DINÂMICA VEICULAR

DINÂMICA VEICULAR

(Parte 1 de 4)

Ground Vehicle Dynamics Ground Vehicle Dynamics

Originally published in German. Popp, K. and Schiehlen,W. (1993). Fahrzeugdynamik. B.G. Teubner, Stuttgart. ISBN 3-519-02373-3

Karl Popp and Werner Schiehlen

Ground Vehicle Dynamics In cooperation with Matthias Kröger and Lars Panning

Prof. Dr.-Ing. Werner Schiehlen Institute of Engineering and Computational Mechanics University of Stuttgart Pfaffenwaldring 9 70569 Stuttgart, Germany E-mail: schiehlen@itm.uni-stuttgart.de

In cooperation with:

Prof. Dr.-Ing. Matthias Kröger Institute of Machine Elements, Design and Manufacturing Technical University Bergakademie Freiberg Agricolastrasse 1 09596 Freiberg, Germany E-mail: kroeger@imkf.tu-freiberg.de

Dr.-Ing. Lars Panning Institute of Dynamics and Vibration Research Leibniz Universitaet Hannover Appelstrasse 1 30167 Hanover, Germany E-mail: panning@ids.uni-hanno

ISBN 978-3-540-24038-9 e-ISBN 978-3-540-68553-1 DOI 10.1007/978-3-540-68553-1 Library of Congress Control Number: 2010920023 c© 2010 Springer-Verlag Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of rial is concerned, specifically the rights of translation, reprinting, reuse of illustrations, r broadcasting, reproduction on microfilm or in any other way, and storage in data banks. cation of this publication or parts thereof is permitted only under the provisions of the Copyright Law of September 9, 1965, in its current version, and permission for use must be obtained from Springer. Violations are liable to prosecution under the German Copyri

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Preface

The book Ground Vehicle Dynamics is the revised English edition book Fahrzeugdynamik originally published in German back in 1993. the preparation of this English edition the first author Karl Popp away far too early. In his spirit, and for his memories, two members Popp’s research group at the Leibniz University Hanover, Matthias and Lars Panning, agreed to contribute to the ongoing work on this ed the book. However, it took more time than originally planned and Ma Kroger moved to the Technical University Bergakademie Freiberg as h the Institute of Machine Elements, Design and Manufacturing.

Vehicle dynamics deals with the mechanical modeling as well as the ematical description and analysis of vehicle systems. The aim of this is a methodologically based introduction to the dynamics of ground systems. The different kinds of vehicles like automobiles, rail cars or m ically levitated vehicles are not considered one by one but the dyna common problems of all these vehicle systems are treated from a unifo point of view. This is achieved by a system oriented approach. The ation of meaningful mathematical models allows simulations of motio parameter studies well in advance of setting up a first prototype. The tr shorter periods for the development and to larger numbers of vehicle v demands from the engineer comprehensive computations, the fundamen of which are presented in this book.

The fundamental concept of this book is based on a modularizatio vehicle subsystems with standardized interfaces. In the first vital pa models of vehicles, guidance and suspension systems as well as guidew tems are presented, they are mathematically described in detail and they assembled to complete vehicle-guideway systems. The second metho cally oriented part is devoted to the performance criteria driving sta driving safety and durability. Then, it follows a review on the computa tional methods for linear and nonlinear vehicle systems. The sophistica theoretical methods related to the demanding problems in vehicle dyna ics are applied in the third part to longitudinally, laterally and ver

The many problems included in the book show mainly simple applica of the theory presented, and their solutions will support the reader in understanding the theory and the fundamentals of railway and road v

The book is devoted on the one hand to students of applied mechanics system theory as well as mechanical engineering and automotive engineering. The book will support lectures on vehicle systems and provide a view general behavior of ground vehicles. On the other hand the book illustr engineers joining or working with a vehicle company, or one of their supplier advanced methods which are the basis of software tools widely used to industry. Thus, the book may contribute to continuing education. Mo the systematic methodologically based approach is a good example for divisions of mechanical engineering and mechatronics.

Compared to the successful German edition which has been out of p some years, the English edition has been only slightly revised. New ref are added throughout the book, in Chapter 2 the recursive formalisms multibody dynamics are discussed, in Chapter 6 the revised ISO Sta 2631 is considered, in Chapter 7 the standard time integration codes pro by Matlab are evaluated and in Chapter 10 a planar half-car model has included to fil the gap between the quarter-car model and the complex model. Since the book is now written in English, some German keywo added in the appendix. This may help the German reader to identify easily the technical terms for subjects in which she or he is interested.

The authors and contributors of the book acknowledge the cont support of the Institute of Engineering and Computational Mechanics University of Stuttgart headed by Peter Eberhard. We thank our co-w from Hanover and the many students from Stuttgart, for typing for tables and text as well as for drawing the figures. Moreover, thanks a to members of the Institute of Engineering and Computational Mecha proofreading the manuscript. In particular, Daniel Garcıa Vallejo, a p from the University of Seville, Spain contributed to the final editoria on the book during his stay at the Institute. Finally it has to be p out that the cooperation with Petra Jantzen, Dieter Merkle and Christoph Baumann from Springer-Verlag was excellent.

Freiberg Matthias Hanover Lars P Stuttgart Werner Sc October 2009

Anregung unseres verehrten Lehrers, Herrn Prof. Dr. Dr. - Ing. E.h. K. nus. Berucksichtigung fanden die Ergebnisse zahlreicher neuerer, zum gemeinsamer Forschungsarbeiten. Vor allem aber ist die mehr als zehnj¨ Lehrerfahrung der Verfasser aus Vorlesungen uber Fahrzeugdynamik Technischen Universitat Munchen, der Universitat Hannover und der versitat Stuttgart eingeflossen. Hilfreich zur Aufbereitung des umfang Stoffes waren ferner die bei der Durchfuhrung des Kurses ”Dynamics o Speed Verhicles” am Internationalen Zentrum fur Mechanik (CISM) in gesammelten Erfahrungen.

Die Fahrzeugdynamik befaßt sich mit der mechanischen Modellier sowie der maithematischen Beschreibung und Analyse von Fahrzeug men. Ziel dieses Buches ist es, eine methodenorientierte Einfuhrung Dynamik landgestutzter Fahrzeugsysteme zu geben. Dabei werden nic einzelnen Fahrzeugarten wie Kraftfahrzeuge, Schienenfahrzeuge oder netschwebebahnen nebeneinander betrachtet, sondern die allen Fahrzeug temen gemeinsamen dynamischen Probleme unter einheitlichen Gesich ten behandelt. Dies ist durch eine systemtheoretische Betrachtung moglich. Die Bereitstellung aussagekraftiger mathematischer Modelle Bewegungssimulationen und Parameterstudien lange bevor der erste typ gebaut wird. Der Trend zu kurzeren Entwicklungszeiten und eine Variantenvielfalt verlangen heute vom Ingenieur umfassende Berechn fur die dieses Buch die Grundlagen vermitteln soll.

Das Grundkonzept des vorliegenden Buches beruht auf einer Mo isierung der Fahrzeugteilsysteme mit standardisierten Schnittstellen. sten zentralen Teil werden die Modelle fur Fahrzeuge, Trag- und Fuhrsysteme sowie Fahrwege im einzelnen begrundet, mathematisch ausfuhrlich besc und zu Gesamtmodellen fur Fahrzeug-Fahrweg-Systeme zusammeng Der zweite, methodenorientierte Teil wird durch die Beurteilungskr Fahrstabilitat, Fahrsicherheit, Fahrkomfort und Bauteil-Lebensdauer geleitet. Anschließend folgt die Darstellung der Berechnungsmethoden lineare und nichtlineare Fahrzeugsysteme. Die der anspruchsvollen gabenstellung ent-sprechenden theoretischen Verfahren werden im Teil am Beispiel einfacher Longitudinal-, Lateral- und Vertikalbe gen verdeutlicht. Ein Anhang mit Ergebnissen aus der Theorie optima Mehrgroßenregelsysteme tragt dem Trend zu aktiven Fahrzeugkompo Rechnung. Eine Vielzahl aufeinander abgestimmter und in die einzelnen tel eingestreuter Beispiele mit ausfuhrlichen Losungen sollen das Verst der Theorie erleichtern und die Anschauung fordern.

Das Buch wendet sich einerseits an die Studierenden der Angew

Mechanik und Systemtheorie sowie der Fahrzeugtechnik. Es soll insb dere Vorlesungen uber spezielle Fahrzeugsysteme unterstutzen und den fur allgemeine Zusammenhange scharfen. Andererseits zeigt es dem

Die Verfasser danken Herrn Dipl.-Ing. R. Austermann und Herrn Dipl.-Ing P. Eberhard fur die sorgfaltige Durchsicht der Druckfahnen sowie Her Pietsch fur die Erstellung der Reinzeichnungen und Bilder. Dank g ferner den vielen Helfern beim Schreiben des Manuskripts. Schließlic unser Dank dem Verlag B. G. Teubner fur die erwiesene Geduld und stets erfreuliche Zusammenarbeit.

Hannover K Stuttgart W. Sc Sommer 1992

List of Problems and their Solutions
1 System Definition and Modeling
2 Vehicle Models
2.1 Elements of Multibody Systems
2.2 Kinematics
2.2.1 Frames of Reference for Vehicle Kinematics
2.2.2 Kinematics of a Rigid Body in an Inertial Frame
Frame
2.2.4 Kinematics of Multibody Systems
2.3 Dynamics
2.3.1 Inertia Properties
2.3.2 Newton-Euler Equations
2.3.3 Principles of d’Alembert and Jourdain
2.3.4 Energy Considerations and Lagrange’s Equations
2.4 Equations of Motion for Multibody Systems
2.5 Formalisms for Multibody Systems
2.5.1 Non-recursive Formalisms
2.5.2 Recursive Formalisms
2.5.2.1 Kinematics
2.5.2.2 Newton-Euler Equations
2.5.2.3 Equations of Motion
2.5.2.4 Recursion
3 Models for Support and Guidance Systems
3.1 Models for Passive Spring and Damper Systems
3.2 Models of Force Actuators

Contents 2.2.3 Kinematics of a Rigid Body in a Moving Reference 3.2.1 Models of Magnetic Actuators . ... .. .. .. .. .. .. .. ..

3.4.2 Definition of the Rigid Body Slip
3.4.3.1 Linear Law of Contact Forces
3.4.3.2 Contact Forces Considering Saturation
3.4.4 Contact Forces of Elastic Tires on a Rigid Road
3.4.4.1 The Brush Model
3.4.4.2 Contact Forces for Pure Lateral Slip
3.4.4.3 Contact Force for Pure Longitudinal Slip
3.4.4.4 Linear Contact Force Law
Longitudinal and Lateral Slip
4 Guideway Models
4.1 Models for Elastic Guideways
4.1.1 Models for Periodically Pillared Beams
Vibrations
4.1.3 Models for Continuously Bedded Beams
4.2 Perturbation Models for Rigid Guideways
4.2.1 Mathematical Description of Stochastic Processes
4.2.2 Models for Unevenness Profiles
4.2.3 Models for Vehicle Excitation Processes
5 Models for Vehicle-Guideway-Systems
5.1 State Equations of the Subsystems
5.2 State Equations of the Complete System
6 Assessment Criteria
6.1 Driving Stability
6.2 Ride Comfort
6.2.1 Deterministic Excitation
6.2.2 Stochastic Excitation
6.2.3 Shape Filter for the Human Perception
Whole-body Vibration
6.3 Ride Safety

3.4.1 Rolling of Rigid and Deformable Wheels . . . . . . . . . . 3.4.3 Contact Forces for Elastic Wheels on Elastic Rails . 3.4.4.5 Contact Forces for Simultaneous 4.1.2 Modal Analysis of Beam Structures for Bending 6.2.4 Revised Standards for Human Exposure to 6.4 Durability of Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2.1 Stability
7.2.2 Frequency Response Analysis
7.2.3 Random Vibration
7.2.3.1 Spectral Density Analysis
7.2.3.2 Covariance Analysis
7.3 Nonlinear Systems
7.3.1 Harmonic Linearization
7.3.2 Statistical Linearization
7.3.3 Investigation of Linearized Systems
7.4 Optimization Problems
8 Longitudinal Motions
8.1 Elastic Wheel
8.2 Entire Vehicle
8.3 Aerodynamic Forces and Torques
8.4 Driving and Braking Torques
8.5 Driving Performance
9 Lateral Motions
9.1 Handling of Road Vehicles
9.1.1 Elastic Wheel
9.1.2 Vehicle Model
9.1.3 Steady-state Cornering
9.1.4 Driving Stability
9.1.5 Experimental Studies
9.2 Driving Stability of Railways
9.2.1 Equation of Motion of a Railway Wheelset
9.2.2 Stability of a Free Wheelset
10 Vertical Motions
10.1 Principles of Vehicles Suspension
10.2 Random Vibrations of a Two Axle Vehicle
10.3 A Complex Vehicle Model
10.4 Magnetically Levitated Vehicles
Appendix: Optimal Control of Multivariable Systems
A.1 Mathematical Model
A.2 Task Formulation and Structure Issues
A.3 Structure and Properties of Controllers
A.4 Controller Design
A.6 Observer Design
A.6.1 Observer Design with Pole Assignment
Criterion

A.5 Structure and Properties of Observers ... .. .. .. .. .. .. .. .. A.6.2 Optimal Observer Due to a Quadratic Integral

Systems
Appendix: Key Words
B.1 English - German
B.2 Deutsch - Englisch
References
2.1 Longitudinal motion of an automobile16
2.2 Rotation matrix for a railway wheelset23
2.3 Angular velocity of a rigid body26
2.4 Angular velocity of a railway wheelset27
2.5 Relative motion during cornering30
2.6 Kinematic rolling of a cylinder35
2.7 Kinematic hunting of a railway wheelset38
2.8 Inertia tensor of a railway wheelset46
2.9 Equations of motion of a railway wheelset51
vehicle56
2.1 Lagrangian equations of motion for a differential gear63
of an automobile69
2.13 Equations of motion of a drawbar trailer81
with f = 10 degrees of freedom89
3.1 Mathematical model of a layered leaf spring104
3.2 Slip for a conical wheel127
contact135
3.4 Contact forces considering approximated saturation140
railway wheelset143
3.6 Contact forces for a road vehicle165

List of Problems and their Solutions 2.10 Inertia forces at a magnetically levitated (maglev) 2.12 Equations of motion for the bounce and pitch vibrations 2.14 Application of Neweul formalism on a vehicle model 3.3 Contact area and contact forces for wheel-rail 3.5 Contact forces and linear equations of motion for a 3.7 Contact forces and linear equations of motion for a drawbar trailer . . .. .. .. .. ... .. .. .. .. .. .. ... .. .. .. .. .. .. 170

controlled automobile221
6.1 Stability of a system of second order226

5.1 State equations of the vertical motion of an actively

7.2 Unbalance excitation of wheel vibrations248
7.3 Random vibrations of a single wheel252
7.4 Harmonic linearization of self-excited vibrations255
7.5 Harmonic linearization of a forced oscillator256
8.1 Control process of a vehicle wheel265
8.2 Acceleration of an automobile272
9.1 Driving stability of a road vehicle284

System Definition and Modeling

Ground vehicles ystems are composed by the vehicleb ody, the propu guidance ands uspension devices, andt he guideway, see Fig.1 .1. These ponents are interactingd ynamically with each other.A st he vehicle eling on the guideway, internal propulsion ands uspension forces as external disturbances are acting on the vehicleb ody. Furthermore, the tion oft he vehiclea ffects passengers andg oods carried on the vehic dynamical analysis oft his interplayo f orces andm otions ist he sub vehicled ynamics.T hisa nalysisr equires ani ntegrated treatment ofa l components interacting with each other.

The basis ofa theoretical analysis of vehicles ystems isa n appr mathematical model adapted to the given engineeringt ask.T he qua the results achievable on as ystem’sd ynamical behaviord epends on derlyingm odel. Therefore, the mathematical model has to be as deta possiblet o represent accurately andc ompletely all the essential pro oft he vehicles ystem. On the other hand, the model has to be as simp possiblet o allow efficienta ndf ast simulations oft he vehiclem otionse tial fort he manufacturers competing on ag lobal market.T hese confli requirements show that modeling isa difficulte ngineeringt ask,i n par forc omplex systems like today’s vehicles.

Fort he modeling it isa dvisablet o decompose the total system systems, e.g. the components shown in Fig.1 .1, with interfaces forf orces motionsc learly defined.T hen, the subsystems may be modeled separate andc omposed by modular assembly to the mathematical modelo ft he system.T he modular concept allowsd ifferentm odeling approaches individual subsystems,i t offers the required flexibility ford esignv ar and it is essential fort he lucidityo f large complex systems. On principle, there are two procedures known form odeling, see Fi

• the empirical approach, and • the axiomatic approach.

Guideway Guideway

Vehicleb ody disturbances

Propulsion, guidance, suspension devices disturbances andg oods Vehicle

Fig. 1.1. Vehicle system components

The empirical approach isb ased on measurements with prototy components or vehicles, respectively, which are processed byi dentifica methods resulting in am athematical model. Fort his purpose the syst der consideration ise xcited with known test signalsa ndt he respon recorded.F romt he input-output relation the parameters oft he mat ical model are identified consideringt he a-prior-knowledge on the str oft he system.T hism ethod isc alled parametric identification. Ift her information on the structure oft he system available, then ab lack box lem isg iven. In this case the structure oft he system and its parameter to be found, too, andt he method isc alled non-parametric identifica For linear time-invariants ystems reliable identification methods are able,L jung (1999) and Pintelon and Schoukens (2001). In vehicled yn the frequency response methodp roved to be successful but the cov methodc an be also applied.

The axiomatic approach results directly in am athematical model by cation off undamental, already mathematically described physical prin Thism odel emphasizes the structure oft he system as well as relationsb the system parameters.B ut some of the parameter values like dampin ficients may remain unknown. Fort he application oft he fundamental ples on complex systems it is oftenn ecessary to dealw ith more simples of idealized elements which are called physical models.T he already discussed

PhysicalPrototype model

State equations ofg lobal system of subsystem Mathematicalm odel

Physical principles identification

Measurements,

Subsystem

Fig. 1.2. Approaches of modeling: empirical (left) and axiomatic (right) conflicts related to mathematical modelinga pply to physical modelin The modelsh avet o be as simplea sp ossiblea nda s detailed as required

The mathematical description ofc omplex dynamical systems requ general ac ombination ofb oth approaches.W henever possible, the l pensivea xiomatic approach will be applied.T he missing parameter havet o be foundb y experiments, andt he final results shouldb e va empirically. The modeling isc onsidered to be satisfactoryi ft he theo predictionsc oincide with the experimental findings.

(Parte 1 de 4)

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