Tyre models for shimmy analysis : from linear to nonlinear Citation for published version (APA): Ran, S. (2016). Tyre models for shimmy analysis : from linear to nonlinear. Technische Universiteit Eindhoven. Document status and date: Published: 11/01/2016 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 27. Sep. 2021 ; Tyre Models for Shimmy Analysis: from linear to nonlinear Shenhai Ran 2016 Technische Universiteit Eindhoven Shenhai Ran (2015). Tyre models for shimmy analysis: from linear to nonlinear. Ph.D. thesis, Eindhoven University of Technology, Eindhoven, the Netherlands. A catalogue record is available from the Eindhoven University of Technology Library. ISBN:978-90-386-3987-1 Typeset with LATEX 2ε . Cover design: Yuexu Chen, Eindhoven, the Netherlands. Reproduction: CPI Koninklijke Wöhrmann, Zutphen, the Netherlands. Copyright ©2015 by Shenhai Ran. All rights reserved. No part of the material protected by this copyright notice may be produced or utilised in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without the prior written per- mission of the author. Tyre models for shimmy analysis: from linear to nonlinear PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. F.P.T. Baaijens, voor een commissie aangewezen door het College voor Promoties, in het openbaar te verdedigen op maandag 11 januari 2016 om 16.00 uur door Shenhai Ran geboren te Tangshan, China Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de promotiecommissie is als volgt: voorzitter: prof.dr. L.P.H. de Goey 1e promotor: prof.dr. H. Nijmeijer copromotor: dr.ir. I.J.M. Besselink leden: prof.dr.ir. M. Steinbuch prof.dr.ir. P.W.A. Zegelaar prof.dr. M. Plöchl Technische Universität Wien prof dr. ir. A. de Boer Universiteit Twente Het onderzoek dat in dit proefschrift wordt beschreven is uitgevoerd in overeenstem- ming met de TU/e Gedragscode Wetenschapsbeoefening. Contents Contents i Nomenclature v Summary ix 1 Introduction 1 1.1 Thewheelshimmyphenomenon . 1 1.2 Motivationandobjectives . 4 1.3 Research approach . 5 1.4 Contributions.............................. 7 1.5 Outlineofthethesis . .. .. 7 2 Fundamentals of shimmy and tyre behaviour 9 2.1 Thehistoryofshimmyresearch . 9 2.2 Tyre models for vehicle dynamics analysis . 12 2.2.1 Approachesoftyremodelling . 13 2.2.2 Slipandsignconventionsintyremodelling . 16 2.3 Thestretchedstringtyremodel . 19 2.3.1 VonSchlippe/Kluiters . 23 2.3.2 Pacejkastraighttangent . 25 2.3.3 Comparison of transfer functions . 26 2.4 Linearstabilityanalysis. 28 2.4.1 Suspension with a yaw degree of freedom . 29 2.4.2 Suspension with yaw and lateral degrees of freedom . 30 2.5 Discussion................................ 33 3 Energy balance and tyre motion during shimmy 35 3.1 Energy considerations for shimmy analyses . 35 3.2 Energyflowmethod .......................... 36 3.2.1 Energy transfer through tyre . 37 i Contents 3.2.2 Energy criterion with sinusoidal motion . 38 3.3 Wheelmotionduringshimmy . 42 3.3.1 Equivalentwheelmotion . 43 3.3.2 Motionandenergytransfer . 44 3.4 Discussion................................ 45 4 Nonlinear tyre characteristics and modelling 49 4.1 Nonlineartyremodels . 49 4.1.1 Forceandmoment ....................... 49 4.1.2 Contactpatchdynamics . 51 4.1.3 Turnslip ............................ 53 4.1.4 Rigidringdynamics . 54 4.1.5 Modelnamingconvention . 57 4.2 Baselinecharacteristics . 57 4.2.1 Tyremodels........................... 58 4.2.2 Trailingwheelsuspension . 58 4.2.3 Equationsofmotion . 60 4.3 Bifurcation analysis and MatCont toolbox ............... 62 4.4 Conclusions............................... 64 5 Shimmy analysis with contact patch dynamics 65 5.1 Influence of the relaxation length . 65 5.1.1 Suspension with only yaw degree of freedom . 66 5.1.2 Suspension with lateral flexibility . 71 5.2 Influenceofturnslip . .. .. 72 5.3 Discussion................................ 81 6 Shimmy analysis with rigid ring dynamics 83 6.1 Dynamicsofthetyrebelt . 84 6.1.1 Frequencyresponse. 84 6.1.2 Theshimmyenergy . .. .. 86 6.2 Shimmyanalysis ............................ 89 6.2.1 Vibrationmodes ........................ 89 6.2.2 Bifurcationanalysis. 93 6.3 Discussion................................ 97 7 Conclusions and recommendations 99 7.1 Conclusions............................... 99 7.2 Recommendations ........................... 101 Appendix I Simplified Magic Formula 103 ii Contents Appendix II Parameters for tyre models 107 Bibliography 111 Societal summary 119 Acknowledgements 121 Curriculum Vitae 123 iii Nomenclature Symbol Description Units a half of the tyre contact length m A amplitude of sinusoidal yaw input rad 2 cc tyre lateral carcass stiffness per unit of the length N/m cv tyre lateral stiffness N/m cβ tyre yaw stiffness Nm/rad C contact centre between the tyre and road - Cfα cornering stiffness N/rad Cf φ turn slip stiffness of the lateral force Nm/rad Cmα self-aligning stiffness Nm/rad 2 Cmφ turnslipstiffnessofthealigningmoment Nm /rad dby lateral damping of the tyre belt Ns/m dbψ yaw damping of the tyre belt Nms/rad dbγ camber damping of the tyre belt Nms/rad dc lateral tyre carcass damping Ns/m dcy residual lateral damping of the contact patch Ns/m dcψ residual yaw damping of the contact patch Nms/rad dy lateral suspension damping Ns/m dψ yaw suspension damping Nms/rad e mechanical trail m Ek kinetic energy J Fd driving force at wheel centre N Ft tension force in the string N Fy tyre lateral force N Fya lateral force at axle centre from the wheel N Fyc tyre lateral force at the contact patch N Fr resulting lateral force at wheel centre N Fz tyre vertical force N Fzo nominal tyre vertical force N v Nomenclature Hy,x (s) transfer function: input x, output y - Hmn magnitude of the transfer function Hmn (s) - 2 Iaz rim moment of inertia about the vertical axis kgm 2 Ibx tyre belt moment of inertia about the longitudinal axis kgm 2 Iby tyre belt moment of inertia about the lateral axis kgm 2 Ibz tyre belt moment of inertia about the vertical axis kgm 2 Icz contact patch moment of inertia about the vertical axis kgm 2 Isz suspension moment of inertia about the vertical axis kgm 2 Iwz wheel moment of inertia about the vertical axis kgm 2 Iz total yaw moment of inertia of the wheel and suspension kgm j complex variable, j2 = 1 - − kby lateral stiffness of the tyre belt N/m kbψ yaw stiffness of the tyre belt Nm/rad kbγ camber stiffness of the tyre belt Nm/rad kc lateral tyre carcass stiffness N/m kcy residual lateral stiffness of the contact patch N/m kcψ residual yaw stiffness of the contact patch Nm/rad ky lateral suspension stiffness N/m kψ yaw suspension stiffness Nm/rad m total mass of the wheel and suspension kg ma mass of the rim including part of the tyre kg mb mass of the tyre belt kg mc mass of the contact patch kg mw mass of the wheel kg Mr resulting self-aligning at wheel centre Nm Mz tyre self-aligning moment Nm Mza self-aligning moment at axle from the wheel Nm Mzc tyre self-aligning moment at the contact patch Nm Mzr residual aligning moment Nm Re effective rolling radius m Rl loaded tyre radius m R0 unloaded tyre radius m Rφ path radius of turn slip m s Laplace variable - S imaginary slip point - t time s tp pneumatic trail m t0 initial time of the integration s = 1 T time period of vibrations, T f s U potential energy J v lateral deformation of the tyre string m vi Nomenclature v1 lateral deformation of the string leading contact point m v2 lateral deformation of the string trailing contact point m V forward velocity m/s Vc velocity of the contact patch m/s Vcx longitudinal velocity of the contact patch m/s Vcy lateral velocity of the contact patch m/s Vsx longitudinalslidingvelocity m/s Vsy lateral sliding velocity