Railroad vehicle dynamics: a computational approach

45814_C000.fm Page i Thursday, June 14, 2007 1:05 PM RAILROAD VEHICLE DYNAMICS A Computational Approach 45814_C000...

Author: Ahmed A. Shabana | Khaled E. Zaazaa | Hiroyuki Sugiyama

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RAILROAD VEHICLE

DYNAMICS A Computational Approach

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RAILROAD VEHICLE

DYNAMICS A Computational Approach

Ahmed A. Shabana Khaled E. Zaazaa Hiroyuki Sugiyama

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-4581-9 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Shabana, Ahmed A., 1951Railroad vehicle dynamics : a computational approach / Ahmed A. Shabana, Khaled E. Zaazaa, Hiroyuki Sugiyama. p. cm. Includes bibliographical references and index. ISBN 978-1-4200-4581-9 (alk. paper) 1. Railroad cars--Dynamics--Mathematics. 2. Railroad cars--Mathematical models. I. Zaazaa, Khaled E. II. Sugiyama, Hiroyuki, 1974- III. Title. TF550.S48 2007 625.201’51--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

2007060396

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Table of Contents Preface.......................................................................................................................xi Acknowledgments....................................................................................................xv Chapter 1 1.1

1.2 1.3

1.4

1.5

1.6 1.7 1.8 1.9

Railroad Vehicles and Multibody System Dynamics......................................2 1.1.1 Generality .............................................................................................2 1.1.2 Nonlinearity..........................................................................................4 1.1.3 Implementation of Railroad Vehicle Elements....................................6 Constrained Dynamics .....................................................................................9 Geometry Problem .........................................................................................11 1.3.1 Differential Geometry ........................................................................12 1.3.2 Rail and Wheel Geometry .................................................................14 Contact Theories ............................................................................................17 1.4.1 Creep Forces ......................................................................................17 1.4.2 Wheel/Rail Creep Theories................................................................18 General Multibody Railroad Vehicle Formulations....................................... 18 1.5.1 Constraint Contact Formulation.........................................................19 1.5.2 Elastic Contact Formulation ..............................................................20 Specialized Railroad Vehicle Formulations...................................................20 Linearized Railroad Vehicle Models .............................................................23 Motion Stability .............................................................................................24 Motion Scenarios ...........................................................................................27 1.9.1 Hunting...............................................................................................28 1.9.2 Steady Curving...................................................................................28 1.9.3 Spiral Negotiation ..............................................................................30 1.9.4 Twist and Roll ....................................................................................30 1.9.5 Pitch and Bounce ...............................................................................31 1.9.6 Yaw and Sway....................................................................................31 1.9.7 Dynamic Curving...............................................................................31 1.9.8 Response to Discontinuities...............................................................32

Chapter 2 2.1 2.2

Introduction ........................................................................................1

Dynamic Formulations ....................................................................35

General Displacement ....................................................................................36 Rotation Matrix ..............................................................................................37 2.2.1 Direction Cosines...............................................................................38 2.2.2 Simple Rotations ................................................................................41 2.2.3 Euler Angles.......................................................................................41 2.2.4 Euler Parameters ................................................................................45

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2.3

Velocities and Accelerations ..........................................................................49 2.3.1 Velocity Vector ...................................................................................49 2.3.2 Acceleration Vector ............................................................................50 2.3.3 Generalized Orientation Coordinates.................................................51 2.3.4 Singular Configuration.......................................................................53 2.4 Newton-Euler Equations ................................................................................58 2.5 Joint Constraints.............................................................................................62 2.5.1 Spherical Joint....................................................................................62 2.5.2 Revolute Joint.....................................................................................63 2.5.3 Cylindrical Joint.................................................................................64 2.5.4 Prismatic Joint....................................................................................65 2.6 Augmented Formulation ................................................................................66 2.7 Trajectory Coordinates...................................................................................70 2.7.1 Velocity and Acceleration ..................................................................72 2.7.2 Equations of Motion ..........................................................................74 2.8 Embedding Technique....................................................................................76 2.8.1 Coordinate Partitioning and Velocity Transformation.......................77 2.8.2 Elimination of the Constraint Forces.................................................78 2.8.3 Reduced-Order Model........................................................................78 2.9 Interpretation of the Methods ........................................................................80 2.9.1 Kinematic and Dynamic Equations ...................................................80 2.9.2 Augmented Formulation ....................................................................83 2.9.3 Embedding Technique........................................................................84 2.9.4 D’Alembert’s Principle ......................................................................85 2.10 Virtual Work...................................................................................................86 Chapter 3 3.1

3.2

3.3 3.4 3.5 3.6

3.7

Rail and Wheel Geometry...............................................................89

Theory of Curves ...........................................................................................90 3.1.1 Arc Length and Tangent Line............................................................90 3.1.2 Curvature and Torsion........................................................................91 Geometry of Surfaces ....................................................................................92 3.2.1 Tangent Plane and Normal Vector .....................................................94 3.2.2 First Fundamental Form.....................................................................95 3.2.3 Second Fundamental Form ................................................................96 3.2.4 Normal Curvature...............................................................................99 3.2.5 Principal Curvatures and Principal Directions ................................100 Rail Geometry ..............................................................................................103 Definitions and Terminology .......................................................................106 Geometric Description of the Track ............................................................108 Computer Implementation ...........................................................................111 3.6.1 Track Segment Types.......................................................................112 3.6.2 Linear Representation of the Segments...........................................112 3.6.3 Derivatives of the Angles.................................................................114 Track Preprocessor.......................................................................................116

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3.8

3.7.1 3.7.2 3.7.3 3.7.4 Wheel

Chapter 4 4.1

4.2 4.3

4.4

5.2

5.3 5.4

5.5 5.6 5.7 5.8

Contact and Creep-Force Models ................................................127

Hertz Theory ................................................................................................128 4.1.1 Geometry and Kinematics ...............................................................128 4.1.2 Contact Pressure...............................................................................133 4.1.3 Computer Implementation ...............................................................138 Creep Phenomenon ......................................................................................140 Wheel/Rail Contact Approaches..................................................................145 4.3.1 Exact Theory of Rolling Contact.....................................................146 4.3.2 Simplified Theory of Rolling Contact .............................................147 4.3.3 Dynamic and Quasi-Static Theory ..................................................147 4.3.4 Three- and Two-Dimensional Theory..............................................147 Creep-Force Theories...................................................................................147 4.4.1 Carter’s Theory.................................................................................147 4.4.2 Johnson and Vermeulen’s Theory ....................................................149 4.4.3 Kalker’s Linear Theory ....................................................................150 4.4.4 Heuristic Nonlinear Creep-Force Model .........................................153 4.4.5 Polach Nonlinear Creep-Force Model .............................................154 4.4.6 Simplified Theory.............................................................................156 4.4.7 Kalker’s USETAB ............................................................................159

Chapter 5 5.1

Track Preprocessor Input .................................................................117 Numerical Integration ......................................................................118 Track Preprocessor Output ..............................................................120 Use of the Preprocessor Output during Dynamic Simulation ........121 Geometry ..........................................................................................123

Multibody Contact Formulations.................................................161

Parameterization of Wheel and Rail Surfaces.............................................162 5.1.1 Track Geometry ...............................................................................163 5.1.2 Wheel Geometry ..............................................................................165 Constraint Contact Formulations .................................................................165 5.2.1 Contact Constraints ..........................................................................166 5.2.2 Constrained Dynamic Equations .....................................................167 Augmented Constraint Contact Formulation (ACCF) ................................168 Embedded Constraint Contact Formulation (ECCF) ..................................171 5.4.1 Position Analysis..............................................................................172 5.4.2 Equations of Motion ........................................................................173 Elastic Contact Formulation-Algebraic Equations (ECF-A).......................174 Elastic Contact Formulation-Nodal Search (ECF-N)..................................177 Comparison of Different Contact Formulations..........................................178 Planar Contact ..............................................................................................179 5.8.1 Intermediate Wheel Coordinate System ..........................................181 5.8.2 Distance Traveled.............................................................................182

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5.8.3 5.8.4

Profile Parameters ............................................................................184 Coupling between the Surface Parameters ......................................185

Chapter 6

Implementation and Special Elements ........................................187

6.1

General Multibody System Algorithms.......................................................188 6.1.1 Constrained Dynamics .....................................................................188 6.1.2 Penalty and Constraint Stabilization Methods ................................189 6.1.3 Generalized Coordinates Partitioning ..............................................191 6.1.4 Identification of the Independent Coordinates ................................194 6.2 Numerical Algorithms — Constraint Formulations ....................................194 6.2.1 Augmented Constraint Contact Formulation (ACCF) ....................195 6.2.2 Embedded Constraint Contact Formulation (ECCF) ......................201 6.3 Numerical Algorithms — Elastic Formulations..........................................205 6.3.1 Elastic Contact Formulation Using Algebraic Equations (ECF-A)....206 6.3.2 Elastic Contact Formulation Using Nodal Search (ECF-N) ..............208 6.4 Calculation of the Creep Forces ..................................................................210 6.5 Higher Derivatives and Smoothness Technique ..........................................211 6.6 Track Preprocessor.......................................................................................214 6.6.1 Change in the Length Due to Curvature .........................................216 6.6.2 Use of the Preprocessor Output during Dynamic Simulation ........218 6.7 Deviations and Measured Data....................................................................219 6.7.1 Track Deviations ..............................................................................220 6.7.2 Measured Track Data .......................................................................222 6.7.3 Track Quality and Classes ...............................................................223 6.8 Special Elements ..........................................................................................225 6.8.1 Translational Spring-Damper-Actuator Element .............................227 6.8.2 Rotational Spring-Damper-Actuator Element .................................230 6.8.3 Series Spring-Damper Element .......................................................231 6.8.4 Bushing Element ..............................................................................232 6.9 Maglev Forces ..............................................................................................236 6.9.1 Electrodynamic Suspension (EDS)..................................................236 6.9.2 Electromagnetic Suspension (EMS) ................................................237 6.9.3 Modeling of Electromagnetic Suspensions .....................................237 6.9.4 Multibody System Electromechanical Equations............................240 6.10 Static Analysis..............................................................................................242 6.10.1 Augmented Constraint Contact Formulation...................................242 6.10.2 Embedded Constraint Contact Formulation ....................................244 6.10.3 Line Search Method.........................................................................245 6.10.4 Continuation Method .......................................................................246 6.11 Numerical Comparative Study.....................................................................247 6.11.1 Simple Suspended Wheelset ............................................................247 6.11.2 Complete Vehicle Model..................................................................248

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Chapter 7 7.1

7.2

7.3 7.4

7.5 7.6 7.7

General Displacement ..................................................................................256 7.1.1 Trajectory Coordinate System .........................................................256 7.1.2 Body Coordinate System .................................................................258 7.1.3 Generalized Trajectory Coordinates ................................................259 Velocity and Acceleration ............................................................................260 7.2.1 Velocity of the Center of Mass........................................................260 7.2.2 Acceleration of the Center of Mass.................................................261 7.2.3 Angular Velocity and Acceleration ..................................................262 Equations of Motion ....................................................................................264 Trajectory Coordinate Constraints...............................................................265 7.4.1 Numerical Example..........................................................................266 7.4.2 Use of the Cartesian Coordinates ....................................................269 Single-Degree-of-Freedom Model ...............................................................272 Two-Degree-of-Freedom Model ..................................................................277 Linear Hunting Stability Analysis ...............................................................280 7.7.1 Model 1 ............................................................................................287 7.7.2 Model 2 ............................................................................................288

Chapter 8 8.1 8.2

8.3 8.4 8.5 8.6 8.7 8.8

Specialized Railroad Vehicle Formulations .................................255

Creepage Linearization .................................................................291

Background ..................................................................................................291 Transformation and Angular Velocity..........................................................295 8.2.1 Matrix Identities...............................................................................295 8.2.2 Definition of the Angular Velocity ..................................................296 Euler Angles.................................................................................................298 Linearization Assumptions...........................................................................300 Longitudinal and Lateral Creepages............................................................301 Spin Creepage ..............................................................................................305 Newton-Euler Equations ..............................................................................306 Concluding Remarks....................................................................................309

Appendix A

Contact Equations.......................................................................313

Appendix B

Elliptical Integrals.......................................................................319

References .............................................................................................................321 Index......................................................................................................................333

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Preface The methods of computational mechanics have been used extensively in modeling many physical systems, including machines, vehicles, mechanisms and robotics, space structures, and biomechanical and biological systems, among many others. Multibody system techniques, in particular, have been used successfully in the study of various and fundamentally different applications. This success can be attributed to the generality and flexibility of these techniques, which facilitate tailoring the general formulations to a specific application. The aim of this book is to present a computational multibody system approach that can be used to develop complex models of railroad vehicle systems that include significant details. One of the important features that distinguish railroad vehicle systems from other multibody system applications is the vehicle/rail (guide) interaction. Special force or kinematic constraint elements must be included in the multibody system algorithm if the vehicle/rail interaction is to be accurately modeled. To accomplish this goal, interesting geometric problems that are particular to railroad vehicle systems must be addressed and solved. By considering these additional geometric variables, which are required to describe the vehicle/rail interaction, multibody system formulations can be modified and improved for use in the analysis of detailed railroad vehicle models. This book presents several computational multibody system formulations and discusses their computer implementation. The computational algorithms based on these general formulations can be used to develop general- and special-purpose railroad vehicle computer programs for use in the analysis of railroad vehicle systems, including the study of derailment and accident scenarios, design issues, and performance evaluation. The book focuses on the development of fully nonlinear formulations, supported by an explanation of the limitations of the linearized formulations that are frequently used in the analysis of railroad vehicle systems. This book is designed for an introductory course on railroad vehicle dynamics that is suitable for senior undergraduate and first-year graduate students. The students are expected to have knowledge of dynamics at the intermediate level and also have knowledge of basic vector and matrix algebra. This book can also be used as a reference by researchers and practicing engineers, who commonly use generalpurpose multibody system computer programs in the analysis, design, and performance evaluation of railroad vehicle systems. In this book, it is assumed that the components of the railroad vehicle system that have distributed inertia are rigid bodies; the generalization of the presented formulations to the case of flexible body dynamics is straightforward, as described in previous publications by the authors of this book. The chapters of the book are organized to guide the reader from basic concepts and definitions through a final understanding of the utility of fully nonlinear multibody system formulations in the analysis of railroad vehicle systems.

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Chapter 1 provides a brief introduction of basic concepts and definitions. The motivation for using multibody system approaches in the analysis of railroad vehicles is first discussed. The reader is then introduced to the subject of constrained dynamics, the geometry problem, contact theory, and general and specialized multibody railroad vehicle formulations. Linearized vehicle models and motion stability and scenarios are among the topics discussed in this chapter. Chapter 2 reviews the analytical methods used to develop general multibody system formulations. The generalized coordinates are defined, and methods for describing the general displacement of a rigid body in space are presented. The motion description is expressed in terms of a rotation matrix that defines the orientation of the rigid body in the global coordinate system. Newton-Euler equations that can be used to obtain the equations of motion of the three-dimensional rigid bodies are presented, and the reader learns how to use these equations to develop general formulations that can be used to solve for the body accelerations. In particular, the augmented Lagrangian formulation and the embedding techniques are discussed. The augmented Lagrangian formulation leads to a large system of equations that has a sparse matrix structure, while the embedding technique leads to a minimum set of strongly coupled equations. Chapter 3 discusses the geometry problem that is fundamental in railroad vehicle dynamics. The theory of curves and surfaces and the local invariant geometric properties are defined. Important definitions related to the rail and track geometry are introduced, and the reader learns how to describe and construct a track with an arbitrary shape using three input parameters that are used by the railroad industry. The wheel geometry is also introduced in this chapter, and the equations used to describe the wheel surface in terms of two geometric parameters are presented. Contact mechanics is fundamental in the analysis of railroad vehicle systems, and it is necessary to describe the vehicle/rail interaction. Chapter 4 reviews the contact theories. In particular, Hertz contact theory is discussed in detail, since it is widely used in the analysis of the wheel/rail contact. Creep forces are developed as the result of the wheel/rail contact, and several creep-force theories are discussed. Some of these theories are linear, while the others are nonlinear theories. Chapter 5 shows how to model the wheel/rail contact in multibody system formulations. Several formulations are presented. In some of these formulations, the wheel and rail surfaces are assumed to remain rigid. The contact between the wheel and the rail is described using kinematic algebraic constraint equations that do not allow for wheel/rail separation or penetration. The methods that employ constraint equations to describe the contact between the wheel and the rail are called constraint contact formulations. Other multibody formulations presented in this chapter assume that the wheel and rail surfaces can experience small local deformation in the contact region. The normal force of the wheel/rail interaction is obtained by using a compliant force model. In this case, wheel/rail separation and penetration are allowed. Methods that do not impose kinematic constraints to describe the contact are called elastic contact formulations. Both the constraint and elastic contact formulations are discussed in detail and compared in Chapter 5. Chapter 6 discusses the computer implementation of the formulations presented in this book as well as the formulation of special elements that are particular to

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railroad vehicle systems. The reader learns how to solve numerically the resulting system of differential and algebraic equations that govern the motion of the multibody vehicle systems. Special elements, such as the magnetic levitation forces used in magnetically levitated trains (Maglev), are also discussed. Numerical examples are presented at the end of this chapter to compare the results obtained using the different nonlinear formulations presented in Chapter 5. General and specialized formulations are often used in the analysis of railroad vehicle systems. General-purpose computer codes provide the flexibility and generality for building very detailed models and for exploiting advanced flexible body capabilities in a straightforward manner. Special-purpose computer codes, on the other hand, exploit the special features and characteristics of the railroad vehicle systems. Chapter 7 introduces the trajectory coordinates, which can be used to develop specialized formulations for railroad vehicle systems. The dynamic equations of motion are developed in terms of these trajectory coordinates. The resulting equations are used to obtain reduced-order models that provide insight into the stability and dynamics of railroad vehicles. Many of the existing formulations used in railroad vehicle dynamics employ linearization of the kinematic and dynamic equations. Chapter 8 examines the effect of this linearization. Using results from the literature, it is shown that linearization of the creep velocities can lead to significant errors in the prediction of the longitudinal and lateral forces. Such forces are important because they are factors in the calculation of some of the derailment criteria used in the railroad industry. To help in understanding the limits of linearization, the fully nonlinear equations are first developed and used to obtain the linearized kinematic equations. The effect of this linearization on the form of the Newton-Euler equations is also examined.

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Acknowledgments The authors would like to acknowledge the contributions of many colleagues and students to the development of this book. The materials presented in this book summarize research that has been sponsored for several years by the Federal Railroad Administration (FRA). The authors would also like to thank Dr. Magdy El-Sibaie and Mr. Ali Tajaddini of FRA for their support and encouragement during this project. Several researchers and engineers from Volpe National Transportation Systems Center, ENSCO Inc., Center for Automated Mechanics, National Transportation Safety Board, and Northern Illinois University have been involved in this project. The authors gratefully acknowledge the contributions of Professor Behrooz Fallahi, Mr. Erik Curtis, Dr. Alan Kushner, Mr. Brian Marquis, Dr. Kevin Renze, Dr. Jalil R. Sany, Mr. Amit Singh, and Dr. Brian Whitten. The chapters of this book were reviewed by our Ph.D. students Cheta Rathod, Graham Sanborn, and Tariq Sinokrot, whose efforts in proofreading the manuscript are very much appreciated. Thanks to Mr. Chris Guenzler and Mr. Jason Heineman for providing some of the figures for this book. Thanks also to our families for their patience and understanding during the time of preparation of this book. Ahmed A. Shabana Chicago, Illinois Khaled E. Zaazaa Springfield, Virginia Hiroyuki Sugiyama Osaka, Japan

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1

Introduction

Railroad vehicles are among the most widely used methods of transporting passengers and goods. Trains have been used in commerce for more than a century, but the last three decades, in particular, have seen significant progress in rail transportation technology. Some modern trains operate at high speeds to minimize cost and transportation time. As train operating speeds increase, safety and comfort remain paramount concerns. High-speed trains are designed to ensure safe operations by attempting to identify and eliminate the causes of derailment. Comfort, on the other hand, can be achieved by controlling undesirable vibration and noise sources. Modern trains are complex mechanical systems that can be better analyzed and designed using modern computational mechanics techniques. Developing detailed computer models of high-speed railroad vehicles is necessary to study vehicle performance, to improve existing designs and develop new ones, and to develop safety guidelines for different operating and loading conditions. Computer dynamic simulations have, in fact, come to play an integral role in railroad vehicle performance, in safety and accident evaluation, and in the design of new vehicles. Recent advances in the fields of computational mechanics and numerical methods are directly applicable in the nonlinear dynamic analysis of railroad vehicle systems. The aim of this book is to develop computational methods for the dynamic modeling of railroad vehicle systems as a tool for analyzing vibration and stability. The emphasis will be on developing fully nonlinear formulations and computational algorithms. Linearization techniques, which are often employed in developing algorithms for use in studying railroad vehicle system dynamics, are not adopted in this book, although Chapter 8 does examine the effects of linearizing the kinematic and dynamic equations of railroad vehicle systems. This chapter introduces some of the topics, concepts, and definitions that are discussed in the subsequent chapters of this book. First, the advantages of developing and using general multibody system algorithms in the analysis of railroad vehicles are summarized. In the second section of this chapter, the analytical methods that will be used in developing the dynamic equations of motion for railroad vehicles are described. The computational methods of constrained dynamics are highly recommended for use in developing detailed and accurate models for railroad vehicle systems. Two problems distinguish railroad vehicle systems from many other multibody system applications: the geometry and contact problems. The surface geometry of the wheel and rail, as well as the track geometric shape, enter into the formulation of the equations of motion and in the formulation of the wheel/rail contact. These two important geometry and contact problems are discussed in Sections 1.3 and 1.4. In Section 1.5, the implementation of the wheel/rail contact element in general multibody system algorithms is discussed. Several sets of coordinates can be used to develop the equations of motion of railroad vehicle systems. Some of these 1


2

Railroad Vehicle Dynamics: A Computational Approach

coordinates can be used to develop general-purpose computer formulations and codes, while others can be used to develop specialized and less general formulations and codes. This important topic of the coordinate selection is introduced in Section 1.6. It is common in the literature to employ linearization techniques in developing the dynamic equations of motion. However, as train operating speeds increase, one needs to employ fully nonlinear formulations to accurately predict the vehicle dynamic behavior. The issue of the linearization of the kinematic and dynamic equations is introduced in Section 1.7. Sections 1.8 and 1.9 discuss railroad vehicle stability and possible motion scenarios.

1.1 RAILROAD VEHICLES AND MULTIBODY SYSTEM DYNAMICS Multibody system dynamics is a branch of the general field of computational mechanics that is concerned with developing and solving the nonlinear equations that govern the motion of complex physical systems. The components of a multibody system can experience large rotations and displacements. The motion of these components, however, is subjected to kinematic constraints that are the result of mechanical joints and specified motion trajectories. Multibody system techniques are general and have been used in the analysis, design, and performance evaluation of numerous applications including vehicles, machines, space structures, biomechanical and biological systems, robotics and mechanisms, as well as many other applications. The equations of motion are developed in their most general form using the principle of mechanics. These equations are implemented on digital computers in order to develop programs that can automatically and systematically construct and numerically solve the equations of motion of a system that consists of an arbitrary number of bodies and joints. The dynamics of railroad vehicle systems, as will be shown in this book, can be systematically described using computational multibody system algorithms. There are several advantages for adopting multibody system methodologies in the computer-aided analysis of railroad vehicle systems. Among these advantages, which are discussed in this section, are their generality, their ability to systematically solve nonlinear problems, and the straightforward implementation of special railroad vehicle elements.

1.1.1 GENERALITY Multibody system formulations are designed to be general to facilitate development of vehicle models that include significant details. General forcing functions and motion constraints can be systematically introduced into the vehicle’s dynamic equations. Development of accurate and realistic dynamic models of railroad vehicle systems, such as the one shown in Figure 1.1, requires the inclusion of significant details. For example, trains may consist of a large number of cars connected by coupling elements. Each car, sometimes referred to as a vehicle, includes car body, bogies, suspension elements, bushings, bearings, as well as other components. The


Introduction

3

FIGURE 1.1 Railroad vehicles. (Courtesy of AMTRAK.)

bogies, such as the one shown in Figure 1.2, also represent complex systems that include frames and wheelsets that can have independent motions. The wheelsets, which can rotate freely about their own axes, are connected to the frame using primary suspensions, while the frame is connected to the bolster using a pin joint, and the bolster is connected to the car body using the secondary suspensions, as shown in Figure 1.2. The motion of the train is produced as the result of the friction between the rotating wheels and the rails. The forces of dynamic interaction between the wheels and the rails significantly influence the dynamics and stability of railroad vehicles. Clearly, a railroad vehicle system consists of a large number of interconnected components that experience independent relative motion. These components are connected by force elements such as springs, dampers, and bushings as well as joints that impose restrictions on the motion of the system. As will be shown throughout this book, the dynamics of such a complex system can be described using a system of differential and algebraic equations (DAE) that must be solved simultaneously. The subject of multibody systems, a branch of the field of computational mechanics, is devoted, as previously mentioned, to the formulation and numerical solution of the differential and algebraic equations of systems that consist of interconnected bodies. Using multibody system techniques, the equations of motion of railroad vehicle system models that include significant details can be systematically developed and numerically solved. Generality is one of the main advantages of using multibody system algorithms that provide, in addition to the systematic inclusion of general forces and constraints, the capabilities of modeling flexible bodies using the finite element method and other


4


FIGURE 1.2 Example of a bogie.

structural analysis techniques (Seo et al., 2005; Shabana, 1997, 2005). While flexible body dynamics will not be covered in this book, it is important to point out that, in the case of high-speed trains, the effects on the vehicle system dynamics due to deformations of the car bodies, rails, wheelsets, and the pantograph/catenary systems can be significant. By using a computational multibody system approach, the well-developed and advanced flexible multibody system formulations can be exploited in a straightforward manner in the dynamic analysis of railroad vehicle systems (Seo et al., 2005).

1.1.2 NONLINEARITY Multibody system algorithms are based on nonlinear formulations that can be used to accurately model the complex nonlinear behavior of railroad vehicle systems. By using these general nonlinear formulations, there is no need to resort to linearization techniques to obtain a solution for the vehicle dynamic equations. As will be discussed in this book, the use of the linear theory can lead to erroneous results in important simulation scenarios and can also lead to inaccurate prediction of the vehicle critical speed. Many of the books that are devoted to railroad vehicle system dynamics employ linearization techniques in formulating the dynamic equations of motion. In contrast, this text is focused on developing nonlinear formulations and describes the numerical algorithms that are used to solve the resulting system of differential and algebraic equations of motion, thereby obtaining dynamics and stability results that accurately represent the actual system behavior. Nonlinearities in railroad vehicle systems can be geometric or material nonlinearities. Geometric nonlinearity is due to the large rotation of some components of the vehicle system or due to large deformation of some of the elastic elements


Introduction

5

FIGURE 1.3 Nonlinearity due to the track geometry. (Courtesy of AMTRAK.)

and components. For example, each wheelset can have an arbitrary large rotation about its own axis. This large rigid body rotation introduces nonlinearities in the kinematic and dynamic equations, as will be demonstrated in Chapter 2. On the other hand, if the deformation of a suspension element is large, the use of a linear force/displacement relationship does not lead to an accurate model. In the case of large deformations, the forces depend nonlinearly on the displacements. Another source of geometric nonlinearity is the track geometry. Figure 1.3 shows a train that travels on a curved track. Curved-track geometry is a source of nonlinearity. Furthermore, the formulation of the problem of contact between the wheel and the rail must take into account the geometry of the wheel and rail surfaces. The formulation of the contact conditions, as shown in this book, requires the use of nonlinear kinematic relationships expressed in terms of the wheel and rail geometric surface parameters. Material nonlinearities arise when the force/displacement constitutive equations are nonlinear. This can be the case when a component or an element experiences plastic or viscoelastic behavior. For example, the dynamic interaction between the wheel and the rail requires the calculation of creep forces (Johnson, 1985; Kalker, 1990). The general form of the creep-force/displacement relationship is expressed in terms of nonlinear stiffness and viscoelastic coefficients. The calculation of these creep forces is essential in the dynamic analysis of railroad vehicle systems, and the constitutive laws used in the formulation of these forces are examples of material nonlinearities. Other examples of material nonlinearities are the use of nonlinear elastic, plastic, or viscoelastic coefficients in the formulation of the suspension, bearing, and bushing forces. Both geometric and material nonlinearities can be systematically incorporated into the multibody system formulations.


6

1.1.3 IMPLEMENTATION

Railroad Vehicle Dynamics: A Computational Approach OF

RAILROAD VEHICLE ELEMENTS

One important element that distinguishes railroad vehicles from other multibody system applications is the vehicle/rail interaction. There are two types of elements that are currently used in railroad vehicle systems. The first is the wheel/rail contact, while the second is magnetic levitation (Maglev). The wheel/rail element and the Maglev system are shown schematically in Figure 1.4. Generating train motion by using wheels rolling and sliding on rails remains the most widely employed method. As previously mentioned, the wheel/rail interaction is described in terms of creep forces, as well as other forces and kinematic variables, and leads to the known hunting phenomenon, which is a source of significant lateral and yaw oscillations that contribute to vehicle instability, particularly at certain operating speeds. Multibody system algorithms can also be used to study derailment scenarios and develop derailment criteria. One derailment criterion used in the literature is to measure the ratio between the lateral force L and the vertical force V acting on the wheel, as shown in Figure 1.5a. When the lateral force exceeds a certain limit, the momentum generated by this force can cause derailment. The L/V ratio can be predicted using fully nonlinear models based on multibody system algorithms. In practice, however, simplified approaches such as Nadal’s formula (Nadal, 1908) are often used to determine the limit of L/V ratio. Nadal’s formula is based on a simple force balance that can be used to determine the L/V ratio before a derailment occurs. As shown in Figure 1.5a, the lateral and vertical forces L and V that apply on the rail are in balance with the reaction forces N and F that apply on the wheel. Therefore, if the wheel rotates relative to the rail in the sense shown in Figure 1.5, and if the friction coefficient between the wheel and the rail is µ and the wheel flange angle is α, as shown in Figure 1.5, then the L/V ratio is given by the following simple formula: L tan α − µ = V 1 + µ tan α

(1.1)

Because of the sense of the wheel rotation shown in Figure 1.5, the wheel has a positive angle of attack (AOA), which is defined as the angle between the direction

FIGURE 1.4 Vehicle–track interaction.


Introduction

7

FIGURE 1.5 Force balance in the case of a wheel climb.

of the forward velocity of the wheel and the longitudinal tangent to the rail at the contact point. If the L/V ratio exceeds the right-hand side of Equation 1.1, wheel climb occurs. It is important to point out that Nadal’s formula, as defined by Equation 1.1, depends only on the wheel flange maximum contact angle, and it is only valid in the case of positive angle of attack. Therefore, one should in general use more accurate methods that can correctly predict the vehicle derailment. In this book, the general three-dimensional wheel/rail contact theory is discussed using several nonlinear multibody system contact formulations that can be used to study derailment of detailed railroad vehicle models. The problems that can be encountered when the L/V ratio exceeds a certain value are not limited to wheel climbs. For instance, if the lateral force generated by the second point of contact between the wheel flange and the rail is relatively high, this force can cause lateral rail displacement. This rail displacement produces what is known as gage widening that can lead to a wheel/rail separation, as shown in Figure 1.6. Furthermore, if the L/V ratio exceeds a certain limit, a rail rollover can occur, as shown in Figure 1.7. Rail rollover is one of the most common sources of accidents, especially when the vehicle travels over a spiral region of the track. In general, if the L/V ratio is higher than the ratio D/H, where D and H are as shown in Figure 1.7, the two forces L and V generate two opposite moments. If the moment generated by the lateral force is higher than the moment generated by the vertical force, the rail can rotate about its corner. Blader (1989) showed that if the contact point is located at the gage point on the rail, the L/V ratio must be limited to the range 0.73


8


FIGURE 1.6 Gage widening.

to 0.66, depending on the rail shapes. Clearly, as the contact point moves toward the corner of the rail, the acceptable range of the L/V ratio is reduced. Other sources of rail rollover are the wheel shape and the lubrication conditions of the rail and the center plate. Therefore, it is difficult to determine the acceptable limit of the L/V ratio by using simple models. For this reason, there is a need for computational methods that can be used to develop detailed models to study the dynamics and stability of railroad vehicle systems. In Maglev trains, on the other hand, there is no contact between the vehicle and the guide during the train motion, since magnetic forces are used to levitate the vehicle. The use of Maglev trains is limited to special applications, and such trains are still in the experimental stage. For this reason, most of the discussion in this book is focused on developing formulations for wheel/rail contact in railroad vehicle systems. Nonetheless, both wheel/rail contact and Maglev elements, as well as other elements particular to railroad vehicles such as the pantograph/catenary systems (Seo et al., 2005), can be systematically incorporated in multibody system algorithms, as will be demonstrated in this book.

FIGURE 1.7 Geometric parameters used to study rail rollover.


Introduction

9

1.2 CONSTRAINED DYNAMICS The equations of motion of mechanical systems can be formulated using the Newtonian or the Lagrangian approach. In the Newtonian approach, vector mechanics is used to define the forces. If the system consists of bodies connected by mechanical joints, free-body diagrams are constructed to show the joint reaction forces as well as the inertia and applied forces. The Newtonian approach, in which the equilibrium of each body is first studied separately, can be used for relatively simple systems, but it is not suited for the analysis of complex systems such as railroad vehicles. On the other hand, in the Lagrangian approach, which is based on D’Alembert’s principle, scalar quantities such as the virtual work and kinetic and potential energies can be used to develop the body equations of motion. There is no need in this case to study the equilibrium of the bodies in the system separately. The Lagrangian approach, which is based on the known Lagrange-D’Alembert equation, can be used to systematically eliminate the reaction forces or to keep these forces in the final form of the equations of motion, as discussed below. Before using the general multibody system approaches to formulate the nonlinear equations of motion of railroad vehicle systems, a set of coordinates called generalized coordinates that define the configuration of the system components must first be introduced. Different sets of coordinate types can be used as generalized coordinates in the dynamic formulations. In general formulations of railroad vehicle dynamics, two different sets of coordinates are often used: the absolute coordinates and the trajectory coordinates. The trajectory coordinates, which are often used in developing formulations tailored for railroad vehicle systems, are discussed in a later section of this chapter. In the absolute-coordinate formulations, the configuration of a rigid body in the multibody railroad vehicle system is defined using two sets of coordinates. The first set consists of three absolute Cartesian coordinates that define the global position vector of the origin of a selected body coordinate system, as shown in Figure 1.8, while the second set consists of three independent rotation parameters that define the orientation of the body coordinate system with respect to the global frame of reference. Among the sets of rotation parameters that can be used to describe the orientation of the moving body coordinate system are the direction cosines, Euler angles, and Euler parameters. As shown in Chapter 2, the absolute position vector of an arbitrary point on the body can be expressed in terms of the generalized absolute Cartesian and rotation coordinates. Using the absolute position vector, the absolute velocity and acceleration of an arbitrary point on the body can be determined and used with the Newton-Euler equations to obtain the body equations of motion. The motion of the components of the railroad vehicles is subjected to constraints that result from mechanical joints and specified motion trajectories. Thus, the formulation of the nonlinear algebraic equations that describe the motion kinematic constraints becomes necessary in the Lagrangian formulation. Chapter 2 presents the formulation of the kinematic constraints using a set of nonlinear algebraic equations for several commonly used mechanical joints. These constraint equations enter into the formulation of the dynamic equations of motion of the three-dimensional, constrained, multibody vehicle systems. When algebraic


10


FIGURE 1.8 Absolute coordinates.

constraint equations are present with the dynamic differential equations, one has a mixed system of differential and algebraic equations (DAE) that must be solved simultaneously. Two approaches are commonly used to solve the resulting system of differential and algebraic equations: the augmented formulation and the embedding technique. In the augmented formulation, Lagrange multipliers are used to combine the constraint equations with the system differential equations of motion, leading to a larger system of equations that has a sparse matrix structure. The embedding technique, in contrast, uses the kinematic constraint equations to systematically eliminate some of the coordinates, leading to a smaller system of equations that has a dense matrix structure. Both the augmented formulation and the embedding technique are discussed in Chapter 2. The remainder of this section discusses the concept of the system degrees of freedom and the relationship between the kinematic constraints, dependent coordinates, and reaction forces. Railroad vehicle systems consist of a large number of bodies that include wheelsets, bogie frames, car bodies, suspension elements, and other components. These bodies, as previously mentioned, are interconnected by mechanical joints and are subjected to different types of forces and loading conditions. The coordinates introduced to describe the motion of the bodies in the system and formulate the dynamic equations of motion are called, as previously mentioned, the generalized coordinates. If six coordinates (three translation and three rotation coordinates) are used to describe the unconstrained motion of a rigid body in the system, the number of the system generalized coordinates is equal to 6 multiplied by the number of bodies. These generalized coordinates, however, are not independent, since they are related by algebraic equations that represent mechanical joints and specified motion trajectories. The concept of the degrees of freedom is fundamental to the study of the motion of the constrained dynamic systems. The degrees of freedom are defined as


Introduction

11

the independent coordinates that are required to describe the configuration of the system. The number of degrees of freedom, therefore, depends on the number and types of mechanical joints, specified motion trajectories, and the number of coordinates used to describe the unconstrained motion of the bodies. Each algebraic constraint equation can be used to eliminate one coordinate by writing this coordinate in terms of the others. For a system that consists of nb rigid bodies subjected to nc algebraic constraint equations, the number of the system degrees of freedom nd for spatial problems, in which the unconstrained motion of a rigid body is described using six coordinates (three translations and three rotations), is given by nd = 6 × nb − nc

(1.2)

This equation is called the Kutzbach criterion. In this equation, 6 × nb is the total number of coordinates required to describe the unconstrained motion of the system, while the number of the constraint equations nc defines the number of dependent coordinates, which must be equal to the number of linearly independent constraint equations. It is shown in Chapter 2 that in the augmented formulation, one obtains a system of equations that has dimension equal to 6 × nb plus the number of constraint equations nc. This system can be solved for the body accelerations as well as the constraint forces. That is, the number of independent constraint forces is equal to the number of constraint equations, which is equal to the number of dependent coordinates. Preventing the motion in one direction is physically equivalent to introducing one reaction force, which is mathematically equivalent to introducing an algebraic constraint equation and an additional dependent coordinate. In the augmented formulation, the equations of motion are formulated in terms of redundant coordinates and the constraint forces. Since the algebraic constraint equations are not eliminated, one has a system of differential and algebraic equations that must be solved simultaneously. In the embedding technique, on the other hand, one obtains a number of acceleration equations equal to the number of the system degrees of freedom nd. The algebraic constraint equations are used to systematically eliminate dependent coordinates and the associated constraint forces. In this case, the minimum number of equations of motion is obtained, and the resulting system of differential equations does not include constraint forces. These differential equations can be integrated numerically using well-developed numerical differential equation solvers, since the algebraic constraint equations are eliminated. The loss of the sparse matrix structure is one of the main disadvantages of using the embedding techniques. Furthermore, computer codes based on the embedding techniques tend to be less general and less user friendly compared with the general-purpose computer codes that are based on the augmented formulation.

1.3 GEOMETRY PROBLEM The study of the geometry is necessary in the analysis of the wheel/rail contact in railroad vehicle systems. In fact, the computer-aided analysis of railroad vehicle


12


systems is generally divided into two main stages. The first stage is a preprocessing stage in which the track geometry and the wheel- and rail-surface profiles are defined. In the second stage, the equations of motion of the multibody vehicle system are numerically solved. In this second stage, the geometric parameters that define the wheel and rail surfaces and the track geometry enter into the formulation of the contact conditions and the system equations of motion. To develop general and detailed models of railroad vehicle systems, the multibody system algorithms used in the second stage, sometimes called the main processing stage, must be modified to include a wheel/rail contact model. Three steps are employed in the computational algorithm used in the main processing stage to obtain the numerical solution of the wheel/rail contact problem. First, the geometry of the contact surfaces of the wheels and the rails is used to determine accurately the locations of the points of contact between the wheels and the rails. Second, the kinematic variables are defined in terms of the geometric parameters of the wheel and rail surfaces. These variables include normalized kinematic quantities called creepages that measure the relative velocities between the wheels and the rails at the contact points. In the third step, the dynamic or kinetic forces that act on the wheels and the rails as the result of the contact are determined. The accuracy of the numerical solution of the contact problem depends strongly on the accurate prediction of the location of the contact points. The solution for the contact locations requires an accurate representation of the geometry of the wheel and the rail surfaces. This representation can be defined using local surface geometric properties, such as the radii of curvature and the tangent and the normal vectors to the surfaces. These geometric properties are not only important for determining the contact locations, but they are also important (as described in Chapter 4) in determining the forces that represent the interaction between the wheels and the rails. Therefore, basic knowledge of differential geometry is necessary to understand the wheel/rail contact problem. In particular, the theories of curves and surfaces are fundamental in the study of the dynamic interaction between the wheel and the rail. For example, in the case of curved tracks, the arc lengths of space curves are used to describe the distance traveled by the vehicle and to define the orientation of a rail coordinate system at the points of contact. The forces of the dynamic interaction between the wheels of the vehicle and the track can be defined in this rail coordinate system.

1.3.1 DIFFERENTIAL GEOMETRY The theories of curves and surfaces are covered in texts on the subject of differential geometry. A curve is defined as a real vector function that can be uniquely expressed in terms of one parameter, t. That is, the components of the vector function can be determined once this parameter is specified. Using this definition, a curve over the interval a ≤ t ≤ b can be written in the following form:

()

()

y t =  y1 t

()

y2 t

()

y3 t 

T

(1.3)

This equation, which is the parametric representation of a curve, can be used to determine the location of a point on the curve for an arbitrary value of the parameter


Introduction

13

FIGURE 1.9 Space curves.

t. Curves such as the one shown in Figure 1.9 can be parameterized using their arc length s. It will be shown in Chapter 3 that when the arc length is used as a parameter, the derivative of Equation 1.3 with respect to the arc length parameter defines a unit tangent. Further differentiations lead to the definition of invariant local geometric properties for the curve, such as the curvature and torsion. While curves can be described using one parameter, the description of the geometry of a surface requires the use of two independent parameters, as shown in Figure 1.10. Using a Cartesian coordinate system, each point on the surface is assumed to have a unique position vector x that can be defined in the three-dimensional space in terms of the two independent parameters as follows: x(s1, s2 ) =  x1 (s1, s2 )

x2 (s1, s2 )

x3 (s1, s2 ) 

T

(1.4)

where s1 and s2 are the parameters used to describe the surface geometry and are called the surface parameters. In order to apply differential calculus, Equation 1.4 must satisfy certain differentiability requirements, which will be discussed in Chapter 3. As in the case of curves, surfaces have invariant properties that can be

FIGURE 1.10 Geometry of surfaces.


14


used to uniquely define the surface geometry. Among these properties are the first and second fundamental forms of the surface and the Gaussian curvature. Knowing the surface geometry, one can define a tangent plane and a normal to this plane. At the point of contact between the wheel and the rail, one must be able to define the tangent plane and the normal in order to define the tangential creep forces and the normal force that enter into the dynamic formulation of the equations of motion. Furthermore, the geometric properties of the surfaces, such as the principal curvatures, are used to define the geometry and dimension of the contact area, as will be discussed throughout this book.

1.3.2 RAIL

AND

WHEEL GEOMETRY

The dynamic behavior of railroad vehicles depends on the wheel and rail geometry, and for this reason, it is important to accurately describe the wheel and rail geometry to correctly predict the vehicle response. The method used for the description of the surfaces of the wheels and rails should be general to allow representing arbitrary geometry of the surfaces. It is also important to be able to describe mathematically the wheel and rail profiles in a general form. For example, in the case of a straight segment of a track, the surface of the rail can be obtained by translation of the profile curve, as shown in Figure 1.11. This surface can be defined by the parametric equations u r =  s1r

s2r

f (s2r ) 

T

(1.5)

where sr1 is the distance along the rail (arc length) and is defined as the rail longitudinal surface parameter, and sr2 is the rail lateral surface parameter that is used

FIGURE 1.11 Rail surface.


Introduction

15

FIGURE 1.12 Gage and super-elevation.

as an independent variable to describe the rail profile. The complete description of the track geometry requires the use of several definitions that include, for example, the gage and super-elevation, which are shown in Figure 1.12. The gage G is defined as the lateral distance between two points on the inner faces of the right and left rails, while the super-elevation h is defined, as shown in Figure 1.12, as the vertical distance between the right and left rails. These definitions, among others, and their use in formulating the geometric equations of the track are discussed in Chapter 3. It will also be shown that the track geometry can be completely defined using three inputs: the horizontal curvature, which can be defined by projecting the space curve of the rail on the horizontal plane; the development angle, which defines the elevation of the rail; and the bank angle, which defines the track super-elevation. Using these three inputs, a track with a complex shape can be constructed in a straightforward manner, as described in Chapters 3 and 6. An efficient and systematic description of the track geometry is obtained using several standard track segment types shown in Figure 1.13. These segments include tangent (straight), curved, tangent-to-curve entry spiral, and curve-to-tangent exit spiral as well as other segment types, as discussed in Chapter 3. Simple geometry is used to describe the shapes of these segments to facilitate the development of an efficient mathematical formulation to define tracks with complex geometry. In general, as previously mentioned, the computer simulation of the nonlinear dynamics of railroad vehicle systems consists of two main stages. In the first stage, the track geometry and the wheel and rail profiles are defined, while in the second stage, the equations of motion of the railroad vehicle are formulated and solved using the geometry input obtained in the first stage. For the first stage, one often

FIGURE 1.13 Track segments.


16


FIGURE 1.14 Wheel geometry.

develops a preprocessor computer code that can be used to define tracks with arbitrary geometry. The track preprocessor code has input that is based on the definitions and terminologies used by the railroad industry. The output of the track preprocessor is a data file that is used as an input to the main processor computer code that is used to solve the dynamic equations of the multibody railroad vehicle system. Chapters 3 and 6 discuss the structure of a preprocessor computer code that can be used to define the track geometry. The wheel geometry also has a significant effect on the dynamics and stability of railroad vehicles. The critical speed of a vehicle strongly depends on the shape of the profiles of the wheels and rails. The surface of the wheel is a surface of revolution obtained by a complete rotation of the curve that defines the wheel profile about the wheel axis, as shown in Figure 1.14. Therefore, the surface of the wheel can be defined mathematically in a selected Cartesian coordinate system by the following equation:  x 0 + g(s1w )sin s2w    u w (s1w , s2w ) =  y0 + s1w   z − g(s w ) cos s w  0 1 2  

(1.6)

where s1w is the lateral surface parameter that represents the independent variable for the wheel profile g(s1w), and s2w is an angular surface parameter that represents the rotation of the wheel profile about its axis. The variables x0, y0, and z0 are constants. Tapered wheels tend to self-center as compared with cylindrical wheels, reducing the possibility of the contact between the wheel flange and the inner surface of the rail. If cylindrical wheels are used, the flanges wear rapidly due to the repeated contact with the inner side of the rail. Using a simplified model, it can also be shown that, if the taper were to be in the opposite sense, the wheelset would be unstable (Karnopp, 2004; Popp and Schiehlen, 1993). This unstable behavior is discussed in greater detail in Section 1.8.


Introduction

17

1.4 CONTACT THEORIES The wheel/rail contact is another important problem that must be addressed in the analysis of railroad vehicle systems. The contact analysis depends on the geometry of the wheel and rail surfaces, and for this reason it is important to first understand the geometry of the surfaces, as discussed in the preceding section. In railroad dynamics, a standard procedure is followed to determine the contact forces for a given wheel and rail configuration. First, the normal contact force is determined. This is the force normal to the plane tangent to the wheel and rail surfaces at the contact point. Second, using this normal force, the material and geometric properties of the wheel and the rail and the relative velocities between the wheel and the rail, the tangential creep force, and the spin moment can be determined. This section introduces the wheel/rail contact theories that are the subject of more detailed discussion in Chapter 4. In the discussion presented here and in Chapter 4, only the contact between one wheel and one rail is considered in order to focus on the procedure used for solving the contact problem. In the following section and in Chapter 5, the use of the contact formulation in general multibody system algorithms is discussed. Various contact models are used in different computer formulations to describe the wheel/rail interaction. In general, the contact between two rigid bodies can be at a single point or area, depending on the shape of the two bodies. These two types of contact are known as nonconformal and conformal contact, respectively (Johnson, 1985). If the shape of the two bodies is such that the two bodies in the region of contact fit exactly or even closely together, the contact is defined as a conformal contact. If the two bodies, on the other hand, touch at a point or a line, the contact is called nonconformal. If an external load is applied on each body, the two bodies will deform at the contact point to form an area of contact. The contact area in the case of the nonconformal contact is assumed to be small compared with the dimensions of the two bodies. The problem of nonconformal contact between two surfaces was studied by Hertz (1882). Hertz assumed that the area of contact is elliptical. In studying the wheel/rail dynamic interaction, the assumption of nonconformal contact is justified because of the shape of the wheel and rail surfaces. The contact region is assumed to be elliptical, and its dimension can be determined as described in Chapter 4 using the geometry of the two surfaces. To this end, one has to determine the principal curvatures of the two surfaces as well as the principal directions. Given the configurations of the wheel and the rail, several methods can be used to determine the normal contact forces. As will be discussed in the following section and in Chapter 5, the methods for determining the normal contact forces can be based on the assumption that the wheel and the rail surfaces are rigid or on the assumption that the two surfaces can deform in the small region of contact.

1.4.1 CREEP FORCES Due to the elasticity of the bodies and the external applied normal load, some points on the surfaces in the contact region may slip while others may stick when the two bodies move relative to each other. The difference between the tangential strains of the bodies in the adhesion area leads to a small apparent slip. This slip is called


18


creepage and is defined using the relative velocity between the two bodies at the contact point. Creepages generate tangential creep forces and creep spin moment, which play a fundamental role in the dynamics and stability analysis of railroad vehicles. The creep tangential forces and spin moment are generated during the wheel/rail interaction, since the motion of the wheel relative to the rail is a combination of rolling and sliding. These creep tangential forces and spin moment have a significant effect on the steering and stability of the railroad vehicle systems. For this reason, the creep phenomenon cannot be ignored in the analysis of railroad vehicle systems.

1.4.2 WHEEL/RAIL CREEP THEORIES There are several contact theories that have been developed to determine the tangential creep force and the spin moment for a given normal force and material and geometric properties of the wheel and rails. Some of these theories are two-dimensional, while others are three-dimensional; and some are based on linearized models, while the others are based on nonlinear models. Some of the theories are based on closedform expression for the relationship between the creep forces and the creepages, while other theories require numerical interpolation. Most of the creep-force models that are in use are a function of the geometry of the contact area. To determine the dimension and shape of the contact area for a given normal load, most of the models employ Hertz’s contact theory and assume that the contact area has an elliptical shape. Hertz’s contact theory can be used to determine the penetration of the two surfaces in contact as well as the contact ellipse dimension. While Hertz’s theory is developed for a static case and assumes frictionless contact, the use of the Hertz theory to determine the shape and dimension of the contact area is widely accepted by the railroad vehicle research community. Hertz’s contact theory as well as several wheel/rail creep-force theories are discussed in more detail in Chapter 4.

1.5 GENERAL MULTIBODY RAILROAD VEHICLE FORMULATIONS By establishing a procedure for determining the normal and tangential creep forces as well as the creep spin moment, such a procedure can be systematically implemented in a general multibody system algorithm that can be used in the analysis of railroad vehicle models that include significant details. As previously mentioned, the analysis of the wheel/rail interaction requires accurate prediction of the normal contact forces. The normal contact force and the geometric and material properties of the wheel and the rail are required for the evaluation of the creep tangential forces and spin moment. There are two main computational approaches that can be used in the multibody system formulations to predict the location of the contact points on-line and determine the normal contact force when the wheel/rail interaction is considered. In the first approach, the contact between the wheel and the rail is described using kinematic constraint equations that are imposed at the position, velocity, and acceleration levels. In this case, it is assumed that the wheel and rail surfaces do not penetrate. The normal contact force can be determined as the reaction


Introduction

19

force due to imposing the contact constraint equations. When equality constraints are used, it is assumed that there is no separation or penetration between the wheel and the rail. In the second approach, which is based on an elastic force model, the wheel and rail surfaces are allowed to have a small deformation in the contact region. The normal contact force that results from the wheel/rail interaction is predicted using a compliant force model with assumed stiffness and damping coefficients. In this approach, the wheel has six degrees of freedom with respect to the rail, and wheel/rail separation and penetration are possible. Chapter 5 discusses four nonlinear dynamic formulations for the analysis of the wheel/rail contact. Two of these formulations, which are based on the constraint approach, assume that the wheel and rail surfaces are rigid and employ nonlinear algebraic kinematic constraint equations to describe the contact between the wheel and the rail. The other two formulations, which are based on the elastic approach, assume that the wheel and rail surfaces can experience small local deformations, and the contact force is modeled using a compliant force element. The constraint and elastic approaches are conceptually different, and they lead to models with different numbers of degrees of freedom. The basic features of the constraint and elastic contact formulations are discussed in this section. A more detailed discussion is provided in Chapter 5.

1.5.1 CONSTRAINT CONTACT FORMULATION Figure 1.15 shows a wheel and a rail that are in contact. If no penetrations or separations are allowed in the case of nonconformal contact, the wheel can have five degrees of freedom with respect to the rail. These degrees of freedom are two translations in the tangent plane at the contact point and three relative rotations. In this case, it is assumed that there is no relative motion along the normal to the tangent plane at the contact point. This implies that imposing the nonconformal

FIGURE 1.15 Multibody system contact formulation.


20


contact conditions eliminates one degree of freedom; this is the freedom of the relative motion along the normal to the tangent plane at the contact point. Recall that the general description of the geometry of the contact surfaces requires introducing two surface parameters for each body. That is, for each contact between a wheel and a rail, one must define the geometry using four surface parameters. Since four geometric parameters are introduced and one degree of freedom is eliminated, imposing the nonconformal contact conditions requires, in general, the introduction of five kinematic constraint equations that can be used to eliminate the four geometric parameters and one degree of freedom for each contact. It is shown in Chapter 5 that the geometric surface parameters can be treated as nongeneralized coordinates because, when the equations of motion are formulated in terms of the surface parameters, there is no inertia or forces associated with these geometric parameters. Two equivalent constraint formulations that employ two different solution procedures can be developed. The first method leads to a larger system of equations by augmenting the dynamic equations of motion with all the contact constraint equations. In this augmented formulation (discussed previously in Section 1.2), the surface parameters can be selected as degrees of freedom. In the second method, on the other hand, an embedding procedure is used to obtain a reduced system of equations from which the surface parameter accelerations are systematically eliminated (discussed previously in Section 1.2). In this second method, the surface parameters cannot be selected as degrees of freedom.

1.5.2 ELASTIC CONTACT FORMULATION In the formulations based on the elastic approach, the wheel has six degrees of freedom with respect to the rail, and the normal contact force is defined as a function of the penetration using Hertz’s contact theory or using assumed stiffness and damping coefficients. Unlike the constraint contact formulation, the elastic contact formulation allows for wheel/rail separation and penetration. From the solution of the dynamic equations of motion, the position of the wheel with respect to the rail can be determined and used to check whether or not the wheel and rail surfaces penetrate. If penetration occurs, the penetration and its time rate can be used to determine the normal force. Clearly, accurate prediction of the location of the contact points is necessary for the robust implementation of the elastic contact formulations. One of the elastic methods discussed in this book in Chapter 5 is based on a search for the contact locations using discrete nodal points that are used to describe the wheel and rail surface profiles. In the second elastic approach discussed in this book, the contact points are determined by solving a set of nonlinear algebraic equations.

1.6 SPECIALIZED RAILROAD VEHICLE FORMULATIONS Two different strategies are often adopted when developing computational tools for a particular engineering application. The first strategy is based on developing algorithms that are tailored to that particular application. In this case, one can exploit the particular features of the application to optimize the computational algorithm. The disadvantage of using this approach is the difficulties that can be encountered


Introduction

21

when more-general scenarios or physical phenomena are considered. The second strategy is based on modifying existing multibody system algorithms to provide the capabilities for modeling that particular application. This second approach provides the flexibility of dealing with general scenarios and also allows exploiting advanced multibody capabilities that are already parts of the computational algorithms. Both the general and specialized approaches are discussed in this book. In this book, the absolute Cartesian coordinates are used to develop the general computer formulations for the nonlinear railroad vehicle system. This motion description leads to systematic and straightforward implementation of the contact formulations in existing general-purpose multibody system computer algorithms. This description also preserves the sparse matrix structure of the dynamic formulation and permits the exploitation of advanced and well-developed multibody system dynamics capabilities such as body flexibility. However, the absolute Cartesian coordinate description is not the only approach that has been used in the motion description of railroad vehicle systems, as many specialized codes adopt other sets of coordinates. Another set of coordinates that is often used in the specialized railroad vehicle formulations is trajectory coordinates. When these coordinates are used, the motion of an arbitrary body in the railroad vehicle system is defined, as described in Chapter 7, using coordinates that depend on the track geometry. A track coordinate system X irY irZ ir, called the body-trajectory coordinate system, which follows the motion of the body, is introduced in Figure 1.16; that is, each body i has its trajectory coordinate system. The location of the origin and orientation of the body-trajectory coordinate system can be uniquely defined by the arc length coordinate, which represents the distance traveled along the track space curve. This trajectory coordinate system is used to define the configuration of the body in the global coordinate system. To this end, another coordinate system, called the body coordinate system, XirY irZ ir, is introduced for each body, as shown in Figure 1.16. The origin of the body coordinate system is assumed to be attached to the body center of mass, and the body coordinate system is selected such that it has no displacement in the longitudinal direction of motion with respect to the trajectory coordinate system. Since the geometry of the track is assumed to be known, the complete definition of the trajectory coordinate system, which includes translation and orientation parameters, requires only one time-dependent coordinate: the distance traveled. On the other hand, the description of the motion of the body with respect to the trajectory coordinate system requires five time-dependent coordinates: two translations and three angles defined with respect to the trajectory coordinate system. This leads to the following set of trajectory coordinates of an arbitrary body i in the system: pi = [s i

yir

z ir

ψ ir

φ ir

θ ir ]T

(1.7)

where si is the arc length coordinate along the track space curve, and y ir and z ir are, respectively, the coordinates that define the location of the center of mass in the lateral direction and in a direction normal to the plane that contains the track space curve, as shown in Figure 1.16. These two coordinates are defined with respect to


22


FIGURE 1.16 Trajectory coordinates.

the trajectory coordinate system whose location and orientation are defined by the arc length si. The angles ψ ir, φ ir, and θ ir are, respectively, the yaw, roll, and pitch angles that define the orientation of the body with respect to the trajectory coordinate system. The use of the trajectory coordinates, compared with the absolute coordinates, has the advantage of simplifying the formulation of some railroad vehicle constraints and of some forcing functions. The trajectory coordinates, on the other hand, make the implementation of railroad vehicle system formulations in generalpurpose multibody system algorithms more difficult, can lead to the loss of the sparse matrix structure of the dynamic equations, and make exploiting advanced flexible body capabilities less straightforward. Using the generalized trajectory coordinates, one can easily develop different types of reduced-order vehicle models for different types of analysis, as discussed in Chapter 7. Single-degree-of-freedom vehicle models can be developed using the arc length coordinate, and these are often used to develop algorithms for the analysis of the train longitudinal forces (Meng et al., 2005). Such an analysis is important to avoid accidents and improve the operating efficiency for a particular train makeup. The forces in the couplers that join vehicles can be examined to determine whether or not these forces reach critical values that cause coupler failures. A two-degree-of-freedom model can also be developed using the trajectory coordinates to study the lateral stability of railroad vehicle systems, as discussed in Chapter 7. The lateral instability caused by the hunting phenomenon is the result of the coupling between the lateral and yaw displacements and the creep forces. For this reason, a two-degree-of-freedom model that has the lateral displacement and the yaw angle as independent coordinates can be used to examine the hunting instability. In this case, one has the following set of trajectory coordinates for an arbitrary body i: pi = [ yir

ψ ir ]T

(1.8)


Introduction

23

The two-degree-of-freedom model can be developed by imposing constraints on the other four trajectory coordinates. Since the trajectory coordinates are used as the generalized coordinates, such constraint equations become linear functions in the generalized trajectory coordinates, thereby leading to a simple procedure for eliminating the four dependent coordinates and the associated constraint forces from the dynamic equations of motion. As a result, one obtains a minimum set of differential equations of motion expressed in terms of the lateral and yaw trajectory coordinates. Chapter 7 provides a more detailed discussion on the trajectory coordinates and their use in formulating the equations of motion of railroad vehicle systems.

1.7 LINEARIZED RAILROAD VEHICLE MODELS Some existing specialized railroad vehicle system formulations employ linearized kinematic and dynamic equations. It is known that railroad vehicle models are sensitive to such a linearization, and formulations that employ kinematic linearization can predict, particularly at high speeds, significantly different dynamic response compared with models that are based on fully nonlinear kinematic and dynamic equations (Shabana et al., 2006). To examine this problem analytically and numerically quantify the effect of the approximations used in the linearized railroad vehicle models, the fully nonlinear kinematic and dynamic equations must first be obtained. The linearized kinematic and dynamic equations used in some railroad vehicle models can then be obtained from the fully nonlinear model to shed light on the assumptions and approximations used in the linearized models. The assumptions of small angles that are often made in developing railroad vehicle models and their effect on the angular velocity, angular acceleration, and the inertia forces can be investigated. The creepage expressions that result from the use of the assumptions of small angles can be obtained and compared with the fully nonlinear expressions. As previously mentioned, among the parameters required to evaluate the force of interaction between the wheel and the rail are the wheel and rail profile geometry data, the materials of the wheel and the rail, and the creepages that depend on the relative velocities between the wheel and the rail. To determine the longitudinal and tangential creep forces as well as the spin moment, the creepages are multiplied by very high creepage coefficients. Approximations used in the definition of the creepages can, therefore, lead to dynamic and stability results that differ from those predicted using the nonlinear models, particularly at high speeds. A study that compared different computer codes that are based on different formulations, some of which employ linearized creepage expressions, showed that the difference in the predicted critical speed between linearized railroad vehicle models and nonlinear railroad vehicle models can be very significant, exceeding 21 m/sec for some models (Iwnicki, 1999). This finding is significant, since the prediction of the critical speed is one of the main objectives of using railroad vehicle codes and computer formulations. Inaccurate prediction of the critical speed can have serious consequences and can negatively impact the accuracy of predicting derailment and accident scenarios and the evaluation of safety criteria. Chapter 8 examines the approximations used in the linearized creepages. The fully nonlinear expressions are first obtained and then used to derive the linearized


24


creepages. The basic assumptions used to derive the linearized creepages are summarized. It is shown that these assumptions are automatically satisfied if the roll angle is assumed to be zero. The numerical results presented in the literature show that the use of such assumptions can significantly influence the accuracy of the results (Shabana et al., 2006). This explains the significant differences between the railroad vehicle results predicted using computer codes that are based on fully nonlinear formulations and those codes that employ kinematic linearization of the creepages. In particular, the results presented in the literature clearly show that the linearization of the kinematic creepage expressions can lead to significant errors in the values predicted for the longitudinal and lateral creep forces as well as the spin moment (Shabana et al., 2006). These results also show the errors in the lateral and vertical forces that enter into the calculation of the L/V ratio that is used in some derailment criteria.

1.8 MOTION STABILITY One of the important characteristics of the motion of railroad vehicles is the hunting phenomenon. Hunting is defined as the lateral motion of the wheelset with respect to its initial (equilibrium) position. The wheelset consists of two wheels (right and left) that are connected by an axle. In general, the wheel shape is conical, with the largest diameter near the inner face of the rail, as shown in Figure 1.17. With this shape, the wheelset would automatically tend to self-center during the motion, and as a result there is less contact with the flange (Karnopp, 2004). A simple analysis can be used to explain this behavior. To this end, assume that the wheelset shown in Figure 1.17 travels over a track. Due to any excitation that is the result of track deviations or perturbations, the wheelset moves laterally. Assume that, at the initial (equilibrium) configuration, the wheelset has zero lateral position (y = 0) and the right and left wheels have initial rolling radii Rr and Rl , respectively, as shown in Figure 1.17. If the two wheels are identical and symmetrically located with respect to the axle, the two rolling radii Rr and Rl in the initial symmetric configuration are equal and are denoted as R0. The wheel conicity γ is defined by the slope of the

FIGURE 1.17 Wheelset rolling radii.


Introduction

25

wheel profile, as shown in Figure 1.17. As the wheelset moves laterally, the change ∆R in the two rolling radii can be determined using the following equation: ∆R = yγ

(1.9)

It follows that the right and left wheels’ rolling radii are given at any instant of time by Rr = R0 − yγ   Rl = R0 + yγ 

(1.10)

If the wheelset is rotating with a constant angular velocity ω, the velocities Vr and Vl of the right and left wheels are given, respectively, by Vr = Rrω   Vl = Rlω 

(1.11)

where Rr and Rl are as defined in Equation 1.10. The velocity of the center of the wheelset is V and is given by

(

V = Vr + Vl

)

2 = R0ω

(1.12)

If the yaw angle ψ shown in Figure 1.18 is assumed small (tan ψ ≈ ψ), the rate of change of the lateral motion can be determined as follows: y =

dy dy dx = = ψ V = ψ R0ω dt dx dt

FIGURE 1.18 Wheelset hunting motion.

(1.13)


26


In this simple analysis, the rate of change of the yaw angle ψ can be written as

(

)

ψ = Vr − Vl G = −2 yωγ G

(1.14)

where G is as shown in Figure 1.18. Differentiating Equation 1.13 with respect to time and substituting for ψ from Equation 1.14, one obtains  2 R ω 2γ  y+ 0 y=0  G 

(1.15)

This equation, which can be used to describe the lateral motion of the wheelset, is similar to the equation that governs the motion of a simple mass-spring system. If the coefficient of y in the preceding equation is positive, the solution of this equation can be written in the following form: y = A sin(ω nt + C )

(1.16)

where A and C are constants that can be determined using initial conditions, and ωn is the system undamped natural frequency, which is given upon the use of Equation 1.15 by

ωn = V

2γ R0G

(1.17)

The solution of Equation 1.16, which is obtained with the assumption that (ωn)2 is positive, represents a sustained oscillation with constant amplitude. In this case, the wheelset oscillates about its equilibrium position, as shown in Figure 1.19a. Clearly, the coefficient of y in Equation 1.15 is positive if the conicity γ is positive, as shown in Figure 1.17. The period of the oscillation is

λ=

2π 2π = ωn V

R0G 2γ

(1.18)

Equations 1.17 and 1.18 are known as Klingel’s formulas (Klingel, 1883). In the case of a cylindrical wheel, on the other hand, the conicity is equal to zero (γ = 0), and the coefficient of y in Equation 1.15 is equal to zero. The solution of Equation 1.15 in this case is a straight line, as shown in Figure 1.19b, and the wheelset motion due to a lateral disturbance is not oscillatory. In the case of a negative conicity, the coefficient of y in Equation 1.15 is negative, and the solution is an exponentially increasing function of time, as shown in Figure 1.19c. Therefore,


Introduction

27

(a)

(b)

(c)

FIGURE 1.19 Wheelset lateral motion.

based on the simple kinematics presented in this section, the wheel must have a positive conicity to have a stable oscillatory solution. It is important to point out that the simple analysis presented in this section is based on purely kinematic considerations and does not take into account the effect of any forces. In reality, the wheelset is subjected to friction forces due to the difference in the velocities of the wheel and rail at the contact point. This velocity difference is used to define the normalized relative velocities (creepages) that enter into the calculation of the creep forces that act on the wheel. Because of these creep forces, the wheelset lateral motion can be oscillatory about the equilibrium position with amplitude that increases or decreases with time, even in the case of positive conicity. It is also interesting to note by examining Equations 1.13 and 1.14 that, in the case of oscillatory motion, ψ = 0 corresponds to y = 0; that is, the yaw angle is maximum or minimum when the lateral displacement is zero and vice versa. This implies that there is a phase angle of π/2 between the lateral displacement and the yaw angle.

1.9 MOTION SCENARIOS We conclude this chapter with a discussion on some important railroad vehicle motion scenarios. These simulation scenarios are important in the design process of railroad vehicles. Most standard regulatory codes, such as the Federal Railroad Administration Code in the United States, require that a newly designed vehicle must be tested for such motion scenarios before the vehicle is put into service. During railroad vehicle operations, one or more of the following motion scenarios are encountered (Blader, 1989): hunting, steady curving, spiral negotiation, twist and roll, pitch and bounce, yaw and sway, dynamic curving, and response to discontinuities. In the remainder of this section, these motion scenarios are discussed


28


in greater detail. It is important, however, to point out that accurate computer modeling of some of these scenarios requires the use of fully nonlinear dynamic formulations and the use of a three-dimensional contact theory to describe the wheel/rail dynamic interaction.

1.9.1 HUNTING The hunting motion was discussed in the preceding section. As mentioned before, a wheelset hunts due to its shape, and the result is a lateral oscillation coupled with a yaw rotation. The resulting vibration must remain at a certain acceptable level in order to achieve specific comfort and safety requirements. Above a certain operating speed that depends on the railroad vehicle design, the vehicle may experience severe hunting that can be a source of discomfort or even the cause of derailment. The speed at which the railroad vehicle becomes unstable is called the critical speed. Above the critical speed, the vehicle is subjected to significantly higher forces due to the hunting phenomenon and the resulting impact between the wheel flange and the rail. The impact between the wheel flange and the rail is known in the railroad vehicle literature as the second point of contact, assuming that there is a first point of contact between the wheel tread and the rail surface. Below the critical speed, the second point of contact ensures vehicle stability and prevents derailments. However, if the L/V ratio increases beyond a certain limit due to severe hunting, derailment can occur. It is important, however, to mention that some wheel profiles are designed such that no second point of contact occurs, and the wheel will always have one point of contact with the rail. In this case, the forces generated will be at only one point. Regardless of the number of wheel contact points, it is important to know the critical speed of the railroad vehicle. This speed can be determined using the results of computer simulations or by experimental field testing. In general, one expects to find two instability regions for a vehicle; the first one occurs in a low-speed range and is associated with the instability of the car body, while the second occurs at higher speed and is associated with the bogie hunting motion. These two instability regions were identified by Matsudaira, who defined these two regions as primary and secondary hunting (Matsudaira, 1960). It is important to point out that the first type of instability can be easily controlled by using lateral dampers in the secondary suspension to reduce the amplitude of the lateral motion of the car body. The second type of instability, which occurs at higher speeds, is characteristic of the bogie system. This type of instability depends on the bogie suspension design and the wheel geometry (Valtorta et al., 2001).

1.9.2 STEADY CURVING When a vehicle travels over a constant curve (curve with a constant curvature), two important forces must be studied. The first force is the lateral force, which must be in balance with the centrifugal force due to the track curvature. The centrifugal force tends to push the vehicle out of the curve toward what is known as the high rail. Therefore, if the curve has a certain curvature and super-elevation, there is a balance speed at which the component of the centrifugal force is equal to the lateral component


Introduction

29

FIGURE 1.20 Balance speed.

of the gravity force, as shown in Figure 1.20. For instance, if the vehicle is traveling with a speed equal to V over a track that has a radius of curvature R, then at the balance speed, one must have the following relationship: mV 2 cos φ = mg sin φ R

(1.19)

where g is the gravity constant and m is the mass of the vehicle. In general, φ in the preceding equation is equal to the track angle φtrack plus the roll angle of the car, which is the result of the suspension elasticity. Assuming that φ is equal to φtrack and for general track, one can use the following small angle approximation: cos φ ≈ 1,

sin φ =

h G

(1.20)

where h is the track super-elevation and G is the track gage. Therefore, the balance speed is defined as follows:

V=

gRh G

(1.21)

If the vehicle is traveling with a velocity below the balance speed, the vehicle is said to have cant excess. Cant excess is defined as the amount of super-elevation that needs to be reduced so that the current vehicle speed will be equal to the balance speed. On the other hand, if the vehicle is traveling with a speed that is above the balance speed, the vehicle is said to have cant deficiency. Cant deficiency is defined


30


as the amount of the super-elevation that is needed to be increased so that the vehicle current speed will be equal to the balance speed. In most cases, the vehicle must be tested in both cases: cant deficiency and excess. In the case of cant deficiency, the vehicle has high lateral forces that can cause undesirable motion on the higher rail. These lateral forces, if high enough, will produce a wheel climb that can lead to a vehicle derailment. Another type of force that the vehicle is subjected to is the longitudinal force. When the wheelset negotiates a curved track, the outer rail has a larger radius of curvature than the inner rail. This requires the outer wheel to travel larger distance than the inner wheel. In this case, as the wheelset rotates with a constant angular velocity, one of the wheels (outer or inner) or both wheels will slip. The slip can be reduced if the rolling radii of the two wheels are allowed to vary during the wheel motion. This change in the rolling radius is accomplished by using the conical wheel profile. In the case of conical wheels, as the wheelset negotiates a curve, the wheelset will move laterally in the direction of the outer rail. Consequently, the outer wheel will have larger rolling radius and higher velocity in the longitudinal direction as compared with the inner wheel. This reduces the slip and wear, and leads to better curving behavior. Therefore, better curving behavior can be achieved by increasing the wheel conicity. Some modern designs of wheel profiles are not conical. These profiles are designed such that they consist of arcs that are developed based on worn wheels. These profiles have shapes that lead to improvement in stability and reduction of wear. In general, during curve negotiations, the wheelset tends to self-steer over the track as the result of the torque generated by the longitudinal forces acting on the right and left wheels. If the steering moment is not large enough or if the vehicle suspension connections generate an opposite moment that is higher than the moment generated by the longitudinal forces, the vehicle will fail to travel over the curve. This scenario can happen if the vehicle has a very stiff suspension.

1.9.3 SPIRAL NEGOTIATION Similar to the case of constant curve, the vehicle can be subjected to similar forces during spiral negotiation. However, the spiral, unlike the constant curve, does not have a constant curvature or super-elevation. Due to the change in the curvature and super-elevation, spirals are the sections of track where a significant number of derailments occur. In general, the vehicle travels with a constant speed over the spiral region. This speed, for a certain portion of the spiral, can be above the balance speed, leading to a sudden impact at a certain portion of the spiral between the wheel flange and the higher rail. Furthermore, as the vehicle changes its direction over the spiral area, twist is induced in the car body.

1.9.4 TWIST

AND

ROLL

The twist and roll motion can be the result of the response of the vehicle to periodic track cross-level variation. Cross-level variation can be caused by staggered rails or by the vehicle due to wheel lift. In general, it is important to examine the twist and roll scenario, particularly in the case of freight cars. Statistics show that a good


Introduction

31

percentage of freight car derailments are the result of cross-level variation, especially in the case of cars with high centers of gravity. The car-body roll can reach a dangerous level if the roll stiffness used between the car body and the center plate and sidebars is low. The result in this case is an increase in the roll amplitude if the vehicle speed and the track wavelength produce an excitation frequency that coincides with the natural frequency of the vehicle roll. Imposing a limit on the maximum value of the motion amplitude that the vehicle can have is important, and such a limit can be determined by simulations of the twist and roll motion scenarios.

1.9.5 PITCH

AND

BOUNCE

If the track has a vertical perturbation due to profile deviation, large pitch and bounce oscillations of the vehicle can be generated. The pitch motion is defined as the rotation about an axis across the track, while the bounce motion is defined as the vertical motion of the vehicle. If the suspension damping in these motion modes is low, oscillations that persist for a relatively long time occur as the result of track profile deviations. Therefore, it is important to properly select the suspension parameters to reduce the amplitude of these modes of vibration.

1.9.6 YAW

AND

SWAY

The yaw and sway is the response of the vehicle in the lateral and vertical directions due to track perturbation. For example, hunting can cause such a motion scenario. This type of motion leads to either high oscillations or high impact forces when the wheel flange comes into contact with the rail. In most cases, the effect of this motion scenario can be tested by using a track similar to the one shown in Figure 1.21. It is important to examine this dynamic behavior, since it is one of the most common sources of derailment.

1.9.7 DYNAMIC CURVING During curve negotiation, high impulsive lateral forces may be produced between the wheel and the rail as the result of rail irregularities. Computer simulations can be performed to examine the response of the railroad vehicle when it negotiates a curved track. In this case, profile and alignment deviations can be superimposed on a constant curve to obtain the desired track configuration. The acceptable vehicle

FIGURE 1.21 Example of a track that can be used to test yaw and sway motion.


32


FIGURE 1.22 Example of a track that can be used for dynamic curving.

models should lead to forces that are below the margin that causes wheel-climb. The dynamic-curving simulations must be performed for a wide range of the vehicle speed to test the stability and check for cant deficiency and cant excess. Figure 1.22 depicts an example of a track that can be used to test the dynamic-curving scenarios.

1.9.8 RESPONSE

TO

DISCONTINUITIES

Track discontinuities are defined as abrupt changes in the rail position. These discontinuities can occur when there is an abrupt change in track stiffness, misalignment, a change in soil properties, etc. For example, bridge abutments and road crossings can lead to track discontinuities. Such changes, if they are in the vertical direction, can cause a significant pitch and bounce motion of the vehicle. If the discontinuities are in the lateral direction, yaw and sway motion can be generated. Discontinuities, in general, lead to high impulsive tangential and normal forces that can cause derailment. Turnouts (switches and crossings) are used to change the travel direction of the train and are among several factors that can cause motion discontinuities (Kassa et al., 2006; Schupp et al., 2004). In general, the turnout is constructed using a switch panel (point section), a crossing panel (crossing section), and a closure panel (lead rail) that connects the switch panel with the crossing panel, as shown in Figure 1.23. For high-speed trains, a movable swing nose is used (Kono et al., 2005). As the


Introduction

33

FIGURE 1.23 Rail turnouts.

wheel contact switches from the stock rail to the tongue rail in the switch panel or the crossing panel, multiple wheel/rail contacts are possible, as shown in Figure 1.23. These contacts lead to severe impact forces. Furthermore, as the tongue rail changes its shape along the track, the wheel moves from the stock rail to the tongue rail. This transition causes a large disturbance in the wheel motion, and if the tongue rail is not close enough to the rail, derailment may even occur. The simulation of turnout crossings is often required in accident investigations. In this case, a method that can be used to describe the change of the rail profile as a function of the rail longitudinal distance must be adopted. The general-purpose contact formulations presented in Chapter 5 can be used in the simulation of such scenarios.


45814_C002.fm Page 35 Sunday, June 10, 2007 10:23 PM

2

Dynamic Formulations

Two approaches are often used to formulate the dynamic equations of motion of mechanical systems: the Newtonian and the Lagrangian approaches. In the Newtonian approach, vector mechanics is used to develop the dynamic equations. If the system consists of bodies connected by mechanical joints, free-body diagrams are constructed to show the reaction joint forces as well as the inertia and applied forces. The Newtonian approach, in which the equilibrium of each body is first studied separately, can be used for relatively simple systems and is not suited for the analysis of complex systems such as railroad vehicles. In the Lagrangian approach, which is based on D’Alembert’s principle, on the other hand, scalar quantities such as the virtual work and the kinetic and potential energies can be used to develop the body equations of motion. In this case, there is no need to study the equilibrium of the bodies in the system separately. The Lagrangian approach, which is based on the known Lagrange-D’Alembert equation, can be used to systematically eliminate the constraint forces or keep these forces in the final form of the equations of motion. The concept of the generalized coordinates is fundamental in the Lagrangian formulation of the equations of motion. Recall that the unconstrained motion of a rigid body can be described using six coordinates; three coordinates are used to describe the translation of a reference point on the body, and three coordinates are used to define the orientation of the rigid body in space. As will be shown in later sections of this chapter, the order of the finite rotation of rigid bodies is not commutative, and the components of the angular velocity vector of the rigid body are not, in general, the time derivatives of a set of orientation coordinates. Two different parameterizations of the finite rotation are commonly used in the analysis of multibody systems; the first is a set of three Euler angles, and the second is a set of four Euler parameters. In the case of the Euler angle representation, the orientation of a rigid body is defined using three successive rotations, while four parameters are used in the Euler parameter representation to avoid singularities encountered when three independent orientation parameters are used. In this book, the set of Cartesian translation and orientation coordinates is called the set of absolute generalized coordinates. Other sets of generalized coordinates can also be selected, as will be discussed elsewhere in this book. This chapter presents general methods that can be used in the computer formulation of the equations of motion of multibody systems consisting of rigid bodies. The generalized coordinates that define the global position of the origin and the orientation of a selected body reference frame are introduced and used to define the relationships between the angular velocity vector and the time derivatives of the generalized orientation coordinates. Expressions for the global velocity and acceleration vectors of an arbitrary point on the body are developed and used to derive the equations of motion in the three-dimensional space. When the absolute Cartesian

35


36


generalized coordinates are used, one obtains the Newton-Euler equations of motion, in which there is no inertia coupling between the translational and rotational coordinates of the rigid body. In railroad vehicle systems, the absolute Cartesian coordinates that define the configurations of the vehicle components can be related because of mechanical joints or specified motion trajectory constraints. This chapter discusses the formulation of the algebraic joint constraint equations. Using these constraint equations, two different methods for formulating the equations of motion can be used. In the first method, the augmented formulation, the constraint equations augment the system differential equations of motion, leading to a large system that has a sparse matrix structure. In the second method, the embedding technique, the constraint equations are used to systematically eliminate the dependent variables and the constraint forces, leading to a smaller system of dynamic equations expressed in terms of the system degrees of freedom. Both the augmented formulation and the embedding technique are discussed in this chapter. The trajectory coordinates, which are used to develop special-purpose computational algorithms for railroad vehicle systems, are briefly discussed in this chapter. A more detailed discussion of these coordinates is presented in Chapter 7.

2.1 GENERAL DISPLACEMENT In the three-dimensional analysis, the unconstrained motion of a rigid body is described using six independent generalized coordinates: three independent coordinates define the translation of a selected reference point on the body, and three coordinates define the body orientation. For an arbitrary rigid body i, as the one shown in Figure 2.1, the translational motion can be defined using the global position of the reference point O i that is fixed on the body, while the orientation of the body

FIGURE 2.1 Coordinates of rigid body i.



37

can be defined using the direction cosines of the axes of the body coordinate system XiY iZ i. Using this description, the global position vector of an arbitrary point on the rigid body i can be written as ri = R i + A i u i

(2.1)

where Ri is the global position vector of the origin of the body coordinate system defined as R i = [ R xi

Ryi

Rzi ]T

(2.2)

and Ai is a 3 × 3 rotation matrix that defines the orientation of the axes of the body coordinate system with respect to the global coordinate system. The vector u i is the position vector of the arbitrary point on the body with respect to the origin of the body coordinate system and is defined in terms of its components as u i = u xi

uyi

T

uzi  = [ x i

yi

z i ]T

(2.3)

In this equation, u xi = x i , uyi = y i , and uzi = z i are the local coordinates of the arbitrary point defined in the body coordinate system. In rigid body dynamics, the local position vector u i is assumed to be constant and does not depend on time. The location of the origin of the body coordinate system (reference point) can be selected arbitrarily. Nonetheless, it is advantageous to have a centroidal body coordinate system, which has an origin that is rigidly attached to the center of mass of the body. The use of such a centroidal body coordinate system leads to the well-known Newton-Euler equations, which do not include inertia coupling between the translation and rotation of the body.

2.2 ROTATION MATRIX This section discusses several methods for defining the orientation matrix Ai given in Equation 2.1. In particular, three methods are addressed: the direction cosines, Euler angles, and Euler parameters. The method of the direction cosines is rarely used in computational dynamics, since it requires the use of nine parameters that are not independent. In this case, six constraint equations must be imposed. The method of Euler angles, which is widely used, employs three independent parameters that can be used to define the body orientation in space. This method, however, suffers from a singularity problem at certain configurations, and for this reason, the method of Euler parameters, which employs four coordinates to define the body orientation in space, is often used. In this case, one must impose one algebraic constraint equation, since only three independent parameters are required to describe general three-dimensional rotations.


38


2.2.1 DIRECTION COSINES Let ii, ji, and ki be unit vectors along the Xi, Y i, and Z i axes of the body coordinate system, respectively. The components of these three vectors can be defined in the global coordinate system. These unit vectors can be written in terms of their components along the unit vectors i, j, and k along the X, Y, and Z axes of the global coordinate system as follows: i i i ii = α11 i + α12 j + α13 k    i i i ji = α 21 i + α 22 j + α 23 k  i i i i + α 32 j + α 33 k  ki = α 31

(2.4)

The elements αij represent the components of the orthogonal unit vectors ii, ji, and ki along the respective X, Y, and Z axes of the global coordinate system. Since the unit vectors i, j, and k are also orthogonal, the elements αij can be defined using the preceding equation as follows: i α11 = ii ⋅ i

i α12 = ii ⋅ j

i α13 = ii ⋅ k

i α 21 = ji ⋅ i

i α 22 = ji ⋅ j

i α 23 = ji ⋅ k

i α 31 = ki ⋅ i

i α 32 = ki ⋅ j

i α 33 = ki ⋅ k

      

(2.5)

where the nine scalar components αij (i, j = 1, 2, 3) are called the direction cosines. Let ui be a vector whose components are defined in the body coordinate system along the axes Xi, Y i, and Z i by the scalar coordinates xi, yi, and zi. This vector can then be written as follows: u i = x i ii + y i ji + z i k i

(2.6)

Substituting Equation 2.4 into Equation 2.6, one obtains u i = u xi i + uyi j + uzi k

(2.7)

i i i i i i  uxi = α11 x + α 21 y + α 31 z   i i u iy = α12 x i + α 22 yi + α 3i 2 z i   i i i i i  uzi = α13 x + α 23 yi + α 33 z 

(2.8)

where



39

The preceding equations can also be written in a matrix form as follows: ui = Aiui

(2.9)

where u i = [u xi

uyi

uzi ]T

ui = [ x i

yi

z i ]T

and i α11  i Ai = α12  i α13

i   ii ⋅ i α 31   i i α 32  = i ⋅ j i  i α 33  i ⋅ k

i α 21 i α 22 i α 23

ji ⋅ i ji ⋅ j ji ⋅ k

ki ⋅ i   ki ⋅ j  ki ⋅ k 

(2.10)

Since the components of the unit vectors i, j, and k along the X, Y, and Z axes of the global coordinate system are simply defined by i = [1

0

0]T ,

j = [0

0]T ,

1

k = [0

0

1]T ,

(2.11)

the matrix Ai given by Equation 2.10 can be rewritten simply as A i = [ii

ji

ki ]

(2.12)

It is clear from this equation that the columns of the transformation matrix Ai are ii, ji, and ki, which define unit vectors along the Xi, Y i, and Z i axes of the body coordinate system, and the elements of the matrix Ai are the direction cosines that define the components of these unit vectors in the global coordinate system. The analysis presented in this section shows that the transformation matrix that relates the components of vectors defined in the body coordinate system to their global components can be constructed using the nine direction cosines αij (i, j = 1, 2, 3). However, these nine direction cosines are not independent due to the fact that three independent parameters are sufficient to describe the orientation of the rigid body in space. Since the direction cosines represent the components of three orthogonal unit vectors defined in the global coordinate system, the following orthogonality condition must be satisfied by the elements of the transformation matrix Ai: T

T

Ai Ai = Ai Ai = I

(2.13)

where I is a 3 × 3 identity matrix. The preceding equation leads to a total of six independent constraint equations that impose the orthonormality (orthogonal unit vectors) conditions on the vectors ii, ji, and ki, leading to only three independent orientation parameters.


40


EXAMPLE 2.1 The Xi and Y i axes of the coordinate system of a rigid body i are given in the global coordinate system by the vectors [0.0 1.0 1.0]T and [−1.0 1.0 −1.0]T, respectively. Determine the transformation matrix that defines the orientation of body i in the global system. Solution. The unit vectors along the X i and Y i axes can be obtained as follows:

i = i

 0.0   0.0      1.0  =  0.7071 2 2  (1.0) + (1.0)    1.0   0.7071 1

and

j = i

 −1.0   −0.5774      1.0  =  0.5774  2 2 2  (−1.0) + (1.0) + (−1.0)   −1.0   −0.5774  1

The unit vector along the Z i axis must be perpendicular to both vectors ii and ji, and the direction of this vector is determined using the right-hand rule. This unit vector can be determined using the following cross-product:

ki =

i i × ji

ii × ji

Note that, since the vectors ii and ji are unit vectors and mutually orthogonal, the scalar ii × ji is equal to 1. It follows that the unit vector ki along the Z i axis is given by

 −0.8165    k =  −0.4082   0.4082  i

Using the orthogonal triad ii, ji, and ki, the transformation matrix of the coordinate system of body i can be determined as

A i = [i i

ji

 0.0 

−0.5774

−0.8165 

0.5774

 0.7071

−0.5774

−0..4082 

k i ] =  0.7071



0.4082 



41

2.2.2 SIMPLE ROTATIONS The transformation matrix as the result of simple finite rotations of the coordinate system XiY iZ i about the axes of the global coordinate system XYZ can be obtained as a special case of the direction cosines transformation matrix. For example, since the columns of the transformation matrix Ai are unit vectors along the Xi, Y i, and Z i axes of the body coordinate system, a simple rotation θz of the coordinate system XiY iZ i about the global Z axis leads to the following global definition of the unit vectors along the Xi, Y i, and Z i axes:  cos θ z    i =  sin θ z  ,  0    i

 − sin θ z  0      i j =  cos θ z  , k =  0    1  0    i

(2.14)

Using these three orthogonal unit vectors, the transformation matrix Ai as the result of the rotation θz can be defined as  cos θ z  A =  sin θ z  0  i

− sin θ z cos θ z 0

0  0 1 

(2.15)

Similarly, a rotation θy of the coordinate system XiY iZ i about the global Y axis leads to the following rotation matrix:  cos θ y  A = 0  − sin θ y  i

0 1 0

sin θ y   0  cos θ y  

(2.16)

while a rotation θx about the global X axis leads to the following rotation matrix: 1  A = 0 0  i

0 cos θ x sin θ x

0   − sin θ x  cos θ x 

(2.17)

The transformation matrices given in Equations 2.15 to 2.17 for the simple rotations can be used to develop the transformation matrix in the case of moregeneral three-dimensional rotations using the method of Euler angles.

2.2.3 EULER ANGLES In the method of Euler angles, three simple successive rotations are used to define the orientation of the rigid body in space. Using the three Euler angles associated


42


with three independent axes, the orientation matrix Ai given in Equation 2.1 can be defined as the product of three simple rotation matrices as follows: A i = A1i A 2i A 3i

(2.18)

where A ik (k = 1, 2, 3) are, respectively, the rotation matrices defined in terms of the following three Euler angles: θ i = [θ1i

θ 2i

θ 3i ]T

(2.19)

One can choose an appropriate sequence of the three successive rotations to reach any orientation in space. For example, the orientation of the wheel shown in Figure 2.2 can be conveniently described using the following three successive rotations: θ i = ψ i

φi

θ i 

T

(2.20)

Let XYZ and XiY iZ i be, respectively, the global and the wheel-body coordinate systems, which initially coincide. Rotation of the wheel-body coordinate system XiY iZ i by an angle ψ i (yaw) about the Z i axis leads to the rotation matrix  cos ψ i  A1i =  sin ψ i  0 

− sin ψ i cos ψ i 0

0  0 1 

(2.21)

A second rotation φ i (roll) of the wheel coordinate system XiY iZ i about the Xi axis leads to the rotation matrix 1  Ai2 =  0 0 

FIGURE 2.2 Euler angles.

0 cos φ i sin φ i

0   − sin φ i  cos φ i 

(2.22)



43

Finally, the wheel coordinate system XiY iZ i is rotated by an angle θ i (pitch) about the Y i axis. This rotation defines the transformation matrix  cos θ i  A = 0  − sin θ i  i 3

0 1 0

sin θ i   0  cos θ i 

(2.23)

Using the preceding three simple rotations, the orientation of the wheel can be defined in the global coordinate system by substituting Equations 2.21 to 2.23 into Equation 2.18 to obtain the following transformation matrix: Ai =  cos ψ i cos θ i − sin ψ i sin φ i sin θ i  i i i i i sin ψ cos θ + cos ψ sin φ sin θ i i  − cos φ sin θ 

(2.24) − sin ψ i cos φ i cos ψ i cos φ i sin φ i

cos ψ i sin θ i + sin ψ i sin φ i cos θ i   sin ψ i sin θ i − cos ψ i sin φ i cosθ i   cos φ i cos θ i 

As previously mentioned, the three column vectors of the matrix Ai define unit vectors along the Xi, Y i, and Z i axes of the body coordinate system. The components of these three unit vectors are defined in the global coordinate system. It is important to point out that, in general, the order of the finite rotations in the three-dimensional analysis is not commutative, that is, A1i A 2i A 3i ≠ A 3i A i2 A1i . One of the major drawbacks of using three independent parameters such as Euler angles is the existence of singular configurations that result when the three Euler angle axes of rotation become dependent (Roberson and Schwertassek, 1988; Shabana, 2001). For the sequence of Euler angles used in this section, the singular configurations occur when the angle φ i is equal to ±π/2 (−π < φ i ≤ π). In such a case, the axes of rotation associated with the angles ψ i and θ i are parallel, as shown in Figure 2.3, and therefore, the angle ψ i cannot be distinguished from the angle θ i. A similar singularity problem is encountered when using any known method that employs three parameters to describe the orientation of the rigid body in the three-dimensional

FIGURE 2.3 Singular configuration of Euler angles.


44


space; therefore, all Euler angle representations suffer from the singularity problem. As will be discussed in the next section, at the singular configurations, the time derivatives of the three Euler angles cannot be defined in terms of the components of the angular velocity vector.

EXAMPLE 2.2 Obtain the transformation matrix in terms of Euler angles if the sequence of rotation is defined as follows: a rotation φ i about the Z i axis, a rotation θ i about the Xi axis, and a rotation ψ i about the Z i axis. Solution. The rotation matrix resulting from the angle φ i about the Z i axis is given as

 cos φ i  A1i =  sin φ i  0 

− sin φ i

0

cos φ

0



i

1 

0

The rotation matrix resulting from the angle θ i about the Xi axis is given as

1 

A i2 =  0  0

 

0

0

cos θ i

− sin θ i  cos θ i 

sin θ i

Finally, the rotation matrix resulting from the angle ψ i about the Z i axis is given as

 cos ψ i  A i3 =  sin ψ i  0 

− sin ψ i

0

cos ψ

0

0

i



1 

Using the simple rotation matrices obtained in this example for the three successive rotations, the rotation matrix that defines the orientation of the body in the global coordinate system is given by A i = A1i A i2 A i3

 cos ψ i cos φ i − cos θ i sin φ i sin ψ i  =  cos ψ i sin φ i + cos θ i cos φ i sin ψ i  sin θ i sin ψ i 

− sin ψ i cos φ i − cos θ i sin φ i cos ψ i

sin θ i sin φ i 

− sin ψ i sin φ i + cos θ i cos φ i cos ψ i

− sin θ i cos φ i 

sin θ i cos ψ i



cos θ i

One can show that the preceding transformation matrix is orthogonal, that is, AiT Ai = Ai AiT = I.

 



45

2.2.4 EULER PARAMETERS In order to avoid the singularity problem associated with the three-parameter representation, the four Euler parameters are often used in the computer-aided analysis of multibody systems. As shown in Figure 2.4, the change in the orientation of an arbitrary vector r i can be defined using the three components of a unit vector vi along the instantaneous axis of rotation and the angle of rotation θ i about this axis. Using these four parameters, the vector ri, shown in Figure 2.4, can be obtained by transforming the vector r i using a rotation matrix that is a function of the rotation θ i and the three components of the unit vector vi, as follows: ri = A i r i

(2.25)

where the transformation matrix Ai is given as (Roberson and Schwertassek, 1988; Shabana, 2005) Ai = I + v i sin θ i + 2(v i )2 sin 2

θi 2

(2.26)

where I is a 3 × 3 identity matrix, and v i is a skew-symmetric matrix associated with the unit vector vi and is defined as  0  i v =  v3i − vi  2

FIGURE 2.4 Rodriguez formula.

− v3i 0 v1i

v2i   − v1i  0 

(2.27)


46


where v1i , v2i , and v3i are the components of the unit vector vi. Equation 2.26 is known as the Rodriguez formula. Since the components of the unit vector vi must satisfy ( v1i )2 + ( v2i )2 + ( v3i )2 = 1

(2.28)

the rotation matrix is function of only three independent parameters. To demonstrate the use of Equation 2.26, a simple rotation about the global Z axis is considered. In this case, vi = [0 0 1]T and the transformation matrix defined by Equation 2.26 leads to the following simple rotation matrix: 1  i A = 0 0 

0 1 0

 cos θ i  =  sin θ i  0 

0  0   0  + sin θ i 1   0

0   cos θ i − 1   0 0 +    0 0 

− sin θ i 0 0

0  0 1 

− sin θ i cos θ i 0

0 cos θ i − 1 0

0  0 0 

(2.29)

This is the matrix previously obtained using the direction cosines. The four Euler parameters are defined using the angle of rotation θ i and the three components of the unit vector vi as follows:

θ 0i = cos

θi   2  θi  i i θ 3 = v3 sin  2

θi , 2

θ1i = v1i sin

θi θ = v sin , 2 i 2

i 2

(2.30)

Using these four parameters, the generalized orientation coordinates can be defined as θ i = [θ 0i

θ1i

θ 2i

θ 3i ]T

(2.31)

These four Euler parameters are not totally independent, since, as previously mentioned, the orientation of the body in space can be defined using only three independent coordinates. By using Equations 2.28 and 2.30, it can be shown that the four Euler parameters must satisfy the following condition: 3

∑ (θ )

i 2 k

k =0

=1

(2.32)



47

Substituting the expressions of the four Euler parameters given by Equation 2.30 into Equation 2.26, the orientation matrix Ai can be rewritten in terms of Euler parameters as Ai = I + 2θ is (θ 0i I + θ is )

(2.33)

where θ is = [θ1i θ 2i θ 3i ]T and θ is is the skew-symmetric matrix associated with the vector θ is . The preceding equation can be written more explicitly as 1 − 2(θ 2i )2 − 2(θ 3i )2  Ai =  2(θ1iθ 2i + θ 0i θ 3i )  2(θ iθ i − θ i θ i ) 1 3 0 2 

2(θ1iθ 2i − θ 0i θ 3i ) 1 − 2(θ1i )2 − 2(θ 3i )2 2(θ 2iθ 3i + θ 0i θ1i )

2(θ1iθ 3i + θ 0i θ 2i )   2(θ 2i θ 3i − θ 0i θ1i )  (2.34) 1 − 2(θ1i )2 − 2(θ 2i )2 

Using Equation 2.32, Equation 2.34 can also be rewritten as  2(θ 0i )2 + 2(θ1i )2 − 1  A =  2(θ1iθ 2i + θ 0i θ 3i )  2(θ iθ i − θ i θ i ) 1 3 0 2 

2(θ1iθ 2i − θ 0i θ 3i ) 2(θ 0i )2 + 2(θ 2i )2 − 1 2(θ 2iθ 3i + θ 0i θ1i )

i

2(θ1iθ 3i + θ 0i θ 2i )   2(θ 2i θ 3i − θ 0i θ1i )  (2.35) 2(θ 0i )2 + 2(θ 3i )2 − 1

Using the method of Euler parameters, the singularity problem associated with the three-parameter representation is eliminated at the expense of adding an algebraic constraint equation (Equation 2.32).

EXAMPLE 2.3 Use the Rodriguez formula to determine the transformation matrix of body i resulting from a rotation θ i = π/3 about the vector [−2.0 1.0 5.0]T defined in the global coordinate system. Determine the four Euler parameters associated with this rotation. Solution. A unit vector along the axis of rotation is defined as

v = i

 −2.0   −0.3651     1.0  =  0.1826  2 2 2  (−2.0) + (1.0) + (5.0)   5.0   0.9129  1

Using the Rodriguez formula given by Equation 2.26, the transformation matrix is obtained as


48


A i = I + v i sin θ i + 2(v i ) 2 sin 2

1  = 0  0

0 

0

θi 2

1 0

 −0.8667 

+2.0  −0.0667

 −0.3333



0.3651 sin

0 −0.3651

0

−0.0667

−0.3333 

−0.9667

0.1667  sin 2



−0.1667 

0.1667

 0.5667 

−0.8239

−0.0086 

0.5167

 −0.3248

−0.2329

0.3996  0.9167 

=  0.7572

0.1826 

−0.9129

0

  0  +  0.9129 1   −0.1826



π 3

π 6



The four Euler parameters are determined using Equation 2.30 as

θ 0i = cos

θi π = cos = 0.8660 2 6

θ1i = v1i sin

π θi = −0.3651sin = −0.1826 2 6

θ 2i = v2i sin

θi π = 0.1826 sin = 0.0913 2 6

θ 3i = v3i sin

θi π = 0.9129 sin = 0.4564 2 6

The transformation matrix in terms of Euler parameters is determined using Equation 2.34 as

1 − 2(θ 2i ) 2 − 2(θ 3i ) 2  A i =  2(θ1iθ 2i + θ 0i θ 3i )  2(θ1iθ 3i − θ 0i θ 2i )   0.5667  =  0.7572  −0.3248

2(θ1iθ 2i − θ 0i θ 3i ) 1 − 2(θ1i ) 2 − 2(θ 3i ) 2 2(θ 2iθ 3i + θ 0i θ1i )

−0.8239

−0.0086 

0.5167

0.3996  0.9167 

−0.2329

2(θ1iθ 3i + θ 0i θ 2i ) 



2(θ 2iθ 3i − θ 0i θ1i )  1 − 2(θ1i ) 2 − 2(θ 2i ) 2 



This transformation matrix is the same as the one previously obtained in this example using the Rodriguez formula.



49

2.3 VELOCITIES AND ACCELERATIONS The global position vector of an arbitrary point on body i is defined, as previously shown by Equation 2.1, as the sum of two displacement components. The first is due to the translation of the origin of the body coordinate system, while the second is due to the rotation of the body. Using this general displacement, one can derive expressions for the absolute velocity and acceleration vectors of an arbitrary point on the body.

2.3.1 VELOCITY VECTOR The global velocity vector of an arbitrary point on the rigid body can be obtained by differentiating Equation 2.1 with respect to time. This leads to iui r i = R i + A

(2.36)

In rigid body dynamics, the components of the vector u i are constant. The preceding equation can also be written in the following form: i + ω i × ui r i = R

(2.37)

In this equation, ω i = ω 1i

ω 2i

ω 3i 

T

(2.38)

is the absolute angular velocity vector defined in the global coordinate system. The vector ui is defined as ui = Aiui

(2.39)

· The vector Ri is the global velocity of the reference point Oi, while the vector ωi × ui is the result of the rigid body rotation. Since the vectors ωi and ui can also be defined in the body coordinate system, the absolute velocity vector of Equation 2.37 can be written in the following alternative form: i + Ai (ω i × u i ) r i = R

(2.40)

where ω i is the absolute angular velocity vector defined in the body coordinate system and is related to the angular velocity defined in the global coordinate system by the following transformation: ω i = Ai ω i

(2.41)


50


Using the orthogonality property of the orientation matrix given by Equation 2.13 and comparing Equations 2.36 and 2.40, one can show that T i ω i = Ai A

(2.42)

where ω i is the skew-symmetric matrix associated with the vector ω i and is defined as  0  i ω =  ω 3i  −ω i  2

ω 2i   −ω 1i  0 

−ω 3i 0 ω 1i

(2.43)

and ω 1i , ω 2i , and ω 3i are the components of the angular velocity vector defined in the body coordinate system, that is, ω i = ω 1i

ω 2i

ω 3i 

T

(2.44)

Similarly, one can write the following identity for the angular velocity defined in the global coordinate system: i Ai T ω i = A

(2.45)

where  0  ω =  ω 3i  −ω i  2 i

−ω 3i 0 ω 1i

ω 2i   −ω 1i  0 

(2.46)

It is important to note from Equation 2.45 that the components of the angular velocity vector in the three-dimensional analysis cannot be defined, in general, as the time · derivatives of the generalized orientation coordinates. That is, ωi ≠ θi, and the angular velocity vector, therefore, cannot be integrated directly to obtain the orientation coordinates.

2.3.2 ACCELERATION VECTOR The absolute acceleration vector can be obtained by differentiating Equation 2.36 with respect to time. This yields i + A i u i ri = R

(2.47)



51

The acceleration vector can also be written in the following form: i + α i × u i + ω i × (ω i × u i ) ri = R

(2.48)

where α i = [α1i α 2i α 3i ]T is the angular acceleration vector of body i defined in the global coordinate system. The vector αi is the time derivative of the angular velocity vector ωi. The acceleration term αi × ui on the right-hand side of Equation 2.48 is ωi × ui) is called the normal component, called the tangential component, while ωi × (ω which is also known as the centripetal acceleration. If the vectors αi, ωi, and ui are defined in the body coordinate system, Equation 2.48 can be written in the following alternative form:

{

(

i + Ai α i × u i + ω i × ω i × u i ri = R

)}

(2.49)

where α i = Ai α i

(2.50)

It is more convenient in some numerical algorithms to formulate the Newton-Euler equations in terms of α i instead of αi, since the inertia coefficients associated with α i are constant in the case of rigid body analysis.

2.3.3 GENERALIZED ORIENTATION COORDINATES While the angular velocity vector cannot be directly integrated to obtain the orientation coordinates, the absolute angular velocity and acceleration vectors can be written in terms of the time derivatives of the generalized coordinates. Using Equations 2.42 and 2.45, the angular velocity vector can be written in terms of the time derivatives of the orientation generalized coordinates as follows: ω i = Giθ i ,

ω i = Giθ i

(2.51)

where the coefficient matrices Gi and Gi are expressed in terms of the orientation parameters θi. For example, the matrices Gi and Gi , in the case of the Euler angles θi = [ψ i φ i θ i]T based on the sequence Z i,Xi,Y i, can be obtained as 0  G = 0 1  i

cos ψ i sin ψ i 0

− sin ψ i cos φ i   cos ψ i cos φ i  ,  sin φ i 

 − cos φ i sin θ i  G = sin φ i  cos φ i cos θ i  i

cos θ i 0 sin θ i

0  1  (2.52) 0 


52


Using Equations 2.51 and 2.52, the angular velocity vector ωi defined in the global coordinate system can be written in the following vector form: ω i = Giθ i = g1i ψ i + g i2 φ i + g 3i θ i

(2.53)

where g1i =  0

0

g i2 =  cos ψ i

1

T

sin ψ i

g 3i =  − sin ψ i cos φ i

0 

T

cos ψ i cos φ i

      T i  sin φ   

(2.54)

In Equation 2.53, the absolute angular velocity vector is written as a linear combination of three angular velocity vectors associated with ψ i , φ i , and θ i . The vectors g ik (k = 1, 2, 3) represent unit vectors defined in the global coordinate system about which the three successive rotations are performed. Similarly, the columns of the matrix Gi define unit vectors in the body coordinate system along the axes about which the Euler angles are performed. That is, Gi = A i Gi

(2.55)

The matrices Gi and Gi that appear in Equation 2.51 will be repeatedly used in this book, and these matrices also appear in the formulation of the generalized forces and the Jacobian matrix of the kinematic constraint equations.

EXAMPLE 2.4 Determine the matrices Gi and G i in terms of Euler angles if the sequence of rotation is defined as follows: a rotation φ i about the Z i axis, a rotation θ i about the Xi axis, and a rotation ψ i about the Z i axis. Solution. Using Equation 2.51, the absolute angular velocity vector defined in the global coordinate system can be written as ω i = G i θ i = g1i φ i + g i2 θ i + g 3i ψ i where θi = [φ i θ i ψ i ]T and g ik (k = 1, 2, 3) are the columns of the matrix Gi that represent unit vectors defined in the global coordinate system along which the rotations φ i, θ i, and ψ i are performed, respectively. Therefore, the vector g1i is a unit vector along the Z i axis of the body coordinate system at the initial configuration, that is, g1i =  0

0

1

T



53

The vector g i2 is a unit vector along the Xi axis of the body coordinate system after the first rotation φ i is performed, that is, g i2 =  cos φ i

0 

sin φ i

T

The vector g i3 is a unit vector along the Z i axis of the body coordinate system after the first and second rotations φ i and θ i are performed. That is, g i3 =  sin θ i sin φ i

cos θ i 

− sin θ i cos φ i

T

Accordingly, the matrix Gi is obtained as

0  i G = 0 1 

cos φ i

sin θ i sin φ i 

sin φ i

− sin θ i cos φ i 

  

cos θ i

0

On the other hand, the absolute angular velocity vector defined in the body coordinate system can be written as ω i = G i θ i = g1i φ i + g2i θ i + g3i ψ i where gki (k = 1, 2, 3) are the columns of the matrix G i that represent unit vectors defined in the body coordinate system along which the rotations φ i, θ i, and ψ i are performed, respectively. Hence, the vectors g1i , g2i , and g3i are given as follows: g1i =  sin θ i sin ψ i

cos θ i 

sin θ i cos ψ i

g2i =  cos ψ i g3i =  0

0 

− sin ψ i

0

1

T

T

T

It follows that the matrix G i is defined as

 sin θ i sin ψ i  G i =  sin θ i cos ψ i  cos θ i 

0

cos ψ i − sin ψ 0

i



0 1 

2.3.4 SINGULAR CONFIGURATION As previously discussed, a singularity problem is encountered when Euler angles are used in the description of the three-dimensional rotations. For example, the


54


matrices Gi and Gi given by Equation 2.52 in terms of Euler angles become singular when the angle φ i is equal to ±π/2. In this case, the rotations ψ i and θ i are performed about two parallel axes, and distinction between these two angles cannot be made, as shown in Figure 2.3. As a consequence of the singularity of the matrices Gi and Gi , the time derivatives of Euler angles cannot be expressed in terms of the components of the angular velocity vector. Euler parameters, on the other hand, do not lead to singular configurations. In the case of Euler parameters, the matrices Gi and Gi are defined as (Nikravesh, 1988; Shabana, 2001)  −θ1i  Gi = 2  −θ 2i  −θ i  3

θ 0i θ 3i −θ 2i

−θ 3i θ 0i θ1i

θ 2i   −θ1i  , θ 0i 

 −θ1i  Gi = 2  −θ 2i  −θ i  3

θ 0i −θ 3i θ 2i

θ 3i θ 0i −θ1i

−θ 2i   θ1i  (2.56) θ 0i 

One can show that the orientation matrix given by Equation 2.33 can be expressed in terms of the preceding two matrices Gi and Gi as

Ai =

1 i iT GG 4

(2.57)

Furthermore, the matrices Gi and Gi , which are linear in Euler parameters, satisfy several interesting identities that can be utilized in developing the numerical algorithm. Some of these identities are summarized in Table 2.1.

EXAMPLE 2.5 Discuss the singularity problem of Euler angles if the sequence of rotation is defined as follows: a rotation φ i about the Z i axis, a rotation θ i about the Xi axis, and a rotation ψ i about the Z i axis. Solution. Using the matrices Gi and G i obtained in Example 2.4 in terms of Euler angles for the sequence Z i,Xi,Z i, one can show that the determinant of these matrices is given as det G i = det G i = − sin θ i The singularity is encountered when the angle θ i is equal to zero or π (−π < θ i ≤ π). In such a case, the axes of rotation of the angles φ i and ψ i are parallel; therefore, at this singular configuration, one cannot distinguish between these two angles.



55

TABLE 2.1 Identities of Euler Parameters Absolute  −θ1i  G = 2  −θ 2i  −θ i  3 i

θ 0i θ 3i −θ 2i

Local

θ 2i   −θ1i  θ 0i 

−θ 3i θ 0i θ1i

 −θ1i  G = 2  −θ 2i  −θ i  3 i

ω i = Gi θ i

i α i = Gi θ

i α i = Gi θ

Gi θ i = 0

Gi θ i = 0

i θ i = 0 G

Gi θ i = 0

T

Gi Gi = 4I

(

Gi Gi = 4 I4 − θ i θ i

−θ 2i   θ1i  θ 0i 

θ 3i θ 0i −θ1i

ω i = Gi θ i

T

Gi Gi = 4I T

θ 0i −θ 3i θ 2i

T

)

T

(

Gi Gi = 4 I4 − θ i θ i

T θ i = 41 Gi ω i

T

)

T θ i = 41 Gi ω i

Ai = 41 Gi Gi

T

G i = Ai G i T θ i θ i = 0

3

∑ (θ )

i 2 k

=1

k =0

Using Equation 2.51, which is valid for any set of orientation parameters, the global velocity vector defined by Equations 2.37 and 2.40 can, respectively, be expressed in terms of the time derivatives of the generalized coordinates as i − u iGiθ i r i = R

(2.58)

i − Ai u iGiθ i r i = R

(2.59)

and

where u i and u i are the skew symmetric matrices associated with the vectors ui and u i , respectively. Furthermore, the absolute acceleration vector defined by Equations 2.48 and 2.49 in the global coordinate system can also be expressed in terms of the time derivatives of the generalized coordinates as


56


i − u iGiθ i + a i ri = R v

(2.60)

i − Ai u iGiθ i + a i ri = R v

(2.61)

iθ i ] iθ i = Ai [(ω i )2 u i − u iG a iv = (ω i )2 u i − u iG

(2.62)

and

where

iθ i are identically zero when Euler parameters are iθ i and G One can show that G used. However, this is not the case when Euler angles are used.

EXAMPLE 2.6 The orientation of a rigid body i is defined by the following Euler parameters: θ i =  0.8660

0.2182 

−0.4364

0.1091

T

At this configuration, the absolute angular velocity vector in the global coordinate system is given by ω i = 10.0

1.0

3.0 

T

Find the time derivatives of Euler parameters. Solution. The absolute angular velocity vector is defined in terms of the time derivatives of Euler parameters using the equation ω i = G i θ i . However, since the matrix Gi given by Equation 2.56 is a 3 × 4 matrix in the case of Euler parameters, one cannot directly take an inverse of this matrix to write the time derivatives of the Euler parameters in terms of the components of the angular velocity vector. Alternatively, one can premultiply both sides of the equation ω i = G i θ i by the transpose of the matrix Gi and use the identity T

(

T

Gi Gi = Gi Gi = 4 I4 − θ iθ i

T

)

presented in Table 2.1 (Nikravesh, 1988; Shabana, 2001). By doing this, one can show that

(

)

T T T G i ω i = G i G i θ i = 4 I 4 − θ i θ i θ i = 4θ i

T where I4 is the 4 × 4 Tidentity matrix, and θ i θ i = 0 because of the Euler parameter i i constraint equation, θ θ = 1. Using the preceding equation, the time derivatives of Euler parameters can be determined as



57

 −2θ1i  i 1 1 2θ θ i = G i ω i =  0 i 4 4  −2θ 3  i  2θ 2

2θ

T

 −0.2182  1 1.73320 =  4  −0.4364   −0.8728

−2θ 3i 

−2θ 2i

i  ω 1  −2θ 2i   i  ω 2  2θ1i   i   ω3 2θ 0i   

i 3

2θ 0i −2θ1i

 −0.655   10.0    0.8728     5.094  = 1 0 .   −0.494  0.2182     3.0    1.7320   −0.938 

−0.4364 

0.8728 0.4364 1.7320 −00.2182

Furthermore, using the equation ω i = G i θ i , one can show that

(

G i ω i = G i G i θ i = 4 I 4 − θ iθ i T

T

T

)

θ i = 4θ i

This leads to 1 T θ i = G i ω i 4 Therefore, the time derivatives of Euler parameters can be expressed in terms of the global or local components of the angular velocity vectors as 4θ i = G i ω i = G i ω i . T

T

Premultiplying both sides of this equation by the matrix Gi and using the identity T

T

GiGi = GiGi = 4I , one can show that T

T

G i G i ω i = G i G i ω i = 4ω i from which ωi =

1 4

T

GiGi ω i = Aiω i

T 1 where the transformation matrix Ai is given by A i = G i G i , as previously presented 4 in Equation 2.57.


58


2.4 NEWTON-EULER EQUATIONS The dynamic equations of motion of rigid body systems can be written in different forms. One simple form of the equations of motion that is widely used is based on Newton-Euler equations, which do not include inertia coupling between the translational and rotational displacements. To obtain the Newton-Euler equations of motion, the origin of the body coordinate system has to be attached to the center of mass of the body, which is the case with a centroidal body coordinate system. Furthermore, the equations of motion associated with the rotation of the rigid body must be expressed in terms of the angular velocity and acceleration vectors. NewtonEuler equations of motion are given in a matrix form as follows (Greenwood, 1988; Shabana, 2001): i    0  R Fei =     i i i i i  i Iθθ   α   Me − ω × ( Iθθ ω ) 

 mi I   0

(2.63)

where mi is the mass of the rigid body, I is a 3 × 3 identity matrix, Iθθi is the inertia tensor defined with respect to the centroidal body coordinate system, Fei is the resultant of the external forces, and Mie is the resultant of the external moments defined in the body coordinate system. The inertia tensor Iθθi is defined as  I xxi  T Iθθi = u i u i dm i =  I xyi m  i  I xz

∫

I xyi i I yy

I yzi

I xzi   I yzi   I zzi 

(2.64)

where dmi = ρidVi, ρi is the material density, and dVi is the infinitesimal material volume. Since u i = [ x i yi z i ]T , as given in Equation 2.3, the elements of the inertia tensor can be explicitly written as (Greenwood, 1988; Shabana, 2001) I xxi =

∫

I xyi = −

i [( yi )2 + ( z i )2 ]dm i , I yy = m

∫

x i yi dm i , m

∫

I xzi = −

[( x i )2 + ( yi )2 ]dm i  m    I yzi = − yi z i dm i  m (2.65)

[( x i )2 + ( z i )2 ]dm i , I zzi = m

∫

x i z i dm i , m

∫

∫

Table 2.2 shows the mass moments of inertia of some homogeneous solids when a centroidal body coordinate system is used. In the preceding equation, I xxi, I yyi , and I zzi are called the mass moments of inertia, while I xyi, I xzi, and I yzi are called the products of inertia. Equation 2.63 shows that there is no inertia coupling between the translational and rotational coordinates in the Newton-Euler equations. Furthermore, one can choose the orientation of the centroidal body coordinate system such that all products



59

TABLE 2.2 Mass Moments of Inertia of Homogeneous Solids Thin Circular Disk

I xx

1

I yy

4

m( a ) 2 ,

Sphere

1

I zz

2

m( a ) 2

I xx

I yy

Thin Ring

I xx

1

I yy

2

m( a ) 2 ,

I yy

1 12

m( a ) 2

I zz

I xx

83

I yy

320

I zz

1 12 1 12

m( a ) 2

m(l ) 2 ,

m( a ) 2 ,

I yy

m[( a ) 2

(b) 2 ]

m( a ) 2 ,

I zz

2 5

m (a )2

Cylinder

I zz

0

I xx

I yy

1 12

m[3( a ) 2

Thin Plate

I xx

5

Hemisphere

Slender Rod

I xx

2

I zz

( h) 2 ], I zz

1 m( a ) 2 2

Cone

1 12

m(b) 2

I xx I zz

I yy 3 10

3 80

m[4( a ) 2

m( a ) 2

( h) 2 ],


60


of inertia, I xyi, I xzi, and I yzi, are equal to zero. In such a case, the axes of the body coordinate system are called the principal axes, and the mass moments of inertia associated with the principal axes are called the principal moments of inertia. The principal moments of inertia and principal axes are determined by solving the following eigenvalue problem for the inertia tensor:  Iθθi − λ i I  a i = 0  

(2.66)

In this equation, λi is the eigenvalue and ai is the eigenvector. The preceding equation can be solved for three eigenvalues λki (k = 1, 2, 3), which define the principal moments of inertia, and three eigenvectors a ik (k = 1, 2, 3), which define the principal axes or directions associated with the three eigenvalues λki . Because Iθθi is a symmetric tensor, all λki are real and all a ik are mutually orthogonal. Using the fact that the eigenvectors are orthogonal, the matrix of the principal moments of inertia is given as  λ1i  ( Iθθi ) p = Ti Iθθi Ti =  0 0  T

0 λ2i 0

0  0 λ3i 

(2.67)

where the transformation matrix Ti can be defined using the normalized eigenvectors a ik (k = 1, 2, 3) that represent unit vectors along the three mutually orthogonal principal axes. As previously pointed out, since the components of the angular velocity vector are not, in general, the time derivatives of the generalized orientation coordinates, the components of the angular velocity vector cannot be directly integrated to obtain the generalized orientation coordinates. For this reason, it is important in constrained multibody system applications to be able to write the Newton-Euler equations in terms of the generalized coordinates and their time derivatives using Equation 2.51. This leads to the generalized Newton-Euler equations, which can be written as follows:  miRR   0

i    (Qie ) R 0  R =      i i i i mθθ   θ  (Qe )θ + (Qv )θ 

(2.68)

i where the matrices m iRR and mθθ are the generalized mass matrices associated with i the generalized coordinates R and θi, respectively. These matrices are written as

miRR = m i I ,

T

i mθθ = Gi Iθθi Gi

(2.69)



61

The vector (Qiv )θ is quadratic in the generalized velocities and is given by T iθ i  (Qiv )θ = −Gi ω i × ( Iθθi ω i ) + Iθθi G  

(2.70)

iθ i is identically zero when Euler parameters are used as generalized Recall that G coordinates to describe the orientation of the body. The vectors (Qie ) R and (Qie )θ are, respectively, the generalized external force vectors associated with the translational and rotational coordinates and are defined as (Qie ) R = Fei ,

T

(Qie )θ = Gi Mie

(2.71)

Note that the external force and moment vectors can be defined in either the global or body coordinate system using the following transformation: Fei = A i Fei ,

Mie = A i Mie

(2.72)

Furthermore, the generalized external force vector (Qie )θ can also be written using Equations 2.13, 2.55, and 2.72 as T

(Qie )θ = Gi Mie

(2.73)

The generalized Newton-Euler equations can be written in the following compact matrix form: i = Qie + Qiv Mi q

(2.74)

where  mi Mi =  RR  0

i  (Qie ) R  R  0  0  i i i , q , Q = =   i  , Qv =  i    e i i mθθ   (Qe )θ  (Qv )θ  θ 

(2.75)

For a given set of external forces and moments and a given set of initial coordinates and velocities, the generalized Newton-Euler equations in the case of unconstrained i and θ i , which can be directly integrated forward in time body i can be solved for R to obtain the velocities and coordinates at the next time step. Alternatively, one can use the original Newton-Euler equations defined by Equation 2.63 to solve i and α i , which can be integrated to determine the velocities R i and ω i . Howfor R ever, since the angular velocity vector ω i cannot be directly integrated to determine the orientation coordinates, the time derivatives of the generalized orientation coordinates θ i can be determined using the equation ω i = Giθ i . The vector θ i can then be integrated to determine the orientation coordinates.


62


2.5 JOINT CONSTRAINTS Railroad vehicle systems consist of large number of bodies that include wheelsets, frames, car bodies, and rails. These bodies are interconnected by mechanical joints and force elements. In analytical dynamics, the method of treating mechanical joints and specified motion trajectories that impose constraints on the motion of the bodies is different from the method of treating the external forces. Constraint equations reduce the number of degrees of freedom, while external force elements do not affect the number of the system degrees of freedom. The constraint conditions can be, in most cases, formulated as a set of nonlinear algebraic equations that are functions of the generalized coordinates and time. The use of the nonlinear algebraic constraint equations with the system differential equations of motion leads to a system of differential and algebraic equations that must be solved simultaneously. In the case of constrained motion, two approaches can be used to formulate the force-acceleration equations. In the first approach, the augmented formulation, the equations of motion are formulated in terms of redundant coordinates that are related by the constraint conditions. The constraint equations are added to the system differential equations of motion, leading to a system that includes the constraint forces. This augmented formulation defines a relatively large system of algebraic equations that can be solved for the system accelerations and the constraint forces. In the second approach, the embedding technique, the constraint forces are used to systematically eliminate the dependent coordinates and the constraint forces, leading to a reduced system of differential equations associated with the independent coordinates (degrees of freedom). The resulting system of equations of motion can be solved for the accelerations associated with the independent coordinates. In both the augmented formulation and the embedding technique, the formulation of the joint constraint equations is necessary. For this reason, this section is devoted to the formulations of the nonlinear algebraic constraint equations of some of the commonly used mechanical joints. The augmented formulation is discussed in Section 2.6, while the embedding technique is covered in Section 2.8.

2.5.1 SPHERICAL JOINT The spherical joint, sometimes called a ball joint, allows three relative rotational degrees of freedom between bodies i and j, as shown in Figure 2.5. The spherical joint eliminates three relative translational degrees of freedom between bodies i and j. The constraint equations of the spherical joint can be written as C(q i , q j ) = rPi − rPj = 0

(2.76)

where point P is the joint definition point and rPi = R i + A i u iP , as shown in Figure 2.5.

rPj = R j + A j u Pj

(2.77)



63

FIGURE 2.5 Spherical joint.

2.5.2 REVOLUTE JOINT The revolute joint allows one degree of freedom of relative rotation along the joint axis between bodies i and j, as shown in Figure 2.6. The revolute joint, therefore, eliminates five relative degrees of freedom between the bodies. Therefore, in addition to the spherical joint constraints given by Equation 2.76, which constrains the relative translational displacement, two degrees of relative rotations about two axes perpendicular to the joint axis must be eliminated. Let vi and vj be two vectors defined along the joint axis on body i and body j, respectively, as shown in Figure 2.6. The

FIGURE 2.6 Revolute joint.


64


conditions of parallelism of the two vectors vi and vj throughout the motion can be written using the following dot product (Shabana, 2001): v1i ⋅ v j = 0 ,

v i2 ⋅ v j = 0

(2.78)

where (⋅⋅ ) denotes the dot product, and v1i and v i2 are two vectors defined on body i that are orthogonal to vi. The vectors in the preceding equation can be written in terms of their constant components defined in their respective body coordinate systems as v1i = A i v1i ,

v 2i = A i v 2i ,

vj = Aj vj

(2.79)

where Ai and Aj are the transformation matrices that define the orientation of body i and body j, respectively, and v1i , v 2i , and v j are constant vectors that define, respectively, the components of the vectors v1i , v i2 , and vj in the body coordinate systems. Using these definitions, the constraint equations of the revolute joint can be written as rPi − rPj    C(q i , q j ) =  v1i ⋅ v j  = 0  vi ⋅ v j   2 

(2.80)

2.5.3 CYLINDRICAL JOINT The cylindrical joint allows relative translation and rotation between bodies i and j along the joint axis, as shown in Figure 2.7. This joint, therefore, has two degrees of freedom. One degree of freedom is the relative translation along the joint axis, while the other degree of freedom is the rotation about this axis. Consequently, the cylindrical joint eliminates four relative degrees of freedom, leading to four constraint equations. Let vi and vj be two vectors defined along the joint axis on body i and body j, respectively, and let Pi and Pj be two points on body i and body j defined along the joint axis, as shown in Figure 2.7. The conditions of parallelism of the two vectors vi and vj throughout the motion can be written using the dot product as v1i ⋅ v j = 0 ,

v i2 ⋅ v j = 0

(2.81)

Note that v1i and v i2 are two vectors orthogonal to vi. The relative translation between the two bodies in directions perpendicular to the joint axis can be eliminated using the conditions v1i ⋅ rPij = 0 ,

v i2 ⋅ rPij = 0

(2.82)



65

FIGURE 2.7 Cylindrical joint.

where rPij = rPi − rPj

(2.83)

Therefore, the constraint equations of the cylindrical joint can be defined as  v1i ⋅ v j   i j v ⋅v C(q i , q j ) =  2i ij  = 0 v ⋅ r   i1 Pij   v 2 ⋅ rP 

(2.84)

2.5.4 PRISMATIC JOINT The prismatic joint allows one relative translation along the joint axis between bodies i and j, as shown in Figure 2.8. This constraint can be obtained as a special case of the cylindrical joint by eliminating the relative rotation along the joint axis. To this end, two orthogonal vectors hi and hj that are, respectively, defined on body i and j are introduced and used to define a constraint equation that eliminates the relative rotation between the two bodies along the joint axis. Using these two vectors and the constraint equations of the cylindrical joint, one can write the following five constraint equations for the prismatic joint:


66


FIGURE 2.8 Prismatic joint.

 v1i ⋅ v j   i j v 2 ⋅ v  C(q i , q j ) =  v1i ⋅ rPij  = 0  i ij   v 2 ⋅ rP   hi ⋅ h j   

(2.85)

The fifth equation in Equation 2.85 guarantees that there is no relative rotation between the two bodies about the joint axis.

2.6 AUGMENTED FORMULATION The system constraint equations that describe mechanical joints or specified motion trajectories can be written in a general vector form as follows: C(q, t ) = 0

(2.86)

where C is the vector of the system constraint equations, and q is the vector of the system generalized coordinates. These constraint equations can be added to the generalized Newton-Euler equations using the technique of Lagrange multipliers. In such a case, the matrix equation of motion of the constrained multibody system is given by = Qe + Qv − CqT λ Mq

(2.87)



67

where M is the system mass matrix, Qv is the vector of inertia forces that are quadratic in velocity, Qe is the vector of the generalized external forces, Cq is the Jacobian matrix of the constraint equations, and λ is the vector of Lagrange multipliers that are used to define the generalized constraint forces as −CTq λ . Equations 2.86 and 2.87 represent a system of differential and algebraic equations that can be written in the following augmented form (Shabana, 2001): M  Cq

CTq   q  Q + Qv    =  e  0   λ   Qd 

(2.88)

where Qd is the vector resulting from the differentiation of the system constraint equations of Equation 2.86 twice with respect to time, that is, (q, t ) = C q C q − Q d = 0

(2.89)

The number of equations in Equation 2.88 is equal to the total number of the system generalized coordinates n plus the total number of constraint equations nc. Note that the number of the unknown Lagrange multipliers is equal to the number of constraint equations. As a result, the degrees of freedom of the system are equal to n − nc. If the number of bodies in the system is nb, the vectors and matrices that appear in Equation 2.88 can be written in a more explicit form as follows:  M1  M=    0

 q1   2 q q =  ,    n  q b 

M

                  

(2.90)

(Qie ) R   Ri   0  0  i i i , Q , Q , q = = =     i   e v i i i mθθ   (Qe )θ  (Qv )θ  θ 

(2.91)

2

 C1    C2 C =  ,     Cnc 

 Q1e   Q1v  0   2  2   , Q =  Qe  , Q =  Q v  e v         n   nb M nb  Qvb  Qe   ∂C1  1  ∂q  ∂C2  Cq =  ∂q1    ∂Cnc  ∂q1 

∂C1 ∂q2

∂C2 ∂q2

∂Cnc ∂q

2

∂C1 ∂qnb ∂C2 ∂qnb ∂Cnc ∂qnb

in which  mi Mi =  RR  0


68


A physical and straightforward interpretation of the augmented formulation of Equation 2.88 is given in Section 2.9 using a simple planar example. Equation 2.88 can be solved for the accelerations and Lagrange multipliers, which can then be used to determine the generalized constraint forces. Having obtained the acceleration vectors, the independent accelerations can be identified and integrated forward in time to determine the independent coordinates and velocities. The dependent coordinates and velocities can be determined using the constraint equations at the position and velocity levels. The numerical algorithm used to solve the differential and algebraic equations that result from the use of the augmented formulation is discussed in more detail in Chapter 6.

EXAMPLE 2.7 Determine the constraint Jacobian matrix Cq of a spherical joint that connects two bodies i and j. Solution. The constraint equations of the spherical joint are given by Equation 2.76. Using Equation 2.77, these constraint equations can be written as C(q i , q j ) = R i + A i u Pi − R j − A j u Pj = 0 Differentiating this equation with respect to time, one obtains =R i − A i u i G i θ i − R j + A j u j G j θ j = 0 C P P which can also be written as = I C 

− A i u Pi G i

−I

A j u Pj G j  q ij = C qij q ij = 0

where I is a 3 × 3 identity matrix and qij = [qiT qjT ]T. It follows that the constraint Jacobian matrix of the spherical joint between bodies i and j is given by C qij =  I

− A i u Pi G i

−I

A j u Pj G j 

EXAMPLE 2.8 Determine the constraint Jacobian matrix Cq of a cylindrical joint that connects two bodies i and j. Solution. The constraint equations of a cylindrical joint that connects bodies i and j are given by Equation 2.84 as



69

 v1i ⋅ v j   i j v ⋅v C(q i , q j ) =  2i ij  = 0  v1 ⋅ rP   i ij   v 2 ⋅ rP  Note that these four equations can be written using the following two dot-product equations: CI = v ik ⋅ v j = 0,

CII = v ik ⋅ rPij = 0,

(k = 1, 2)

Differentiating the first equation CI with respect to time, one obtains C I = v j ⋅ v ik + v ik ⋅ v j = 0 where v ik = A i (ω i × vki ) = − A i v ki G i θ i v j = A j (ω j × v j ) = − A j v j G j θ j Using these equations, C I can be written as C I =  0

− v j A i v ki G i T



0

− v ik A j v j G j  q ij = CIqij q ij = 0 T



where 0 is a 1 × 3 null vector. Similarly, differentiating the second equation CII with respect to time, one obtains C II = v ik ⋅ rPij + rPij ⋅ v ik = 0 where i − A i u i G i θ i − R j + A j u j G j θ j = 0 rPij = R P P v ik = A i (ω i × vki ) = − A i v ki G i θ i Using these equations, C II can be written as T C II =  v ik



− v ik A i u Pi G i − rPij A i v ki G i T

T

T

− v ik

v ik A j u Pj G j  q ij = CIIqij q ij = 0 T




70

Railroad Vehicle Dynamics: A Computational Approach Therefore, the constrain Jacobian matrix of the cylindrical joint between bodies i and j can be written as

0  0 Cq =  T  v1i  iT  v 2 ij

− v j A i v 1i G i

0

− v1i A j v j G j 

− v j A i v 2i G i

0

− v i2 A j v j G j 

− v1i A i u Pi G i − rPij A i v 1i G i

− v1i

− v i2 A i u Pi G i − rPij A i v 2i G i

− v i2

T

T

T

T

T

T

T

T

 

T

v1i A j u Pj G j 

T

v i2 A j u Pj G j 

T

T



where 0 is a 1 × 3 null vector.

2.7 TRAJECTORY COORDINATES The analysis presented in this chapter has, so far, focused on using the absolute Cartesian coordinates to derive the equations of motion. These coordinates are widely used in developing general-purpose multibody system algorithms. However, it is important to point out that other sets of coordinates can also be used to derive the dynamic equations of motion of railroad vehicle systems. These sets of coordinates can be used to develop specialized formulations that take advantage of particular features of railroad vehicle systems. However, the relationship between any two sets of coordinates can always be established using a proper coordinate transformation. In this section, another set of coordinates, referred to in this book as the trajectory coordinates, is introduced. More detailed discussion on the trajectory coordinates and their use in developing special-purpose algorithms and computer codes for the dynamic analysis of railroad vehicle systems is presented in Chapter 7. The kinematic and dynamic equations can be formulated using the trajectory coordinates as an alternative to the absolute Cartesian coordinates. In this case, the general displacement of body i shown in Figure 2.9 can be described using six trajectory coordinates: the arc length coordinate si defined along the specified trajectory shown in the figure; the lateral and vertical displacements of the body yir and zir relative to a trajectory coordinate system that follows the body, as shown in Figure 2.9; and three rotation angles of the body ψ tr (yaw), φ tr (roll), and θ tr (pitch) that define the orientation of the body with respect to the trajectory coordinate system. These angles can be written in a vector form as θir = [ψ ir φ ir θ ir]T. Given the parameter si, the location of the origin and the orientation of the trajectory coordinate system that follows the motion of the body can be uniquely defined, as shown in Chapter 3. The location of the origin in the global system is defined by the vector Rti = Rti (si), while the orientation of the trajectory coordinate system at this location can be defined using the three Euler angles ψ ti(si), θ ti(si), and φ ti(si) about the three axes Z ti, Y ti, and X ti. (More details on the choice of this sequence of rotations are presented in Chapter 3.) Note that the three Euler angles that define the orientation of the trajectory coordinate system depend solely on the arc length si. These Euler angles can be used to define the following transformation matrix:



71

FIGURE 2.9 Trajectory coordinates.

A ti =  i ti

jti

 cos ψ ti cos θ ti  =  sin ψ ti cos θ ti  sin θ ti 

k ti  − sin ψ cos φ + cos ψ sin θ sin φ ti

ti

ti

ti

cos ψ cos φ + sin ψ sin θ sin φ ti

ti

ti

ti

− cos θ sin φ ti

ti

ti

ti

  cos ψ sin φ − sin ψ sin θ cos φ  ti ti  cos θ cos φ  (2.92)

− sin ψ sin φ − cos ψ sin θ cos φ ti

ti

ti

ti

ti

ti

ti

ti

ti

ti

This transformation matrix defines the orientation of the trajectory coordinate system with respect to the global coordinate system. If the space curve geometry is specified, the angles ψ ti = ψ ti(si), θ ti = θ ti(si), and φ ti = φ ti(si) are known for a given value of the parameter si. That is, θ ti = [ψ ti (s i )

θ ti (s i )

φ ti (s i )]T

(2.93)

is a known vector for a given value of si. Using the trajectory coordinate system, the global position vector of the center of mass of body i can be written as R i = R ti + A ti u ir

(2.94)

where the vector u ir is the position vector of the center of mass of the body with respect to the origin of the trajectory coordinate system. This vector is defined as


72


u ir = [0

y ir

z ir ]T

(2.95)

where yir and zir are the center of mass coordinates in, respectively, the lateral direction and the direction normal to the plane that contains the space curve. These coordinates are defined in the trajectory coordinate system. Note that the location of the center of mass in the longitudinal direction is assumed to be uniquely defined using the arc length si and, therefore, the first element of the vector u ir is identically zero.

2.7.1 VELOCITY

AND

ACCELERATION

Differentiating Equation 2.94 with respect to time, the absolute velocity vector of the center of mass can be written as i i ti u ir + A ti u ir = Lp R i = R ti + A

(2.96)

where pi is the vector of generalized coordinates of body i, defined as pi = [s i

yir

z ir

ψ ir

φ ir

θ ir ]T

(2.97)

 0 

(2.98)

and the matrix Li is a 3 × 6 coefficient matrix given as  ∂R ti ∂A ti  Li =  i + i u ir  ∂s   ∂s

jti

k ti

where 0 is a 3 × 3 null matrix. Using Equation 2.96, the absolute acceleration of the center of mass can be written as i = Li p i + γ iR R

(2.99)

where γ iR includes all the terms that are quadratic in the generalized velocities. Because the body has three relative rotational degrees of freedom with respect to the trajectory coordinate system, the angular velocity of body i can be written in the global coordinate system as ω i = ω ti + ω ir

(2.100)

where ωti is the absolute angular velocity vector of the trajectory coordinate system, and ωir is the angular velocity vector of the body with respect to the trajectory coordinate system. This equation can be rewritten in terms of the generalized velocity vector p i as ω i = Gtiθ ti + AtiGir θ ir = Hi p i

(2.101)



73

where 0  ti G = 0 1 

sin ψ ti − cos ψ ti 0

0  ir G = 0 1 

cos ψ ir sin ψ ir 0

− cos ψ ti cos θ ti   − sin ψ ti cos θ ti  ,  − sin θ ti 

(2.102)

− sin ψ ir cos φ ir   cos ψ ir cos φ ir   sin φ ir 

and Hi is a 3 × 6 coefficient matrix given as  ∂θ ti  Hi =  Gti i  ∂s  

0

0

 ( AtiGir )  

(2.103)

where 0 in this equation is a 3 × 1 zero vector. The angular acceleration of body i in the global coordinate system can be obtained by differentiating Equation 2.101 with respect to time as i + γ αi α i = Hi p

(2.104)

where γ αi includes all the terms that are quadratic in the generalized velocities. The angular acceleration vector can also be written in the body coordinate system as T

i + γ αi α i = Ai α i = H i p

(2.105)

where T

T

T

Hi = Ai Hi = Air Ati Hi ,

T

γ αi = Ai γ αi

(2.106)

and Air is the rotation matrix that defines the orientation of the body coordinate system with respect to the trajectory coordinate system based on the Euler angle sequence Z ir, Xir, Y ir. This matrix is given as Air =  cos ψ ir cos θ ir − sin ψ ir sin φ ir sin θ ir  ir ir ir ir ir  sin ψ cos θ + cos ψ sin φ sin θ ir ir  − cos φ sin θ 

(2.107) − sin ψ ir cos φ ir

cos ψ ir sin θ ir + sin ψ ir sin φ ir cos θ ir 

cos ψ ir cos φ ir

sin ψ ir sin θ ir − cos ψ ir sin φ ir cos θ ir 

sin φ ir



cos φ ir cos θ ir

 


74


Using Equations 2.99 and 2.105, one can write i   Li  i  γ iR  R +  i   i  =  ip  γ α   α  H 

(2.108)

The kinematic analysis presented in this section shows that the use of the trajectory coordinate system requires information on the second derivatives of the Euler angles ψ ti(si), θ ti(si), and φ ti(si) with respect to the arc length parameter si. This is clear from the definition of the angular acceleration vector, which includes the derivative of Hi of Equation 2.103 with respect to time. As is demonstrated in Chapter 3, the Euler angles ψ ti(si), θ ti(si), and φ ti(si) are functions of the second derivatives, since the orientation of the trajectory coordinate system requires knowledge of the curvature. For this reason, the use of the trajectory coordinates requires fourth-order derivatives of the equation of the space curve that defines the track centerline with respect to the arc length parameter si (Rathod and Shabana, 2007).

2.7.2 EQUATIONS

OF

MOTION

The nonlinear equations of motion for body i can be obtained by substituting Equation 2.108 into the Newton-Euler equations of motion. For body i, this is defined as  mi I   0

i    Fei 0  R =     i i i i i  i Iθθ   α   Me − ω × ( Iθθ ω ) 

(2.109)

Writing the absolute accelerations in terms of the trajectory accelerations using Equation 2.108 and premultiplying by the transpose of the matrix of coefficients of i in Equation 2.108, one obtains the following equations of motion expressed in p terms of the trajectory coordinates: i = Qipe + Qipv Mip p

(2.110)

   T T Qipe = Li Fei + Hi Mie   T T Qipv = − m i Li γ iR − Hi [ Iθθi γ αi + ω i × ( Iθθi ω i )]  

(2.111)

where T

T

Mip = m i Li Li + Hi Iθθi Hi

It is clear from this equation that, when the trajectory coordinates are used, the resulting mass matrix does not take as simple a form as the one used in the NewtonEuler equations of Equation 2.109.



75

As in the case of the absolute coordinates, an augmented formulation in terms of the trajectory coordinates can be developed. In this case, the kinematic constraints can be formulated in terms of the trajectory coordinates. The generalized forces associated with these trajectory coordinates must also be evaluated. Using Equation 2.110, the augmented form of the equations of motion in the case of a system that consists of nb bodies can be written in terms of the trajectory coordinates as M p   C p T

T

CTp   p  Q pe + Q pv    =   0   λ   Qd 

(2.112)

T

where p = [ p1 p2 … pnb ]T is the vector of system trajectory coordinates, Mp is the system mass matrix associated with the trajectory coordinates, Qpe is the vector of system applied forces associated with the trajectory coordinates, Qpv is the vector of centrifugal and Coriolis forces, Cp is the Jacobian matrix of the kinematic constraint equations, λ is the vector of Lagrange multipliers, and Qd is the vector resulting from the differentiation of the system constraint equations twice with respect to time. The use of the trajectory coordinates in developing specialized railroad vehicle dynamic algorithms has the advantage of making the formulation of some kinematic constraint equations easier. This is demonstrated by the following example.

EXAMPLE 2.9 Body i shown in Figure 2.9 is subjected to the following kinematic constraints: 1. The body is driven by a function f(t) along a space curve with predefined geometry. 2. The pitch rotation of the body is specified by a function g(t). 3. The body has no relative motion with respect to the trajectory coordinate system along the Z ti axis. Derive the equations of motion of the body i using the augmented formulation and the trajectory coordinates. Solution. Given that the motion of the body along the space curve is specified, one has the following constraint equation expressed in terms of the function f(t): C1 = s i − f (t ) = 0 The equation of the constraint on the pitch rotation of the body can be defined using the function g(t) as C2 = θ ir − g(t ) = 0


76

Railroad Vehicle Dynamics: A Computational Approach Given that the vertical motion with respect to the trajectory coordinate system is constrained, one has C3 = z ir − d 0 = 0 where d0 is the constant Z ti coordinate of the origin of the body coordinate system with respect to the trajectory coordinate system. Using the preceding three equations, the vector of constraint equations can be written in the following form:

 s i − f (t )    C = θ ir − g(t )  = 0  z ir − d 0    Differentiating this equation twice with respect to time, the constraint equation at the acceleration level is given as

 s i − f (t )  = θir − g(t )  = C p C p − Qd = 0    z ir    where the constraint Jacobian matrix Cp and the vector Qd are, respectively, given by

1  Cp = 0  0

0

0

0

0

0

0

0

0

0

1

0

0

0

 1 , 0 

 f (t )    Qd =  g(t )   0 

Note that because the constraint equations are linear in the generalized trajectory coordinates, the constraint Jacobian matrix is constant, and thus the right-hand-side vector of the constraint acceleration equation Qd depends on time only. Accordingly, the augmented form of the equations of motion for this one-body system is given by

 M ip   Cp

i  C Tp   p

 Qipe + Qipv    =   0   λ   Qd 

2.8 EMBEDDING TECHNIQUE In the augmented formulation discussed in the preceding two sections, the equations of motion are expressed in terms of redundant coordinates that are not independent because of the kinematic constraint relationships. For this reason, the constraint



77

forces appear explicitly in the equations of motion. Using the embedding technique, one can systematically eliminate the constraint forces to obtain a minimum set of the equations of motion expressed in terms of the system degrees of freedom. To obtain the minimum set of differential equations of motion and eliminate the constraint forces from these equations, the constraint equations are used to define a velocity transformation matrix that can be used to write the system velocities and accelerations in terms of the independent velocities and accelerations, respectively.

2.8.1 COORDINATE PARTITIONING

AND

VELOCITY TRANSFORMATION

Differentiating the constraint equations with respect to time, one obtains C q q = −C t

(2.113)

This equation can also be written as C qi q i + C q d q d = −C t

(2.114)

In this equation, the vectors qd and qi are, respectively, the vectors of dependent and independent generalized coordinates that are written as q = [q Td

q Ti ]T

(2.115)

Using Equations 2.114 and 2.115, the total vector of the system velocities can be written in terms of the independent velocities as q = B q i

(2.116)

It is important to point out that if the constraints are explicit functions of time (rheonomic constraints), Equation 2.116 should be altered to include an additional term that results from the differentiation of the constraint equations with respect to time. In general, joint constraints are not explicit functions of time, while specified motion trajectory constraints (driving constraints) can be explicit functions of time. The matrix B that appears in Equation 2.116 is called the velocity transformation matrix and is defined using Equation 2.114 as  −C −1 C  B =  q d qi  I  

(2.117)

In the case of complex railroad vehicle systems, the independent coordinates qi can be identified using numerical procedures such as Gaussian elimination and LU


78


factorization. The independent coordinates must be selected such that the constraint Jacobian matrix C q d associated with the dependent coordinates is a nonsingular square matrix (Garcia de Jalon and Bayo, 1994; Shabana, 2001; Wehage, 1980).

2.8.2 ELIMINATION

OF THE

CONSTRAINT FORCES

The velocity transformation matrix can be used to eliminate the generalized constraint forces −CTq λ that appear in the equations of motion (Equation 2.87). It can be shown that BT C Tq = 0

(2.118)

This result can be proved by substituting Equation 2.117 into Equation 2.118, leading to

BT C Tq =  −C qTi C q−dT

C Tq  I   Td  = −C Tqi C q−dT C qT d + C Tqi = 0  C qi 

(2.119)

which shows that indeed BT C Tq = 0 .

2.8.3 REDUCED-ORDER MODEL Using Equation 2.116, the system acceleration vector can be expressed in terms of the independent accelerations as = Bq i + γ q

(2.120)

where the vector γ includes all the terms that are quadratic in the velocities. This vector is given by  −C −1 [(C q ) q + 2Cq t q + Ctt ] γ =  qd q q  0  

(2.121)

Substituting Equation 2.120 into Equation 2.87, premultiplying by the transpose of the velocity transformation matrix B, and using Equation 2.118, one obtains a minimum set of dynamic differential equations of motion expressed in terms of the system degrees of freedom as i = (Qe )i + (Q v )i Mi q where

(2.122)



79

   (Qe )i = BT Qe   T (Qv )i = B (Qv − Mγ )  Mi = BT MB

(2.123)

Note that Equation 2.122 does not include any constraint forces.

EXAMPLE 2.10 For the system of Example 2.9, use the embedding technique to derive the equations of motion in terms of the degrees of freedom. Solution. Since the system is subjected to three constraint equations associated with the longitudinal and vertical displacements and the pitch rotation, the body has three degrees of freedom. These degrees of freedom are the lateral displacement yir, the yaw angle ψ ir, and the roll angle φ ir. Hence, the dependent and independent coordinates can be, respectively, written as p id = [s i

z ir

θ ir ]T ,

ψ ir

p ii = [ y ir

φ ir ]T

Using the preceding equations, the constraint Jacobian matrices are obtained in terms of the independent and dependent coordinates as

1 

C pd =  0  0

0

0

0

1 ,

1



0 

0 

C pi =  0  0

0

0

0

0

0



0 

Note that the constraint Jacobian matrix associated with the dependent coordinates is a nonsingular square matrix. Using the acceleration equation as previously obtained in Example 2.9, the system accelerations can be expressed in terms of the independent accelerations as

 s i   f (t )   ir   ir   y   y   z ir   0  p ii + γ i p i =  ir  =  ir  = Bi ψ   ψ   φir   φir       θir   g(t )  ii = [ where p y ir ψ ir φir ]T and


80


0  1 0 Bi =  0 0   0

0 0 0 1 0 0

 f (t )     0   0  γi =    0   0     g(t ) 

0

 0 0 , 0 1  0 

Using Equation 2.122, one has ii = (Qipe ) i + (Qipv ) i M ipi p where T

M ipi = Bi M ip Bi ,

T

(Qipe ) i = Bi Qipe ,

T

(Qipv ) i = Bi (Qipv − M ip γ i )

2.9 INTERPRETATION OF THE METHODS The augmented formulation and the embedding technique discussed in this chapter are the two basic approaches used to develop general-purpose computer multibody system algorithms. It is clear that the augmented formulation leads to a larger system of equations of motion that has a sparse matrix structure. This system of equations is expressed in terms of redundant coordinates and, therefore, the constraint forces appear explicitly in this system of equations. The use of Lagrange multipliers in the augmented formulation leads to a symmetric coefficient matrix in the acceleration equations. In the embedding technique, on the other hand, the dependent coordinates and the constraint forces are systematically eliminated, leading to a minimum set of equations of motion expressed in terms of the system independent coordinates (degrees of freedom). In this section, a physical interpretation of the augmented formulation and the embedding technique is presented using a simple example. To this end, the planar wheel/rail example shown in Figure 2.10 is considered.

2.9.1 KINEMATIC

AND

DYNAMIC EQUATIONS

The wheel is assumed to have a radius r, mass mw, and mass moment of inertia I w about its center of mass. The rail segment is assumed to be circular with radius R. The wheel, which is subjected to an external force Fw = [Fxw Fyw ] T and a moment M w, is assumed to roll without slipping on the rail and, therefore, there is no friction force due to sliding. Since, in this example, pure rolling of the wheel is considered and the rail is assumed to be fixed, the system has only one degree of freedom. It is clear from Figure 2.10 that the tangent and normal vectors at the contact points are given as follows:



81

FIGURE 2.10 Planar wheel/rail example.

 − cos φ  tr =  ,  sin φ 

 − sin φ  nr =    − cos φ 

(2.124)

The velocity on the wheel at the contact point can be written as w + ω w × uw rcw = R c

(2.125)

In this equation, 0   ω =  0 , θ w    w

 −r sin φ    u =  −r cos φ   0    w c

(2.126)

where θ w is the angular velocity of the wheel. Using the preceding two equations, one obtains w   w w + ω w × u w =  Rx + rθ cos φ  rcw = R c w w  Ry − rθ sin φ 

(2.127)

Since the condition of pure rolling is assumed, the absolute velocity of the wheel at the contact point must be identically zero. That is, rcw = 0 . Therefore, Equation 2.127 leads to the following two constraint equations on the motion of the wheel: R xw + rθ w cos φ = 0   R yw − rθ w sin φ = 0 

(2.128)


82


These two constraint equations that guarantee that the wheel will roll without slipping reduce the number of independent velocities of the system to one. It is also clear from Figure 2.10 that

(

)

(

Rxw = − R − r sin φ ,

)

Ryw = − R − r cos φ

Using these two equations and Equation 2.128, one obtains the following relationship between the derivatives of the two angles θ w and φ:

(

)

rθ w = R − r φ

(2.129)

The constraints of Equation 2.128 can be used to identify two dependent velocities, say R xw and R yw . Using the generalized coordinate partitioning method, the system dependent velocities can be expressed in terms of the independent velocity θ w . Figure 2.11 shows the free-body diagram of the wheel. In this figure, N = [Nx Ny]T is the reaction force at the contact point. The free-body diagram of Figure 2.11 can be used to write the following Newton-Euler equations for the planar wheel: xw = Fxw + N x mwR

   w w w w m Ry = Fy − m g + N y   I wθw = M w + N x r cos φ − N yr sin φ 

(2.130)

In this equation, g is the gravity constant. Equations 2.130 and 2.128 represent the system differential and algebraic equations that include the differential equations of motion and the algebraic constraint equations, respectively. We note that, for given initial conditions for the coordinates and velocities, the three scalar equations in xw , R yw ,θw , Nx , and Ny . Therefore, one needs Equation 2.130 have five unknowns: R the two constraints of Equation 2.128 in order to have five equations that can be solved for the three unknown accelerations and the two components of the reaction

FIGURE 2.11 Free-body diagram.



83

force. We also note that the number of constraint equations is equal to the number of dependent variables, and this number is also equal to the number of the unknown independent constraint forces. In the remainder of this section, this simple planar wheel/rail example is used to provide a physical interpretation of the augmented formulation and the embedding technique.

2.9.2 AUGMENTED FORMULATION In the augmented formulation, the constraint equations at the acceleration level are combined with the system differential equations of motion to obtain a number of equations equal to the number of the unknown accelerations and constraint forces. Differentiating the velocity constraints of Equation 2.128 with respect to time, one obtains sin φ = 0  xw + rθw cos φ − rθφ R   cos φ = 0  yw − rθw sin φ − rθφ R 

(2.131)

Combining Equations 2.130 and 2.131 and rearranging the terms in these equations such that the unknowns are on the left-hand side, one obtains xw − N x = Fxw mwR

  w w w w  m Ry − N y = Fy − m g  w w w I θ − N x r coos φ + N yr sin φ = M   sin φ xw + rθw cos φ = rθφ  R   cos φ yw − rθw sin φ = rθφ R 

(2.132)

This equation can be written in the following matrix form: mw   0  0   1  0 

0 mw 0 0 1

0 0 Iw r cos φ −r sin φ

−1 0 −r cos φ 0 0

xw   F w  0  R x   w   w w  −1   Ry   Fy − m g  r sin φ   θw  =  M w     sin φ  0   N x   rθφ     0   N y   rθφ cos φ 

(2.133)

Equation 2.133 can be written in the form of Equation 2.88 as follows: M  Cq

CTq   q  Q + Qv    =  e  0   λ   Qd 

(2.134)


84


where, in this example, q = [Rxw Ryw θ w]T, Qv in this planar example is equal to zero, and mw  M= 0  0 

0 w

m 0

 Fxw   w w  Qe =  Fy − m g  ,  Mw   

0 1  0  , Cq =  0 I w 

0 1

       − N x   λ=   − N y   

r cos φ   −r sin φ 

sin φ   rθφ Qd =  , rθφ cos φ 

(2.135)

From the definition of λ in Equation 2.135, it is clear that Lagrange multipliers can be interpreted in this example as the negative of the reaction forces. This simple interpretation of Lagrange multipliers is not general, since in other constraint types, the physical meaning of Lagrange multipliers can be different. It is clear that the coefficient matrix in Equation 2.133 has a sparse matrix structure, with many elements equal to zero. This sparse matrix structure is exploited in most computational algorithms used in general-purpose multibody computer codes.

2.9.3 EMBEDDING TECHNIQUE In the embedding technique, a minimum set of equations of motion is obtained by systematically eliminating the dependent accelerations and the constraint forces. The constraints at the acceleration level (Equation 2.131) can be used to write two dependent accelerations in terms of the independent acceleration. If Rxw and Ryw are considered as the dependent accelerations and θw is considered as the independent acceleration, it is clear from Equation 2.131 that sin φ  xw = −rθw cos φ + rθφ R   w w Ry = rθ sin φ + rθφ cos φ 

(2.136)

This equation can be used to write all the system accelerations in terms of the independent acceleration θw as follows: sin φ  xw   −r cos φ  R  rθφ  w     w  =  Ry  =  r sin φ  θ + rθφ cos φ  q  θw   1    0      

(2.137)

This equation can be written in the form of Equation 2.120 as = Bθw + γ q

(2.138)



85

where sin φ   rθφ   γ = rθφ cos φ    0  

 −r cos φ    B =  r sin φ  ,  1   

(2.139)

The equations of motion of Equation 2.130 can also be written in the form of Equation 2.87 as = Qe + Qv − CqT λ Mq

(2.140)

where the vectors and matrices in this equation are as defined in Equation 2.135. Substituting Equation 2.138 into Equation 2.140 and premultiplying by BT, one obtains the following equation of motion, which does not include the constraint forces:

(I

w

)

(

)

+ m wr 2 θw = M w − rFxw cos φ + r Fyw − m w g sin φ

(2.141)

If the external forces and moments are assumed to be zero and if the wheel is replaced by a cylinder with I w = mwr 2/2, Equation 2.141, upon the use of Equation 2.129, reduces to the following familiar equation of the single-degree-of-freedom oscillatory system given by 3( R − r ) φ + g sin φ = 0 2

(2.142)

It is clear that Equation 2.141 does not have any constraint forces, since it can be verified that BT C Tq = 0 in this example, as previously discussed in this chapter.

2.9.4 D’ALEMBERT’S PRINCIPLE The concept used in the embedding technique was first introduced by D’Alembert, who suggested treating the inertia forces in the same manner as the applied forces. Using D’Alembert’s principle, the constraint forces can be eliminated from the equations of motion, leading to a number of equations of motion equal to the number of the system degrees of freedom. For example, if the inertia forces are treated as the applied forces, one can take the moments of the two sets of forces about any point and equate the moment of the inertia forces to the moment of the applied forces. For example, using the free-body diagram in Figure 2.11, the moments of the applied force about the contact point c can be written as

(

)

M w − rFxw cos φ + r Fyw − m w g sin φ


86


while the moment of the inertia forces about the same point is given by xwr cos φ + m w R ywr sin φ I wθw − m w R Substituting for the accelerations Rxw and Ryw from the constraint of Equation 2.136, it is clear that the moment of the inertia forces about the contact point c can be written as I w + m wr 2 θw . Equating the moments of the inertia forces to the moments of the applied forces, one obtains Equation 2.141.

(

)

2.10 VIRTUAL WORK While the concepts upon which the augmented formulation and the embedding technique are based can be demonstrated using a simple example, as shown in the preceding section, the principle of virtual work can be used to provide a systematic and more general derivation of the methods. This principle will be used in Chapter 5 to provide a simple and straightforward derivation of the equations of motion when the system is subjected to contact constraints that require the use of nongeneralized surface parameters. A virtual change in the system coordinates δq is an infinitesimal change that occurs without regard to time change. This virtual change in the coordinates can be used to develop the principle of virtual work in dynamics. To develop this important principle, Equation 2.87 can be rewritten as − Qe − Qv + CqT λ = 0 Mq

(2.143)

Multiplying this equation by the transpose of the vector δq, one obtains the following scalar equation:

(

)

− Qe − Qv + CqT λ = 0 δ qT Mq

(2.144)

This equation is called the Lagrange-D’Alembert equation, and it is the basis for the general development of the augmented formulation and the embedding technique. The Lagrange-D’Alembert equation (Equation 2.144) can also be written as follows:

δ Wi = δ We + δ Wc

(2.145)

where δWi is the virtual work of the system inertia forces, δWe is the virtual work of the system applied forces, and δWc is the virtual work of the system constraint forces. These expressions of the virtual work are given as

(

)

− Qv , δ Wi = δ qT Mq

δ We = δ qT Qe ,

δ Wc = −δ qT CqT λ

(2.146)

Since the equilibrium of the system is considered, and since the reaction forces acting on the interconnected bodies are equal in magnitude and opposite in direction, the virtual work of the constraint forces is identically zero, that is,



87

δ Wc = −δ qT CqT λ = 0

(2.147)

for kinematically admissible virtual displacements δq that satisfy the constraint equations. In the case of specified motion trajectories, which are driving constraints, the virtual work of the resulting constraint forces is also equal to zero, since the coordinates associated with the specified motion are prescribed, and as a result, the virtual changes in these prescribed coordinates are equal to zero. The result of Equation 2.147 can also be proven mathematically by writing the virtual changes in the system coordinates in terms of the virtual changes in the independent coordinates or degrees of freedom. This leads to the following relationship:

δ q = Bδ qi

(2.148)

In this equation, B is the velocity transformation matrix previously defined in this chapter. Substituting Equation 2.148 into the expression of the virtual work of the constraint forces and using the fact that BT C Tq = 0 as previously demonstrated, one obtains the important result of Equation 2.147. Using the results of Equation 2.147, Equation 2.145 can be written as

δ Wi = δ We

(2.149)

This is the principle of virtual work in dynamics, which states that, for a multibody system, the virtual work of the system inertia forces is equal to the virtual work of the system applied forces. Furthermore, the virtual work of the constraint forces is identically zero. The results of the embedding technique can be obtained using Equation 2.149 by substituting Equation 2.148 into Equation 2.149. This leads to a minimum set of equations of motion from which the constraint forces are eliminated (Rosenberg, 1977; Shabana, 2001). It is important to point out that the condition of Equation 2.147 that is used to obtain the principle of virtual work in dynamics given by Equation 2.149 is only valid when the equilibrium of the entire system is considered. If the equilibrium of individual bodies is considered, the virtual work of the constraint forces acting on each body separately is not equal to zero. In this case, the virtual work principle for body i in the system can be written as

δ Wii = δ Wei + δ Wci

(2.150)

In this equation, δWii, δWei, and δWci are, respectively, the virtual work of the inertia, applied, and constraint forces of body i. Note that if body i is subjected to constraints, then δWci ≠ 0, while nb

∑δW

i c

= 0,

i =1

where nb is the total number of bodies in the system.

(2.151)



3

Rail and Wheel Geometry

If the goal is to develop general and detailed models of railroad vehicle systems, multibody system algorithms must be modified to include a wheel/rail contact model. Three steps are employed in the computational algorithm used to obtain the numerical solution of the wheel/rail contact problem. The first is the geometry step, in which the locations of the points of contact between the wheel and the rail are determined. The second is the kinematic step, in which normalized kinematic quantities called creepages that measure the relative velocities between the wheel and the rail at the contact points are determined. In the third step, called the dynamic or kinetic step, the forces that act on the wheel and the rail as the result of the contact are determined. The accuracy of the numerical solution of the contact problem depends strongly on the accurate prediction of the location of the contact points. The solution for the contact locations requires an accurate representation of the geometry of the wheel and the rail surfaces. This representation can be defined using local surface geometric properties such as the radii of curvature and the tangent and normal vectors to the surfaces. These geometric properties are not only important for determining the contact locations (geometry problem), but they are also important, as described in Chapter 4, in determining the forces that represent the dynamic interaction between the wheel and the rail. Therefore, basic knowledge of differential geometry is necessary to understand the wheel/rail contact problem. In particular, the theories of curves and surfaces are fundamental in the study of the dynamic interaction between the wheel and the rail. This chapter discusses topics in differential geometry that are repeatedly used in this book and that are used in the geometric description of wheel and rail surfaces. The spatial representation of the rail geometry in particular is crucial in investigating and predicting the railroad vehicle dynamic response. For example, the study of the curving behavior of a vehicle is important in determining the safe speed of operation. Furthermore, track irregularities that influence the dynamic response of the vehicle can cause ride discomfort or even train derailments. The track geometry is also fundamental in the formulation of the dynamic equations, which, as shown in later chapters of this book, are expressed in terms of generalized coordinates that depend on the track geometry. This chapter discusses the mathematical representation of the rail (or the track) geometry and introduces the definitions and terminologies that are used in the field of railroad vehicle dynamics and that will be used repeatedly throughout this book. The parametric representation of the wheel surface is discussed at the end of the chapter.

89


90


3.1 THEORY OF CURVES To introduce the concept of a curve, a Cartesian coordinate system is used to define the position vector of points in space in terms of three components as y =  y1

y3 

y2

T

(3.1)

A curve is defined as a real vector function that can be uniquely expressed in terms of one parameter, t. That is, the components of the vector function can be determined once this parameter is specified. Using this definition, a curve defined over the interval a ≤ t ≤ b can be written in the following form:

()

()

()

y t =  y1 t

y2 t

()

y3 t 

T

(3.2)

This equation, which is the parametric representation of a curve, can be used to determine the location of a point on the curve for an arbitrary value of the parameter t. The components yi (i = 1, 2, 3) must be differentiable over the interval on which t is defined. An example of the preceding equation is the parameterized differentiable curve y(t) = [rcost rsint αt]T, where r and α are constants. One can show that the trace of this curve is a helix of pitch 2πα on the cylinder x2 + y2 = r 2 (Do Carmo, 1976).

3.1.1 ARC LENGTH

AND

TANGENT LINE

For a given t,

()

()

dy  dy1 t = dt  dt 

dy2 t dt

()

dy3 t   dt  

T

(3.3)

is called the tangent line to the curve at t. In the study of the theory of curves, the existence of the tangent line at every point is essential. If at a point t, dy(t)/dt = 0, the point is called a singular point. The arc length of a curve from point t0 to point t can be obtained using the tangent line as follows: t

s=

∫ dt dt dy

(3.4)

t0

A curve can be parameterized by its arc length s. In this case, one can write y = y(s). Using the parameterization in terms of the arc length, one can show using Equation 3.4 that the tangent vector, t(s) = dy/ds, becomes a unit vector, that is,



91

()

dy = t s =1 ds

(3.5)

It is important to recognize that while any parameter can be used to define the curve, the tangent obtained by differentiation with respect to the parameter is a unit vector only when the curve is parameterized by its arc length. Note that for an arbitrary parameter t, one has dy dy  ds  = dt ds  dt 

(3.6)

In the remainder of this section, we will assume that the curve is parameterized by its arc length s.

3.1.2 CURVATURE

AND

TORSION

If a curve is parameterized by its arc length, the curvature vector is obtained by differentiating the tangent vector. The curvature vector of the curve y(s) is defined as

()

y ′′ s =

d 2 y dt = ds 2 ds

(3.7)

Since the tangent obtained by differentiation with respect to the arc length is a unit vector, the tangent and curvature vectors are orthogonal. This follows from differentiating the equation y′T y′ = 1, which leads to 2y′T y″ = 0, proving the orthogonality of the tangent and curvature vectors. The curvature of the curve at a point s is defined as

()

()

k s = y ′′ s = t ′(s )

(3.8)

Since the tangent vector t(s) has unit length, the curvature k(s) measures the rate of change of orientation of the tangent vector. In other words, the curvature measures how rapidly the curve pulls away from the tangent line. Because of the orthogonality of the tangent and curvature vectors, a unit vector along the curvature vector defines the unit normal to the curve nc (subscript c is used here to indicate a normal to a curve, since n is used throughout this book as the normal to a surface). Therefore, the unit normal to the curve at s is defined as

()

nc s =

( ) = t ′ (s ) k (s ) k (s )

y ′′ s

(3.9)


92


The plane formed by the unit tangent and normal vectors is called the osculating plane at s. The radius of curvature at s is defined as R = 1/k(s). The cross-product of the orthogonal tangent and normal unit vectors defines a vector that is normal to the osculating plane. This new vector is called the binormal vector at s and is given by

() ()

()

b s = t s × nc s

(3.10)

The three orthogonal unit vectors t, nc, and b form a coordinate system called the Frenet frame. The orientation of this frame is defined by the matrix [t nc b]. The t–b plane is called the rectifying plane, and the nc–b plane is called the normal plane. Differentiating Equation 3.10 with respect to s and recognizing that the two vectors t′(s) and nc(s) are parallel, one obtains

()

()

() ()

() ()

()

b′ s = t ′ s × nc s + t s × nc′ s = t s × nc′ s

(3.11)

That is, b′(s) is normal to t(s), and since b(s) is a unit vector, b′(s) is also orthogonal to b(s). It follows that b′(s) is parallel to nc, and one can write b′(s) in the following form:

()

() ()

b′ s = − τ s n c s

(3.12)

where τ is called the torsion. The curvature and torsion completely describe the behavior of the curve in the neighborhood of s. In summary, we have the following equations (Kreyszig, 1991):    n′c = − kt + τb   b′ = − τn c  t ′ = kn c

(3.13)

These equations, which express the derivatives in terms of the tangent, normal, and binormal unit vectors, are called the Serret-Frenet formulas.

3.2 GEOMETRY OF SURFACES The geometry of a surface can be described using two independent parameters. Using a system of Cartesian coordinates x1, x2, and x3, each point on the surface is assumed to have a unique position vector x that can be defined in the three-dimensional space in terms of these two independent parameters as follows: x(s1, s2 ) =  x1 (s1, s2 )

x2 (s1, s2 )

x3 (s1, s2 ) 

T

(3.14)



93

where s1 and s2 are the parameters used to describe the surface geometry and are called the parameters of the surface. If differential calculus is to be applied, Equation 3.14 must satisfy certain requirements of differentiability with respect to s1 and s2. Furthermore, Equation 3.14 must satisfy the following conditions (Goetz, 1970; Kreyszig, 1991): (a) In a bounded s1–s2 domain, each point represented by Equation 3.14 corresponds to just one pair of s1 and s2 in this domain. That is, the mapping of Equation 3.14 is one-to-one. (b) In the bounded domain, the Jacobian matrix  ∂x1  ∂s  1 ∂x   ∂x2 = ∂s2   ∂s1  ∂x3   ∂s1

 ∂x J=  ∂s1

∂x1  ∂s2  ∂x2   ∂s2  ∂x3   ∂s2 

(3.15)

is of rank two. Condition (b) is satisfied if

(

∂x s1, s2 ∂s1

) × ∂x (s , s ) ≠ 0 1

2

∂s2

This condition implies that the two columns of the matrix of Equation 3.15 are linearly independent. For example, assume that Equation 3.14 is used to represent the surface shown in Figure 3.1. One can define a curve x(s1,s2c) for a constant parameter s2 = s2c and another curve x(s1c,s2) for a constant parameter s1 = s1c. Therefore, ∂x(s1 , s2c ) ∂s1 represents the tangent vector to the curve x(s1,s2c) along the s1 coordinate line, and ∂x(s1c , s2 ) ∂s2 represents the tangent vector to the curve x(s1c,s2) along the s2 coordinate line. Therefore, if

(

∂x s1 , s2c ∂s1

) × ∂x ( s

1c

, s2

∂s2

)≠0


94


FIGURE 3.1 Surface mapping.

for arbitrary s1c and s2c, the rank of the Jacobian matrix J is two, and Equation 3.14 represents a surface. The two tangents to the coordinate lines are not necessarily orthogonal or unit vectors. In the wheel/rail contact problem, the tangent and the normal vectors at the contact points on the surfaces enter into the formulation of the kinematic and force relationships. For example, these vectors are used to determine the locations of the points of contact between the wheel and the rail. They are also used to determine the principal curvatures and the principal directions of the surface at a specified point. The principal curvatures and principal directions are used to calculate the shape of the area of contact between the wheel and the rail, as discussed in Chapter 4.

3.2.1 TANGENT PLANE

AND

NORMAL VECTOR

Equation 3.14 represents the surface as a function of the two parameters s1 and s2. One can define a curve on the surface by assuming that these two parameters are related and can be expressed as functions of a single parameter t, that is, s1 = s1(t) and s2 = s2(t), where t is any arbitrary variable. Therefore, Equation 3.14 can be written as follows:

(

)

x s1 (t ), s2 (t ) = y (t )

(3.16)

where y(t) is a regular curve on the surface and dy ∂x ds1 ∂x ds2 = + ≠0 dt ∂s1 dt ∂s2 dt

(3.17)

In writing this equation, it is assumed that ∂x/∂s1 × ∂x/∂s2 ≠ 0, ds1/dt ≠ 0, and ds2/dt ≠ 0. Equation 3.17 defines a tangent to the surface at point P that also belongs to the curve y(t). This tangent is a linear combination of the two linearly independent tangent vectors ∂x/∂s1 and ∂x/∂s2 at point P, as shown in Figure 3.2 and as also



95

FIGURE 3.2 Tangents to a surface.

demonstrated by Equation 3.17. Therefore, the plane that contains the two tangents ∂x/∂s1 and ∂x/∂s2 is the tangent plane to the surface. Consequently, a nonzero vector dy/dt is tangent to the surface at point P if, and only if, it is parallel to the tangent plane at this point. The normal to the surface at point P can be defined as the normal to the tangent plane and can be written as follows: n=

x ,1 × x ,2 x ,1 × x ,2

(3.18)

where x,1 = ∂x/∂s1 and x,2 = ∂x/∂s2. As in the case of curves, the surface can be defined uniquely using certain local quantities called the first and second fundamental forms. These fundamental forms, which were introduced by Gauss and are discussed below, can be used to calculate the curvatures and the principal directions of the surface at an arbitrary point.

3.2.2 FIRST FUNDAMENTAL FORM The first fundamental form of a surface is defined as follows: I = dx ⋅ dx = dx T dx

(3.19)

Since dx = x,1ds1 + x,2ds2, where x,1 = ∂x/∂s1 and x,2 = ∂x/∂s2, Equation 3.19 can be written as follows: I = ( x,1ds1 + x,2 ds2 )T ( x,1ds1 + x,2 ds2 )

(3.20)

This equation can also be written as

( )

I = E ds1

2

+ 2 Fds1ds2 + G ( ds2 )2

(3.21)


96


where E = x,T1 x,1 , F = x,T1 x,2 , G = x,T2 x,2

(3.22)

From Equation 3.21, it is clear that the first fundamental form is a homogenous function of second degree in ds1 and ds2 with coefficients E, F, and G. These coefficients are called the coefficients of the first fundamental form. The first fundamental form can be used to measure distances, angles, and areas on the surface. For example, consider the curve y(t) given by Equation 3.16. The length of this curve over the domain a ≤ t ≤ b is given by b

l=

∫ a

b

=

∫ a

dy dt = dt

b

∫

dx dx dt = ⋅ dt dt

a

b

T

 ds1 ds2   ds1 ds2   x,1 dt + x,2 dt   x,1 dt + x,2 dt  dt

∫ a

2

2

 ds   ds  ds ds E  1  + 2 F 1 2 + G  2  dt dt dt  dt   dt 

(3.23)

Equation 3.23 shows that the arc length on the surface is the integral of the square root of the first fundamental form. The coefficients of the first fundamental form can also be used to determine the angle between the two tangents x,1 and x,2 as cos α =

x ,T1x ,2 = x ,1 x ,2

F

(3.24)

EG

This equation shows that the two tangents x,1 and x,2 to the surface at a point are perpendicular if F = 0

3.2.3 SECOND FUNDAMENTAL FORM The second fundamental form of a surface is defined as follows:

(

II = − dx ⋅ dn = − x,1ds1 + x,2 ds2

( )

= L ds1

2

) (n ds T

,1

1

+ n,2 ds2

)

( )

+ 2 Mds1ds2 + N ds2

(3.25)

2

where L = − x,T1 n,1 , M = −

(

)

1 T x,1 n,2 + x,T2 n,1 , N = − x,T2 n,2 2

(3.26)



97

and n is the normal vector to the surface that is given by Equation 3.18. It is clear from Equation 3.25 that the second fundamental form is also a homogenous function of second degree in ds1 and ds2 with coefficients L, M, and N. These coefficients are called the coefficients of the second fundamental form. Since x,1 and x,2 are perpendicular to n for all s1 and s2, one has the following identities:

(x n) (x n) (x n) (x n) T ,1

T ,1

T ,2

T ,2

= x,T11n + x,T1 n,1 = 0    = x,T12 n + x,T1 n,2 = 0  ,2   T T = x,21n + x,2 n,1 = 0  ,1   = x,T22 n + x,T2 n,2 = 0  ,2  ,1

(3.27)

These identities lead to x,T11n = − x,T1 n,1 , x,T12 n = − x,T1 n,2 , x,T21n = − x,T2 n,1 , x,T22 n = − x,T2 n,2 (3.28) Using the preceding equations, the coefficients of the second fundamental form can be written as follows: L = x,T11n, M = x,T12 n, N = x,T22 n

(3.29)

where x,ij = ∂2x/(∂si∂sj). Using the preceding equation, one can show that the second fundamental form can be written as II = d 2 x ⋅ n

(3.30)

where

( )

d 2 x = x,11 ds1

2

( )

+ 2 x,12 ds1ds2 + x,22 ds2

2

(3.31)

To explain the physical meaning of the second fundamental form, suppose that a point P is on the surface x and a point Q is a point in the neighborhood of point P on the surface x, as shown in Figure 3.3. The length of the components of the vector u between points P and Q projected onto the normal n is defined as follows: d = uT n

(3.32)


98


FIGURE 3.3 Second fundamental form of a surface.

If point P is defined at x(s1, s2) and point Q is defined at x(s1 + ds1, s2 + ds2), Equation 3.32 can be written as follows:

(

)

T

d = u T n = x(s1 + ds1 , s2 + ds2 ) − x(s1 , s2 ) n

((

) ( ))

2  1 =  dx + d 2x + O ds1 + ds1 2 

= dx T n +

((

2

T

  n 

(3.33)

) ( ))

2 1 2 T d x n + O ds1 + ds1 2

2

Since dxTn = 0 by virtue that dx is parallel to the tangent plane, one has

d=

((

1 2 T d x n + O ds1 2

) + ( ds ) ) = 21 II + O (( ds ) + ( ds ) ) 2

2

1

2

2

1

1

(3.34)

It is clear from this equation that if higher-order terms are neglected, the second fundamental form is twice the projection of the vector u onto n, and the absolute value of the second fundamental form is twice the distance projected along the normal from Q to the tangent plane at P. Clearly, if the surface is flat (planar), this projected distance is zero for any arbitrary pairs of points. Equation 3.33 is called the osculating paraboloid at P. The coefficients of the second fundamental form can be used to determine the nature of the surface in the neighborhood of P as follows: The The The The

surface surface surface surface

is is is is

called called called called

elliptic at point P if LN − M2 > 0 hyperbolic at point P if LN − M2 < 0 parabolic at point P if LN − M2 = 0 planar at point P if L = N = M = 0



99

3.2.4 NORMAL CURVATURE Let y = y(s1 (t ), s2 (t )) be a regular curve Ln defined on the surface x = x(s1, s2). The normal curvature vector to Ln at point P denoted by Kn is defined as the projection of the curvature vector K of the curve defined by y onto the normal n to the surface at point P and is given by

(

)

Kn = K ⋅ n n

(3.35)

Note that Ln can be defined as the intersection of a plane that contains the tangent to Ln and the normal vector, as shown in Figure 3.4. The component of the curvature vector of Ln along the normal to the surface at P is called the normal curvature and is defined as kn = K ⋅ n

(3.36)

It is important to mention that, from Equation 3.36, the sign of kn depends on the direction of the normal n. A positive sign of kn means that the normal vector n is directed toward the curvature center. It is also clear from Equation 3.18 that the sign of n depends on the manner in which the surface parameters are ordered and, therefore, the choice of the order of the surface parameters must be taken into consideration. In the wheel/rail contact problem, it is important to know the location of the curvature center and whether it lies inside or outside the wheel or the rail, as will be discussed in the following chapter. Recall that the curvature vector of the curve Ln at a point P on the surface x is given by K=

FIGURE 3.4 Surface curvature.

dt dt = ds dt

dx dt

(3.37)


100


where t is the tangent vector to Ln at P, and s is the arc length parameter. Since t is orthogonal to n, one has

( )

d T t n =0 dt which leads to T

 dt  T dn  dt  n = −t dt

(3.38)

Substituting Equation 3.37 into Equation 3.36 and using Equation 3.38, one obtains

kn =

L ( ds1 dt )2 + 2 M ( ds1 dt )( ds2 dt ) + N ( ds2 dt )2 E ( ds1 dt )2 + 2 F ( ds1 dt )( ds2 dt ) + G ( ds2 dt )2

(3.39)

This equation shows that kn depends on the ratio (ds1/dt)/(ds2/dt), which is the direction of the tangent line to Ln at point P, and also depends on the first and second fundamental coefficients that are functions of the coordinates of point P. Therefore, all the curves through point P that are tangent to the same line through P have the same normal curvature. As the normal curvature to the curve Ln at point P depends on the location of point P and the direction of the tangent line at point P, the normal curvature can be written as

kn =

L ( ds1 )2 + 2 Mds1ds2 + N ( ds2 )2 II = , I E ( ds1 )2 + 2 Fds1ds2 + G ( ds2 )2

( ds1 )2 + ( ds2 )2 ≠ 0

(3.40)

The first fundamental form I is positive definite, since it is a measure of the square of a distance. It follows from the preceding equation that the sign of kn depends on the sign of the second fundamental form II. Clearly for a planar point, kn = 0 in all directions, while for an elliptic point, kn ≠ 0 and has the same sign as ds1/ds2. In the case of a hyperbolic point, kn can be positive, negative, or zero, depending on ds1/ds2, while for a parabolic point, kn has the same sign and is zero for a direction for which the second fundamental form II is equal to zero.

3.2.5 PRINCIPAL CURVATURES

AND

PRINCIPAL DIRECTIONS

The normal curvature is called the principal curvature if its value is maximum or minimum. Therefore, the principal curvatures can be determined by solving the following two equations: ∂kn ∂kn = =0 ∂ ds1 ∂ ds2

( )

( )

(3.41)



101

Using Equation 3.40, one can show that Equation 3.41 leads to two scalar equations that can be written in the following matrix form:  L − kn E   M − kn F

M − kn F   ds1  0    =   N − kn G   ds2  0 

(3.42)

This equation has a nontrivial solution if, and only if, the determinant of the coefficient matrix is equal to zero, that is,  L − kn E det   M − kn F

M − kn F  =0 N − kn G 

(3.43)

− ( EN + GL − 2 FM ) kn + LN − M 2 = 0

(3.44)

or

( )

( EG − F 2 ) kn

2

The two roots of this quadratic equation determine the principal curvatures k1 and k2 that can be used in Equation 3.42 to determine the principal directions. The mean curvature Km at a point P is defined as the average of the principal curvatures: Km =

(

1 k1 + k2 2

)

The Gaussian curvature KG at P is defined as KG = k1k2. The Gaussian curvature is an invariant property of the surface. The principal curvatures and principal directions are used in Chapter 4 in Hertz contact theory to determine the dimension of the area of contact between the wheel and the rail.

EXAMPLE 3.1 A surface is represented by x = s1i + 2s1s2 j + s2 k where i, j, and k are the unit vectors along the axes of a Cartesian coordinate system. Determine the principal curvatures and the principal directions of this surface at point (s1,s2) = (1,0). Solution. Using the surface equation, one has x,1 = i + 2s2 j,

x,2 = 2s1 j + k,

x,11 = 0,

x,12 = 2 j,

x,22 = 0


102


n=

x ,1 × x ,2 x ,1 × x ,2

=

( 2s i − j + 2s k ) ( 2s ) + 1 + ( 2s ) 2

1

2

2

2

1

The coefficients of the first fundamental form are

( ),

E = x T,1x ,1 = 1 + 4 s2

2

( )

F = x T,1x ,2 = 4 s1s2 , G = x T,2 x ,2 = 1 + 4 s1

2

The coefficients of the second fundamental form are

L = x T,11n = 0,

M = x T,12 n =

−2

( ) + ( 2s )

1 + 2s1

2

, 2

N = x T,22 n = 0

2

Because LN − M2 < 0 for all s1 and s2, the surface is hyperbolic. The principal curvatures of this surface are the roots of Equation 3.44 and are given by

k1,2 =

− b ± b 2 − 4 ac 2a

where a = EG − F 2 ,

b = − ( EN + GL − 2 FM ),

c = LN − M 2

At the given point

E = 1, F = 0, G = 5, L = 0, M = −2

5 , N = 0, a = 5, b = 0, c = −0.8

Therefore, the principal curvatures are given as

k1 = 0.4, k2 = −0.4 and the principal directions are

 1   1   ,    −0.4472   0.4472 



103

3.3 RAIL GEOMETRY The dynamic behavior of railroad vehicles depends on the track geometry, and for this reason, it is important to accurately describe the rail geometry to be able to correctly predict the vehicle response. Using the methods of differential geometry discussed in the preceding section, the rail surface can be defined in a parametric form. The description of the surface geometry must be general to allow the use of the parametric equations to represent arbitrary rail profiles that can be provided analytically, in tabulated form, or from direct measurements. For example, in the case of a straight segment of the track, the surface of the rail can be obtained by translation of the profile curve, as shown in Figure 3.5. This surface can be defined by the parametric equations  X = s1  Y = s2  Z = f (s2 ) 

(3.45)

where s1 is the distance along the rail (arc length) and is defined as the rail longitudinal surface parameter, and s2 is the rail lateral surface parameter that is used as an independent variable to describe the rail profile. If the rail profile, for example, has a sinusoidal shape, the function f(s2) can be defined analytically as follows:  πs  f (s2 ) = h cos  2   w 

(3.46)

where h is the height and w is the width of the rail head. The function f(s2) can also represent measured rail profile. In this case, this function can be defined using a spline function that depends on s2.

FIGURE 3.5 Rail surface and its coordinate system.


104


FIGURE 3.6 Coordinate systems used to define the track geometry.

In the case of curved rails, representation of the position and orientation of the rail cross section (profile frame) as a function of the longitudinal surface parameter s1 is required. To account for different possible scenarios, such as a variation in the gage or relative rotations of the rails, it is necessary to define the surface of each rail (right and left) in the track independently. It is also important to be able to represent each rail as a separate body that can have independent motion and can be subjected to independent loading conditions and kinematic constraints. Such a general representation can be achieved by introducing several coordinate systems, as shown in Figure 3.6. First, a global coordinate system XYZ that is assumed to be fixed in time is introduced. The track, as shown in Figure 3.6 can be defined by two bodies l and r that represent, respectively, the left and right rails. The frames XrYrZr and X lY lZ l are, respectively, the right and left rail body coordinate systems that can have parallel axes and a common origin before displacement. When the right rail, for example, has a contact with one of the wheels at point Pr, a profile frame for this contact, XrpYrpZrp, is introduced, as shown in Figure 3.6 and Figure 3.7. The profile frame XrpYrpZrp translates along the right rail space curve and rotates about its origin. If the Xrp axis of the profile frame is assumed to be along the tangent to the space curve, the location of the contact point Pr can be simply defined with respect to the profile frame as u rp = 0

s2

( )

f s2 

T

(3.47)

If the profile of the rail cross-section is changing along the space curve, Equation 3.47 takes the following general form: u rp = 0

s2

(

)

f s1, s2 

T

(3.48)



105

FIGURE 3.7 Position of the contact point.

Therefore, the location of a point Pr on the rail surface in the body coordinate system XrYrZr is given by the vector u r , as shown in Figure 3.7, as follows (Berzeri et al., 2000): u r = R rp + A rp u rp

(3.49)

where R rp is the vector that defines the location of the origin Orp in the body r coordinate system, and A rp is the matrix that defines the orientation of the profile frame with respect to the body coordinate system. Note that in Equation 3.49, R rp and A rp depend only on the rail longitudinal surface parameter s1, while u rp depends on the rail lateral surface parameter s2 and can also depend on the rail longitudinal surface parameter s1 in the more general case, as previously discussed and shown in Equation 3.48 and Figure 3.8. In this more general case, the rail profile changes along the track, and Equation 3.49 can be written as

FIGURE 3.8 Contact points and surface parameters.


106


(

)

( )

( ) (

u r s1, s2 = R rp s1 + A rp s1 u rp s1, s2

)

(3.50)

Equation 3.49 or, equivalently, Equation 3.50 will be used in Chapter 5 to define the conditions of contact between the wheel and the rail. Depending on how the geometric surface parameters are treated, one can obtain different forms of the equations of motion, as discussed in Chapter 5. The description presented in this section also allows for the special case in which the right and left rail can be treated as one body. In this case, one track frame X tY tZ t, shown in Figure 3.6, can serve as the body coordinate system for both rails, while the vector R rp and the matrix A rp can be defined as functions of the longitudinal surface parameter of each rail space curve. In the method described in this book, the right and left rail space curves are assumed to be independent, regardless of whether or not the two rails are treated as one body or two separate bodies.

3.4 DEFINITIONS AND TERMINOLOGY This section discusses some of the basic definitions and terminology used in railroad vehicle dynamics, particularly at the preprocessing stage in which the track geometry is defined. Gage The gage G is defined as the lateral distance between two points on the heads of the right and left rails, as shown in Figure 3.9. These two points are located at a distance of 5/8 in. (14 mm) from the top of the rail head. In North America, the standard gage value varies from 56 to 57.25 in. Super-elevation The super-elevation h is defined as the vertical distance between the right and left rail, as shown in Figure 3.9. Curvature The curvature is different from zero in the case of curved track and is defined as the value of the angle ψ required to obtain a 100-ft-length chord AB of constant radius RH in the horizontal plane, as shown in Figure 3.10. Grade The grade is defined as the ratio (percentage) between the vertical elevation and the longitudinal distance.

FIGURE 3.9 Definition of the gage and super-elevation.



107

FIGURE 3.10 Definition of the curvature using 100-ft chord.

Cant angle The cant angle is defined as the rotation of each rail about its longitudinal axis, as shown in Figure 3.11.

FIGURE 3.11 Definition of the cant angle.

Profile The profile is defined as the vertical deviation of the rail space curve, as shown in Figure 3.12. Alignment The alignment is defined as the lateral deviation of the rail space curve, as shown in Figure 3.12.

FIGURE 3.12 Rail deviations (profile and alignment).


108


3.5 GEOMETRIC DESCRIPTION OF THE TRACK The coordinates of an arbitrary point on the rail surface can be determined in terms of the coordinates of a point on the rail space curve and the orientation of a coordinate system at this point. This section presents a computational method that can be used to determine the locations of points on the rail space curve and the angles that define the orientation of the profile frames at these points based on measured railroad industry inputs. The method can be applied to the left rail space curve and its profile frame X lpY lpZ lp, the right rail space curve and its profile frame XrpYrpZrp, or the track centerline and the track frame X tY tZ t. Using the typical variables measured by the railroad industry, one can develop a systematic procedure for constructing the track geometry. First, the definitions used in the industry are presented, and analytical interpretations of some of these definitions are provided before concluding this section. For the sake of simplicity of the presentation in this section, some of the conventionally used superscripts that indicate body numbers are dropped. In the railroad industry, the geometry of the track is defined using the following three variables (Berzeri et al., 2000; Dukkipati and Amyot, 1988; Rathod and Shabana, 2007): 1. Projection, which defines the planar curve obtained by projecting the reference line on the horizontal plane 2. Development, which defines an elevation angle θ 3. Super-elevation, which defines a bank angle φ that represents the rotation of the profile frame about the tangent to the reference line For instance, the projection of segment AB of a space curve on the horizontal plane can be represented by A0B0 (projection) as shown in Figure 3.13. Assume that the arc length of the actual curve is denoted as s, while the arc length of the projected curve is denoted as S. At point P0 on the horizontal plane, the radius of curvature is RH. The following relationship between the horizontal curvature CH and the

FIGURE 3.13 Projection of the space curve on the horizontal plane.



109

FIGURE 3.14 Sign convention for positive rotations.

rotation angle ψ about the vertical axis holds (Berzeri et al., 2000; Rathod and Shabana, 2007): dψ = CH dS =

dS RH

(3.51)

The relation between the arc length s and the projected arc length S is given by

ds =

dS cos θ

(3.52)

where the development angle θ is as shown in Figure 3.13. Analytical derivation of the preceding two equations is given later in this section. In the railroad industry, the track is defined by providing: the horizontal curvature CH as a function of the projected arc length S, the vertical development angle θ as a function of the arc length s, and the bank angle φ as a function of the projected arc length S. The signs of these quantities follow a standard convention. The curvature at an arbitrary point is positive if the y-coordinate of the center of the curvature is positive with respect to the profile frame. In other words, the curvature is positive if it is the result of a positive rotation ψ. The curvature angle ψ, the vertical development angle θ, and the bank angle φ are positive when the rotations are in the directions shown in Figure 3.14. Note that for the angles θ and φ, a sign convention different from the right-hand rule is followed in order to be consistent with the measurements made by the industry. As a result, the vertical development angle is positive if, for a positive step increment ds, a positive increment dz is obtained, as shown in Figure 3.13; and a positive bank angle would accommodate a curve with a positive curvature. By using the functions CH (S), θ(s), and φ(S), the reference line and the orientation of its coordinate system can be completely defined. For instance, the rotation about the vertical axis can be obtained as S1

∫

ψ = ψ 0 + CH ( S )  dS = ψ ( S )

(3.53)

S0

where ψ0 and S0 are the values of ψ and S at point A shown in Figure 3.13. Knowing the three Euler angles ψ, θ, and φ that follow the sequence Z, -Y, -X, one can define the following transformation matrix that defines the orientation of the profile frame:


110

 cos ψ cos θ  A =  sin ψ cos θ  sin θ 


− sin ψ sin φ − cos ψ sin θ cos φ   cos ψ sin φ − sin ψ sin θ cos φ   cos θ cos φ 

− sin ψ cos φ + cos ψ sin θ sin φ cos ψ cos φ + sin ψ sin θ sin φ − cos θ sin φ

(3.54) If x = [x y z]T is the vector that defines the coordinates of an arbitrary point on the space curve, the unit tangent vector is defined as t = dx/ds, as was shown previously. If the first column of the transformation matrix in Equation 3.54 is considered as the unit tangent to the space curve at s, the location of an arbitrary point on the space curve can be defined using the equation dx = tds, which, upon integration and using Equation 3.52, leads to  x = x o + cos ψ S  dS = x S   S0   S  y = yo + sin ψ S  dS = y S   S0  s   z = zo + sin θ s  ds = z s  s0  S

∫

( )

( )

∫

( )

( )

∫

()

(3.55)

()

Furthermore, the projected arc length S can be defined using Equation 3.52 as S1

∫

S = S0 + cos θ (s)  ds = S (s)

(3.56)

S0

The analysis presented in this section shows that if the inputs CH, θ, and φ are given, the locations of the points on the space curve and the orientation of the profile frame can be completely defined. In the analysis presented in the preceding section, the tangent vector t to the space curve is taken as the first column of the transformation matrix of Equation 3.54. A vector of length ds along the unit tangent can be written as  cos ψ cos θ    ds = ds  sin ψ cos θ   sin θ   

(3.57)



111

The projection of this vector on the horizontal plane is given by

(

)

t h = ds − ds T k k

(3.58)

where k = [0 0 1]T is a unit vector along the vertical axis defined in the global coordinate system. Using the preceding two equations, one can show that

( )

2

t Th t h = ds cos 2 θ

(3.59)

This equation defines the length of th as the projected infinitesimal arc length dS = ds cosθ, which shows that no approximation is made in the relationship given in Equation 3.52 (Rathod and Shabana, 2006). The vector th of Equation 3.58 that is the result of the projection of the curve on the horizontal plane can be written in a more explicit form as:  cos ψ cos θ    t h =  sin ψ cos θ    0  

(3.60)

The unit tangent and the curvature vectors for the projection of the curve on the horizontal plane are  cos ψ    t H =  sin ψ  ,  0   

 − sin ψ  ∂ψ   kH = cos ψ  ∂S     0 

(3.61)

This equation shows that the horizontal curvature is ∂ψ/∂S, as given by Equation 3.51.

3.6 COMPUTER IMPLEMENTATION The development presented in the preceding section shows that some of the basic quantities that define the curved rail are expressed as functions of either the arc length s or the projected arc length S. It is necessary at this point to express all quantities in terms of a single parameter. Dukkipati and Amyot (1988) selected the actual arc length s that represents the physical distance traveled by the wheelset. This choice is more convenient when the equations of motion are formulated using the trajectory coordinates. In this case, a coordinate system moving along the rail is used in developing some specialized railroad vehicle formulations. In some of the general multibody formulations presented in this book, however, there is no clear advantage in using a certain independent parameter, and the projected arc length S can be selected as well. However, to be consistent with the choice usually made in the literature, the actual arc length s is used as the independent parameter, while the


112


value of the projected arc length S can be obtained by solving Equation 3.56. This value can then be used in Equation 3.55, which depends explicitly on S.

3.6.1 TRACK SEGMENT TYPES In general, it is not possible to obtain closed-form solutions of Equations 3.53, 3.55, and 3.56 using the input quantities CH, φ, and θ. In reality, the rail is divided into segments, and the input variables are assumed to vary linearly within each segment. Only a small number of segment types is considered in railroad applications. Different segment types are described below (Berzeri et al., 2000). Tangent: A tangent denotes a straight segment. Therefore, the horizontal curvature CH is always equal to zero at all points of this segment. Curve: A curve denotes a circular arc. Therefore, the horizontal curvature CH is assumed constant along this segment. Tangent-to-curve entry spiral: This segment connects a tangent to a curve. Therefore, in this case, the horizontal curvature CH is equal to zero at the beginning and is equal to the inverse of the radius of curvature at the end. The variation from the initial value (zero) to the final value is assumed to be linear with respect to the projected arc length S. Curve-to-tangent exit spiral: This segment connects a curve to a tangent. Therefore, in this case, the horizontal curvature CH is equal to the inverse of the radius of the curve at the beginning and is equal to zero at the end. The variation from the initial value to the final value is assumed to be linear with respect to the projected arc length S. Curve-to-curve spiral: This segment connects two curves. Therefore, the horizontal curvature CH varies linearly with respect to the projected arc length S between the values that it takes at the two ends of the segment. In addition, each segment type described above can include a linear variation of the vertical development angle or a linear variation of the bank angle.

3.6.2 LINEAR REPRESENTATION

OF THE

SEGMENTS

The segment definitions and assumptions described above can be used to obtain a simple mathematical description of the rail. Given a segment whose end points are A and B and using the assumption that the horizontal curvature varies linearly, the horizontal curvature at any point within the segment can be defined as CH =

C1 ( S − S0 ) − C0 ( S − S1 ) S1 − S0

(3.62)

where C0, C1, S0, and S1 are the values of the curvature and the projected arc length at points A and B, respectively. Using Equation 3.53 and the linear form for the horizontal curvature of Equation 3.62, one has



ψ =ψ0 +

113

1  C1 S − S0 S1 − S0  2

(

)

2

−

2 C0 C S − S1  + 0 S1 − S0 2  2

(

)

(

)

(3.63)

The angle θ is related to the vertical curvature Cv through the equation dθ = Cv ds

(3.64)

Note that the assumption of linear relationship between θ and s implies that the vertical curvature is assumed to be constant within each segment and equal to Cv =

θ1 − θ 0 s1 − s0

(3.65)

where θ0 and θ1 are the values of the vertical development angle at the end points A and B, respectively, and s0 and s1 are the values of the arc length at the same points. Consequently, the angle θ is given by

θ = θ 0 + Cv (s − s0 )

(3.66)

Finally, the linearity of the bank angle function φ(S) implies that

φ=

φ1 ( S − S0 ) − φ0 ( S − S1) S1 − S0

(3.67)

where φ0 and φ1 are the values of the bank angle at points A and B, respectively. The linear relationship between the angle θ and the arc length s allows obtaining a closed-form expression of the function S(s). Using Equations 3.52 and 3.64, it is possible to write dθ C = v dS cos θ

(3.68)

This equation leads to (Berzeri et al. 2000)  1 1  S0 + sin θ (s)  − sin θ 0 Cv Cv S (s ) =   S + (s − s ) cos θ 0  0

if Cv ≠ 0

(3.69)

if Cv = 0

The value S obtained from this equation is then used in Equations 3.62, 3.63, and 3.67.


114


3.6.3 DERIVATIVES

OF THE

ANGLES

The formulation of the wheel/rail contact problem, as will be seen from the analysis presented in Chapter 5, requires calculation of the tangent and normal vectors of the surfaces as well as higher derivatives of these vectors with respect to the surface parameters. Using some of the new finite element methods such as the absolute nodal coordinate formulation (Berzeri et al., 2000; Shabana, 2005), these higher vector derivatives can be evaluated without the need to evaluate higher derivatives of the angles. In some specialized railroad vehicle formulations, however, due to the nature of the coordinates used, higher derivatives of the angles must be evaluated (Rathod and Shabana, 2006). In the remainder of this section, expressions for the derivatives of the angles that are required in some specialized railroad vehicle dynamic formulations are presented. Differentiating Equation 3.63 with respect to s leads to  dCH cos 2 θ − CH Cv sin θ   dS  dCH Cv sin θ cos θ − CH Cv2 cos θ  ψ ′′′ = −3  dS

ψ ′ = CH cos θ , ψ ′′ =

(3.70)

In most railroad vehicle applications, the curvature CH and its derivatives, the angle θ and the vertical curvature Cv , are assumed to be small. If these assumptions are used, Equation 3.70 shows that the first derivative ψ ′ is small and is of the same order of magnitude as CH , while ψ ″ and ψ are infinitesimal quantities of the third order. In Equation 3.66, it is assumed that the development angle θ is a linear function of the arc length s. Therefore, its first derivative with respect to s is constant and is given by

θ ′ = Cv

(3.71)

It is clear from this equation that the higher derivatives θ ″ and θ are equal to zero. In Equation 3.67, the bank angle is defined as a linear function of the projected arc length S. Using Equations 3.52 and 3.64, the derivatives of φ with respect to s can be written as  φ −φ   φ −φ   φ −φ  φ ′ =  1 0  cos θ , φ ′′ = −  1 0  Cv sin θ , φ ′′′ = −  1 0  Cv2 cos θ (3.72)  S1 − S0   S1 − S0   S1 − S0  If φ, θ, and Cv are assumed small, the first derivative φ′ is a small quantity of the same order of magnitude as φ, while higher derivatives become of the third order. The expressions of the derivatives of φ, θ, and ψ with respect to S and s are presented in Table 3.1 and Table 3.2. In these tables, the exact expression is compared with the expression obtained by retaining only first-order terms. The simplification



115

TABLE 3.1 Derivatives of the Orientation Angles with Respect to S Variable

Exact Expression

First-Order Approximation

dψ/dS d2ψ/dS2 d3ψ/dS3 dθ/dS d2θ/dS2 d3θ/dS3 dφ/dS d2φ/dS2 d3φ/dS3

CH dCH /dS 0 Cv /cosθ Cv2[sinθ/cos3θ] Cv3[1 + 2sin2θ/cos5θ] (φ1 − φ0)/(S1 − S0) 0 0

CH dCH /dS 0 Cv 0 0 (φ1 − φ0)/(S1 − S0) 0 0

TABLE 3.2 Derivatives of the Orientation Angles with Respect to s Variable dψ/ds d2ψ/ds2 d3ψ/ds3 dθ/ds d2θ/ds2 d3θ/ds3 dφ/ds d2φ/ds2 d3φ/ds3

Exact Expression CH cosθ [dCH /dS]cos2θ − CHCvsinθ −3[dCH /dS]Cvsinθcosθ − CH Cv2cosθ Cv 0 0 [(φ1 − φ0)/(S1 − S0)]cosθ −[(φ1 − φ0)/(S1 − S0)]Cvsinθ −[(φ1 − φ0)/(S1 − S0)]Cv2cosθ

First-Order Approximation CH dCH /dS 0 Cv 0 0 (φ1 − φ0)/(S1 − S0) 0 0

is made by assuming that the quantities CH, Cv , φ, and θ and their derivatives are small (Berzeri et al., 2000). In some specialized railroad vehicle formulations, such as the one presented in Chapter 7, it is necessary to differentiate the transformation matrix A of Equation 3.54 with respect to the surface parameter s. The first derivative of A can be obtained by using the chain rule of differentiation as A′ = Aψ ψ ′ + Aθθ ′ + Aϕφ ′

(3.73)

where Aα = ∂A/∂α for α = ψ, θ, φ, and a prime denotes a differentiation with respect to the surface parameter s. The expression for the derivatives ψ ′, θ ′, and φ′ are given in Table 3.2. Further differentiation with respect to the surface parameter leads to


116


A′′ = Aψ ψ ′′ + Aθθ ′′ + Aϕφ ′′ + Aψψ ψ ′ 2 + Aθθθ ′ 2 + Aφφφ ′ 2 + 2 Aψθψ ′θ ′ + 2 Aψφψ ′φ ′ + 2 Aθφθ ′φ ′

(3.74)

In this equation, no simplifications are made. Clearly, further differentiation would result in an even more complicated expression for the third derivative A″′. Table 3.2 shows that when the input quantities CH, θ, and φ are small, the derivatives of the orientation angles contain terms of the third order. It is possible to show that by neglecting all the terms of the third order and higher, one obtains (Berzeri et al., 2000) A′′ = Aψ ψ ′′ + Aψψ ψ ′ 2 + Aθθθ ′ 2 + Aφφφ ′ 2 + 2 Aψθψ ′θ ′ + 2 Aψφψ ′φ ′ + 2Aθφθ ′φ ′ (3.75) and A′′′ = 3( Aψψ ψ ′ + Aψθθ ′ + Aψφφ ′)ψ ′′

(3.76)

Furthermore, by neglecting all the terms of the second order and higher, that is, by retaining only first-order terms, one obtains the result A′′ = Aψ ψ ′′ ,

A′′′ = 0

(3.77)

Depending on the simulation scenario, a first-order or second-order approach may be justified, leading to a simplification of the equations of motion.

3.7 TRACK PREPROCESSOR The computer simulation of the nonlinear dynamics of railroad vehicle systems consists of two stages. In the first stage, the track geometry, based on the description presented in the preceding sections, is defined. In the second stage, the equations of motion of the railroad vehicle are solved using the track geometry input obtained in the first stage. For the first stage, one often develops a preprocessor computer code that can be used to define tracks with arbitrary geometry. The track preprocessor code has input that is based on the definitions and terminology used by the railroad industry. The output of the track preprocessor is a data file that is used as an input to the main processor computer code used to solve the dynamic equations of the multibody railroad vehicle system. In this section, the structure of the preprocessor computer code that can be used to define the track geometry is discussed. The preprocessor can be designed to read a simple and standard input and use the input information to generate an output that can be used by the dynamic simulation code. Based on the analysis presented in the preceding sections, one can recognize the following basic tasks that can be performed by the track preprocessor (Berzeri et al., 2000):



117

1. Read input information that defines the track segments. 2. Define the functions CH (S), θ(s), and φ(S) introduced in the preceding sections. Use this information to determine the third Euler angle ψ as described in the preceding section. 3. Calculate the coordinates x, y, and z of the nodes on the track centerline by evaluating numerically the integrals presented in the preceding sections. 4. Define the space curves of the right and left rails using a finite number of nodal points. 5. Create an output in a format that can be read by the dynamic simulation code.

3.7.1 TRACK PREPROCESSOR INPUT In order to define the input to the track preprocessor, each rail is divided into segments. The input to the preprocessor provides information about the geometry of each track segment. This information includes the length, the curvature, the superelevation, and the grade at the end points (nodes) of each segment. In the railroad industry, this information is given in the form shown in Table 3.3. The first column of this table contains the node number. The second column shows the distance of each node from a selected origin, measured in feet along the track centerline. The curvature is reported in the third column of Table 3.3, and generally in North America its value is given using a 100-ft chord definition. Using Figure 3.10 and the information given in the third column of Table 3.3, the value of the curvature for a given segment is obtained using the following equation: CH =

sin(ψ 2) 50′

(3.78)

TABLE 3.3 Typical Railroad Input Entries Used to Define a Curved Track Node No.

Distance (ft)

Curvature (deg)

Super-Elevation (in.)

Grade (%)

1 2 3 4 5 6 7 8 9 10

0 100 150 450 500 650 720 1020 1145 1195

0 0 5 5 −3 −3 7 7 0 0

0 0 1.5 1.5 −1 −1 2 2 0 0

0 0 0 0 0 0 0 0 0 0


118


FIGURE 3.15 Top view of the track given in Table 3.3.

The values of the super-elevation h for each node of the track are reported in the fourth column of Table 3.3. To find the value of the bank angle φ, it is necessary to know the value of the gage G. Using Figure 3.9, it is straightforward to obtain the relationship sin φ =

h d

(3.79)

Finally, the grade is defined in the fifth column of Table 3.3. The values reported in Table 3.3 correspond to the curved track shown in Figure 3.15. Based on the definitions given in the preceding sections, the first segment is a tangent; the second segment is a tangent-to-curve entry spiral; the third segment is a curve; the fourth segment is a curve-to-curve spiral; the fifth segment is a curve; the sixth segment is again a curve-to-curve spiral; the seventh segment is a curve; the eighth segment is a curve-to-tangent exit spiral; and the ninth segment is a tangent. It is possible to determine the values of the functions CH , θ, and φ at any point within the segment using Equations 3.62, 3.66, and 3.67 and a linear interpolation. The value of the angle ψ is given by Equation 3.63. Finally, using Equation 3.69, it is possible to write all these quantities as a function of the same variable s. The values of these variables can be used with numerical integration to define the right and left rail space curve, as described below.

3.7.2 NUMERICAL INTEGRATION After obtaining the orientation angles as functions of the arc length s, the coordinates defined by the integrals of Equation 3.55 can be evaluated. In principle, the results of the integration define the coordinates of each point on the space curve with respect



119

FIGURE 3.16 Top view of a curve segment.

to the body coordinate system X lY lZ l for the left rail or XrYrZr for the right rail. Since closed-form solutions of the coordinate integrals cannot be obtained in general, numerical integration methods such as trapezoidal and Simpson rules or Gaussian quadrature can be used (Atkinson, 1978). In the case of curved tracks, the lengths of the segments of the right rail differ from the lengths of the segments of the left rail. Since the input data are based on the track centerline, the length of the segments of the right and left rails must be adjusted during the process of the numerical integration. Figure 3.16 shows a segment of a curved track. In this figure, the value of the gage is exaggerated to show the difference between the lengths of the arcs AB, ArBr, and AlBl as shown in Figure 3.17. It is clear from this figure that the right rail segment length can differ significantly from the left rail segment length. This difference, if not taken into consideration when the track geometry is defined, can have a significant effect on the accuracy of the numerical solution when the equations of motion of the multibody railroad vehicle system are integrated. A method for adjusting the length of the segments of the right and left rails is described in Chapter 6.

FIGURE 3.17 Length of the right and left rail segments.


120


3.7.3 TRACK PREPROCESSOR OUTPUT The output of the track preprocessor includes information that can be used to define the space curve of the rails and the track centerline. The output of the track preprocessor has, as a minimum, the following information for each node: 1. The value of the arc length parameter s of the space curve at the node 2. The orientation angles θ, φ, and ψ of the space curve at each node 3. The location of each nodal point with respect to the initial reference frame that will be used as the body rail or track frame in the multibody simulation code This information can be used to determine the properties of the space curve at a specified arc length parameter s by using interpolation between the nodes during the dynamic simulations. However, as will be shown in Chapter 5, some multibody system formulations presented in this book require the evaluation of higher-order derivatives of the tangents and the normal with respect to the arc length parameter s in order to correctly define the contact constraints and forces. These higher derivatives, which can be up to the third derivative in some formulations, can be obtained using a finite element formulation, as previously mentioned, or they can be expressed in terms of the derivatives of Euler angles. Euler angle derivatives can be provided as part of the track processor output, or they can be obtained during the dynamic simulation by differentiation of the interpolation functions. If the higher derivatives are to be included as part of the output, the output of the track processor can also include the following information: 1. The longitudinal tangent vector given in Table 3.4 and its first and second derivatives (Note that the longitudinal tangent is the first column of the transformation matrix given by Equation 3.54.) 2. The first, second, and third derivatives of the orientation angles given in Table 3.2.

TABLE 3.4 Track Longitudinal Tangent and Its Derivatives with Respect to s Coordinate

t1

dt1/ds

d 2 t1/ds2

x

cosψ cosθ

−ψ ′sinψ cosθ − θ ′cosψ sinθ

y

sinψ cosθ

ψ ′cosψ cosθ − θ ′sinψ sinθ

z

sinθ

θ′cosθ

−ψ ″sinψ cosθ − (ψ ′)2cosψ cosθ − θ ″cosψ sinθ − (θ ′)2cosψ cosθ + 2ψ ′θ ′sinψ sinθ ψ ″cosψ cosθ − (ψ ′)2sinψ cosθ − θ ″sinψ sinθ − (θ ′)2sinψ cosθ − 2ψ ′θ ′cosψ sinθ θ″cosθ − (θ′)2sinθ



121

Whether or not higher derivatives are included in the track preprocessor output, this output can include, as previously mentioned, three different sets of data: one set for the right rail, one set for the left rail, and one set for the track centerline.

3.7.4 USE OF THE PREPROCESSOR OUTPUT SIMULATION

DURING

DYNAMIC

The output data of the track preprocessor are expressed in terms of the arc length parameter of the space curve s = s1. The complete representation of the rail surface, however, requires the use of two surface parameters, as described in Section 3.2. It is assumed that the profile of the rail is known and can be described using the function f(s2), where s2 is the second (lateral) rail surface parameter. If the rail profile is constant along the track, Equation 3.50 can be written as a function of the two surface parameters s1 and s2 as follows: u r (s1, s2 ) = R rp (s1 ) + A rp (s1 )u rp (s2 )

(3.80)

Using the information included in the track preprocessor output, one can use linear or cubical interpolation to define R rp (s1 ) and A rp (s1 ) and their derivatives with respect to s1, as described in the preceding section. The vector u rp (s2 ) represents the location of the contact point with respect to the profile coordinate system, as shown in Figure 3.7, and is given as follows: u rp (s2 ) = 0

s2

f (s2 ) 

T

(3.81)

Using Equation 3.80, one can define the two tangent vectors and the normal as follows:  ∂u rp dR rp dA rp (s1 ) rp = + u (s2 )  ds1 ds1 ∂s1   rp rp du (s2 ) ∂u  t2 = = A rp (s1 )  ∂s2 ds2   n = t1 × t2    t1 =

(3.82)

where t1 and t2 are the two tangent vectors at a point on the rail that has the two surface parameters s1 and s2 as coordinates. These two tangent vectors, which are defined with respect to the rail coordinate system (body coordinate system) and are not necessarily orthogonal, represent the tangent plane at this point. The normal vector n is defined using the third equation in Equation 3.82 as the cross-product of the two tangent vectors.


122


EXAMPLE 3.2 For the track described in Table 3.3, determine the location and orientation of the profile frame at the nodes of the track centerline, assuming that the track has a gage value equal to 1.42 m. Neglect the effect of the head width. Solution. For an arbitrary node i, one can write

 0.0254 hi   G 

φi = sin −1 

i = 1, 2, … , 10

rad,

where h is the given super-elevation in inches at node i, and G is the gage. The angle ψi can be written as

ψi =

π cri 180

where cri is the given curvature in degrees at the node. The angle θi can be defined as

 GRi   100 

θ i = sin −1 

i = 1, 2, …, 10

rad,

where GRi is the given grade percentage. The horizontal and vertical curvatures are given by

CHi =

sin(ψ i 2)

Cv i =

θ i − θ i −1 si − si −1

i = 1, 2, … , 10

1/m,

50 × 0.3048

1/m,

i = 1, 2, … , 10

Because the grade is equal to zero for all nodes, one has Si = Si + (si − si −1 ) cos θ i ,

ψ i = ψ i −1 +

CHi + CHi −1 2

i = 1, 2, … , 10

(S − S ), i

i −1

i = 1, 2, …, 10

Finally, integrating Equation 3.55 numerically, the following are the locations and orientations of the profile frame at each node:



123

Node no.

Distance, s (m)

CH (m−1)

θ (deg)

φ (deg)

ψ (deg)

x (m)

y (m)

z (m)

1 2 3 4 5 6 7 8 9 10

0 30.48 45.72 137.16 152.40 198.12 219.46 310.90 349.00 364.24

0.0 0.0 0.00286 0.00286 −0.00172 −0.00172 0.00401 0.00401 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.02932 0.02932 −0.01953 −0.01953 0.03909 0.03909 0.0 0.0

0.0 0.0 0.02181 0.28353 0.29225 0.21372 0.23813 0.60442 0.68073 0.68073

0.0 30.48 45.72 135.84 150.43 194.68 215.52 298.50 328.70 340.54

0.0 0.0 0.11 13.98 18.39 29.83 34.40 71.58 94.79 104.38

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

3.8 WHEEL GEOMETRY The surface of the wheel is a surface of revolution obtained by a complete rotation of the curve that defines the wheel profile about the wheel axis, as shown in Figure 3.18. Therefore, the surface of the wheel can be defined mathematically by the following equation:  x0 + g ( s1 ) sin s2    u( s1 , s2 ) =  y0 + s1   z − g ( s ) cos s  1 2  0

(3.83)

where s1 is the lateral surface parameter that represents the independent variable for the wheel profile g(s1), and s2 is an angular surface parameter that represents the rotation of the wheel profile about its axis. The variables x0, y0, and z0 are the position coordinates of the origin of the profile coordinate system with respect to the wheelset coordinate system or with respect to the wheel coordinate system if the two wheels are not rigidly connected, as in the case of flexible axles. The values x0 and z0 can be selected equal to zero, and y0 can be selected to be half of the back-to-back distance between the two wheels of a wheelset.

FIGURE 3.18 Wheel surface.


124


Equation 3.83 shows that only one function, g(s1), is required to define the wheel surface. It is convenient to use a general procedure based on a spline function representation, as will be discussed in Chapter 6. In this case, measured data can be used to define the wheel profile. Knowing the profile, one can determine the tangent and normal vectors at an arbitrary point on the wheel surface. These vectors and their derivatives with respect to the surface parameters are required to determine the contact location and forces, as will be discussed in Chapter 5. The two tangent vectors that can be used to represent the contact plane and the normal vector defined with respect to the wheelset coordinate system can be written as follows: t1 =

∂u , ∂s1

t2 =

∂u , ∂s2

n = t1 × t2

(3.84)

where u is the vector that defines the location of the contact point with respect to the wheelset coordinate system and has the components given by Equation 3.83, t1 and t2 are the two tangent vectors that define the tangent plane at the contact point, and n is the normal vector.

EXAMPLE 3.3 For the wheel surface defined by Equation 3.83, determine the two tangent vectors and the normal vector and their derivatives with respect to the two surface parameters at an arbitrary point on the wheel surface. Solution. The two tangent vectors and the normal vector are given as follows:

t1 = ∂u ∂s1

u x y z

x0 + g(s1)sins2 y 0 + s1 z0 – g(s1)coss2

(dg(s1)/ds1)sins2 1.0 –(dg(s1)/ds1)coss2

t2 = ∂u ∂s2 g(s1)coss2 0.0 g(s1)sins2

n = t1 × t2 g(s1)sins2 –g(s1)(dg(s1)/ds1) –g(s1)coss2

The derivatives of the tangent and normal vectors can be determined as follows:

First Derivative with Respect to s1 ∂ t1 ∂s1 x y z

∂ t2 ∂s1

(d2 g(s1)/ds12)sins2 0.0 2

2

–(d g(s1)/ds1 )coss2

∂n ∂s1

(dg(s1)/ds1)coss2

(dg(s1)/ds1)sins2

0.0

–(dg(s1)/ds1)2 – g(s1)(d2 g(s1)/ds12) –(dg(s1)/ds1)coss2

(dg(s1)/ds1)sins2



125

First Derivative with Respect to s2 ∂ t1 ∂s2 x y z

∂ t2 ∂s2

(dg(s1)/ds1)coss2 0.0 (dg(s1)/ds1)sins2

∂n ∂s2

–g(s1)sins2 0.0 g(s1)coss2

g(s1)coss2 0.0 g(s1)sins2

Second Derivative with Respect to s1 ∂2 t1 ∂s1 x

2

∂2 t2 ∂s1

2

∂2 n ∂s1

y

(d3 g(s1)/ds13)sins2 0.0

(d2 g(s1)/ds12)coss2 0.0

z

–(d3 g(s1)/ds13)coss2

(d2 g(s1)/ds12)sins2

2

(d2 g(s1)/ds12)sins2 –3(dg(s1)/ds1)(d2 g(s1)/ds12) –g(s1)(d3 g(s1)/ds13) –(d2 g(s1)/ds12)coss2

Second Derivative with Respect to s2 ∂2 t1 ∂s2

2

x y z

∂2 t2 ∂s2

∂2 n ∂s22

–g(s1)coss2 0.0 –g(s1)sins2

–g(s1)sins2 0.0 g(s1)coss2

2

–(dg(s1)/ds1)sins2 0.0 (dg(s1)/ds1)coss2

Second Derivative with Respect to s1 and s2 ∂2 t1 ∂s1∂s2 x y z

(d2g(s1)/ds12)coss2 0.0 (d2g(s1)/ds12)sins2

∂2 t2 ∂s1∂s2 –(dg(s1)/ds1)sins2 0.0 (dg(s1)/ds1)coss2

∂2 n ∂s1∂s2 (dg(s1)/ds1)coss2 0.0 (dg(s1)/ds1)sins2



4

Contact and Creep-Force Models

The formulation of the contact forces that describe the dynamic interaction between the wheel and the rail is one of the fundamental problems that must be addressed in developing a multibody system formulation for railroad vehicle systems. Various contact models are used in different computer formulations to describe the wheel/rail interaction. In this chapter, the rigid body contact is discussed, with particular emphasis on Hertz theory and the formulations of the wheel/rail creep forces. The definitions and concepts introduced in this chapter will be used in the formulation of the normal and tangential forces that describe the wheel/rail interaction. The contact between two rigid bodies can be at a single point or area, depending on the shape of the two bodies. These two types of contact are known as nonconformal or conformal contact, respectively. If the shape of the two bodies is such that the two bodies in the region of contact fit exactly or even closely together, the contact is defined as a conformal contact. If the two bodies, on the other hand, touch at a point or a line, the contact is called nonconformal. If an external load is applied on each body, the two bodies will deform at the contact point to form an area of contact. The contact area in the case of the nonconformal contact is small as compared with the dimensions of the two bodies. In 1882, Heinrich Hertz presented a contact theory that accounts for the shape of the surfaces in the neighborhood of the contact area (Hertz, 1882). Hertz assumed that the area of contact is elliptical. In wheel/rail dynamics, the assumption of nonconformal contact is often used, since the shapes of the wheel and rail surfaces are significantly different. In this case, the use of Hertz theory to examine the contact geometry and the maximum stresses can be justified. Due to the elasticity of the bodies and the externally applied normal load, some points on the surfaces in the contact region may slip while others may stick when the two bodies move relative to each other. The difference between the tangential strains of the bodies in the adhesion area leads to a small apparent slip. This slip is called creepage and is defined using the kinematics of the two bodies. Creepages generate tangential creep forces and creep spin moment; all will sometimes be referred to collectively in this book as the creep forces. For example, in the case of the wheel/rail contact, tangential forces and spin moment are generated, since the motion of the wheel relative to the rail is a combination of rolling and sliding. These creep forces and moment have a significant effect on the steering and stability of railroad vehicle systems. In this chapter, Hertz theory, which is often used in the study of the wheel/rail contact problem, is first discussed. The creepages, the normalized relative velocities that enter into the calculation of the creep forces, are then defined. It is shown that most creep-force models are expressed in terms of the creepages. Different creep-force

127


128


theories can be used; some of these theories are based on linear models, while the others employ nonlinear force-creepage relationships. The chapter concludes with a discussion of some of the creep theories that are used in railroad vehicle system formulations.

4.1 HERTZ THEORY In 1882, Hertz introduced a contact theory that accounts for the shape of the surfaces in the neighborhood of the area of contact between nonconformal bodies (Hertz, 1882). In this theory, it is assumed that the contact area is, in general, elliptical. This assumption was based on an observation of interference fringes at the contact of two glass lenses (Johnson, 1985). In the Hertz theory, it is further assumed that each body is an elastic half-space loaded over a small elliptical region. The assumption of elastic half-space implies that the concentrated contact stresses can be treated separately from the general distribution of the stresses in the two bodies due to their shape and the way in which they are supported. This assumption is valid when the dimensions of the contact area are small compared with the dimensions of the two bodies and the relative radii of curvature of the two surfaces. Hertz theory is based on the assumption of small deformation of two elastic bodies due to the static compression, and it neglects the effect of the friction forces. For the wheel/rail contact problem, Hertz theory is the most commonly used theory to determine the shape of the contact area and the local deformation of the wheel and rail surfaces at the contact region. The assumptions used in Hertz theory can then be summarized as follows: 1. The surfaces of the bodies are continuous and nonconformal. 2. The strains are small. 3. The stress resulting from the contact force vanishes at a distance far from the contact area. 4. The surfaces are frictionless. 5. The bodies are elastic, and no plastic deformation occurs in the contact area. In the case of wheel/rail contact, most researchers assume that these assumptions are valid.

4.1.1 GEOMETRY

AND

KINEMATICS

Following these assumptions, Hertz assumed that if two bodies i and j are in contact as shown in Figure 4.1, the shape of the surface of each body in the region close to the origin can be defined as follows: z i = Ai ( x i )2 + Bi ( y i )2 + C i ( x i y i ) +   z j = A j ( x j )2 + B j ( y j )2 + C j ( x j y j ) + 

(4.1)



129

FIGURE 4.1 Two bodies in contact.

where Ak, Bk, and C k (k = i, j) are constants that depend on the body geometry, and x k and y k (k = i, j) are the axes of the two bodies at the contact plane. The gap between the two surfaces at the origin can then be defined as h = zi − z j

(4.2)

This equation, upon the use of Equation 4.1, leads to h = A( x )2 + B( y )2 + Cxy +

(4.3)

In this equation, A, B, and C are constants. The axes x k and y k (k = i, j) and x and y can be chosen such that the terms x ky k (k = i, j) and xy are zero. In this case, Equations 4.1 and 4.3 can be written, after neglecting higher-order terms, as follows:      1  1 j 2  j j 2 z = − (x ) + (y )   2 R2j  2 R1j  zi =

1 1 ( x i )2 + ( y i )2 i 2 R1 2 R2i

(4.4)

and h = A( x ) 2 + B( y ) 2 =

1 1 ( x )2 + ( y )2 2 R1 2 R2

(4.5)

where R1k and R2k (k = i, j) are, respectively, the principal radii of curvature of the surfaces of bodies i and j at the origin; and R1 and R2 are the principal relative radii


130


FIGURE 4.2 Contact plane.

of curvature. That is, ∂2 z k /∂x k = 1/R1k and ∂2 z k /∂y k = 1/R2k . Note that the axes x i and x j are not in general parallel, and their orientation can differ by an angle ψ. These two axes can differ from the common x-axis by arbitrary angles α and β, as shown in Figure 4.2a. Using Equation 4.2 and the transformation of the axes x i and x j to the x-axis, the term C in Equation 4.3 is defined as follows: 2

C=

2

1 1 1 1 1 1 − j  sin 2β −  i − i  sin 2α j  2  R1 R2  2  R1 R2 

(4.6)

Therefore, Equation 4.5 is satisfied if C in Equation 4.6 is equal to zero, and this leads to 1 1 1 1 1 1 − j  sin 2α =  i − i  sin 2β  j 2  R1 R2  2  R1 R2  Based on the preceding equation, one can draw the triangle shown in Figure 4.2b. This triangle can be used to define the following equation:

B− A=

1 1 1 1 1 1 − i  cos 2α +  j − j  cos 2β  i 2  R1 R2  2  R1 R2  (4.7) 2

2

 1 1  1 1 1  1 1  1 1 = − i  +  j − j  + 2  i − i   j − j  cos 2ψ  i 2  R1 R2   R1 R2   R1 R2   R1 R2  and

A+ B =

1 1 1 1 1 + i + j + j i  2  R1 R2 R1 R2 

(4.8)



131

EXAMPLE 4.1 To show how Equation 4.5 can be obtained from Equation 4.4, one may write these equations in the following form: T

T

z i = u i Ci u i ,

z j = u j C ju j ,

h = u T Cu

where

xi  ui =  i  , y 

x j  x , u=  j y  y  

uj = 

and

1  i 1 R1 Ci =  2 0 

 1  j  , C j = − 1  R1 2 1  0 i  R2   



1   , C = 1  R1 2 1  0 j  R2  

0 



0 

0 



1 



R2 

It is also clear that u i = Ai u i ,

u j = A ju j

where

 cos α

− sin α 

 sin α

cos α 

Ai = 

 cos β

sin β 

 − sin β

cos β 

Aj = 

,



and

x j  x i  j , u =  j i y  y 

ui =  It follows that

T

T

h = z i − z j = u i Ci u i − u j C j u j T

T

= u T Ai Ci Ai u − u T A j C j A j u

(

T

T

)

= u T Ai Ci Ai − A j C j A j u


132


which can be written as h = z i − z j = u T Cu = A( x ) 2 + B( y) 2 + Cxy where

( )

T

T

C = Cij = A C A − A C A i

i

i

j

j

j

is a symmetric matrix whose elements are defined using the preceding equation as

C11 =

1 1 1 1 1 2  2 2 2  i cos α + i sin α + j cos β + j sin β  2  R1 R2 R1 R2 

C12 = C21 = C22 =

  1 1  1 1 1  j − j  sin 2β −  i − i  sin 2α  4  R1 R2   R1 R2  

1 1 1 1 1 2 2 2 2   i sin α + i cos α + j sin β + j cos β  2  R1 R2 R1 R2 

This leads to A = C11 , B = C22 , and C = C12 + C21 = 2C12 . Note that the coefficient given by Equation 4.6 is obtained as C = 2C12 . If the x and y axes are chosen to be the principal directions, the matrix C must be diagonal, and its diagonal elements are the principal values. For this condition to be satisfied, one must have C = 2C12 = 0, that is,

C = 2C12 =

1 1 1 1 1 1 − sin 2β −  i − i  sin 2α = 0 2  R1j R2j  2  R1 R2 

which is the same as equating the expression of Equation 4.6 to zero.

It is clear from Equation 4.5 that the contours of the gap h between the two bodies are ellipses with axes ratio equal to ( A/B). To determine the size of the contact ellipse, assume that an external normal load Fn is applied to press the two bodies against each other. Due to the pressure applied on the bodies, the surfaces of body i and j will be displaced vertically by the distance ui and u j, respectively, as shown in Figure 4.3. If the two points Pi and Pj coincide, the total deformation can be given as follows: ui + u j + h = δ i + δ j

(4.9)

where h is given by Equation 4.2. Therefore, if the two bodies are deformed and the total deformation is given as δ = δ i + δ j, the following equation, after substituting h from Equation 4.5 into Equation 4.9, must be satisfied: u i + u j = δ − A( x )2 − B( y)2

(4.10)



133

FIGURE 4.3 Contact bodies under externally applied normal load.

4.1.2 CONTACT PRESSURE Since the contact area is assumed small compared with the dimensions of the two bodies, one can consider the two bodies in contact as semi-infinite. The contact pressure can be assumed to satisfy the following requirements for the equilibrium of the two bodies (Goldsmith, 1960): 1. The total applied force Fn must be equal to the total resisting force generated by the vertical component of the pressure p in the contact area, that is, Fn =

∫∫ pdxdy .

2. The components of displacement vanish at infinity, that is, the displacement at a distance away from the contact region can be neglected. 3. The normal stresses outside the contact region are assumed to be zero. 4. The normal stresses acting on the two bodies are in balance within the contact region. 5. The shear stresses τxz and τyz along the surfaces of the bodies are zeros. These conditions can be satisfied by assuming that the pressure p is a quadratic function of x and y; therefore, the pressure distribution in the contact area is assumed to take the following form: 2

p = p0

 x  y 1−   −    a  b

2

(4.11)

In this equation, p0 is a constant, and a and b are the lengths of the ellipse semiaxes. As shown in Appendix A, the pressure p produces a displacement u given by (Goldsmith, 1960; Johnson, 1985; Love, 1944)


134


u=

1− ν 2 ( L − M ( x ) 2 − N ( y) 2 ) πE

(4.12)

where E is Young’s modulus of elasticity, ν is the Poisson’s ratio, and

π p0 ab M= 2 N=

L=

π p0 ab 2 π p0 ab 2

∞

∫ 0

∞

∫ 0

∞

∫ 0

   (a 2 + w)3 (b 2 + w) w   2  dw π p0 b  a  = 2 2   Ee − Ke     (a 2 + w)(b 2 + w)3 w e a  b    dw  = π p0 bKe 2 2  (a + w)(b + w) w  dw

=

π p0 b Ke − Ee e 2a 2

(

)

(4.13)

where Ee and Ke are the complete elliptical integrals of argument e = 1 − b 2 /a 2 , b < a, and are given in Appendix B. Using Equation 4.12, the displacement ui + u j can be written as u i + u j = ( L − M ( x )2 − N ( y)2 ) /π E ij

(4.14)

1 1 − (ν i )2 1 − (ν j )2 = + E ij Ei Ej

(4.15)

where

Since the pressure distribution is semi-ellipsoidal, the total normal load Fn is given by Fn =

2 p0π ab 3

(4.16)

Using Equations 4.11 and 4.16, one obtains (Goldsmith, 1960; Hertz, 1882; Love, 1944) 2

p=

 x  y 3Fn 1−   +   2π ab  a  b

2

(4.17)

Using Equations 4.10 and 4.14, ui + u j can be written as follows:

δ − Ax 2 − By 2 =

L − M ( x ) 2 − N ( y) 2 π E ij

(4.18)



135

Equating the coefficients in Equation 4.18, one has

δ=

A=

B=

L pb = 0ij Ke ij πE E

(4.19)

(

M pb = ij 02 2 Ke − Ee ij πE E ea

N pb = ij 02 2 ij πE E ea

)

(4.20)

 a  2    Ee − Ke   b  

(4.21)

Using these equations, one obtains B R1 (a /b)2 Ee − Ke = = A R2 Ke − Ee 1 AB = 2

1 p b = 0 R1R2 E ij a 2e 2

((a/b) E − K )( 2

e

e

     Ke − Ee  

(4.22)

)

The contact ellipse semi-axes are defined as follows:

(

(

)

)

(4.23)

(

(

)

)

(4.24)

a = m 3π Fn K1 + K 2 4 K3

b = n 3π Fn K1 + K 2 4 K3

13

13

where K1 and K2 are constants that depend on the material properties of the two bodies and are given as follows:

K1 =

( )

1− νi

π Ei

2

,

K2 =

( )

1− ν j

2

πE j

(4.25)

In Equations 4.23 and 4.24, K3 is a constant that depends on the geometric properties of the two bodies and is defined as follows:

K3 = A + B =

1 1 1 1 1 + + + 2  R1i R2i R1j R2j 

(4.26)


136


TABLE 4.1 Hertz Coefficients m and n θ (deg)

m

n

θ (deg)

m

n

0.5 1 1.5 2 3 4 6 8

61.4 36.89 27.48 22.26 16.5 13.31 9.79 7.86

0.1018 0.1314 0.1522 0.1691 0.1964 0.2188 0.2552 0.285

10 20 30 35 40 45 50 55

6.604 3.813 2.731 2.397 2.136 1.926 1.754 1.611

0.3112 0.4125 0.493 0.530 0.567 0.604 0.641 0.678

θ (deg)

m

n

60 65 70 75 80 85 90

1.486 1.378 1.284 1.202 1.128 1.061 1.0

0.717 0.759 0.802 0.846 0.893 0.944 1.0

Source: Hertz, H., Über die berührung fester elastische Körper und über die Harte, Verhandlungen des Vereins zur Beförderung des Gewerbefleisses, Leipzig, Nov. 1882.

The coefficients m and n in Equations 4.23 and 4.24 are given by Hertz in Table 4.1 as functions of the angular parameter θ for the values of θ between 0° and 180° (Hertz, 1882), where θ is defined as

(

θ = cos−1 K 4 /K3

)

(4.27)

where K4 = B − A =

2 2 (4.28)  1 1  1 1  1 1 1  1 1 + cos + − − − 2 − 2 ψ  Ri Ri   R j R j  2  R1i R2i   R1j R2j  1 2 1 2

In the computer implementation of this formulation, the coefficients m and n can be interpolated for a given value of the angle θ using the entries of Table 4.1. These coefficients are needed to calculate the semi-axes a and b. One can use linear or cubic spline interpolation to determine these coefficients using the values of Table 4.1. An alternative approach to the numerical interpolation is to develop closed-form expressions for the coefficients m and n as functions of θ. The following closedform equations were proposed by Berzeri (Shabana et al., 2001): m = Am tan(θ − π 2) +

    + Dn sin θ  

Bm + Dm θ Cm

1 n= + Bnθ Cn An tan(θ − π 2) + 1

(4.29)



137

TABLE 4.2 Coefficients Used for the Closed-Form Functions m and n Coeff.

Value

Coeff.

Value

Am Bm Cm Dm

−1.086419052477 −0.106496432832 1.350000000000 1.057885958251

An Bn Cn Dn

−0.773444080706 0.256695354565 0.200000000000 −0.280958376499

Source: Shabana, A.A., Berzeri, M., and Sany, J.R., ASME Journal of Dynamic Systems, Measurement, and Control, 123, 168, 2001. With permission.

where the value of θ is given in radians, and the coefficients Ak, Bk, Ck, and Dk (k = m,n) are given in Table 4.2 (Shabana et al., 2001). Equation 4.29 provides a good approximation for m and n. Note also that the first equation of Equation 4.29 captures the asymptotic behavior of the function m when θ approaches zero. Using Equations 4.19 and 4.24, the Hertz force law can be defined as follows: Fn = K hδ 3 2 =

(

4β

3 K1 + K 2

)

A+ B

δ3 2

where β is a constant and is given in Table 4.3 (Goldsmith, 1960).

TABLE 4.3 Hertz Coefficient β for Hertz Force A/B

β

1.0 0.7041 0.4903 0.3333 0.2174 0.1325 0.0718 0.0311 0.00765

0.3180 0.3215 0.3322 0.3505 0.3819 0.4300 0.5132 0.6662 1.1450

(4.30)


138


4.1.3 COMPUTER IMPLEMENTATION In the case of the wheel/rail contact problem, the wheel (body i) and the rail (body j) have the radii of curvature shown in Figure 4.4. In this figure, R1i and R1j are the principal rolling radii of the wheel and the rail, respectively, and R2i and R2j are the principal transverse radii of curvature of the wheel and the rail, respectively. The radius of curvature of a body is considered to be positive if the corresponding center of curvature is within the body, as shown in Figure 4.4 (Garg and Dukkipati, 1988). Knowing the normal contact force or the deformation of the two bodies (penetration), the rolling radii and the principal transverse radii of the wheel and the rail (which can be determined from the geometry, as described in the preceding chapter), and the coefficients m and n (which can be determined using interpolation or the closed-form equations), one can determine the contact ellipse semi-axes using Equations 4.24 and 4.25. It is important to distinguish between the longitudinal and the lateral semi-axes, since the creep forces are functions of the dimensions of the contact ellipse, as will be shown in the next section. In general, one can determine the direction of the contact ellipse based on the radii of curvature for the two bodies in contact. The following rule can be applied: if 1 1 1 1 + j ≥ i + j, i R1 R1 R2 R2 then the transverse semi-axis of the contact ellipse (in the y direction) is greater than or equal to the longitudinal semi-axis. Conversely, if 1 1 1 1 + j ≤ i + j, i R1 R1 R2 R2 then the transverse semi-axis (in the y direction) is less than or equal to the longitudinal semi-axis. Using Equations 4.24 and 4.26 and Table 4.1, it is clear that the larger dimension of the contact ellipse is associated with the coefficient m.

FIGURE 4.4 Wheel and rail radii of curvature.



139

EXAMPLE 4.2 Each of the two wheels of a single wheelset traveling on a tangent track is assumed at a given time to have a normal load equal to 45 kN. The right wheel rolling radius is assumed at this time to be 0.457 m. The wheel profile is assumed to be conical, and the transverse radius of curvature of the rail is equal to 0.254 at this configuration. Assuming the wheelset has zero yaw rotation, determine the dimensions of the contact ellipse and the maximum contact pressure if the wheel and the rail are made of steel. Solution. Since the wheel and the rail are made of steel, one can assume the following material properties for the wheel and the rail: Poisson’s ratio ν is 0.28, and Young’s modulus of elasticity is 2.1 × 1011 N/m2. Since the wheel is conical, R2w is equal to ∞, and since the rail is tangent, the longitudinal rolling radius R1r is equal to ∞. Equations 4.23 and 4.26 lead to

K1 =

K2 =

K3 =

1 1



2  R1w

+

( )

1− νw

2

π Ew

( )

1− νr

2

π Er

1 R2w

+

=

1 R1r

=

+

1 − (0.28) 2 2.1 × π × 1011

1 − (0.28) 2 2.1 × π × 1011

= 1.396 × 10 −12

m 2 /N

= 1.396 × 10 −12

m2/ N

1 

1 1 1  −1  = 2  0.457 − 0 + 0 + 0.254  = 3.064 m  

R2r 

Since the wheelset has zero yaw rotation, then ψ = 0, and one has

K4 =

=

1  1

2  R1w 1 

−

1 

2

2

 1  1 1  1  1 1  +  r − r  + 2  w − w   r − r  cos 2 ψ w   R1 R2   R1 R2  R2   R1 R2  2

2

   1  1  1  − 0 +  0 − + 2 −0 0− cos(0)    0.457   0.254  2  0.457 0.254 

= 0.875

1

m -1

Using Equation 4.27, the argument for the m and n coefficients can be evaluated as

(

)

θ = cos−1 K 4 /K3 = 1.281 rad


140


Using the closed-form equations given in Equation 4.29 and Table 4.2,

m = 1.306,

n = 0.813

From Equations 4.24 and 4.25

(

)

13

= 0.599 × 10 −2 m

(

)

13

= 0.373 × 10 −2 m

a = m 3π Fn K1 + K 2 4 K3  b = n 3π Fn K1 + K 2 4 K3  It is clear that

1 1 1 1 + < + R1w R1r R2w R2r Therefore, the longitudinal contact ellipse semi-axis dimension is equal to a, and the transverse semi-axis dimension is equal to b. The maximum contact pressure can be determined from Equation 4.17 as follows:

p0 =

3 Fn = 96.075 MPa 2π ab

The contact area is equal to πab = 0.703 × 10−4 m2.

4.2 CREEP PHENOMENON The relative motion between two bodies i and j that are in contact can be the result of rolling and sliding motion. In the general case of rolling and sliding, the two bodies have different velocities vi and vj at the contact point and different angular velocities ω i and ω j. The relative angular velocity along the normal to the surfaces at the contact point is called the spin. If the velocities vi and vj at the contact point are not equal, the rolling motion is accompanied by sliding. If ω i and ω j are not equal, the motion is accompanied by rolling and/or spin. When rolling occurs without sliding or spin, the motion is considered to be pure rolling. In the case of the contact of two elastic bodies subjected to external applied normal load, some contact points on the contact surface may slip, while other points may stick. The difference between the tangential strains of two bodies in the adhesion area leads to a small slip that is called creepage. The creepage is, therefore, due to a combination of elastic deformation and friction. This phenomenon was recognized in 1926 by Carter (Carter, 1926).



141

FIGURE 4.5 Wheel rolling over a rail.

Consider the case of a wheel rolling on a rail, as shown in Figure 4.5. Let t1r and t r2 , as shown in Figure 4.6, be the unit orthogonal tangents to the rail at the contact point in the longitudinal and lateral directions, and let nr be the unit normal to the surfaces at the contact point, that is, nr =

t1r × t 2r t1r × t 2r

(4.31)

The absolute velocity of the wheel w is assumed to be vw. The magnitude of the wheel velocity along the longitudinal tangent defined at the contact point is given by T

V = v w t1r

(4.32)

The creepages are normalized relative velocities that are defined as follows: (v w − v r )T t1r   V  w r T r  (v − v ) t 2  ζy =  V  w r T r  (ω − ω ) n  ϕ= V 

ζx =

FIGURE 4.6 Contact frame.

(4.33)


142


These creepage expressions are fundamental in the kinematic and force analysis of the wheel/rail interaction. In the case of pure rolling with no lateral oscillations, one can show that

ζx = ζy = 0

(4.34)

In the multibody system formulations, the global velocity of an arbitrary point on an arbitrary rigid body i can be defined, as described in Chapter 2, as follows: i + ω i × ui r i = R

(4.35)

where the vector Ri is the global velocity vector of the origin of the body coordinate system, ui is the local position vector of the arbitrary point on body i defined in the global frame, and ω i is the absolute angular velocity vector of the body coordinate system defined in the global coordinate system. This angular velocity vector is given as ω i = ω xi

ω iy

ω zi 

T

(4.36)

If a wheel w is in contact with a rail r at point P whose global position is defined using the coordinates of the two bodies by the two vectors rPw and rPr , respectively, the global velocity vector of the contact point can be defined in terms of the coordinates of the two bodies as follows: w + ω w × uw  rPw = R P  r + ω r × ur  rPr = R P 

(4.37)

The velocities of Equations 4.36 and 4.37 can be used in Equation 4.33 to define the creepages in terms of the generalized coordinates and velocities of the two bodies as follows:

ζx =

(rPw − rPr ) ⋅ t1r , V

ζy =

(rPw − rPr ) ⋅ t r2 , V

ϕ=

(ω w − ω r ) ⋅ n r V

(4.38)

These definitions of the creepage are the general expressions used in the nonlinear analysis of multibody railroad vehicle systems. Note that these expressions are functions of the geometry of the rail at the contact point. These creepage expressions, however, are linearized and simplified in some specialized railroad vehicle system formulations. The effect of this linearization is discussed in more detail in Chapter 8.



143

EXAMPLE 4.3 A wheelset, denoted as body w, is traveling on a tangent track as shown in Figure 4.7. At the initial configuration, the location of the right wheel contact point with respect to the wheelset coordinate system is given by u rw = [0 –a –r0]T, where a is the lateral position of the right wheel geometric center and is assumed to be equal to 0.75 m, and r0 is the initial rolling radius and is assumed to be equal to 0.457 m. The wheelset is assumed to have a forward velocity of 10 m/s. The orientation of the wheelset is defined with respect to a global frame by the Euler angles θw = [ψ w φ w θ w]T = 0 at the initial configuration. The global angular velocity vector of the wheelset is given by ωw = [ω xw ω yw ω zw ]T . It is assumed that the angle between the lateral tangent and the wheelset axis at the contact point is defined by the contact angle δk for contact k (k = r, l), as shown in Figure 4.7 (subscripts r and l are used here to denote right and left contact, respectively). Determine the creepage values for the right wheel contact at this configuration assuming that δr = 0.025 rad. Solution. The orientation of the wheelset can be defined with respect to the global coordinate system using the transformation matrix given in Chapter 2. The sequence of rotation is assumed as follows: a rotation ψ w about the Zw axis, a rotation φ w about the Xw axis, and a rotation θ w about the Yw axis. Accordingly, the rotation matrix of the wheelset w is given as follows: Aw =

 cos ψ w cos θ w − sin ψ w sin φ w sin θ w  w w w w w  sin ψ cos θ + cos ψ sin φ sin θ w w  − cos φ sin θ

− sin ψ cos φ w

cos ψ cos φ w

sin φ

w

w

cos ψ sin θ + sin ψ sin φ cos θ

w

sin ψ sin θ − cos ψ sin φ cos θ

w

w

w

1  A = I = 0  0

FIGURE 4.7 Wheel set frame.

w

w

0

0

1

0

0



1 

w

w

cos φ cos θ

w

At the initial configuration, one has

w

w

w

w

w

   


144


The global position vector of the center of mass of the wheelset at the given configuration can be written as R w =  Rxw

Ryw

T

Rzw  =  0

0

0.457 

T

The global velocity of the right wheel geometric center is given by w + ω w × (Aw u w ) r0w = R 0 where u 0w = [0 –a 0]T = [0 –0.75 0]T is the local position vector of the geometric center of the right wheel defined in the body coordinate system. Assuming pure rolling at the initial configuration, the angular velocity can be written as



V



r0

ω w = 0



T

0  =  0

21.882



0 

T

Thus,

10  10      r =  0  + 0 =  0   0   0  w 0

The global velocity vector of the contact point is w + ω w × (Aw u w ) =  0 rPw = R P 

0

0 

T

The tangents and the normal can be defined in the global coordinate system as follows:

 t1r

1 

n r  = A w  0

t r2

 0

0 

0 cos δ r − sin δ r



sin δ r  cos δ r 

Note that at the initial configuration, Aw = I. Assuming that the rail is fixed, the creepage expressions are defined as follows:

ζx =

rPw ⋅ t1r V

,

ζy =

rPw ⋅ t r2 V

,

ϕ=

ω w ⋅ nr V

w ⋅ t r . Using the data and the results of this example, one can show that where V = R 1 the creepages are given by



ζx =

145

0 10

= 0, ζy =

0 10

=0

and

ϕ=

V  r  sin δ r 0 V

=

sin δ r r0

=

sin(0.025) 0.457

= 0.0547

As a result, the longitudinal and lateral creepages are zero, while the spin creepage is not equal to zero. This spin creepage is called geometric spin and is attributed to the component of the wheel angular velocity along the spin axis. This component is defined, as shown in the preceding equation, by ϕ = (sin δr)/r0, which is function of the geometric parameters δr and r0 only.

4.3 WHEEL/RAIL CONTACT APPROACHES When a wheel travels on a rail, the creepages at the contact point generate tangential forces that play an important role in the steering and stability of railroad vehicles. According to Hertz contact theory, the contact area, as described in Section 4.1, is assumed to be elliptical. Due to the compressive forces in the contact region, the wheel and the rail will deform. The deformations can be defined in the coordinate system XrcYrcZrc, whose origin is defined at the contact point, as shown in Figure 4.5. If the two bodies move relative to each other in the presence of Coulomb’s friction, tangential forces are developed in the contact region. Tangential forces are also developed if the bodies spin relative to each other. In general, the contact area is divided into two regions; the adhesion region, where the surface particles of the bodies do not slide relative to each other, and the slip region, where there is sliding. Based on the assumptions stated in Section 4.1, Hertz theory does not consider the shear traction Ft = [Ftx Fty]T between the two is defined in the global coordinate bodies in the contact region. The true slip w system as the relative velocity on the tangent plane. In most railroad vehicle formulations, for simplicity, the true slip is determined based on pure rigid body kinematics. In this case, this slip component is defined as follows:

( )

(

T

)

x , y = r wr − r wr n r n r w

(4.39)

In this equation, nr is the normal to the surface at the contact point, and r wr is the relative velocity at the contact point defined as r wr = r w − r r

(4.40)


146


The components of the slip given by Equation 4.39 can be defined in a coordinate system defined at the contact point using the following projection:  ⋅ t1r   rPwr ⋅ t1r  x, y =  w1  =  w w =  2 ⋅ t 2r  rPwr ⋅ t r2  w w

( )

(4.41)

where n r ⋅ t1r = n r ⋅ t 2r = 0 . The preceding tangential slip defined on the contact plane can be used to define the tangential traction Ft using Coulomb’s law as T

Ft =  Ftx

Fty  ≤ µ p;

µ pw Ft = − ; w

= 0 (adhesion area )  w     ≠ 0 (slip area ) w  

(4.42)

where µ is the friction coefficient and p is the contact pressure. Therefore, the longitudinal and lateral creep forces and the spin creep moment can be expressed as follows:

Fx =

∫∫

Fy =

∫∫

M=

∫∫

     Fty dx dy     ( xFty − yFtx ) dx dy   Ftx dx dy

(4.43)

The creep forces and the spin moment depend on the creepages, the contactellipse dimension, and the normal force. The relationship between the longitudinal, lateral, and spin creepages and the longitudinal and lateral forces and spin moments are governed by the creep-force law (Kalker, 1990). Some creep-force law classifications are discussed below (Garg and Dukkipati, 1988).

4.3.1 EXACT THEORY

OF

ROLLING CONTACT

In the exact theory of rolling contact, the constitutive law is obtained by deriving the traction-displacement relationship using the general elasticity theory. In this case, the wheel and the rail near the contact region are treated as elastic half-space, an assumption that is employed in most railroad vehicle formulations (Kalker, 1979).



4.3.2 SIMPLIFIED THEORY

OF

147

ROLLING CONTACT

In the simplified theory of rolling contact, the traction-displacement constitutive law takes a simple form, which is given by (Kalker, 1973): u wr = u w − u r = [ L x Ftx

Ly Fty ]T

(4.44)

where uw and ur are the displacements of the two bodies at the contact point (both components are defined in the same coordinate system), and Lx and Ly are, respectively, the compliant parameters in the longitudinal and lateral directions. These compliant coefficients depend on the material and geometric parameters of the two bodies in the contact region (Kalker, 1973).

4.3.3 DYNAMIC

AND

QUASI-STATIC THEORY

If the inertial effects are included in the formulation of the rolling contact theory, the theory is called dynamic. On the other hand, if the inertial effect is neglected, the theory is called quasi-static. The inertial effect becomes significant if the speed is higher than 500 km/h (Kalker, 1979). Therefore, in the railroad vehicle dynamics, the quasi-static theory is used instead of the dynamic theory.

4.3.4 THREE-

AND

TWO-DIMENSIONAL THEORY

If the displacement-force relationship depends on all three coordinates x, y, and z, the theory is said to be three-dimensional theory. On the other hand, if the displacement-force relationship is independent of the lateral coordinate, the theory is called two-dimensional. Therefore, in the two-dimensional theories, the effect of the spin creepage is not considered, and this limits the use of such two-dimensional theories in railroad vehicle dynamics formulations.

4.4 CREEP-FORCE THEORIES Several creep-force theories have been developed and applied to solve the wheel/rail contact problem. Kalker has provided surveys of wheel/rail contact theories (Kalker, 1979, 1980, 1991). This section presents brief descriptions of some of these theories. It is important, however, to mention that some of the theories presented in this section are no longer being used in wheel/rail contact computer codes that are based on nonlinear three-dimensional multibody system formulations. They can be used, however, for linear or special models.

4.4.1 CARTER’S THEORY In 1926, Carter introduced the first two-dimensional creep theory (Carter, 1926). In this theory, Carter introduced a closed-form relationship between the longitudinal creepage and the tangential force. In this theory, Carter approximated the shape of


148


FIGURE 4.8 Contact area according to Carter.

the contact area by a two-dimensional uniform rectangular strip. It was assumed that the wheel and rail surfaces can be represented, respectively, by cylinder and thick plate. The radius of the wheel is assumed to be larger than the circumferential length of the contact area and, therefore, the problem can be treated as an infinite elastic medium bounded by a plane with a local pressure distribution and tangential traction in the contact area. Carter showed that the difference between the circumferential velocity of the driven wheel and the translation velocity of the wheel at the contact point is not equal to zero when acceleration or braking couple is suddenly applied to the wheel. This difference increases in absolute value with increasing couple until a Coulomb maximal value is reached. The behavior in the contact region, based on Carter’s theory, can be described using a simple model of a wheel traveling in the positive direction of the x-axis. Let A be the point at which the first contact occurs, and A′ be the point of departure, as shown in Figure 4.8. Let ABA′ represent the curve of the limiting value of the tangential traction, and let ADCA′ be the actual curve of the tangential traction that starts at A and never exceeds the limiting curve ABA′, as shown in Figure 4.8. Beyond point C, the surfaces slip with limiting tangential traction, as the pressure between the surfaces is insufficient to prevent the movement. Carter’s closed-form solution for the relation between longitudinal creepage and tangential force is given by (Carter, 1926): f=

(

) R lN   2 ( λ + 2G )  1−

πG λ + G

w

  1− q 

q

(4.45)

where ƒ is the tractive force per unit creepage in the longitudinal direction; q is the ratio F/Fx, where F is the total tractive effort of the wheel and Fx is the tangential force in the longitudinal direction; G and λ are, respectively, the material modulus of rigidity and Lame’s constant; Rw is the wheel rolling radius; l is the equivalent length of the contact in the transverse direction of the rail and is equal to (4b/3), where b is the contact ellipse semi-axis dimension in the lateral direction; and N is the total normal force. Equation 4.45 can be written as follows (Kalker, 1991): − kζ x + 0.25k 2ζ x ζ x Ft  = µ N l −sign(ζ ) x 

k ζx ≤ 2 k ζx > 2

(4.46)



149

FIGURE 4.9 Comparison between Carter’s law and linear law.

where Ft is the tangential force per unit lateral length exerted at the contact on the wheel; µ is the coefficient of friction; Nl is the total normal force per unit lateral length exerted on the wheel; ζx is the creepage and is equal to 2(Vt − Vc)/(Vt + Vc), where subscripts t and c refer, respectively, to tangential forward and circumferential velocities; and k is Carter’s creepage coefficient, which is equal to 4Rw /µa, where Rw is the wheel rolling radius and a is semi-axis dimension of the contact ellipse in the rolling direction (longitudinal). Figure 4.9 shows the difference between Carter’s law and the linear law. In general, Carter’s theory is capable of predicting the frictional losses in a locomotive driving wheel. It is clear from Equation 4.46 that the tractive force per unit creepage depends on the tractive effort.

4.4.2 JOHNSON

AND

VERMEULEN’S THEORY

In 1958, Johnson (Johnson, 1958a, 1958b) extended Carter’s theory to the threedimensional case of two spheres without spin. Later, Vermeulen and Johnson extended the method to smooth half-spaces without spin (Vermeulen and Johnson, 1964). In this theory, the contact surface between the two rolling bodies transmitting a tangential force is asymmetrically divided into two regions: the slip region and the stick or no-slip region. The adhesion area was assumed to be elliptical. The area of adhesion is assumed to touch the leading edge of the contact ellipse, as shown in Figure 4.10. According to Johnson and Vermeulen, the total resulting tangential force F = [Fx Fy]T can be determined as follows:   1  3    1 − τ  − 1 ζ i + η j for τ ≤ 3, τ ( 1 / ) F   3   = µN  −(1 / τ ) ζ i + η j for τ > 3, 

(

(

)

)

(4.47)


150


FIGURE 4.10 Contact area according to Johnson and Vermeulen.

where ζ is the normalized longitudinal creepage, defined as (πabGζx)/µNφ; η is the normalized lateral creepage, defined as (πabGζy)/µNψ ; τ = ζ 2 + η 2 ; i = [1 0]T; j = [0 1]T; N is the normal force; G is the modulus of rigidity; ζx is longitudinal creepage; ζy is the lateral creepage; a and b are the dimensions of the contact ellipse semiaxes in the rolling and lateral directions, respectively; µ is the coefficient of friction; and

φ = Be − ν ( De − Ce )   ψ = Be − ν (a / b)2 Ce 

for a ≤ b, e =

1 − a /b

φ = (b / a)  De − ν ( De − Ce )      ψ = (b / a)  De − νCe  

for a ≥ b, e =

1 − b /a

( )

( )

2

2

          

In this equation, Be, De, and Ce are the complete elliptical integrals of argument e, and ν is the Poisson’s ratio. Vermeulen and Johnson compared the results obtained from Equation 4.47 with results of measurements. The difference between the formula and measurement results was attributed to the assumption of an elliptical no-slip region. It is important to note that this theory is valid in the case of no spin. In railroad dynamics, since the spin effect is important, the Vermeulen and Johnson theory is rarely used in the computer formulation of the creep forces.

4.4.3 KALKER’S LINEAR THEORY Kalker suggested that, for very small creepages, the area of slip is very small and its effect can be neglected (Kalker, 1967). Therefore, the adhesion area can be assumed to be equal to the area of contact. Kalker described the behavior of the contact point as follows. Along a line parallel to the direction of rolling, the particle starts to penetrate, and as the slip is very small (no-slip condition), traction starts to build up. As the particle leaves the contact area, the traction becomes zero.



151

Therefore, the true slip given by Equation 4.39 vanishes everywhere in the contact area, that is,

( )

x, y = 0 w

(4.48)

Integrating this equation with respect to x

∫ w ( x, y ) dx = u( x, y)

(4.49)

where u(x,y) is the tangential displacement difference and is determined by assuming that the traction is continuous at the leading edge of the contact area. Using the traction-displacement relationship based on the general elasticity theory and integrating the traction over the contact area (Equation 4.43), the linear relation between the creepages and the creep forces and moment are obtained as c11  Fx      Fy  = −Gab  0  M     0

0

 ζ x    abc23  ζ y   abc33   ϕ  0

c22 − abc23

(4.50)

where ζx, ζy , ϕ are the longitudinal, lateral, and spin creepages, respectively; a is the contact ellipse semi-axis dimension in the rolling direction; b is the contact ellipse semi-axis dimension in the lateral direction; G is the modulus of rigidity; and cij are creepage coefficients that depend only on Poisson’s ratio and the ratio of the semi-axis of the contact ellipse, as shown in Table 4.4 (Kalker, 1990). Kalker also introduced the following definitions:

G=

1 1 1  + r ,  w 2 G G 

ν 1  νw νr  =  +  G 2  G w Gr 

(4.51)

where G is an average shear modulus of rigidity of the wheel w and the rail r, and ν is a combined Poisson’s ratio of the wheel and the rail. In 1984, Kalker calculated the creepage and spin coefficients when the relative slip is small but the contact area is not necessarily elliptic. These calculations were made with the aid of the program CONTACT. The error was found to be less than 5%, which led to the conclusion that the coefficients given in Table 4.4 are accurate enough for the analysis of the wheel/rail contact problem. The linear theory is extensively used in railroad vehicle dynamics. Haque et al. (Haque et al., 1979) developed a program that can be used to interpolate for the creep coefficients. In this program, linear interpolation is used to obtain the coefficients using the data given in Table 4.4.


152


TABLE 4.4 Kalker’s Creepage and Spin Coefficients c11

c23 = −c32

c22

c33

g

ν=0

0.25

0.5

ν=0

0.25

0.5

ν=0

0.25

0.5

ν=0

0.25

0.5

(a/b) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2.51 2.59 2.68 2.78 2.88 2.98 3.09 3.19 3.29

3.31 3.37 3.44 3.53 3.62 6.72 3.81 3.91 4.01

4.85 4.81 4.8 4.82 4.83 4.91 4.97 5.05 5.12

2.51 2.59 2.68 2.78 2.88 2.98 3.09 3.19 3.29

2.52 2.63 2.75 2.88 3.01 3.14 3.28 3.41 3.54

2.53 2.66 2.81 2.98 3.14 33.1 3.48 3.65 3.82

0.334 0.483 0.607 0.720 0.827 0.930 1.03 1.13 1.23

0.473 0.603 0.715 0.823 0.929 1.03 1.14 1.25 1.36

0.731 0.809 0.889 0.977 1.07 1.18 1.29 1.40 1.51

6.42 3.46 2.49 2.02 1.74 1.56 1.43 1.34 1.27

8.28 4.27 2.96 2.32 1.93 1.68 1.50 1.37 1.27

11.7 5.66 3.72 2.77 2.22 1.86 1.60 1.42 1.27

(b/a) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

3.4 3.51 3.65 3.82 4.06 4.37 4.84 5.57 6.96 10.7

4.12 4.22 4.36 4.54 4.78 5.10 5.57 6.34 7.78 11.7

5.2 5.3 5.42 5.58 5.8 6.11 6.57 7.34 8.82 12.9

3.40 3.51 3.65 3.82 4.06 4.37 4.84 5.57 6.96 10.7

3.67 3.81 3.99 4.21 4.50 4.90 5.48 6.40 8.14 12.8

3.98 4.16 4.39 4.67 5.04 5.56 6.31 7.51 9.79 16.0

1.33 1.44 1.58 1.76 2.01 2.35 2.88 3.79 5.72 12.2

1.47 1.57 1.75 1.95 2.23 2.62 3.24 4.32 6.63 14.6

1.63 1.77 1.94 2.18 2.50 2.96 3.70 5.01 7.89 18.0

1.21 1.16 1.10 1.05 1.01 0.958 0.912 0.868 0.828 0.795

1.19 1.11 1.04 0.965 0.892 0.819 0.747 0.674 0.601 0.526

1.16 1.06 0.954 0.852 0.751 0.650 0.549 0.446 0.341 0.228

Note: g = 0, c11 = π2/4(1 – ν); c22 = π2/4; c23 = –c32 = π g {1 + ν(0.5Λ + ln 4 – 5)}/3(1 – ν); Λ = ln(16/g2); and c33 = π2/16(1 – ν)g. Source: Kalker, J.J., Three-Dimensional Elastic Bodies in Rolling Contact, Kluwer, Dordrecht, Netherlands, 1990. With permission.

EXAMPLE 4.4 At a certain configuration, a wheel has contact ellipse semi-axis dimensions a = 6.523 × 10−3 m and b = 2.298 × 10−3 m. Assume that the wheel and the rail have Poisson’s ratio equal to 0.24 and modulus of rigidity equal to 8 × 1010 N/m2. At this configuration, the longitudinal creepage is equal to zero, the lateral creepage is 0.01, and the spin creepage is 0.1016 m−1 Determine the creep forces. Solution. The contact ellipse semi-axis ratio a/b = 2.838. Using Kalker’s table, the creep coefficients are determined as c11 = 5.8468,

c22 = 5.8192,

c23 = 3.6574,

c33 = 0.72518



153

Using Equation 4.51, the tangential creep forces and moment can be determined as

 c11  Fx      Fy  = −Gab  0   M  0

 ζ x  0       abc23  ζ y  = −  71511.292 N    −168.4938 N ⋅ m  abc33   ϕ 

0

0

c22 − abc23

4.4.4 HEURISTIC NONLINEAR CREEP-FORCE MODEL In general, the wheel/rail contact is a highly nonlinear problem. Using linear models for the force-creepage relationship can lead to errors. Two sources of the nonlinearity in the wheel/rail contact problem are: 1. Nonlinear wheel/rail geometric functions 2. Adhesion limits on the force-creepage relationship The creep-force models discussed thus far in this section are not based on fully nonlinear theory and employ simplifying assumptions. In Johnson and Vermeulen’s theory, the spin creepage is ignored. This assumption limits the application of the theory to the case of pure longitudinal and lateral creepages. The effect of the spin creepage is important, especially in the case of flange contact. Kalker’s linear theory is limited to the case of small creepages. White et al. (1978) and Shen et al. (1983) suggested a new approximate heuristic nonlinear theory based on Kalker’s linear theory. In this theory, the saturation law of Johnson and Vermeulen is used, and the effect of spin creepage on the creep forces is considered. The longitudinal and lateral creep forces are first calculated using Kalker’s linear theory, defined in Equation 4.50, as follows: c11  FxK   K  = −Gab   0  Fy 

ζ   x  ζ y  abc23    ϕ  0

0 c22

(4.52)

where FxK and FyK are the forces evaluated using Kalker’s linear theory. The resultant creep force obtained from the linear model is calculated as follows:

FL =

(F ) + (F ) K x

2

K y

2

(4.53)

According to Coulomb friction law, the magnitude of the resultant creep force cannot exceed the pure slip value µN. Using Johnson and Vermeulen’s theory, the resultant force FL is limited by the nonlinear value FL as follows (Shen et al., 1983):


154


2 3    µ N  FL  − 1  FL  + 1  FL   ,     27  µ N   FL =   µ N  3  µ N     µ N ,

FL ≤ 3µ N

(4.54)

FL > 3µ N

Therefore, the creep-force reduction coefficient is defined as follows:

ε=

FL FL

(4.55)

The nonlinear creep-force model is given by  Fx   Fx   =ε   Fy   Fy 

(4.56)

It is important to note that the spin-creepage contribution to the lateral creep forces is included in computing the creep force given by Equation 4.52. This theory gives more realistic values for creep forces outside the linear range than Kalker’s linear theory (Shen et al., 1983). Shen et al. (1983) showed that the results obtained from the heuristic model are in good agreement with Kalker’s simplified theory. However, in the case of high values of spin, the heuristic theory leads to unsatisfactory results (Kalker, 1991).

4.4.5 POLACH NONLINEAR CREEP-FORCE MODEL In 1999, Polach introduced an algorithm for the computation of the wheel/rail creep forces (Polach, 1999). Polach assumed that the shape of the contact area is elliptic. In Hertz contact theory, the maximum stress distribution is equal to σ. Therefore, the maximum tangential stress at any arbitrary point is

τ max = µσ

(4.57)

where µ is the coefficient of friction and is assumed to be constant in the whole contact area. Assuming that the relative displacement between the bodies in the adhesion area increases linearly from one side on the edge of the contact A to the other side C, as shown in Figure 4.11, the tangential stress increases linearly with the distance from the leading edge. As the tangential stress reaches its maximum value, sliding takes place. Due to the longitudinal and lateral creepage ζx and ζy, respectively, the tangential force is given by Fpx = −

 2µ N  ε + tan −1 ε   2 π  1+ ε 

(4.58)



155

FIGURE 4.11 Normal and tangential stress distribution for Polach theory. (Courtesy of Polach, O., Vehicle System Dynamics Supplement, 33, 728, 1999. With permission.)

where N is the normal contact force, and ε is the gradient of the tangential stress in the area of adhesion and is given by

ε=

1 Gπ abCh νc 4 µN

(4.59)

where G is the modulus of rigidity, and Ch is a constant that depends on Kalker’s coefficients as follows:  ζ   ζ  Ch =  c11 x  +  c22 y  ν  ν  2

2

(4.60)

and ν is the magnitude of the creepage and is given by

ν = ζ x2 + ζ y2

(4.61)

Using a modified lateral creepage that accounts for the effect of the spin creepage φ, Equation 4.61 can be written as:

ν c = ζ x2 + ζ yc2

(4.62)

where the modified lateral creepage ζyc is given by ζ y ; ζ y + ϕa ≤ ζ y  ζ yc =  ζ y + ϕ a ; ζ y + ϕ a > ζ y 

(4.63)


156


Assuming that the moment effect caused either by the spin creepage or by the lateral creepage is small compared with other moments acting on the system, the lateral tangential force that accounts for the effect of the spin is given by Fpy = −

9 aµ NK 1 + 6.3(1 − e − a b )  16

(4.64)

where K is a constant defined as  δ3 δ2 1 1 (1 − δ 2 )3 K = εy  − + − 2 6  3  3

(4.65)

and δ is given by

δ=

(ε y )2 − 1

(4.66)

(ε y )2 + 1

where εy is the gradient of the tangential stress and is given by

εy =

 c23ζ yc 8 Gb ab  3 µ N  1 + 6.3 1 − e − a b 

(

)

  

(4.67)

where c23 is Kalker’s coefficient. Finally, the creep forces are given as follows:     ζy ϕ Fy = Fpx + Fpy  νc νc  Fx = Fpx

ζx νc

(4.68)

Polach’s algorithm yields accurate prediction of the tangential contact forces, and it has been implemented in some computer codes developed recently for the dynamic analysis of railroad vehicle systems. A FORTRAN code based on this theory has been published in the literature (Polach, 1999).

4.4.6 SIMPLIFIED THEORY As briefly discussed in the previous section, one can approximate the relationship between the tangential surface traction Ft = [Ftx Fty]T and the tangential surface displacement uwr = uw − ur using compliant parameters Lx and Ly as follows: u wr = u w − u r = [ L x Ftx

Ly Fty ]T

(4.69)



157

In such a case, the complex expressions that result from the use of the general elasticity theory can be simplified by using the compliant parameters. In the case of steady-state rolling, the slip inside the contact region is assumed to be zero (Kalker, 1979), that is, ζ − ϕ y  ∂u wr x, y = V  x =0 w −V ∂x ζ y + ϕ x 

( )

(4.70)

Substituting Equation 4.69 into Equation 4.70, one has ζ − ϕ y   Lx (∂Ftx /∂x )  x, y = V  x w −V  =0 ζ y + ϕ x   L y (∂Fty /∂x ) 

( )

(4.71)

Integrating the preceding equation with respect to x, the tangential traction can be obtained as  x (ζ x − ϕ y) + k ( y)   Lx  Ftx     = x (ζ y + 12 ϕ x ) + l ( y)   Fty    Ly  

(4.72)

where k(y) and l(y) are the integration constants that are arbitrary functions of y. These integration constants are determined by using the condition that the tangential traction is equal to zero at the leading edge of the contact ellipse. This leads to the following equation:   ( x − a( y))(ζ x − ϕ y)   Lx  Ftx     = 2 2 1  Fty  ζ y ( x − a( y)) + 2 ϕ ( x − a( y) )    Ly  

(4.73)

where a(y) is given by a( y) = a 1 − ( y /b)2 . Substituting Equation 4.73 into Equation 4.43, one obtains

Fx =

∫ ∫

Fy =

∫ ∫

b

a( y)

−b

− a( y)

b

−b

Ftx dx dy = −

8a 2bζ x 3Lx

8a 2bζ y π a 3bϕ − Fty dx dy = − 3L y 4 Ly − a( y) a( y)

      

(4.74)


158


where the spin moment is neglected in the preceding equation. On the other hand, Kalker’s linear theory, obtained using the general elasticity theory, as shown in Equation 4.50, leads to the following longitudinal and lateral force-creepage relationship Fx = −Gabc11ζ x Fy = −Gabc22ζ y − G (ab)1.5 c23ϕ

   

(4.75)

Note that cij are creepage coefficients given by Table 4.4. By equating Equations 4.74 and 4.75, the compliant parameters Lx and Ly are given by Lx =

8a 3Gc11

π a2 8a L y1 = , Ly2 = 3Gc22 4G abc23

      

(4.76)

where two different definitions Ly1 and Ly2 are given for Ly so that the creep forces obtained using the simplified theory will be equal to those obtained using Kalker’s linear theory. As a result, while only two compliant parameters are originally defined in the constitutive model given by Equation 4.69, the three compliant parameters Lx, Ly1, and Ly2 are required, since Ly1 and Ly2 differ significantly. For this reason, the following single compliant parameter L is introduced (Jacobson and Kalker, 2001):

L=

Lx ζ x + L y1 ζ y + L y 2 ab ϕ (ζ x )2 + (ζ y )2 + abϕ 2

(4.77)

where L = Lx ;

ζy = ϕ = 0

L = L y1;

ζx = ϕ = 0

L = Ly2;

ζx = ζy = 0

     

(4.78)

It is important to note that the analytical solution based on the simplified theory can be obtained when the effect of spin moment is ignored, while a numerical method is required for a more general case of wheel/rail contact problems. A program called FASTSIM that is based on the simplified theory was developed by Kalker (Kalker, 1982) and has been widely used in railroad vehicle computer programs. In this program, the contact surface is discretized into several strips, and the creep forces



159

are calculated by incrementing the tangential tractions from one strip to another. The complete algorithm of FASTSIM can be found in the literature (Kalker, 1982).

4.4.7 KALKER’S USETAB USETAB is one of the available codes that can be used to accurately predict the wheel/rail creep forces. Since this program is designed for a general wheel/rail contact problem, the coefficient table becomes very large to account for all of the possible contact geometry. USETAB has two sets of tables. In the first set, the creepage coefficients given in the literature (Kalker, 1990) are listed. In the second set, the Hertzian creep-force law constants that were generated by numerous runs of the CON93 program developed by Kalker are listed. The inputs to USETAB are the contact ellipse semi-axis dimensions, the creepages, the equivalent modulus of rigidity, the Poisson’s ratio, and the coefficient of friction. The output of USETAB is the creep forces and spin moment defined in the contact coordinate system. The estimated error resulting from the use of USETAB is approximately 1.5% compared with the fully nonlinear theory, while this error for FASTSIM, which is the code currently used in many programs for the calculation of the creep forces, is approximately 15%.


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5

Multibody Contact Formulations

As discussed in the preceding chapter, if one wants to evaluate the tangential creep forces and spin moment, one must know the normal contact force as well as other wheel/rail material and geometric parameters. There are two main approaches that can be used in the multibody system formulations to determine the normal contact force when the wheel/rail interaction is considered. In the first approach, the contact between the wheel and the rail is described using kinematic constraint equations. The normal contact force can be determined as the reaction force due to the imposition of the contact constraint equations. When equality constraints are used, it is assumed that there is no penetration or separation between the wheel and the rail. In the second approach, a compliant force with assumed stiffness and damping coefficients is used to describe the wheel/rail contact. In this approach, the wheel has six degrees of freedom with respect to the rail and the wheel/rail separation is possible. In this chapter, four nonlinear dynamic formulations for the analysis of the wheel/rail contact are discussed. Two of these formulations employ nonlinear algebraic kinematic constraint equations to describe the contact between the wheel and the rail (constraint approach), while in the other two formulations, the contact force is modeled using a compliant force element (elastic approach). In the formulations based on the elastic approach, as previously mentioned, the wheel has six degrees of freedom with respect to the rail, and the normal contact force is defined as a function of the penetration using Hertz’s contact theory or using assumed stiffness and damping coefficients. One of the elastic methods discussed in this chapter is based on a search for the contact locations using discrete nodal points. In the second elastic approach discussed in this chapter, the contact points are determined by solving a set of algebraic equations. In both elastic methods, the contact points are determined on-line instead of using a look-up table, which is a commonly used method for determining the location of the contact points in some specialized railroad vehicle dynamic algorithms. In the formulations based on the constraint approach, on the other hand, the case of a nonconformal contact is assumed, and nonlinear kinematic constraint equations are used to impose the contact conditions. This approach leads to a model in which the wheel has five degrees of freedom with respect to the rail. In the constraint approach, unlike the elastic approach, wheel penetration and lift are not permitted, and the normal contact forces are calculated as constraint forces. Two equivalent constraint formulations that employ two different solution procedures are discussed in this chapter. The first method leads to a larger system of equations by adding all the contact constraint equations to the dynamic equations of motion, while

161


162


in the second method, an embedding procedure is used to obtain a reduced system of equations from which the surface parameter accelerations are systematically eliminated. In the constraint formulations presented in this chapter, three-dimensional contact conditions are imposed. That is, the contact conditions are expressed in terms of four geometric parameters that define the wheel and rail surfaces. In some other constraint formulations that exist in the literature, the contact is treated as a two-dimensional problem and only two geometric parameters are used to describe the wheel and rail profiles. This planar contact approach, however, has been used for the most part with the trajectory coordinates. As will be discussed in Chapter 7, the use of the trajectory coordinates requires a higher order of differentiability and smoothness as compared with the use of the absolute Cartesian coordinates employed in this chapter. Furthermore, the trajectory coordinates, as previously mentioned, are more suited for the development of specialized railroad vehicle dynamic formulations. Nonetheless, the planar contact formulation is discussed in the last section of this chapter for completeness. The focus in this chapter will be mainly on the formulation of the multibody system equations of motion. It will be shown how the wheel/rail contact model can be systematically included in the multibody system formulations when the constraint and elastic approaches are used. Computational algorithms and solution procedures that can be used with the formulations presented in this chapter are discussed in Chapter 6.

5.1 PARAMETERIZATION OF WHEEL AND RAIL SURFACES To accurately determine the location of the point of contact between two bodies, a complete parameterization of the surfaces must be used. In general, as discussed in Chapter 3, a set of four surface parameters can be used to describe the geometry of the two surfaces in contact, as shown in Figure 5.1. The surface parameters can be written in a vector form as s = s1i s2i s1j s2j 

T

(5.1)

where superscripts i and j denote bodies i and j, respectively. Using these parameters, the location of the contact point P can be defined, respectively, in the coordinate systems of bodies i and j as

u

i P

(s , s ) i 1

i 2

( ( (

) ) )

 x i s1i , s2i    =  y i s1i , s2i  ,    i i i  z s , s 1 2   

u

j P

(s , s ) j 1

j 2

( ( (

) ) )

 x j s1j , s2j    =  y j s1j , s2j     j j j  z s , s 1 2   

(5.2)

The tangents to the surface at the contact point are defined in the body i coordinate system as



163

FIGURE 5.1 Schematic representation of two bodies in contact.

t1i =

∂u iP , ∂s1i

t2i =

∂u iP ∂s2i

(5.3)

and the normal vector as n i = t1i × t2i

(5.4)

The parameterization used in Equation 5.2, as well as the tangent and normal vectors defined in Equations 5.3 and 5.4, can be used to describe the geometry of the wheel and rail surfaces as described below.

5.1.1 TRACK GEOMETRY It was shown in Chapter 3 that the surface geometry of the rail r can be described in the most general form using the two surface parameters s1r and s2r , where s1r represents the rail arc length and s2r is the surface parameter that defines the rail profile, as shown in Figure 5.2. The surface parameters s1r and s2r are defined in a profile coordinate system XrpYrpZrp, as shown in Figure 5.3. The location of the origin and the orientation of the profile coordinate system, defined respectively by the vector R rp and the transformation matrix A rp, can be uniquely determined using the surface parameter s1r . Using this description, the global position vector of an arbitrary point on the surface of the rail r can be written as follows: rr = R r + A r (R rp + A rp u rp )

(5.5)

where Rr is the global position vector of the origin of the rail body coordinate system XrYrZr, Ar is the transformation matrix that defines the orientation of the rail coordinate


164


FIGURE 5.2 Surface parameters.

system, and u rp is the local position vector that defines the location of the arbitrary point on the rail surface with respect to the profile coordinate system. The location and orientation of the rail profile coordinate system depends only on the distance traveled along the track s1r , while the local position of an arbitrary point on the rail surface at any section depends on the value of s2r , that is, R rp = R rp (s1r ), A rp = A rp (s1r ), u rp = 0 where f is the function that defines the rail profile.

FIGURE 5.3 Track geometry.

s2r

f (s2r ) 

T

(5.6)



165

FIGURE 5.4 Wheel geometry.

5.1.2 WHEEL GEOMETRY The geometry of the wheel surface can be described using the two surface parameters s1w and s2w . These surface parameters are defined in a wheelset coordinate system XwYwZw. The surface parameter s1w defines the wheel profile, and s2w is an angular parameter that represents the rotation of the contact point, as shown in Figure 5.2. The location of the origin and the orientation of the wheel coordinate system are defined, respectively, by the vector Rw and the transformation matrix Aw. Using this description, the global position vector of an arbitrary point on the surface of the wheel w can be written as follows: r w = R w + Aw u w

(5.7)

where u w is the local position vector that defines the location of the arbitrary point on the wheel surface with respect to the wheel coordinate system. For example, in the case of the right wheel of a wheelset, this vector is defined as

( )

u w =  g s1w sin s2w 

− L + s1w

( )

− g s1w cos s2w  

T

(5.8)

where g is the function that defines the wheel profile, and L is the distance between the origin of the wheelset coordinate system and point Q of the wheel, as shown in Figure 5.2 and Figure 5.4.

5.2 CONSTRAINT CONTACT FORMULATIONS In this chapter, two constraint contact formulations and two elastic contact formulations that can be used to study the wheel/rail dynamic interaction are discussed.


166


Some of these three-dimensional contact formulations differ conceptually in the way the geometric surface parameters are treated. As shown in the preceding section, the geometry of the surfaces of the wheel and rail can be described using the surface parameters that define the location of the contact points. In the constraint formulations, the contact between two surfaces can be described using a set of nonlinear algebraic equations that must be imposed at the position, velocity, and acceleration levels. One may choose not to eliminate the surface parameters, leading to an augmented form of the equations of motion expressed in terms of Lagrange multipliers associated with the kinematic contact constraint equations. This augmented constraint contact formulation will be referred to in this chapter as ACCF. Alternatively, one may choose to systematically eliminate the surface parameters using the contact constraints, leading to an embedding formulation of the equations of motion that does not explicitly include the surface parameter accelerations. This embedded constraint contact formulation will be referred to in this chapter as ECCF. In both formulations, the nonlinear algebraic contact constraint equations are necessary in the formulation of the dynamic equations of motion. This is despite the fact that the two formulations (augmented and embedded) lead to significantly different forms of the dynamic equations.

5.2.1 CONTACT CONSTRAINTS When two rigid bodies come into contact as shown in Figure 5.1, two types of nonconformal kinematic contact conditions need to be satisfied. First, two points (contact points) on the two surfaces must coincide; and second, the two surfaces must have the same tangent planes at the contact point. These two conditions define the following five constraint equations that are required to describe the nonconformal contact between the wheel and rail: k

rPw − rPr    C k (q w , q r , s wk , s rk ) =  t1w ⋅ n r  = 0  t w ⋅ nr    2

(5.9)

or, equivalently, k

 t1r ⋅ (rPw − rPr )   r w r   t 2 ⋅ (rP − rP )  C k (q w , q r , s wk , s rk ) = n r ⋅ (rPw − rPr )  = 0   w r  t1 ⋅ n   t w ⋅ nr  2  

(5.10)

where superscript k denotes the contact number; superscripts r and w denote rail and wheel, respectively; qw and qr are, respectively, the generalized coordinates of the wheel and rail; and t1, t2, and n are the two tangents and the normal to the surface



167

at the contact point. In Equation 5.10, the three relative displacement constraints (point constraints defined by the first three scalar equations) are defined in a rail coordinate system at the contact point. The first three point constraints of Equation 5.9, on the other hand, are defined in the global coordinate system.

5.2.2 CONSTRAINED DYNAMIC EQUATIONS As discussed in Chapter 2, the principle of virtual work in dynamics for a system that consists of interconnected bodies states that the virtual work of the system’s inertia forces is equal to the virtual work of the system’s externally applied forces (Shabana, 2001). This principle can be stated mathematically as follows:

δ Wi = δ We

(5.11)

where δWi is the virtual work of the system inertia forces, and δWe is the virtual work of the applied forces. In the preceding equation, the virtual work of the constraint forces is not included, and it is identically equal to zero, since the dynamic equilibrium of the multibody system is considered. The virtual work of the inertia forces and the virtual work of the applied forces can, in general, be written for a multibody system as

(

)

− Qv , δ Wi = δ qT Mq

δ We = δ qT Qe

(5.12)

where q is the vector of the system generalized coordinates, M is the system mass matrix, Qv is the vector of centrifugal and Coriolis inertia forces, and Qe is the vector of the system applied forces. Therefore, the principle of virtual work of Equation 5.11 leads to

(

)

− Q = 0 δ qT Mq

(5.13)

where Q = Qv + Qe. In Equation 5.13, the elements of the vector of generalized coordinates q are not independent because of the kinematic constraint equations. The vector of the kinematic algebraic constraint equations, which include the contact constraints and other constraints that describe mechanical joints and specified motion trajectories, can be written in a vector form as

( )

C q, s = 0

(5.14)

where s represents the vectors of surface parameters that describe the wheel and rail surface geometry and that are used to formulate the contact constraints as previously described in this section. Taking a virtual change in the constraint equations leads to

δ C = Cqδ q + C sδ s = 0

(5.15)


168


In this equation, Cq and Cs are the constraint Jacobian matrices resulting from the differentiation of the constraint equations with respect to the vectors q and s, respectively. It follows from the preceding equation that

(

)

λ T Cqδ q + C sδ s = 0

(5.16)

for any vector λ . The vector λ is called the vector of Lagrange multipliers. Adding Equations 5.13 and 5.16, one obtains

(

)

+ CTq λ − Q + δ s T CTs λ = 0 δ qT Mq

(5.17)

It is important to note from this equation that there are no inertia and generalized forces associated with the surface parameters used to describe the wheel/rail surface geometry, and for this reason, these parameters are treated as nongeneralized coordinates. Differentiating Equation 5.14 twice with respect to time, the constraint equations at the acceleration level can be written as + C ss = Q d Cq q

(5.18)

where Qd is a quadratic velocity vector resulting from the differentiation of the constraint equations twice with respect to time. Equations 5.17 and 5.18 are the basis for the two constraint formulations that will be discussed in the following two sections.

5.3 AUGMENTED CONSTRAINT CONTACT FORMULATION (ACCF) In the constraint contact formulations, the system differential equations of motion and the wheel/rail algebraic contact constraint equations are solved simultaneously for the system generalized coordinates and the system surface parameters in order to correctly account for the kinematic and dynamic couplings between the wheel/rail generalized coordinates and the nongeneralized surface parameters. One can combine the vector of generalized and nongeneralized coordinates in one vector, which can be written as follows: p = q T

s T 

T

(5.19)

In terms of this vector, Equation 5.17 can be written as follows:

(

)

δ pT H + CTp λ = 0

(5.20)



169

In this equation, − Q  Mq H= ,  0 

C p = C q

C s 

(5.21)

Because of the algebraic kinematic constraints, the elements of the vector p are not independent. The number of dependent coordinates is equal to the number of kinematic constraint equations, nc. If the total number of coordinates including the nongeneralized surface parameters is n, the number of independent coordinates (degrees of freedom) is nd = n – nc. One can then select a set of independent coordinates pi, which may include nongeneralized surface parameters, and write the total vector of coordinates p as follows: p = p Ti

p Td 

T

(5.22)

In this equation, pd is the vector of dependent coordinates that has dimension nc. Using this coordinate partitioning, Equation 5.20 can be written as δ pTi 

 Hi + CTp λ  i =0 δ pTd   T  H d + C pd λ 

(5.23)

In this equation, subscripts i and d refer to vectors and matrices associated with independent and dependent coordinates, respectively, and CPi and CPd are the constraint Jacobian matrices associated with the independent and dependent coordinates, respectively. Since the number of constraint equations that are assumed to be linearly independent is equal to the number of dependent coordinates, the dependent coordinates can be selected such that the constraint Jacobian matrix CPd is a square nonsingular matrix. The preceding equation can also be written in the following form:

(

)

(

)

δ pTi Hi + CTpi λ + δ pTd H d + CTpd λ = 0

(5.24)

Using the fact that the constraint Jacobian matrix CPd is a square nonsingular matrix, one can choose to define the arbitrary vector λ as the solution of the following system of algebraic equations: H d + CTpd λ = 0

(5.25)

Furthermore, since the elements of the vector pi are assumed to be independent, the first part of Equation 5.24 leads to Hi + CTpi λ = 0

(5.26)


170


Combining the preceding two equations, one obtains H + CTp λ = 0

(5.27)

Using the definitions of the matrices and vectors in this equation, which are given previously in this section, it is clear that Equation 5.27 can be written as M  0 C  q

0 0 Cs

CqT   q   Q     T   C s   s  =  0  0   λ  Qd  

(5.28)

The solution of this equation contains not only the generalized accelerations and Lagrange multipliers, but also the second time derivatives of the nongeneralized surface parameters. These augmented equations of motion are solved for the generalized and nongeneralized accelerations and Lagrange multipliers. Equation 5.28 ensures that the generalized and nongeneralized accelerations automatically satisfy the contact constraints at the acceleration level. Having obtained the acceleration vectors, the independent accelerations, which may include the generalized and nongeneralized variables, can be identified using the constraint Jacobian matrix. These independent accelerations can be integrated forward in time to determine the independent coordinates and velocities. The dependent generalized and nongeneralized coordinates can be determined by solving the constraint equations at the position level using an iterative Newton-Raphson solution procedure. The constraint equations at the velocity level can be solved to determine the dependent generalized and nongeneralized velocities. A more detailed discussion on the numerical algorithm used with the augmented constraint contact formulation is presented in Chapter 6. It is important to emphasize again that in the ACCF method, generalized and nongeneralized coordinates (surface parameters) can be selected as the independent variables. The state equations associated with these independent variables are identified and integrated forward in time. This is one of the important differences between the ACCF method and the ECCF method, discussed in the following section, where the surface parameters are systematically eliminated and thus cannot be selected as independent coordinates or degrees of freedom in the ECCF method. It is clear that the solution of Equation 5.28 must satisfy the following equation: CTs λ = 0

(5.29)

We also note that only the contact constraints or specified motion trajectories can be functions of the nongeneralized surface parameters; other joint constraints do not depend on these geometric parameters. Each contact introduces five independent contact constraints and five Lagrange multipliers associated with these constraints.



171

These five constraint equations and five Lagrange multipliers must satisfy the preceding equation. Since there are only four surface parameters, the preceding equation for each contact represents a system of four scalar equations in five unknowns. If the constraint equations are linearly independent, the rank of the constraint Jacobian matrix associated with the surface parameters for each contact will be four. Therefore, Equation 5.29, when applied to each contact, implies that there is only one independent Lagrange multiplier; that is, the five contact constraints, which require introducing four nongeneralized surface parameters and eliminate only one degree of freedom, can be used to determine only one independent constraint (reaction) force. This force can be used to determine the normal contact force that enters in the formulation of the tangential creep forces.

5.4 EMBEDDED CONSTRAINT CONTACT FORMULATION (ECCF) In the embedded constraint contact formulation (ECCF), the surface parameters (nongeneralized coordinates) are systematically eliminated from the equations of motion. This leads to a smaller system of dynamic equations of motion, from which the contact constraint forces (except for the normal contact force) are eliminated. Before providing the details of this elimination, further discussion of the relationship between the dependent coordinates and the constraint forces is needed. To this end, Equation 5.29 for a contact k is written as follows: C ks λ k = 0 T

(5.30)

where T

s k =  s1wk s2wk s1rk s2rk  ,

λ k =  λ1k λ2k λ3k λ4k λ5k 

T

(5.31)

As previously pointed out, the five Lagrange multipliers associated with each contact are not totally independent. Because of the dimension and rank of the T coefficient matrix C ks in Equation 5.30, only one Lagrange multiplier is independent, as discussed in the preceding section. This result is consistent with the fact that the contact constraints eliminate one degree of freedom, and the normal contact force can be expressed in terms of one Lagrange multiplier only. In the embedded constraint contact formulation discussed in this section, the final form of the dynamic equations of motion includes only one Lagrange multiplier associated with each wheel/rail contact. To arrive at this form, the four surface parameters associated with each contact are treated as dependent coordinates, and these geometric parameters do not explicitly appear in the final form of the equations of motion, as is the case in the augmented constraint contact formulation.


172


5.4.1 POSITION ANALYSIS To systematically eliminate the four surface parameters associated with each contact, four contact constraint equations are selected such that their Jacobian matrix associated with the four surface parameters is a nonsingular square matrix. Since the degree of freedom eliminated by the contact constraints can be (without loss of generality) the relative motion along the normal to the surface at the contact point, the four constraint equations are chosen from Equation 5.10 by excluding the third equation that defines the relative motion along the normal to the tangent surface at the contact point. This third equation will be added to the system differential equations of motion in order to determine the normal constraint contact force. To this end, the vector of constraint equations associated with contact k is written in the following partitioned form: T C k (q w , qr , s wk , s rk ) = C dk 

T C nk  

T

(5.32)

where k

 t1r ⋅ (rPw − rPr )   r  t 2 ⋅ (rPw − rPr )  dk w r wk rk  C (q , q , s , s ) = =0  t w ⋅ nr  1   w r  t 2 ⋅ n 

(5.33)

C nk (q w , q r , s wk , s rk ) = n rk ⋅ (rPwk − rPrk ) = 0

(5.34)

Assuming that the Jacobian matrix associated with the constraints of Equation 5.33 is nonsingular, these four constraint equations can be used to express the four surface parameters in terms of the generalized coordinates of the wheel and the rail. However, since the wheel/rail generalized coordinates must satisfy the fifth contact constraint (Equation 5.34) as well as other kinematic constraints imposed on the motion of the multibody vehicle system, the position analysis involves, at the position level, two coupled solution stages, each of which requires the use of the iterative Newton-Raphson procedure. For a given set of wheel/rail generalized coordinates, Equation 5.33 is solved iteratively using a Newton-Raphson algorithm to determine the surface parameters that enter into the formulations of Equation 5.34. Using these calculated surface parameters, Equation 5.34 and other nonlinear kinematic constraints imposed on the motion of the multibody system are also iteratively solved using a Newton-Raphson algorithm to determine the system generalized coordinates. This iterative process continues until convergence is achieved for both stages. This is discussed in greater detail in Chapter 6.



5.4.2 EQUATIONS

OF

173

MOTION

The four contact constraints of Equation 5.33 do not prevent the penetration along the normal to the surfaces at the contact point. Consequently, the fifth constraint of Equation 5.34 must be used to ensure that such a penetration does not occur. The embedded form of the dynamic equations of motion are developed by substituting Equation 5.32 into Equation 5.17 to obtain the following variational equation of motion:

(

T

T

)

(

T

T

)

+ Cqd λ d + Cqn λ n − Q + δ s T C ds λ d + C ns λ n = 0 δ qT Mq

(5.35)

where C ds and C ns are the Jacobian matrices associated with the contact constraints of the types presented in Equations 5.33 and 5.34, respectively; and λd and λn are the Lagrange multipliers associated with these constraint equations. Note that the virtual change in the system surface parameters can be expressed in terms of the virtual change in the wheel/rail coordinates using Equation 5.33, as follows:

δ s = Bδ q

(5.36)

where B = −(C ds )−1Cqd is the velocity transformation matrix associated with the contact constraints. Substituting Equation 5.36 into Equation 5.35, the principle of virtual work for this system can then be written in terms of the virtual changes in the generalized coordinates only as

(

)

T   + Cqn + C ns B λ n − Q = 0 δ qT  Mq  

(5.37)

It is important to note in Equation 5.37 that the generalized contact forces associated with the contact constraints of Equation 5.33 are systematically eliminated from the equations of motion using the following identity:

( )

T T T  Cqd λ d + BT C sd λ d = Cqd λ d +  − C sd 

−1

T

T  Cqd  C sd λ d = 0 

(5.38)

Furthermore, the acceleration kinematic equation s = Bq + Bq

(5.39)

which is the result of Equation 5.36, can be used to eliminate s from the second derivative of the constraint of Equation 5.34 with respect to time. Following a procedure that can be considered as a special case of the procedure used in the


174


preceding section to derive the augmented constraint contact formulation, one can show by using Equations 5.37 and 5.39 that the system equations of motion, after eliminating the nongeneralized surface parameters, can be written as M  J

  Q  JT   q   =   0   λ  Qdn 

(5.40)

where J, an implicit function of s, is the Jacobian matrix of the constraints of Equation 5.34 and other constraints imposed on the motion of the multibody system; λ is the vector of the system Lagrange multipliers, which include only Lagrange multipliers associated with the constraints of Equation 5.34 and other noncontact constraints imposed on the motion of the system; and Qdn is a quadratic velocity vector that results from the differentiation of the constraint equations twice with respect to time. This vector is a function of the surface parameters s and their first time derivatives. It is clear from the analysis presented in this section that, in the embedded constraint contact formulations (ECCF), the surface parameters are treated as dependent variables and are systematically eliminated from the equations of motion. For this reason, only wheel/rail accelerations are integrated forward in time. It should also be noted that only one Lagrange multiplier associated with the contact constraint of Equation 5.34 is used in Equation 5.40. This Lagrange multiplier defines the normal contact force used to define the longitudinal, lateral, and spin creep forces. Furthermore, since the surface parameters and the dependent Lagrange multipliers are systematically eliminated without making any assumptions, the embedded constraint contact formulation (ECCF), as defined by Equation 5.40, and the augmented constraint contact formulation (ACCF), as defined by Equation 5.28, are in principle equivalent, despite the fact that these two formulations require the use of different solution procedures and numerical algorithms, as will be discussed in Chapter 6.

5.5 ELASTIC CONTACT FORMULATION-ALGEBRAIC EQUATIONS (ECF-A) The two constraint contact formulations discussed in the preceding two sections do not allow penetration or separation between the two bodies in contact. Imposing the contact conditions eliminates one degree of relative motion between the wheel and the rail. In the elastic approach, on the other hand, no kinematic contact constraints are imposed; the wheel has six degrees of freedom with respect to the rail; and small penetrations at the contact points are allowed. In the elastic contact formulations, a compliant force element that consists of stiffness and damping forces is used to determine the normal contact force. The location of the points of contact can be determined using look-up tables, or they can be determined on-line using a discrete nodal search or by solving a set of algebraic equations. In this and the following section, two elastic contact formulations are discussed. In the first, called ECF-A, the locations of the contact points are determined by solving a set of algebraic equations. In the second elastic contact method, called ECF-N, the locations of the contact points are determined using a nodal search.



175

In the first elastic method discussed in this section, the location of the contact points is determined by first solving a set of algebraic equations. For each contact, four algebraic equations (Equation 5.33) are solved to determine the four parameters that describe the geometry of the wheel and the rail surfaces. After determining the four surface parameters that satisfy the algebraic nonlinear equations, the distance between the surfaces along the normal is evaluated using a fifth equation (Equation 5.34) to determine whether or not there is a contact. The method presented in this section allows for the definition of the wheel and rail surfaces using spline function representations, thereby enabling the use of measured profile data for the wheel and the rail. It is important to point out that, while the algebraic equations used in this method are the same as the contact constraints presented in the preceding sections, the method presented in this section (ECF-A) cannot be considered as a constraint contact formulation because the algebraic equations: (a) are not imposed at the velocity and acceleration levels, (b) allow for penetrations and separations, and (c) do not introduce a Lagrange multiplier, as is used in the constraint method, to determine the normal contact force. In the elastic methods discussed in this section, the normal contact force is determined using a compliant force model that has stiffness and damping coefficients. To determine the location of the contact point using the elastic contact approach, one can define the following four algebraic equations (Escalona, 2002; Pombo and Ambrosio, 2003; Shabana et al., 2005): t1r ⋅ r wr = 0   t r2 ⋅ r wr = 0   t1w ⋅ n r = 0   t 2w ⋅ n r = 0 

(5.41)

where t1i and t i2 (i = w, r) are, respectively, the tangents to the wheel and rail surfaces at the potential contact point; rwr = rw − rr is the vector that defines the relative position of the point on the wheel with respect to the point on the rail; and nr is the normal to the rail surface. Note that the first two equations in the preceding equation are the same as the first two equations in Equation 5.10, while the last two equations in the preceding equation are the same as the last two equations in Equation 5.10. Because the tangent and normal vectors are functions of the surface parameters, and assuming that the generalized coordinates of the wheel and the rail are known, one can rewrite the preceding set of algebraic equations in a vector form as E ( s) = 0

(5.42)

where E is the vector of nonlinear algebraic equations that can be solved using an iterative Newton-Raphson algorithm for the surface parameters that define potential


176


nonconformal contact points. This requires evaluating the Jacobian matrix of the algebraic equations and iteratively solving the following system for each contact to determine Newton differences associated with the surface parameters:  r w  t1 ⋅ t1    t r2 ⋅ t1w   w  ∂t1 ⋅ n r  ∂s1w  w  ∂t 2 ⋅ n r  ∂s w  1

t1r ⋅ t 2w

∂t1r wr r r ⋅ r − t1 ⋅ t1 ∂s1r

t r2 ⋅ t 2w

∂t r2 wr r r ⋅ r − t 2 ⋅ t1 ∂s1r

∂t1w r ⋅n ∂s2w

∂n r w ⋅ t1 ∂s1r

∂t 2w r ⋅n ∂s2w

∂n r w ⋅ t2 ∂s1r

∂t1r wr r r  ⋅ r – t1 ⋅ t 2  ∂s2r   t1r ⋅ r wr  ∂t r2 wr r r  s1w  ⋅ r – t2 ⋅ t2   w   r wr  r ∂s2   s2  t2 ⋅ r  − =   s r  r  t w ⋅ nr  ∂n w 1   1w r    ⋅ t 1   s2r  ∂s2r  t 2 ⋅ n   r ∂n w  (5.43) ⋅ t2  ∂s2r 

In this equation, s1w , s2w , s1r , and s2r are the Newton differences. Convergence is achieved when the norm of the violation of the algebraic equations or the norm of the Newton differences is less than a specified tolerance. Having determined the vector of the surface parameters, the penetration can be calculated using the third equation of Equation 5.10 as

δ = r wr ⋅ n r

(5.44)

If the surfaces penetrate, normal contact forces can be calculated using Hertz's contact theory, while the creep forces can be calculated using one of the creep-force models discussed in Chapter 4. The generalized normal and creep forces associated with the system generalized coordinates are determined and introduced to the multibody system dynamic equations of motion as generalized external forces. In the evaluation of the normal contact force, as an alternative to the use of the Hertzian component that is a function of the indentation, a damping force proportional to the time derivative of indentation can also be included. An expression of the normal force that can be used is given by (Shabana et al., 2004) F = Fh + Fd = − K hδ 3 2 − Cδ δ

(5.45)

where δ is the indentation, Fh is the Hertzian (elastic) contact force, Fd is the damping force, Kh is the Hertzian constant that depends on the surface curvatures and the elastic properties, and C is a damping constant. The velocity of indentation δ is evaluated as the dot product of the relative velocity vector between the contact points on the wheel and rail and the normal vector to the surface at the contact point. The reason for including the factor δ in the damping force is to guarantee that the contact force is zero when the indentation is zero.



177

5.6 ELASTIC CONTACT FORMULATION-NODAL SEARCH (ECF-N) As an alternative for using the ECF-A method that employs algebraic equations to define the location of the contact points, one can define the profile of the wheel and rail using discrete nodal points (Shabana et al., 2004, 2005). The distance between these nodes can be calculated to determine the points on the wheel and the rail that may come into contact. The use of this method has the advantage that it does not require a certain degree of smoothness of the surfaces. It has, however, the disadvantage that the change in the lateral surface parameters of the wheel and rail is not smooth, since the contact is assumed to occur at discrete nodal points. When the contact jumps from one node to the neighboring one, a small jump in the relative velocity between the wheel and the rail at the contact point is expected. This small jump leads, in some examples, to discontinuity of the creepages. Since the creepage coefficients that enter into the calculation of the creepage forces are very high, the discontinuous change in the contact locations resulting from the use of the nodal search leads to high impulsive forces. This problem, however, can be solved by using interpolation to determine the contact points instead of using discrete points. As discussed in the literature (Shabana et al., 2004), the nodal search for the contact points consists of the following three steps: 1. Calculation of the rail arc length s1r traveled by the wheel. The parameter s1r defines the rail cross-section in which the points of contact are located. 2. Calculation of the wheel angular parameter s2w . The parameter s2w defines the wheel diametric section in which the points of contact are located. 3. Search for the contact points. In this phase, the rail parameter s2r and the wheel parameter s1w of the points of contact are determined. This phase of the search starts once the sections of the wheel and rail in which the contact points are located have been determined. The exact position of the contact points is determined in this phase of the search. To determine the arc length traveled by the wheel, a selected point Q on the center of the wheel, as shown in Figure 5.4, is used first to determine the rail space curve parameter s1r . It is assumed that the rate of change of the rail parameter s1r is equal to the projection of the velocity of this point on the tangent along the longitudinal rail direction, that is, T

s1r = rQw ⋅ t1r

(5.46)

where rQw is the global velocity vector of point Q, and t1r is the longitudinal tangent to the rail. The preceding differential equation is solved simultaneously with the differential and algebraic equations of the multibody system to determine s1r , which is used for the search for the point of contact between the wheel and the rail. Clearly, one needs to introduce a number of arc length first-order differential equations


178


(Equation 5.46) equal to the number of wheels in the dynamic model. Note that the ECF-A method does not require introducing these first-order differential equations, since the arc length traveled by the wheel is readily available from the solution of the algebraic equations (Equation 5.42). To determine the points of contact between the wheel and the rail, the global position vectors of the nodal points that define the wheel and rail profiles must be determined (Shabana et al., 2004). The distance between the points on the wheel and the points on the rails can be calculated and used with a user-specified tolerance criterion to determine the points of contact. Since this search can lead to a large number of contact points, an optimized procedure that improves the computational efficiency can be adopted. The contact points can be grouped in batches. A batch is a collection of a sequence of pairs of points on the wheel and rail that have penetration. While the algorithm can allow for an arbitrary number of contact batches, a limit on the number of contact batches can be imposed in the numerical implementation. The two points (one on the wheel and one on the rail) that lead to the maximum indentation can be selected as the points of contact for any given batch. The number of points of contact between the wheel and the rail is equal to the number of the contact batches. That is, the algorithm used to search for the contact points can be designed to allow for multiple contacts between the wheel and the rail. Once the contact points are determined, an elastic force model similar to the one presented for the ECF-A method can be used. Before concluding this section, it is important to mention that Equation 5.46 is not the only method that can be used to determine the rail longitudinal surface parameter s1r . Another possible method is to use the wheel absolute Cartesian coordinates and solve a system of nonlinear algebraic equations instead of differential equations to determine s1r . The use of a simple first-order differential equation, however, can be more efficient as compared with the solution of nonlinear algebraic equations, which can be prone to convergence problems.

5.7 COMPARISON OF DIFFERENT CONTACT FORMULATIONS As discussed previously in this chapter, there are two different approaches that are commonly used to solve the multibody contact problems: the constraint and elastic contact approaches. These two approaches lead to different mathematical models for determining the normal contact force. In the constraint approach, the nonconformal contact conditions (Equations 5.9 or 5.10) are imposed on the motion of the system, and the normal contact force is predicted as a constraint force obtained using the technique of Lagrange multipliers associated with the contact constraints (Equations 5.28 and 5.40). In this case, no wheel/rail separations or penetrations are allowed. The study of the wheel/rail separation scenarios using the constraint approach requires a special mathematical treatment, since in this case one must deal with a system with variable kinematic structure. This may require the use of a generalized impulse-momentum approach, as discussed in the literature (Khulief and Shabana, 1986).



179

On the other hand, in the elastic approach, no contact constraint conditions are imposed. The normal contact force is defined using a compliant force model, which is a function of the indentation between the two contact surfaces (Equation 5.45) and the assumed stiffness and damping coefficients. In other words, the normal contact force in the elastic approach is defined as an external force by allowing small elastic deformation of the contact surfaces. Using this approach, the wheel/rail separation scenarios can be easily examined, since the system degrees of freedom do not change due to contact or separation. For this reason, the elastic approach can be used effectively in modeling many simulation scenarios such as derailment and flange contact. It is well known, however, that the use of the compliant contact force model in the elastic approach introduces high frequencies in the solution of the system equations of motion. This problem is not generally encountered when the constraint formulations are used. Using the constraint approach, different contact formulations that require the use of different numerical solution procedures can be developed. In this chapter, two general multibody system constraint contact formulations are discussed. In the augmented constraint contact formulation (ACCF), the equations of motion are expressed explicitly in terms of the nongeneralized surface parameters. The second time derivatives of the nongeneralized surface parameters can be chosen as independent accelerations that can be integrated forward in time using direct numerical integration methods. In the embedded constraint contact formulation (ECCF), on the other hand, the nongeneralized surface parameters are systematically eliminated from the equations of motion using the velocity transformation given by Equation 5.36. The surface parameters in the ECCF are always considered as dependent coordinates that are determined by solving Equation 5.33. Two elastic contact formulations are also discussed in this chapter. The main difference between the two formulations is the method used to determine the location of the contact points. The form of the equations of motion remains the same in both formulations. In the first elastic method (ECF-A), algebraic equations are solved using an iterative Newton-Raphson procedure to determine the locations of the contact points. These algebraic equations are not imposed as constraints at the position, velocity, or acceleration levels; no Lagrange multipliers are introduced; and no degrees of freedom are eliminated. As demonstrated in this chapter, this method requires a certain degree of smoothness in the definition of the wheel and rail surfaces. This degree of smoothness is not required when using the second elastic formulation, ECF-N, which is based on nodal search techniques. The ECF-A method, however, leads to smoother solution as compared with the ECF-N, which employs discrete wheel/rail profile points. Table 5.1 provides a comparison of the four contact formulations discussed in this chapter. Computational algorithms based on the four contact formulations are discussed in more detail in Chapter 6, where examples are also presented to compare the results obtained using different formulations.

5.8 PLANAR CONTACT In three-dimensional problems, the location of the points of contact between the wheel and the rail can be determined by solving a two-dimensional contact problem.

Elastic

Constraint Embedded constraint contact formulation (ECCF) Elastic contact formulation-algebraic equations (ECF-A) Elastic contact formulation-nodal search (ECF-N)

Method Augmented constraint contact formulation (ACCF)

Wheel Separation

Requires special mathematical treatment

Yes

Normal Contact Force

Lagrange multiplier (constraint force)

Compliant force

No restriction

Smooth

E(s) = 0

Nodal search

Smooth

Contact Surface Smooth

C(q,s) = 0 (two stages)

Contact Search C(q,s) = 0

M   J

M  0 C  q

CTq   q   Q      CTs   s  =  0  0   λ  Qd  

= Q Mq

  Q  JT   q   =   0   λ  Qdn 

0 Cs

0

Equations of Motion

180

Approaches

TABLE 5.1 Contact Formulations





181

This is the case of planar contact analysis, in which the kinematic coupling between some of the geometric surface parameters is neglected. The use of this planar contact method leads to approximate prediction of the locations of the contact point, as will be later explained in this section. In the planar contact formulation, the contact conditions are formulated in terms of only two surface parameters that define the wheel and rail profiles. The other two surface parameters, the rail arc length and the rotational wheel parameter, are assumed to be known. If a constraint contact formulation is used in this case of planar contact, one needs only three contact constraint equations to determine the two unknown profile surface parameters and also eliminate one degree of freedom. The same three algebraic equations used as constraints in the constraint contact formulation can also be used as the basis for developing an elastic contact formulation by following the procedure described in Section 5.5. In this section, the formulation of the planar contact conditions is discussed.

5.8.1 INTERMEDIATE WHEEL COORDINATE SYSTEM As discussed in Chapter 2, two different sets of coordinates are often used to formulate the kinematic and dynamic equations of the railroad vehicle systems: absolute Cartesian coordinates and trajectory coordinates. Recall that the vector of trajectory coordinates is given for the wheel as p w = [s w

y wr

z wr

ψ wr

φ wr

θ wr ]T

(5.47)

where sw is the rail arc length that defines the location of the origin of the wheel trajectory coordinate system introduced in Chapter 2; ywr and zwr define the relative position of the wheel center of mass with respect to the wheel trajectory coordinate system; and ψ wr, φ wr, and θ wr are the Euler angles that define the orientation of the wheel with respect to the trajectory coordinate system. The sequence of rotations used for Euler angles is Zw, Xw, Yw, as discussed in Chapter 2. If the equations of motion of the railroad vehicle system are formulated using the trajectory coordinates, all the elements of the vector of Equation 5.47 are known at a given instant of time. On the other hand, if the equations are formulated in terms of the absolute Cartesian coordinates, one can always develop the relationships between the absolute coordinates and the trajectory coordinates and use these relationships to determine the elements of the vector pw of Equation 5.47. These relationships are discussed in greater detail in Chapter 7. Therefore, in this section, it is assumed that the elements of the vector pw of Equation 5.47 can be determined regardless of the set of coordinates used in the formulation of the dynamic equations of the railroad vehicle system. In the planar contact formulation, one introduces an intermediate wheel coordinate system that does not experience the pitch rotation θ wr of the wheel about its Yw axis. The orientation of this intermediate wheel coordinate system can be defined in the wheel trajectory coordinate system using the two Euler angles ψ wr and φ wr. The transformation matrix that defines the orientation of the intermediate wheel coordinate system in the global coordinate system can then be written as follows:


182


A wi = Atw A z A x =  a1wi

a 2wi

a 3wi 

(5.48)

where Atw is the transformation matrix that defines the orientation of the wheel trajectory coordinate system introduced in Chapter 2 with respect to the global coordinate system, and  cos ψ wr  A z =  sin ψ wr  0 

− sin ψ wr cos ψ wr 0

1 0   0  , Ax = 0 0 1  

0

  − sin φ  cos φ wr  0

cos φ sin φ wr

wr

wr

(5.49)

It follows that the transformation matrix that defines the orientation of the intermediate wheel coordinate system is given by  cos ψ wr  A = A  sin ψ wr  0  wi

tw

− sin ψ wr cos φ wr cos ψ wr cos φ wr sin φ wr

sin ψ wr sin φ wr   − cos ψ wr sin φ wr   cos φ wr 

(5.50)

At this point, it is important to point out that, if the absolute Cartesian coordinates are used, one can simply obtain the transformation matrix Awi using one simple matrix multiplication. Let Aw be the transformation matrix that defines the orientation of the wheel coordinate system in the global coordinate system. The matrix Awi can then be obtained from Aw as A wi = A w A Ty

(5.51)

where the matrix Ay accounts for the pitch rotation about the wheel Y w axis and is given by  cos θ wr  Ay =  0  − sin θ wr 

0 1 0

sin θ wr   0  cos θ wr 

(5.52)

The columns of the transformation matrix Awi of Equations 5.50 or 5.51 define unit vectors along the axes of the intermediate wheel coordinate system.

5.8.2 DISTANCE TRAVELED It is important to differentiate between the coordinate sw that defines the position of the origin of the wheel trajectory coordinate system and the arc length s1r of the rail space curve at which the contact occurs. In some railroad vehicle formulations, track



183

coordinate systems that travel with a constant velocity are introduced to define the configuration of the vehicle components. The distance traveled by the wheel s1r is determined as the product of this constant velocity and time. In addition to the unnecessary complications that arise from using such a motion description, the use of a track coordinate system that travels with a constant velocity has a major disadvantage when simulations of braking are considered (Handoko, 2006). Since the rails can have deviations, it is important to use an accurate method that is independent of s w or the component forward velocity to determine the coordinate s1r . In this section, the discussion focuses on two planar contact methods that can be used to predict the surface parameter s1r at which the contact occurs. In the first method, the kinematic coupling between the parameter s1r and the two parameters s2r and s1w is neglected. In the second method, which is briefly discussed before concluding this section, the kinematic coupling between the parameter s1r and the two parameters s2r and s1w is considered. This second planar contact method is more accurate, since the three parameters s1r , s2r , and s1w are determined simultaneously by solving a set of nonlinear algebraic equations. Recall that the location of an arbitrary point on the rail space curve can be defined in the global coordinate system as

( )

r0r = R r + A r u r0 s1r

(5.53)

In this equation, s1r is the arc length of the rail space curve, Rr is the global position vector of the rail body coordinate system, and u r0 (s1r ) is the vector that defines the position of the arbitrary point on the rail space curve with respect to the rail body coordinate system, as shown in Figure 5.5. To determine the arc length s1r at which

FIGURE 5.5 Planer contact condition.


184


the vector r0r of the preceding equation has a zero component along the Xwi axis of the wheel, one must solve the following equation for the arc length s1r : T

(

)

a1wi ror − R w = 0

(5.54)

In this equation, Rw is the vector that defines the global position vector of the origin of the wheel body coordinate system, and a1wi is a unit vector along the Xwi axis of the intermediate wheel coordinate system. The vector a1wi is the first column of the transformation matrix Awi defined by Equation 5.50 or, alternatively, by Equation 5.51. Using Equation 5.53, the preceding equation can be written as a1wi

T

(R

r

)

+ Ar u r0 − R w = 0

(5.55)

In the case of arbitrary curved track, this equation is a nonlinear function of the rail arc length s1r . Assuming that the wheel and the rail generalized coordinates are known, the preceding equation can be solved iteratively using a Newton-Raphson procedure. In this procedure, the following algebraic equation must be iteratively solved:

(a

wi T 1

)

Ar t1r ∆s1r = − a1wi

T

(R

r

+ Ar u ro − R w

)

(5.56)

where ∆s1r is the Newton difference, and t1r = ∂u ro ∂s1r is the longitudinal tangent of the rail space curve defined in the rail body coordinate system. The preceding scalar equation can be solved for ∆s1r as follows:

{

 a1wi T R r + Ar u ro − R w ∆s1r = −  T a1wi Ar t1r 

}  

(5.57)

Convergence is achieved if the Newton difference is smaller than a specified tolerance or if Equation 5.55 is satisfied. Note that the procedure described in this section for determining s1r is an alternative for using Equation 5.46, which requires introducing a first-order differential equation.

5.8.3 PROFILE PARAMETERS Knowing the coordinate s1r , the planar contact conditions can be imposed to determine the location of the point of contact between the wheel and the rail. It is assumed that the profile of the wheel is defined using the parameter s1w , while the profile of the rail is defined using the parameter s2r . Clearly, when the wheel comes into contact with the rail, one degree of freedom is eliminated. By introducing the intermediate wheel coordinate system, one can impose only three constraint equations, which can be used to eliminate one degree of freedom and determine the profile surface



185

parameters s1w and s2r . The first two conditions ensure that the distance between the wheel and the rail at the contact point in the plane Y wiZ wi is equal to zero. The third condition ensures that the normal to the rail at the contact point is perpendicular to the tangent to the wheel at that point. This third condition is required when the assumption of nonconformal contact is used. These three contact conditions can be written as follows:

( (

) )

t r2 R r + Ar u r − R w − Awi u wi = 0    rT r r r w wi wi n R + A u − R − A u = 0  rT w  n t1 = 0  T

(5.58)

In this equation, t r2 and nr are the lateral tangent and normal to the rail profile at the contact point, respectively; and t1w is the tangent to the wheel at the contact point. If the generalized coordinates of the wheel and rail are given and s1r is assumed to be known, u r in the preceding equation is a function of s2r only while u wi is a function of s1w only. Therefore, two of the nonlinear equations in the preceding equation can be used to determine the profile parameters s1w and s2r , while the remaining equation can be used to either eliminate one degree of freedom by expressing one coordinate in terms of the other wheel coordinates or to introduce a compliant force model to describe the wheel/rail interaction, as described in Section 5.5. The first and third equations in Equation 5.58 are the two equations that are often used to solve for the profile surface parameters s1w and s2r , while the second equation is the one that is used to eliminate the degree of freedom that represents the relative motion along the normal to the rail at the contact point. This second equation can alternatively be used to define a penetration that enters into the formulation of a compliant force model in an elastic contact approach. As previously pointed out, the constraint approach requires imposing the constraints at the position, velocity, and acceleration levels, and as a result, there are Lagrange multipliers associated with these constraints when the augmented formulation is used. An embedded constraint contact formulation can also be used in the case of planar contact by systematically eliminating the surface parameters, as described in Section 5.4.

5.8.4 COUPLING

BETWEEN THE

SURFACE PARAMETERS

It is clear from the analysis presented thus far in this section that, when the planar contact is used, the search for the two profile surface parameters is independent of the other two parameters s2w and s1r . In the planar contact analysis, the parameter s2w is assumed to be equal to the pitch angle θ wr, while the parameter s1r can be determined using an independent search, as described in this section. These two parameters s2w and s1r are assumed to be fixed while searching for the profile parameters. This approach is different from the three-dimensional contact analysis in which the four surface parameters are solved for simultaneously using nonlinear coupled algebraic equations, as previously described in this chapter. It is important to point out


186


that the planar contact method described here does not guarantee that the two points on the wheel and the rail coincide; that is, these two designated contact points do not have the same global position vector. These two points can differ by a longitudinal shift. To ensure that there is no longitudinal shift between the two points on the wheel and the rail, Equation 5.55 must be replaced by the following equation: t1r

T

(R

r

)

+ Ar u r − R w − Awi u wi = 0

(5.59)

In this equation, t1r is the longitudinal tangent at the potential contact point. Therefore, to guarantee that there is no longitudinal shift, the following three equations must be solved simultaneously to determine s1w , s1r , and s2r : t1r

T

t r2

T

(R + A u − R (R + A u − R r

r

r

r

r

r

T

n r t1w = 0

) )

− Awi u wi = 0    w − Awi u wi = 0    

w

(5.60)

The penetration can be calculated using the following equation:

δ = nr

T

(R

r

+ Ar u r − R w − Awi u wi

)

(5.61)

This modified planar contact formulation, which is a special case of the threedimensional formulation given by Equation 5.41, ensures that — in the case of a yaw angle, variable vehicle velocity, or in the case of a curved track — there is no longitudinal shift between the two points of contact on the wheel and the rail. If a contact constraint formulation is used by replacing Equation 5.61 with a constraint equation, the two contact points coincide, and both will have the same global position vector.


6

Implementation and Special Elements

The preceding chapter presented several three-dimensional formulations for solving the wheel/rail contact problem. These formulations, as demonstrated in this chapter, can be implemented in general multibody system computer algorithms that are designed to solve a system of differential and algebraic equations. In railroad dynamics, the multibody vehicle model can have an arbitrary number of bodies whose motions are subjected to kinematic constraints in addition to the wheel/rail contact elements. Therefore, it is important to discuss the general computational multibody algorithms before discussing the implementation of the special wheel/rail contact elements in these algorithms. Consequently, the first section of this chapter focuses on the structure of the multibody system computational algorithms and the numerical procedures that can be used to develop a general-purpose multibody system computer code. Sections 6.2 and 6.3 describe the implementation of the wheel/rail contact elements in multibody system algorithms. This implementation is based on the constraint and elastic contact formulations presented in Chapter 5 for determining the wheel/rail contact locations and forces. Section 6.4 outlines the procedure used to calculate the creep forces, and Section 6.5 explains the need for the use of higher derivatives with respect to the surface parameters in the contact formulations. Sections 6.6 and 6.7 explain the structure of the track preprocessor based on the geometric description presented in Chapter 3. Section 6.7 focuses on the computer description of rail irregularities, such as deviations, as well as the use of measured track data. Section 6.8 develops some force elements that are frequently used in railroad vehicle dynamic models. The use of some of these elements is not limited to railroad vehicle system applications; indeed, such force elements can also be found in other multibody system applications. Magnetically levitated vehicles (Maglev), which are supported by magnetic forces, are another example of guided trains. Maglev vehicles are still in an experimental stage, but they are in service in some countries that have the potential for developing very high speed trains. Section 6.9 gives a brief introduction to the Maglev systems. One of the important problems encountered in the computer-aided analysis of large-scale railroad vehicle models is the accurate determination of the initial static equilibrium configuration. For this reason, computational approaches that can be used to determine the initial static equilibrium configuration of railroad vehicle models are discussed in Section 6.10. This chapter concludes with numerical examples of railroad systems that demonstrate the computer implementation and the use of the multibody wheel/rail contact formulations discussed in Chapter 5.

187


188


6.1 GENERAL MULTIBODY SYSTEM ALGORITHMS Multibody system computational algorithms are designed to solve the differential and algebraic equations (DAE) that govern the constrained motion of physical systems. The capability of solving differential and algebraic equations is one of the main features that distinguish the general computational multibody system algorithms from other algorithms that are used to develop special-purpose computer codes. This section reviews the basic equations used in the multibody system formulations and explains the use of these equations to develop a computational algorithm for solving the differential and algebraic equations of the multibody system.

6.1.1 CONSTRAINED DYNAMICS Two basic methods were introduced in Chapter 2 for the computer formulation of the dynamic equations of motion of the multibody railroad vehicle systems. In the first method, the augmented formulation, the equations of motion are formulated in terms of redundant coordinates and Lagrange multipliers; in the second method, the embedding technique, the kinematic constraint equations are systematically eliminated from the dynamic equations of motion, which are expressed in terms of the system degrees of freedom. In the case of the augmented formulation, the equations of motion of the system (as discussed in Chapter 2) can be written in the following matrix form (Shabana, 2005): M  Cq

CTq   q   Q    =   0   λ  Qd 

(6.1)

where M is the system symmetric mass matrix, Cq is the Jacobian matrix of the is the system generalized accelerations vector, λ is system constraint equations, q the vector of Lagrange multipliers associated with the system constraints, Q is the generalized force vector, and Qd is the vector that results from the differentiation of the kinematic constraint equations twice with respect to time and absorbs the quadratic terms in the first time derivatives of the generalized coordinates. In Equation 6.1, the mass matrix M and the constraint Jacobian matrix Cq are functions of the system coordinates q; and the vectors Q and Qd can be functions of the system generalized coordinates and velocities q and q as well as time. Therefore, given the initial conditions, which are defined by the initial values of the vectors q and q at time t = 0, Equation 6.1 can be constructed and solved for the unknown accelerations and the vector of Lagrange multipliers λ. Since the system is subjected to conq straint conditions, the coordinates and velocity vectors q and q must satisfy the constraint equations at the position and velocity levels. That is, the coordinate vector must satisfy the following constraint conditions at any time t, including t = 0: C(q, t ) = 0

(6.2)

Differentiating this equation with respect to time, it is clear that the velocity vector must satisfy the following conditions at any time t, including t = 0:



189

C q q = −C t

(6.3)

where Ct = ∂C/∂t is the vector of the partial derivatives of the constraints of Equation 6.2 with respect to time. The vector Ct is identically equal to zero if all the constraint equations are not explicit functions of time. Note that Equation 6.1 ensures that the constraint equations are satisfied at the acceleration level. Methods that can be used to avoid violations of the kinematic constraints at the position and velocity levels will be discussed later in this section. The solution of Equation 6.1 defines the vector of the generalized accelerations at a given time t. This vector can be integrated numerically to define the system q generalized velocities and coordinates q and q at t = t + ∆t, where ∆t is the integration time step. To enable the use of well-developed integration methods designed to solve nonlinear first-order differential equations, the following state vector and its time derivatives are defined: q  y =  , q 

q  y =    q

(6.4)

Given the vector y at time t, it is clear that the solution of Equation 6.1 can be used to define the vector y at time t, which is required to advance the integration to determine y at time t + ∆t. The vector y can be used with standard numerical integration methods designed to solve first-order differential equations to determine the system generalized velocities and coordinates at time t = t + ∆t. These coordinates at time and velocities can be used in Equation 6.1 to evaluate the accelerations q t + ∆t. This process continues until the end of the simulation is reached. It is important, however, to point out that, due to the numerical error that results from the numerical integration of the accelerations, the coordinates and velocities do not necessarily satisfy the constraint equations at the position and velocity levels, respectively. The violations in the constraint equations at the position and velocity levels can be significant, particularly for long simulations of systems that consist of many interconnected bodies that are subjected to severe loading conditions, such as in the case of railroad vehicles. Robust multibody system algorithms are designed to avoid the violation of the kinematic constraint equations and to ensure that these constraint equations are satisfied at both the position and velocity levels. Two techniques are often used to ensure that the numerical solution of the multibody system equations satisfies the constraint equations. These two techniques are based on the penalty and constraint stabilization methods, and the generalized coordinates partitioning method. Both of these different techniques are discussed below.

6.1.2 PENALTY

AND

CONSTRAINT STABILIZATION METHODS

Among the methods used in the multibody system literature to avoid violations of the constraint equations are the penalty and the constraint stabilization methods. The penalty method is based on the concept of introducing a force function that reduces the violation in the constraint. For example, if C(q,t) = 0 is the vector of the constraint functions, one can define a force vector that takes the following form:


190


, where K and D are matrices of stiffness and damping coefficients, F = −KC − DC respectively. Note that by a proper selection of the matrices K and D, the force F when added to the vector Q of Equation 6.1 will introduce a restoring force if the constraint equations are violated at the position or velocity level. The results obtained by the penalty method have been compared with the results of other methods that are used to ensure that the constraint equations are not violated (Ozaki and Shabana, 2003a, 2003b). In the constraint stabilization method, which borrows a concept from the vibration and feedback control theories (Nikravesh, 1988; Shabana, 1996a), velocity- and coordinate-dependent terms are introduced to satisfy the constraint equations. For example, a simple constraint on a coordinate y that requires this coordinate to remain zero all the time can be written at the acceleration level as y = 0 . The numerical integration of this acceleration equation, due to numerical errors, does not, in general, guarantee that y = y = 0 at an arbitrary time t. This system can be stabilized by introducing damping and stiffness terms and then writing the constraint equation at the acceleration level as y + 2α y + β 2 y = 0. In this equation, 2α y and β 2y are the terms introduced to achieve stability, and α and β are assumed coefficients that can be selected in order for y and y to approach zero with a minimum level of oscillations. This simple concept can be generalized and applied to the constraint equations. Recall that if one has the correct solution, the constraint equations and their derivatives can be written as follows:

( )

( )

( )

q, t = C q, t = 0 C q, t = C

(6.5)

Following the argument made for the single equation, the constraints can be stabilized by writing the constraint equations at the acceleration level instead of q, t = 0 as (Baumgarte, 1972) C

( )

+ 2α C + β 2C = 0 C

(6.6)

= Q de Cq q

(6.7)

− β 2C Qde = Qd − 2α C

(6.8)

This equation can be written as

In this equation,

Using Equation 6.7, the augmented form of the equations of motion (Equation 6.1) can be modified and written as M  Cq

CTq   q    Q   =  2  0   λ  Qd − 2α C − β C 

(6.9)



191

The difference between the penalty method and the constraint stabilization method is clear from this equation. In the penalty method, the generalized force vector Q is changed, while in the constraint stabilization method discussed in this section, the constraint equations are changed. Based on the constraint stabilization method discussed in this section, the following simple algorithm for solving the dynamic equations of motion of the multibody system can be proposed: 1. At the initial time t = t0, the initial conditions defined by the initial values of q and q can be used to evaluate the mass matrix M, the Jacobian matrix of the constraints Cq, the vector of applied forces Q, and the vector Qde of Equation 6.8. 2. By selecting the coefficients α and β, Equation 6.9 can be constructed and the vector of Lagrange muland solved for the acceleration vector q tipliers λ at the specified time t. Lagrange multipliers can be used to define the generalized constraint forces. can be integrated numerically to determine the 3. The acceleration vector q coordinate and velocity vectors q and q at time t = t + ∆t. 4. If the end of the simulation time is reached, stop; otherwise, go to Step 2 of the algorithm and continue until the end of the simulation time is reached. The accuracy of the solution obtained using this algorithm depends, as documented in the multibody system dynamics literature, on the values of the coefficients α and β. In general, if α and β are not equal to zero, the solution of the system predicted by the algorithm outlined above oscillates around the correct value. In general, α and β can be selected to take values between 1 and 10. The coefficients α and β can also be chosen such that the system is critically damped (Nikravesh, 1988).

6.1.3 GENERALIZED COORDINATES PARTITIONING Numerical problems can be encountered in many applications when the penalty or the constraint stabilization method is used. Another alternative that can be used to develop a robust numerical algorithm that ensures that the constraints are not violated is the method of the generalized coordinates partitioning (Wehage, 1980). This method, which is widely used in multibody system algorithms and computer codes, ensures that the constraints are satisfied at the position, velocity, and acceleration levels. In the generalized coordinate partitioning method, a set of independent coordinates is identified and used to define the associated differential equations that can be integrated to determine the independent velocities and coordinates. Knowing the independent generalized coordinates (degrees of freedom), one can determine the dependent generalized coordinates using the kinematic constraint equations. Knowing the independent velocities, the constraint equations at the velocity level can be used to determine the dependent velocities. In order to explain this procedure, consider a virtual change of the system generalized coordinates δq. In this case, the constraints of Equation 6.2 lead to


192


Cqδ q = 0

(6.10)

In this equation, the constraint Jacobian matrix Cq has the number of rows equal to the number of the constraint functions nc, and the number of columns is equal to the number of the system generalized coordinates n. If the constraint equations are linearly independent, the Jacobian matrix Cq has a full row rank. In this case, one can identify a set of independent coordinates equal to (n − nc) and write the total vector of the system generalized coordinates in the following partitioned form: q  q= i q d 

(6.11)

where qi and qd are the vectors of independent and dependent coordinates, respectively. Note that the number of dependent coordinates is equal to the number of constraint equations. Therefore, if the independent coordinates (degrees of freedom) are known, the nonlinear constraint equations of Equation 6.2 can be solved for the dependent coordinates using an iterative Newton-Raphson algorithm. By applying this method, it is guaranteed that the constraint equations are satisfied at the position level. Using this coordinate partitioning, Equation 6.10 can be written as follows: C qi δ q i + C q d δ q d = 0

(6.12)

Since the number of the dependent coordinates is equal to the number of the system constraint equations, the Jacobian matrix C q d associated with these coordinates is a square matrix with dimension nc × nc, while the Jacobian matrix C qi associated with the independent coordinates has dimension nc × (n − nc). If the constraint equations are independent, the system degrees of freedom can be selected such that the matrix C q d is nonsingular. In this case, the virtual change in the dependent coordinates can be expressed in terms of the virtual change of the independent coordinates as follows (Shabana, 2005):

δ qd = Cq−1d (Cqi δ qi )

(6.13)

Differentiating the constraint equations with respect to time and using a similar procedure, one can obtain the following relationship for the dependent and independent velocities: C qi q i + C q d q d = −C t

(6.14)

The dependent velocities can then be written in terms of the independent velocities as follows: q d = −C q−1d (C qi q i + C t )

(6.15)



193

When the generalized coordinate partitioning method is used, one needs to integrate only the state equations associated with the independent coordinates. Therefore, the following state vector and its derivative are formed: q  y =  i , q i 

q  y =  i  i  q

(6.16)

can be determined for a given q and q Note that the total acceleration vector q using Equation 6.1. The independent accelerations can be selected and the vector y can be formed. By numerically integrating y , one obtains the independent coordinates and velocity vectors qi and q i , respectively. Knowing the independent coordinates qi, the dependent coordinates can be determined as previously mentioned by solving the nonlinear constraint equations of Equation 6.2 using an iterative Newton-Raphson algorithm. Similarly, knowing the independent velocities q i , the dependent velocities can be determined using the linear velocity relationship of Equation 6.15. Therefore, by using the generalized coordinate partitioning method, it is guaranteed that the constraint equations are satisfied at the position and velocity levels. Furthermore, by determining the accelerations as the solution of Equation 6.1, it is also guaranteed that the constraint equations are satisfied at the acceleration level. In summary, the following computational algorithm based on the generalized coordinate partitioning method can be proposed: 1. At the initial time t = t0, the initial conditions defined by the initial value of q and q can be used to evaluate the mass matrix M, the Jacobian matrix of the constraints Cq, the vector of applied forces Q, and the vector Qd that appear in Equation 6.1. It is assumed that the initial conditions satisfy the kinematic constraint conditions imposed on the motion of the system. 2. Equation 6.1 can be constructed and solved for the acceleration vector q and the vector of Lagrange multipliers λ at the specified time t. Lagrange multipliers can be used to define the generalized constraint forces. i associated with the system degrees of 3. The independent accelerations q freedom are identified and used to define the vector y of Equation 6.16. This vector can be integrated numerically to determine the independent coordinate and velocity vectors qi and q i at time t = t + ∆t. 4. Knowing the independent coordinates qi, the nonlinear constraint equations of Equation 6.2 can be solved for the dependent coordinates qd using an iterative Newton-Raphson algorithm. 5. Knowing all the coordinates q and the independent velocities q i from the numerical integration of y , the dependent velocities can be determined by solving the system of linear equations of Equation 6.15. 6. At this stage, all the coordinates and velocities are known. If the end of the simulation time is reached, stop; otherwise, go to Step 2 of the algorithm and continue until the end of the simulation time is reached. It is clear that this algorithm is more computationally intensive than the algorithm based on the penalty or the constraint stabilization methods.


194


6.1.4 IDENTIFICATION

OF THE INDEPENDENT

COORDINATES

In the generalized coordinate partitioning method, the coordinates are partitioned as independent and dependent. Therefore, identifying the independent coordinates is required in order to use the generalized coordinate partitioning method. For large and complex systems that include many bodies and joints, the identification of the independent coordinates by inspection can be a very difficult or even impossible task. In this case, one must resort to numerical methods to select the system degrees of freedom. There are several methods that can be used to determine the independent coordinates using the Jacobian matrix of the kinematic constraints. For example, the Gaussian procedure with full or partial pivoting can be used to identify the independent and dependent coordinates by performing on the rectangular constraint Jacobian matrix a sequence of elementary operations that defines a nonsingular square submatrix. The columns of this nonsingular matrix are associated with the dependent coordinates, while the remaining columns are associated with the independent coordinates. For example, if a system has n coordinates and nc constraint equations, the constraint Jacobian matrix Cq is rectangular and has the dimension nc × n. Each column in this matrix is associated with an element of the vector of generalized coordinates q. Performing the Gaussian procedure, which consists of elementary row and column operations on the rectangular Jacobian matrix Cq, changes the order of the elements in the vector q. The Gaussian procedure eventually leads to an identity nonsingular submatrix that has dimension nc × nc associated with a reordered set of coordinates, the dependent coordinates qd; while the remaining n − nc columns are associated with another set of reordered coordinates, the independent coordinates qi. For example, if a system has four constraint equations (nc = 4) and six coordinates (n = 6), the number of independent coordinates is 2. The coordinates are initially arranged in the following order [1 2 3 4 5 6]. After the Gaussian procedure is performed on the constraint Jacobian matrix, the first four columns of the resulting matrix will form an nc × nc = 4 × 4 identity matrix. Assume that the Gauss procedure results in the following reordering of the columns and coordinates [1 4 5 6 2 3]. Then, the system dependent coordinates are the coordinates 1, 4, 5, and 6, while the system independent coordinates are the coordinates 2 and 3.

6.2 NUMERICAL ALGORITHMS — CONSTRAINT FORMULATIONS Several methods for the wheel/rail contact problem were presented in the preceding chapter. Some of these methods are based on the constraint contact formulation, while the others are based on the elastic contact formulation. In particular, the following two constraint methods were discussed: • •

Augmented constraint contact formulation (ACCF) Embedded constraint contact formulation (ECCF)

These methods require the use of different solution procedures and numerical algorithms to determine the location of the contact points and normal forces that act at



195

these points. In this section, the implementation of these two methods in a general multibody system computational algorithm is discussed. It is important, however, to point out that, while these methods use different schemes to determine the location of the contact points and normal forces, all these methods can employ the same procedure to determine the tangential contact forces, as will be discussed in Section 6.4. Furthermore, the rail and wheel geometric description discussed in Chapter 3 is the same for all these methods.

6.2.1 AUGMENTED CONSTRAINT CONTACT FORMULATION (ACCF) In this method, as explained in Chapter 5, five scalar contact constraint equations are used to describe the wheel/rail interaction. These five constraint equations for an arbitrary contact k are C p  rPw − rPr      C k (q w , q r , s w , s r ) =  C1  =  t1w ⋅ n r  = 0  C   t w ⋅ nr    2  2

(6.17)

where superscripts r and w denote rail and wheel, respectively; qw and qr are, respectively, the generalized coordinates of the wheel and rail; sw and sr are the vectors of surface parameters; and t1, t2, and n are the two tangents and normal to the surface at the contact point, respectively. In addition to the contact constraint equations, railroad vehicle models may include mechanical joints and specified motion trajectories. In this case, the total vector of the system constraint equations can be written as

(

)

C q, s, t = 0

(6.18)

where q is the vector of the system generalized coordinates, s is the vector of the system surface parameters, and t is time. As discussed in Chapter 5, after differentiating the preceding equation twice with respect to time, the constraint equations at the acceleration level can be written as + C ss = Q d Cq q

(6.19)

where Cq and Cs are the sub-Jacobians of the constraint equations associated, respectively, with the generalized coordinates q and the surface parameters s; and Qd is a vector that absorbs quadratic terms in the first derivatives of the generalized coordinates and the surface parameters The augmented form of the equations of motion (Equation 6.1) can then be modified to account for the contact conditions as follows: M  0 C  q

0 0 Cs

CqT   q   Q     T   C s   s  =  0  0   λ  Qd  

(6.20)


196


and s as well as the These equations can be solved to determine the accelerations q vector of Lagrange multipliers λ. In the constraint contact formulations, the constraint equations including the contact constraints must be satisfied at the position, velocity, and acceleration levels. When the constraint contact formulation is used, it is necessary to determine the sub-Jacobians Cq and Cs as well as the quadratic velocity terms that result from differentiating the constraint equations twice with respect to time. For example, the first vector constraint equation (three scalar equations) in Equation 6.17 can be written more explicitly as follows: C p (q, s) = R w + Aw u w − R r − Ar u r

(6.21)

where Rw and Rr are the global position vectors of the origins of the body coordinate systems of the wheel and the rail, respectively; Aw and Ar are the transformation matrices that define the orientation of the wheel and the rail body coordinate systems, respectively; and u w and u r are the local position vectors of the contact point with respect to the wheel and the rail coordinate systems, respectively. Recall that Rw, Rr, Aw, and Ar in the preceding equation are functions of the body generalized coordinates q only, while u w and u r are functions of the wheel and the rail surface parameters. The sub-Jacobian matrices associated with the preceding constraint equations can be written as

(C )

=  I

( )

 ∂u w =  Aw w ∂s 

p q

Cp

s

− Aw u wG w − Ar

Ar u r Gr       

−I ∂u r   ∂s r 

(6.22)

where G is the matrix that relates the angular velocity vector defined in the body coordinate system to the time derivatives of the orientation coordinates, as discussed in Chapter 2. Note that in the preceding equation ∂u i  ∂u i = ∂s i  ∂s1i

∂u i   i  = t1 ∂s2i  

t2i  ,

i = w, r

(6.23)

where t1i and t2i are the vectors tangent to the surface at the contact point, as shown in Chapter 3. A similar procedure can be applied to the last two constraint equations of Equation 6.17. One can verify that the sub-Jacobian matrices associated with these two constraint equations are given by

(C )

= 0 

(C )

∂ tl w = [n A A ∂s w

l q

l s

rT

− n r Ar A w tl wG w T

rT

w

T

− tl w A w Ar n r Gr     , r w T w T r ∂n  tl A A ]  ∂s r 0

T

T

l = 1, 2 (6.24)



197

Similarly, the quadratic velocity vector Qd that results from the differentiation of the constraint equations twice with respect to time can be evaluated. A simple procedure that can be used to determine Qd is to differentiate the constraint equations twice with respect to time. The differentiation leads to two types of terms: the first type includes second derivatives of the generalized coordinates and surface parameters, while the second type includes terms that are quadratic in the velocities. These quadratic velocity terms can be grouped together and moved to the right-hand side of the equation to define the vector Qd. This process can be demonstrated using the constraint vector of Equation 6.21. To this end, one can write the second time derivative of this constraint equation in the following form:  w (q, s) = R w − A w u wG wθ w + A w ∂u s w + Qw C p p  ∂s w   

( )

 r r − Ar ∂u sr − Qr + Ar u r Gr θ p  ∂s r   

( )

d

r −R (6.25)

d

=0

In this equation, 



(Q ) = (G θ ) × u + ω × (ω × u ) + 2ω ×  A ∂∂us s  i p

i i

i

i

i

i

i

i

i

i

i

d

2 2 ∂2u i ∂2u i   ∂2u i + A  i 2 s1i + 2 i i s1i s2i + i 2 s2i  , ∂ s s s ∂ ∂ s ∂  1 2 2  1

( )

i

( )

(6.26) i = w, r

The quadratic velocity vector (Qp)d can then be defined as follows:

(Q )

p d

( ) + (Q )

= − Qwp

d

r p

(6.27)

d

and the other vectors that appear in Equation 6.26 are defined as follows: ω i = Giθ i ,

u i = Ai u i ,

i = w, r

(6.28)

where ωi (i = w, r) is the angular velocity vector defined in the global coordinate system. Following a similar procedure, one can show that the quadratic velocity vectors that result from the differentiation of the last two constraints of Equation 6.17 twice with respect to time are given by

(Q )

l d

(

)

T T T = − n r blw + 2tlw n r + t lw br ,

l = 1, 2

(6.29)


198


In this equation,  ∂t w   tlw = ω w × t lw + A w  lw  s w   ∂s    r    ∂ n n r = ω r × n r + Ar  r  s r   ∂s  

(6.30)

where l = 1, 2, and the vectors bw and br are given, respectively, by w wθ w × t w + ω w × ω w × t w + 2ω w ×  A w ∂ tl s w  blw = G l l   w ∂s  

(

)

(

 ∂2 t w + A w  wl 2 s1w  ∂s1

( )

2

)

2  ∂2 t w ∂2 t w + 2 w l w s1w s2w + wl 2 s2w  ∂s1 ∂s2 ∂s2 

( )

(6.31)

and r r θ r × n r + ω r × ω r × n r + 2ω r ×  Ar ∂n s r  br = G   r ∂s  

(

)

(

 ∂2n r + Ar  r 2 s1r  ∂s1

( )

2

)

( )

∂2n r ∂2n r + 2 r r s1r s2r + r 2 s2r ∂s1 ∂s2 ∂s2

2

  

(6.32)

It is clear from the formulation of the quadratic velocity vector that the constraint contact formulation requires the evaluation of the third derivatives of vectors with respect to the surface parameters. This is clear, since, for example ∂2 tmw ∂slw

2

=

∂2  ∂u w  , 2  w  ∂slw  ∂sm 

m = 1, 2; l = 1, 2

(6.33)

and ∂2n r ∂slr

2

=

(

∂2 t1r × t r2 ∂slr

2

),

l = 1, 2

(6.34)

where tlr =

∂u r , ∂slr

l = 1, 2

(6.35)



199

This section focuses on the contact constraints, but railroad vehicle models may include other joints that are standard elements of the constraint library of generalpurpose multibody computer codes. The formulation of the constraint equations associated with these joints, as well as their Jacobian matrices and quadratic velocity vectors, can be found in the literature (Shabana, 2001). The constraint Jacobian matrix and the quadratic velocity vector that result from the differentiation of the constraint equations twice with respect to time can be used with the system mass matrix and the vector of applied forces to construct Equation 6.20. This equation can be solved for the accelerations and the vector of Lagrange multipliers. Lagrange multipliers can be used to predict the constraint forces, including the normal contact constraint forces. As discussed in Chapter 5, it is clear that the solution of Equation 6.20 must satisfy the following equation: CTs λ = 0

(6.36)

For an arbitrary contact, the preceding equation has four scalar equations, since there are four surface parameters associated with each contact, while the vector of Lagrange multipliers associated with this contact has five elements. The preceding equation, therefore, can be used to define one independent Lagrange multiplier; other Lagrange multipliers can be expressed in terms of this independent multiplier. That is, there is only one independent constraint force associated with the degree of freedom eliminated by the five contact constraint equations. This independent constraint force can be used to define the normal contact force. In the numerical algorithm described in this section, the normal contact force obtained by solving Equation 6.20 is stored and used in the next time step to calculate the tangential creep forces and spin moment. Another important issue that must be considered when developing a computational algorithm to solve the wheel/rail contact problem is the accurate determination of the initial values of the surface parameters and their time derivatives. In the augmented constraint contact formulation, surface parameters can be selected with other independent coordinates as the optimum set of the system degrees of freedom. In this case, it is important to provide accurate values of these parameters and their time derivatives, since other dependent coordinates are expressed in terms of the degrees of freedom. While the initial values of the surface parameters (contact location) can be easily determined using a CAD program, determining the initial values of the first derivatives of the surface parameters is not as simple. In order to determine the derivatives of the surface parameters at the initial configuration, one can use the available known initial values of the time derivatives of the coordinates (velocities). Using the contact constraint equations and the procedure described in Chapter 5, the initial values of the surface parameters can be determined using the following equation: s 0 = Bq 0

(6.37)


200


where B = −(C ds )−1Cqd is the velocity transformation matrix, which can be obtained using the contact constraints. Based on the discussion presented in this section, a computational multibody system algorithm can be developed based on the augmented constraint contact formulation (ACCF). This algorithm is summarized as follows: 1. At the initial configuration, t = t0, accurate values of the surface parameters s, the coordinates q, and the velocities q are provided. 2. Using the initial values, the matrix B of Equation 6.37 is evaluated and used to determine the initial values of the time derivatives of the surface parameters s . 3. If the generalized coordinate partitioning method is used, the constraint Jacobian matrix [Cq Cs] is evaluated. A set of independent coordinates or degrees of freedom is identified based on the numerical structure of the constraint Jacobian matrix. Note that surface parameters may be selected as degrees of freedom. If the penalty method or the constraint stabilization method is used, this step is skipped. 4. Given the independent coordinates, the nonlinear constraint equations are solved iteratively using a Newton-Raphson algorithm to determine the dependent coordinates and surface parameters. This step guarantees that the constraint equations are satisfied at the position level. If the penalty method or the constraint stabilization method is used, this step is skipped. 5. Given the independent velocities, which may include time derivatives of the surface parameters, the constraint equations at the velocity level C q q + C s s = −C t is solved for the dependent velocities using the generalized coordinate partitioning method, as previously described in this chapter. This step guarantees that the constraint equations are satisfied at the velocity level. If the penalty method or the constraint stabilization method is used, this step is skipped. 6. Using the system coordinates and velocities that satisfy the constraint equations at the position and velocity levels, the mass matrix M, the vector of applied forces Q, the constraint Jacobian matrix, and the quadratic velocity vector that arise from differentiating the constraint equations twice with respect to time are evaluated and used to construct the augmented form of the equations of motion (Equation 6.20). The vector Q includes the tangential creep forces, which are calculated using the procedure described in Section 6.4. 7. Equation 6.20 is solved for the generalized coordinate and surface parameter accelerations as well as the vector of Lagrange multipliers. Lagrange multipliers associated with the contact constraints are used to determine the normal contact force, which is stored to be used in the next time step to determine the tangential creep forces and moments. 8. The independent accelerations, which may include second derivatives of the surface parameters, are identified and used to define the state equations associated with the system degrees of freedom. These independent state



201

equations are integrated forward in time to determine the independent coordinates and velocities at time t = t + ∆t. 9. If the end of the simulation time is reached, stop; otherwise, go to Step 4 and continue until the end of the simulation is reached. It is clear from this algorithm that the use of the generalized coordinate partitioning method requires adopting a more sophisticated procedure as compared with the use of the penalty method or the constraint stabilization method. In the generalized coordinate partitioning method, the independent variables must be identified, and only the state equations associated with these independent variables are integrated forward in time to determine the independent coordinates and velocities. One of the main features that distinguish the augmented constraint contact formulation from the embedded constraint contact formulation is the fact that, in the augmented constraint contact formulation, the system degrees of freedom or the independent coordinates can include surface parameters, since these degrees of freedom are identified based on the numerical properties of the constraint Jacobian matrix. In fact, for the three-degree-of-freedom unsuspended wheelset with a prescribed forward velocity, the optimum set of degrees of freedom is found to include only surface parameters, and no other generalized coordinates are selected by the algorithm outlined above among the three degrees of freedom. It is also clear from this algorithm that an approximate value of the normal contact force is used to determine the tangential creep forces and spin moment at a given time step. The normal force is stored from the previous time step to avoid the use of an iterative procedure performed on Equation 6.20 to determine the accelerations, Lagrange multipliers, and the creep forces and moments. The creep forces and moments are a function of the normal force, and all these forces enter into the formulation of Equation 6.20. Since the relationship between these forces is nonlinear, an iterative Newton-Raphson algorithm can be used to determine these forces and, at the same time, solve for the accelerations. However, numerical experimentation with several railroad vehicle models showed that storing the values of the normal forces from the previous time step and using these values to avoid applying the iterative Newton-Raphson procedure on Equation 6.20 lead to accurate results. Comparison was also made with the results obtained using the elastic contact formulations, which do not require the use of an iterative procedure to determine the normal and creep forces (Shabana et al., 2005). This comparison showed very good agreement between the results of the algorithm outlined above and the results obtained using the elastic contact formulations.

6.2.2 EMBEDDED CONSTRAINT CONTACT FORMULATION (ECCF) This method, as discussed in the preceding chapter, is conceptually the same as the augmented constraint contact formulation. The contact constraints are imposed at the position, velocity, and acceleration levels. The basic difference between the two methods is that, in the embedded constraint contract formulation, the surface parameters are systematically eliminated. This difference, which necessitates the use of a


202


different numerical solution procedure, forces the numerical algorithm to select the system degrees of freedom from the elements of the generalized coordinate vector q. In this case, as discussed in the preceding chapter, one constraint between the wheel and the rail is added to the system differential equations of motion in order to determine the normal force. Recall from the analysis presented in the preceding chapter that this equation is given for a contact k by C nk (q w , q r , s wk , s rk ) = n rk ⋅ (rPwk − rPrk ) = 0

(6.38)

where all the variables that appear in this equation are as defined in Chapter 5. This equation is added to the system equations of motion, and the Lagrange multiplier associated with this constraint is used to determine the normal contact force. The other four contact constraint equations that are used to eliminate the surface parameters (as presented in Chapter 5) are k

 t1r ⋅ (rPw − rPr )   r  t ⋅ (rw − rr ) C dk (q w , q r , s wk , s rk ) =  2 w P r P  = 0  t ⋅n  1   w r  t 2 ⋅ n 

(6.39)

Using the preceding two equations, it was shown in the preceding chapter that the augmented form of the system equations of motion can be written as follows: M  J

  Q  JT   q   =   0   λ  Qdn 

(6.40)

where J, an implicit function of the vector of the system surface parameters s, is the Jacobian matrix of the constraint equations, including the constraint of Equation 6.38 and other constraints imposed on the motion of the multibody system; λ is the vector of the system Lagrange multipliers that include only Lagrange multipliers associated with the constraints of Equation 6.38 and other noncontact constraints imposed on the motion of the system; and Qdn is a quadratic velocity vector that results from the differentiation of the constraint equations twice with respect to time. This vector is a function of the surface parameters s and their first time derivatives. Recall that for a virtual change in the system variables, Equation 6.38 yields

δ C nk = Cqnkδ q + Csnkδ s k = 0

(6.41)

The Jacobian matrices that appear in this equation can be written as follows:



T Cqnk =  n r 

 T C =  n r t1w  nk s

nr

T

( − A u G ) w

rT w 2

n t

w P

w

wr T P

r

−nr

 ∂n r   ∂s r   1

203

{n ( A u G ) + r (− A n G )} 

T

rT

wr T P

r

r

r P

 ∂n r    ∂s r    2  

r

wr T P

k

r

r

r

    (6.42)

k

where rPwr = rPw − rPr and all the other symbols that appear in this equation are again as defined in Chapter 5. Equation 6.39 can be used to write the virtual changes of the surface parameters in terms of the virtual changes in the generalized coordinates. Substituting the results into Equation 6.41, the virtual changes in the surface parameters can be eliminated. Similarly, by differentiating Equation 6.39 once and twice with respect to time, one can write the first and second time derivatives of the surface parameters in terms of the first and second time derivatives of the generalized coordinates. As demonstrated in Chapter 5, these two relationships can be written, respectively, as follows: s = Bq ,

s = Bq + Bq

(6.43)

In particular, one can show that the second relationship in this equation can be obtained from the second derivative of the constraints of Equation 6.39, which can be written as follows: k

dk C

 t1r ⋅ rPwr + t1r ⋅ rPwr + 2 t1r ⋅ rPwr  r wr r wr  r + 2 t2r ⋅ rPwr  t ⋅ r + t ⋅ =  2w P r 2w P r =0  t ⋅ n + t ⋅ n w r  1 1 + 2 t 1 ⋅ n  w r w r  + 2 t2w ⋅ n r   t 2 ⋅ n + t 2 ⋅ n

(6.44)

The preceding equation can also be used to determine the quadratic velocity vector, as described in the case of the augmented constraint contact formulation. Given the generalized coordinates q, Equation 6.39 can be solved using an iterative NewtonRaphson procedure to determine the surface parameters. Given the vector of generalized velocities q , the constraints of Equation 6.39 at the velocity level can be solved to determine the time derivatives of the surface parameters. This is equivalent to using the linear system of algebraic equations in the velocities given by the first equation in Equation 6.43. The second equation in Equation 6.43 is also a linear system of algebraic equations in the accelerations. This equation can be used to eliminate the second time derivatives of the surface parameters from the second time derivative of the constraint of Equation 6.38, which is given by r + 2 n r ⋅ rPwr = 0 rPwr + rPwr ⋅ n C nk = n r ⋅

(6.45)


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As mentioned in Chapter 5, the use of this procedure implies that the geometric surface parameters are always treated as dependent variables and are systematically eliminated from the equations of motion. For this reason, only independent generalized accelerations are integrated forward in time. It is, perhaps, important at this point to reiterate that only one Lagrange multiplier associated with the contact constraint of Equation 6.38 is used in Equation 6.40. This Lagrange multiplier defines the normal contact force used to define the longitudinal, lateral, and spin creep forces. One can then outline the following numerical solution procedure based on the embedded constraint contact formulation: 1. At the initial configuration, t = t0, accurate values of the surface parameters s, the coordinates q, and the velocities q are provided. Note that the surface parameters in the embedded constraint contact formulation cannot be selected as degrees of freedom. 2. Using the initial values of the coordinates, the matrix B of Equation 6.37 is evaluated and used to determine the initial values of the time derivatives of the surface parameters s . 3. Given the values of the generalized coordinates q, Equation 6.39 can be solved using an iterative Newton-Raphson algorithm to determine the surface parameters. The surface parameters can be substituted into Equation 6.38, thereby writing this equation solely in terms of the generalized coordinates q. 4. If the generalized coordinate partitioning method is used, and because the surface parameters are eliminated from the constraint equations, the constraint Jacobian matrix J is evaluated. A set of independent coordinates or degrees of freedom is identified based on the numerical structure of the constraint Jacobian matrix. Note that in the embedded constraint contact formulation, the surface parameters cannot be selected as degrees of freedom. If the penalty method or the constraint stabilization method is used, this step is skipped. 5. Given the independent coordinates qi, the nonlinear constraint equations are solved iteratively using a Newton-Raphson algorithm to determine the dependent coordinates qd. Using qi and qd, Equation 6.39 is solved iteratively for the surface parameters. This two-stage iterative procedure continues until the obtained generalized coordinates and surface parameters satisfy the constraint equations. This step guarantees that the constraint equations are satisfied at the position level. If the penalty method or the constraint stabilization method is used, this step is skipped. 6. Given the independent velocities q i that do not include time derivatives of the surface parameters, the constraint equation at the velocity level is solved for the dependent generalized velocities q d using the generalized coordinate partitioning method, as previously described in this chapter. 7. Using the vector q = [q Ti q Td ]T , the first equation in Equation 6.43 can be solved for the time derivatives of the surface parameters This step and the



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previous step guarantee that the constraint equations are satisfied at the velocity level. If the penalty method or the constraint stabilization method is used, these two steps are skipped. 8. Using the second equation in Equation 6.43, the second derivatives of the surface parameters can be substituted into Equation 6.45, thereby writing . this equation solely in terms of the vector of generalized accelerations q 9. Using the system coordinates and velocities that satisfy the constraint equations at the position and velocity levels, the mass matrix M, the vector of applied forces Q, the constraint Jacobian matrix, and the quadratic velocity vector that arise from differentiating the constraint equations twice with respect to time are evaluated and used to construct the augmented form of the equations of motion (Equation 6.40). as well as the 10. Equation 6.40 is solved for the generalized accelerations q vector of Lagrange multipliers. Lagrange multipliers associated with the contact constraints are used to determine the normal contact force, which is stored to be used in the next time step to determine the tangential creep forces and moments. 11. The independent accelerations that do not include second derivatives of the surface parameters are identified and used to define the state equations associated with the system degrees of freedom. These independent state equations are integrated forward in time to determine the independent coordinates and velocities at time t = t + ∆t. If the end of the simulation time is reached, stop; otherwise, go to Step 3 and continue until the end of the simulation is reached. Clearly, the main difference between this algorithm and the algorithm based on the augmented constraint contact formulation is the systematic elimination of the surface parameters. By doing so, the surface parameters are always treated as dependent variables.

6.3 NUMERICAL ALGORITHMS — ELASTIC FORMULATIONS In the preceding chapter, two elastic contact formulations were discussed. In these elastic formulations, the contact conditions are not imposed as constraints and, therefore, no degrees of freedom are eliminated as the result of the wheel/rail dynamic interaction. The elastic contact formulations that allow for wheel/rail separation and penetration require the use of simpler numerical algorithms as compared with the constraint contact formulations. The following two methods were discussed in detail in Chapter 5: • •

Elastic contact formulation using algebraic equations (ECF-A) Elastic contact formulation using nodal search (ECF-N)

This section discusses the computer implementation of these two methods.


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6.3.1 ELASTIC CONTACT FORMULATION USING ALGEBRAIC EQUATIONS (ECF-A) This method, as discussed in the preceding chapter, is not considered as a constraint method, since no Lagrange multipliers are associated with the contact conditions and no degrees of freedom are eliminated as the result of the wheel/rail interaction. In this case, the system equations of motion take a form similar to Equation 6.1. As described in Chapter 5, the location of the contact point is determined by solving the nonlinear algebraic equations of Equation 6.39 for the four surface parameters that define the geometry of the wheel and rail surfaces. To this end, the formulation of the following equation is required for each contact when the iterative NewtonRaphson algorithm is used:  r w  t1 ⋅ t1    t r2 ⋅ t1w   w  ∂t1 ⋅ n r  ∂s1w  w  ∂t 2 ⋅ n r  ∂s w  1

t1r ⋅ t 2w

∂t1r wr r r ⋅ r − t1 ⋅ t1 ∂s1r

t r2 ⋅ t 2w

∂t r2 wr r r ⋅ r – t 2 ⋅ t1 ∂s1r

∂t1w r ⋅n ∂s2w

∂n r w ⋅ t1 ∂s1r

∂t 2w r ⋅n ∂s2w

∂n r w ⋅ t2 ∂s1r

∂t1r wr r r  ⋅ r – t1 ⋅ t 2  ∂s2r   t1r ⋅ r wr  ∂t r2 wr r r  s1w  ⋅ r − t2 ⋅ t2   w   r wr  r ∂s2   s2  t2 ⋅ r  − =   s r  r  t w ⋅ nr  ∂n w 1   1w r    ⋅ t 1   s2r  ∂s2r  t 2 ⋅ n   r ∂n w  ⋅ t2  ∂s2r  (6.46)

The coefficient matrix in this equation is the Jacobian matrix of the algebraic equations of Equation 6.39, while the vector on the right-hand side represents the values of the algebraic equations for given values of the surface parameters. Note that in this section, the phrase algebraic equations is adopted instead of the phrase constraint equations to make clear that this method is not a constraint method that requires certain constraint functions be imposed at the velocity and acceleration levels and, therefore, that no degrees of freedom are eliminated and no Lagrange multipliers are introduced. The preceding equation is iteratively solved until convergence is achieved. Convergence is achieved when the norm of the violation of the algebraic equations or the norm of the Newton differences is less than a specified tolerance. The speed of convergence depends on the initial guess used for the surface parameters. Having determined the vector of the surface parameters that define the location of the contact points on the wheel and the rail, the value of the penetration can be calculated using the following equation:

δ = r wr ⋅ n r

(6.47)

The normal contact forces can be calculated using Hertz's contact theory, while the creep forces can be calculated using one of the creep-force models discussed in



207

Chapter 4. As presented in Chapter 5, the following expression of the normal force can be used (Shabana et al., 2004): F = Fh + Fd = − K hδ 3 2 − Cδ δ

(6.48)

Using the elastic contact formulation based on the algebraic equations (ECF-A), the following numerical procedure, which can be implemented in a general multibody system computational algorithm, can be summarized: 1. At the initial time t = t0, the initial values of the generalized coordinate and velocity vectors q and q , respectively, are provided. An initial guess of the surface parameters must also be provided. 2. The independent generalized coordinates are identified, and the constraint equations and the generalized coordinate partitioning method are used to adjust the dependent generalized coordinates. This ensures that the constraint equations, which do not include any contact constraints, are satisfied at the position level. If the penalty or the constraint stabilization method is used, this step is skipped. 3. The vector of generalized coordinates is used to solve Equation 6.39 for the surface parameters using a Newton-Raphson algorithm that requires formulating the matrix equation given in Equation 6.46. This second Newton-Raphson solution stage, in the case of the generalized coordinate partitioning method, is independent of the first Newton-Raphson stage discussed in Step 2 of the algorithm. This is an important difference between the position analysis performed in this method and the position analysis performed in the embedded constraint contact formulation discussed in the preceding section. Furthermore, the surface parameters must be determined regardless of whether or not the generalized coordinate partitioning method is used. 4. The independent generalized velocities are used to determine the dependent generalized velocities by solving the system of linear equations C q q = −C t using the generalized coordinate partitioning method. This step guarantees that the constraint equations are satisfied at the velocity level. Note that in this step, the time derivatives of the surface parameters are not required. If a penalty method or the constraint stabilization method is used, this step is skipped. 5. Equation 6.47 and the value of the surface parameters determined in Step 3 are used to determine the value of the penetration for each contact. If the value of the penetration is positive, determine the normal contact force using Equation 6.48; otherwise, skip the calculations of the normal force. 6. In the case of positive penetration, the normal force, the material properties, and the geometry of the wheel and the rail surfaces can be used to determine the creep forces and moments that can enter into the formulation of the dynamic equations as external forces.


208


7. Since all the coordinates and velocities are known, the mass matrix M, the vector of applied forces Q, and the quadratic velocity vector Qd of Equation 6.1 can be evaluated. Equation 6.1 can then be constructed and solved and the vector of Lagrange multipliers λ . for the acceleration vector q 8. Lagrange multipliers can be used to determine the generalized constraint forces, while, in the case of the generalized coordinate partitioning method, the independent accelerations can be identified and integrated forward in time to determine the independent coordinates and velocities. In the case where a penalty method or the constraint stabilization method is used, all the accelerations are integrated to determine all the generalized coordinates and velocities. 9. If the end of the simulation time is reached, stop; otherwise, go to Step 2 and continue until the end of the simulation time is reached. While a set of algebraic equations must be solved in the ECF-A method, the computational algorithm outlined above makes clear the basic differences between this method, which does not eliminate degrees of freedom due to the contact, and the ECCF method discussed in the preceding section. It is also clear that, in the elastic contact formulation, there is no need to store the normal contact force from the previous time step for the creep-force calculations, as is the case in the constraint contact formulation. Obviously, one can always use a hybrid method in which the surface parameters or, equivalently, the locations of the contact points are determined using a constraint method, while the normal force is determined using an elastic approach. The use of this hybrid method, however, can be questioned, since in the case of wheel/rail separation, the motions of the wheel and the rail should be totally independent.

6.3.2 ELASTIC CONTACT FORMULATION USING NODAL SEARCH (ECF-N) This formulation uses a method similar to the one used in the ECF-A method for calculating the normal force. However, the two methods differ in the way the location of the contact point is determined. In the ECF-N method, the contact location is determined based on a search using the relative distance between nodal points that represent the wheel and the rail profiles instead of solving a set of algebraic equations, as in ECF-A. Therefore, the numerical procedure used for the ECF-N method is similar to the numerical procedure used for the ECF-A method. The only difference is replacing the solution of the algebraic equations with an optimized search procedure to determine the two nodal points on the wheel and the rail surfaces that have the maximum indentation. As mentioned in Chapter 5, the nodal search calculation is performed in a two-dimensional plane for simplicity. Furthermore, since the longitudinal distance traveled by the wheel at the contact point needs to be determined, an additional differential equation can be introduced for each contact to determine the distance traveled on the rail, and this information is then used in constructing the projection plane used for the nodal search. This first-order differential equation is given by



s1r = rQwT ⋅ t1r

209

(6.49)

where s1r is the rail longitudinal surface parameter, rQw is the absolute velocity of a point on the wheel, and t1r is the tangent to the rail in the longitudinal direction. The following numerical algorithm based on the ECF-N can be proposed: 1. At the initial time t = t0, the initial values of the generalized coordinate and velocity vectors q and q , respectively, are provided. An initial guess of the surface parameters must also be provided. 2. The independent generalized coordinates are identified, and the constraint equations and the generalized coordinate partitioning method are used to adjust the dependent generalized coordinates. This ensures that the constraint equations, which do not include any contact constraints, are satisfied at the position level. If a penalty method or the constraint stabilization method is used, this step is skipped. 3. The vector of generalized coordinates and the distance traveled by the wheel s1r are used to search for the location of the contact points. The result of this search can be used to determine the rail and wheel profile parameters s2r and s1w at the points of contact. 4. The independent generalized velocities are used to determine the dependent generalized velocities by solving the system of linear equations C q q = −C t using the generalized coordinate partitioning method. This step guarantees that the constraint equations are satisfied at the velocity level. Note that, in this step, the time derivatives of the surface parameters are not required. If the penalty method or the constraint stabilization method is used, this step is skipped. 5. Equation 6.47 and the value of the surface parameters determined in Step 3 are used to determine the value of the penetration for each contact. If the value of the penetration is positive, the normal contact force is calculated using Equation 6.48; otherwise, the calculations of the normal force are skipped. 6. In the case of positive penetration, the normal force, the material properties, and the geometry of the wheel and the rail surfaces can be used to determine the creep forces and moments that can enter into the formulation of the dynamic equations as external forces. 7. Since all the coordinates and velocities are known, the mass matrix M, the vector of applied forces Q, and the quadratic velocity vector Qd of Equation 6.1 can be evaluated. Equation 6.1 can then be constructed and solved and the vector of Lagrange multipliers λ. for the acceleration vector q 8. Lagrange multipliers can be used to determine the generalized constraint forces, while, in the case of the generalized coordinate partitioning method, the independent accelerations can be identified and integrated forward in time to determine the independent coordinates and velocities. In cases where the penalty method or the constraint stabilization method is used, all the accelerations are integrated to determine all the generalized coordinates and velocities.


210


9. With the accelerations, the first-order differential equations defined by Equation 6.49 are integrated to determine the distance traveled by the wheels. The number of these first-order differential equations is equal to the number of wheel/rail contacts. The numerical solution of these equations defines the parameter s1r used in Step 3. 10. If the end of the simulation time is reached, stop; otherwise, go to Step 2 and continue until the end of the simulation time is reached. Numerical experimentation showed that Equation 6.49 provides an accurate estimate for the parameter s1r . While this equation represents an efficient method to solve for the parameter s1r , this is not the only method that can be adopted. Clearly, if the generalized coordinates are known, one can write the relationship between these coordinates and any other set of coordinates such as the trajectory coordinates discussed in Chapters 2 and 7. This is a nonlinear relationship that can be used with a Newton-Raphson algorithm to solve for the trajectory coordinates that include the distance traveled by the wheel. This approach, however, requires the solution of at least three nonlinear algebraic equations for each contact. Equation 6.49, therefore, represents a more efficient and robust method for obtaining the parameter s1r .

6.4 CALCULATION OF THE CREEP FORCES In all the computational algorithms discussed so far in this chapter, one method can be used for the calculation of the creep forces. Despite the fact that different methods can be used to determine the normal forces, the same methods of calculating the principal curvatures, the normalized velocity creepages, and the dimension of the contact ellipse that enter into the calculation of the creep forces can be employed for all contact formulations and can be common in all computational algorithms discussed in this chapter. The calculated creep forces can be introduced to the dynamic equations of motion as generalized applied forces using the vector Q of Equations 6.1, 6.20, or 6.40, depending on the contact formulation used to describe the wheel/rail interaction. The main steps for calculation of the creep forces can be summarized as follows: 1. Knowing the location of the point of contact, the velocity of the contact point on the wheel and the rail can be determined (as discussed in Chapter 2) using the following equation: i + ω i × ui , r i = R

i = w, r

(6.50)

where Ri is the global position vector of the origin of the body coordinate system, ωi is the absolute angular velocity vector of the body defined in the global coordinate system, and ui is the position of the contact point with respect to the origin of the body coordinate system defined in the global system.



211

2. Knowing the location of the contact point, the vectors tangent and normal to the surfaces at the contact points can be determined. The longitudinal and lateral tangent vectors to the rail, t1r and t r2 , respectively, and the normal vector nr at the contact point can be determined using the procedure described in Chapter 3. 3. Using the absolute velocities of the wheel and the rail and the tangent and normal vectors, the normalized velocity creepages can be calculated as described in Chapter 4 using the following equations:

ζx =

(r w − r r ) ⋅ t1r , V

ζy =

(r w − r r ) ⋅ t r2 , V

ϕ=

(ω w − ω r ) ⋅ n r V

(6.51)

where V is the forward velocity of the wheel. 4. Using the methods of differential geometry discussed in Chapter 3, the principal radii of curvature of the wheel and rail surfaces at the contact point can be determined. 5. Using the normal force, the wheel and the rail material properties, and the principal curvatures of the wheel and rail surfaces at the contact point, Hertz contact theory can be used to determine the dimension of the semiaxes of the contact ellipse. 6. The information obtained in the previous steps enters into the formulation of the creep forces. The creep forces can be determined using any of the creep theories discussed in Chapter 4. As discussed in Chapter 4, there are several creep theories in the literature, some of which are linear while the others are nonlinear, and some are two dimensional while the others are three dimensional.

6.5 HIGHER DERIVATIVES AND SMOOTHNESS TECHNIQUE The solution of the contact problem using the methods discussed in this chapter and the preceding chapter requires, particularly in the case of the constraint formulations, evaluating higher derivatives of position vectors of points on the wheel and rail surfaces with respect to their geometric surface parameters. For example, the calculations of the dimensions of the contact ellipse semi-axes, which are necessary for all methods, require evaluating the principal curvatures and principal directions at the contact points, as explained in Chapter 4. The evaluation of the principal curvatures and principal directions requires the evaluation of second derivatives of vectors with respect to the surface parameters. Furthermore, determining the location of the contact points requires evaluating the tangent vectors and their derivatives with respect to the surface parameters, as discussed in this book. Similarly, one can show that when the constraint contact formulations are used, third derivatives of position vectors with respect to the surface parameters need to be evaluated. It is necessary to use an accurate procedure for evaluating these derivatives to correctly


212


determine the location of the contact points and forces. Note that in the computer implementation of the formulation discussed in this book, the wheel and rail profiles are described using spline functions defined using discrete nodal points. Accurate evaluation of the higher derivatives of the profile functions with respect to the surface parameters is required and, therefore, lower-order polynomials must be excluded. Note that the use of a cubic spline leads to a linear second derivative and to a constant third derivative within each interval. This approximation can be acceptable when a sufficiently large number of nodes (interpolation points) is used. To obtain a better description of the third derivatives, a spline of order higher than three can be used. The use of even-order splines, however, is in general not recommended because of several drawbacks ranging from the lack of symmetry in the boundary conditions to poor convergence properties (Ahlberg et al., 1967). Consequently, the best choice after a cubic spline is a fifth-order spline. One can still use the cubic spline and develop a procedure that leads to a better approximation of the second derivatives. For example, one can use a three-layer spline by building second and third spline functions on top of the original cubic spline (Shabana et al., 2001). As shown in Figure 6.1, a first cubic spline function S1(x) is obtained using the nodes (xi,yi), (i = 1,…,n) and the derivatives y1′ and y n′ at the end points. As a result, the spline S1(x) gives a good representation of the original function and its derivatives. This good approximation is used to generate a new set of data points, ( xi , yi′), (i = 1, …, n). The second cubic spline S2 is obtained using this new set of data points plus the derivatives y1′′ and y n′′ at the end points. Following the same procedure, it is possible to define a third spline S3 that yields a better approximation for the third derivatives. This method clearly does not affect the order of convergence of the higher derivatives, as this depends on the first spline S1(x). As is the case in many practical problems, the reference function is not known in closed form, and the main concern is the continuity of the higher derivatives. The three-layer spline proposed in this section defines smooth second and third derivatives starting with a few interpolating points. The best choice for the number of the spline layers used to represent the profiles depends on the number of data points available. If only a small number of measurement points is available, then it might be necessary to use a twolayer or three-layer spline representation to obtain a better approximation for the higher derivatives. The procedure described in this section can be used during the dynamic simulations to evaluate the tangents and their derivatives. Cubic spline subroutines are well documented and are available in the literature (Press et al., 1992). The spline representation for the profiles of the wheel and the rail allows the use of data obtained from direct measurements. However, these data points will contain all the irregularities coming from the manufacturing process and the noise from measurement and other sources. In the case of using the assumptions of nonconformal contact, it is reasonable to assume that the irregularities and the measurement noise do not have an effect on the contact force on a macro scale. However, the presence of small irregularities can have an adverse effect on the calculations of the higher derivatives of the curve, leading to a high level of numerical noise that can be the source of convergence problems (Shabana et al., 2001). Therefore, it is necessary to use smoothing techniques with the data obtained from measurements before using these data in a spline function representation. An efficient



213

FIGURE 6.1 Scheme for the generation of two- and three-layer spline. (From Shabama A.A., Berzeri, M., and Sany, J.R., 2001, Numberical procedure for the simulation of wheel/rail contact dynamics, ASME Journal of Dynamic Systems, Measurement, and Control, 123, 169–178. With permission.

way to obtain a smooth curve based on measured data is to use one of the smoothing cubic spline functions presented in the literature (Shikin and Plis, 1995). These spline functions are defined in terms of continuous cubic polynomials that have continuous first and second derivatives in the domain of definition [a,b] = [x1,xn], and they minimize the function b

n

∫

j( f ) = [ f ′′( x )]2 dx +

∑ ρ1 [ f (x ) − y ]

2

i

i =1

a

i

(6.52)

i

where (xi,yi) are the coordinates of node i, and ρi ≥ 0 are given weight functions (i = 1, …, n). The resulting spline will pass through the points (xi,yi + δi), where δi decreases as ρi increases. If the weight coefficients are chosen to be all equal to zero, then the smoothing spline becomes an interpolating spline. One possible algorithm that utilizes the spline smoothing procedure presented by Shikin and Plis (1995) can be summarized as follows: 1. Make change of coordinates by using a cumulative chordal distance sj calculated from the original measured points Pi (xi,yi), (i = 1, …, n) as j

sj =

∑P

P.

i −1 i

i=2

2. From the two sets (si,xi) and (si,yi), the curves x = x(s) and y = y(s) are obtained using the smoothing code given by Shikin and Plis (1995). 3. After this process, the new xi and yi values are combined to form a new curve y(x). 4. A third smoothing spline with smaller weighting coefficients is applied as a final refinement. This spline function generates the data points that will be used in the actual dynamic simulation. This representation has two advantages. The first is that the curves obtained are typically smoother than the curve y(x), and give a better description for the arches


214


of the original curve y(x), particularly when a curve has very high values of the first derivative or, in the case a cubic polynomial, would not provide a good fit. The second advantage is that, using two smoothing splines, it is possible to correct measurement errors that can be present not only in the coordinates yi, but also in the coordinate xi.

6.6 TRACK PREPROCESSOR Chapter 3 presents a detailed description of the geometry and the main parameters and segments used to define a track. This description can be used to numerically construct the track geometry in a form that can be utilized by computational multibody system algorithms. Two steps are followed to efficiently utilize the track geometry data. In the first step, the preprocessor stage, the track is numerically described based on specific inputs that are used and provided by the railroad industry, as discussed in Chapter 3. The track preprocessor, based on these inputs, defines the track geometry using a number of nodal points that are specified by the user. The preprocessor defines the locations of the nodes and three Euler angles that define the orientation of a coordinate system attached to these nodes. The preprocessor then generates a data file that is used as an input to the main processor used in the dynamic analysis. The distance traveled by a wheel can be predicted using the contact formulations discussed in this book. This defines the longitudinal arc length parameter that can be used, in turn, to define the segment of the rail within which the contact point lies. Using the arc length and knowing the two nodes that define this track segment, an interpolation scheme can be adopted to determine the geometric parameters that enter into the formulation of the contact conditions. These geometric parameters include the tangents and normal vectors as well as their higher derivatives, depending on the contact formulation used. It is important to point out that the track data file generated by the track preprocessor can be very large, particularly in cases where simulations are performed over a long distance. As discussed in Chapter 3, the input data to the track preprocessor include variables such as the track gage, the horizontal curvatures, the super-elevation, the grade, and the right and left rail cant angles. Specifically, the output of the track preprocessor has, as a minimum, the following information for each node: 1. The value of the arc length parameter s of the space curve at the node 2. The orientation angles θ, φ, and ψ that define the orientation of a coordinate system at the nodes on the space curve 3. The location of each nodal point with respect to the initial reference frame that will be used as the body frame of the rail or the track in the multibody system simulation code Within a segment, a polynomial whose coefficients are functions of the nodal variables given in the track preprocessor output can be formulated in the main processor to define the position vector and its derivatives between the two nodes of this segment. As another alternative, one can write the tangent and normal vectors as well as their derivatives explicitly in terms of Euler angles and their derivatives.



215

Euler angle derivatives can be provided as part of the track preprocessor output, or they can be obtained during the dynamic simulation by differentiation of the interpolation functions. If the higher derivatives of Euler angles are to be included as part of the output, the output of the track preprocessor can also include the following information: 1. The longitudinal tangent vector given in Table 3.4 of Chapter 3 and its first and second derivatives 2. The first, second, and third derivatives of the orientation angles given in Table 3.2 of Chapter 3 The disadvantage of using this alternative, in which higher derivatives of Euler angles are provided as part of the output of the track preprocessor, is the large size of the data file. In general, the output of the track preprocessor must have three different sets of data: one set for the right rail, one set for the left rail, and one set for the track centerline. If higher derivatives are to be part of the track preprocessor output, the computational algorithm presented in Chapter 3 can be slightly modified. The computational algorithm used in the track preprocessor to define the track geometry in terms of higher derivatives can then be summarized as follows: 1. Read the basic track input data, which includes the node numbers, the distance of the nodes from a specified origin, the curvature, the superelevation, the gage, and the grade. 2. Determine the functions CH(S), θ(s), and φ(S) by the following respective equations: CH =

C1 ( S − S0 ) − C 0 ( S − S1 ) , S1 − S0

θ = θ 0 + Cv (s − s0 ), and

φ=

φ1 ( S − S0 ) − φ0 ( S − S1) , S1 − S0

where all the symbols used in these equations are as defined in Chapter 3. Use this information to determine the third Euler angle ψ from the equation

ψ =ψ0 +

1  C1 S − S0 S1 − S0  2

(

)

2

−

2 C0 C S − S1  + 0 S1 − S0 , 2  2

which also was obtained in Chapter 3.

(

)

(

)


216


3. Calculate the coordinates x, y, and z of the nodes on the track centerline by evaluating numerically the integrals presented in Chapter 3 and given by the following equations: S

∫

( )

( )

( )

( )

()

()

x = x 0 + cos ψ S  dS = x S , S0

S

∫

y = y0 + sin ψ S  dS = y S , S0

and s

∫

z = z0 + sin θ s  ds = z s . s0

4. Define the space curves of the right and left rails using a finite number of nodal points. 5. Generate the derivatives of the longitudinal tangent and its derivatives up to the third derivatives using the information in Table 3.4 of Chapter 3. 6. Create an output in a format that has the track node locations and the tangent vector and its derivatives as a function of the track arc length s.

6.6.1 CHANGE

IN THE

LENGTH DUE

TO

CURVATURE

The input data to the track preprocessor describe mainly the track centerline. Given the gage and other input parameters, the geometry of the right and left rails can be constructed. In the case of tangent tracks with no deviations, the length of the right and left rails is the same as the length of the track centerline. In the case of a curved track, on the other hand, the arc lengths of the space curves of the track centerline and the right and left rails differ. As described in Chapter 3, the location of the nodal points on the space curves are obtained using numerical integration. The results of the numerical integration define the coordinates of each point of the track centerline with respect to the body coordinate system: O lX lY lZ l for the left rail or OrXrYrZr for the right rail. To account for the change in the arc length of the right and left rails due to the curvature, another term in addition to the coordinates of the track centerline needs to be calculated using numerical integration. Figure 6.2 shows a curve segment of a track. The value of the gage is exaggerated in this figure to illustrate the difference between the lengths of the arcs AB, ArBr, and AlB l. The length of the arc ArBr can be calculated by considering the fact that, at an arbitrary point on this arc, the horizontal radius of curvature will be equal to the radius RH and half the horizontal distance ∆R between the rails. The value of the horizontal distance can be calculated as shown in Figure 3.9 of Chapter 3 as



217

FIGURE 6.2 Top view of a curve segment.

∆R = d cos φ

(6.53)

where the distance d between the rails is given as the sum of the gage G and the width of the rails w. From this equation, it is clear that the infinitesimal arc length dSr for the right space curve is given by    1  1 dS r =  RH + d cos φ  dψ =  1 + CH d cos φ  dS 2    2 

(6.54)

When no irregularities are considered, the rail profile frames and the track frame corresponding to the same cross-section have the same orientation angles. Therefore, the equation S1

∫

S = S0 + cos θ (s)  ds = S (s) S0

obtained in Chapter 3 also applies to both dsr and dSr, so that one obtains ds r =

 dS r  1 = 1 + CH d cos φ  ds cos θ  2 

(6.55)

The difference between dsr and ds can be used to determine the total length lr of the segment (arc) ArBr that differs from the length l of AB by the quantity s1

∫

1 ∆l = CH d cos φ ds 2 s0

(6.56)


218


FIGURE 6.3 Lengths of the curve segments.

The same difference, with opposite sign, applies to the length ll of the segment AlB l. Therefore, the length l r and l l of the right and left segments, respectively, can be calculated as l l = l − ∆l , l r = l + ∆l

(6.57)

Figure 6.3 shows the difference between the length l of the arc AB and the length l r of the arc ArB r. The arc length s measured on the track centerline and the arc length sr measured on the space curve of the right rail are also shown. The output of the track preprocessor must show the value of the arc length s for the right and left rails, which is updated according to Equation 6.57.

6.6.2 USE

PREPROCESSOR OUTPUT DYNAMIC SIMULATION

OF THE

DURING

The output data of the track preprocessor are expressed in terms of the arc length parameter of the space curve s = s1. However, the complete representation of the rail surface requires the use of two surface parameters, as described in Section 3.2 of Chapter 3. It is assumed that the profile of the rail is known and can be described using the function f(s2), where s2 is the second (lateral) rail surface parameter. (The superscript r that refers to the rail is dropped here for simplicity.) If the rail profile remains the same along the track, and if s1 and s2 are the parameters that define the location of a contact point, the location of the contact point on the rail with respect to the rail body coordinate system can be written as a function of the two surface parameters s1 and s2 (as shown in Chapter 3) as follows: u r (s1, s2 ) = R rp (s1 ) + A rp (s1 )u rp (s2 )

(6.58)

where R rp is the vector that defines the location of the origin Orp of the profile coordinate system in the body r coordinate system, and A rp is the matrix that defines the orientation of the profile frame with respect to the body frame, as discussed in Chapter 3. Using the information included in the track preprocessor output, one can use linear or cubic interpolation to define R rp (s1 ) and A rp (s1 ) and their derivatives



219

with respect to s1, as previously described in this chapter. The vector u rp (s2 ) , which represents the location of the contact point with respect to the profile coordinate system (as shown in Chapter 3) is given as follows: u rp (s2 ) = 0

s2

f (s2 ) 

T

(6.59)

Using Equation 6.58, one can define the two tangent vectors and the normal vector as follows:  ∂u rp dR rp dA rp (s1 ) rp = + u (s2 )  ds1 ds1 ∂s1   rp rp du (s2 ) ∂u  rp = A (s1 ) t2 =  ∂s2 ds2   n = t1 × t2    t1 =

(6.60)

where t1 and t2 are the two tangent vectors at a point on the rail that has the two surface parameters s1 and s2 as coordinates. These two tangent vectors, which are defined with respect to the rail coordinate system (body coordinate system) and are not necessarily orthogonal, represent the tangent plane at this point. The normal vector n is defined using the third equation in Equation 6.60 as the cross-product of the two tangent vectors. As previously explained, the tangent and normal vectors and their derivatives enter into the formulation of the contact forces and kinematic constraints.

6.7 DEVIATIONS AND MEASURED DATA Thus far the discussion has been focused on the analytical description of the track geometry. In reality, tracks can have irregularities known as deviations. It is important to include the effect of the deviations in the description of the track geometry to accurately examine the response of railroad vehicle systems. As discussed in Chapter 1, new vehicles are tested using specific simulation scenarios that involve certain standard track irregularities designed to excite a certain mode of motion. This section presents a brief description of track deviation functions that can be used in railroad vehicle dynamic simulations, and the implementation of these functions in the track preprocessor is discussed. Another important requirement in railroad dynamics is to examine the response of a vehicle over real tracks that are defined using field measured data. A track preprocessor must be designed to read these data and use this information to construct the track geometry. This section proposes a procedure for the use of field measured data that can be implemented in the track preprocessor.


220


6.7.1 TRACK DEVIATIONS Track deviations are defined either in the vertical or lateral direction. The vertical deviations are known as profile deviations, while the lateral deviations are known as alignment deviations. Profile and alignment deviations can have arbitrary shapes. In railroad dynamics, the analytical track deviations are defined using a set of functions that are shown in Table 6.1 and that are defined based on irregularities

TABLE 6.1 Track Deviation Functions Deviation

Shape

Function

Cusp

y

Bump

y

Jog

y

Plateau

y

Trough

y

Ae

Ae

ks

1 2 ks

2

Aks (1 4k 2 s 2 )

A2 1

Ak

8

ks

1/ k

2

s2



221

TABLE 6.1 (continued) Track Deviation Functions

y

Sinusoid

y

Damped Sinusoid

A sin ks

Ae

ks

cos ks

Source: Hamid, A. et al., Analytical Description of Track Geometry Variations, FRA report DOT/FRA/ORD-83/03.1, USDOT, Federal Railroad Administration, Washington, DC, 1983.

observed from the measured track data (Hamid et al., 1983). Clearly, some of these functions are exponential, which means that their starting and ending points are at infinity. One can assume that these functions are applied on the positive section of its abscissa. Therefore, the starting point of the function will be at s = 0 in Table 6.1, and it will increase along the track, assuming that the end of the track is at the point of infinity. In general, these functions can be used either in the vertical or the lateral direction. Since it is easier to define these functions with respect to the local space curve frame, and since the track preprocessor defines the locations of the nodes with respect to the body coordinate system, the track deviations must also be defined with respect to the body coordinate system. For a given deviation function, one can write a vector u that defines the coordinates of a point on the space curve with respect to the track frame, as shown in Figure 6.4. If the deviation is in the lateral direction, the vector u takes the following form: u = 0

f (s r )

0 

T

(6.61)

where ƒ(sr) is a function selected from Table 6.1. The coordinates of the rail space curve can be changed using the definition of the vector u . This vector can be defined with respect to the body coordinate system as follows: u = A rp u

(6.62)


222


FIGURE 6.4 Rail deviation.

where A rp, as discussed in Chapter 3, is the orientation matrix of the rail profile coordinate system with respect to the body coordinate system. The change in the orientation of the space curve due to the deviation can be determined as follows:

∆β = tan −1

df (s r ) ds r

(6.63)

where β = ψ or θ, depending on whether the deviation is lateral or vertical, respectively. Therefore, one can implement the following numerical procedure in the track preprocessor to account for the effect of the deviations: 1. After generating the track nodes, the numbers of the nodes at which the track deviation starts and ends are determined. 2. The space curve orientation matrix A rp at the nodes that represent the domain of the deviation is evaluated. 3. Depending on the deviation type, the vector u is evaluated. 4. Evaluate the vector u and define the change in the angle according to Equations 6.62 and 6.63. 5. Superimpose the vector u on the track nodal position vectors. 6. Based on the deviation type, update the track nodal rotations in the deviation region. Using this simple algorithm, the track preprocessor can be designed to construct track geometry with arbitrary deviations.

6.7.2 MEASURED TRACK DATA Measured track data are important in examining the vehicle dynamic response. In most accident investigations, dynamic simulations based on field measured track



223

data must be performed to identify the cause of the accident. Measured track data can be provided in different formats. In general, measured track data include the point locations along the track, the gage, the curvature, the super-elevation, right and left rail profile deviations, and right and left rail alignment deviations. A simple and straightforward procedure for implementing the field measured data in a track preprocessor is to use the measured data, including the deviations, to define the standard inputs to the track preprocessor. In this case, the gage is variable and must be defined at each node. For simplicity, it is recommended to use a fixed mean value for the gage and use the actual gage to define alignment deviation. In this case, the alignment due to the gage variation can be equally divided between the right and left rails. Furthermore, since the measured track data are often provided as raw data, it is recommended to use a smoothing process similar to the one previously discussed in this chapter. In some standards, the gage and super-elevation in the measured data can also be filtered using a moving-average window. The width of this window depends on the format of the data.

6.7.3 TRACK QUALITY

AND

CLASSES

Tracks are classified based on the maximum allowable deviations (irregularities). As previously mentioned in this section, the track irregularities can be classified into two categories: alignment and profile. In some cases, due to load changes across the ties, the track experiences a change in the cross level known as warp (twist), which is more accurately defined as the rate of the change in cross level. In general, there are nine classes of track in the U.S. (Track Safety Standards Part 213 subpart A-F and G). Each class is defined by a gage limit and by maximum allowable deviations for alignment and profile. Based on the vehicle type and design, each class corresponds to a maximum allowable vehicle speed. Table 6.2 shows the maximum allowable speed for each class of track. In general, lower classes are used for freight trains, but in some cases a passenger train can be operated on tracks of lower classes.

TABLE 6.2 Track Classes (Track Safety Standards Part 213, Subpart A to F and G) Maximum Allowable Speed (mph)

Track Class

Freight Train

Passenger Train

1 2 3 4 5 6 7 8 9

10 25 40 60 80 … … … …

15 30 60 80 90 110 125 160 200


224


TABLE 6.3 Gage Limit for Each Track Class (Track Safety Standards Part 213, Subpart A to F and G) Track Class

Minimum Gage Limit (in.)

Maximum Gage Limit (in.)

1 2 3 4 5 6 7 8 9

56 56 56 56 56 56 56 56 56.25

58 57.75 57.75 57.50 57.50 57.25 57.25 57.25 57.25

The data of Table 6.2 show that higher track classes correspond to higher track quality that allows trains to operate at higher speeds. For track classes 6 to 9, freight trains are allowed to travel at passenger train speeds if specific requirements are met (Track Safety Standards Part 213 subpart A-F and G). Note that the gage depends on the maximum allowable track alignment irregularities. Table 6.3 shows the gage limit for each class of track. In general, the change in the gage is limited to a maximum value of 0.5 in. within 31 ft for track classes 6 and higher. Table 6.4 and Table 6.5 show the maximum allowable alignment and profile deviations for track classes 6 and higher, respectively. In Table 6.4 and Table 6.5, δ1, δ2, and δ3 are the maximum allowable deviation from uniform profile on either rail at the middle of 31-, 62-, and 124-ft chords, respectively.

TABLE 6.4 Alignment Limit for Each Track Class (Track Safety Standards Part 213, Subpart G) Three or More Nonoverlapping Deviations

Single Deviation

Track Class

δ1 (in.)

δ2 (in.)

δ3 (in.)

δ1 (in.)

δ2 (in.)

δ3 (in.)

6 7 8 9

0.5 0.5 0.5 0.5

0.75 0.5 0.5 0.5

1.5 1.25 0.75 0.75

0.375 0.375 0.375 0.375

0.5 0.375 0.375 0.375

1.0 0.875 0.5 0.5

Note: δ1, δ2, and δ3 are the maximum allowable deviation in alignment on either rail at the middle of 31-, 62-, and 124-ft chords, respectively.



225

TABLE 6.5 Profile Limit for Each Track Class (Track Safety Standards Part 213, Subpart G) Three or More Nonoverlapping Deviations

Single Deviation

Track Class

δ1 (in.)

δ2 (in.)

δ3 (in.)

δ1 (in.)

δ2 (in.)

δ3 (in.)

6 7 8 9

1.0 1.0 0.75 0.5

1.0 1.0 1.0 0.75

1.75 1.50 1.25 1.25

0.75 0.75 0.50 0.375

0.75 0.75 0.75 0.50

1.25 1.0 0.875 0.875

Note: δ1, δ2, and δ3 are the maximum allowable deviation from uniform profile on either rail at the middle of 31-, 62-, and 124-ft chords, respectively.

As mentioned previously, a track can have certain deviations that can be represented by using closed-form functions. These functions are cusp, bump, jog, plateau, trough, sinusoid, and damped sinusoid. These functions, which are sometimes called signatures, can be used to represent track alignment or profile deviations. Hamid (Hamid et al., 1983) defined the parameters that can be used to represent these functions based on actual measurements. The values of these parameters are given in Table 6.6. Hamid et al. (1983) also provided a list of the possible occurrences for each of these track deviation functions, a summary of which is given in Table 6.7. In general, a single cusp occurs as a profile irregularity due to the joint between welded rails. Bumps can be alignment or profile irregularities and can be found simultaneously on both rails. In general, a bump has a smoother shape and covers a longer distance than a cusp. In the case of bridges, bumps can be considered as profile irregularities, while spirals can have alignment bumps at a certain distance from their start. A jog can occur due to variations in the track stiffness, as in the case of the connection between soft track and bridge. Plateau is caused by variations of track stiffness or the wear of the high rail. In some cases, a combination of cusp and plateau can be found prior to the spiral. Such a combination is dangerous, since it causes a rapid change in the wheel load. Trough is the result of poor drainage or localized soft subgrade. A sinusoid is found in the case of bridge or reverse curves where no tangent segment is used to separate between the two curves. A damped sinusoidal occurs usually on a single rail, and it is usually the result of a significant change in the track stiffness due to switches, grade crossings, and curves.

6.8 SPECIAL ELEMENTS In addition to the geometry and contact problems, there are some special elements that distinguish railroad vehicles from other multibody system applications. This section discusses the formulations of several force elements that are often used in railroad vehicle models. Among these elements are the translational and rotational

0.8–1.4 0.8–1.4 … 0.8–1.3 … … 0.5–1.0

Cusp Bump Jog Plateau Trough Sinusoid Damped sinusoid 0.5–0.3 0.5–2.8 0.5–3.3 1.2–1.6 1.4–2.2 0.8–1.2 1.0–2.2 0.011–0.103 0.009–0.083 0.006–0.025 0.025–0.027 0.013–0.029 0.033–0.020 0.013–0.015 0.9–3.0 1.0–3.0 1.6–2.8 0.6–1.0 … … 0.9–1.2

0.031–0.095 0.017–0.031 0.020–0.05 0.026–0.04 … … 0.051–0.061

Cross Level k (ft−1) A (in.)

Range of Values Alignment k (ft−1) A (in.)

0.9–3.0 0.5–4.0 0.5–5.0 0.9–3.0 0.7–2.0 1.0–1.5 …

A (in.)

0.016–0.095 0.013–0.065 0.008–0.045 0.009–0.033 0.020–0.025 0.020–0.025 …

Profile k (ft−1)


0.016–0.061 0.031–0.040 … 0.029–0.08 … … …

Gage A (in.) k (ft−1)

226

Deviation Function

TABLE 6.6 Parameters of Analytical Representations of Track Irregularities





227

TABLE 6.7 Occurrence of Track Deviation Functions Deviation Function

Occurrence

Cusp

Joints, turnouts, interlocking, sun kinks, buffer rail, insulated joints in continuous welded rail (CWR), splice bar joint in CWR, piers at bridge

Bump

Soft spots, washouts mud spots, fouled ballast, joist, spirals, grade crossings, bridges, overpasses, loose bolts, turnouts, interlinking

Jog

Spirals, bridges, crossings, interlocking, fill-cut transitions

Plateau

Bridges, grade crossing, areas of spot maintenance

Trough

Soft spots, soft and unstable subgrades, spirals

Sinusoid

Spirals, soft spots, bridges

Damped sinusoid

Spirals, turnouts, localized soft spot


spring-damper-actuator elements, the series spring-damper element, and the bushing element. Railroad vehicles, for example, have two types of suspensions, primary and secondary suspensions. The primary suspension is used to connect bodies such as bogie frames or equalizer bars to the wheelset. The secondary suspension is used to connect the car body to the bogie or trucks. These suspensions are a combination of translational spring-damper-actuator elements, rotational (torsional) spring damper elements, friction elements, etc. In some vehicles, the car can be connected to the bogie by a joint such as a pin (revolute) joint or a cylindrical joint. The wheels are mounted on the frames using bearing elements. Figure 6.5 shows examples of elements that can be used in modeling railroad vehicles. Chapter 2 presents several joints that can be used in the modeling of railroad vehicles. This section, on the other hand, discusses the formulations of several force elements that can be used in modeling the suspension of a railroad vehicle.

6.8.1 TRANSLATIONAL SPRING-DAMPER-ACTUATOR ELEMENT This element can consist of a spring, a damper, and an actuator. The coefficients used in this element formulation to define the force can be linear or nonlinear functions of the relative motion and velocity of the two bodies connected by this element. Consider a spring-damper-actuator element that connects two bodies i and j as shown in Figure 6.6. The force element is connected to body i at point Pi that has a local position vector u iP with respect to the body coordinate system. It is also connected to body j at point Pj that has a local position vector u Pj with respect to the body j coordinate system, as shown in Figure 6.6. The spring constant is assumed to be k, the damping coefficient is c, and the actuator force acting along a line


228


FIGURE 6.5 Elements of railroad vehicle models.

FIGURE 6.6 Translational spring-damper-actuator element.

connecting points Pi and Pj is fa. The spring has an undeformed length equal to l0. Therefore, the force acting along a line connecting points Pi and Pj can be written as follows:

(

)

Fs = k l − l0 + cl + f a

(6.64)

where l and l are the deformed spring length and its time derivative, respectively. The length l is given by T

l = rPij = rPij rPij

(6.65)

where rPij is the position vector of point P i with respect to point P j and is given by



229

rPij = rPi − rPj = R i + A i u iP − R j − A j u Pj

(6.66)

where Ri and R j are the global position vectors of the coordinate systems of bodies i and j, respectively, and Ai and Aj are the transformation matrices that define the orientation of the coordinate systems of the two bodies. The virtual work of the force Fs of Equation 6.64 is given by

δ W = − Fsδ l

(6.67)

where δl is the virtual change in the spring length given by Equation 6.65. This virtual change can be written as follows: T

δl =

rPij δ rPij

(r r ) ij p

T

ij p

1 T = rPij δ rPij l (6.68)

T

=

rPij  i δ R − u iPGiδθ i − δ R j + u PjG jδθ j  l 

where u iP and u Pj are the skew symmetric matrices associated with the vectors A i u iP and A j u Pj , respectively; θ i and θ j are the orientation parameters of the two bodies; and Gi and G j are the matrices that relate the angular velocity vector to the time derivatives of the orientation parameters of the two bodies. Therefore, the virtual work of Equation 6.67 can be written as follows:

δ W = − FsrˆPij δ R i − u iPGiδθ i − δ R j + u PjG jδθ j  T

(6.69)

where rˆPij = rPij l is a unit vector along the line that connects points P i and P j. Equation 6.69 can also be written in the following form: T

T

T

T

δ W = QiR δ R i + Qθi δθ i + Q Rj δ R j + Qθj δθ j

(6.70)

where QiR , Qθi , Q Rj , and Qθj are the generalized forces associated with the coordinates of bodies i and j, respectively, and are given by    i i T i T ij Qθ = FsG u P rˆP    Q Rj = FsrˆPij  T T Qθj = − FsG j u Pj rˆPij  QiR = − FsrˆPij

(6.71)


230


Therefore, the generalized spring-damper-actuator force vectors acting on bodies i and j, respectively, can be defined using the preceding equations as follows:  Qi  Qi =  Ri  ,  Qθ 

Q j  Q j =  Rj   Qθ 

(6.72)

These vectors can be introduced to the right-hand side of Equation 6.1 as generalized forces associated with the generalized coordinates of the two bodies.

6.8.2 ROTATIONAL SPRING-DAMPER-ACTUATOR ELEMENT While the translational spring-damper-actuator element produces rectilinear force, the rotational spring-damper-actuator element produces torque. The coefficients that enter into the formulation of the rotational spring-damper-actuator element can be linear or nonlinear functions of the relative rotations and relative angular velocities of the two bodies connected by this element. Consider a rotational spring-damperactuator element that connects two bodies i and j as shown in Figure 6.7. The spring constant is assumed to be kθ , the damping coefficient is cθ , and the actuator torque acting in the direction of the rotation axis defined by the unit vector n ij is Tθ. Therefore, the torque exerted on body i as the result of the rotation θ ij with respect to body j about the spring axis is

(

)

Ts = kθ θ ij − θ 0ij + cθθ ij + Tθ

(6.73)

In this equation, θ oij is the relative rotation between the two bodies about the axis of rotation when the spring is undeformed. The virtual work of the torque Ts is given by

δ W = −Tsδθ ij

FIGURE 6.7 Rotational spring-damper-actuator element.

(6.74)



231

In the analysis presented in this section, it is assumed that the relative rotation θ ij between bodies i and j is small. In this case, it can be shown that the relation between the infinitesimal virtual rotation vector δβ i about the axes of the global Cartesian coordinate system and the virtual change in the generalized orientation coordinates δθi are related as follows (Shabana, 2001):

δ β i = Giδθ i

(6.75)

where Gi is the matrix that relates the angular velocity vector to the time rate of the orientation coordinates of the body, as discussed in Chapter 2. Using the preceding equation, one can write the virtual change of the relative rotation δθ ij between bodies i and j about the axis nij as T

T

δθ ij = n ij (δ β i − δ β j ) = n ij (G jδθ j − G jδθ j )

(6.76)

It follows that the virtual work of Equation 6.74 can be written as follows: T

δ W = −Ts n ij (G jδθ j − G jδθ j )

(6.77)

This equation can be written compactly as T

T

δ W = Qθi δθ j + Qθj δθ j

(6.78)

One can, therefore, use the preceding two equations to define the generalized forces associated with the two bodies that result from the rotational spring-damperactuator element as   0 Qi =  , i T ij  −TsG n 

 0  Qj =   j T ij TsG n 

(6.79)

These forces can be used in Equation 6.1 as generalized applied forces. Note that this force element has no components associated with the translation coordinates of the two bodies.

6.8.3 SERIES SPRING-DAMPER ELEMENT The series spring-damper element is widely used in railroad vehicle system applications. In this element, the spring and the damper are connected in series, as shown in Figure 6.8. In the case of the series connection, the force in the spring and the damper is the same and is denoted as Fs. One can therefore write the following equation for the force Fs: Fs = kδ s = cδd

(6.80)


232


FIGURE 6.8 Series spring-damper element.

where δs is the deformation in the spring, and δd is the relative velocity between the two ends of the damper defined by points P1 and P2, as shown in Figure 6.8. The total relative displacement between the two end points P1 and P3 of the series springdamper element is denoted as δt and can be written as

δt = δs + δd

(6.81)

δt = δs + δd

(6.82)

It follows that

It is clear from Equation 6.80 that Fs = kδs . Using this equation with Equations 6.80 and 6.82, one obtains the following differential equation: dFs k dδ + Fs = k t dt c dt

(6.83)

Since the positions and velocities of the end points of the series spring-damper element are functions of the generalized coordinates and velocities of the two bodies connected by this element, the solution of Equation 6.83 can be obtained on-line to determine the spring force Fs. Following a procedure similar to the one used for the translation spring-damper-actuator element, the generalized forces due to the series spring-damper element can be formulated.

6.8.4 BUSHING ELEMENT The bushing element is commonly used in vehicle applications, including railroad vehicles. In most applications, the bushing element is made of rubber and it produces



233

FIGURE 6.9 Bushing element.

stiffness and damping forces in different directions. In the formulation presented in this section, it is assumed that the bushing element can have different values of linear or nonlinear stiffness and damping coefficients in different directions. In the model developed in this section, a bushing coordinate system that is assumed to be rigidly attached to one of the bodies connected by the bushing element (say body j) is introduced, as shown in Figure 6.9. This coordinate system can be defined in the initial configuration based on data provided by the user for the location of two points on the body. Using the position vectors u Pj1 and u Pj2 of two points P1j and P2j on body j, one axis of the coordinate system of the bushing can be defined as follows:

n

j

(u =

j P1

− u Pj2

u −u j P1

)

(6.84)

j P2

where n j is one of the bushing axes defined in the body j coordinate system, as shown in Figure 6.9. This axis can be defined in the global coordinate system as nj = A j n j , where Aj is the transformation matrix that defines the orientation of the coordinate system of body j. The axis n j can be used to define the directional properties of the bushing element. Using this axis, one can construct the other two axes of the bushing coordinate system and define its orientation with respect to the body j coordinate system using the following transformation matrix: A bj =  t1j

t2j

n j 

(6.85)

where t1j and t2j are the two unit vectors that complete the three orthogonal axes of the bushing coordinate system. Assuming that body j is a rigid body, the bushing


234


coordinate system defined with respect to the global frame can be determined as follows: A bj = A j A bj

(6.86)

If the bushing (a) is assumed to be connected to body i at point Pi, which is defined by the position vector u iP with respect to the body i coordinate system, and (b) is connected to body j at point P1j , which is defined by the position vector u Pj1 with respect to the body j coordinate system, then one can define the following vector of relative position between the two points Pi and P1j : rij = R i + A i u iP − R j − A j u Pj1

(6.87)

i u i − R j − A ju j r ij = R i + A P P1

(6.88)

It follows that

Using Equation 6.86, the preceding two equations, and assuming that points P i and P1j initially coincide, one can define the following bushing deformation and rate of deformation vectors in the bushing coordinate system: T δ bij = Abj r ij

T

δ bij = A bj r ij ,

(6.89)

The rotational deformation of the bushing is defined by the relative orientation of the bushing coordinate systems of bodies i and j as follows: T

A bij = A bj A bi

(6.90)

where Abj is the orientation matrix of the bushing coordinate system of body j (as previously defined by Equation 6.86), while Abi is the orientation matrix of the bushing coordinate system of body i that is defined as A bi = A i A bi . Note that the local orientation matrix of the bushing coordinate system of body i is defined at the initial configuration as T

A bi = Ai A bj

(6.91)

t =0

Assuming that the relative rotations between the two bodies connected by the bushing element are small, the relative rotation matrix Abij can be used to extract three relative rotations, θ xbij , θ ybij , and θbij z , defined in the bushing coordinate system, i.e., θ bij = θ xbij

θ ybij

T

θ zbij  .



235

The relative angular velocity between the two bodies defined in the bushing coordinate system can also be determined as follows: T

(

ω bij = A bj ω i − ω j

)

(6.92)

where ω i and ω j are the absolute angular velocity vectors of bodies i and j, respectively, defined in the global coordinate system. The bushing stiffness and damping coefficients are often determined using experimental testing, and these coefficients are defined generally in the bushing coordinate system. In the multibody formulations, the bushing coefficients are defined using stiffness and damping matrices such as  k xx  Kr =  kyx   kzx

k xy kyy kzy

k xz   kyz  ,  kzz 

c xx  C r =  c yx   c zx

c xy c yy c zy

c xz   c yz   c zz 

(6.93)

where Kr and Cr are the translational stiffness and damping matrices, respectively, defined with respect to the bushing coordinate system. The rotational stiffness matrix Kθ and damping matrix Cθ can be written in a similar form. In most cases, the diagonal form of the stiffness and damping matrices is used for simplicity. In terms of these matrices, the force vector defined in the bushing coordinate system can be written as follows:  FRb  K r  b =   Mθ   0

0   δ bij  Cr  +  Kθ   θ bij   0

0   δ bij    Cθ  ω bij 

(6.94)

This force vector can be defined in the global coordinate system as  FRb   A bj FRb   b  =  bj b   Mθ   A Mθ 

(6.95)

Using the principle of virtual work, the generalized bushing forces associated with the generalized coordinates of the two bodies can be obtained as follows:    i iT iT b iT b Qθ = G u P FR − G Mθ   Q Rj = FRb   T T T Qθj = −G j u Pj1 FRb + G j Mθb   QiR = − FRb

(6.96)


236


These generalized forces are introduced to the system equations of motion using the vector Q of Equation 6.1.

6.9 MAGLEV FORCES The contact between the wheels and the rails is one of the main factors that limit increasing the speed of railroad vehicle systems. The vehicle critical speed depends on the creep forces, which significantly influence the hunting motion. On the other hand, the noncontact driving concept used in magnetically levitated (Maglev) trains eliminates this restriction and allows for increasing the train operational speed. It is known that magnetic suspensions require less power than conventional railroad vehicle systems, and since there is no contact or friction in the magnetic levitation, the noise and wear can be reduced, resulting in improved level of comfort, less maintenance, and environmentally acceptable mass-transportation systems (Dukkipati, 2000). There are two different types of magnetic suspension systems currently used in Maglev vehicles: electrodynamic suspension (EDS) and electromagnetic suspension (EMS), as shown in Figure 6.10. In this section, the fundamental principles of the magnetic levitation system are briefly reviewed, and the modeling issues that arise in developing multibody Maglev vehicle models are discussed.

6.9.1 ELECTRODYNAMIC SUSPENSION (EDS) In the electrodynamic suspension system, the vehicle is lifted by the magnetic forces that act on the vehicle and the guideway, as shown in Figure 6.10a. Since the electromagnetic field is developed as the vehicle moves, the flux produced by the onboard coils induces currents in the passive coils or nonmagnetic sheets on the guideway. As a result, the induced currents produce a magnetic flux that opposes the magnetic flux of the onboard electromagnet, producing repulsive forces between the vehicle and the guideway. Since the repulsive forces are produced as the vehicle moves above the passive coil on the guideway, the vehicle cannot be lifted unless a certain speed is achieved and, therefore, the electrodynamic suspension system requires

FIGURE 6.10 Maglev suspensions.



237

FIGURE 6.11 Electromagnetic suspension.

wheels at standstill and at low speed. For this reason, this suspension system is not suitable for an urban mass transportation that requires frequent stops at several stations.

6.9.2 ELECTROMAGNETIC SUSPENSION (EMS) In the electromagnetic suspension system, the vehicle is lifted by the forces of attraction produced between the electromagnet built on the vehicle and the ferromagnet on the guideway, as shown in Figure 6.10b. The iron core of a magnetic circuit is excited by a current-carrying coil, and as a result, the core on the vehicle is attracted to the ferromagnetic rail. As will be explained later in this section, the electromagnetic suspension is statically unstable in the sense that the attractive force increases as the air gap between the pole face of the electromagnet and the ferromagnet decreases. For this reason, an appropriate levitation control needs to be developed and implemented to achieve statically and dynamically stable suspension characteristics. Since the levitation force is independent of the vehicle speed, the vehicle can be lifted at low speed and at standstill when the electromagnetic suspension system is used. This suspension system, therefore, is suitable for urban mass transportation systems that require low operating speed and frequent stops at several stations. Note also that eddy currents that tend to decrease energy efficiency are induced in the case of high-speed operation.

6.9.3 MODELING

OF

ELECTROMAGNETIC SUSPENSIONS

The levitation forces of electromagnetic suspension systems are discussed here in an effort to develop a model for these forces that can be implemented in the computational multibody system algorithms. As shown in Figure 6.11, let the length of the pole face be a, the width be b, the permeability of the free space be µ0, and the flux density across the air gap be Ba. The force of attraction between the pole surface and the ferromagnet is defined by (Sinha, 1987)

Fz =

( Ba )2 A µ0

(6.97)


238


where A = ab. The flux density can be expressed in terms of the reluctance of the mutual flux RM as Ba =

Fmmf ARM

(6.98)

where Fmmf is a magnetomotive force, which can be defined as Fmmf = N i(t )

(6.99)

In the preceding equation, N is the number of turns of the coil, and i(t) is the current in the coil. Substituting Equation 6.98 into Equation 6.97 and using Equation 6.99, one obtains Fz =

1  N i(t )  µ0 A  RM 

2

(6.100)

The reluctance of the mutual flux, denoted as RM, consists of the reluctance of the air gap, electromagnet, and the ferromagnet, which can be respectively defined as (Dukkipati, 2000) RM (d z ) =

2d z w + 2h + 2a w + 2a + + µ0 ab µc ab µ pa1b

(6.101)

where dz is the distance between the pole face and the ferromagnet, i.e., dz = dz(t); a1 is the thickness of the ferromagnet; w and h are the dimensions shown in Figure 6.11; and µc and µp are, respectively, the permeability of the electromagnet and the ferromagnet. Assuming that µi ≡ µc ≈ µp and a1 ≈ a, the preceding equation can be reduced to RM (d z )

(

2 d z (t ) + r0 µ0 A

)

(6.102)

where r0 is given as r0 =

µ0 (2a + w + h) µi

(6.103)

Substituting Equation 6.102 into Equation 6.100, one has 2

µ N 2 A  i(t )  Fz (d z , i) = 0 ≤ ( Fz )max 4  d z (t ) + r0 

(6.104)



239

FIGURE 6.12 Characteristics of the levitation force.

It can be seen from the preceding equation that the attractive (levitation) force increases as the current increases, as shown in Figure 6.12a. On the other hand, since the attractive (levitation) force is defined as a reciprocal function of the air gap distance dz(t), the levitation force decreases as the air gap increases, as shown in Figure 6.12b, leading to inherently unstable suspension characteristics that are a function of the air gap. For this reason, the electromagnetic suspension requires an appropriate feedback control system to achieve stable suspension characteristics. It should also be noted at this point that the flux density across the air gap Ba has saturation characteristic, and therefore, the preceding attractive force should be less than the maximum force (Fz)max that is given by ( Fz )max =

( Ba )2max A µ0

(6.105)

where (Ba)max is the maximum flux density that is given as (Dukkipati, 2000) ( Ba ) max =

PM B0 PM + PL

(6.106)

and B0 in this equation is in the range of 1.5 ≈ 2.0 Wb/m2; PM is the permeance of mutual flux, given as PM = 1/RM; and PL is the permeance of the leakage flux given as b  πa  PL = µ0 h  + ln  1 +  2 w   w

(6.107)

Having obtained the electromagnetic force, one can determine the generalized electromagnetic forces of body j associated with the reference coordinates Rj and the orientation coordinates θ j as T

j j (Qmf ) R = Fmfj , (Qmf )θ = G j (u Pj × Fmfj )

(6.108)


240


where Fmfj is the vector of the electromagnetic force defined in the global coordinate system, Gj is the matrix that relates the global angular velocity of body j to the time derivatives of the orientation coordinates, and u Pj is the vector of the point of application of the force defined in the global coordinate system. Since it is convenient to define the magnetic force with respect to the body coordinate system, the following transformation is used: Fmfj = A j Fmfj , u Pj = A j u Pj

(6.109)

where Fmfj and u Pj are, respectively, the electromagnetic force vector and the vector of the point of application of the force defined in the body coordinate system, and are given as Fmfj = [0

0

Fz j (d zj , i)]T , u Pj = [ x Pj

y Pj

j

z P ]T

(6.110)

The generalized electromagnetic forces of Equation 6.108 can simply be added to the generalized forces on the right-hand side of the multibody system equations of motion as M  Cq

CTq   q  Qe + Qv + Qmf    =   0   λ   Qd 

(6.111)

where M is the system mass matrix; Qv is the vector of inertia forces that are quadratic in the velocities; Qe is the vector of the generalized external forces; Cq is the Jacobian matrix of the constraint equations; λ is the vector of Lagrange multipliers; and Qd is the vector resulting from the differentiation of the system constraint equations twice with respect to time.

6.9.4 MULTIBODY SYSTEM ELECTROMECHANICAL EQUATIONS Recall that the electromagnetic force given by Equation 6.104 is function of the current i(t) as well as the air gap dz (t) defined by the system generalized coordinates q. As a result, one needs to solve the preceding multibody equations simultaneously with electronic circuit equations expressed for i(t) as v = Ri(t ) +

d ( L (t )i(t )) dt

(6.112)

= Ri(t ) + L (t )i(t ) + i(t ) L (t ) where v is a supplied power voltage, and L is the inductance of the electromagnetic suspension defined by  1 1  L (t ) = N 2  +  RL RM 

(6.113)



241

where the reluctance of the leakage flux RL = 1/PL can be obtained using Equation 6.107, while the reluctance of the mutual flux RM is given by Equation 6.102. As a result, the total inductance L(t), which can be written as a function of the air gap dz(t), is obtained as L (d z ) = LL +

µ0 N 2 A 2(d z (t ) + r0 )

(6.114)

where LL = N 2PL = N 2/RL. To achieve stable electromagnetic suspension characteristics, the supplied power voltage is controlled by the position feedback such that the air gap remains constant (Sinha, 1987). In such a case, the power voltage is defined as a function of the air gap dz and its velocity as v ( d z , dz ) = v0 + f ( d z , dz )

(6.115)

where ν0 is the power voltage at the static equilibrium configuration given for a gap distance in operation, and f is a control function that depends on dz or dz . Substituting Equation 6.114 into Equation 6.112, one has the following equation for i(t ) :  µ0 N 2 A dz (t ) µ0 N 2 A   LL + 2(d (t ) + r )  i (t ) = − Ri(t ) + 2(d (t ) + r )2 i(t ) + v  0  z z 0

(6.116)

Using the preceding equation, the current for all the electromagnetic suspensions in the multibody vehicle can be written in the following vector form: i(t ) = Z (i, q, q )

(6.117)

where i = [i1, …, inm]T and nm is the number of the electromagnetic suspensions, and Z is a vector function that depends on the currents as well as the system generalized coordinates and velocities. Using Equations 6.111 and 6.117, the equations of motion of an electromagnetic-type Maglev vehicle can be written as M  C q   0

CTq 0 0

0 q  Q(ii, q, q )      0   λ  =  Qd (q, q )   I   i   Z(i, q, q ) 

(6.118)

where I is an identify matrix, and Q = Qe + Qv + Qmf . The preceding augmented multibody electromechanical equations of motion can be used to determine variable derivatives that can be integrated forward in time to determine the generalized coordinates q and the velocities q as well as the current i. A numerical algorithm similar to the one discussed at the beginning of this chapter can be used to obtain the solution and check on the violation of the kinematic constraints.


242


6.10 STATIC ANALYSIS One of the important problems encountered in the computer-aided analysis of railroad vehicle models is the accurate determination of the initial static equilibrium configuration. When the contact constraint formulations are used to model the wheel/rail interaction, the resulting nonlinear static equilibrium equations are functions of the generalized coordinates and the surface parameters. If the elastic contact formulations are used, on the other hand, the form of equations of motion remains the same as the standard form of the multibody equations of motion, since no additional constraints associated with the wheel/rail contact are introduced. In this case, the geometric surface parameters are used only to determine on-line the coordinates of the contact points that enter into the calculations of the generalized contact forces. Using the form of the equations of motion in the case of the elastic contact formulations, the static equilibrium equations, assuming that the velocities and accelerations are zero, can be written as follows: CT λ − Q(q, t )  g ( z) =  q =0  C(q, t ) 

(6.119)

where the generalized force vector Q includes the elastic contact forces, and the vector z = [qT λT]T consists of the unknown generalized coordinates q and the Lagrange multipliers λ associated with the joint and specified motion trajectory constraints. The preceding equation, whose solution is discussed in the literature (Shabana, 2001, 2005) is not expressed explicitly in terms of the surface parameters, and there are no Lagrange multipliers associated with the wheel/rail contact forces. This, however, is not the case when the constraint contact formulations are used. This section discusses the use of the augmented and embedded constraint contact formulations in the static equilibrium analysis of railroad vehicle systems. The numerical algorithms used to solve the resulting nonlinear algebraic equations are also summarized. To improve the convergence of the iterative Newton-Raphson algorithm used to solve the nonlinear algebraic static equilibrium equations, the line search method and the continuation method are often used. These two methods are discussed before concluding this section.

6.10.1 AUGMENTED CONSTRAINT CONTACT FORMULATION Using the principle of virtual work, the variational form of the static equilibrium equation of a railroad vehicle system subjected to contact constraints can be written as follows:

δ qT (CTq λ − Q) + δ s T CTs λ = 0

(6.120)

where the vectors q and s are, respectively, the vectors of the system generalized coordinates and surface parameters; C(q,s,t) is the vector of the system constraint



243

functions that include the contact constraints as well as mechanical joints and specified motion trajectories; Q is the vector of the system generalized forces; Cq and Cs are, respectively, the constraint Jacobian matrices resulting from the differentiation with respect to the vectors q and s; and λ is the vector of Lagrange multipliers associated with joint and contact constraints. Equation 6.120 is a special case of the variational form of the dynamic equations of motion obtained in Chapter 5, when the velocities and accelerations are assumed to be zero. Using the constraint equations C(q,s,t) = 0 and following a procedure similar to the one used to derive the augmented form of the dynamic equations of motion, one obtains the following nonlinear algebraic static equilibrium equations: CTq λ − Q(q, t )    g ( z) =  CTs λ =0  C(q, s, t )   

(6.121)

These equations can be solved using an iterative Newton-Raphson procedure to determine the vector of unknown variables z = [qT sT λT ]T. In a Newton-Raphson algorithm, the following equation is iteratively solved for the Newton differences: K∆z ( k ) = − g ( k )

(6.122)

where ∆z(k) is the vector of Newton differences and (CTq λ − Q)q  K =  (CTs λ )q  Cq 

(CTq λ )s (CTs λ )s Cs

CqT   CTs   0 

(6.123)

is the consistent Jacobian matrix defined as K = ∂g/∂z. In Equation 6.123, the subscripts indicate differentiation with respect to the vectors q and s. Using Newton differences at iteration k, the solution vector z is updated as follows: z ( k +1) = z ( k ) + ∆z ( k )

(6.124)

where the index k denotes the iteration number. Recall, as discussed in Chapter 5, that the generalized force vector Q is a function of the generalized coordinates q only. The analytical expressions of the Jacobian matrices Cq and Cs in Equation 6.123 in the case of the contact constraints were presented previously in Section 6.2. A computational algorithm for the static equilibrium analysis of railroad vehicle systems based on the augmented constraint contact formulation can be summarized as follows:


244


1. An initial estimate of the surface parameters s = s0 and the generalized coordinates q = q0 is made. 2. Using the first equation in Equation 6.121, the vector of Lagrange multipliers λ can be determined by solving the equation CqCTq λ = CqQ(q, t ), where the coefficient matrix C q C Tq is a nonsingular square matrix. 3. The residual vector g(z) for the given solution vector z is evaluated. If the norm of the residual vector is less than the specified tolerance, convergence is achieved and the solution for the static equilibrium analysis is successfully obtained and is defined by the vector z. If convergence is not achieved, go to Step 4. 4. Evaluate the Jacobian matrix K given by Equation 6.123 and solve the system of algebraic equations of Equation 6.122 for Newton differences ∆z(k). Use the Newton differences to update the vector z as described by Equation 6.124 and go to Step 3.

6.10.2 EMBEDDED CONSTRAINT CONTACT FORMULATION The embedded constraint contact formulation can be used to systematically eliminate the nongeneralized surface parameters from the static equilibrium equations, as demonstrated previously in the more general case of dynamics. It can be shown that the use of the embedded constraint contact formulation leads to the following variational static equilibrium equation:

δ qT ( J T λ − Q) = 0

(6.125)

where the matrix J, which is an implicit function of s, is the embedded contact constraint Jacobian matrix; λ is the vector of the Lagrange multipliers that include only independent Lagrange multipliers associated with the normal contact forces; and other noncontact constraints imposed on the motion of the system that are defined by the vector Cn(q, s(q), t) = 0. Recall that, in the case of the embedded constraint contact formulation, the surface parameters s are expressed as functions of the vector of generalized coordinates q. It can be shown in this case that the static equilibrium equations are defined as  J T λ − Q(q, t )  g ( z) =  n =0  C (q, s(q), t ) 

(6.126)

This system of nonlinear algebraic equations can be solved for the vector of unknowns z = [qT λT]T using a Newton-Raphson algorithm. In this case, the consistent Jacobian matrix K associated with Equation 6.126 is given as follows: ( J T λ − Q)q + ( J T λ )s B K= J 

JT   0 

(6.127)



245

where B, as discussed in Chapter 5, is the velocity transformation matrix that results from the use of the embedded constraint contact formulation and that relates the virtual change in the surface parameters to the virtual change in the generalized coordinates as δs = Bδq. This equation clearly shows that the change in the generalized coordinates q leads to change in the dependent surface parameters s; therefore, the surface parameters that are not included in the solution vector z must be iteratively updated at each Newton step by solving, for given values of the generalized coordinates q, the following nonlinear contact constraint algebraic equations for all contacts in the system:  t1r ⋅ (rPw − rPr )   r  t ⋅ (rw − rr ) C d (q w , q r , s w , s r ) =  2 w P r P  = 0  t ⋅n  1   w r  t 2 ⋅ n 

(6.128)

A computational algorithm for the static equilibrium analysis based on the embedded constraint contact formulation can be summarized as follows: 1. An initial estimate of the generalized coordinates q = q0 is made. 2. Using the vector q, Equation 6.128 is solved using an iterative NewtonRaphson algorithm to determine the surface parameters s. The surface parameters are required to evaluate the contact constraint equations and the constraint Jacobian matrix. 3. Using the first equation of Equation 6.126, Lagrange multipliers λ are determined by solving the algebraic equations JJT λ = JQ(q,t). 4. The residual vector g(z) for the given vector z is evaluated. If the norm of the residual vector is less than the specified tolerance, convergence is achieved and the static equilibrium configuration is defined by the vector z. Otherwise, proceed to Step 5. 5. Evaluate the Jacobian matrix K defined in Equation 6.127 and use this matrix to solve the system of algebraic equations of Equation 6.122 for Newton differences ∆z(k). Use the Newton differences to update the vector z as described by Equation 6.124 and go to Step 4.

6.10.3 LINE SEARCH METHOD In most railroad vehicle system applications, the static equilibrium equations given by Equation 6.121 or Equation 6.126 are highly nonlinear as the result of the nonlinearity of the kinematic constraint equations and the forcing functions. For this reason, difficulties can be encountered in achieving convergence using the conventional Newton-Raphson procedures. The line search method is an approach that can be used to improve the stability and convergence of the Newton-Raphson procedure by employing a backtracking algorithm. In this method, the solution vector is not necessarily updated using the full Newton step to ensure that the norm of the vector g


246


is continuously reduced at every step. In the line search method, the solution vector is updated as follows: z( k +1) = z( k ) + η ∆z( k ) ,

0 a, as shown in Appendix B. Note that, since the pressure distribution is semi-ellipsoidal, the total normal load Fn is given by Fn =

2 p0π ab 3

(A.19)



APPENDIX B

Elliptical Integrals The complete elliptical integrals presented in Chapter 4 are given as follows: π 2

Be =

∫ 0

π 2

cos 2 w 1 − e 2 sin 2 w

dw =

∫ 0

cos 2 w cos2 w − g 2 sin 2 w

π 2

Ce =

∫

−

dw

3

sin 2 w cos2 w(1 − e 2 sin 2 w) 2 dw

0

π 2

∫

De =

0

sin 2 w 1 − e 2 sin 2 w

dw

π 2

Ee =

∫

1 − e 2 sin 2 w dw

0

π 2

Ke =

∫ 0

1 1 − e sin 2 w 2

dw

where e = 1 − a 2 /b 2 , b > a and g = a/b It is clear that the elliptical integrals are related to each other as follows: Ke = 2 De − e 2Ce

Ee = (2 − e 2 ) De − e 2Ce

Be = De − e 2Ce

De = ( Ke − Ce ) /e 2

Be = Ke − De

Ce = ( De − Be ) /e 2

The values of the elliptical integral as a function of g are given in Table B.1.

319


320


TABLE B.1 Complete Elliptical Integrals g

Be

Ce

De

Ee

Ke

e2

0

1.0

–2 + ln(4/g)

–1 + ln(4/g)

1.0

+ln(4/g)

1.00

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

0.9964 0.9889 0.9794 0.9686 0.9570 0.9451 0.9328 0.9205 0.9081 0.8959 0.8838 0.8719 0.8603 0.8488 0.8376 0.8267 0.8159 0.8055 0.7953

2.3973 1.7352 1.3684 1.1239 0.9463 0.8105 0.7036 0.6170 0.5460 0.4863 0.4360 0.3930 0.3555 0.3235 0.2955 0.2706 0.2494 0.2289 0.2123

3.3877 2.7067 2.3170 2.0475 1.8442 1.6827 1.5502 1.4388 1.3435 1.2606 1.1879 1.1234 1.0656 1.0138 0.9669 0.9241 0.8851 0.8490 0.8160

1.0049 1.0160 1.0315 1.0505 1.0723 1.0965 1.1227 1.1507 1.1802 1.2111 1.2432 1.2763 1.3105 1.3456 1.3815 1.4181 1.4554 1.4933 1.5318

4.3841 3.6956 3.2964 3.0161 2.8012 2.6278 2.4830 2.3593 2.2516 2.1565 2.0717 1.9953 1.9259 1.8626 1.8045 1.7508 1.7010 1.6545 1.6113

0.9975 0.9900 0.9775 0.9600 0.9375 0.9100 0.8775 0.8400 0.7975 0.7500 0.6975 0.6400 0.5775 0.5100 0.4375 0.3600 0.2775 0.1900 0.0975

1.0

π 0.7864= --4

π 0.19635 = -----16

π 0.7864 = --4

π 1.5708 = --2

π 1.5708 = --2

0.00


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45814_Idx.fm Page 333 Tuesday, June 5, 2007 2:24 PM

Index A Absolute acceleration vector, 51, 55 Absolute angular velocity vector, 51, 292, 299, 301 Absolute coordinates, 9, 10, 21, 70 Absolute nodal coordinate formulation, 114 Acceleration of center of mass, 261–262 Acceleration kinematic equation, 173 Acceleration vector, 50–51, 51, 68, 188, 274 Adhesion region, of contact area, 145 Algebraic equations, 174, 175, 205, 206–208, 269 elastic contact formulations with, 174–176 static equilibrium, 243 Alignment, 107 limit by track classes, 224 Alignment irregularities, 223 Alignment variation, 32 Angle derivatives, 114–116 Angle of attack (AOA), 6 Angular acceleration vector, 73, 262–263, 307 Angular velocity creepage linearization, 295 matrix identities and, 295–296 Angular velocity vector, 51, 72, 140, 257, 262–263, 293 in creep theory, 142 defining in body coordinate system, 262 as linear functions of time derivatives, 297 and linearization assumptions, 300 with trajectory coordinate system, 271 of wheel, 81 Applied forces vector of, 199 virtual work of, 87, 167 Arc length, 216, 255 in curve theory, 90–91 as function of time, 267 and projected arc length, 109 Augmented constraint contact formulation (ACCF), 166, 168–171, 179, 180, 194, 195–201 in static analysis, 242–244 surface parameters in, 199

Augmented formulation, 10, 36, 62, 66–68, 76, 188 examples, 68–70 interpretation of methods, 83–84

B Backtracking algorithm, 245 Balance speed, 28, 29 Ball joint, 62 Binormal vector, 92 Body coordinate system, 21 angular velocity vector in, 262, 273 and general displacement, 258–259 Body trajectory coordinate system, 21, 255 Bogie frames, 3, 4, 10 Bolster, 3 Bounce, 27, 31 Bump deviations, 220, 225, 226, 227 Bushing element, 227, 232–236 rotational deformation of, 234 Bushing stiffness, 235

C Cant angle, defined, 107 Cant deficiency, 29–30 Carter's creepage coefficient, 149 Carter's theory comparison with linear law, 149 of creep forces, 147–149 Cartesian coordinates, 255, 270 complex expressions using, 266 constraint Jacobian matrix with, 271–272 Jacobian matrix associated with, 270 time derivatives of, 271 use with trajectory coordinate constraints, 269–272 vs. trajectory coordinate-based formulations, 265 Center of mass, 256, 272 absolute accelerations of, 306 absolute velocity vector of, 273 acceleration of, 261–262 global position vector of, 258 velocity of, 260–261

333


334


Centripetal acceleration, 51 Chain rule of differentiation, 115 Closed-form functions, 140 in Hertz contact theory, 136–137 Closure panel, 32 Coefficients, of second fundamental form, 97 Complete vehicle model, 248–253 Cone, mass moments of inertia, 59 Conformal contact, 17, 127 Conicity. See Wheel conicity Constant curve, 28 Constant forward velocity constraint, 265 Constrained dynamics, xii, 1, 9–11, 167–168, 188–189 degrees of freedom in, 10 Constraint approach, to wheel/rail contact, 161 Constraint contact formulations, 19–20, 165–166 constrained dynamic equations, 167–168 contact constraints, 166–167 Constraint equations, 67, 206 and degrees of freedom, 62 Constraint forces, elimination through embedding techniques, 78 Constraint formulations augmented constraint contact formulation (ACCF), 195–201 embedded constraint contact formulation (ECCF), 201–205 numerical algorithms, 194–195 Constraint Jacobian matrix, 68, 76, 79, 199, 204 with Cartesian coordinates, 271–272 Constraint library, 199 Constraint stabilization method, 189–191, 200 difference from penalty method, 191 Contact angle, and force balance, 7 Contact area adhesion region and slip region, 145 assumption of elliptical shape, 145 on elastic half-space, 314 elliptical shape of, 128 per Carter, 148 per Johnson/Vermeulen, 150 shape of, 128 Contact bodies, 129 under externally applied normal load, 133 schematic representation, 163 Contact-constraint conditions, 179 Contact constraints, 166–167, 173 Contact ellipse semi-axes, 135, 139, 152, 282 Contact equations, 313–317 Contact formulations, comparison of, 178–179, 180 Contact frame, 141 Contact models, 127–128 and Hertz theory, 128–140 wheel/rail contact approaches, 145–147

Contact plane, 130 Contact point, 177 absolute velocity vector of, 302 determining location of, 249 global velocity vector of, 144 location of, 104, 105 and surface parameters, 105 tangents to surface at, 162–163 velocities at, 140 Contact pressure in Hertz theory, 133–137 maximum, 139 CONTACT program, 151 Contact theories, 1, 17, 128 creep forces, 17–18 three-dimensional, 28 wheel/rail creep theories, 18 Continuation method, 242 in static analysis, 246–247 Coordinate partitioning, 169, 192 Coordinate systems, in track geometry, 104 Coulomb's friction, 145, 153 Coupling. See also Kinematic coupling between surface parameters, 185–187 Creep-force models, 127–128 Carter's theory, 147–149 and Hertz theory, 128–140 heuristic nonlinear model, 153–157 Johnson and Vermeulen's theory, 149–150 Kalker's linear theory, 150–153 Kalker's USETAB, 159 Polach nonlinear model, 154–156 simplified theory, 156–159 theories of, 147 Creep-force reduction coefficient, 154 Creep forces, 5, 17–18, 18, 22, 127, 140–142, 206, 207, 209, 277 calculation of, 210–211 example, 143–145 longitudinal and lateral, 146 simplified, 285 Creep spin moment, 18, 127 Creepage, 127, 140 for right wheel contact, 143–145 small area of slip, 150 Creepage coefficients, 292 Kalker's, 152 Creepage linearization, 291 background, 291–295 Euler angles and, 298–300 linearization assumptions, 300–301 longitudinal and lateral creepages, 301–304 Newton-Euler equations and, 306–309 spin creepage, 305–306 transformation and angular velocity, 295–298


Index Critical damping, 191 Critical speed, 4, 28 comparative computer code calculations, 297 consequences of inaccurate predictions, 291 role of wheel/rail contact in, 236 and wheel conicity, 289 Cross-level variation, 30, 32 as cause of freight car derailments, 31 Crossing panel, 32 Crossings, 32 Cubic splines, 212 Curvature, 13, 269 in curve theory, 91–92 defined, 106, 107 radius of, 92 surface, 99 Curve segment lengths of, 218 Curve smoothing, 213. See also Smoothness technique Curve theory, 12, 89, 90 arc length and tangent line in, 90–91 curvature and torsion in, 91–92 Curve-to-curve spiral, 112 Curve-to-tangent exit spiral, 15, 112 Curves parametric representation, 12, 90 rail segment type, 112 Cusp deviations, 220, 225, 226, 227 Cylinder, mass moments of inertia, 59 Cylindrical joints, 65 constraint Jacobian matrix, 68 joint constraints, 64–65 Cylindrical wheels, conicity in, 26

D D'Alembert's principle, 35 interpretation of methods, 85–86 Damped sinusoid deviations, 221, 225, 226, 227 Damping coefficients, 235 Damping matrix, 235 Deformation, 133, 138. See also Local deformation Degrees of freedom, 11, 77, 161, 174, 202. See also Independent generalized coordinates in constrained dynamic systems, 10 for cylindrical joints, 64 eliminating with two-degree-of-freedom model, 277–278 elimination by imposing nonconformal contact conditions, 20 and number of equations of motion, 85 reduction by constraint equations, 62

335 single-degree-of-freedom model, 272–276 with trajectory coordinate system, 272 Dependent coordinates, 11, 77 Derailment, 1 due to cross-level variation, 31 L/V ratio and, 308, 310 Derailment criteria, 6 Development angle, 15, 108, 109, 114 Deviations, 219 and measured data, 219 track deviations, 220–222 Differential and algebraic equations (DAE), 3, 62, 188 Differential geometry, 12–14 Direction cosines, 9, 37, 38–39 Discontinuities, response to, 27, 32–33 Driving constraints, 77 Dynamic curving, 27, 31–32 Dynamic equations linearization of, 23 Dynamic formulations, 35–36 augmented formulation, 66–70 embedding technique, 76–80 general displacement, 36–37 interpretation of methods, 80–86 joint constraints, 62–66 Newton-Euler equations, 58–61 rotation matrix, 37–48 trajectory coordinates, 70–76 velocities and accelerations, 49–57 virtual work, 86–87

E Elastic approach, 175 to wheel/rail contact, 161 Elastic bodies, 128 in Hertz theory, 128 Elastic contact formulations, 20, 165, 179, 180, 249 using algebraic equations, 174–176, 205, 206–208 using nodal search, 177–178, 205, 208–210 Elastic deformation, 140. See also Deformation Elastic force model, 19 Elastic formulations, numerical algorithms, 205 Elastic half-space, 146 contact area, 314 Electrodynamic suspension (EDS), 231–232 Electromagnetic suspension (EMS), 236, 237 inductance of, 240 modeling of, 237–240 Elliptic surface, 98 assumptions in Hertz contact theory, 145


336


Elliptical contact area, in Hertz theory, 128 Elliptical integrals, 319–320 Embedded constraint contact formulation (ECCF), 166, 171, 174, 179, 180, 185, 194, 201–205, 204, 248 equations of motion, 173–174 position analysis, 172 in static analysis, 244–245 surface parameter elimination in, 201 Embedding technique, 10, 11, 36, 62, 76–78, 77, 162, 188 and elimination of constraint forces, 78 reduced-order model, 78–79 Equality constraints, 161 Equations of motion, 15, 75, 116, 180 for arbitrary body, 274–276 for ECCF, 173–174 as generalized trajectory coordinates, 264 Newton-Euler, 264 specialized formulations, 264–265 Euler angles, 9, 37, 41–44, 70, 74, 120, 214, 256, 292, 299 creepage linearization and, 298–300 singular configuration, 43 time derivatives, 257 Euler parameters, 9, 37, 45–47 identities of, 55 Even-order splines, 212 External force vectors, generalized, 61

F FASTSIM program, 158 Federal Railroad Administration Code, 27 Free body diagram, 82 First derivatives, 114 in wheel geometry, 124–125 First fundamental form, 14 in surface geometry, 95–96 Flange contact, 268 Flexible bodies, modeling with finite element method, 3 Force balance, in wheel climb, 7 Force of attraction, in Maglev trains, 237 Force vector, 188, 235 Forward velocity, 303 Fourth-order derivatives, 74 Frenet frame, 92 Frictionless surfaces, in Hertz theory, 128

G Gage, 15 defined, 106

Gage limits, by track class, 224 Gage variation, 32 Gage widening, 7, 8 Gap contours, in Hertz theory, 132 Gaussian curvature, 14, 101 Gaussian elimination, 77 Gaussian procedure, with full or partial pivoting, 194 General displacement, 36–37, 256 body coordinate system, 258–259 generalized trajectory coordinates, 259–260 trajectory coordinate system, 256–257 General elasticity theory, 146 Generalized coordinates, 9, 10, 35, 168, 191, 200, 202, 204 partitioning method, 201 using Euler parameters, 46 Generalized coordinates partitioning, 189, 191–193 Generalized external force vectors, 61, 240 Generalized force vectors, 280 Generalized impulse-momentum approach, 178 Generalized mass matrix, 265, 280 Generalized Newton-Euler equations, 60 Generalized orientation coordinates, 51–52 Generalized trajectory coordinates, 259–260, 269 Generalized velocity vector, 72, 209 Geometric profile assumptions, 281 Geometry problems, 1, 11–12 differential geometry, 12–14 rail and wheel geometry, 14–16 Global position vector, 37, 71, 144, 163, 297 of arbitrary body point, 259 of body center of mass, 258 of center of mass, 259 Global velocity, 144 Grade, defined, 106

H Hemisphere, mass moments of inertia, 59 Hertz coefficients, 136 for Hertz force, 137 Hertz contact theory, 128, 145, 206 computational example, 139–140 computer implementation, 138–140 contact pressure in, 133–137 and creep-force models, 18 geometry and kinematics, 128–130, 132 Hertz force law, 137 Heuristic nonlinear creep-force model, 153–154 High rail, 28 High-speed trains, 1


Index Higher-order derivatives, 120, 203, 216 of constraint equations, 197 or profile functions, 212 of position vectors, 211 and smoothness technique, 211–214 with trajectory coordinate-based formulations, 265 Horizontal curvature, 15 relationship to rotation angle about vertical axis, 109 Hunting, 24, 27, 28 and two-degree-of-freedom model, 277 Hunting frequency, 287 Hunting phenomenon, 6 effects of wheel/rail contact on, 236 and lateral instability, 22 and motion stability, 24 Hybrid methods, 208 Hyperbolic surfaces, 98

I Implementation, 187 Independent generalized coordinates, 77, 191, identification of194 Independent generalized velocities, 207 Independent velocities, 200 Inertia mass moments of, 58, 59, 86 principal moments of, 60 products of, 58 vector of quadratic, 67 virtual work of, 87, 167 Inertia coupling, between translational and orational displacement, 58 Initial static equilibrium configuration, 242 Instability phenomenon, 289 Intermediate wheel coordinate system, 181–182, 184, 293, 300 Euler equations in, 307 transformation matrix defining, 182

J Jacobian matrix, 93, 169, 188, 196, 202 of kinematic constraint equations, 52, 270 Jog deviations, 220, 225, 226, 227 Johnson and Vermeulen's theory contact area in, 150 of creep forces, 149–150 Joint constraints, 62 cylindrical joints, 64–65 prismatic joints, 65–66

337 revolute joints, 63–64 spherical joint, 62, 63

K Kalker's coefficient, 156 Kalker's linear theory, 153, 158, 282 of creep forces, 150–152 creepage and spin coefficients, 152 example, 152–153 Kalker's USETAB, 159 Kinematic constraint equations, 18, 25, 74, 162, 266 avoiding violation of, 189 interpretation of methods, 80–83 linearization of, 23 with trajectory coordinates, 265 Kinematic coupling, 183 between surface parameters, 185–186 Kinematic linearization, 291 Kinematic step, in rail/wheel geometry, 89 Kinematic variables, 12 Kinetic forces, 12 Klingel's formulas, 26, 287 Kutzbach criterion, 11

L L/V ratio, 6 in derailment criteria, 310 problems in wheel climbs, 7 use in derailment criteria, 308 Lagrange-D'Alembert equation, 9, 35, 86 Lagrange multipliers, 10, 66, 84, 168, 170, 171, 200, 202, 205, 208, 209, 240 vector of, 67 Lagrangian formulation, 9, 35 Lame's constant, 148 Lateral contact force, error in, 294, 311 Lateral creepages, 301–304, 304 nonlinear vs. linearized formulations, 304 Lateral force, 282, 283, 284 ratio to vertical force, 6 Lateral instability, and hunting phenomenon, 22 Lateral motion, 26 Lateral oscillations, 6 Lateral stability, 288 Levitation force, characteristics of, 239 Line search method, 242 in static analysis, 245–246 Linear constraints, 266 Linear hunting stability analysis, 280–287


338


Linearization techniques, 1 assumptions, 300–301 Linearized creepage expressions, 23, 284, 306 Linearized kinematic/dynamic equations, 23 Linearized vehicle models, xii, 23–24 Local deformation, in contact theory, 128 Longitudinal contact force, nonlinear vs. linearized formulations, 309 Longitudinal creep forces, 285, 301–304, 304 nonlinear vs. linearized formulations, 303 Longitudinal force, 30, 282, 284 Longitudinal shift, 186 Longitudinal tangent vector, 120 LU factorization, 78

M Maglev forces, 236 electrodynamic suspension (EDS), 231–232 electromagnetic suspension (EMS), 237 modeling of electromagnetic suspensions, 237–240 multibody system electromechanical equations, 240–241 Magnetic levitation forces, 6 Magnetomotive force, 238 Mass matrix, 208 Mass moments of inertia, 58, 59 Material modulus of rigidity, 148 Material nonlinearities, 4 Matrix identities, 295–296 Measured data, 222–223 deviations and, 219 Measurement errors, correcting with smoothing splines, 214 Motion amplitude, limiting maximum value of, 31 Motion scenarios, 27–28 dynamic curving, 31–32 hunting, 28 pitch and bounce, 31 response to discontinuities, 32–33 spiral negotiation, 30 steady curving, 28–30 twist and roll, 30–31 yaw and sway, 31 Motion stability, 24–27 Motion trajectories, 62 Multibody contact formulations, 161–162, 187 augmented constraint contact formulation (ACCF), 168–171 comparison of, 178–179 constraint contact formulations, 165–168 elastic contact formulation-algebraic equations (ECF-A), 174–176

elastic contact formulation-nodal search (ECF-N), 177–178 embedded constraint contact formulation (ECCF), 171–174 parameterization of wheel and rail surfaces, 162–165 planar contact, 179–186 Multibody railroad vehicle formulations constraint contact formulation, 19–20 elastic contact formulation, 20 general, 18–19 Multibody system algorithms, 188 constrained dynamics, 188–189 constraint stabilization method, 189–191 generalized coordinates partitioning, 191–193 identification of independent coordinates, 194 for Maglev forces, 240–241 penalty method, 189–191 Multibody system contact formulation, 19 Multibody system dynamics, 2 generality, 2–4 implementation of railroad vehicle elements into, 6–8 nonlinearity, 4–5

N Nadal's formula, 6 Natural frequency, 26 Newton differences, vector of, 243 Newton-Euler equations, xii, 9, 58–61, 260, 264, 291, 293 creepage linearization and, 306–309 effect of linearization on, xiii generalized form, 60 of motion, 36 Newton-Raphson algorithm, 170, 175, 179, 184, 201, 203, 204, 210, 245, 246, 247 in static analysis, 242 Newtonian approach, 9, 35 Nodal points, relative distance between, 208 Nodal search, 174, 179, 205, 208–210 elastic contact formulation using, 177–178 Nonconformal contact, 17, 127, 166 in Hertz theory, 128 Noncontact driving concept, 236 Nongeneralized coordinates, 168 surface parameters as, 171 Nonlinear constrained multibody formulation, 288 Nonlinear creep-force model, 154, 293 Nonlinear dynamic analysis, 1, 161 Nonlinear equations of motion, 74, 292 for spin creepage, 305 vs. linearized, 291


Index Nonlinear force-creepage relationships, 128 Nonlinearity of multibody system dynamics, 4–5 sources in wheel/rail contact problem, 153 Normal component, 51, 176 Normal contact force, 201, 206, 268 estimating for wheel/rail interactions, 161 nonlinear vs. linearized formulations, 308 right wheel of front wheelset, 253 Normal curvature, 99–100 Normal plane, 92 Normal stress distribution, 133, 155 Normal vector, 89, 121, 124 in surface geometry, 94–95 Normalized relative velocities, creepage as, 141 Numerical algorithms augmented constraint contact formulation (ACCF), 195–201 constraint formulations, 194–195 elastic contact formulation using algebraic equations (ECF-A), 206–208 elastic contact formulation using nodal search (ECF-N), 208–210 elastic formulations, 205 embedded constraint contact formulation (ECCF), 201–205 Numerical comparative study, 247 complete vehicle model, 248–253 simple suspended wheelset, 247–248

O Orientation angles derivatives of, 115 first, second, and third derivatives of, 120 Orthagonality condition, 39 Oscillating plane, 92 Oscillation period, 26

P Pantograph/catenary systems, 8 Parabolic surfaces, 98 Parameterization track geometry, 163–164 of wheel and rail surfaces, 162–163 wheel geometry, 165 Penalty method, 189–191, 200 comparison with generalized coordinate partitioning method, 201 difference from constraint stabilization method, 191

339 Penetration, 205, 207 calculating for ECF-A, 176, 186 determining with surface parameters, 209 Pitch, 27, 31, 256 Pitch angles, 22 Pitch motion, 31, 75 Pitch rotation, 181 Pitch velocity, 300, 303 Planar contact, 162, 179–181 condition of, 183 intermediate wheel coordinate system, 181–182 modified formulation, 186 Planar surfaces, 98 Plateau deviations, 220, 225, 226, 227 Poisson's ratio, 134, 282 Polach nonlinear creep-force model, 154–156 Position analysis, 172 Pressure distribution in Hertz theory, 133 semi-ellipsoidal, 134 Primary hunting, 28 Primary suspensions, 3, 227 Principal axes, 60 Principal curvatures, 14, 100–101 Principal directions, 100–101 Principal moments of inertia, 60 Principal relative radii of curvature, 129–130 Principal transverse radii, 138 Prismatic joints, joint constraints, 65–66 Products of inertia, 58 Profile, 223 defined, 107 Profile deviations, 220 Profile frame location and orientation example, 122–123 transformation matrix defining orientation of, 109–110 Profile irregularities, 223 Profile limits, by track class, 225 Projection, of space curve on horizontal plane, 108 Pure rolling, 81

Q Quadratic velocity vector, 197, 198, 203, 270 Quasi-linearization technique, 289

R Radius of curvature, 92, 129–130, 138 Rail arc length, calculating during nodal search, 177


Rail deviations, 107, 222 Rail elevation, 15 Rail geometry, 14–16, 89, 103–106, 207 computer implementation, 111–116 and curve theory, 90–92 and surface geometry, 92–102definitions and terminology, 106–107 and track geometry, 108–111 track preprocessor and, 116–123 Rail lateral surface parameter, 14, 103 Rail longitudinal surface parameter, 14, 103 Rail profile geometry, 23 Rail radii of curvature, 138 Rail rollover, 7 geometric parameters in, 8 Rail segments, length of, 119 Rail space curve, changing coordinates of, 221 Rail surface and coordinate system, 103 parameterization of, 162–163 track geometry, 163–164 Rail turnouts, 32, 33 Railroad input entries, defining curved track, 117 Railroad vehicle elements, 228 implementation in multibody-system dynamics, 6–8 Railroad vehicles, 1–2, 3 and constrained dynamics, 9–11 contact theories, 17–18 general multibody formulations, 18–20 geometry problem, 11–17 linearized models, 23–24 motion scenarios, 27–33 motion stability and, 24–27 and multibody system dynamics, 2–8 specialized formulations, 20–23 Rectifying plane, 92 Reduced-order model, 78–79 Redundant coordinates, 80 Relative angular velocity, 140 Revolute joints, joint constraints, 63–64 Rheonomic constraints, 77 Right rail, data sets for, 121 Right wheel contact, creepage values for, 143–145 Rigid bodies contact among, 127 coordinates of, 36 equations of motion for multibody systems of, 35 global position vector of, 37 orientation of, 36–37, 56–57 translational motion of, 36 velocity vector, 49 Rodriguez formula, 45, 46

Roll angles, 22, 256 Rolling contact, 141 exact theory of, 146 simplified theory of, 147 Rolling radii, 138 Root loci, 288 Rotation matrix, 37, 41, 258, 259, 298 direction cosines, 38–40 Euler parameters, 45–48 simple rotations, 41 Euler angles, 41–44, 42 Rotation parameters, 9 Rotational spring-damper-actuator element, 225, 227, 230–231

S Scalar equations, 196 Scalar quantities, and Lagrangian approach, 35 Second derivatives, 211 in wheel geometry, 125 Second fundamental form, 14 in surface geometry, 96–98 of surfaces, 98 Second-order derivatives, 74 Second point of contact, 28 Secondary hunting, 28 Secondary suspensions, 3, 227 Series spring-damper element, 227, 231–232 Shear stresses, in Hertz theory, 133 Signatures, 225 Simple rotations, 41 Simple suspended wheelset, 247–248 Simplified creep-force theory, 156–159 Single-degree-of-freedom model, 272–274 Singular configuration, 53–54, 55–56 of Euler angles, 43 Singular points, 90 Singularity problem avoiding with Euler parameters, 45, 47 with Euler angles, 44, 53–54 Sinusoid deviations, 221, 225, 226, 227 Skew-symmetric matrix, using Euler parameters, 45 Slender rod, mass moments of inertia, 59 Slip region of contact area, 145 vanishing for small creepage forces, 151 Smoothing splines, 214 Smoothness technique, higher derivatives and, 211–214 Space curves, 13 Special elements, 187, 225–227


Index bushing element, 232–236 rotational spring-damper-actuator element, 230–231 series spring-damper element, 231–232 translational spring-damper-actuator element, 227–230 Specialized railroad vehicle formulations, 20–23, 255–256 equations of motion, 264–265 general displacement, 256–260 linear hunting stability analysis, 280–289 single-degree-of-freedom model, 272–276 trajectory coordinate constraints, 265–272 two-degree-of-freedom model, 277–280 velocity and acceleration, 260–263 Sphere, mass moments of inertia, 59 Spherical joints, 63 constraint Jacobian matrix, 68 joint constraints, 62 Spin coefficients, 140, 151 Kalker's, 152 Spin creep moment, 146, 282, 284 Spin creepage, 301, 305–306 nonlinear vs. linearized formulations, 295 Spin moment, 17 nonlinear vs. linearized formulations, 310 Spiral negotiation, 27, 30 Spline representations, 213 Stability, and wheel geometry, 16 Static analysis, 242 augmented constraint contact formulation, 242–244 continuation method, 246–247 embedded constraint contact formulation in, 244–245 line search method, 245–246 Static compression, 128 Static equlibrium, algebraic equations, 243 Steady curving, 27, 28–30 Stock rail, 33 Stress distribution, normal and tangential, 155 Successive rotations, and Euler angles, 41–44 Super-elevation, 15, 108, 117 defined, 106 Surface curvature, 99 Surface gap, 129 Surface geometry, 13, 89, 92–94 first fundamental form, 95–96 normal curvature in, 99–100 principal curvatures and principal directions in, 100–102 second fundamental form, 96–98 tangent plane and normal vector in, 94–95 Surface mapping, 94 Surface parameters, 13, 93, 163, 164, 171

341 in ACCF, 199 coupling between, 185–186 as dependent variables, 204, 205 determining from iterative Newton-Raphson procedure, 203 determining value of penetration using, 209 eliminating from equations of motion, 171, 185, 201, 204, 244 eliminating in elastic contact formulations, 166 as independent variables, 170 second derivatives of, 205 selecting as degrees of freedom, 200 Surfaces theory, 12 Suspended wheelset, 247–248, 286, 287, 292 model, 288 model data, 294 and track deviation, 293 Suspension elements, 10 Sway, 27, 31 Switch panel, 32 Switches, 32 Symmetric coefficient matrix, 80 System accelerations, in two-degree-of-freedom model, 279 System mass matrix, 67, 199 System symmetric mass matrix, 188

T Tangent, rail segment type, 112 Tangent line, in curve theory, 90–91 Tangent plane in surface geometry, 94–95 at wheel contact point, 124 Tangent-to-curve entry spiral, 15, 112 Tangent vectors, 121 Tangential component, 51 Tangential contact force, nonlinear vs. linearized formulations, 309 Tangential creep forces, 127, 201 Tangential forces, 145 Tangential stress distribution, 155 Thin circular disk, mass moments of inertia, 59 Thin plate, mass moments of inertia, 59 Thin ring, mass moments of inertia, 59 Three-dimensional contact theory, 28, 147, 162, 166 Three-dimensional rotations, and singularity problems, 53–54 Three-layer spline, generation of, 212 Time derivatives of absolute generalized Cartesian coordinates, 271


342


and absolute vectors, 51 angular velocity vectors as functions of, 297 of Euler angles, 54, 257 of Euler parameters, 56 of nongeneralized surface parameters, 170 of orientation parameters, 258 Tongue rail, 33 Torsion, 13 in curve theory, 91–92 Total applied force, in Hertz theory, 133 Track centerline, 121 coordinates of, 117 Track classes, 223–225 alignment limit by, 224 gage limit by, 224 profile limit by, 225 Track deviation functions, 219, 220–221 Track deviations, 220–222 and measured track data, 222–223 and track quality/classes, 223–225 Track geometry, 15, 164 coordinate systems in, 104 descriptive, 108–111 Track irregularities, parameters of, 226 Track perturbation, 31 Track preprocessor, 116–117, 214–216 input, 117–118 length change due to curvature, 216–218 numerical integration, 118–119 output, 120–121 use of output during dynamic simulation, 121–123, 218–219 Track quality, 223–225 Track Safety Standards, 223 Track segments, 15 linear representation, 112–113 Track super-elevation, 15 Traction displacement relationship, 146, 150–151 Trajectory coordinate constraints, 265–266, 269, 270 numerical example, 266–268 and use of Cartesian coordinates, 269–272 Trajectory coordinates, 21, 22, 36, 70–72, 75, 162, 181, 210 advantages and disadvantages, 255 constraints on, 265–272 and general displacement, 256–257 generalized, 259–260 velocity and acceleration, 72–74 Transformation matrix, 258, 297, 298–299 and orthagonality condition, 39 and simple rotations, 41 using Euler angles, 43 using Euler parameters, 45

Translational spring-damper-actuator element, 225, 227–230, 228 Translational stiffness, 235 Traveled distance, of wheelset, 267 Tread contact, 268 and force balance, 7 Trough deviations, 220, 225, 226, 227 Turnouts, 32 Twist and roll, 27, 30–31 Two-degree-of-freedom model, 27–280, 281, 286 Two-dimensional contact theory, 147, 179 Two-layer spline, generation of, 212

U Unit vectors, 38 and direction cosines, 38 in simple rotations, 41 USETAB software, 159

V Vehicle dynamic response, 222 Vehicle-track interaction, 6 Vehicles, defined for railroad, 2–3 Velocities and accelerations, 260 acceleration vector, 50–51 angular, 262–263 center of mass acceleration, 261–262 center of mass velocity, 260–261 generalized orientation coordinates, 51–53 singular configuration, 53–57, 55–56 trajectory coordinates, 72–74 velocity vector, 49–50 Velocity, of center of mass, 260–261 Velocity transformation matrix, 77, 264, 269, 280 Velocity vector, 49–50 with single-degree-of-freedom model, 273 Vertical bump, track with, 252 Vertical contact forces, 268 error in, 294, 310 Vertical displacement, of trailing wheelset, 250, 253 Vertical force, ratio of lateral force to, 6 Vertical motion, constraints on, 76 Virtual work, 86–87, 167, 230, 242

W Wheel angular parameter, calculating for nodal search, 177 Wheel conicity, 24, 283 effect on degree of stability, 289


Index Wheel geometry, 14–16, 16, 23, 89, 123–125, 165, 207 computer implementation, 111–116 and curve theory, 90–92 definitions and terminology, 106–107 first derivatives, 124–125 second derivatives, 125 and surface geometry, 92–102 Wheel profile, 184 Wheel radii of curvature, 138 Wheel/rail contact, 6 approaches to, 145–147 dependence of stability on, 292 dynamic and quasi-static theory of, 147 elastic vs. constraint approaches to, 161 exact theory of rolling, 146 model of, 12, 89 points of, 178 simplified theory of, 147 three- and two-dimensional theory, 147 vs. Maglev forces, 236 Wheel/rail creep theories, 18 Wheel/rail separation, 7, 178, 205 Wheel rolling radius, 148 Wheel separation, 180

343 Wheel surface mathematical definition, 123 parameterization of, 162–163 track geometry, 163–164 Wheel velocity, at contact point, 81 Wheelset lateral displacement, 249 single, 281 traveled distance, 267 Wheelset frame, 143 Wheelset hunting motion, 25 Wheelset rolling radii, 24, 25 Wheelsets, 10 lateral motion in, 25

Y Yaw, 22, 27, 31, 256, 277 Yaw angle, 25 rate of change, 26 of wheelset, 249 Yaw moment, 282, 283, 285 Yaw oscillations, 6 Young's modulus of elasticity, 134, 316


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