices. In this analysis, flows turbulent, incompressible, and isothermal were assumed. The governing equations involved are the Reynolds-averaged Navier–Stokes equations. Three types of k–epsilon models were employed to solve the turbulence. One commercial software for computational fluid dynamics, which uses a finite volume method in the discretization of the governing equations, was used to obtain the solutions. The simulations’ results were compared to experimental data for the pressure and showed a satisfactory concordancy, aiming to demonstrate the applicability of the computational model developed. Keywords Venturi · Numerical analysis · Turbulence modeling

1 Introduction The Venturi is a device that was first employed to solve simple problems and improve hydraulics’ knowledge. Years after its development, it started to be used in some industrial applications, among them the flow measurement. Currently, the Venturi principle is employed for various technological purposes in different processes, N. M. Caetano · L. E. M. Lima (B) Federal University of Technology–Paraná, Ponta Grossa, Paraná 84017-220, Brazil e-mail: [email protected] N. M. Caetano e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_7

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highlight the air purifiers, gas–solid injectors, and jet pumps, among others. The industrial applications of the Venturi vary from minor adjustments in its geometry to more complex engineering adaptations. The Venturi has a circular or polygonal cross-sectional area, depending on your utilization, and it consists of three sequential parts: the convergent, the throat, and the divergent. The Venturi geometric characteristics allow pressure recovery occurs from inlet to outlet. Certain phenomena such as the vena contracta are not present due to the gradual reduction of areas. Thus, the flow is regular throughout the Venturi when the flow rate is relatively low. However, for higher flow rates, separation points can occur after the Venturi throat, which causes the eddies generation, as well as a counterflow, doing that the pressure does not fully recover [1, 2]. Devices based on the Venturi principle are capable of performing the transformation of pressure energy into velocity energy. This characteristic is used extensively by industry to improve air quality and also define its moisture, receiving the name of Venturi aeration. Another example is the gas–solid injectors, where the air circulating in the Venturi takes solid particles that are inserted in the throat region, carrying them to the desired locale [3–7]. For the same principle, the air purifiers are generally employed in the food industry to remove impurities and regulate the temperature, using a fluid that is injected in the throat region and afterward dispersed in the form of droplets in the environment to be purified [8, 9]. In the lift jet pumps’ technology, the Venturi design requires a higher engineering level. This device employs the Venturi principle in the lift and flow of submerging fluids such as water and oil. The operation of a lift jet pump is relatively simple. At the entrance of the Venturi-type device, is injected a power fluid, which sucks the surrounding working fluid into a chamber where the mixing occurs. The power and working fluids are mixed and then transported to the surface through the duct using the thrust resulting from power fluid. Models for jet pumps currently are derived under the assumption that the power and well fluids are incompressible liquids that, in many cases, are assumed to have equal densities. The need for increasing reservoir gas fraction rates, beyond those observed using conventional gas-lift methods, was the principal catalyst for this high volume lift system. Hydraulic jet pumping (HJP) can be a suitable alternative to the electric submersible pumping (ESP). The jet pump also can be employed to remove loose sand particles in low-pressure wells and absorption wells, as well as presents a good alternative for well’s completion, formation damage, and hydraulic fracturing [10–15]. The cavitating flows through a venturi channel were investigated numerically by [16]. The governing equations were solved based on the unsteady thermal nonequilibrium multi-fluid model. They observed the generation of the two-dimensional (2D) structures in the cavitating flow. Also, they found that the production of superfluid’s counterflow against the normal flow is due to an increase in the gas phase volume fraction. Ghassemi and Fasih [17] also studied the cavitating Venturis with three different throat diameters to investigate the Venturi size effect on its mass flow rate and to analyze the Venturis’ performances. In their experiments, were measured the mass flow rates under different downstream and upstream pressure conditions with time-varying downstream pressure. They found that the Venturi size does not

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affect its expecting function to keep mass flow rate constant. Also, they have shown that applying a discharge coefficient and using only upstream pressure the cavitating Venturi can work as a flowmeter, with a high degree of accuracy in a wide range of mass flow rates. Cruz-Maya et al. [18] investigated the effects of the viscous stresses in the boundary layer, the dimensionless wall temperature in the throat nozzle, and the flow field curvature at the nozzle nucleus on the discharge coefficient of a Venturi. For this, they used one commercial code for computational fluid dynamics (CFD). The correlation achieved was validated through a direct comparison with others obtained previously from experimental data, presenting a maximum relative deviation of 0.2%. Ahmadvand and Talaie [19] simulated the droplet dispersion through a cylindrical Venturi scrubber using one CFD code based on the Eulerian approach. They employed the standard k– model to solve the flow turbulence. The comparison between the results and experimental data showed a better agreement for the droplet concentration distribution. Using CFD techniques, Hollingshead et al. [20] observed that the discharge coefficients decrease rapidly with decreasing Reynolds number (Re) for Venturi flowmeter. Their results provide an improved understanding of differential flowmeters operating at low Re values. Also, they demonstrate the CFD usefulness in predicting discharge coefficient trends at these conditions. Blocken et al. [21] used CFD methods to gain insight into the aerodynamic performance of a Venturi-shaped roof (called VENTEC roof). The three-dimensional (3D) steady Reynolds-averaged Navier–Stokes (RANS) equations, as well as the renormalization group (RNG) k– turbulence model, were solved in their simulations. They performed a detailed analysis of the influence of the so-called Venturi-effect, as well as the wind-blocking effect, on the aerodynamic performance of the VENTEC roof. Their results indicate that due to the wind-blocking influence, the higher contraction ratio does not produce the best aerodynamic performance and higher negative pressure, which is a counterintuitive conclusion. He and Bai [22] investigated the application of five turbulence models to predict the flow field in a Venturi meter over-reading in wet gas. They found that the standard k– turbulence model was in better agreement with the experimental data. Their study gained a more comprehensive understanding of Venturi meter wet gas over-reading and provided a reference for the design of a wet gas Venturi meter prototype. Wang et al. [23] studied the roughness influence on the discharge coefficient of axisymmetric sonic Venturi nozzles, systematically. In the turbulent core region, was employed the standard k– model. In the boundary layer region, was adopted standard wall functions. Their CFD simulations provided relations between discharge coefficient and relative roughness. Using CFD analysis, Sanghani and Jayani [24] studied geometrical parameters’ optimization for Venturi meter, aiming to obtain a minimum pressure drop. The optimal combination of the geometrical parameters found was as follows: convergent cone angle of 17°, the divergent cone angle of 7°, the diameters ratio (or beta ratio) of 0.75, and the throat length of 0.007 mm. In this case, the pressure drop was 40599.4 Pa in the Venturi meter.

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Using CFD and laboratory data, Sharp et al. [25] demonstrated the relationship between the recovery cone angle on the classical Venturi meter design with the associated head loss. Their results showed that the optimum recovery cone angle to minimize head loss is a function of diameters ratio and Reynolds number. Moreover, the developed computational codes provide acceptable cone angles ranges, which becomes possible to find an optimal recovery cone angle to minimize permanent pressure loss. Jing et al. [26] presented CFD simulations of the two-phase flow of wet gas in the Venturi tube under high-pressure conditions. Their study employed the discrete phase model (DPM) and the standard k– turbulence model. Their results showed that the over-reading values increase with the Lockhart–Martinelli parameters and the gas volume flow rate, but decrease with the increase of the pressure. Also, they present results for the flow field, the pressure distributions, and the droplet concentration. This work presents a numerical study of the flow characteristics in Venturi-type devices. This study aims to analyze the flow pressure recovery effect in a Venturi, which is relevant to reduce head loss in a system. The flow was considered turbulent, incompressible, and isothermal. For this study, one CFD commercial software, which uses a finite volume method in the discretization of the governing equations, was employed. The results achieved were compared to pressure experimental data for several Reynolds numbers values, to demonstrate the applicability of the computational model developed.

2 Model This section is composed of two parts, namely. The first part (Sect. 2.1) shows the RANS equations definitions that constitute the numerical model. The second part (Sect. 2.2) presents the turbulence models descriptions, which were applied to close the turbulent viscous stress term of the RANS equations.

2.1 RANS Equations The RANS equations are time-averaged Navier–Stokes equations written for the instantaneous velocity decomposed in mean value and fluctuation, i.e., u i = u i + u i . Equations (1) and (2) express the transport equations (mass and momentum, respectively) in steady-state for an incompressible Newtonian fluid [27]: ∂u j =0 ∂x j

(1)

 ∂  ρu i u j +  p δi j − τ itotj = 0 ∂x j

(2)

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where xi and u i represent the position and velocity vectors, respectively, δi j is the Kronecker delta operator, and ρ is the density. The modified pressure,  p , and the total (laminar and turbulent) viscous stress, τ itotj , are determined using Eqs. (3) and (4), respectively:  p = p + ρgxi

(3) 

turb τ itotj ≡ τ ilam j + τ i j = (μ + μt )

∂u j ∂u i + ∂x j ∂ xi

 (4)

where p is the static pressure, g is the gravitational acceleration, and μ and μt are the molecular and turbulent dynamic viscosities, respectively.

2.2 Turbulence Models The turbulence models employed in this work for the flow simulation in a Venturi device are three well-known types of k– models, often available in most CFD software: standard, RNG, and realizable. In the standard k– model [28], the turbulent kinetic energy, k, is derived from the exact equation, while its dissipation rate, , is obtained by reasoning about physical phenomena. The RNG k– model [29, 30] is derived using statistical techniques called renormalization group theory and has an extra term in the dissipation rate that optimizes the results accuracy for fluids subjected to tension quickly. The realizable k– model [31] differs from the standard to contain an alternative formulation that presents a modified equation for the dissipation rate. This modified equation was derived from the exact equation for the motion to the main area of vorticity change. The general form of the three k– turbulence models employed presents two transport equations. The first for the turbulent kinetic energy k, Eq. (5), and the second for its dissipation rate , Eq. (6):     μt ∂k ∂ ∂ ρku j − μ + = Pk + Pb − YM − ρ (ρk) + ∂t ∂x j σk ∂ x j

(5)

    ∂ μt ∂ ∂ ρu j − μ + = λ1 + λ2 ρ (ρ) + ∂t ∂x j σ ∂ x j

(6)

where σk and σ are the turbulent Prandtl numbers for k and , respectively, and these constants are given in Table 1, depending on the k– model type. For steady-state, the time derivatives in Eqs. (5) and (6) can be eliminated, i.e., ∂/∂t = 0. The turbulent (or eddy) dynamic viscosity, μt , is computed by combining k and  as expressed by Eq. (7): k2 (7) μt = ρCμ 

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where Cμ is a constant, and its value is given in Table 1, depending on the k– model type. The kinetic energy production term due to the mean velocity gradients, Pk , is given by Eq. (8): ∂u j Pk = −ρu i u j ≡ μt S 2 (8) ∂ xi where S (≡ 2Si j Si j ) is the modulus of the mean rate-of-strain tensor considering the Boussinesq hypothesis, i.e., Si j = (1/2)(∂u i /∂ x j + ∂u j /∂ xi ). The kinetic energy production term due to buoyancy, Pb , is given by Eq. (9): Pb = βgi

μt ∂ T Pr t ∂ xi

(9)

where T is the absolute temperature, Pr t is the turbulent Prandtl number for the turbulent heat flux in the energy conservation equation obtained by time-averaged

Table 1 Parameters and constants of the three k- turbulence models employed Parameter or k- turbulence model constant Standard RNG Realizable C1 C1 C2 C2 ∗ C2 C3 C3 Cμ σk σ η η0 β0 A0 As

0 1.44 0 1.92 – 1

0 1.42 0   ∗ + C η3 (1 − η/η ) / 1 + β η3 C2 μ 0 0 1.68 1

max [0.43, η/ (η + 5)] 1.44 1.9 0 – 0

a

a

a

0.09 1.0 1.3 – – – – –

0.0845 0.7194 0.7194 Sk/ 4.38 0.012 – –

U∗

–

–

(A0 + As U ∗ k/)−1 1.0 1.2 Sk/ – – 4.04 √ 6 cos φ b

i j i j c Si j Si j +

0 ≤ C3 ≤ 1, but an approximation that satisfies both limits is C3 = tanh |v/u|, where v and u are the velocity components parallel and √

perpendicular to the gravitational vector, respectively [33] b φ = (1/3) cos−1 6Si j S jk Ski /S 3 c i j = i j − 2i jk ωk and i j = i j − i jk ωk , where i j is the mean rate-of-rotation tensor viewed in a rotating reference frame with the angular velocity ωk a

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processes [27], and gi is the gravitational acceleration vector. For the standard and realizable k– models, the Pr t default value is 0.85. For the RNG k– model, a formula derived analytically by the RNG theory is used to compute the Pr t value [29, 30]. Although Pr t ≈ 0.9 is often employed. By definition, the thermal expansion coefficient is β = −(1/ρ)(∂ρ/∂ T ) p . For isothermal flow, the kinetic energy production term due to buoyancy can be eliminated, i.e., Pb = 0. The dilatation dissipation term, YM , is modeled by Eq. (10) [32]: YM = 2ρMt2

(10)

where the turbulent Mach number is given as M = k/a 2 , by definition, and a t √ (≡ γ RT , for an ideal gas) is the speed of sound, with γ being the specific heats’ ratio, and R is the gas constant. For incompressible flow, the dilatation dissipation term can be eliminated, i.e., YM = 0. The parameters λ1 and λ2 are introduced in this work to generalize the Eq. (6) so that it serves for the three k– turbulence models employed. Equations (11) and (12) determine λ1 and λ2 , respectively: λ1 = C1 (C3 Pk + C3 Pb ) 

 k

C2 − C2 λ2 = C 1 η − 1 + Cμ ν/νt

(11) 

 k

(12)

where ν and νt are the molecular and turbulent kinematic viscosities, respectively, and the other parameters and constants present in Eqs. (11) and (12) are defined in Table 1, depending on the k– model type.

3 Experimental Apparatus and Data The experimental apparatus utilized in this work was produced by EDIBON International–Technical Teaching Equipment–with purposes of the study and measurement of the head loss in pipes and accessories [34]. Figure 1 shows a schematic representation of the experimental facility and its components. A centrifugal pump (0.37 kW power, 30–80 l/min flow rate, at 20.1–12.8 m head) supplies water to the experimental circuit. After pass through the test circuit, the water flows downward to a collecting tank of 165 l capacity. The water exits this tank to feed the centrifugal pump in a closed-loop. The water flow rate measurement uses a rotameter calibrated at ±100 l/h uncertainty operating within the range of 600– 6000 l/h. The Venturi tube is of transparent acrylic and its dimensions are 180 mm length, 32 mm larger section, and 20 mm smaller section (also shown in Fig. 1). The pressure taps are localized at three axial positions of the Venturi tube (inlet, throat, and outlet), as seen in Fig. 1. Two U-tube manometers (0–1000 mmH2 O) are applied

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Venturi (test section) Inlet

Water reservoir

Outlet 60 mm

Rotameter (flow meter)

20 mm

50 mm

Q 20 mm Throat

32 mm Convergent

Valve

P1

P3

P2

U-tube manometers

Centrifugal pump

32 mm Divergent

Bourdon pressure gauge

Water recycling circuit

Fig. 1 Schematic representation of the experimental apparatus and its components

to measure the pressure differences between inlet and throat, as well as between outlet and throat. A Bourdon pressure gauge (0–2.5 bar) measures the pressure at outlet relative to the atmosphere. At approximately ±20 mbar was estimated the pressure uncertainty. The environment conditions of the tests were at nearly atmospheric pressure of 1021 mbar and a temperature of 18◦ C. The barometrical pressure reading comes from the weather service provided by the Paraná Technology and Environmental Monitoring System (Simepar), available at http://www.simepar.br/. Table 2 shows the experimental test grid, which consists of one set with 14 values of water flow rates, Q, measured using the rotameter shown in Fig. 1. The flow

Table 2 Experimental test grid Q (l/h) Re 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200

6552 8737 10,921 13,105 15,289 17,473 19,657 21,842 24,026 26,210 28,394 30,578 32,762 34,947

p1 (Pa)

p2 (Pa)

p3 (Pa)

108,149 108,198 110,208 112,237 116,257 120,306 124,031 128,433 132,522 138,590 144,561 150,747 156,885 164,943

108,051 108,012 109,943 111,865 115,747 119,639 123,551 127,374 131,237 137,080 142,874 148,678 154,462 162,247

108,100 108,100 110,100 112,100 116,100 120,100 124,100 128,100 132,100 138,100 144,100 150,100 156,100 164,100

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rates range encompasses the region of turbulent flow occurrence, i.e., Re > 4000. In Table 2, the quantities p1 , p2 , and p3 correspond to the absolute pressures measured at the inlet, throat, and outlet of the Venturi tube.

4 Numerical Procedure The 2D computational domain used in the simulations corresponds to the internal region of the Venturi geometry defined in Fig. 1, using a boundary condition of symmetry around the revolution z-axis (axial coordinate). The input parameters used in the simulation are the boundary conditions of inlet velocity, based on Q values, and of outlet absolute pressure p3 , obtained experimentally, see Table 2. For the Venturi’s wall surface, it was considered the no-slip boundary condition. The CFD commercial software ANSYS® Fluent® Release 18.0 was employed for the analysis performed. All simulations were performed for steady-state, using a finite volume method [35] with a structured and non-uniform mesh. The pressure–velocity coupling was solved using the semi-implicit method for pressure linked equations (SIMPLE) algorithm [35]. The “Enhanced Wall Treatment” option was selected that provides consistent solutions for all y + values and is recommended when using the k–  turbulence model for general single-phase fluid flow problems [36]. The following schemes have been considered in the simulations performed: • Second-order for pressure. • Second-order upwind for momentum. • First-order upwind for turbulent kinetic energy and its dissipation rate. The ANSYS® Meshing™ tool was applied to generate the computational mesh automatically, see Fig. 2. Close to the surface, where all gradients are most significant, further mesh refinement was needed. The domain rest was enough refined using the “Smooth Transition” software mode to provide satisfactory results convergence coupled with computational cost. The mesh quality was evaluated using the “Skewness” parameter, which measures how equilateral the mesh element is, being an apt indicator of the mesh quality and suitability [36]. All simulations are performed in the Computational Research Laboratory, using a computer with the following characteristics:

Outlet

Inlet Wall

Symmetry axis

Fig. 2 2D computational mesh used

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• Microsoft Windows® 7 (64-bit) operating system. • Intel® Core™ i7–3770 (3.4 GHz) processor. • 8 GB RAM.

5 Results and Discussion This section is composed of two parts, namely. The first part (Sect. 5.1) presents the numerical–experimental comparisons for the absolute pressure p, as well as for the pressure difference p, considering the three k– turbulence models employed in this work: standard, RNG, and realizable. The second part (Sect. 5.2) presents the quantities distributions (or fields) obtained using the standard k– model: absolute pressure p, velocity vectors u i , turbulent kinetic energy k, and turbulent dynamic viscosity μt .

5.1 Numerical–Experimental Comparisons Figures 3, 4, and 5 show the comparisons between numerical and experimental values for the absolute pressure p as a function of the flow rate Q. These results correspond to the three pressure taps distributed through Venturi: inlet, p1 , throat, p2 , and outlet, p3 . The results shown in Figs. 3, 4, and 5 show a satisfactory agreement between numerical simulation and experimental data. The results of the three k– turbulence models presented a relative deviation modulus smaller than 1% compared with the experimental data for the p values. Figures 6, 7, and 8 show the comparisons between numerical and experimental values for the pressure difference p as a function of the Reynolds number Re. These results correspond to the three pressure differences distributed through Venturi: inlet and throat, p1−2 , outlet and throat, p3−2 , and inlet and outlet, p1−3 . The results shown in Figs. 6, 7, and 8 also show a satisfactory agreement between numerical simulation and experimental data. The results of the three k– turbulence models also presented a relative deviation modulus smaller than 1% compared with the experimental data for the p values. Table 3 shows the relative deviations moduli between numerical and experimental values for the pressure differences p as a function of the Reynolds number Re. The relative deviations moduli, as well as their root mean square (RMS), were smaller than 1% for the three k– turbulence models employed, as seen in Table 3. As expected, the relative deviations moduli values tend to increase as the Reynolds number increases, as seen in Figs. 6, 7, and 8. Despite this, the lowest pressure differences are obtained between inlet and outlet, demonstrating the pressure recovery through the Venturi, as predicted in the literature.

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p1 (num.) p1 (exp.) p2 (num.) p2 (exp.) p3 (num.) p3 (exp.)

160 150

p / (kPa)

131

140 130 120 110 100 500

1000

1500

2000 Q / (l/h)

2500

3000

3500

Fig. 3 Comparison between numerical (num.) and experimental (exp.) values for p as a function of Q, using the standard k– model

170

p1 (num.) p1 (exp.) p2 (num.) p2 (exp.) p3 (num.) p3 (exp.)

160

p / (kPa)

150 140 130 120 110 100 500

1000

1500

2000

2500

3000

3500

Q / (l/h) Fig. 4 Comparison between numerical (num.) and experimental (exp.) values for p as a function of Q, using the RNG k– model

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170

p1 (num.) p1 (exp.) p2 (num.) p2 (exp.) p3 (num.) p3 (exp.)

160

p / (kPa)

150 140 130 120 110 100 500

1000

1500

2000

2500

3000

3500

Q / (l/h) Fig. 5 Comparison between numerical (num.) and experimental (exp.) values for p as a function of Q, using the realizable k– model

3.5 3

Δp / (kPa)

2.5

Δp1−2 (num.) Δp1−2 (exp.) Δp3−2 (num.) Δp3−2 (exp.) Δp1−3 (num.) Δp1−3 (exp.)

2 1.5 1 0.5 0 5000

10000

15000

20000

25000

30000

35000

Re Fig. 6 Comparison between numerical (num.) and experimental (exp.) values for p as a function of Re, using the standard k– model

Numerical Assessment of the Pressure Recovery …

3.5 3

Δp / (kPa)

2.5

133

Δp1−2 (num.) Δp1−2 (exp.) Δp3−2 (num.) Δp3−2 (exp.) Δp1−3 (num.) Δp1−3 (exp.)

2 1.5 1 0.5 0 5000

10000

15000

20000

25000

30000

35000

Re Fig. 7 Comparison between numerical (num.) and experimental (exp.) values for p as a function of Re, using the RNG k– model

3.5 3

Δp / (kPa)

2.5

Δp1−2 (num.) Δp1−2 (exp.) Δp3−2 (num.) Δp3−2 (exp.) Δp1−3 (num.) Δp1−3 (exp.)

2 1.5 1 0.5 0 5000

10000

15000

20000

25000

30000

35000

Re Fig. 8 Comparison between numerical (num.) and experimental (exp.) values for p as a function of Re, using the realizable k– model

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Table 3 Relative deviations moduli, in %, between numerical and experimental values for pressure differences p as a function of the Reynolds number Re Re k- turbulence model Standard RNG Realizable p1−2 p3−2 p1−3 p1−2 p3−2 p1−3 p1−2 p3−2 p1−3 6552 8737 10,921 13,105 15289 17,473 19,657 21,842 24,026 26,210 28,394 30,578 32,762 34,947 RMS

0.05 0.05 0.11 0.14 0.18 0.20 0.28 0.26 0.32 0.35 0.46 0.45 0.43 0.52 0.31

0.04 0.06 0.08 0.10 0.12 0.14 0.23 0.22 0.30 0.34 0.39 0.46 0.50 0.58 0.31

0.01 0.01 0.03 0.03 0.06 0.06 0.05 0.04 0.02 0.01 0.07 0.01 0.07 0.06 0.04

0.05 0.05 0.11 0.14 0.18 0.20 0.28 0.26 0.32 0.35 0.46 0.45 0.43 0.52 0.31

0.03 0.05 0.08 0.09 0.11 0.13 0.22 0.20 0.28 0.32 0.37 0.43 0.47 0.50 0.28

0.01 0.01 0.03 0.04 0.07 0.07 0.06 0.06 0.03 0.03 0.09 0.02 0.04 0.01 0.05

0.05 0.05 0.10 0.13 0.18 0.20 0.27 0.26 0.31 0.34 0.45 0.44 0.42 0.51 0.30

0.03 0.04 0.06 0.06 0.07 0.08 0.15 0.12 0.17 0.21 0.28 0.31 0.36 0.43 0.21

0.02 0.01 0.04 0.08 0.11 0.12 0.13 0.14 0.14 0.13 0.18 0.14 0.06 0.08 0.11

These results are suitable because the Venturi geometry is relatively straightforward, and hence the three k– turbulence models used tend to show satisfactory performance. Lastly, the results obtained using the three k– turbulence models for flow simulations in the Venturi-type device are quite similar.

5.2 Quantities Distributions Figures 9 and 10 show the distributions (or fields) of the absolute pressure p and the velocity vectors u i through the Venturi at flow rate Q of 3200 l/h. These results show a pressure values reduction at the throat, and subsequent pressure values recovery at the outlet. In opposition, the velocity values increase at the throat and decrease at the outlet. This behavior is expected in this type of device [1, 2]. Figures 11 and 12 show the distribution (or fields) of the turbulent kinetic energy k and the turbulent dynamic viscosity μt through the Venturi at flow rate Q of 3200 l/h. These results show an increase in the turbulent kinetic energy values after the throat. This increase in the turbulent kinetic energy values explains the vortices formation in the divergent section after the throat. This can also be justified by an increase in the turbulent dynamic viscosity values at this divergent section.

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Fig. 9 Distribution of p, in Pa, through the Venturi at Q = 3200 l/h

Fig. 10 Distribution of u i , in m/s, through the Venturi at Q = 3200 l/h

For all other Q values presented in Table 2, the simulations results are similar to those obtained with Q = 3200 l/h, with proportional values for each of the flow quantities shown in Figs. 9, 10, 11 and 12.

6 Conclusions This work presented a numerical analysis of the turbulent flow in a Venturi-type device, where the pressure recovery phenomenon occurrence was verified. Three different types of k– turbulence models are employed to obtain numerical results: standard, RNG, and realizable. These three turbulence models presented satisfactory

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Fig. 11 Distribution of k, in m2 /s2 , through the Venturi at Q = 3200 l/h

Fig. 12 Distribution of μt , in kg/(m s), through the Venturi at Q = 3200 l/h

results for the Venturi topology once this geometry is relatively straightforward. Furthermore, the computational model validation showed a sufficient level of agreement with experimental results. The Venturi tube presents higher values of the turbulent kinetic energy and turbulent dynamic viscosity in the divergent part. These characteristics can be applied to maximize the mixture level after the throat in some industrial applications of this device type. This study allowed a better understanding of the behavior of turbulent flow in a Venturi device. In future works, complex phenomena can be introduced in this computational model, for example, multiphase flow, particle injection, cavitation, among others. Acknowledgements The authors thank the Federal University of Technology–Paraná for the resources made available.

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20. Hollingshead CL, Johnson MC, Barfuss SL, Spall RE (2011) Discharge coefficient performance of Venturi, standard concentric orifice plate, V-cone and wedge flow meters at low Reynolds numbers. J Pet Sci Eng 78(3–4):559–566. https://doi.org/10.1016/j.petrol.2011.08.008 21. Blocken B, van Hooff T, Aanen L, Bronsema B (2011) Computational analysis of the performance of a Venturi-shaped roof for natural ventilation: Venturi-effect versus wind-blocking effect. Comput Fluids 48(1):202–213. https://doi.org/10.1016/j.compfluid.2011.04.012 22. He D, Bai B (2012) Numerical investigation of wet gas flow in Venturi meter. Flow Meas Instrum 28:1–6. https://doi.org/10.1016/j.flowmeasinst.2012.07.008 23. Wang C, Ding H, Zhao Y (2014) Influence of wall roughness on discharge coefficient of sonic nozzles. Flow Meas Instrum 35:55–62. https://doi.org/10.1016/j.flowmeasinst.2013.11.007 24. Sanghani C, Jayani D (2016) Optimization of Venturimeter geometry for minimum pressure drop using CFD analysis. Recent Trends Fluid Mech 3:31–35 25. Sharp ZB, Johnson MC, Barfuss SL (2018) Optimizing the ASME Venturi recovery cone angle to minimize head loss. J Hydraul Eng 144(1):04017057:1–04017057:9. https://doi.org/ 10.1061/(ASCE)HY.1943-7900.0001387 26. Jing J, Yuan Y, Du S, Yin X, Yin R (2019) A CFD study of wet gas metering over-reading model under high pressure. Flow Meas Instrum 69:101608:1–101608:10. https://doi.org/10. 1016/j.flowmeasinst.2019.101608 27. Tannehill JC, Anderson DA, Pletcher RH (1997) Computational fluid mechanics and heat transfer. Series in computational and physical processes in mechanics and thermal sciences, 2nd edn. Taylor & Francis Group, Washington, DC 28. Launder BE, Spalding DB (1974) The numerical computation of turbulent flows. Comput Methods Appl Mech Eng 3(2):269–289. https://doi.org/10.1016/0045-7825(74)90029-2 29. Yakhot V, Orszag SA (1986) Renormalization group analysis of turbulence I. Basic Theory. J Sci Comput 1(1):3–51. https://doi.org/10.1007/BF01061452 30. Yakhot V, Orszag SA, Thangam S, Gatski TB, Speziale CG (1992) Development of turbulence models for shear flows by a double expansion technique. Phys Fluids A: Fluid Dyn 4(7):1510– 1520. https://doi.org/10.1063/1.858424 31. Shih TH, Liou WW, Shabbir A, Yang Z, Zhu J (1995) A new k- eddy viscosity model for high reynolds number turbulent flows. Comput Fluids 24(3):227–238. https://doi.org/10.1016/ 0045-7930(94)00032-T 32. Sarkar S, Lakshmanan B (1991) Application of a Reynolds stress turbulence model to the compressible shear layer. AIAA J 29(5):743–749. https://doi.org/10.2514/3.10649 33. Henkes RAWM, Van Der Vlugt FF, Hoogendoorn CJ (1991) Natural-convection flow in a square cavity calculated with low-Reynolds-number turbulence models. Int J Heat Mass Transf 34(2):377–388. https://doi.org/10.1016/0017-9310(91)90258-G 34. EDIBON (2012) Practical exercises manual of the AFT unit. EDIBON international–technical teaching equipment, Madrid, 1st edn. Manual of the experimental unit for study and measurement of head loss in pipes and accessories 35. Patankar SV (1980) Numerical heat transfer and fluid flow. Series in computational methods in mechanics and thermal sciences, 1st edn. Hemisphere Publ, Co., New York, NY 36. ANSYS (2017) ANSYS Fluent Tutorial Guide. ANSYS Inc, Canonsburg, PA. Release 18

Dynamics of Mechanical and Structural System

Delamination Fault Compensation in Composite Structures Luke Megonigal, Foad Nazari, Amirhassan Abbasi, T. Haj Mohamad, and C. Nataraj

Abstract In this study, a methodology is presented to use smart materials for compensation of the negative influences of delamination on the natural frequencies of composite structures. Natural frequency as a physical characteristic of these structures should be designed and kept within a specific range. Delamination, being one of the most common and important types of faults in composite structures, strongly affects the natural frequencies. Based on the fault specifications (i.e., number, severity, location, etc.) the numerical values of the natural frequencies would be different. During the operation of these structures, it is normally impractical or expensive to fix the delamination fault, but it is desirable to keep the structure’s functioning intact. In this study, we present a methodology to partially activate pre-embedded Shape Memory Alloys (SMAs) and keep the natural frequencies of delaminated composite structures in the desired ranges. Keywords Shape memory alloy · Fault compensation · Delamination · Composite materials

1 Introduction Recent advances in engineering designs and consequent logistical limitations have opened a new horizon in the development of high performing materials for a wide range of applications. The main idea has been to improve the structural properties of materials like weight, stiffness and damping without compromising on cost or performance. Along this line, a major part of recent research has focused on altering the microstructure of the materials to achieve structural improvements. From these efforts have emerged a new area of active or multifunctional materials which can be designed to have several desirable structural properties. It has been shown that appropriate microstructural modifications in such materials can result in multi-functional abilities like actuation, sensing and control in a single structure [1]. L. Megonigal · F. Nazari (B) · A. Abbasi · T. Haj Mohamad · C. Nataraj Villanova Center for Analytics of Dynamic Systems (VCADS), Villanova University, Villanova, PA 19085, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_8

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Smart materials are a subgroup of such active/multifunctional materials with the potential of detecting an external excitation source like temperature and magnetic field, and responding reversibly. The response will be a function of magnitude of the external excitation and is achieved by changing the physical or mechanical properties. The excitation can be implemented by a wide range of sources like temperature, magnetic field, electricity, light intensity, moisture, etc. [1]. Shape memory alloys (SMAs) are a subset of smart materials that can reverse a pre-defined shape upon going under excitation of an external source like temperature. It is done by microstructural changes that result in transition between martensite and austenite phases directly and inversely. This transition and corresponding changes in the material properties can be used for vibration control objectives. In addition, the recovery ability of SMAs is of significance in actuator design or fiber reinforcement of composite structures [1]. Such reinforced composite structures embedded with SMA fibers are used in many technological fields, especially in aerospace applications. The importance and effectiveness of this tool has attracted the attention of many researchers and they have attempted to look at this strategy from different perspectives. One of the main topics worth discussing is how these SMA fibers affect the operation of the structures. For this purpose, they have modeled the SMA reinforced structures numerically [2] and analytically [3, 4]; then, they have used those models to analyze the system from different aspects, including, but not limited to deflection [5], free vibration [3, 5], buckling [6, 7] and structural acoustics [5]. To design the SMA properly, one needs to analyze how the SMA fiber parameters, like orientation, location, relative volume fraction and so on, affect the structural performance in each of the above-mentioned aspects [8, 9]. Carrying out these analyses has resulted in interesting achievements. As an example, Baz et al. [10] studied the SMA wires of the NITINOL type and discovered that optimal tuning of SMA wires can make the system stiffer and less susceptible to buckling. In complementary work, Chen and Levy [9] used SMA layers as actuators in a composite structure and proposed a vibration control approach based on the optimal linear control theory. To control the static and dynamic behavior of composite structures with embedded SMA fiber actuators, two main techniques are generally used, namely active properties tuning (APT) and active strain energy tuning (ASET) [11, 12]. Both APT and ASET are effective techniques to control the behavior of structures. In APT technique, the SMA fibers are elongated elastically, and are then inserted in the polymer matrix. In other words, the fibers have not experienced any plastic deformation prior to placement. Actuation of the inserted SMAs is performed by heating the fibers. To do so, an electrical current is passed through the SMA fibers which effects the change from martensite phase to austenite. As a result of this transformation, the Young’s modulus of the SMAs are increased in a controlled manner which results in increasing the effective Young’s modulus of the whole composite structure [13]. To summarize, APT controls the structural response by changing the stiffness. For ASET technique the process is a bit different as the SMA fiber are embedded into the matrix after inelasticity elongation [13]. In this technique, during electrical heating and phase transformation of the SMA fibers, the embedded pre-strained fibers attempt to return to their original shape contraction and the recovery stress arises.

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To control the response of the structure, ASET mostly relies on the recovery force exerted by a pre-strained SMA fiber along the length of fibers. This force causes the change in the energy balance and hence the modal response of the structure [14]. In almost all cases, ASET present more versatility of control than APT [14]; however, ASET technique has very limited applications in engineering practice. Mostly because in ASET technique, independent of structural boundary conditions, additional constraints need to be imposed for the SMA wires to produce tensile recovery stresses during activation of SMA wires. Otherwise, during the activation process, the produced recovery stress is compressive and can cause a significant reduction in critical load and natural frequency of the structure which is not desirable. Furthermore, activation of SMAs in the ASET technique can cause high recovery stresses which can damage the structure [15]. One of the times that SMAs can play an important role in structural control is when natural frequency tuning is of interest. To take advantage of SMAs in free vibrational behavior of composites, one needs to see first how these fibers influence the structure from this aspect. For analyzing the natural frequencies and mode shapes of a composite plate with embedded SMA wires, Barzegari et al. [2] developed a numerical approach. They studied the effects of SMA wires’ axial loads, consequent to temperature variations, on the plates’ natural frequencies. An analytical closed form solution for free vibrational response of a composite plate reinforced with SMA fibers was also proposed by Mahabadi et al. [3]. They used Hamilton principle for deriving the equations of the motion based on the first-order shear deformation theory (FSDT). They studied effects of many parameters such as volume fraction, pre-strain, orientation and location of SMA fibers on the natural frequencies of composite plates. As reported in [16, 17], activation of SMA fibers can increase the natural frequency of composite structures. The magnitude of this increase can be managed to keep the natural frequency in the desired range when, for any reason, the structure is experiencing a natural frequency drop. This reduction can be due to different reasons like thermal excitation, [18], damage [19], etc. In fact, in many cases, structural faults developed from stress cycling and unexpected external forces pose dramatic problems for composite materials. Even though a composite material may be initially designed to meet vibrational specifications, structural faults such as delamination continually threaten to decrease the natural frequency of a composite structure in real-time. The literature widely establishes that delamination decreases the natural frequencies of composites [20]. Physics establishes that the stiffness of a composite reduces as delamination propagates and uncouples the displacement between adjacent layers. The magnitude of these decreases depends upon the specifications of the delamination. In [20] it is reported that delamination at the mid-plane or near the edges as well as delamination which cover more than 50% of a layer’s surface area strongly affect the first natural frequency. To model the delaminated composite structure in this study, COMSOL finite element software was used. Modeling delamination in layered materials needs three parameters: location, size and direction, to govern the extent of a delamination’s effect on a composite. COMSOL’s Thin Elastic Layer Interface allow users to specify stiffness between layers in composite structures independent of their thickness. Thus,

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a regional approach to delamination modeling with COMSOL successfully decouples the displacement of layered surface areas adjacent to a delamination. In other words, the stiffness relating two surface areas above and below a delamination is zero. Hence, the stiffness of the overall composite decreases and the natural frequency immediately decreases. Composites for which natural frequency tolerances are very strict have the most to risk with a delamination. Although much research has focused on modeling and studying the effects of SMA materials on response of composite structures, the idea of using SMA materials for fault compensation of these structures still is far from mature. In recent years, only a few studies have used various SMA activation patterns to tune the properties of composite structures [21, 22]. However, to the best of our knowledge, no study has been reported in the literature on employing the functionally activation of SMA layers to compensate for the effect of delamination, when, in fact, this is one of the most common faults in composite structures.

2 Fault Compensation Approach In this section, we describe the fault compensation procedure of this study to show how it is used to compensate the effect of delamination in composite materials. As mentioned in the previous section, SMA fibers, depending on their specifications, can affect the system response, differently. Location, orientation, number, geometry, material, activation technique and temperature of SMAs are among the factors that can cause this difference. When the system is faced with a fault, depending on the fault characteristics, the change in the free vibration response would be different. In the case of the delamination fault, depending on the size, location, direction and other aspects of the fault, the level of reduction in the first natural frequency would be different. Also, over time, the fault characteristics may change, for example, the delamination may propagate. The question is: how can we use SMA to not only increase but also tune the natural frequency of the system and keep it in the desired range over the time. The main idea of this paper is to use the existing difference in the influence of any pre-embedded SMA fibers of the structure to tune and the natural frequency value, and to keep it tuned. It is patently clear that not only a low natural frequency but also a high natural frequency would not be desirable. Therefore, we should always use the full capacity of the embedded SMAs to increase the natural frequency. In other word, to compensate the effects of the fault, here delamination, we might rely on activation of a fraction of the SMAs. However, it does also matter which SMAs need to be activated. So, we can have different activation patterns, where each one can cause a specific increment value in the natural frequency. Each pattern has a specific function which determines which SMA fiber should be activated and which one should be kept passive. As activation and de-activation of SMAs usually is relatively easy to carry out in practice, over time, as the fault specification changes, the operator of

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the structure can switch from one pattern to another to keep the natural frequency of faulty structure in the acceptable range as close as possible to the healthy condition.

3 Case Study In this section, we present a case study to demonstrate how the aforementioned procedure can be applied to a practical case and how effective it is in compensation of the effects of delamination in composite structures. In this example, we considered an eight layered rectangular composite beam with layers of equal thicknesses and reinforced with the SMA fibers with the ply stacking sequence of [0/90]-2s . A 2D schematic view of the SMA reinforced beam with delamination fault is illustrated in Fig. 1. To avoid unnecessary complexity, here it is assumed that SMAs in each layer can be all fully passive or all fully active. The material properties of the structure that was analyzed in this study is tabulated in Table 1. Dimensions of the considered beam are 127 × 12.7 × 1.016 mm3 . As can be seen in Fig. 1, it is assumed that the delamination occurred in midspan between the 3th and 4th layers of the structure and along the thickness. The nondimensionalized length of delamination (the delamination length divided by the beam length) is 0.4. The compensation results of this study are presented in nondimentionalized form (Eq. 1). μ=

ω f ault y ωintact

(1)

where, ω stands for the first natural frequency of the structure. In order to compute the properties of the composite beam in this study, the micromechanical rule of mixture is used. The elements of stiffness matrix elements for the SMA reinforced composites according to these models are calculated as Eq. (2). kepox y = 1 − ksma ;

Fig. 1 A schematic view of delaminated composite beam

Value

Property

3.43

26.3

67.0

0.35

0.3

SMA (Aus)

Epoxy

SMA (Mar)

Epoxy

SMA (Aus)

Poisson Ratio

Elasticity Modulus (GPa)

Table 1 Material properties of SMA reinforced composite structure

0.3

SMA (Mar)

1250.0

Epoxy

6448.1

SMA (Aus)

Density (Kg/m3 ) 6448.1

SMA (Mar)

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E 1 = E sma ksma + E epox y kepox y ; E 2 = 1/(ksma /E sma + kepox y /E epox y ); E2 = E3; v12 = ksma vsma + kepox y vepox y ; v21 = v12 (E 2 /E 1 ); v23 = v12 ; G 12 = 1/(ksma /G sma + kepox y /G epox y ); G 23 = E 2 /(2(1 + v23 )); G 13 = G 12 ;

(2)

where v, E and G denote the Poisson’s ratio, elastic modulus and shear modulus, respectively. Also, κsma and κepox y stand for the volume fraction of SMA fibers and epoxy, respectively. The density in this micromechanical models is calculated as follows: ρ = ρsma κsma + ρepox y κepox y ;

(3)

To compensate for the effect of delamination on the natural frequency of the structure, 17 different activation patterns are analyzed. The number and configuration of the active and passive layers in these patterns are different, and so they change the natural frequencies of the structure differently. In this study we are looking for the activation pattern for which the corresponding natural frequency is closest to the natural frequency of the structure in a healthy condition. Table 2 shows the effects of considered activation patterns on μ. As can be seen, pattern #0 presents the healthy condition and so μ = 1. Also, pattern #1 shows that in the case without SMA activation the delamination decreases μ by 4.7%. The effects of other SMA activation patterns on μ illustrate that pattern # 14 can compensate for the natural frequency reduction the most, so it can make the natural frequency of the damaged structure closer to the healthy one. Furthermore, the results of patterns # 2 and # 3 show that activation of SMA in only one of two outer layers increased the natural frequency of the structure respectively 9.9% and 8.8% more than the reduction caused by delamination (4.7%). On the other hand, result of pattern # 16 shows that activation of all layers except outer layers can increase the structure’s natural frequency just 2.5% more than the reduction caused by delamination. Another interesting point is that the direction of the SMA fibers which are activated matters very much. Considering pattern # 5 to 7 shows that pattern #6 results in much more compensation than patterns # 5 and 7. It is mainly because that, in this pattern, the activated SMAs are along the length of the beam while in patterns #5 and #7 they are along the width of the structure.

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Table 2 Effect of SMA activation patterns on μ (A: Active SMA, P: Passive SMA)

Pattern Number

Health Status

1

2

3

4

5

6

7

8

0

Intact

P

P

P

P

P

P

P

P

1

Faulty Faulty

P

P

P

P

P

P

P

P

95.3

A

P

P

P

P

P

P

P

109.9

Faulty

P

P

P

P

P

P

P

A

108.8

4

Faulty

P

A

P

P

P

P

P

P

95.6

5

Faulty

P

P

P

P

A

P

P

P

95.3

6

Faulty

P

P

P

P

P

A

P

P

98.3

7

Faulty

P

P

P

P

P

P

A

P

95.6

8

Faulty

P

A

P

P

A

P

P

P

95.7

2 3

SMA Activation Pattern

μ(%) 100.0

9

Faulty

P

A

P

P

P

A

P

P

98.6

10

Faulty

P

A

P

P

P

P

A

P

95.9

11

Faulty

A

P

P

P

P

P

P

A

125.7

12

Faulty

P

A

P

P

A

A

P

P

98.7

13

Faulty

P

A

P

P

A

P

A

P

95.9

14

Faulty

P

A

P

P

A

A

A

P

98.9

15

Faulty

P

A

A

P

A

A

A

P

102.4

16

Faulty

P

A

A

A

A

A

A

P

102.5

17

Faulty

A

A

A

A

A

A

A

A

131.8

4 Conclusion In this study, a multi-layer composite structure with a delamination fault was modeled using COMSOL finite element software. The APT technique to embed the SMA fibers is used and the effects of different SMA activation patterns on the natural frequency of the structure were analyzed. Finally, it was illustrated that functional patterns for activation of the SMAs in different layers of composite structure can be employed to compensate for the reduced natural frequency due to delamination. The pattern that made the natural frequency of the delaminated structure closest to the healthy condition selected as the best pattern. As the delamination in composite structures may be propagated over time, the proper activation pattern can also be changed easily to keep the natural frequency of the structure in a desired range. This research can be continued and extended by using an optimization technique to optimally activate the shape memory alloys in different layers and to automatically minimize the difference between the natural frequencies of the healthy and damaged structures. The authors of this paper are currently working towards this objective.

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Acknowledgements This work is supported by the US Office of Naval Research under the grants ONR N0014-15-1-2311 and ONR N00014-19-1-2070 with Capt. Lynn Petersen as the Program Manager. We deeply appreciate this support and are humbled by ONR’s enthusiastic recognition of the importance of this research.

References 1. Rao A, Srinivasa AR, Reddy JN (2015) Design of shape memory alloy (SMA) actuators. Vol 3. Springer 2. Barzegari MM et al (2012) Effect of shape memory alloy wires on natural frequency of plates. J Mech Eng Autom 2(1):23–28 3. Mahabadi RK, Shakeri M, Pazhooh MD (2016) Free vibration of laminated composite plate with shape memory alloy fibers. Latin Am J Solids Struct 13(2):314–330 4. Malekzadeh K, Mozafari A, Ghasemi FA (2014) Free vibration response of a multilayer smart hybrid composite plate with embedded SMA wires. Latin Am J Solids Struct 11(2):279–298 5. Rogers C, Liang C, Jia J (1991) Structural modification of simply-supported laminated plates using embedded shape memory alloy fibers. Comput Struct 38(5–6):569–580 6. Katariya PV, Das A, Panda SK (2018) Buckling analysis of SMA bonded sandwich structure– using FEM. In: IOP conference series: materials science and engineering. IOP Publishing 7. Kuo S-Y, Shiau L-C, Chen K-H (2009) Buckling analysis of shape memory alloy reinforced composite laminates. Compos Struct 90(2):188–195 8. Zak A, Cartmell M, Ostachowicz W (2003) Dynamics of multilayered composite plates with shape memory alloy wires. J Appl Mech 70(3):313–327 9. Chen Q, Levy C (1996) Active vibration control of elastic beam by means of shape memory alloy layers. Smart Mater Struct 5(4):400 10. Baz A et al (1995) Control of the natural frequencies of nitinol-reinforced composite beams. J Sound Vib 185(1):171–185 11. Rogers C, Liang C, Li S (1991) Active damage control of hybrid material systems using induced strain actuator. In: 32nd structures, structural dynamics, and materials conference. American Institute of Aeronautics and Astronautics 12. Paine JSN, Rogers CA (1994) Active materials and smart structures. Bellingham, WA. p 358 13. Rogers C, Robertshaw H (1988) Engineering Science Preprints 25, ESP25. 88027, Society of Engineering Sciences. Shape memory alloy reinforcement composites 14. Zweben CH (2000) Comprehensive composite materials. Vol 1. Elsevier 15. Ostachowicz W, Cartmell M, Zak A (2001) Statics and dynamics of composite structures with embedded shape memory alloys. In: Proceedings of the international conference on structural control and health monitoring 16. Lau Kt, Zhou L-m, Tao X (2002) Control of natural frequencies of a clamped-clamped composite beam with embedded shape memory alloy wires. Composite Struct 58:39–47 17. Tsai X-Y, Chen L-W (2002) Dynamic stability of a shape memory alloy wire reinforced composite beam. Compos Struct 56(3):235–241 18. Dehkordi MB, Khalili S (2015) Frequency analysis of sandwich plate with active SMA hybrid composite face-sheets and temperature dependent flexible core. Compos Struct 123:408–419 19. Abolbashari MH, Nazari F, Rad JS (2014) A multi-crack effects analysis and crack identification in functionally graded beams using particle swarm optimization algorithm and artificial neural network. Struct Eng Mech 51(2):299–313 20. Della CN, Shu D (2007) Vibration of delaminated composite laminates: a review. 60(1):1–20 21. Nazari F, Hosseini SM, Abolbashari MH (2019) Active tuning and maximization of natural frequency in three-dimensional functionally graded shape memory alloy composite structures using meshless local Petrov-Galerkin method. J Vib Control 25(15):2093–2107

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Space Robotics and Associated Space Applications Ijar M. da Fonseca

Abstract This chapter deals with space robotics and its associated space applications. In order to contextualize the robotics in the space exploration scenario, this paper presents an overview of the space age and the first robotic probes preceding the landing of the astronauts on the Moon. In addition, the paper approaches and discusses space robotics fundamental concepts, classification, and safety-critical aspects for space robotics applications. The space robotics state of the art presented in this article, focuses on the International Space Station (ISS) Extravehicular Activities (EVA), the planetary explorations (like the current Mars exploration), JAXA’s Hayabusa mission that rendezvoused and landed on the 25143 Itokawa asteroid, and in the ESA/DLR’s Rosetta spacecraft carrying its Philae robotic module which landed on the 67P/Churyumov-Gerasimenko comet. The article includes a discussion of on-orbit servicing and rendezvous & docking/berthing (RVD/B). Finally, the paper presents the modeling techniques to derive the equations of motion for manipulator-like spacecraft. Keywords Space robotics · Robot manipulators · Rendezvous and docking/berthing · On-orbit servicing (OOS)

1 Introduction The space robotics characterized all the space missions in the beginning of the space era and was a key issue to the evolving of the space exploration. In the beginning of the space race between the former Soviet Union (URSS) and the USA, the satellites were robot spacecraft or probes orbiting the Earth that evolved in an amazing progress to the Moon, other planets and, the Sun and finally, asteroids and comets. Remarkably, the space journey reached the interstellar region with Voyager 1 and 2 after visiting all the most distant planets of the solar system. The robot spacecraft evolved systematically, first to the Earth orbit by sending to ground control centers alive signals and Earth I. M. da Fonseca (B) Technological Institute of Aeronautics (ITA), DCTA, São José dos Campos, S.P, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_9

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images as photographed from space. In the next, the space race included sending cosmonauts (term used by the soviets) and astronauts (term used by the USA). The idea was sending humans to space and then starts space walk tests to study the capability of man to work and survive in space. The meaning of such walk was time that the astronaut/cosmonaut stayed floating outside the spacecraft, connected by an umbilical cable. However, before humans went into space in the 1960s, the two leaders of space race sent animals to orbit aiming investigation of biological effects of spaceflights. The soviets got the first by sending to orbit a dog named Laika (November 1957). It became the first animal to orbit the Earth, paving the way for human spaceflight during the upcoming years. Later, in 1961, the US sent an ape to orbit, Ham, selected as the chimpanzee to test the safety of space flight on the ape body. After the tests with animal, humans took their way to orbit implementing investigation of several effects of spaceflight on the human body and space walk tests to investigate the body reactions in the microgravity and the capability of the space suits to protect the man against the hostile radiation and temperature outside the spacecraft. Next step it was flights to the Moon and the planetary exploration. The first goal was the Moon. Here we note that the U.S. and the former Soviet Union took different approaches to implement the Moon landing. The soviet option was to land robot spacecraft on the Moon and then return them to Earth without cosmonauts. The U.S. opted to land the men on the Moon and bring them safely back home. The common requirements for both countries to accomplish their goals were the study of the Moon’s surface and the finding of an appropriate and safe site to land, in addition to return from there with soil samples. The use of robot spacecraft by the soviets clearly shows that they would not have to risk cosmonauts’ life. On the other the hand U.S. manned lunar missions would have to face that risk. Before the man landed the Moon, several robot spacecraft sent toward the Moon aimed gathering data that would be used to the mapping of the lunar surface and the choosing a safe-landing site. The first robot spacecraft were impact probes. The goals were photographing or recording videos and sending them to ground control stations before crashing on the Moon’s surface. The results were the mapping of the surface and determining safe land-sites as planned. The Soviets and Americans accomplished these objectives. Missions then extended to robotic probes landing on the Moon to test landing on pre-selected sites. The soviet robotic space mission to land the Mon and return to Earth did require some operations that would be necessary to the U.S. manned mission. One the U.S. planned lunar mission requirements was the capability to implement undocking, rendezvous and re-docking operations of the LM. The implantation of these operations occurred in two phases of the flight, the descent to the Moon and the lifting from its surface to rendezvous an re-dock with the lunar orbiter CM. Before Apollo-11 mission implementation, those RVD techniques were subject of studies and several orbital tests. The strong competition between soviets and the U.S. to get firsts during the space race appears clearly in the conquest of the Moon. The soviets objective was to battle the U.S. by arriving on the Moon, collecting lunar samples and returning them to Earth without any on-board cosmonauts. Had them succeed the U.S. mission would

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lose much of its bright in the space race. However, the soviets, whom had the most of the first in the space race since the launching of the Sputnik satellite in 1957, would not win the game this time. When the astronauts landed successfully on the Moon the soviet robot probe Luna-15 was also arriving, but it failed to land and crashed in the Moon. That was the space race in the time of Cold War. The soviets had collected the most of the all-first in the space race since 1957 and lost perhaps this most important event of the space race. The U.S. planed, executed and implemented the most challenging and complicated, not to say the difficult, project ever, to get that first by landing a man on the moon. The Apollo-11 mission obtained very important outcomes as landing the first man on the moon, first human work on the moon, collecting Moon’s soil samples and returning them to Earth, implementing operations of the LM undocking, rendezvous and re-docking with the Apollo-11 CM lunar orbiter, and returning back home successfully. The LM never returned to Earth. After the Appolo-11 mission, NASA sent several manned missions to the Moon. Meanwhile other plans were in place such as exploring other planets of our solar system and development of space stations. The idea of using robot manipulators, a space shuttle as well astronauts/cosmonauts for on-orbit servicing implementation was part of the then future projects. Also, during the period that preceded the man landing on the Moon, the soviet got an interesting first by sending robotic probes to Venus, including a robotic probe landing on the planet. The U.S. sent also spacecraft to the planet. Their competition continued from Venus to Mars. Both of them obtained important outcomes as close images of the planets, atmosphere composition, surface data, and planet temperatures. The planet surface mappings and possible landing sites paved plans for the next projects of space exploration. The space stations came about and this time the competition between the two giants of the space exploration starts merging to cooperative projects for space activities. The Skylab Orbital Workshop was a temporary U.S. space station. It consisted of a 22-foot-diameter cylindrical Orbital Workshop (OWS) with two wings-like solar arrays, a cylindrical Airlock Module (AM), a multiple docking adapter (MDA), and a truss-mounted Apollo Telescope Mount (ATM) with four solar arrays arranged in a windmill formation. Uncrewed, the launching inserted the space station into 435 mKm low Earth orbit. A micrometeoroid shield tore away from the OWS, damaging the station during launch. The incident took one of the main solar arrays and jammed the other main. This deprived Skylab of most of its electrical power and removed protection from intense solar heating, threatening to make it unusable. Three subsequent missions (Skylab 2, Skylab 3, and Skylab 4) delivered three-astronaut crews to the space station. The first crew deployed a replacement heat shade and freed the jammed solar panels to save Skylab. A maintaining operation of such magnitude was the first OOS ever performed in space. A four mission (Skylab 4) was a rescue mission to bring back home the astronauts. The delay in the Space Shuttle project became a problem to NASA’s plan to extend the station life. The re-boost of the space station was not possible and Skylab’s orbit decayed and it burned up in the atmosphere on July 11, 1979, over the Indian Ocean. In 1993, the U.S. effort turns to the International Space Station (ISS), a multinational collaborative project involving 5 space agencies: NASA (U.S.), Rosccosmos

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(Russia), JAXA (Japan), ESA (Europe), and CSA (Canada). It is a modular station assembled in space, following ideas by von Brown and his conceptions of space station designs in the late 1950s. This kind of activity belongs to the set of OOS [5], a subject of the next section. The ISS is still operating in space in an integrated effort of several countries, enabling a series of scientific research in astrobiology, astronomy, meteorology, physics, and other fields, in addition to several engineering experiments in the orbital microgravity environment. The station allowed the consolidation of robot manipulator applications and other types of robots as well. There are experiments never thought presently implemented in the station, such as a greenhouse for growing vegetables in an Earth pressure, temperature, an atmosphere. Intergovernmental treaties and agreements establish the ownership and use rights of the ISS. To close this section it is worthy no point out where we are in the space exploration started in the late 1950s. NASA has some robots operating in Mars and preparing to start the Moon and Mars colonization. All the planets in solar system received a visiting probe for image taking and scientific studies. Included are several planetary moons and the rocks of the Saturn’ rings. Since the 1990s, space probes visited 16 minor celestial bodies: asteroids, dwarf planets, and Kuiper belt objects. The Hubble Space Telescope orbits the Earth and it has imaged several asteroids and observed distant galaxies. Spacecrafts has also visited 9 comets: Giacobini–Zinner, Halley, Grigg–Skjellerup, Borrelly, Wild 2, Tempel 1, C/2006 P1 (McNaught), Hartley 2, Churyumov–Gerasimenko. Voyager 1 and 2 are in spacecraft have travelling in space for almost 44 years. Launched in 1977 both spacecraft have reached the interstellar space. The spacecraft visited all the most distant giant planets of the solar system, sending images to Earth. The Voyagers are the most distant human-made objects from Earth. Part of the Voyager missions objectives was a study of the outer Solar System. The spacecraft are about 22.2 billion km from Earth as of March 12, 2020 and their communication with Earth is still alive. Outcomes from Voyager probes include the study of the weather, magnetic fields of Saturn and Jupiter and the rings of Saturn. Voyager 1 was the first probe to provide detailed images of the moons Saturn and Jupiter and the first spacecraft to cross the heliopause and enter the interstellar medium a region beyond the influence of the Solar System, joining Voyager. Voyager 2 has begun to provide the first direct measurements of the density and temperature of the interstellar plasma. Each Voyager space probe carries a gold-plated audio-visual disc in case intelligent life-forms from other planetary systems find either of the spacecraft. The discs carry photos of the Earth and its life forms, scientific information, spoken greetings from people, and a collection of music. It includes are also a miscellany of sounds of Earth as the sounds of whales, a baby crying, waves breaking on a shore.

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2 Space Robotics and on-Orbit Servicing (OOS) The orbital dynamics includes the orbital and the attitude motion. The orbital robotics refers to the robot motion in orbital environment of microgravity, subjected to the external forces and toques. The space robotics classifies as orbital and planetary robotics. Orbital robotics takes place in the microgravity environment where prevails the combined effects of Sun, Moon and other planets gravitational forces, strong temperature variation, electromagnetic effects, atmospheric drag, and huge cosmic radiation. In the next the paper discusses the orbital robotics issues Due to microgravity, any artificial or natural orbiting body experiences lack of weight and floats, as they were weightless. In such scenario, even low forces or torque can translate and/or rotate large loads. In orbital mechanics, the orbiting motion a space vehicle is the motion of its center of mass (CM) about its attractor (planet or other celestial body). The CM position can changes if • any mass component of the vehicle moves from the original position, breaking the nominal mass distribution taken into account to define the original CM; • Fuel and/or water consumption affects the mass configuration about the original CM; • In case of undocking or docking and berthing to or de-berthing from a space vehicle In this study, the focus is the case of robot manipulator moving from its base under joint control commands. In general, the robot manipulator is at rest (off) and part of the satellite, counting on for the nominal orbiting CM position. When the robot arm starts moving, the CM of the subsystem changes to a new position. However, as the position of the original CM cannot change unless control forces are applied, the system moves in the contrary direction of manipulator arm, keeping orbiting CM in the same position. As the robot moves back with the satellite the manipulator end effectors fails to capture its target. To avoid the failure the robot control commands must compensate the system back motion when implementing the robot arm path planning. This is to say that it is necessary to plan the arm path taking into account the spacecraft back motion to avoid the arm missing its target. However, another problem affects the path planning. The space vehicle orientation (attitude motion) can change in response to the torque commanded to the manipulator joints. The reaction to the commanded torque acts on the satellite and it moves in rotation, taking away the end effectors from the capture point in the target. It is not easy to design an AOCS (attitude and orbit control subsystem) to compensate the change in the CM position and at same time keep the satellite in its nominal attitude. In fact, this problem requires analyzing the relative orbit and attitude motion. The berthing operation involves RVD between two orbiting space vehicles followed by berthing (grasping our capture operation). The relative attitude implies synchronizing the attitude motion of both spacecraft so that the relative attitude is zero. The same is valid fort the relative position. When the relative velocity is zero, the position of the

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CM of both spacecraft stays at the same distance from each other. To close this subject it is necessary to discuss the manipulator workspaces. Manipulators can capture or grasp only objects that are inside its workspace. This is to say that the AOCS must have the capability to synchronize the relative attitude in zero and then, identify and track the berthing point to have it inside the manipulator workspace when relative velocity reaches zero. In short, the requirements for a successful grasping are • The grasping point shall be inside the manipulator workspace • Relative attitude between both spacecraft shall be zero • The Relative orbital rate between both spacecraft shall be zero The idea behind the OOS [7] relates to space robotics and RVD/B operations [1, 9, 13]. It involves providing in-orbit services to spacecraft and space stations such as refueling, maintenance, exchange of astronauts and scientific experiments in space stations, assembling parts of large space structures in space, cleaning around the orbit to prevent collisions with space debris, and extending life of satellites. The idea remounts von Browns’ conceptual projects of space stations. According to his ideas, the assembling of Space Stations by astronauts with support of robot manipulators, a space shuttle for travelers to and from Earth would be feasible and necessary to assembly space station in orbit. The U.S. learned much on this with the Skylab experience. Astronauts send to that station repaired damages that happened during the launching of the Skylab. RVD/B is the mean to approach one spacecraft to another (rendezvous) and dock (docking port required) and berthing (using robot manipulators). The sequence of RVD phases preceding docking or berthing occurs in the scenario of microgravity and encompasses the problems discussed about floating, relative motion and non-fixed base for robots [5, 12]. The implementation of the Rendezvous & Docking technique remounts the first years of the space era in the 60s [10, 11]; RVD/B orbital operation was a key issue to accomplish the landing of man in the Moon. Several orbital tests of RVD/B implemented by NASA and Russian consolidated the implementation a sequences of docking, undocking and re-docking of two space vehicles in orbit, executed by astronauts and cosmonauts and some executed automatically (en.wikipedia.org/wiki/Space_rendezvous) Interviewed by Doug Ward/Elk Lake, Michigan—June 29, 1999. The success of the tests allowed the NASA Apollo 11 to separate from its lunar module (LM) with two astronauts, which landed on the Moon. A third astronaut stayed inside the Command Module (CM) orbiting the Moon and waiting for the LM return from the moon. After accomplishing successfully the landing and the planned activities on the Moon surface, the astronauts lifted off the LM that rendezvoused and re-docked to the orbiter CM. Then the astronauts entered back to the CM, discarded the LM, and returned to Earth. The lunar mission operation of RVD counted on the astronaut participation. Later, in the 1997 the Japanese implement an orbital test of RVD/B by using for first time a robot manipulator for berthing operation. The Engineering Test Satellite VII (ETS-VII), developed and launched by National Space Development Agency of Japan (NASDA) successfully orbited the Earth carrying out important and innovative experiments on orbital robotics including a 2 meter-long, 6 DOF robot manipulator. The ETS-VII was an un-manned spacecraft. The experimental orbital tests

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accomplished remarkable outcome in space robotic application in RVD technology. It was the world’s first satellite to be equipped with a robotic arm, https://en.wik ipedia.org/wiki/ETS-VII-cite_note-fujipress-2 and the first unmanned spacecraft to conduct autonomous rendezvous & docking operations successfully. Japan was one of the partners of ISS project development and implementation, giving a very important contribution ISS with the module JEM (Japanese Experiment Module) which includes a 10 m long robot manipulator system (JEMRMS). The automated rendezvous and the concept of OOS pushed the development of RVD/B for OOS to the ISS. The first space tug for OOS was the Space Shuttle. A partially reusable low Earth orbital spacecraft system, it was operated from 1981 to 2011. Astronauts took off the Earth to ISS and then came back to Earth using the Space Shuttle. The Shuttle missions provided OOS by taking water, fuel, experiments, replacing astronauts and so on, via Space Shuttle. The use of Shuttle robotic manipulator as cargo helped the assembling of the ISS by transporting large structures in space for years. The Shuttle also provided OOS by grasping and repairing satellite and then putting them to their orbit. The Shuttle was equipped with long robotic manipulator. Some satellites were taken aboard the Shuttle, repaired and the, put back on their orbit. It was remarkable the repairing of Hubble Telescope in 1993. It was amazing to watch the astronauts alive in the TV repairing the space telescope. The concept of OOS applications and the RVD/B operations implemented by astronauts evolved to automatic RVD with support from ground, and then to autonomous operation. Future missions to colonize the Moon and Mars shall enforce new developments in the RVD/B and planetary robotic activities toward completely autonomous control subsystems. The RVD requires the target spacecraft to have docking ports. In this case, the control must identify the docking port and track it very accurately in the close proximity [2] phase to reach simultaneously the relative prescribed pose with relative velocity very close to zero to execute a soft docking safely. In case of berthing, a robot manipulator grasps the target and then the on-board operator commands the telerobots to complete the docking operation safely. The requirements for close proximity are the same as that for docking. The berthing may also be a target vehicle capability. In this case, the chaser spacecraft approach to the target reaching zero relative velocity when it is inside the target manipulator workspace. The target arm then captures the chaser and completes the docking safely by using telorobot capability of the target on-board control subsystem. This grasping operation counts on the participation of on-board astronaut. Nevertheless, the docking or capture techniques, the chaser must synchronize its orbital and attitude motions to that of the target. A RVD/B is cooperative when the operation involves interactions with the target spacecraft, a ground station, and a data relay satellite, to accomplish the docking or berthing. It may not require actions of astronauts. If the target is alive, the control commands may come for data relay satellites or ground station control centers. If the target spacecraft has a known geometry but it is death and without docking ports, it is still possible to implement a berthing operation. To accomplish such goal, a possible solution is the use of cameras and an on-board computer with capability to process and analyze the images to find a place on target to grasp it. This case still classifies as

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cooperative berthing because the chaser receives the input on the chaser geometry. In case of debris with unknown geometry, the removing operation classifies as a noncooperative and the chaser on-board computer would have to include capabilities to grasp autonomously the target and take it to a parking orbit or simply take it away from the spacecraft trajectory. For the case of unmanned chaser to grasp unknown asteroids and comets, the chaser’s AOCS (Attitude Orbit Control Subsystem) must include capability to rendezvous and land in the target without support of ground station or data relay satellites. However, for known celestial trajectories, help can still come from an Earth ground station, mainly in the case the rendezvous and landing mission includes a target orbiter and a landing module. During the module descent the target orbiter send/receives information from Earth and then command the lading module. If the target surface is unknown, the orbiter takes close images from it and sends them to ground station. The images are processed and analyzed to define the proper landing site. Then the control station sends the information to orbiter that by its turn commands the landing site to the descent module. This was the case of Rosetta mission and its lander module to the comet 67P/Churyumov–Gerasimenko (67P). One important concept in the area of space robotics is space telerobot [12]. This concept refers to machines that perform physical tasks in the space environment. The machines go from space manipulators to robonauts and planetary/lunar rovers. Robonauts are the ultimate NASA robots with capability to mimic humans’ arm motion. The astronauts may be using leap motion controller to teach robonauts to follow and learn their arm motion. Actually, the Robonauts are already being tested and used in the ISS to support astronauts by performing the EVA. Inside the ISS they helps astronauts in the IVA. The robonauts conception was that of human being image as they have shape similar to humans. They possess articulated legs, articulated arms and five fingers hands. By having hands similar to humans they can manipulate tools that man uses. Their legs contain more degrees-of-freedom than humans’ lags and ends with hands instead of feet. This feature allows the robonauts to grasp and hold firmly outside on the ISS structure, while performing EVA. Telerobots are machines that operate both autonomously, when no operator commands are need for the task or in conjunction with an operator who monitor and send commands to the remote site. The main reason for using telerobots is to perform tasks that would somehow be difficult, risky and even impossible for humans in the orbital environment or on the surface of celestial bodies. Most of the tasks for telerobots are related to OOS. They shall perform activities of • • • • • • • •

maintenance, repairing, refueling, replacing spacecraft’s modules, bringing experiments to/from space vehicles, replacing experiments, capturing debris for cleaning orbits, capturing satellites to extend their life,

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• executing EVA • helping astronauts with intra vehicular activities (IVA), • transporting and assembling structures in orbit. Telerobots are to support a lot of work in space, be it in orbit or another celestial body. In a near future, the telerobots will be a strategic component of the Moon and Mars colonization. A space telerobot involves a local site or station from where the operator interacts with the remote site. The local site includes the interface from where the operator receives the remote site data, analyzes them to understand the task in operation and commands the telerobot. Figure 1 shows the data flow of a space telerobot illustrating the local and remote sites, the operator to remote site interface, and the major components of the space telerobot. The use of long robot arms for on-orbit large load transportations brings about another complication, the structural flexibility of the manipulator links. This is a problem known as CSI (Control Structure Interaction). The vehicle control action excites the arm links flexible modes of vibration during orbital robotic operations. The arm links flexible vibrations may lead the end effectors to miss the grasping point in the target. For this case, no implemented efficient solution is available. However, researcher’s studies suggest that the use of viscoelastic and PZT materials could be a good solution for the control problem involving structural flexibility. From the point of view RVD/B dynamics analysis, there are other complicated issues, such

Fig. 1 Space telerobot system data flow (From [12])

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mathematical modeling and computer simulations of the relative motion between chaser and target spacecraft. The paper discuss in next the planetary robotics issues. The planetary robotics deals with robot motion on the surface of planets or other celestial body. Because of celestial bodies may have different gravitational forces, temperatures, radiation levels and different surfaces features compared to Earth, the planetary robot must be designed properly for the environment it will work. For the lunar regolith surface, the robot moves well over wheels. This is not the case for robot mobility on Mars surface. The Mars planetary rovers use a combination of wheels and articulated legs to in the area covered by Rockies, regolith, and sand of the Martian planet. If the surface is swamp-type, ice-type, and liquid-type then it is necessary to find the proper devices for the robot mobility. The planetary robots do not face microgravity problems of the orbital robot operations. Temperature, atmosphere, radiation and electromagnetic effects of planetary surfaces also affect the material, joints functioning, and efficiency of robotic systems. The landing in certain celestial bodies like comets and asteroids where the local gravity is very weak the robot stability may require some kind of control device to keep the vehicle stable on the surface. The case of landing in asteroids and comets is in a gray zone between orbital robotics and planetary robotics and did not receive yet any classification en terms of space robotics. Maybe this case will receive a classification as asteroids/comets robotics. In short, spacecraft rendezvous followed by landing in an unknown celestial body requires command control and trajectory monitoring from ground control centers or from orbiting spacecraft about the target body. In such case, a previous knowledge of the body orbit, shape, surface features, local gravity and temperature among others is a key issue for the landing successful accomplishment. To illustrate this, consider the landing of the Rosetta’s Philae module on the 67P/ChuryumovGerasimenko comet. After traveling 10 years eight months in space, Rosetta received ground command to free its lading module toward the surface of the comet nucleus (November 12, 2014). However, it was not a straightforward command since the comet shape, surface features, size, and other important aspects were unknown, a meaningful selection of the landing site would require information that Rosetta would send to Earth. Rosetta had appropriate set of sensors, cameras, and other instruments to provide the comet’s data to the Earth control center. Rosetta accomplished this data by orbiting and taking close images of the comet. Receiving and processing the data received from Rosetta the control center determined the comet’s shape, surface characteristics, rotation and orientation, temperature variation, local gravity, Sun illumination, and analyzed the trajectories and landing sites for Philae landing. The control center still detected a last minute problem with a thruster that failed to fire. However, after checking that there was not possibility to fix it, the control center decided to command the landing. Philae had stability problems when its anchoring harpoons failed to deploy and that failed thruster could not support the landing. After bouncing off the surface twice, Philae achieved the first-ever nondestructive landing on a comet nucleus, in spite of its final and uncontrolled touchdown that left it in a non-optimal position and orientation. Despite the landing problems, the robot probe’s instruments obtained and transmitted to Earth the first images from surface of

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a comet and made the first direct analysis of a comet, sending back data that allowed scientist to determine the composition of the comet’s surface for the first time.

3 Mathematical Modeling Issues and Computer Simulations One important issue involved in planning space robotic missions is the dynamics analysis. The output of such analysis is previews on stability, control law/techniques performance, safety critical problems that may lead to mission failures as for example, collision avoidance, among other. In this scenario, the mathematical modeling and computer simulations play a very important role. Mathematical modeling is the activity of obtaining the mathematical model (the equations of motion) that represents the body or object dynamics in some environment. Such environment can be the orbital microgravity or space environment, ground and atmospheric environment of Earth and other planets or celestial bodies as well, the submarine environment where the hydrodynamics forces are dominants, and so on. The environment acts on the dynamics with torque and/or forces (dissipation forces, and magnetic forces, gravitational forces. The equations of dynamics are second order differential equations representing the body motion. Sometimes control forces/torques conducts the dynamics to guide the system dynamics in a desired or prescribed way. The mathematical model must be as accurate as possible aiming a consistent dynamics analysis. In this study, the focus is on the mathematical modeling and control. The control is a wide area of research and engineering that involve the methods, techniques and procedures to keep the system dynamics under a controlled motion. The mathematical model of robot-like spacecraft is, of course the same as any spacecraft, a set of differential equations combining two types of motion, the orbit and the attitude dynamics. The orbit refers to the path of the spacecraft center of mass about the Earth and the attitude refers to the rotational motion of the spacecraft about its center of mass and relates the orientation of the satellite with respect to (w.r.t.) some reference system of axis. For satellite-mounted manipulator or manipulator-like spacecraft, it is necessary to take into account the following • the manipulator is not mounted on an inertially fixed base, but rather than in a movable base, affecting the capability of the robot arm to reach the commanded position in space unless the path panning compensate the change in the CM position of the manipulator when it operates in space. The CM changes in response to the commanded control/forces the arm joints; • the orientation (attitude) of the manipulator-like spacecraft changes due to the joints control commanded torque, unless the vehicle attitude and orbit control subsystem (AOCS) maintains the attitude in the nominal specification while the manipulator works; • control commands during the phase of close proximity to a space target are computed on the base of relative motion data;

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• the path planning to take the robot end effectors to a point in the target must consider that, in the microgravity environment, the arm links motion causes a shift of the manipulator CM. In response to this, the system moves back in contrary motion of the robot arm to keep the system CM in the same position. It is worth to note that changes in the on-orbit CM result in changes of the orbit and depends on the orbit control through an increment of velocity, v, or on a gravity assist maneuver. Accidents as collision with a significantly sized debris or during RVD/B orbital maneuver may also cause orbit changes in addition to destruction of the less massive body. This explains why, in absence of any orbit control commands, the orbiting robot manipulator moves somehow to maintain the orbiting CM in the same position. If the path planning does not compensate such motion, the manipulator end effectors miss the target. There are different approaches to derive the equations of orbital dynamics. The Newton-Euler’s method is a vector approach to obtain the orbit and attitude equations of motion. Working with vectors frequently requires body-free diagrams to compute reactions forces/torques and solve an algebraic system of equation to derive the differential equations of motion. In this approach, vector senses impacts the equations. It is necessary to avoid signal errors while deriving the equations of motion once they may compromise the consistence of the resulting equations. The Euler’s equations are three first order differential equations in the components of the angular velocity vector. The Euler’s equations are three first order differential equations in the components of the angular velocity vector. Those components are non-linear functions of the attitude angles and rates. The integration of the Euler’s equations does not result in angular coordinates. For the dynamics analysis it is necessary to derive the set of first order kinematics differential equations and integrate those equations in conjunction with the Euler’s equations to obtain angular position and rates. There are singularities in the Euler angles no matter which sequence of rotation chosen. To avoid the problem, the recommendation is to use quaternion to describe the attitude differential equations and after integration convert the quaternion back to attitude angles and rates. Reference [6] shows the dynamics analysis for a satellite by using quaternion instead of Euler’s angles. The Euler’s equations in vector format is I ω˙ + ω × I ω = N

(1)

where ω is the angular velocity vector, I is the inertia tensor, and N is the torque vector. In matrix notation the system equation is [I ]{ω} ¯ + [ω][I ¯ ]{ω} = {N } ⎤ I11 −I12 I13 where [I ] = ⎣ I21 I22 I23 ⎦ is the matrix of inertia I31 I32 I33 ⎡

(2)

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⎡

⎤ 0 −ω3 ω2 ¯ = ⎣ ω3 0 −ω1 ⎦ is the skew symmetric matrix in the components of [ω] −ω2 ω1 0 angular velocity ⎧ ⎫vector, ω ⎨ ω1 ⎬ {ω} = ω2 is the angular velocity vector ⎩ ⎭ ω ⎧ 3⎫ ⎨ ω˙ 1 ⎬ {ω} ˙ = ω˙ 2 is the angular acceleration vector ⎩ ⎭ ω˙ 3 From the kinematics the equation in terms of the Euler’s angles, considering the sequence of rotation 3, 2, 1 is ⎧ ⎫ ⎡ ⎤⎧ ⎫ 1 0 − sin θ2 ⎨ θ˙1 ⎬ ⎨ ω1 ⎬ ⎦ ⎣ = 0 cos θ1 sin θ1 cos θ2 θ˙ ω ⎩ ˙2 ⎭ ⎩ 2⎭ θ3 ω3 0 − sin θ1 cos θ1 cos θ2

(3)

⎧ ⎫ ⎨ θ˙1 ⎬ By solving Eq. (6) by θ˙2 ⎩˙ ⎭ θ3 ⎧ ⎫ ⎡ ⎤−1 ⎧ ⎫ 1 0 − sin θ2 ⎨ ω1 ⎬ ⎨ θ˙1 ⎬ ω θ˙2 = ⎣ 0 cos θ1 sin θ1 cos θ2 ⎦ ⎩ 2⎭ ⎩˙ ⎭ θ3 0 − sin θ1 cos θ1 cos θ2 ω3 ⎧ ⎫ ⎡ ⎤⎧ ⎫ cos θ2 sin θ1 sin θ2 cos θ1 sin θ2 ⎨ ω1 ⎬ ⎨ θ˙1 ⎬ 1 ⎣ = θ˙ 0 cos θ1 cos θ2 − sin θ1 cos θ2 ⎦ ω2 ⎩ ⎭ ⎩ ˙2 ⎭ cos θ2 θ3 0 sin θ1 cos θ1 ω3

(4)

(5)

The Euler’ equations can be written in terms of the attitude angles an rates by writing {ω}, ˙ {ω}, ¯ and { ω} in terms of Eq. (3). Of course, {ω} ˙ is obtained by taking the time derivative of Eq. (3). As shown in the kinematic differential equation [4], there are singularities for θ2 = n π2 , n = 1, 3, 5 . . . . To solve singularity associated with kinematic differential equations it is usual to write the attitude equations in function of the quaternion. For the sake of illustration consider the Eqs. (7–8) where the Euler’s modified equations are written in term of quaternion, considering a diagonal inertia matrix. The problem formulation considers a satellite in a transient phase when a 10 meters long mast with a 10 kg tip mass deploys in orbit to change the moments of inertia properly for passive gravity-gradient stabilization. For consistence, the dynamics includes the gravity-gradient torque. For details of the problem formulation see [6] and for quaternion formulation see [14].

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Izz− I22 12μ q − q q q + q q ω − ω (q )(q ) 1 2 3 4 1 3 2 4 2 3 Ix x R3

I11 − I33 12μ 2 1 2 ω˙ 2 = q1 + q4 − (q1 q3 + q2 q4 ) − ω1 ω3 − I22 R3 2



I22 − I11 12μ 2 1 2 q1 + q4 − ω˙ 2 = (q1 q2 − q3 q4 ) − ω1 ω2 − I33 R3 2 ω˙ 1 =

(6) I˙22 ω2 I22

(7)

I˙33 ω3 I33

(8)

where q1 , q2 , q3 andq4 are the components of the quaternion. μ is the gravitational constant for the Earth R is the orbit radius  μ is the orbital rate. R3 The time derivative of the moments of inertia appearing in Eqs. (7–8) are due to the variation of the inertias during the mast deployment. Note that the Euler’s equations, as shown in Eqs. (1–2), are valid for rigid spacecraft or other rigid bodies referred to an inertial system. Many space vehicles are not rigid bodies. There may be flexible components as solar panels, antennas and mobile components such as robot links in operation, reaction wheels, or masts that deploy in orbit changing the mass distribution causing the CM and other mass properties do change. In addition, fuel consumption affects the inertia properties. For example, in case of flexible vibration, fuel consumption, robot arm motion, etc. the inertia matrix vary w.r.t. to time and the Euler’s equations become as Eqs. (6–7) a has the general form I ω˙ + I˙ω + ω × I ω = N

(9)

For a robot-like spacecraft the tensor of inertia, I become dependent of the manipulator link angles that vary w.r.t. time. Then the term I˙ω appears in the Euler’s equations. Sometimes the term I˙ω appears only during a transient phase of the vehicle dynamics. For example, the transient in the dynamics analysis presented in [6] refers to the time spam during a 10 meters long mast deployment. After the mast deployment, the term I˙ω becomes zero. The mast deployment changes the system dynamics because of the changes in the inertia properties. It is necessary to take into account this fact during the launching phase, when stages separate from the launcher. The same occurs during docking and undocking operations. When a chaser spacecraft docks with a target in space, the resulting composed system presents a inertia properties with changes in the inertia tensor and the CM. In a prescribed orbit, the CM is fixed. Therefore, if there is a new configuration of masses the complete system moves to synchronize the mass distribution w.r.t. the orbiting CM. For robot-like spacecraft in action to grasp a target, the control commands must take into account the system synchronization motion otherwise the robot end effectors miss the target.

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Another optional approach to derive the Euler’s equations is iterative formulation presented by [3]. The starting point is to consider the manipulator as a chain of bodies (links), each one capable of motion relative to its neighbors. Then it is reasonable to compute the velocity of each link, starting by the robot base. i

ωi+1 = i ωi + ii+1 R θ˙i+1 i+1 Zˆ i+1

θ˙i+1 i+1 Zˆ i+1 i+1 i R ωi+1 i

(10)

⎧ ⎫ ⎨ 0 ⎬ = 0 ⎩ ⎭ θi+1

  = i+1 R i ωi + ii+1 R θ˙i+1 i+1 Zˆ i+1 i

then i+1

ωi+1 = i+1 R i ωi + θ˙i+1 i+1 Zˆ i+1 i

The angular rate of the link i + 1 in frame {i + 1} (upper script) is i+1

i+1 ˆ ωi+1 = i+1 R i ωi + θ˙i+1 Z i+1 i

(11)

The angular velocity of the link i + 1 is the angular velocity of previous link (i) plus its own rotation. Here ii+1 R is the rotation matrix relating frames {i} and {i +1}. The time derivative of Eq. (10) is i+1

i+1 ˆ i+1 ˆ ω˙ i+1 = i+1 R i ω˙ i + i+1 R i ωi × θ˙i+1 Z i+1 + θ¨i+1 Z i+1 i i

(12)

Then the Euler’s equations in the iterative format become Ci

I ω˙ i + ω i × Ci I ω i = N i

(13)

where Ci indicates that the frame {i} is defined in the center of mass of each link of the manipulator. One of the most used formulations to derive space manipulator equations of dynamics and its associated kinematics differential equations is the LagrangeHamilton variational approach. The approach bases on the system energy. It consists of defining the vector positions of an elemental w.r.t. the system CM of each component of the spacecraft. Then take the time derivatives of the position vectors to obtain the velocities. Finally, write the system kinetic energy. There may be other type of energy such the potential energy. If it is the case, derive the potential energy. Next step writhe the Lagrange’s L. Finally, use the Lagrange’s formula to derive the equations of motion for the manipulator.

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d dt



∂L ∂ q˙i

−

∂L ∂qi

= Qi

(14)

where L is the Lagrange’s function given by T - V (kinetic T and potential Energies, V ). In this formula, the qi is the generalized coordinate θi . If prismatic joint is considered then the kinetic energy includes the generalized coordinate di (manipulator standard notation for prismatic joint variables). For manipulator-like spacecraft, the mathematical model involves the attitude motion coupled with the arm variables. The Langrage’s formula for quasi-coordinates applies to obtain the Euler’s modified equations     ∂T d ∂T − [ω] ¯ = {N } dt ∂ω ∂ω

(15)

where 

∂T ∂ω

 =

⎧ ⎫ ∂T ⎪ ⎪ ⎨ ∂ω1 ⎬ ∂T

∂ω2 ⎪ ⎩ ∂T ⎪ ⎭

⎡

∂ω3

⎤ 0 −ω3 ω2 ¯ = ⎣ ω3 0 −ω1 ⎦ [ω] −ω2 ω1 0 1 Tn = T p + T1 + T2 + T3 . . . + Tn 2 i=1 n

T =

where  = manipulator link, p means the manipulator-like spacecraft base or platform). Consider Fig. 2 for a brief discussion of the work [7]. The figure illustrates a manipulator-like spacecraft that orbit the Earth under gravity-gradient torque). For this case study no potential energy is considered and the Lagrangian function reduces to L = T. This system the Kinetic energy derived in [7] for the model shown in Fig. 2 is T = Tp +

3 

Ti

(16)

i

Tp = T1 =

1 1 1 I px ωx2 + I px ω2y + I px ωz2 2 2 2

(17)

   2   1  Ix1 c2 θ1 + I y1 s 2 θ1 ωx2 + Ix1 s 2 θ1 + I y1 c2 θ1 ω2y + Iz1 ωza + θ˙1 (18) 2

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Fig. 2 Manipulator-like spacecraft showing joints, links, frames and the RPY frame (From da Fonseca [7])

1 1 (I y2 s 2 θ1 c2 θ2 + Iz2 s 2 θ2 )(ωx + sθ1 θ˙2 )2 + (I y2 s 2 θ1 s 2 θ2 2 2 1 +Iz2 c2 θ2 )(ω y − cθ1 θ˙2 )2 + I y2 c2 θ1 (ωza + θ˙1 )2 + Iz2 2 −I y2 s 2 θ1 )sθ2 cθ2 (ωx ω y − ωx cθ1 θ˙2 + ω y sθ1 θ˙2 − sθ1 cθ1 θ˙22 ) −I y2 sθ1 cθ1 cθ2 (ωx + sθ1 θ˙2 )(ωza + θ˙1 ) + I y2 sθ1 cθ1 sθ2 (ω y − cθ1 θ˙2 )(ωza + θ˙1 ) (19) T2 =

 2 1 I y3 s 2 θ1 c2 (θ2 + θ3 ) + Iz3 s 2 (θ2 + θ3 ) ωx + sθ1 (θ˙2 + θ˙3 ) 2 1 + [Iz3 c2 (θ2 + θ3 ) + I y3 s 2 θ1 s 2 (θ2 + θ3 )][ω y − cθ1 (θ˙2 + θ˙3 )]2 2

T3 =

1 + I y3 c2 θ1 (ωza + θ˙1 )2 − [(Iz3 − I y3 s 2 θ1 )s(θ2 + θ3 )c(θ2 + θ3 )][ωx + sθ1 (θ˙2 2 +θ˙3 )][ω y − cθ1 (θ˙2 + θ˙3 )] − I y3 sθ1 cθ1 c(θ1 + θ3 )[ωx + sθ1 (θ˙2 + θ˙3 )](ωza +θ˙1 ) + I y3 sθ1 cθ1 s(θ2 + θ3 )[ω y − cθ1 (θ˙2 + θ˙3 )](ωza + θ˙1 ) (20) where xyz coincides with the principal axis of inertia of the spacecraft platform (p) when the manipulator is not operating. In the nominal configuration, the initial position is zero for the manipulator link angles and rates. I px , I py , and I pz are the principal moments of inertia for the spacecraft platform; i i = 1,2,3 are links of the robotic manipulator

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Ixi i = 1,2,3 are the moments of inertia w.r.t. x − axis for the links i , i = 1,2,3; I yi i = 1,2,3 are the moments of inertia of w.r.t. y − axis Izi i = 1,2,3 are the moments of inertia of w.r.t. z − axis T  is angular velocity vector of the spacecraft. The components of the ω x ω y ωz vector are the angular rates. sθi and cθi , i = 1,2 are the sinθi and cosθi , respectively s(θ2 + θ3 ) and c(θ2 + θ3 ) are sin(θ2 + θ3 ) and cos(θ2 + θ3 ), respectively. θ˙i i = 1,2,3 are the rates of the robotic manipulator ωza = ωz + n, where n is orbital rate. This algebraic manipulation is just to compute the kinetic energy and then use Eqs. (14) and (15) obtain explicitly the differential equations for the rotational motion in orbit. As the robot-like spacecraft orbits the Earth, it is necessary to derive the associated equations of the orbital motion. Such equations apply during the close proximity approach of the space robot w.r.t. its target in orbit. That is to say that these equations refer to the relative orbital motion between the space robot and its target. The Newton’s Method is the formulation used to derive the mathematical model for this case. [14] derive these equations as x¨ − 2n y˙ − 3n 2 x = 0

(21)

y¨ + 2n x˙ = 0

(22)

z¨ + n 2 z˙ = 0

(23)

The external torque N associated to the Euler’s modified equation Eq. (9, 13, 15) must also be modeled. Particularly, for the mathematical modeling of RVD/B it is necessary to include in the mathematical model, the reaction forces and torques arising from the manipulator motion. [3] presents this feature of the manipulator motion in the iterative Newton-Euler’s dynamics formulation. For a more complete dynamics analysis via computer simulation of motion it is necessary to design a control law including a proper control technique. For the case approached in [7], the dynamics analysis and control implemented by computer simulation is a huge and complicated problem formulation, including the complete set of non linear equations. For details refers do that paper.

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4 Conclusion This work presents a brief historic of the space exploration to contextualize the space robotics. The article approaches the space robotics subject and its associated applications for on-orbit and planetary activities. The discussion of orbital robots points the key issues to have robot efficiency when working in the microgravity environment. For planetary exploration, the article points the planets and other celestial body surfaces as the main problem that robot rovers must be able to overcome. This study presents the concept, applications, and operation of telerobots, pointing them as very important for the OOS and as strategic for colonization of Moon and Mars in a near future. The work also points the microgravity and planetary surfaces features as the main performance problems for robot tasks on-orbit and on the surface of planets. The paper presents some consideration about the huge radiations, high temperature, atmosphere, weather, and features of planets/moons surfaces and their associated effects on robots efficiency in performing tasks. The work discusses the classification of the robots according to their OOS and planetary exploration tasks, including asteroids and comet missions. In the context of orbital applications, the paper presents the RVD/B and OOS as the main orbital robot applications. In addition, it presents the important concept, description and data flow of telerobots. Finally, the article presents the most used approaches for mathematical modeling and presents some examples of two research papers published by the author of this article. Acknowledgements This work is partially supported Capes, INCT/CNPq, AEB, UFPR.

References 1. Arantes Jr, G (2011) Rendezvous with a Non-cooperating Target, PhD Thesis dissertation, International University of Bremen, IUB, Germany 2. Arantes Jr G, Rocco EM, Da Fonseca IM, Theil S (2010) Far and proximity maneuvers of a constellation of service satellites and autonomous pose estimation of customer satellite using machine vision. Acta Astronautica. https://doi.org/10.1016/j.actaastro.2009.11.022 3. Craig Johon J. (2005) Introduction to robotics Mechanics and control, 2005 Pearson Education, Inc. Pearson Prentice Hall Pearson Education, Inc. Upper Saddle River, NJ 07458c 4. Da Fonseca IM, Nacib Pontuschka M, Luiz Bertoze Lima G (2019) Kinematics for spacecrafttype robotic manipulators. Kinematics, IntertechOpen, https://www.intechopen.com/onlinefirst/kinematics-for-spacecraft-type-robotic-manipulators 5. Da Fonsecal IM et al (2015) Interaction between motions of robotic manipulator arms and the non-fixed base in on-orbit operations. J Aerosp Technol Manag (Online). https://doi.org/10. 5028/jatm.v7i4.458 6. Da Fonseca IM (1985) Anais VIII Congresso Brasileiro de Engenharia Mecânica, COBEM 85, Instituto Tecnológico de Aeronáutica, São José Dos Campos, SP, 10-13 de Dezembro de 1985 7. Da Fonseca IM, Goes LCS, Seito N, Da Silva Duarte MK, De Oliveira EJ (2017) Attitude dynamics and control of a spacecraft like a robotic manipulator when implementing on-orbit servicing. Acta Astronautica 137:490–497. https://doi.org/10.1016/j.actaastro.2016.12.020 8. En.wikipedia.org, Space Rendezvous and docking of spacecraft https://en.wikipedia.org/wiki/ Space_rendezvous#cite_note-2, July 2019

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9. Fehse W (2003) Automated rendezvous and docking of spacecraft. Cambridge University Press, Cambridge. ISBN: 9780511543388. https://doi.org/10.1017/cbo9780511543388 10. Gatland K (1976) Manned spacecraft, Second Revision. Macmillan Publishing Co., Inc., New York, pp 117–118. ISBN 0–02-542820-9 11. Hall Rex, Shayler David J (2001) The rocket men: Vostok & Voskhod, the first soviet manned spaceflights. Springer-Praxis Books, New York, pp 185–191 12. Skaar SB, Ruoff CF (1994) Teleoperation and robotics in space, progress in astronautics and aeronautics. Vol. 161, Published by the American Institute of Aeronautics and Astronautics, Inc 13. Wertz JR, Bell R (2003) Autonomous rendezvous and docking technologies status and prospects, SPIE AeroSense Symposium, Paper No. 5088–3 Space Systems Technology and Operations Conference, Orlando Florida, April 21–25 2003 14. Wie BS (2014) Vehicle dynamics and control. 2nd edn. American Institute of Aeronautics and Astronautics, Inc. 970 p

Stick-Slip Phenomenon: Experimental and Numerical Studies Ingrid Pires and Hans Ingo Weber

Abstract The stick-slip phenomenon is the most severe stage of the torsional oscillations present in most drilling routines. It results in drilling operations inefficiency and damages the drilling equipment. Stick-slip arises from the strong nonlinear interaction between drill strings and borehole formation. In this work, we study the torsional oscillations presented by slender structures like drill strings. First, we present a low dimensional model of the system used for the analysis. The model is used for numerical investigation. Then we describe the experimental setup used. The setup is capable of reproducing the torsional behavior experienced by real drilling systems. Finally, we investigate the influence of the nonlinear control parameters in the system dynamics employing the experimental data collected and the numerical simulations.

1 Introduction The excessive vibration of the drilling system leads to drilling operations inefficiency and damages the drilling equipment. Therefore, a good understanding of the system dynamics under different system conditions is necessary. The literature addressing drill string dynamics and experimental studies is vast [1–3]. Ideally, the entire drilling system should rotate at a constant speed. Although, due to the drill string slenderness, torsional vibration is present in most drilling operations. The stick-slip phenomenon is the critical stage of torsional vibrations when the nonlinear interactions cause a complete arrest of the drill bit until it is suddenly released. Other modes of vibration are present during drilling as axial and lateral oscillations. The uncoupling of these three modes of vibration is considered in some studies as a simplifying hypothesis. I. Pires (B) · H. I. Weber Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil e-mail: [email protected] H. I. Weber e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_10

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In this paper, we focus our attention on torsional vibration. This work utilizes a test rig designed to offer a torsional behavior comparable to the behavior presented by real drilling systems. The experimental rig has simple brake devices to introduce friction to the system, disturbing the rotating motion. The rig operates under different combinations of parameters, which permits us to experimentally observe different types of torsional responses. For the numerical investigation, we model the rig as an actuated torsional pendulum and the disturbance as nonlinear friction. This work aims to analyze the torsional behavior of the experimental system, investigating the influence of the nonlinear control parameters in the type of system response. We use, for this purpose, experimental and numerical bifurcations diagrams. The paper is divided as follows. Section 2 presents the dynamical model developed in [7]. In Sect. 3, we explain the experimental setup used to mimic drill string oscillations and present some experimental results. Section 4 concentrates on the numerical and experimental studies and the comparison between them. Lastly, in Sect. 5, we summarize the main conclusions of this work.

2 Physical and Mathematical Model The authors derived the physical and mathematical model of the three degrees-offreedom model in [7]. The model presented in Fig. 1 idealizes the mechanical subsystem as a torsional pendulum. Although simple, the torsional pendulum model can describe the stick-slip phenomenon well. The mechanical subsystem is composed of disc D2, and the shaft connecting it to the DC-motor. Disc D2 has a moment of inertia J2 . The shaft torsional stiffness is denoted by k2 , and the linear damping is denoted by d2 . The electric subsystem is modeled as a voltage source connected in series with a resistor and an inductor. Therefore, the governing differential equations of the model showed in Fig. 1 are [7]:

R

L

+ V

i

τm , θ˙m J2

e -

J3

θ˙3

k2

d2

θ˙2

Fig. 1 Schematic representation of the experimental setup with DC motor, disc, and shaft [7]

Stick-Slip Phenomenon: Experimental and Numerical Studies

J2 θ¨2 + d2 (θ˙2 − θ˙3 ) + k2 (θ2 − θ3 ) = −T f 2 , d2 (θ˙3 − θ˙2 ) + k2 (θ3 − θ2 ) = η(K T i − Cm η θ˙3 − T f − Jm η θ¨3 , di L + Ri + K E η θ˙3 = V, dt

173

(1)

here θ2 , θ˙2 , and θ¨2 are angular displacement, angular velocity, and angular acceleration of D2, respectively, and T f 2 is the resistive friction torque on disc D2. In Eq. (1), i denotes the DC-motor electric current, and L and R are the armature inductance and resistance, respectively. The angular velocity θ˙m is the velocity of the DC-motor inertia, Jm . Cm is the speed regulation; K T , the constant motor torque; K E , the voltage constant; and T f , the internal friction torque. The input voltage is t V = κ p (ωr e f − θ˙3 ) + κi 0 (ωr e f − θ˙3 )dt, where κ p and κi are proportional constant and integral constant, respectively, and ωr e f is the reference velocity of the system. Equation 1 constitutes a set of three coupled differential equations, along with the nonlinearity associated with the friction torque. The resistive friction torque, T f 2 , is explained in detail in [7]. In Sect. 4 we present the numerical results obtained with this dynamical model.

3 Experimental Rig This section presents concisely the experimental rig developed at Pontifícia Universidade Católica do Rio de Janeiro [5]. The rig was designed to reproduce the torsional oscillations experienced by real drill strings isolating it from the other modes of vibration. Figure 2 shows a picture of the test rig, which has been introduced in [5]. As we can see in Fig. 2 consists of a horizontal apparatus with an ENGEL GNM5480-G6.1 DC motor, two solid discs, D1 and D2, and two low-stiffness shafts,

Fig. 2 Experimental setup with DC-motor, two discs, D1 and D2, and two low-stiffness shafts [7]

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Fig. 3 Brake devices equipped with a load cells [7]

connecting the DC-motor to the discs and transmitting motion. There are two brake devices to induce friction torque, placed in discs D1 and D2 (Fig. 3). The friction torque leads the system to experience torsional vibrations and stick-slip phenomenon. In this work, we limit our analysis to a reduced system constituted of the DC-motor, the intermediary disc, and the flexible shaft connecting them. It is noteworthy that the drilling process is controlled by many parameters including Weight On Bit (WOB) and rotary speed [4]. Then it is important to work on a rig that operates in different conditions, enabling the study of how parameters changes influence the system response. In our experiments, we can vary either the normal force between the brake device and pin and the reference velocity imposed by the DC motor. Different sensors are used in the experimental setup, enabling us to measure the important parameters and variables of the system dynamics. These include three rotary encoders to measure the motor and discs speed, and two load cells to measure de normal force exerted by the brake devices. We run and record the experiments for various sets of system parameters (reference velocity and normal force). We use the LabVIEW based Data Acquisition System (DAQ) to observe real-time measurements and to save the data for post-processing. Figure 4 displays an example of time histories of a test for a certain combination of parameters where the existence of torsional vibrations with the stick-slip phenomenon is evident. The rig is driven from the motor with an angular reference velocity of 55 RPM, and the normal contact force applied to the disc is equal to 50 N . Graphs (a) and (b) in Fig. 4 depict the friction torque and the normal contact force. In graph (c), we can observe the oscillation of the disc angular velocity. We note from this last graph oscillations of a peak to peak amplitude of 110 RPM, twice the reference value. As expected, we see that the value of friction torque decreases with the increase in speed and increases with the decrease in speed [6].

Stick-Slip Phenomenon: Experimental and Numerical Studies

175

Fig. 4 Torsional oscillations occurring in the reduced experimental rig for ωr e f = 55 R P M and N2 = 50 N

4 Nonlinear Analysis In the previous section, we pointed out that the test rig may exhibit different dynamical behaviors depending on its parameters, as well as drilling systems. This section examines the numerical and experimental torsional vibrations experienced by the rig, with special attention to the stick-slip phenomenon. This severe stage of torsional oscillations exists for different system conditions; in other words, different combinations of the control parameters (normal contact force and reference angular velocity, N2 , and ωr e f , respectively). Moreover, the system exhibits different stickslip responses depending on the control parameters combination. To observe the influence of the reference angular velocity on the torsional vibrations of the rig, we varied it in the range from 5 to 50 RPM, in steps of 2.5 RPM. Then we plotted a bifurcation diagram to identify the type of response for different ωr e f . Figure 5 presents both experimental and numerical bifurcation diagrams.

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Fig. 5 Bifurcation diagram with respect to reference angular velocity, ωr e f , for N2 = 10 N

Fig. 6 Bifurcation diagram with respect to normal contact force, N2 , for ωr e f = 55 R P M

From the two bifurcation diagrams depicted in Fig. 5, we see that as ωr e f increases so do the amplitudes of stick-slip oscillations until it vanishes and we can only observe torsional oscillations. However, in the experimental results, we observe interchanging stick-slip and oscillatory solutions regions. In Fig. 5 one may notice that the stick-slip phenomenon happens for values of ωr e f until 47.5 R P M. We acquired the system responses displayed in Fig. 5 for the same value of the normal contact force between the brake device and the disc D2, N2 = 10 N . From Fig. 5 we can conclude that stick-slip phenomenon vanishes for high values of ωr e f . Similarly, we repeated the analysis to observe the influence of normal force, N2 , on the type of system response. We varied it from 0 to 50 N in steps of 2.5 N for ωr e f = 55 RPM. We used the same parameters of torque friction curve for all of the numerical simulations, as T f,2 depends on N2 [7]. In the previous section, we observed that N2 varies throughout the test (Fig. 4), although, for simplicity, we assume it to be constant here. Figure 6 presents the experimental and numerical bifurcation diagrams with respect to N2 . From Fig. 6, we notice that for the lower values of N2 there are torsional oscillations. As N2 increases so do the amplitudes of torsional oscillations and stick-slip phenomenon appears. It happens for both experimental and numerical tests. Also, one may notice that varying the normal contact force there are no interchanging solutions regions.

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5 Concluding Remarks In this contribution, we presented an analysis of the torsional behavior of an experimental rig. The test rig developed at Pontifícia Universidade Católica do Rio de Janeiro exhibits torsional dynamic behavior similar to real drilling systems. For the numerical studies, we mathematically model the rig as an actuated torsional pendulum. Hence, we investigated the influence of the control parameters, reference angular velocity, and normal contact force, on the type of response of the system. The analysis intended to identify in which conditions the stick-slip phenomenon is more likely to appear. From the results presented in this work, we conclude that the stick-slip phenomenon occurs for low values of reference velocity and high values of normal contact force. Future work will include developing new methods of control to suppress the stick-slip phenomenon.

References 1. Srivastava S, Teodoriu C (2019) An extensive review of laboratory scaled experimental setups for studying drill string vibrations and the way forward. J Petrol Sci Eng 106272 2. Patil PA, Teodoriu C (2013) A comparative review of modelling and controlling torsional vibrations and experimentation using laboratory setups. J Petrol Sci Eng 112:227–238 3. Wiercigroch M, Kapitaniak M, Vaziri V, Nandakumar K (2018) Complex dynamics of drillstrings: theory and experiments. In: MATEC Web of Conferences. VETOMAC XIV. Lisbon, Portugal 4. Bourgoyne AT Jr (1991) Applied drilling engineering. Society of Petroleum Engineers, Dallas, US 5. Cayres BC (2013) Numerical and experimental analysis of nonlinear torsional dynamics of a drilling system. Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brasil 6. Leine R, Nijmeijer H (2004) Dynamics and Bifurcations of non-smooth mechanical systems. In: Lecture notes in applied and computational mechanics. Springer, Wiley, Netherlands 7. Pires I, Cayres BC, Pamplona D, Weber HI (2019) Torsional friction-induced vibrations in slender rotating structures. In: Proceedings of the 15th IFToMM World Congress. Krakow, Poland

Determination of Wheel-Rail Interaction Forces of Railway Vehicles for Evaluation of Safety Against Derailment at Running on Twisted Tracks Ion Manea, Marius Ene, Ion Girnita, Gabi Prenta, and Radu Zglimbea

Abstract Locomotives must provide a high degree of security and safety in running on any kind of railways, including twisted tracks. Safety against derailment on twisted tracks is evaluated by the limit value of the ratio between transverse guiding force, Y and vertical wheel force, Q. The paper presents a combined method for safety evaluation against derailment on twisted tracks by performing tests on a twisted test stand and on a curved and flat test railway. On the stand with twisted track is measured the minimum forces, Qa,min on the wheels of vehicle. On a curved and flat test railway, with a curve of 150 m, are measured the guiding forces of the leading wheelset. The evaluation of safety against derailment is made by calculation of ratio, (Y/Q)a , for the outer wheel of the leading wheelset. The paper presents the LE Measuring Wheelset designed for electric locomotives and calibrated for simultaneous measurements of the wheel-rail interaction forces in the vertical and transverse directions. Are also presented the tests performed by RRA Bucharest for certification, according to European Technical Specifications for Interoperability, of the LEMA 6000 kW electric locomotive, produced by Softronic Craiova-Romania, from the point of view of the safety against derailment at running on twisted tracks. Keywords Railway vehicles · Wheel-rail interaction forces · Twisted tracks

1 Introduction Locomotives are complex traction railway vehicles with an important role in ensuring of freight and passenger transport. Given the operating conditions, the huge price of the locomotive itself and the importance of carried load (including operating personnel and passengers), as well as the incalculable consequences of a possible railway accident, locomotives must provide a high degree of security and safety in I. Manea (B) · I. Girnita · G. Prenta · R. Zglimbea SC Softronic SRL, Calea Severinului, No. 40, Craiova, Dolj, Romania e-mail: [email protected] M. Ene Romanian Railway Authority-RRA, Calea Grivitei, No. 393, sector 1, Bucharest, Romania © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_11

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running on any kind of railway, including twisted tracks. Demonstration of the degree of safety in running is mandatory for all new or renewed railway vehicles, before putting into circulation [1]. Harmonization of the railway traffic in European space is regulated by the Technical Specifications for Interoperability (TSI) which establishes the relations between the essential requirements applicable to the railway transport systems and the European standards. Thus, all new railway vehicles must comply with TSI requirements starting with June 2019 [2]. The existence of a high track twist in the railway network is possible and they are the result of transition layout between leveled track and canted track as well as cross-level deviations. The degree of safety against derailment is described by the limit value of the ratio (Y/Q) between transverse guiding forces, Y, and vertical wheel forces, Q. The limit value of the ratio (Y/Q) for initiating flange climbing is influenced by flange angle and the friction forces between flange and rail. The EN 14,363:2016 standard regulates the procedures and levels for certification by tests regarding to the safety against derailment for railway vehicles running on the twisted tracks. For universal applications, the EN 14,363:2016 standard regulates assessing of the safety against derailment for railway vehicles running on twisted track through two methods: 1. Tests on twisted track: The evaluation of safety against derailment is carried out by measuring vertical, Q, and transverse, Y, forces when vehicle running on the twisted track. 2. Tests on a twisted stand and on a curved and flat track. The evaluation is carried out by: – measurement of the minimum wheel force of the vehicle on the stand with twisted track; – measurement of the guiding forces on a curved and flat railway consists of a tangent followed by curve with a radius of curvature of 150 m, without transition, twist, and cant; – calculation of the ratio, (Y/Q)a , for the outer wheel of the wheelset in the leading position. The paper presents a measuring wheelset, designed for electric locomotives and calibrated for the simultaneous measurement of the wheel-rail interaction forces, Y and Q. Also, the article presents the tests performed on the test stand with twisted track and in the current curved and flat railway with the curvature radius, R = 150 m. The tests have been carried out by RRA Bucharest on the electric locomotive LEMA 6000 kW produced by Softronic Craiova-Romania, in view of TSI certification for international traffic admission in the European railway network.

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2 Tests on Curved and Flat Railway 2.1 LE Measuring Wheelset for Measurement of Wheel-Rail Interaction Forces Measurements of the wheel and rail interaction forces using a full-disk wheel is a very difficult problem because it is practically impossible to find areas where can be possible to separate the two types of forces, Y and Q. To avoid this inconvenient, it was realized the LE Measuring Wheelset with spokes to measure the interaction forces with rail of the electric locomotive LEMA 6000 kW. The spokes of the measuring wheelset were equipped with strain gauges, in full-bridge configuration, being transformed into measuring transducers for Y and Q forces, and were calibrated as measuring transducers as such. Each wheel is equipped with 24 spokes, of which 12 are designed as transducers for transverse forces, Y, and 12 as transducers for vertical forces, Q. The spokes for measurement of Y forces have an ellipticalshapes with the major axis oriented perpendicular to the wheel plane. The spokes for measurement of Q forces are thinner and have a round-shapes. To select the place for the strain gauges application on the spokes it was used the ANSYS finite elements program. To increase the sensitivity of the transducers so made, the successive spokes Q were connected in antiphase and the successive Y spokes were connected in phase. In this way, when running a full wheel length under constant vertical and horizontal loads, the voltage signal supplied by the vertical spokes has an almost sinusoidal shape, and the voltage signal provided by the transverse spokes has an almost constant shape. To be used as transducer for measurements of independent Y and Q forces, the LE Measuring Wheelset was calibrated on a stand that allows the successive and independent application of the vertical and transverse forces on each spoke. Loading of the spokes was made with a 150 kN hydraulic cylinder and the force was measured with a calibrated cell type U2B-200 kN. Figure 1 shows the calibration stand with LE Measuring Wheelset installed for the application of calibration procedure. Each spoke behaves like a linear force transducer, Fig. 2 representing the force characteristic of a spoke, the same kind of calibration charts being obtained for all spokes. The interpolation factors, A and B, were determined for each spoke, in Table 1 being represented the interpolation factors for all spokes of the LE Measuring Wheelset. It can be seen that spokes have no identical characteristics, requiring the measurement of the angle of each spoke, taking as reference a given landmark. For measurement of the spokes angle, an angle transducer was mounted on the axis of the measuring wheelset and as a reference, was taken the vertical line passing through the axis of the wheelset (Figs. 3 and 4). Taking into account the particularities of the system for measurement of Y and Q forces, results that the measurement of wheel-rail interaction forces cannot be done in real-time, but only by post-processing of the acquired data.

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150000

F (N)

Fig. 1 Calibration stand with LE Measuring Wheelset installed for the calibration procedure

Calibration chart. W1. Sp1 y = -600753x + 381943

100000

50000

0 0.4

Output signal (V) 0.45

0.5

0.55

0.6

Fig. 2 Calibration chart of spoke 1 from wheel 1 of LE Measuring Wheelset

The tests for measurement of the Y and Q forces involve the analogical measurement and recording of the following parameters: 4 voltage signals from the strain gauges and 1 analog voltage signal from the angle transducer. Data acquired over the entire duration of the tests are stored in the numerical data files for subsequent post-processing. Under TestPoint software, it has developed a program that performs the following tasks: – When the recorded signals are passed, are determined the maximums and minimums of the voltage signals recorded from the vertical spokes, for both wheels W1 and W2; – At each determined maximum/minimum, reads the angle and, using data from Table 1, determines which spoke is interacting with the track. Reads the interpolation parameters A and B for both vertical and horizontal spokes, those closest to the angle read.

46.70

15/16

165.60

14.60

13/14

23/24

345.00

11/12

137.20

315.30

9/10

21/22

284.10

7/8

75.70

255.40

5/6

107.10

225.70

3/4

19/20

196.90

1/2

17/18

Angle (degree)

Spoke

423.75

−377.25

592.15

414.36

453.92

−695.43

−713.54

400.23

−414.92

651.29 407.71

400.50

465.00 403.57

400.34

−440.39

686.15

−726.17 442.85

402.49

369.16

−578.03

−414.82

408.38

650.88

419.51

483.98

−760.08

−454.24

426.41

711.48

428.67

382.65

−415.72

−601.90

AW1_OT (kN/V)

652.41

BW1_V (kN)

AW1_V (kN/V)

231.72

226.65

223.00

220.76

218.97

219.06

218.96

220.08

223.43

229.49

233.31

234.48

BW1_OT (kN)

Table 1 Linear interpolation factors, A and B, of the LE Measuring Wheelset

−593.64

662.65

−752.67

705.22

−704.27

689.19

−725.56

664.71

−603.78

635.83

−541.60

678.13

AW2_V (kN/V)

−102.84

115.55

−131.24

123.56

−123.27

120.05

−126.49

114.84

−104.09

109.54

−92.72

116.67

BW2_V (kN)

438.08

429.91

422.61

404.01

381.14

369.19

375.19

402.06

431.80

441.95

441.38

440.73

AW2_OT (kN/V)

19.40

19.09

18.78

17.97

16.95

16.46

16.74

17.94

19.17

19.54

19.43

19.49

BW2_OT (kN)

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Interpolation factors for vertical spokes 1000

AW1_V(kN/V)

BW1_V(kN)

500 0 -500 -1000

1

3

5

7

9

11

13

15

17

19

21

23

Fig. 3 Interpolation factors for the vertical spokes of LE Measuring Wheelset, wheel 1

Interpolation factors for transverse spokes 600

AW1_OT(kN/V)

BW1_OT(kN)

400

200

0

1

3

5

7

9

11

13

15

17

19

21

23

Fig. 4 Interpolation factors for the transverse spokes of LE Measuring Wheelset, wheel 1

– By linear interpolation, it is determined the value of the interaction forces Y and Q. The particularity of the measurements with wheels with spokes is that from continuous measurements result in discrete values of forces when the wheels passing over the track. The main characteristics of the LE Measuring Wheelset: gauge, 1.435 m; diameter, 1.25 m; equivalent conicity, tanγe = 0.49; measurement error, 4%.

2.2 Device for Measurement of the Flange Angle The flange angle between the track and outer wheel of the leading wheelset is very important in the assessment of the safety against derailment at running on the twisted track.

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WA 200

Displacement Transducer WA 200 L=1.163 m

A

D

O

B

α

Fig. 5 Device for measurement of the flange angle

For measurements of the flange angle, has been achieved a special device consisting of a frame which covers the outer wheel of the LE Measuring Wheelset, in the leading position, which measures the angle between the tangent to the track in the contact point and wheel. Figure 5 shows the device for measuring the flange angle. In the front, the frame is caught by a slide on a horizontal rod that is fixed by the track cleaner, mounted, in turn, on the front of the bogie frame. Parallel, and in solidarity with the horizontal sliding rod, is mounted an inductive WA200 displacement transducer, with a measuring range of 200 mm. The mobile part of the transducer WA200 is firmly fixed to the front of the frame, while the housing of WA200 is firmly fixed to the track cleaner. In this way, the WA200 displacement transducer measures the distance from the center of the slide to the reference position of the track considered in alignment. The rear end of the frame slides through a roller on the track. The frame slides over the wheel, and at a wheel rotation with an angle α, around the contact point with the track, the front end of the frame moves left–right with a distance D from a hypothetical position of the wheel in alignment. The frame can move front-back keeping to the constant the OA cathetus.

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2.3 Software and Measuring Equipment During the tests, were used the following equipment and software: – LE Measuring Wheelset, gauge: 1.435 m, diameter: 1.250 m, equivalent conicity: 0.49; – Angle transducer type Ri360P1, measuring range: 0° ÷ 360°, Iout : 4 ÷ 20 mA; – Displacement transducer type WA200, measuring range: 200 mm; – Locomotive Speed Measuring and Recording Equipment, type IVMS; – Data acquisition system type LAN-XI 3053 B-120, 12 input channels, 24 kHz, 24bit; – Laptop, Dell Precision M6800: 32 GB RAM, Processor i7-4930MX; – Pulse Labshop, Bruel and Kjaer, real-time data acquisition software; – Pulse Reflex, Bruel and Kjaer, post-processing software. – TestPoint, post-processing software.

2.4 Tests on Curved and Flat Railway A number of critical problems appear during the measurement of the Y and Q forces: the LE Measuring Wheelset is in rotation motion; the area where the strain gauges are positioned is heavily electromagnetically disturbed; the space between the axle of the LE Measuring Wheelset and the traction motor is very limited and does not allow positioning of the wireless transmission modules, etc. Due to these problems, it was realized two devices containing each two galvanic isolated modules type MB 38 for strain gauge in full-bridge configuration, powered by supercapacitor battery. Both modules are mounted on a wheel and supply the measuring full bridge strain gauge configurations mounted on the wheels, being an interface that provides both supplying of strain gauge and protection against electromagnetic disturbances. To perform the tests on the curved and flat railway, in view to assess the safety against derailment when vehicle running on twisted tracks, it was chosen a railway that complies with the standard imposed conditions. The test railway must consist of an alignment followed by a curve, in the horizontal plane, with a radius of curvature of 150 m. The railway shall reflect the normal condition of typical railway considering the rail profile, track gauge and state of maintenance. A railway that complies with the mentioned conditions is the access rail to the Locomotive Travelers Depot in Northern Railway Station – Bucharest. The tests were carried out on the electric locomotive LEMA 6000 kW series 91 53 0 480,040–1, equipped in the leading position with the LE Measuring Wheelset, Figs. 6 and 7. The direction of running and direction of curvature were chosen in such a way that the wheel with the lowest wheel load (see part 3) of the leading wheelset to be in

Determination of Wheel-Rail Interaction Forces of Railway Vehicles …

Fig. 6 LEMA 6000 kW series 91 53 0 480,040–1 ready for tests on curved and flat railway

Fig. 7 Detail for LE Measuring Wheelset and device for measurement of the flange angle

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Fig. 8 Time history of characteristic parameters for tests on curved and flat railway

the outer part of the curve. During the tests the tested locomotive was not in traction nor braking, being pushed by a diesel locomotive with a constant speed of 5 km/h. During the tests, were measured and calculated the following parameters (see Fig. 8): – – – – –

Guiding forces on wheels 1/2 (black and blue traces): Y1/2 (N); Vertical forces on wheels 1/2 (red and green traces): Q1/2 (N); Locomotive speed: Speed (km/h); Flange angle of the outer wheel (magenta trace): Alfa (degrees); Angle of the LE Measuring Wheelset with vertical direction: Angle (degrees).

Figure 8 shows the time history of characteristic parameters for tests on the curved and flat railway. The bottom part displays show the values of the parameters at the moment selected by the two cursors. The right part displays show the average values of parameters between the two cursors. The tests were performed both for the situation when the driving cab 1 was in front and the driving cab 2 was in front. For both situations the device for measurement of the flange angle was installed on the outer wheel of the leading wheelset. Every time the tests were carried out three times, in order to statistically processing of the measured data. For the full curve circulation, the average and maximum values of the characteristic parameters were determined. Table 2 presents the means of average values and maximum values of characteristic parameters determined for running on the curved and flat railway.

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Table 2 Characteristic parameters for running on curved and flat railway Type

Q_W1out(N) Y_W1out(N) Q_W1inn(N) Y_W1inn(N)

Driving cab 1 in Average values front Max. values Type

132,231.2

51,387.97

95,617.82

144,211.2

64,505.34

106,822.6

36,296.65

Q_W6out(N)

Y_W6out(N)

Q_W6inn(N)

Y_W6inn(N)

62,050.55

95,338.01

34,772.78

78,231.8

106,243.3

36,998.23

Driving cab 2 in Average Values 132,634.7 front Max. values 144,698

32,899.44

3 Tests on Stand with Twisted Tracks To evaluate safety against derailment, it is necessary to know the minimal vertical wheel force during the negotiation of the twisted track. This may be determined on an appropriate test stand which is able to simulate the track twist corresponding to the simultaneous occurrence of the twist on a bogie wheelbase as well as the twist on bogie center distance. To determine the locomotive specific parameters in view to evaluation of safety against derailment at running on the twisted tracks, for CO–CO type locomotive, the test stand must allows that at least the supports of the two wheelsets of one bogie may be lifted and lowered to simulate the imposed twist track. The LEMA 6000 kW electric locomotive is a CO–CO type, having two bogies, each with three wheelsets, with the following characteristics: – – – –

total weight:1260.7 kN; wheelset load: 210.12 kN; bogie wheelbase: 2a + = 4.35 m; bogie center distance: 2a ∗ = 10.6 m.

For LEMA 6000 kW locomotive the twists corresponding to bogie wheelbase, 2a + , and to the bogie center distance, 2a ∗ , are given by the following relationships: For the bogie wheelbase: g + = 1.1 ∗



   15 15 + 2 = 5.993o/oo = 26.07 mm + 2 = 1.1 ∗ 2a + 4.35

(1)

where: g + (o/oo ), represents the value of the limit twist, amplified by 10%; 2a + = 4.35 m, represents the bogie wheelbase. For bogie center distance: g∗ =

15 15 + 2 = 3.415o/oo = 36.2 mm +2= 2a ∗ 10.6

where: g ∗ (o/oo ), represents the value of the limit twist;

(2)

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2a ∗ = 10.6 m, represents the bogie center distance. During the tests on the stand with twisted tracks, the zjk displacements of the wheels and the corresponding vertical load, Qjk , of each wheel must be measured. The purpose of the tests is to determine the following parameters: – Q0,j , the average wheel forces of each wheelset for rails in alignment (twist g = 0); – Qjk , the deviation of wheel forces from Q0 due to the combined bogie and body twists; – Qjk, min , the minimum wheel force due to the combined bogie and body twists. The tests on stand with twisted tracks were carried out within the Softronic Craiova, on the certificated locomotive scale equipped with devices for simulating the twisted tracks. The devices for simulating the twisted tracks were designed to simulate all possible combinations of the bogies and body twists, Fig. 9 representing the test sequences. For calculation of the characteristic parameters of the tests, special devices for controlled and independent lifting of the locomotive wheels were made, under each wheel being introduced calibrated gauges, with different heights, to satisfy all the twist conditions of the bogie and body. The height of the gauges was calculated in such a way that, with one bogie suspended horizontally on the track, all the twist conditions, corresponding to the all combination twists can be achieved at the second bogie. With reference to the Softronic LEMA 6000 kW electric locomotive, it is assumed that are made the twists of the track under the wheels of wheelsets 1, 2 and 3 of bogie I, while the wheels of wheelsets 4, 5 and 6 of bogie II remain in a horizontal plane. First, apply the twist corresponding to the bogie center distance 2a ∗ by lifting the wheels W11, W21, W31 and lowering the wheels W12, W22, W32, until the testing twist of the body, g ∗ , is achieved. Further, from this position, an additional twist is applied, corresponding to the bogie wheelbase,2a + , by simultaneous handling the wheels W11 and W32 (lifting) and wheels W12 and W31 (lowering), until the testing twist of the bogie, g + , is achieved, with an increase of 10%. With this sequence, the minimum load on wheels Qmin appears on wheel W12. In order to obtain a closed hysteresis diagram, the twist on the bogie wheelbase is reduced to zero, then the twist on the bogie center distance is reduced to zero. For determination of the minimum force on the wheel W11, they shall proceed in the same manner. Table 1 gives the values of the vertical heights of the wheels (relative to the upper surface of the rail) during the tests that combine both the bogies and body twists. Figure 10 presents the sketch of a calibrated gauge, for gauge with a height of 36 mm. Figure 11 presents the mounting sketch of the calibrated gauges under a wheelset, during the tests on stand with twisted tracks. Figure 12 shows the LEMA 6000 kW locomotive during the tests on stand with twisted tracks, and Fig. 13 shows a detail of mounting the calibrated gauges under wheels. The supports for the calibrated gauges

Determination of Wheel-Rail Interaction Forces of Railway Vehicles …

41 1 W41

W51 W61 W 61 W 51 1 W42 W62 W52

31

21

W21 W31 W21 1 W31 1 W12 W32 W22

11

W11 1

191

I

21

11

31 32

22

12

31

21

11

32

22

12

II

III

31

21

11

31

21

32

22

12 11

32

22

12

IV

V 32

22

12

Fig. 9 Test sequence that combines the bogie and body twists

8

Calibrated gauge 36 + 26 mm 47

62

15

15

8

15

Fig. 10 Sketch of a calibrated gauge

are sized so as not influence the measurement error of the locomotive scale. For each test configuration, described in Table 3 by, zjk displacements, the vertical wheel load, Qjk , is measured. The vertical wheel force on level track for the tested wheelset is calculated as:

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Calibrated gauge

B

1346

52

52

1450

B = 26; 39; 52; 57; 62; 75; 88 mm

Fig. 11 Mounting sketch of calibrated gauges for tests on stand with twisted tracks

Q 0, jk =

Q 0, jk,min + Q 0, jk,max 2

(3)

where: Q 0, jk,min , is the minimum wheel force at g ∗ = 0 and g + = 0 within hysteresis diagram; Q 0, jk,max , is the maximum wheel force at g ∗ = 0 and g + = 0 within hysteresis diagram. Mean wheel force for the tested wheelset is calculated with relation: Q 0, j =

Q 0, j1 + Q 0, j2 2

(4)

Wheel force deviation Q jk , due to tested effects is evaluated from the loading diagram at the twist (g ∗ + g + ). The minimum wheel forces are calculated through the relationship: Q jk,min = Q 0, jk − Q jk

(5)

The tests on the stand with twisted tracks were performed for both bogies, aiming to the determination of the minimum wheel load of the extreme, front and rear, wheelsets. Table 4 presents the minimum values of vertical wheel forces for both wheels of extreme wheelsets.

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Fig. 12 Electric locomotive LEMA 6000 kW during the tests on stand with twisted tracks

4 Evaluation of the Safety Against Derailment at Running on Twisted Tracks The degree of safety against derailment at running on twisted tracks is described by the maximum value of ratio (Y/Q) between transverse guiding force, Y and vertical wheel force, Q. Calculation of the maximum value of the ratio, (Y/Q)a,max is done for the wheel of the leading wheelset located on the outside part of the curve. Analysis of safety against derailment is done for both extreme wheelsets 1 and 6, corresponding to the cases when the locomotive goes with the driving cabs 1 or 2 in the front position. The following formula is evaluated for each tested wheelset:

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Fig. 13 Mounting of the calibrated gauges under wheels during the tests on stand with twisted tracks Table 3 Vertical displacements of LEMA 6000 kW wheels for tests on stand with twisted tracks Ord z_ W11 z_ W21 z_ W31 z_ W12 z_ W22 z_ W32 z_ W41−z_ W62 (mm) (mm) (mm) (mm) (mm) (mm) (mm) 1

31

31

31

31

31

31

31

2

49

49

49

13

13

13

31

3

62

49

36

0

13

26

31

4

49

49

49

13

13

13

31

5

36

49

62

26

13

0

31

6

49

49

49

13

13

13

31

7

31

31

31

31

31

31

31

8

13

13

13

49

49

49

31

9

0

13

26

62

49

36

31

10

13

13

13

49

49

49

31

11

26

13

0

36

49

62

31

12

13

13

13

49

49

49

31

13

31

31

31

31

31

31

31

Determination of Wheel-Rail Interaction Forces of Railway Vehicles …

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Table 4 The minimum wheel forces determined during the tests on twist test rig Q 0,1 le f t

Wheel force on level track for the wheel W1left

104.25 kN

Q 0,1 right

Wheel force on level track for the wheel W1right

101.6 kN

Q 0,6 le f t

Wheel force on level track for the wheel W6left

106 kN

Q 0,6 right

Wheel force on level track for the wheel W6right

102.3 kN

Q 0,1 right/le f t

Mean wheel force on W1right and W1left

102.925 kN

Q 0,6 right/le f t

Mean wheel force on W6right and W6left

104.15 kN

Q 1le f t,min

Minimum wheel force on W1left

3.025 kN

Q 1right,min

Minimum wheel force on W1right

3.275 kN

Q 6le f t,min

Minimum wheel force on W6left

9.05 kN

Q 6right,min

Minimum wheel force on W6right

9.75 kN

Note left and right refers to the position of the wheels when looking to the driving cab I



Y Q

 = ja

Y ja,med Q jk,min + Q j H

(6)

where: Y ja,med , is the quasistatic guiding force evaluated by the tests described in Sect. 2; Q jk,min , is the minimum wheel force evaluated by the tests described in Sect. 3; Q j H , is the change of wheel force due to the moment of the sum of guiding forces:  h  Q j H = Y ja + Y ji 2b A

(7)

h = 0.66m, is the effective height above the rail of the primary lateral suspension; 2b A = 1.5m, is the lateral distance of wheel-rail contact points. Above, the index “a” appoints the outer wheel and the index “i” appoints the inner wheel. The locomotive is considered to be safe against derailment at running on twisted tracks if the following condition is fulfilled for both wheelsets in leading position: 

Y Q



 ≤

a,max

Y Q

 (8) lim

  According to EN 14,363:2016 the recommended limit value for the ratio 

Y Q

Y Q

is:

 = 1.2

(9)

lim

Taking into account the values of the characteristic parameters determined by tests on the stans with twist tracks and on the curved and flat railway, presented in

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  Sects. 2 and 3, the values of the ratio

Y Q

a,max

for the outer wheel of wheelsets 1

and 6 are determined: 

   Y12,med Y Y 51387.97 = = = = 1.085 Q 1a,max Q 12 3025 + (64505.33 + 36296.64) 0.66 Q 12,min + (Y12 + Y11 ) 2bh 1.5 A     Y62,med Y Y 62050.55 = = = = 1.038 Q 6a,max Q 62 9050 + (78231.80 + 36998.23) 0.66 Q 62,min + (Y62 + Y61 ) 2bh 1.5

(10) (11)

A

Results that the electric locomotive LEMA 6000 kW, from the point of view of safe against derailment at running on twisted tracks, corresponds to the requirements of the EN 14,363:2016 standard and can, therefore, be admitted for traffic in European railways.

5 Conclusions 5.1 Wheel-rail interaction forces of railway vehicles is an important parameter for evaluation of safety against derailment at running on twisted tracks. The LE Measuring Wheelset is a good tool for simultaneous and independent measurement of transverse, Y, and vertical, Q, interaction forces. The main characteristics of the LE Measuring Wheelset are: gauge, 1.435 m; diameter, 1.25 m; equivalent conicity, tanγe =0.49; measurement error, 4%. 5.2 The article presents the tests performed by Romanian Railway Authority-RRA with the Softronic electric locomotive LEMA 6000 kW on the twisted stand and on the curved and flat railway in view to evaluating the safety against derailments at running on the twisted tracks. Following the performed tests, it follows that the LEMA 6000 kW electric locomotive meets the requirements of the European Technical Specifications for Interoperability and can be admitted in the European railway network.

References 1. EN 14363: 2016, Railway applications. Testing and Simulation for the acceptance of running characteristics of railway vehicles. Running Behaviour and stationary tests 2. Directive 2016/797 of the European Parliament on the interoperability of the rail system within the European Union, Official Journal of the European Union 3. UIC 518:2009, Testing and approval of railway vehicles from the point of view of their dynamic behavior—Safety—Rail fatigue—Running behavior 4. Manea I, Sebesan I, Ene M, Matache M, Arsene S (2017) System for measurement of interaction forces between wheel and rail for railway vehicles. MATEC Web Conf 137:01006. https://doi. org/10.1051/matecconf/201713701006

A Small-Scale Dynamometer Roller Analysis by Laval Rotor Approach Maria Augusta M. Lourenço, Fabricio L. Silva, Ludmila C. A. Silva, Jony J. Eckert, and Fernanda C. Corrêa

Abstract In the transport industry scenario, evaluate vehicle performance is essential to development. This assessment is frequently made in roller dynamometers, which enables reliable and controlled experiments. To simplify the development process, a small-scale dynamometer has been developed and allows reduced cost tests and less time spent. Therefore, this work analyses the roller behavior, one of its essential pieces, approaching it by a Laval rotor and evaluating the displacements of its shaft centers. The results presented that, under the working conditions, the values of the center displacements are very low and can be disregarded and the rollers do not reach the resonance. Keywords Laval rotor · Chassis dynamometer · Small-scale vehicles · Roller

1 Introduction Nowadays, the automotive industry aims the development of sustainable mobility solutions, due to the market competition and more stringent environmental regulations [1, 2]. Therefore, the concept design of vehicles became a crucial step in vehicle development [1]. One alternative to performing an early evaluation of the new vehicle concept is the use of small-scale prototypes, aiming to facilitate the controller development, identify possible issues and avoiding the high cost of the real scale prototype [3].

M. A. M. Lourenço (B) · F. L. Silva · L. C. A. Silva · J. J. Eckert Integrated Systems Laboratory (LabSIn), Integrated Systems Department (DSI), University of Campinas (UNICAMP), 200 Mendeleyev Street, Campinas 13083970, SP, Brazil e-mail: [email protected] F. C. Corrêa Electrical Engineering Department, Federal Technological University of Paraná (UTFPR), Monteiro Lobato Avenue, Jardim Carvalho Ponta Grossa 84016210, PR, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_12

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Fig. 1 Integrated system laboratory—LabSIn chassis dynamometer [10]

Besides the cost advantages of producing small-scale vehicle prototypes, also its performance needs to be evaluated to ensure that it represents a similar vehicle on a real scale. Usually, chassis dynamometers are used to measuring the vehicle traction force, torque and power, enabling the evaluation of the vehicle performance [4], using rollers inertia and actuators that emulate loads as aerodynamic drag and climbing resistance. Such equipment is used by the automotive industry, especially regarding vehicle fuel consumption and generated tailpipe emissions, providing controlled and reliable experimental conditions, which can be replicable ensuring the reproducibility of the performed experiments [5]. The dynamometer inertia and its actuators absorb the vehicle output power, converting it in kinetic energy and/or electricity or heat, that is dissipated [6–8]. One example of a full-scale dynamometer with controlled actuators is the LabSIn bench shown in Fig. 1. However, these full-size vehicle testing facilities are very expensive, regarding its installation, operation, and maintenance [9]. In this work, we present a small-scale dynamometer, as shown in Fig. 2, designed to evaluate vehicle prototypes up to 17 kg, propelled in the 4 × 2 or 4 × 4 drivetrain configurations. As in the real scale dynamometer, the small one also has 8 rollers, that convert the kinetic energy of the vehicle mass displacement in equivalent inertia rotational energy. Therefore, each vehicle wheel is supported by 2 rollers as shown in Fig. 2. This double tire-roller contact is a common assemble in a dynamometric bench, including small-scale dynamometers as the one developed by Fajri et al. [11] to test a small-scale hybrid electric vehicle. Once the inertial rollers are essential components of the developed small-scale dynamometer, this work aims to evaluate its behavior by a Laval rotor approach for its shaft and roller, analysing the displacements of its center for the working conditions.

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199

Fig. 2 The 3D model of the small-scale dynamometer with a vehicle prototype

2 Methodology As seen in Fig. 3, in Laval rotor analysis, the rotor is placed in the center of a bisupported shaft and it is unbalanced. Its gravity center is illustratively indicated by G according to the disk geometric center (u y , u z ), where e is the eccentricity and θ is its unbalance angle [12]. According to Krämer [12], the differential equations (1)–(3) describe the Laval rotor behavior, where T [Nm] is the torque applied to the system, J [kg m2 ] is the moment of polar inertia (roller and shaft), m [kg] is the mass (roller and shaft), K [N/m] is the stiffness of the shaft and c [Ns/m] is the damping coefficient (10−4 K). The unbalance e [m] estimated is 10−3 /m and the gravity acceleration g considered is 9.81 m/s2 . Jw [kg m2 ] is the inertia of the wheel in contact with the roller.

Fig. 3 Laval rotor and its projection on the y-z plane

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(J + 0.5Jw )θ¨ = T − TR + K u z (e cos θ ) − K u y (e sin θ) − cu˙ y (e sin θ) + cu˙ z (e cos θ)

(1)   m u¨ y + cu˙ y + K u y = me θ¨ sin θ + θ˙ 2 cos θ − N y + FT y − R y

(2)

  m u¨ z + cu˙ z + K u z = me θ¨ cos θ + θ˙ 2 sin θ − mg − Nz − FT z + Rz

(3)

As seen in the Eqs. (1)–(3), for the dynamometer analysis, forces, torques and inertias where added at the original Laval rotor model to represent the specific application. The mechanical efforts, detailed in the following sections, are the roller tractive force FT [N], the rolling resistance force R [N], roller torque TR [N] and the vehicle weight, represented by the normal force N [N]. The differential equations (1)–(3) were solved using the ode45 function from MatLab™software. For the first iteration, the system was considered starting from rest, where the initial conditions were considered null.

2.1 Roller Inertia As mentioned before, the determination of the roller inertia is fundamental to ensure an accurate representation of the vehicle displacement in the dynamometer. The first step is to calculate the required inertia Ir eq [kg m2 ] that meets the longitudinal translational kinetic energy of the vehicle mass m veh [kg] according to vehicle velocity V [m/s] and the angular rotation of the roller θ˙ [rad/s] as shown in Eq. (4). Notwithstanding, it is important to mention that the remaining vehicle rotational components presented in the vehicle powertrain does not need to be included in the roller inertia, once their dynamic behavior remains unaltered in the vehicle operation. 1 1 m veh V 2 = Ir eq θ˙ 2 2 2

(4)

Disregarding the small contribution of the tire slip in the contact, which is usually lower than 2% [4, 13], the roller speed θ˙ can be defined by Eq. (5) as a function of the roller radius rr [m] and V . V θ˙ = (5) rr Therefore, the value of Ir eq is defined by the combination of Eqs. (4) and (5), as shown in Eq. (6). (6) Ir eq = m veh rr2 Once the proposed bench configuration has two rollers in contact on each tire, the resulting Ir eq value must be divided among the 8 rollers resulting in the inertia of each roller shaft. Due to a limitation in the axes distance between each pair of rollers,

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that cannot be larger than 80 mm, the maximum value of rr is fixed as 35 mm and its inertia value is defined by the roller length L r [m]. The inertia of each roller shaft Is [kg m2 ] must be subtracted from the Ir eq value, in order to define the length of the rollers L r [m]. The shaft radius rs [m] also is considered in the calculation of the cylinder mass m r [kg] (Eq. 7) and its inertia Ir [kg m2 ] (Eq. 8), according to the density of the material used in the roller ρr [kg/m3 ].   m r = ρr L r π rr2 − rs2 Ir =

(7)

 1  2 m c rr + rs2 2

(8)

Combining the Eqs. 7 and 8, it is possible to define the length of each roller L r as shown in Eq. 9.   I 2 r8eq − Is   Lr = (9) ρr π rr4 − rs4

2.2 Case Study In this paper, a 10 kg vehicle is used to perform the analyses. Table 1 presents the resulting values of mass, inertia, and geometry of the dynamometer rollers, its respective shafts, and the vehicle prototype features. The Laval disk is the bench roller and the supports are rigid bearings, as shown in Fig. 4. The traction force at the roller FT is the input operating torque T divided by the roller radius rr (Eq. 10). The T torque is defined according to the transmission

Table 1 Dynamometric and vehicle parameters Shaft Radius rs Length Mass Inertia Is Young module Roller Rollers center distance Radius rr Length L r Mass m r Inertia Ir Density ρr Vehicle prototype Mass m veh Wheel inertia Jw

8 mm 150 mm 0.435 kg 1.309 ×10−4 kg m2 200 GPa 80 mm 35 mm 82 mm 2.35 kg 1.515 ×10−3 kg m2 7860 kg/m3 10 kg 2.119 ×10−3 kg m2

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Fig. 4 A simplified view of the rollers and its forces

ratio Nt of the gears that connect each wheel to its respective electric motor, as seen in Eq. (11). Moreover, the electric motor torque TE M is defined according to the curve shown in Fig. 5, which varies as a function of the electric motor speed θ˙E M (Eq. 12). It is important to mention that the tire contact was considered as a rigid body to facilitate the development, but for better precision, the effects of tire deformation in the future must be considered and the particularities of this type of contact. TE M N t rr

(10)

T = TE M N t

(11)

θ˙E M = θ˙ Nt

(12)

FT =

The analysed vehicle prototype is propelled by four 5 W electric motors, each one coupled to its receptive wheel by a gear reducer of Nt = 2. Once the maximum torque is defined as 1.1 Nm, the resulting torque curve (Fig. 5) is generated according to the procedure described in [14, 15]. The rolling resistance R was calculated based on the equation presented by Genta and Morello [16], which varies according to the vehicle weight and compensated the scale of the vehicle. Still, Eckert et al. [4], through experimental evaluations, highlight that the rolling resistance increases in dynamometers with double contact rollers, as compared to the literature equations developed to simulate tire-ground rolling resistance. Therefore, a correction factor FC = 1.4 is considered based on these experimental results [4]. Furthermore, only a single roller of the dynamometer is analysed, therefore, the overall resistance force is also divided by the 8 rollers,

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Fig. 5 Electric motor torque curve

assuming that this load is equally distributed in all contact points. The rolling resistance force R is presented by Eq. (13) and the resultant torque TR that acts in the roller is given by Eq. 14. R=

 FC  7 × 10−6 V 2 + 0.015 m veh g 8 TR = Rrr

(13) (14)

3 Results Figure 6 shows the center displacements in the Y and Z axis for the operating condition. For both directions, it is almost instantly stabilized and the amplitudes are under 0.002 mm. In Y case, the center displacement values oscillate almost around zero and the small offset of 0.0008 mm is due to the lateral component of the vehicle prototype weight. For the Z case, this oscillation is below zero with a 0.0012 mm offset, because of the deflection due to the weight of the system and the prototype vertical component. After 0.1 s, the rotations are also stabilized, as shown in Fig. 6. This way, the maximum rotation is 8.6 rad/s, with a working frequency of 1.37 Hz, which is far from its natural frequency 288 Hz. This way, the system operates in safe conditions and does not reach the resonance frequency. In order to perform broader analysis, a hypothetical case where the resonance effect is reached was also evaluated, as shown in Fig. 7. In this case, the centers displacements are notable at its natural frequency rotation (288 Hz), in both axis, as expected. It is worth emphasizing these extreme conditions will not be reached under the operating situation, since the electric motor limits the torque input.

M. A. M. Lourenço et al. a) Rotation and displacement in Y

10 -3

1

10

0.5 0

5

-0.5 -1

0

0.02

0.04

0.06

0.08

Rotation (rad/s)

Center displacement in Y (mm)

204

0 0.1

b) Rotation and displacement in Z

10 -3

0

10

-0.5 5

-1 -1.5 -2

0

0.02

0.04

0.06

0.08

Rotation (rad/s)

Center displacement in Z (mm)

Time (s)

0 0.1

Time (s)

Fig. 6 Center displacements in the Y and Z axis for the operating condition

300

0.815

200

0.81 100

0.805 0.8

0

0.5

1

1.5

2

2.5

3

Rotation (rad/s)

Center displacement in Y (mm)

a) Rotation and displacement in Y 0.82

0

Time (s) 300

0.3 0.295

200

0.29 100

0.285 0.28

0

0.5

1

1.5

2

2.5

Time (s)

Fig. 7 Center displacements in the Y and Z axis under the resonance effect

3

0

Rotation (rad/s)

Center displacement in Z (mm)

b) Rotation and displacement in Z

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4 Conclusions The use of small-scale prototypes to predict behaviors and perform early evaluation tests is promising, while avoiding the real scale expenses. Aiming to increase the development of sustainable mobility, this work showed a Laval rotor approach for a small-scale chassis dynamometer roller. The analysis showed a very small variation of the roller center for the operating conditions, in both axis (Y and Z), not affecting the functioning of the system and being able to be disregarded in future analysis. It also reported the hypothetical behavior under resonance to reach extreme conditions and perform a more complete analysis. It is important to highlight that the rotation of the electric motor does not reach the system natural frequency, and the resonance situation does not occur. Therefore, this work showed that the small-scale chassis dynamometer is robust and allowing the reliable bench usage for small-scale vehicles test. Acknowledgements The authors thank the Support Fund for Teaching, Research and Extension (FAEPEX), the Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES), the National Council for Scientific and Technological Development (CNPq), the University of Campinas (UNICAMP) and the São Paulo Research Foundation (FAPESP) for scholarships and financial support.

References 1. Holjevac N, Cheli F, Gobbi M (2019) A simulation-based concept design approach for combustion engine and battery electric vehicles. Proc Inst Mech Eng Part D J Automobile Eng 233(7):1950 2. Holjevac N, Cheli F, Gobbi M (2019) Multi-objective vehicle optimization: comparison of combustion engine, hybrid and electric powertrains. Proc Inst Mech Eng Part D J Automobile Eng p 0954407019860364 3. Velasquez A, Higuti V, Guerrero H, Milori D, Magalhães D, Becker M (2016) Helvis-a smallscale agricultural mobile robot prototype for precision agriculture. In: 13th International conference of precision agriculture. International Society of Precision Agriculture, St. Louis, Missouri, USA, p 17 4. Eckert JJ, Bertoti E, dos Santos Costa E, Santiciolli FM, Yamashita RY, e Silva LCdA, Dedini FG (2017) Experimental evaluation of rotational inertia and tire rolling resistance for a twin roller chassis dynamometer. Technical report, SAE Technical Paper 5. Bertoti E, Eckert J, Yamashita R, Silva L, Dedini F (2017) Experimental characterization of a feedforward control for the replication of moving resistances on a chassis dynamometer. In: International symposium on multibody systems and mechatronics. Springer, pp 379–388 6. Figliola RS, Beasley DE (2015) Theory and design for mechanical measurements 7. Giakoumis EG (2017) Driving and engine cycles. Springer, pp 315–345 8. Ye L, Yang G, Li D (2014) Analytical model and finite element computation of braking torque in electromagnetic retarder. Front Mech Eng 9(4):368 9. Kardasz M, Kazerani M (2016) Systematic electric vehicle scaling for test bed simulation. In: 2016 IEEE transportation electrification conference and expo (ITEC) 10. Bertoti E, Yamashita RY, Eckert JJ, Santiciolli FM, Dedini FG, Silva LC (2018) Application of pattern recognition for the mitigation of systematic errors in an optical incremental encoder. In: International conference on rotor dynamics. Springer, pp 65–78

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11. Fajri P, Ferdowsi M, Lotfi N, Landers R (2016) Development of an educational small-scale hybrid electric vehicle (hev) setup. IEEE Intell Transp Syst Mag 8(2):8 12. Krämer E (2013) Dynamics of rotors and foundations. Springer Science & Business Media 13. Nicolazzi LC, Rosa ED, Leal LDCM (2001) Uma introdução à modelagem quase-estática de veículos automotores de rodas. Brasil: Publicação interna do GRANTE-Depto de Engenharia Mecâncica da UFSC 14. Tong W (2014) Mechanical design of electric motors. CRC Press 15. Eckert JJ, Silva LCDAE, Costa EDS, Santiciolli FM, Corrêa FC, Dedini FG (2019) Optimization of electric propulsion system for a hybridized vehicle. Mech Based Design Struct Mach 47(2):175 16. Genta G, Morello L (2009) The automotive chassis, vol 1. Springer

Dynamics and Control

Optimal Control for Path Planning on a 2 DOF Robotic Arm with Prismatic and Revolute Elastic Joints Jose A. G. Luz Junior, Angelo M. Tusset, Mauricio A. Ribeiro, and Jose M. Balthazar

Abstract This work presents the mathematical description, simulation and results for a dynamic analysis of a robotic arm with two degrees of freedom, one revolutionary and one a prismatic. The revolute link was considered with an elastic joint and the prismatic link was considered an ideal link. The SDRE technique was used to control the elastic joint to eliminate errors in position, vibration and to guide the arm through a specific path for position and speed. The work still has a wide variety of possibilities to improvement on the dynamic representation of the prismatic joint with some flexibility, a more detailed mathematical representation on the elastic joint control and explore more possibilities beyond SDRE control that can provide more precision on the speed control. Keywords Robotics · SDRE control · Elastic joint

1 Introduction Robotics research plays a big role in modern engineering generating interest in almost all branches, from medicine to aerospace. The incorporation of robotics in complex surgery and medical procedures is a field of medicine with great advances because precision, efficiency and repetition without loss of quality are examples of how this advance has happened [4]. The possibility of carrying out medical procedures at the

J. A. G. Luz Junior · J. M. Balthazar UNESP—Universidade Estadual Paulista Júlio Mesquita Filho, São Paulo, Brazil e-mail: [email protected] J. M. Balthazar e-mail: [email protected] A. M. Tusset (B) · M. A. Ribeiro (UTFPR)—Federal University of Technology, Paraná, Brazil e-mail: [email protected] M. A. Ribeiro e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_13

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same time, with more precision and reducing physical work are considered fundamental for modern medicine [13]. The improvement on productivity, effectiveness and risk reduction of human activities are examples of those real applications and improvement that robotics provides [1]. Robotic manipulators are one of the many structures that a robot can be made. The manipulators are autonomous structures characterized by the composition of links and joints, a fixed spatial reference and a great concern on the end-effect, referred as tool in some cases. Another particularity of manipulators is the concern over the tool used by the robot to make his function. The “end-effector” is the term use to describe the tool used by the manipulator and is the interface between the robot arm and the application. Grippers, pincers, welders and pick and place structures are examples of tools use as “end-effector” by robotic manipulators [5]. In addition to medical areas, robotics has been used as a tool for other areas of engineering. This is the case with the application of some control and learning techniques, such as Neural Networks, to improve robotic application. Solutions provided by the combination of these two fields have presented excellent results at points where robotics began to see its theoretical and conceptual limit [10]. Using Fuzzy and Neuro Fuzzy to provide the description for the kinematics is a prove and funded approach to structure with kinematic redundancy and impossible algebraic solution [2]. Dealing with parametric error on the robotic variables using neural networks represent a great improvement on projects optimization [3]. In this work is considered the SDRE (State-Dependent Riccati Equation) control, SDRE control is being applied in many robotic manipulators systems due to its easy application and advantage of considering the influence of nonlinearities of the system, and is no need to linearize the system to be applied [6–9, 12].

2 Equations of Motion There are several possible approaches to the existence of elastic articulation in the robotic arm, such as analyzing the small elastic deformations in the articulation between the link and the actuator, considering that the articulation is made of elastic material or the gear in the articulation has elastic behavior. A damping constant c and stiffness constant k relates the link position and the motor position. As presented on the Fig. 1, the equations of motion were express using cartesian coordinate. Where x represents the first link, θ2 the position of the motor and θ3 link position.

2.1 Mathematical Model The main addition on the traditional dynamic model is the elastic description between the DC motor and the second link. When moving, the elastic component on the joint between the DC Motor and the link makes that initially the θmotor is bigger than θlink

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Fig. 1 Robotic arm system using cartesian coordinates and the elastic joint

but makes that when the DC Motor stopping the motion on the link still stand and θlink is bigger than θmotor . To describe this system mathematically, there are two main approaches: considering the forces on the system or describe the energy on the system [11]. Lagrangian mechanical is based on the description of the amount of energy in the system through the potential and kinetics energy components, the Eqs. (1) and (2). The kinetic and potential energy can be expressed as the sum of these two components on each body of the system described individually. This means that the system can be divided into n bodies and each of them has a kinetic and potential equation that together describe the whole system energy. 1 m n vn2 2 n=1 N

Z=

P = mn hn g

(1) (2)

On the robotic system there are two bodies, the first and the second link each one with his characteristics that differentiate one to each other. The prismatic link has no elastic component, so de dynamic description follows the canonical Eqs. (2) and (1), for potential and kinetics energy. However, the second link, the revolute one, present the elastic joint so this means an additional component on the energy. The kinematic and potential energy of the whole system is shown in Eqs. (4) and (5). The first link was mass m 1 , the second m 2 and the DC motor m m . The second link was Center of Mass Cm , the Inertia in relation to its own center of rotation Jl , Eq. (3), and Il is the Inertia in relation to the CC Motor and Link barycenter. For the potential energy (g) is the gravity constant. Jl = Il + m 2 Cm2 K =

  1  2 m 2 xC ˙ m θ˙3 cos(θ3 ) + (m 1 + m 2 + m m )x˙ + Jl θ˙3 + Jm θ˙2 2

(3) (4)

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P=

1 k(θ3 − θ2 ) − m 2 d2 g cos(θ3 ) 2

(5)

The Lagrangian is based on the kinematic and potential energy to calculate the equations of motion, as shown in Eq. (6), and the Euler-Lagrange Eq. shown in Eq. (7) uses the Lagrangian to express the relationship between all the components of force in the system, the conservative and non-conservative forces. L=K−P

(6)

∂L d ∂L − = τi dt ∂ q˙i ∂qi

(7)

where: q1 = x, q2 = θ2 and q3 = θ3 . Substituting Eqs. (3–5) in Eqs. (6–7), the equations are obtained:     m 2 Cm x˙ θ˙3 sin(θ3 ) + τ3 cos(θ3 ) − Cm m 2 Cm θ˙3 sin(θ3 ) + τ1   x¨ = Cm m 2 cos(θ3 )2 − M (θ3 − θ2 )k θ¨2 = Jm + Il   m 2 Cm θ˙3 m 2 Cm cos(θ3 ) − M θ˙1 sin(θ3 ) + m 2 Cm cos(θ3 )τ1 − Mτ3   θ¨3 = m 2 Cm m 2 cos(θ3 )2 − M

(8) (9)

(10)

where:M = m 1 + m 2 + m m . For numerical simulations, the following parameters will be considered:   m1 =       0.1 kg , m 2 = 0.1 kg , m m = 0.442 kg , Cm = 0.25[m], k = 0.2 N m , Jm =     2.8 × 10−6 kgm2 and Il = 8.33 × 10−3 kgm . And initial condition: x0 = 0θ20 = 0 and θ30 = 0.

2.2 Numerical Simulation for Without Control Case Figure 2 shows an example of dynamic behavior without control. Some points were taken into account to certify the effectiveness of the calculated dynamic equations. The speed of the links stabilizing at zero, the elastic joint having vibration and the positions go to the maximum extent of x and the only stable point for the second link.

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Fig. 2 Dynamic behavior of the robotic arm without control

3 Control Strategy by SDRE Control Considering the controlled system (Eqs. 8–10) in the form of state equations: ˙ AX + BU X= Being: ⎡

0 ⎢0 ⎢ ⎢0 ⎢ A(x) = ⎢ ⎢0 ⎢ ⎣0 0

1 a22 0 0 0 a62

0 0 0 −k Jm +Il

0 0

0 0 1 0 0 0

0 0 0 −k Jm +Il

0 0

⎡ ⎤ 0 0 ⎢b a26 ⎥ ⎢ 21 ⎥ ⎢ 0 0 ⎥ ⎢ ⎥ ⎥ and B(x) = ⎢ 0 ⎥ ⎢ 0 ⎢ ⎥ ⎣ 0 1 ⎦ a66 b61

(11)

0 0 0 1 Jm +Il

0 0

⎤ 0 b23 ⎥ ⎥ 0 ⎥ ⎥ ⎥. 0 ⎥ ⎥ 0 ⎦ b63

where: a22 =

−m 22 Cm2 cos(θ3 ) sin(θ3 )θ˙3 (Jm +Il ) , m 2 Cm2 (Jm +Il )( M−cos(θ3 )2 )

a26 =

−m 22 Cm2 (Jm +Il ) sin(θ3 )(cos(θ3 )θ˙1 +1) , m 2 Cm2 (Jm +Il )( M−cos(θ3 )2 )

b21 =

b63 =

−M(Jm +Il ) , m 2 Cm2 (Jm +Il )( M−cos(θ3 )2 )

−m 2 Cm cos(θ3 )(Jm +Il ) , m 2 Cm2 (Jm +Il )( M−cos(θ3 )2 )

a66 =

−m 2 Cm2 (Jm +Il ) sin(θ3 )( M θ˙1 +m 2 cos(θ3 )) , m 2 Cm2 (Jm +Il )( M−cos(θ3 )2 )

a62 =

b23 =

−m 2 Cm2 M sin(θ3 )θ˙3 (Jm +Il ) , m 2 Cm2 (Jm +Il )( M−cos(θ3 )2 ) m 2 Cm2 (Jm +Il ) , m 2 Cm2 (Jm +Il )( M−cos(θ3 )2 )

b61 = −b23 .

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⎡

⎤ τ1 And: U = ⎣ τ2 ⎦ is the feedback control, is defined as follows: τ3 U = −R−1 BT Pe

(12)

where: e = [X − X∗ ], X represents the states of the system and X∗ the desired states. Function P is obtained by solving the Ricatti equation defined as follows: AT P + PA − PBR−1 BT P + Q = 0

(13)

The quadratic performance measure for the feedback control problem is given by: 1 J= 2

∞ (eT Qe + UT RU)dt

(14)

t0

where Q and R are positive definite matrices. Another important factor to consider is that the matrix cannot violate the controllability of the system. The controllability matrix is given by:   M = Bn×m An×n Bn×m · · · An−1 n×n Bn×m

(15)

3.1 Numerical Simulation for a Desired Set Point Figure 3 shown the answer using the DC motor limitation, considering X∗ the desired states, Q and R positive⎡definite matrices: ⎤ 102 0 0 0 0 0 ⎢ 0 102 0 0 0 0 ⎥ ⎡ ⎤ ⎡ ⎤ ⎥ ⎢ 100 0.2 ⎥ ⎢ 0 0 1 0 0 0 ⎥ ⎢ X∗ = ⎣ π6 ⎦,Q = ⎢ ⎥ and R = 0.1⎣ 0 1 0 ⎦. ⎢ 0 0 01 0 0 ⎥ π ⎥ ⎢ 001 6 ⎣ 0 0 0 0 103 0 ⎦ 0 0 0 0 0 102 Figure 4 shown the answer using the DC motor limitation, considering X∗ the desired states, Q and R⎡positive definite matrices: ⎤ 103 102 102 102 102 102 ⎢ ⎡ ⎤ ⎡ ⎤ 0 0 0 ⎥ ⎥ ⎢ 0 10 0 100 0.2 ⎥ ⎢ 0 0 2500 0 0 0 ⎥ ⎢ X∗ = ⎣ π6 ⎦,Q = ⎢ ⎥ and R = 0.1⎣ 0 1 0 ⎦. ⎢ 0 0 0 10 0 0 ⎥ π ⎥ ⎢ 001 6 ⎣ 0 0 0 0 105 0 ⎦ 0 0 0 0 0 102

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Fig. 3 Robotic Arm with optimal control for controlling the elastic joint taken the DC Motor AK555/11.1PF12R83CE-V2 limitation into account

Fig. 4 SDRE control with the fastest stabilization, less overshoot and error on the steady state

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The answer on the Fig. 4 shown the fastest, with minimum overshoot and stabilization error.

3.2 Numerical Simulation for a Desired Trajectory The polynomial shown on the Eq. (16) is the one used to the make the path for position and for speed, q is the vector for the position path, q˙ the vector for speed. The coefficients ai j are calculated during the Path Planning from the initial to final point. q(t) = ai5 t 5 + ai4 t 4 + ai3 t 3 + ai2 t 2 + ai1 t + ai0

(16)

q(t) ˙ = 5ai5 t 4 + 4ai4 t 3 + 3ai3 t 2 + 2ai2 t + ai1

(17)

Figure 5 shown the controlled arm using those polynomial paths using the weight matrices (16) and (17), Q and R positive definite matrices: ⎡ ⎤ 103 0 0 0 0 0 ⎢ 0 1 0 0 0 0⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ 100 0.2 ⎢ ⎥ 2 ⎢ 0 0 10 0 0 0 ⎥ X∗ = ⎣ π6 ⎦,Q = ⎢ ⎥ and R = 0.1⎣ 0 1 0 ⎦. ⎢ 0 0 0 10 0 0 ⎥ π ⎢ ⎥ 001 6 ⎣ 0 0 0 0 103 0 ⎦ 0 0 0 0 0 1

Fig. 5 Path Planning for the robotic arm using the SDRE control

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4 Conclusion Several conditions were tested to make sure that dynamic representation could be used to represent this system, such as exploring the range limits for the first link, multi-rotations for the second link and increases and decreases on the elastic joint parameters, kn and dn . Methodologies like Denavit-Hartenberg and Newton-Euler can be tested to find a better mathematical description that takes into account the robot’s work load. The dynamic equations based on the cartesian coordinates and the Lagrangian mechanical, respectively, were capable of represent all the component and can be used to the next steps without any adjustments. The Fig. 2 shown the result without control, the Dynamics behavior, and exemplifies that. The Path Planning on the Fig. 4 and the fixed-point control on the Fig. 3 are examples that the elastic joint can be controlled using the SDRE control. The matrices R and Q can be tuned, as shown on the Figs. 3 and 4, to provide a fastest stabilization or a slow path based on the DC motor limitations. Introducing the appropriate elastic joint control mechanisms and replacing them with the genetic force used up to that point is one of the main next steps of this work. New representations can be tested on the elastic joint and can lead to new state space matrices, making the system easily controlled. The implementation of a State Observer to eliminate some sensors is another possible addition to the system. Acknowledgements The authors acknowledge the CAPES, FAPESP and CNPq, both Brazilian research funding agencies. In addition, the authors thank the organizing committee of the 15th International Conference on Vibration Engineering and Technology of Machinery (VETOMAC XV), where part of this work was presented.

References 1. Cathpole K, Perkins C, Bresee C, Solnik MJ, Sherman B, Fritch J, Gross B, Jagannthan S, Hakami-Majd N, Avenido R, Anger JT (2016) Safety, efficiency and learning curve in robotic surgery: a human factory analysis. Surg Endosc 30(9):349–376 2. Duka A (2015) AFINS based solution to the inverse kinematics of a 3dof planar manipulator. Procedia Technol 19:526–533 3. Guo Y, Chen L (2008) Adaptive neural network control of coordinated motion of dual-arm space robot system with uncertain parameters. Appl Math Mech 29:1131–1140 4. Itik M, Salamci MU, Banks SP (2010) SDRE optimal control of drug administration in cancer treatment. Turk J Elec Eng Comp Sic 18(5):715–729 5. Lewis FL, Dawson DM, Abdallah CT (2004) Robot manipulator control – theory and practice, 2nd edn. Marcel Dekker, New York 6. Lima JJ, Balthazar JM, Rocha RT, Janzen, FC, Bernardini D, Litak G, Tusset AM, Bassinello D (2019) On positioning and vibration control application to robotic manipulators with a nonideal load carrying. Shock and Vibration: 1–14 7. Lima JJ, Tusset AM, Janzen FC, Piccirillo V, Nascimento CB, Balthazar JM (2016) Brasil MRLF (2016) SDRE applied to position and vibration control of a robot manipulator with a flexible link. J Theor Appl Mech (Warsaw) 54:1067–1078

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8. Lima JJ, Tusset AM, Janzen FC, Piccirillo V, Nascimento CB, Balthazar JM, Brasil MRLF (2014) Nonlinear state estimation and control applied to a manipulator robotic including drive motor. Math Eng Sci Aerosp: Transdiscipl Int J 5: 413–425 9. Luz Junior, JAG, Tusset AM, Janzen FC, Rocha RT, Balthazar JM, Nabarrete A (2018) Optimal control for robot manipulators with three-degress-of-freedom. In: Jan Awrejcewicz. (Org.). Springer Proceedings in Mathematics & Statistics. 1ed., pp 135–149. Springer International Publishing, Poland 10. Ogawa T, Kanada H (2010) Solution for Ill-posed inverse kinematics of robot arm by network inversion. J Robot:1–9 11. Spong MW, Hutchinson S, Vidysagar M (2005) Robot modeling and control, 1a edn. John Wiley & Sons, New York 12. Tusset AM, Ribeiro MA, Lenz WB, Balthazar JM (2018) Control chaos strategy applied to a free-joint nonholonomic manipulator. Math Eng, Sci Aerosp: Transdiscipl Int J 9:415–421 13. Winder JS, Juza R, Sasaki J, Rogers AN, Pauli EM, Haluck RS, Estes SJ, Lyn-Sue JR (2016) Implementing a robotics curriculum at an academic general surgery training program: our initial experience. J Robotic Surg 10(3):209–13

Numerical and Experimental Analysis of a Hybrid (Passive-Adaptive) Vibration Control System in a Cantilever Beam Under Broadband Excitation Maurizio Radloff Barghouthi, Eduardo Luiz Ortiz Batista, and Eduardo Márcio de Oliveira Lopes Abstract The present work focuses on a hybrid vibration control approach which is implemented and investigated through simulation and experimental trials. This approach is based on combining a passive control treatment, consisting of a constrained layer of viscoelastic material, with an active control arrangement composed of sensors, actuators, and an adaptive control unit that uses the FxNLMS algorithm. Such a hybrid control system is applied to a cantilever metallic beam subjected to a stochastic broadband excitation. A distinctive conclusion of this work is that the socalled feedback filter, which is part of the FxNLMS-based adaptive control unit, has shown to be detrimental to the control action in the case of the considered excitation. Keywords Adaptive vibration control · FxNLMS algorithm · Hybrid vibration control · Viscoelastic material

1 Introduction Vibration control consists of a set of techniques designed to reduce undesirable vibrations that occur in mechanical systems due to various sources of disturbance. Such techniques are usually characterized by considering three different classes: passive, active, and hybrid control. The hybrid vibration control is the combination M. R. Barghouthi · E. M. de O. Lopes Federal University of Paraná, GVIBS/UFPR, Campus Universitário Centro Politécnico s/n, Paraná, Brazil e-mail: [email protected] E. M. de O. Lopes e-mail: [email protected] E. L. O. Batista (B) Federal University of Santa Catarina, LINSE/EEL/CTC/UFSC, Campus Universitário Trindade s/n, Santa Catarina, Brazil e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. Balthazar (ed.), Vibration Engineering and Technology of Machinery, Mechanisms and Machine Science 95, https://doi.org/10.1007/978-3-030-60694-7_14

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of passive and active control techniques [7, 10, 11]. By working simultaneously with these two control strategies, the global (hybrid) system offers the strength and practicality of the passive action, in addition to the flexibility and adjustability of the active/adaptive action [7]. The hybrid vibration control approach considered in the present work employs a feedforward strategy coupled with magnetic actuators for the active control and a constrained layer of viscoelastic material for the passive control. Regarding the passive control, it is well known that the addition of a viscoelastic layer to a metallic mechanical system usually has a small impact on the global mass, since the density of viscoelastic materials is typically much lower (between 5 and 6 times) than the density of metallic materials. Moreover, the impact on the system stiffness is usually low too, since no significant restorative forces are created. On the other hand, a very pronounced effect is expected regarding damping, due to the high values of loss factor observed for viscoelastic materials. In order to further enhance the effect of adding damping, it is possible to: (1) add a constraining metal sheet over the viscoelastic material; and (2) insert the layer of viscoelastic material in the region of greater structural deformation with a thickness close to that of the vibrating structural part [8]. Active control techniques typically consist of generating a destructive interference between the acceleration signal generated by the control action and the acceleration signal generated by the original source of disturbance in a given position of the mechanical system. In order to generate the control force in the feedforward strategy, the control unit requires two types of information: (1) a vibration reference signal ar (t) correlated with the disturbance source; and (2) the vibration error signal ae (t) at the performance position of the mechanical system. As shown below, these signals ˆ are processed – for instance, by means of a static feedback filter F[n], a static ˆ secondary-path filter S[n], and an adaptive FxNLMS-based filter W [n]—aiming to generate the control action [4, 6]. The present chapter is dedicated to the performance analysis of a hybrid vibration control system applied to a cantilever metallic beam under a stochastic broadband ˆ disturbance, with and without the use of the F[n] feedback filter. In the following sections, some basic concepts and the employed methodology are firstly addressed. Then, the results are presented and discussed and the conclusions are drawn.

2 Basic Concepts This section addresses the basic concepts regarding the system paths, the feedback effect, and the formulation to emulate a hybrid vibration control system.

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2.1 System Paths Figure 1 shows a hybrid control scheme applied to a cantilever beam, which is the one under focus in the present work [1, 2]. Four distinct signal paths are highlighted in Fig. 1, namely, the primary path, the secondary path, the feedback path, and the reference path. From a systems engineering point of view, these paths can be represented by transfer functions. The transfer function relating the excitation generated by the disturbance source with the vibration at the performance point (measured by the error accelerometer) represents the primary path. The transfer function that relates the excitation generated by the control source with the vibration at the performance point represents the secondary path. The feedback path is described by the transfer function which associates the excitation generated by the control source to the vibration at the reference point (measured by the reference accelerometer). Finally, the reference path is described by the transfer function which associates the excitation generated by the disturbance source to the vibration at the reference point.

Passive Control

Disturbance Magnetic Transducer

fp(t) fc(t)

Reference Accelerometer

Amplifier

Error Accelerometer

Control Magnetic Transducer

ar(t)

ae(t)

p(t) Amplifier c(t) ar[n] p[n]

Disturbance Source

+_

Primary path Secondary path

ˆ F[n]

D x[n]

W[n]

Feedback path

c[n]

w[n]

ˆ S[n]

D w[n+1] x'[n]

Adaptive Algorithm

NI Platform

Computer

Fig. 1 Hybrid control scheme

e[n]

Reference path

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2.2 Feedback Effect In adaptive vibration control, as a general rule, the reference signal ar (t) should be correlated only with the disturbance source p(t) [4]. However, in practice, the measuring system will also acquire the dynamic effect of the control source c(t) via the feedback path (see Fig. 1). This superposition of effects is defined, in the context of adaptive control of vibrations, as feedback effect. One way of circumventing the feedback effect (i.e., remove the portion coming from the feedback path from signal ar (t)) is by using a static FIR (finite impulse ˆ response) feedback filter, denoted here by F[n], which is designed to model the ˆ feedback path. The goal of this filter is to generate a digital estimate b[n] of the signal b(t) coming from the feedback path, which is then subtracted from ar [n] (digital version of ar (t)). As a result, a digital estimate x[n] of the reference signal n(t) can be obtained, as illustrated in Fig. 2.

2.3 Digital Emulation of Mechanical Systems 2.3.1

Emulation of Linear Single-Degree-of-Freedom Mechanical Systems

Control Unit

c[n]

D/A

D/A

p(t)

c(t)

Reference Path

Feedback Path

n(t)

+

Disturbance p[n] Source

+

Figure 3 presents the general approach regarding the digital emulation of linear single-degree-of-freedom mechanical systems (LSDOFMS). The impulse response h(t) of the system is sampled over a time interval t resulting in the pulsed impulse response h[n]. The Z transform is applied to h[n] in order to obtain H [z], which is implemented by means of an IIR architecture. The equation of motion for an LSDOFMS with mass m, natural frequency ωn , and damping factor ζ (in which the assumption of viscous damping is implicit) can be written as

b(t)

ar(t)

A/D

^ F[n]

ar[n]

+_

^ b[n]

Fig. 2 Feedback effect cancellation

(t)

Mechanical h(t) System

A/D

Fig. 3 Approach for digital emulation of an LSDOFMS

h[n]

Z

H[z]

x[n]

Numerical and Experimental Analysis of a Hybrid … h(t)

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h[n] 0