Fundamentals of rail vehicle dynamics: guidance and stability

FUNDAMENTALS OF RAIL VEHICLE DYNAMICS ADVANCES IN ENGINEERING Series Editors: Fai Ma, Department of Mechanical Engine...

Author: A. H. Wickens

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FUNDAMENTALS OF RAIL VEHICLE DYNAMICS

ADVANCES IN ENGINEERING Series Editors: Fai Ma, Department of Mechanical Engineering, University of California, Berkeley, U.S.A. Edwin Kreuzer, Department of Mechanics and Ocean Engineering, Technical University Hamburg-Harburg, Hamburg, Germany

FUNDAMENTALS OF RAIL VEHICLE DYNAMICS GUIDANCE AND STABILITY

A.H. WICKENS Loughborough University, UK

Wickens, A. H. Fundamentals of rail vehicle dynamics : guidance and stability / A.H. Wickens. p. cm. -- (Advances in engineering ; 6) Includes bibliographical references and Index. ISBN 90-265-1946-X 1. Railroads--Cars--Dynamics. I. Title. II. Advances in engineering (Lisse, Netherlands) ;6 TF550.W53 2003 625.2’01’5313--dc21 2003045675

This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publishers. Although all care is taken to ensure the integrity and quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: Swets & Zeitlinger Publishers www.swets.nl

ISBN 0-203-97099-3 Master e-book ISBN

ISBN 90 265 1946 X (Print Edition)

Contents Arrangement of Book ..........................................................................................ix Preface ................................................................................................................. xi 1. Basic Concepts.................................................................................................. 1 1.1 Introduction ......................................................................................... 1 1.2 The Railway Wheelset......................................................................... 2 1.3 Creep ................................................................................................... 6 1.4 Stability................................................................................................ 8 1.5 Guidance............................................................................................ 10 1.6 Suspension, Performance and Criteria............................................... 13 1.7 Suspension, Articulation and Curving............................................... 15 References ............................................................................................... 17 2. Equations of Motion ....................................................................................... 19 2.1 Introduction ....................................................................................... 19 2.2 Freedoms and Constraints ................................................................. 19 2.3 Wheel Rail Geometry ........................................................................ 21 2.4 Contact Mechanics ............................................................................ 32 2.4.1 Elasticity and Friction..................................................................... 32 2.4.2 Laws of Friction ............................................................................. 32 2.4.3 Contact Between Wheel and Rail ................................................... 32 2.4.4 Creep .............................................................................................. 33 2.4.4.1 Background ................................................................................. 33 2.4.4.2 Formulation of the Creep Problem .............................................. 34 2.4.4.3 Creep Forces for Small Creepages .............................................. 36 2.4.4.4 Creep Forces for Arbitrary Creepages......................................... 37 2.4.4.5 An Approximate Theory for Arbitrary Creepages....................... 37 2.4.4.6 Heuristic Approximations............................................................ 39 2.4.4.7 Non-Hertzian Effects................................................................... 39 2.4.5 Transient Effects............................................................................. 40 2.5 Creepages .......................................................................................... 40 2.6 Contact Forces ................................................................................... 42 2.7 Kinematics of the Wheelset ............................................................... 44 2.8 Equations of Motion .......................................................................... 46 2.9 Constrained Motion ........................................................................... 48 2.10 Equations of Motion for Small Displacements................................ 52 2.11 Equations of Motion for a Two-Axle Vehicle................................. 61 References ............................................................................................... 66

vi

CONTENTS

3. Dynamics of the Wheelset .............................................................................. 71 3.1 Introduction ....................................................................................... 71 3.2 The Unrestrained Wheelset ............................................................... 71 3.3 Root Locus for Small Motions of the Restrained Wheelset .............. 76 3.4 Instability and Feedback.................................................................... 81 3.5 Amplitude Dependent Behaviour and Limit Cycles.......................... 84 3.6 Energy Balance.................................................................................. 88 3.7 Dynamic Aspects of Guidance .......................................................... 89 3.8 Alternative Methods of Guidance...................................................... 94 References ............................................................................................. 103 4. Guidance of the Two-Axle Vehicle .............................................................. 107 4.1 Introduction ..................................................................................... 107 4.2 Properties of the Stiffness Matrix .................................................... 109 4.3 Steering on Large Radius Curves .................................................... 111 4.4 Response to Cant Deficiency on Large Radius Curves ................... 116 4.5 The Conflict Between Steering and Stability .................................. 118 4.6 Motion on Sharper Curves............................................................... 121 4.7 Response to Misalignments ............................................................. 127 4.8 Flange Forces and Derailment ......................................................... 130 References ............................................................................................. 131 5. Dynamic Stability of the Two-Axle Vehicle ................................................ 133 5.1 Introduction ..................................................................................... 133 5.2 Equations of Motion ........................................................................ 135 5.3 Stiff and Flexible Vehicles .............................................................. 142 5.4 The Flexible Vehicle with Zero Suspension Damping.................... 147 5.5 Damping and the Long Wheelbase Flexible Vehicle ...................... 151 5.6 Stability of the Flexible Vehicle in General .................................... 153 5.7 The Application of Cross-Bracing and Yaw Relaxation ................. 159 5.8 The Stiff Vehicle or Bogie............................................................... 160 5.9 The Three-Piece Bogie .................................................................... 169 References ............................................................................................. 169 6. The Bogie Vehicle ........................................................................................ 173 1 Introduction ........................................................................................ 173 2 Equations of Motion ........................................................................... 175 3 Dynamics of the Conventional Bogie Vehicle ................................... 177 4 Steering and Stability of Multi-Axle Vehicles in General.................. 182 5 Steering and Stability of a Generic Bogie Vehicle ............................. 185 6 Application to Specific Configurations .............................................. 190 7 Stability of Bogie Vehicles with Steered Wheelsets .......................... 197 8 Simple Bogie Model........................................................................... 199 9 Stability of Simple Bogie Model ........................................................ 201 References ............................................................................................. 206

CONTENTS

vii

7. The Three-Axle Vehicle ............................................................................... 209 7.1 Introduction ..................................................................................... 209 7.2 Steering and Stability of Three-Axle Vehicles ................................ 210 7.3 Steering with Unequal Conicities .................................................... 215 7.4 Stability of Vehicle with Uniform Conicity .................................... 217 7.5 Stability with Unequal Conicities.................................................... 225 7.6 Dynamic Response .......................................................................... 230 References ............................................................................................. 232 8. Articulated Vehicles ..................................................................................... 235 8.1 Introduction ..................................................................................... 235 8.2 Steering and Stability....................................................................... 238 8.3 Application to Specific Configurations ........................................... 244 8.4 Stability of an Articulated Three-Axle Vehicle............................... 246 8.5 Stability and Response of an Articulated Four-Axle Vehicle.......... 252 References ............................................................................................. 260 9. Unsymmetric Vehicles.................................................................................. 261 9.1 Introduction ..................................................................................... 261 9.2 Stability Theorems for Rigid and Semi-Rigid Vehicles .................. 263 9.3 Unsymmetric Rigid Vehicle ............................................................ 265 9.4 Steering of a Vehicle with Unsymmetric Inter-Wheelset Structure 267 9.5 Stability of a Two-Axle Articulated Vehicle................................... 272 9.6 The Influence of Elastic Stiffness on Stability ............................... 276 9.7 Applications of Unsymmetry........................................................... 279 References ............................................................................................. 281 Index ................................................................................................................. 283

Arrangement of the book Sections are numbered serially within each chapter. If reference is made to a section within the chapter containing the section, the section number is cited as a single number. Otherwise, a section is identified by two numbers separated by a decimal point, the first number referring to the chapter in which the section appears, and the second identifying the section within the chapter. Equations are numbered serially within each section. If reference is made to an equation within the section containing the equation, the equation number is cited as a single number. If reference is made elsewhere in the same chapter then the equation number is cited as a two-figure number and if reference is made in another chapter all three numbers-chapter, section and equation are cited. Figures and tables are numbered by chapter.

Preface The fundamental method of guidance of the railway vehicle is the coned and flanged wheelset. Whilst facilitating guidance in curves, coning can give rise to sustained lateral oscillations, termed hunting. This oscillation induces forces which can cause damage to both vehicle and track and there can be, at least, discomfort to the passenger and, at worst, the risk of derailment. Inadequate steering on curves can have similar consequences. This book concentrates on the resulting problem of the conflict between guidance and stability and its resolution by proper design of the suspension connecting the wheels and car body of the railway vehicle. The invention of the wheelset, the progressive development of the bogie and the various schemes of articulation which have been developed over the years in order to resolve the design conflict between stability and steering, all predate the theory of railway vehicle dynamics. Engineering insight brought railway technology a long way but empirical methods were not adequate once the railway renaissance started and train speeds increased. A fundamental change in railway technology took place in which the empirical evolution of railway bogies was replaced by a more scientific and numerate approach. This approach has been very successful; for example, not only has stable operation of steel wheel on steel rail vehicles been demonstrated at speeds of over 500 km/h (more than double the speed of the fastest train fifty years ago) but the analytical and predictive capabilities now available have stimulated a rising tide of innovative designs. The detailed modelling of the dynamics of railway vehicles is made possible by the several excellent computer packages that are available, which provide sufficiently detailed and validated mathematical models that can be used with confidence in engineering design and development. These models permit the simulation of the actual motion on a specified stretch of track so that the performance of a specific design can be analysed, or a particular incident recreated. Thus, by simulation the overall performance of a vehicle can be checked. Realism is, of course, essential in design but equates to complexity, and computer output must be tempered with understanding and scepticism. It is important, therefore, that fundamental principles are well understood. This book is concerned with the fundamental principles of guidance and stability, which are a consequence of the mechanics of wheel-rail interaction as embodied in the equations of motion. For research purposes, where the objective is to achieve an understanding of an innovative system or a particular problem, simple models can be very useful and can provide productive insights. Analytical studies which describe the mechanics of various phenomena by the simplest model possible can be used to explore new suspension and vehicle design concepts.

PREFACE

xii

Attention will be concentrated on the configuration and parametric design of the bogie, in relation to steering, dynamic response and stability. Therefore the treatment of the various configurations of vehicle do not simply concentrate on a current typical set of parameters but attempt to consider the consequences of the complete range of parameters open to the designer. By this approach, it is possible to see why much of current practice, though it pre-dates the availability of theory, is the way it is. Moreover, it becomes clear why many innovations failed in the past. Because an important consequence of a more analytical approach is to separate out the dynamic properties of a system from the detailed design of its components the latter will not be discussed. Moreover, the application of active controls to steering and ride control (including body tilting) will not be covered. Active systems will play a large part in the future and those working in the field will require a sound grounding in passive systems. As the emphasis is on ride quality and guidance, a frequency range of roughly 0 15 Hz is of prime interest. This makes it possible to consider that, in general, wheelsets and track (except in the areas of contact) are rigid and that car bodies are without flexibility. This means that some significant phenomena are not discussed here. Moreover, simple forms of suspension elements are assumed. The more straightforward problems of response in the vertical plane, or in the longitudinal direction are not addressed. The basic concepts are described in Chapter 1. A detailed discussion of the equations of motion follows in Chapter 2 in which a compromise has been made between the mathematical rigour of some investigators and the ad-hoc use of Newton’s Laws of others. In this Chapter, though an engineering approach has been followed, great reliance has been placed on the careful derivations of Professor de Pater. The following Chapters deal with the single wheelset and then with progressively more complex configurations of vehicle. If possible, simple analytical results have been derived as these, if available, provide the best basis for understanding the mechanics of the systems involved. All numerical results have been obtained using standard commercially available software for numerical computation. It is a pleasure to acknowledge the stimulus and help I have received over the years from colleagues, too numerous to mention here, at British Rail Research, Loughborough University and through the International Association of Vehicle System Dynamics. A. H. Wickens Idridgehay, 2002

1 Basic Concepts 1.1 Introduction The railway train running along a track is one of the most complex dynamical systems in engineering. It has many degrees of freedom, the interaction between wheel and rail involves both complex geometry of wheel tread and rail head and nonconservative forces generated by relative motion in the contact area, and there are many non-linearities. The long history of railway engineering provides many practical examples of dynamical problems which have degraded performance and safety. The two essential features of operation, running in a train of vehicles and guidance by the track, cause problems which are unique to railways. Inadequate guidance on curves results in high lateral forces between wheel and rail, rapid wear of wheels and rails and the possibility of derailment. Dynamic and static instabilities, and excessive response to track irregularities and other features of track geometry, can result in poor ride quality and high stresses and can contribute to derailment. Operation in a train involves the control of forces acting between the vehicles in the train as the propulsive and braking forces are varied in response to the train traversing hills and valleys. High frequency interaction between wheel and rail can lead to damage to the contacting surfaces and corrugation of the rails, and excessive noise and vibration. The dynamics of the railway vehicle represents a balance between the forces acting between the wheel and the rail, the inertia forces and the forces exerted by the suspension and articulation. Of these, the basic characteristics of the wheel-rail interface such as friction, geometry, and the elasticity in the contact area are hardly under the control of the designer. But the configuration, suspension and forms of articulation can be varied over a wide range of possibilities, limited mainly by the degree of complexity considered acceptable for each application. The objective of suspension design is, therefore, to control the motion of the railway vehicle so that good ride quality is achieved, at the same time dynamic loads and the tendency to derail are reduced to acceptable levels, whilst running on track with geometry that is economically acceptable. In a complete model of the dynamics of a railway vehicle, the vehicle is considered to be assembled from wheelsets, car bodies and intermediate structures which are flexible, and which are connected by components such as springs and dampers. Similarly, the vehicle is considered to run on a track which has a complex structure

2

RAIL VEHICLE DYNAMICS

with elastic and dissipative properties. Each major component has six rigid body degrees of freedom plus additional degrees of freedom representing the elastic distortion of the component. In the latter case, these additional degrees of freedom might represent a finite element model of the structure or a series of natural modes of vibration. The track can be modelled as a continuous structure with a moving interface at the points of contact, where the interaction between wheel and rail is dependent on the relative motion. This kind of model, with varying assumptions, is provided by various computer software packages which are used in the engineering design and analysis of railway vehicles. So, one objective of the study of the dynamics of railway vehicles is the development of sufficiently detailed and validated mathematical models that permit the simulation of the actual motion, on a specified stretch of line, so that the performance of a specific design can be analysed, or a particular incident recreated. Thus, by simulation, the overall performance of an existing or projected vehicle can be checked and design decisions made. A second objective of the study of railway vehicle dynamics is to develop analytical or numerical models describing the mechanics of various phenomena by the simplest model possible. These can be used to explore new suspension and vehicle concepts and to develop a basis for physical understanding and insight. Ideally, not only analysis but synthesis is required in which various possibilities for design are exposed. Simpler models are typically generated by simplifying assumptions and in this book, concerned with guidance and stability, these are that • the vehicle has a longitudinal plane of symmetry (parallel to the direction of motion on straight track) making it possible, under certain conditions, to separate equations governing those motions which are symmetric with respect to the plane of symmetry from those which govern anti-symmetric motions; • variations in longitudinal motion are not considered so that the vehicle moves at constant forward speed; • the motions of interest are at low frequencies and, in most cases, flexibility of components can be neglected. It is the objective of this chapter to explain the basic concepts of stability and guidance of railway vehicles as a preliminary to more detailed mathematical analysis.

1.2 The Railway Wheelset The basic unit of a railway vehicle is the wheelset, Figure 1.1. The conventional wheelset of today has the following features: it consists of two wheels fixed on a common axle, so that each wheel rotates with a common angular velocity and a constant distance between the two wheels is maintained. Flanges are provided on the inside edge of the treads and the flange-way clearance allows, typically, ± 7−10 mm of lateral displacement to occur before flange contact. Whilst many wheelsets commence life with purely coned treads, typically coned at 1/20 or 1/40, these treads wear rapidly in service, so that the treads come to possess curvature in the transverse direction. Similarly, rails also possess curvature in the transverse direction. All these

BASIC CONCEPTS

3

Figure 1.1 Railway wheelset.

features contribute to the behaviour of the railway vehicle as a dynamic system, and it is important to consider their purpose. The conventional railway wheelset has a long history [1] and seems to have evolved by a process of trial and error. Naturally, in the pioneering days of the early railways most attention was concentrated on reducing rolling resistance so that the useful load that could be hauled by horses could be multiplied. Another major problem was the lack of strength and resistance to wear of the materials then available. Moreover, the level of adhesion between rolling wheel and the track was unknown. As a result, many possibilities were tried. An obvious step was to fit wheels with cylindrical treads. However, if the wheels are fixed on the axle and the treads are intended to be cylindrical very slight errors in parallelism would induce large lateral displacements which would be limited by flange contact. There is no guidance until flange contact and thus a wheelset with cylindrical treads tends to run in continuous flange contact. The position of the flange, either inside or outside the rails, was controversial well into the nineteenth century. Nor was there agreement as to whether the wheels should be rigidly fixed to an axle or free to revolve on the axle, though the usual practice seemed to be that wheels were fixed to the axle. The play allowed between wheel flange and rail was initially minimal. In the early 1830s the flangeway clearance was opened up with the objective of reducing the lateral forces between wheel and rail. A further important point is that the geometry of the wheel and rail as it has evolved is particularly favourable for the method of switching which involves a minimum of moving parts and only small gaps in the running surfaces of the rails. It is not known when coning of the wheel tread was first introduced. It would be natural to provide a smooth curve uniting the flange with the wheel tread, and wear of the tread would contribute to this. Moreover, once wheels were made of cast iron, taper was normal foundry practice. The purpose of coning was partly to reduce the rubbing of the flange on the rail, and partly also to ease the motion of the vehicle in curves. A wheelset with coned wheels in a curve can maintain a pure rolling motion if it moves outward and adopts a radial position. Redtenbacher [2] provided the first theoretical analysis in 1855 which is illustrated in Figure 1.2. From the geometry in this figure it can be seen that there is a simple geometric relationship between the

4


R

2λ

R C D

A

O

B l

y

l

OAB = OCD ( r0 - λ y )/( R - l ) = ( r0 + λ y )/( R + l ) y = r0 l / R λ Figure 1.2 Redtenbacher’s formula for the rolling of a coned wheelset on a curve.

lateral movement of the wheelset on a curve y, the radius of the curve R, the wheel radius r0, the lateral distance between the points of contact of the wheels with the rails 2l and the conicity λ of the wheels in order to sustain pure rolling. In practice a wheelset can only roll round moderate curves without flange contact, and a more realistic consideration of curving requires the analysis of the forces acting between the vehicle and the track. It can be seen, in broad terms, why the wheelset adopted its present form. If the flange is on the inside the conicity is positive and as the flange approaches the rail there will be a strong steering action tending to return the wheelset to the centre of the track. If the flange is on the outside the conicity is negative and the wheelset will simply run into the flange and remain in contact as the wheelset moves along the track. Another factor is the behaviour in sharp curves. If the flange is on the inside then the lateral force applied by the rail to the leading wheelset is applied to the outer wheel and will be combined with an enhanced vertical load. As explained later, this diminishes the risk of derailment. With outside flanges the lateral force applied by the rail applied to the inner wheel which has a reduced vertical load and thus the risk of derailment is increased. These factors can be easily demonstrated with the aid of model wheelsets [3]. Thus, it can be seen that for small displacements from the centre of straight or slightly curved track the primary mode of guidance is conicity and it is on sharper

BASIC CONCEPTS

5

curves and switches and crossings that the flanges become the essential mode of guidance. Though this appears to be a modern view, in 1838 Brunel [4] wrote The flanges are a necessary precaution but they ought never to touch the rail and therefore they cannot be said to keep the wheels on the rails. They ought not to come into action except to meet an accidental, lateral force. A railway with considerable curves might be travelled over with carriages at any velocity and with wheels without flanges. The wheels are made conical, the smaller circumference at the outer edge. The pair of wheels are fixed to the axle and thus if anything throws the wheels in the slightest degree to one side the wheel is immediately rolling on a larger circumference than the other and the tendency to roll back is introduced. The carriage is kept always in the middle of the track. A beautiful arrangement. As a concept, this view led to many significant improvements in the design of railway vehicle suspensions in the 20th century. Coning of the wheel tread was well established by 1821. George Stephenson in his Observations on Edge and Tram Railways [5] stated that It must be understood the form of edge railway wheels are conical that is the outer is rather less than the inner diameter about 3/16 of an inch. Then from a small irregularity of the railway the wheels may be thrown a little to the right or a little to the left, when the former happens the right wheel will expose a larger and the left one a smaller diameter to the bearing surface of the rail which will cause the latter to loose ground of the former but at the same time in moving forward it gradually exposes a greater diameter to the rail while the right one on the contrary is gradually exposing a lesser which will cause it to loose ground of the left one but will regain it on its progress as has been described alternately gaining and loosing ground of each other which will cause the wheels to proceed in an oscillatory but easy motion on the rails.

y = a sin ω t

s = Vt

d y ω y ω r0 l 1 =− = = 2 R ds 2 V2 V Rλ 2

2

2

Figure 1.3 Derivation of Klingel’s formula for the kinematic oscillation of a wheelset from Redtenbacher’s formula in Figure 1.2.

6


This is a very clear description of what is now called the kinematic oscillation, as shown in Figure 1.3. Thus, if a wheelset is rolling along the track and is displaced slightly to one side, the wheel on one side is running on a larger radius and the wheel on the other side is running on a smaller radius. Because the wheels are mounted on a common axle one wheel will move forward faster than the other because its instantaneous rolling radius is larger. Hence, if pure rolling is maintained, the wheelset moves back into the centre of the track − a steering action is provided by the coning. However, the wheelset overshoots the centre of the track and the result is the kinematic oscillation. In 1883 Klingel gave the first mathematical analysis of the kinematic oscillation [6] and derived the relationship between the wavelength Λ and the wheelset conicity λ, wheel radius r0 and lateral distance between the contact points between wheels and rails 2l as

Λ = 2π (r0l/λ)1/2 This simple formula follows purely from the geometry of Figure 1.3, and is consistent with Redtenbacher’s formula for the wheelset in a curve. Since distance along the track s = Vt where V is the forward speed and t is time, Klingel’s formula shows that, as the speed is increased, so will the frequency of the kinematic oscillation. Very little else can be deduced about the dynamical behaviour of railway vehicles which must come from a consideration of the forces acting.

1.3 Creep Pure rolling rarely takes place, and wheels and rails are not rigid. The normal load between wheel and rail causes local elastic deformation and an area of contact, the contact patch, is formed. In the case where the surfaces of the wheels and rails are smooth and have constant curvature in the vicinity of the contact patch, Hertz [7] showed that the contact patch was elliptical in shape, and the distribution of normal pressure between wheel and rail over the contact patch is semi-ellipsoidal. If a longitudinal force is applied to the wheel, so that it is braked, a deviation from the pure rolling motion occurs. The deviation in relative velocity divided by the forward speed of the wheel is referred to as the longitudinal creepage. Similarly, lateral creepage is defined as the (incremental) relative lateral velocity divided by the forward speed. In addition, relative angular motion between wheel and rail about the normal to the contact patch is referred to as spin. If the longitudinal creepage is small, it is accommodated by elastic strains in the vicinity of the contact patch. As the wheel rotates, unstrained material enters the contact patch at its leading edge. As the material moves through the contact patch, the relative velocity between the wheel and rail equals the rate of change of strain so that the surfaces are locked together. The magnitude of the resulting longitudinal tangential stress increases linearly with distance from the leading edge. Similarly, lateral creepage gives rise to lateral tangential stresses. Both longitudinal and lateral creepage therefore generate forces which are directly proportional to the corresponding creepage. When there is spin, the pattern of elastic strain is more complicated. In this case, as the material moves

BASIC CONCEPTS

7

(a) slip

locked region σz

σx = µσz

(b)

(c)

slip

locked

Figure 1.4 The contact patch between wheel and rail (a) elevation showing locked region of adhesion at leading edge and region of slip at trailing edge (b) normal pressure σz and tangential traction applied by wheel to rail σx (c) contact patch in plan view.

through the contact region the relative velocity between wheel and rail is directly proportional to the distance from the centre of the contact region and therefore the strain field becomes curved. As a consequence, a lateral force is generated (the couple about the common normal is small and may be safely neglected). As the creepages and spin increase, the tangential stresses increase, and where these stresses exceed the normal pressure multiplied by the coefficient of friction, slipping takes place. The result is that the area of adhesion at the front of the contact patch in which the surfaces are locked together progressively reduces as the creepage increases, Figure 1.4. The relationship between the creep force and creepage is then as shown in Figure 1.5(a). For sufficiently large creepage, slipping takes place over the whole contact patch and the creep force is equal to the normal force multiplied by the coefficient of friction. If both longitudinal and lateral creep occur simultaneously then for small creepages the creep forces can be superposed, but for larger creepages in the area of slipping the tangential stresses act in a direction opposite to the local resultant relative velocity. The result is that then all the creep forces are influenced by both lateral and longitudinal and lateral creepages and spin. Though the lateral force is proportional to spin for small values of the spin, for large values of the spin slipping takes place over a large part of the contact patch and the lateral force reduces to zero. The relationship between lateral force and spin is therefore as shown in Figure 1.5(b). It was Carter's [8] introduction of the creep mechanism into the theory of lateral dynamics that was the crucial step in developing a realistic model of the wheelset.

8


15000

(a)

(b)

-T2 (N)

-T1 (N)

0

8000

0

γ1

0.015

0

0

ω3

6

Figure 1.5 (a) Variation of longitudinal creep force T1 with longitudinal creepage γ1 showing the limiting value µT3 where µ = 0.3 is the coefficient of friction and T3 = 39000 is the normal force (b) Lateral creep force T2 as function of spin ω3 (zero longitudinal and lateral creep).

In the light of the creep theory the action of conicity can be considered as follows If a wheelset is rolling along a track and is displaced laterally, the rolling radii are different on the two wheels. Noting that the wheels are fixed on a common axle, The tread velocities are fractionally different and so longitudinal creep is generated. The corresponding creep forces are equivalent to a couple which is proportional to the difference in rolling radii or conicity, and which tends to steer the wheelset back into the centre of the track. This is the basic guidance mechanism of the wheelset. In addition, when the wheelset is yawed, a lateral creep force is generated. In effect, this coupling between the lateral displacement and yaw of the wheelset represents a form of feedback, and this introduces the possibility of dynamic instability.

1.4 Stability An important feature of a railway vehicle is that, in addition to the vertical suspension connecting the wheelsets to the vehicle body, there are lateral and longitudinal springs and dampers. This is illustrated by the plan-view of a two-axle vehicle shown in Figure 1.6. The purpose of the plan-view suspension is to stabilise the tendency of the wheelsets to oscillate and to facilitate the motion of the vehicle in curves. Because of the action of the creep forces the motion of a rolling wheelset incorporated in a vehicle is significantly different from Klingel’s description. At low forward speeds successive overshoots decrease in magnitude as the vehicle moves along the track, eventually to travel in a straight line down the centre of the track. The vehicle is dynamically stable, because following a slight disturbance it returns to its original path. At high forward speeds successive overshoots grow as the vehicle moves along the track, for in this case it is dynamically unstable.

BASIC CONCEPTS

9

The mechanism of this instability may be appreciated by considering the simple vehicle of Figure 1.6 in the special case where the vehicle body has very high inertia and is assumed to move forward at constant speed but does not undergo any lateral motion. At any forward speed the tendency of the wheelset is to oscillate at the frequency of the kinematic oscillation. Since this frequency is proportional to speed, then at low speeds the inertia forces will be small and the main component of the resultant force acting on the wheelset is the restoring force provided by the springs connecting the wheelsets to the vehicle body. In order to balance this force, creep must be developed and this will cause a progressive reduction in lateral displacement as the wheelset pursues its oscillatory path. At high speeds the inertia forces will dominate, as the frequency is correspondingly high. In this case, creep must be developed which will cause a progressive increase in the lateral displacement of the wheelset during its lateral oscillation. It follows that there is a speed at which the successive overshoots neither grow nor decay, the wheelset then, and only then, tracing out a sinusoidal path. Klingel’s solution for pure rolling then emerges as a special case of the dynamics of the wheelset. In general, a railway vehicle is stable at low speeds so that following a disturbance the vehicle, together with its wheelsets, will undergo a decaying oscillation, and the vehicle will return to the centre of the track. As the speed is increased, the decay rate of the oscillation is reduced. In most cases, at a sufficiently high speed the oscillations following a disturbance grow and eventually lead to a limit cycle oscillation where the amplitude is limited by either contact of the flanges with the rails or slipping of the wheels on the rails. The energy required to sustain this oscillation clearly comes from the energy of the forward motion. This fully developed limit cycle oscillation is generally termed “hunting”. The lowest vehicle speed at which sustained oscillations can occur is known as the critical speed. Early measurements of hunting, made in the 1960s, showed that twoaxle freight vehicles of then current design had critical speeds as low as 30 km/h [9]

Figure 1.6 Plan view of two-axle railway vehicle showing lateral and longitudinal suspension springs.

10


Hunting Limit Cycle

H Vc

V Figure 1.7 Typical variation of lateral force H acting on wheelset as a function of forward speed V for a two-axle vehicle subject to instability.

and this was a contributory cause of many derailments. Similarly a typical passenger bogie vehicle of the same era had a critical speed of about 110 km/h which though not dangerous was the cause of bad riding [10]. In practice, large lateral forces can be experienced by a hunting vehicle, as indicated in Figure 1.7. The first successful theoretical prediction and experiment on the track is described in [11]. The danger posed by hunting at high speeds was appreciated by Matsudaira who was the first to study the effects of the suspension on stability [12], and was graphically demonstrated by the lateral distortion of the track during high speed trials of a locomotive by French Railways in 1955 [13]. Earlier, the Sevenoaks accident in which a locomotive was derailed at high speed was explained by Carter [14] as a form of static instability in which the wheelbase buckled under the action of the creep forces. Thus the guidance offered by the coning of the wheels is the source of potential instability. This is the fundamental conflict in the design of the running gear of a railway vehicle.

1.5 Guidance Guidance is the ability of a vehicle to follow the geometric layout of the track. Ride is the ability of a vehicle to minimise the dynamic response, in terms of stresses and accelerations, to the layout of the track. The actual track layout will consist of the design layout, largely determined by geographical and operational factors and superimposed irregularities or lack of accuracy of real track. The most important aspect of guidance is the behaviour of vehicles in curves. The application of Redtenbacher’s formula shows that a wheelset will only be able to move outwards to the rolling line if either the radius of curvature or the flangeway

BASIC CONCEPTS

11

Y

C S

S

Figure 1.8 Two-axle vehicle in curve showing the relative magnitude of the forces acting between wheel and rail together with the forces applied to the wheelsets by the suspension S, the centrifugal force C and horizontal component of the normal force between wheel and rail Y.

clearance is sufficiently large. Moreover, as discussed above, for stability the wheelsets in a vehicle must be constrained in some way. Thus in many curves, the wheelsets are not able to take up a radial position, and typically the attitude of a two-axle vehicle in plan view is as shown in Figure 1.8. Because a wheelset is constrained by the longitudinal and lateral stiffnesses connecting it to the rest of the vehicle, it balances a yaw couple applied to it by the suspension by moving further out in a radial direction so as to generate equal and opposite longitudinal creep forces, and it will balance a lateral force by yawing further. In sharp curves the flange will be in contact with the rail. In the case of the leading wheelset of a vehicle, the flange force will be reacted by the creep forces generated by the wheels creeping laterally towards the inside of the curve, Figure 1.8. The first essentially correct description of curving was given by Mackenzie [15] in 1883. His discussion was based on sliding friction, and neglects coning, so that it is most appropriate for sharp curves, where guidance is provided by the flanges. Whilst new wheel profiles are often purely coned on the tread, usually to an angle of 1:20, treads wear rapidly and assume a hollow form, Figure 1.9. For a worn profile the variation of the difference in rolling radius between the right hand and left hand wheels as the wheelset is displaced laterally can be very complicated. A useful concept is the equivalent conicity which is the linearised slope of the difference in

12


Figure 1.9 Geometry for typical worn wheel rolling on worn rail, showing the movement of the contact point as the wheel is displaced laterally.

(a )

(b )

Figure 1.10 Normal and lateral tangential forces acting on wheelset (a) in central position (b) in laterally displaced position, illustrating the gravitational stiffness effect.

BASIC CONCEPTS

13

rolling radius plotted against lateral displacement. In practice, the equivalent conicity of a worn wheelset is much greater than that of a purely coned wheelset. This increased conicity enables a wheelset to roll round a curve of much smaller radius than is possible for a coned wheel. The influence of the hollow wheel tread gives rise to additional, significant forces acting between wheel and rail. When the wheelset is rolling on equal radii in the central position, the contact plane is inclined to the horizontal at a small angle as shown in Figure 1.10(a). When the wheelset is displaced laterally, contact is made at new points and the inclination of the contact planes is changed; on one wheel it is increased and on the other it is reduced, as shown in Figure 1.10(b). As the normal reactions between wheel and rail, which support the weight carried by the wheelset are similarly inclined, it is found that there is a lateral resultant when these normal reactions are resolved horizontally. This lateral restoring force is proportional to lateral displacement and is referred to as the gravitational stiffness. A second effect due to the hollow wheel tread arises from the lateral force generated by spin [16]. As the inclination of the contact plane changes with lateral displacement, the resolved component of the angular velocity of the wheelset due to its rolling motion, taken about an axis normal to the contact plane, gives rise to changes in spin creepage. This generates a lateral force which is proportional to lateral displacement and which is of the opposite sign to that of the gravitational stiffness force. For small displacements, typically the total contact stiffness (gravitational + spin) is reduced to about 20% of the gravitational stiffness, and is consequently rather small. However, as mentioned above, the lateral force due to spin becomes small for the large values of spin achieved in flange contact, and there is a large restoring force due to flange contact. Heumann [17] suggested that profiles approximating to the fully worn should be used rather than the purely coned treads then standard. After re-profiling to a coned tread, tyre profiles tend to wear rapidly so that the running tread normally in contact with the rail head is worn to a uniform profile. This profile then tends to remain stable during further use, and is largely independent of the original profile and of the tyre steel. Similarly, rail head profiles are developed which also tend to remain stable after the initial period of wear is over. These results suggested that vehicles should be designed so as to operate with these naturally worn profiles, as it is only with these profiles that any long-term stability of the wheel-rail geometrical parameters occurs. Moreover, a considerable reduction in the amount of wear is possible by providing new rails and wheels with an approximation to worn profiles at the outset and this has now become common practice in many countries.

1.6 Suspension, Performance and Criteria For satisfactory performance, a railway vehicle must meet certain criteria. The most fundamental of these is concerned with the possibility of derailment. Figure 1.11 shows a yawed wheelset in flange contact with a rail. There are initially two points of contact between wheel and rail, one on the wheel tread and the other on the flange. The latter point of contact lies ahead of the former. The onset of flange-climbing

14


F

Q N

α

Y

F b a

Q

Figure 1.12 Flange contact when wheelset is yawed at angle α with rail, showing first point of contact on tread a and second point of contact with flange b.

derailment occurs when the vertical load Q is carried entirely by the point of contact on the flange, and so the derailment limit is defined by the minimum value of the lateral reaction Y. If the component of the tangential force in the transverse vertical plane is denoted by F, and N is the normal reaction, the balance of forces in Figure 1.12 shows that Y N tan α − F < Q N + F tan α The minimum value of Y/Q occurs when F is a maximum so that by the laws of friction F cannot exceed µN where µ is the coefficient of friction, hence tan α − µ Y = Q 1 + µ tan α This derailment criterion is due to Nadal, but like many succeeding studies, its analysis was based on suspect assumptions. However, the application of the laws of creep by Gilchrist and Brickle [18] has shown that Nadal’s formula is correct for the most pessimistic case when the angle of attack is large and the longitudinal creep on the flange is small. It will be apparent that derailment is most likely when a large lateral force occurs simultaneously with a reduced vertical load on a wheel, typically in a curve with a significant vertical irregularity, or a high degree of track twist. Another mode of failure occurs when the lateral forces imposed by the vehicle are sufficient to shift the track laterally. An empirical criterion for this has been given by Prud’homme [19] and is

BASIC CONCEPTS

15

Y = 10 + W/3 where W is the axle-load, and both Y and W are measured in kN. These are extreme conditions. More generally, the curving ability of a railway vehicle is considered good if the angle of yaw of each wheelset is small, flange contact is avoided on all but sharp curves, the lateral forces between wheel and rail are low and the energy expended in wheel rail contact is small. A measure of curving performance that is often used in practice is the degree of flange wear experienced in service and empirical relationships have been established between rates of wear and the energy expended in the contact area. After the essential stability and curving requirements are met, it is considerations of ride quality which dominate the detailed design of railway vehicle suspension systems. The layout of curves is defined by the maximum cant of the track, and vehicle speed, so that the lateral acceleration applied to passengers is within acceptable limits. Moreover, the length of the transition between straight and curved track, and in some cases the shape, is determined by limits on the time rate of change of cant deficiency. So far, the dynamics of railway vehicles running on perfectly aligned straight or curved track has been considered. A further important problem is that of the dynamic response of vehicles to track irregularities. Track irregularities arise from specific features such as switch and crossing work or from the continuous distribution of roughness which must be controlled by maintenance of the track. In the latter case the irregularities may be considered as randomly distributed. The dynamic response, in terms of the acceleration level on the car body, to this stochastic input will characterise the ride quality of the vehicle. Detailed international standards for passenger comfort criteria exist which define frequency weighting characteristics describing human response to vertical and lateral vibration, which are used in the specification and assessment of vehicles. The excursions of the car body on the suspension must remain within the structural clearance gauge of the route on which the vehicle is to run.

1.7 Suspension, Articulation and Curving It can be seen that a vehicle with a perfect suspension would be stable at all operational speeds, would negotiate curves by minimising the forces acting between wheel and rail and in traversing irregular track would minimise the acceleration levels in the car body and the stresses applied to both vehicle and track structures. These requirements are usually conflicting and compromise, informed by analysis, is required in design. Not only are the parameters that are associated with wheel-rail contact, both geometrical and frictional, not under the control of the designer or operator, but they are not known exactly and can vary over a wide range. It follows that practical designs must be very robust in relation to such parameters. On the other hand, there is enormous scope for the design of the suspension system in terms of the way in which the wheelsets and car bodies in a train are connected.

16


It has long been the objective of vehicle design to incorporate wheel and steering arrangements that permit a vehicle to follow a chosen path or a track by a motion which involves pure rolling of the wheels, apart from the necessary transmission of traction forces and the reaction of centrifugal force. In the first place pure rolling might be achieved by a choice of configuration of the wheels and the way in which they are articulated. Such rolling motions may or may not be statically or dynamically stable. Additionally the introduction of creep, in response to inertia and suspension forces may stabilise or destabilise the system. It follows that there are many possible mechanisms of guidance and stability which can be considered. There have been many attempts to provide an alternative to the railway wheelset and reference will be made to some of these in Chapter 3. However, in using the conventional railway wheelset, there are many ways to improve performance in curves by making a vehicle more flexible in plan view, thus encouraging the axles to take up a moreor-less radial position in curves. It will be shown later that a two-axle vehicle that is capable of radial steering on a uniform curve will be dynamically unstable at all speeds so that the design of a two-axle vehicle requires a compromise between stability and curving. This is the subject of Chapters 4 and 5. As discussed in Chapters 6 and 7 for a vehicle with three or more axles it is possible to arrange the suspension so that radial steering and dynamic stability are both achieved. One approach is to provide elastic or rigid linkages directly between wheelsets in a vehicle. This can be referred to as self-steering as the vehicle body is not involved. Alternatively, a linkage system can be provided which allows the wheelsets to take up a radial position but provides stabilising elastic restraint from the vehicle body. This is so-called forced steering as it can be considered that the vehicle body imposes a radial position on the wheelsets. There are many designs in which there is articulation of the vehicle bodies of a vehicle or train. Articulation, in the present context, describes an arrangement in which the relative motion between the vehicle bodies is used to influence the stability and guidance of the vehicle. In many cases the interaction between the vehicles in a train is minimised by the form of coupling between the vehicles, so that longitudinal forces can be transmitted between car bodies, but the coupler is capable of transmitting little or no lateral force or yaw couple. In this case it is a good approximation to treat each vehicle as if it were isolated and the lateral dynamics of each vehicle can be considered to be largely independent of that of the rest of the train. In an articulated vehicle the connections between vehicles form an essential part of the running gear. A common design feature is to link the relative angle between vehicle bodies to yaw of the wheelsets. As is discussed in Chapter 8, such designs improve curving performance and other aspects of vehicle design but can exhibit a wide spectrum of various hunting instabilities. All the configurations discussed so far have been symmetric fore-and-aft. Unsymmetric configurations make it possible, in principle, to achieve a better compromise between curving and dynamic stability, at least in one direction of motion. But additional forms of instability can occur as discussed in Chapter 9. The equations of motion are fundamental for all configurations of vehicle, and the derivation of these for a wheelset and a simple two-axle vehicle is discussed in Chapter 2.

BASIC CONCEPTS

17

References 1. Wickens, A.H.: The dynamics of railway vehicles-from Stephenson To Carter. Proc. I. Mech. E. 212, Part F (1998), pp. 209-217. Gilchrist, A.O.: The long road to solution of the railway hunting and curving problems. Proc. I. Mech. E. 212, Part F (1998), pp. 219-226. 2. Redtenbacher, F.J.: Die Gesetze des Locomotiv-Baues. Verlag von Friedrich Bassermann, Mannheim, 1855, p. 22. 3. Wickens, A.H.: Dynamics and the advanced passenger train. Speaking of Science 1977, Proceedings of The Royal Institution of Great Britain, 50 (1978), pp. 33-65. 4. Vaughan, A.: Isambard Kingdom Brunel − Engineering Knight Errant. John Murray, London, 1992, p. 102. 5. Dendy Marshall, C.F.A.: History of British Railways Down to the Year 1830. Oxford University Press, Oxford, 1938, p. 165. 6. Klingel.: Uber den Lauf der Eisenbahnwagen auf Gerarder Bahn. Organ Fortsch. Eisenb-wes. 38 (1883), pp. 113-123. 7. Timoshenko, S.P.: A History of the Strength of Materials, Mcgraw-Hill, New York, 1953, p. 348. 8. Carter, F.W.: The electric locomotive. Proc. Inst. Civ. Engs. 221, 1916, pp. 221252. 9. Pooley, R.A.: Assessment of the critical speeds of various types of four-wheeled vehicles. British Railways Research Department Report E557, 1965. 10. King, B.L.: The measurement of the mode of hunting of a coach fitted with standard double-bolster bogies. British Railways Research Department Report E439, 1963. 11. Gilchrist, A. O., Hobbs, A.E.W., King, B.L. and Washby, V.: The riding of two particular designs of four wheeled vehicle. Proc. I. Mech. E. 180 (1965), pp. 99113. 12. Matsudaira, T.: Hunting problem of high-speed railway vehicles with special reference to bogie design for the New Tokaido Line. Proc. I. Mech. E. 180 (1965), pp. 58-66. 13. Knothe, K. and Bohm, F.: History of stability of railway and road vehicles. Vehicle System Dynamics, 31 (1999), pp. 283-323.

18


14. Carter, F.W.: The running of locomotives, with reference to their tendency to derail. Inst. Civil Engs, Selected Engineering Paper, No. 81, 1930. 15. Mackenzie, J.: Resistance on railway curves as an element of danger. Proc. Inst. Civ. Engs. 74 (1883), pp. 1-57. 16. Johnson, K.L.: Effect of spin upon the rolling motion of an elastic sphere on a plane. Trans. A. S. M.E. Ser. E, 80 (1958), pp. 332-338. 17. Heumann, H.: Zur Frage des Radreifen-Umrisses. Organ Fortschr. Eisenb.-wes. 89 (1934), pp. 336-342. 18. Gilchrist, A.O. and Brickle, B.V.: A re-examination of the proneness to derailment of a railway wheelset. J. Mech. Eng. Sci. 18 (1976), pp. 131-141. 19. Birmann, F.: Theoretical and experimental solutions of track problems for high speeds. Monthly Bulletin of the International Railway Congress Association 45 (1968), pp. 391-460.

2 Equations of Motion 2.1 Introduction The basic physical phenomena involved in the dynamics of railway vehicles have been described in Chapter 1. Equations of motion governing the stability and dynamic response of vehicles will now be derived which encompass the essential features of the wheel-rail geometry, the frictional forces acting between wheel and rail and the elastic and damping forces generated by the suspension. As attention will be confined to the dynamics at low frequencies, the wheelset and track are assumed to be rigid apart from local elasticity in the contact patch between wheel and rail, and the contributions of the local deflections near the contact patch to the overall motion of the wheelset are neglected. The wheelset, which is assumed to be axisymmetric about the axle centreline, is considered to be constrained to run along the track at constant speed. The track is arbitrarily curved in plan view and may be canted. The kinematics of the wheelset is considered first, and this is followed by a discussion of wheel rail geometry. An evaluation of the creep forces acting between wheel and rail makes it possible to formulate equations of motion of a freely running wheelset. This is followed by the derivation of the equations of motion of a complete two-axle vehicle in which the action of the suspension is taken into account.

2.2 Freedoms and Constraints The track possesses curvature in a horizontal plane with radius R0, cant or crosslevel φ0, and can be displaced locally through a lateral displacement y0, Figure 2.1. R0, φ0, and y0 vary with the distance s along the track. The wheelset reference frame Oxyz is attached to the centreline of the undistorted track, Figure 2.1, and moves along the track at the speed of the vehicle V. Thus the irregularities y0 are measured from this centreline. The origin of this set of axes is located at the centre of mass of the wheelset when the wheelset is central on the track. Ox lies along the tangent to the track centreline, Oy lies along the wheelset axle centreline when the wheelset is central and lying in the radial direction on the curve, and Oz is mutually perpendicular. The coordinates X, Y and azimuth Ψ of the origin of the frame Oxyz, Figure 2.1,

20


X

Y

O*

O

Ψ x

x*

y*

ψ

y

O G

φ0

y

φ y*

z

z*

Figure 2.1 Wheelset axis systems and coordinates.

with reference to an axis system fixed in the earth are given by dX/ds = cosΨ

(1)

dY/ds = sinΨ

(2)

dΨ/ds = 1/R0

(3)

A second set of axes O*x*y*z* has origin at the centre of mass of the wheelset. O*y* coincides with the axle centreline, O*x* is perpendicular to O*y* and O*z* is mutually perpendicular. To specify the orientation of the frame O*x*y*z* it will be convenient to select successive rotations, yaw ψ, and roll φ, about the carried axes Oz*, Ox*. Thus the rotations are taken about the position the axes have taken following the previous rotation. The rotation of the wheelset about the carried axis Oy* is denoted by θ. The displacements of the wheelset centre of mass O* relative to Oxyz are denoted

EQUATIONS OF MOTION

21

by the vector x0 with components ux, uy and uz, but as O* is in the plane yOz and both O and O* move forward at constant speed V x0 = [ 0 uy

uz ]

(4)

and the longitudinal position along the track s = Vt. Thus the position and orientation of a wheelset can be defined in terms of the six variables, s, the lateral and vertical displacements uy and uz and three rotations, yaw ψ, roll φ, and θ. As the area of contact between wheel and rail is small compared with the dimensions of the track contact between the wheelset and the rails can be considered to take place ordinarily at two points. As will be discussed later, two point contact gives rise to two constraint equations which makes it possible to eliminate two of the above coordinates. It will be convenient to eliminate the vertical displacement and roll angle of the wheelset as independent coordinates, so that they become simply dependent functions of lateral displacement and yaw. As the vehicle speed is constrained to be constant, the system has three degrees of freedom.

2.3 Wheel Rail Geometry As discussed in Chapter 1 the most important geometrical characteristics of the wheel rail geometry are (a) the variation of rolling radius with lateral displacement as this governs the conicity effect and (b) the variation of the slope at the contact point with lateral displacement as this governs the gravitational stiffness effect. For a typical wheel and rail combination, Figure 2.2, both profiles have curvature which varies continuously across the rail head and wheel tread and are defined by

ζw = f (ηw)

ζr = g (ηr)

(1)

ζr

ηr Figure 2.2 Typical wheel and rail profiles relative to the rail coordinates ζr , ηr (mm).

22


δl

ηl

δr

ηr

ζl

ζr uy-y0-r0φ ηwr

ζ wr

B

-uz-lφ A

ζ rr

δ0

C central

ηrr

displaced

Figure 2.3 Wheel rail geometry (right-hand). A is the origin of the rail axes ζr, ηr, B is the origin of the wheel axes ζw, ηw, so that A and B are coincident when the wheelset is central. When the wheelset is displaced, contact takes place at C.

where ηw, ζw are the wheel coordinates and ηr, ζr are the rail coordinates, the profiles being the same for the right-hand and left-hand wheels, Figure 2.3. In order to derive equations of motion, the position of the contact points, and the slopes and curvatures at these contact points, as functions of the wheelset lateral displacement and yaw are required. It will be assumed that the cross-sectional geometry of the wheel-rail system does not vary with distance along the track. It follows that the cross-sectional geometry is independent of s and θ. The wheels and rails will be assumed to be rigid in so far as their mutual geometry is concerned. When the wheelset is in the central position on the track, and is not yawed, the angle made by the contact plane with the horizontal is δ0 and the tread circles of the wheels have the same radius r0. When the wheelset is displaced laterally, the angles made between the contact planes and the axle centreline at the new points of contact are δwr and δwl. Similarly, the radii of the tread circles become rr and rl. The position of the contact points is determined by noting that the wheel and rail contact points must occupy the same position in space, the angles made by the contact planes at the points of contact must be the same for wheel and rail, and the contacting bodies cannot penetrate each other. As the angle of yaw of a wheelset is small in most ordinary circumstances, the two-dimensional case where the influence of yaw is neglected will be considered. Consider the right hand wheels and rails, Figure 2.3. When the

EQUATIONS OF MOTION

23

wheelset is in the central position the contact point is A. The centreline of the track is displaced laterally by y0 from the reference axis from which uy is measured. When the wheelset centre of mass is displaced laterally through a distance uy from the reference axis the wheelset rotates about a longitudinal axis through a small angle φ and so the lateral displacement at the contact point is uy - y0 - r0φ, and contact is made at a new point C. If the lateral movement of the contact point on the right hand wheel is ηwr and that on the rail is ηrr then uy - y0 - r0φ - ηwr + ηrr = 0

(2)

Similarly, considering the vertical movement of the wheelset ζwr at the right hand rail uz + lφ + ζwr - ζrr = 0

(3)

Also, if if δrr denotes the angle between the rail axes and the contact plane

φ - δwr + δrr = 0

(4)

The three corresponding equations for the left hand wheel are uy - y0 - r0φ + ηwl - ηrl = 0

(5)

uz - lφ + ζwl - ζrl = 0

(6)

φ + δwl - δrl = 0

(7)

In addition, the slopes at the contact points are given by tan δwr = dζwr /dηwr

(8)

tan δwl = dζwl /dηwl

(9)

tan δrr = dζrr /dηrr

(10)

tan δrl = dζrl /dηrl

(11)

For profiles specified by (1), equations (1) to (11) can be solved to yield the contact positions and slopes, the vertical displacement and roll angle as functions of uy. Measuring equipment has been developed to measure wheel and rail profiles with the necessary accuracy, [1, 2, 3]. The solution has been implemented in computer programs and typically uses a Newton-Raphson iterative procedure to solve the nonlinear algebraic equations. The first step is to determine the points of contact when the wheelset is central. This then defines a new origin for the wheel rail geometric data. Then equations (1) to (11) are solved for the contact points. Once the contact points have been established the various geometrical characteristics can be

24


0

1 φ (mr)

uz (mm)

0

-0.2

-1 -10

0 r

10 ηr (mm) 0 -10 -10

-0.4 -10

10

10

20 ηw (mm) 0 10

0

0

-20 -10

r l 0

10

2 ζr (mm) 1

r

ζw (mm) 2

r

0

l

0

l

-10

10 0 uy (mm)

-10

0 10 uy (mm)

Figure 2.4 Roll angle φ, vertical displacement uz and location of contact points as a function of lateral displacement for the wheel rail combination of Figure 2.2.

determined. Detailed discussions of the analytical aspects of wheel rail geometry have been given by de Pater [4] and Yang [5]. As an example, for the wheelset and rail combination shown in Figure 2.2 the variation of φ, uz and the location of the contact points is shown in Figure 2.4. Figure 2.5 shows the variation of rolling radius, contact angle and the transverse curvatures of the rail and wheel with lateral displacement. The examples shown refer to worn wheel and rail profiles and in this case yield quite smooth characteristics. However, in practice, many rail and wheel profile combinations yield characteristics with major discontinuities. Though, as the wheelset is displaced laterally, the rotation φ and vertical displacement uz are small it will be seen later that their derivatives with respect to uy play an important part in the equations of motion. Expressions for these derivatives will now be derived.

EQUATIONS OF MOTION

25

δ

r

r (mm)

r

l

l

l

l Rw (mm)

Rr (mm)

r

r

Figure 2.5 Variation of rolling radius, contact slope, and radii of curvature with lateral displacement for the wheel rail combination of Figure 2.2.

I l - rrtanδwr

l - rltanδwl

δuy

δz δφ

rl

rltanδwl

δu y

rr

rrtanδwr

Figure 2.6 Tilting and vertical displacement of wheelset due to lateral displacement. Wheelset rotates about instantaneous centre I.

26


Differentiating (3) with respect to uy dζ dζ du z dφ +l = rr − wr du y du y du y du y =

dζrr dηrr dζ wr dηwr − dηrr du y dηwr du y

and using (8) and (9) = tan δrr

dηrr dηwr − tan δ wr du y du y

and as φ is small ⎛ dη dη ⎞ = tan δrr ⎜⎜ rr − wr ⎟⎟ du y ⎠ ⎝ du y and differentiating (2) and substituting ⎛ dφ ⎞ ⎟ = − tan δrr ⎜⎜ 1 − r0 du y ⎟⎠ ⎝

(12)

Similarly ⎛ du z dφ dφ ⎞ ⎟ −l = tan δrl ⎜⎜ 1 − r0 du y du y du y ⎟⎠ ⎝

(13)

Hence from (12) and (13)

φy =dφ/duy = - ( tanδrl + tanδrr)/(2l - r0 tanδrr - r0 tanδlrl)

(14)

zy = duz/duy = l ( tanδrl - tanδrr)/(2l - r0 tanδrr - r0 tanδrl ) (15) which also follow from the geometry of Figure 2.6. For small displacements from the central position

φy = - σ/l ( 1 - r0 δ0 / l )

zy = -εuy/l ( 1 - r0 δ0 / l )

(16)

where

σ = δ0

ε = (δrr - δrl)l/2uy

(17)

ε is the parameter that determines the change of inclination of the normal force between wheel and rail as the wheelset is displaced laterally and therefore influences the gravitational stiffness. σ is the roll parameter. Figure 2.7 shows the variation of the geometrical derivatives φy and zy. Numerical values of the parameters for the wheel rail combination of Figure 2.2 are

EQUATIONS OF MOTION

27

r0 = 0.4500 m, δ0 = 0.0493 and l = 0.7452 m. If uy* is the lateral displacement of the wheelset in the plane of the original points of contact, then from Figure 2.6

u *y = 2luy/(2l - r0 tanδwr - r0 tanδwl)

(18)

because the wheelset is rotating about the point I. For small displacements this becomes u *y = u y/( 1 - r0 δ0 / l ) (19) From (19) the difference between the lateral displacement of the wheelset at the axle and at the contact points for the profiles of Figure 2.2 is only 3%. 0.4

0

φy (1/m)

0.2

zy

-0.2

0 -0.4

-0.2 -0.4 -10

0

10

-0.6 -10

uy (mm)

0

10

uy (mm)

Figure 2.7 Variation of derivatives zy and φy with lateral displacement for the wheel rail combination of Figure 2.2.

For small displacements equations (1)-(11) have a simple solution. In the vicinity of the point of contact Rw and Rr are the radii of curvature of the wheel tread and rail head respectively. When the wheelset is displaced laterally through a distance uy it can be seen from Figure 2.3 that the lateral displacement of the contact point on the rail is, approximately,

ηrr = Rr ( δrr - δ0 )

(20)

to the first order in δ0 and δrr The lateral displacement of the contact point on the wheel is

ηwr = Rw ( δwr - δ0 )

(21)

Substituting in (2) from (20), (21) and (4) and noting from (16) that φ = φyuy, yields for the right hand side

28


δrr , δrl = δ0 ± ε0uy/l

(22)

where, since to first order (1+ r0 δ0 / l)-1 = (1 - r0 δ0 / l)

ε0 = l ( 1 + Rwδ0 /l)/ (Rw - Rr)(1 - r0 δ0 / l)

(23)

and

δwr , δwl = δ0 ± ε 0* uy/l

(24)

where

ε 0* = l ( 1 + Rrδ0 /l)/ (Rw - Rr)(1 - r0 δ0 / l)

(25)

For example, for the wheel rail combination of Figure 2.2, ε0 = 6.423 and ε0∗ = 6.372. Note that ε0 - ε0∗ = σ. For conical wheels, Rw → ∞ , ε0∗ = 0 and

ε0 = δ0 /( 1 - r0 δ0 / l )

(26)

For profiled or worn wheels Rw δ0 / l 3. Repeating the arguments used above it is found that the number of independent elements in E is equal to (R2 - 5R + 6)/2 for an unsymmetric vehicle and to (R2 - 4R + 4)/4 (R even) or to (R2 - 4R + 3)/4 (R odd) for a symmetric vehicle. Dynamic stability requires that all the real parts of the eigenvalues of | As2 + (B/V + D)s + C + E | = 0

(11)

are negative. As discussed in Chapters 3 and 5, in general, as speed is increased, stability is lost at a bifurcation speed VB at which at least one eigenvalue becomes purely imaginary. A necessary condition for the existence of a nonzero critical speed is that the eigenvalues of the system at low speeds |(B/V + D)s + C + E | = 0

(12)

all have negative real parts, as discussed in Chapter 3. If all N wheelsets are identical,


184

and if there were no structural connections between the various rigid bodies of the system, E and D would be null, and equations (12) would reduce to N uncoupled sets of binary equations where N is the number of wheelsets. There would be N identical eigenvalues s = ± iV(λ/lr0)1/2 corresponding to a kinematic oscillation of each of the identical wheelsets. The elements of the corresponding eigenvectors qk will be yi =βi

1/2

ψi = iβi(λ/lr0)

all other qi = 0

(13)

where the βi are arbitrary in magnitude so that N quantities remain undetermined. Now if Eqk =0

(14)

equation (12) will be satisfied, and there will be at least one undamped mode at low speeds involving a kinematic oscillation of the wheelsets. The condition that equation (14) is satisfied is that P > N where P is the degeneracy of the E matrix. If P ≤ N then no solution corresponding to a kinematic oscillation can exist, and thus P ≤ N is a necessary but not sufficient criterion for dynamic stability. A necessary condition for a vehicle with wheelsets of equal conicity to be dynamically stable and capable of perfect steering in the case of zero cant deficiency is therefore 3≤P≤N

(15)

The case where the wheelsets have unequal conicities is discussed in Chapter 9 where it is shown that a small margin of stability can be obtained but is generally of little practical importance. Now consider further the behaviour of the system at low speeds. Denoting the generalised coordinates associated with the wheelsets by qs and the remaining coordinates by qv so that for the bogie vehicle qv = [ yb φb ψb yc φc ψc yd φd ψd ]T

(16)

qs = [ y1 ψ1 y2 ψ2 y3 ψ3 y4 ψ4 ]T

(17)

the equations of motion can be written in the partitioned form ⎡ Ars s 2 + ( Brs / V + Drs ) s + Crs + Ers ⎢ Dus s + Eus ⎣⎢

⎤ ⎡ q s ⎤ ⎡0 ⎤ ⎥⎢ ⎥ = ⎢ ⎥ Auv s 2 + Duv s + Euv ⎦⎥ ⎣qv ⎦ ⎣0⎦ Drv s + Erv

(18)

Making the substitution s = VD and letting V tend to 0 yields the two sets of uncoupled equations governing the motion at low speeds [BrsD + Crs + Ers - ErvEuv-1Eus ] qs = 0 or

(19)

THE BOGIE VEHICLE

185

+ Crs + Ers* ] qs = 0

[BrsD

(20)

where Ers* = Ers - ErvEuv-1Eus is the deflated stiffness matrix involving only the wheelset coordinates and [ Auvs2 + Duvs + Euv ] qv = 0

(21)

indicating that at low speeds the eigenvalues of the system fall into two distinct sets. Firstly, a set associated with the motions of the wheelsets, modified by the influence of the quasi-static elastic interaction between the wheelsets through the elastic connections to the bogie frames and car body. Secondly, a set associated with the natural modes of the vehicle body oscillating on the suspension as if the wheelsets were fixed. This generalises the results already found for the two-axle and bogie vehicles. The eigenvalues of (20) are the same as those of (12) and so the condition for stability at low speeds is dependent on E*. Similarly, the equations governing steady motion in curves, equations (7), can be replaced by [ Crs + Ers* ] qs = Qrc

(22)

The conditions for stability and perfect curving discussed above therefore also apply to the reduced system involving only the wheelset freedoms, and it follows that the number of independent elements in E* is correspondingly reduced. A similar approach has been followed by de Pater [18, 19] with some significantly different details.

6.5 Steering and Stability of a Generic Bogie Vehicle This general approach is now applied to bogie vehicles in which rather general forms of suspension are incorporated so as to meet the dual requirements of steering and stability. As the complete vehicle is assumed to have a transverse plane of symmetry the structure of the stiffness matrices is simplified. Clearly, reversal of the direction of motion should result in identical equations of motion if account is taken of the sign convention chosen for the generalised coordinates. The complete stiffness matrix takes the form bb

E

E= (1)

cb

E

db

E

bc

E

cc

E

dc

E

bd

E

cd

E

dd

E

It will be assumed that there is no direct connection between the leading and trailing bogies, so that bdE and dbE are null. The relationship between the coordinates of the leading bogie and the trailing bogie is therefore defined by


186

qb = Tqd

(2)

where T = 0 except that T16 = T33 = T44 = T61 = -1 and T27 = T55 = T72 = 1. Hence e66 -e67 e77

e63

e61

-e62

-e73 -e74 e75 -e71

e72

e33 dd

E =

e64 -e65

-e32

e34 -e35

e31

e44 -e45

e41 -e42

(3)

(sym) e55 -e51

e52

e11

-e12 e22

e86 -e87

e83

e84 -e85

e81

-e82

e96 -e97

e93

e94 -e95

e91

-e92

cd

E =

(4)

-e10,6 e10,7 -e10,3 -e10,4 e10,5 -e10,1 e10,2

From the above discussion, if there were direct connections between the leading and trailing bogies and their wheelsets the total number of parameters would be 64 assuming that the rigid body conditions were satisfied and the vehicle is symmetric. This number is much reduced if there are no direct connections between the leading and trailing bogies. Further reductions in the number of disposable parameters will occur if it is assumed that the bogies are themselves symmetric and also if the curving condition is imposed. Fully generic schemes, such as those discussed in Chapter 4, could be the starting point for an analysis of the bogie vehicle. However, a more direct approach is to derive the simplest arrangements which are capable of both perfect steering and stability as fully generic schemes are unnecessarily complex. The spirit of the discussion so far, concerning the resolution of the conflict between steering and stability, is that the vehicle should be made as flexible as possible in plan view, consistent with the need for stability. Whilst the range of configurations with the maximum number of parameters are of interest for two and three-axle vehicles, the situation for a four-axle vehicle is quite different. As discussed above, in the case of the vehicle with four axles under consideration here, a fully generic configuration meeting the criterion expressed by equation (3.15) would have a very large number of parameters, far more than is needed for practical configurations which should be as simple and as flexible as possible.

THE BOGIE VEHICLE

187

If, however, attention is confined to low speeds or quasi-static conditions then the equations of motion can be reduced to the form of (4.20) and (4.22), as discussed in Section 3, involving only 8 wheelset coordinates. If the curving condition, equation (4.10), is applied in addition to the rigid body conditions, equations (4.5) and (4.6), the stiffness matrix E* can be expressed in the form of equation (4.4). There will therefore be a maximum of 9 and a minimum of 5 independent parameters in E*. It will be shown below how this suggests an approach to the generation of generic configurations for the steered bogie vehicle. Noting that fully generic arrangements may be useful in generating design variants, the further development of the argument takes as its starting point the simpler form of arrangement shown in Figure 6.2. The arrangement is similar to conventional practice, except that provision is made for asymmetry in both geometry and in the magnitude of the stiffnesses. The solution of equations (4.10) will initially be discussed for this simplified generic arrangement. Assuming zero cant deficiency, equations (4.10) become y1{ y1 + h1ψ1 - yb + d1ϕb - (h + h1)ψb } = 0

(5)

kψ1( ψ1 - ψb) = 0

(6)

ky2{- yb + d1ϕb + (h + h2)ψb + y2 - h2ψ2} = 0

(7)

kψ2( ψb - ψ1 ) = 0

(8)

kyb{ yb + d2ϕb + h5ψb - yc + d3ϕc - (c + h5)ψc} = 0

(9)

kψb( ψb- ψc ) = 0

(10)

kϕ1 ϕb = 0

(11)

kϕb( ϕb - ϕc ) = 0

(12)

kϕ2 ϕb = 0

(13)

ky1{ y4 - h1ψ4 - yd + d1ϕd + (h + h1)ψd } = 0

(14)

kψ1( ψ4 - ψd ) = 0

(15)

ky2{- yd + d1ϕd - (h + h2)ψd + y3 + h2ψ3} = 0

(16)

kψ2( ψd - ψ3 ) = 0

(17)

kyb{ yd + d2ϕd - h5ψd - yc + d3ϕc + (c + h5)ψc} = 0

(18)

kψb( ψd - ψc ) = 0

(19)


188

kϕ1 ϕd = 0

(20)

kϕb( ϕd - ϕc) = 0

(21)

kϕ2 ϕd = 0

(22)

From equations (11) to (13) and (20) to (22) ϕb = ϕc = ϕd = 0. From equations (6), (8), (15) and (17) the wheelsets can only take up a radial position if kψ1= kψ2=0. Similarly, from equations (10) and (19) kψb =kψd = 0. Therefore for zero creep all inter-body yaw stiffnesses must be zero. If h1 and h2 are fixed then equations (5) and (7) define yb and ψb, and equations (14) and (16) define yd and ψd. If h5 is fixed then equations (9) and (18) define yc and ψc. If h1, h2 and h5 are zero then yb = ( y1 + y2 )/2

ψb = ( y1 - y2 )/2h

(23)

yd = ( y3 + y4 )/2

ψd = ( y3 - y4 )/2h

(24)

yc = ( yb+ yd)/2

ψc = ( yb - yd )/2c

(25)

so that in this case the attitude of the vehicle is completely defined by the lateral positions of the wheelsets yi. In the case of motion on a uniform curve, the analysis of Chapter 4 shows that in order to achieve zero lateral and longitudinal creep each wheelset moves outwards and aligns itself radially. Because of symmetry yd = yb, ϕd = ϕb, ψd = -ψb, y4 = y1, y3 = y2, ψ4 = -ψ1, ψ3 =-ψ2 and ψc = 0. For example, in the case of linear conicity and the outer and inner wheelsets are identical, the lateral displacements are y1 = y4 = - lr0/λR + (c + h)2/2R

y2 = y3 = - lr0/λR + (c - h)2/2R

(26)

and the yaw displacements are

ψ1 = -ψ4 = (c + h)/R

ψ2 = -ψ3 = (c - h)/R

(27)

and the displacements of the car body and bogie frames are given from (23-25) as yb = yc = yd = -lr0/λR + (c2 + h2)/2R

ψb = − ψd = c/R

ψc = 0

(28)

In the general case, where h1, h2 and h5 are not zero, Figure 6.7 indicates the geometry in which the wheelsets adopt a radial position with radial displacements appropriate to their conicity, and the frames and car body take up the positions connecting the wheelsets without strain of the suspension elements. Thus, under the conditions considered here, perfect steering is obtained because of the combined action of creep and conicity which is the primary mechanism of guidance. Additional elastic restraint is needed in order to stabilise the kinematic oscillations of the wheelsets. For the system with the elastic stiffness matrix E as derived above it

THE BOGIE VEHICLE

189

Figure 6.7 Geometry of basic bogie vehicle on curve: a bogie frame, b unstrained spring connecting bogie frame to car body (corresponding to kyb ), c and d unstrained springs corresponding to primary lateral suspensions ky1 and ky2.

may be easily verified that there are 4 pairs of imaginary eigenvalues at low speeds corresponding to undamped kinematic wheelset oscillations. This is consistent with the necessary criterion for stability, equation (4.15), derived above. As the degeneracy of E is 8, additional stiffnesses are required to reduce the degeneracy to 4 so that the criterion may be satisfied. Four additional stiffnesses must be provided, and the corresponding rows in the compatibility matrix a must satisfy the rigid body conditions expressed by equations (4.5) and (4.6) and the steering condition expressed by equation (4.10). In addition they must be linearly independent from the existing rows of a. Since the corresponding rows of the compatibility matrix will involve the wheelset yaw angles they can be considered as defining "steering laws", a term used in the literature. These sub-matrices of k and a relating to the additional elastic elements will be denoted by ks and as for convenience. Various possibilities are considered in the next section. These results are consistent with the above discussion concerning the number of independent parameters in E. The basic system, without the additional stiffnesses corresponding to the steering laws, has 3 independent stiffnesses ky1, ky2 and kyb, and a minimum of 5 are required. The two additional stiffness parameters will each appear twice in the 4 additional rows of as, two rows for the leading bogie and two rows for the trailing bogie. (The roll stiffnesses do not affect the degeneracy of E). Since no elastic forces can be generated in perfect steering, as expressed by equations (4.10), it is clear that the above solutions hold even if all the elastic stiffnesses are very large. For the configuration under discussion, in addition to equations (4.1), (4.2) and (4.9) being satisfied, symmetry implies that there is a fourth solution Eqd = 0 where qd is antisymmetric so that yd = -yb, ϕd =− ϕb, ψd =ψb, y4 = -y1, y3 = -y2, ψ4 =ψ1, ψ3 =ψ2 and yc = ϕc = 0. It can be seen that when the elastic stiffnesses are all large the system acting as a free body would have four degrees of freedom, rigid body lateral translation, rigid body yaw and symmetric and antisymmetric bending as a mechanism, corresponding to degeneracy of four in the elastic stiffness matrix, Figure 6.8.

190


Figure 6.8 Mechanism modes for basic bogie vehicle: a rigid body lateral translation, b rigid body yaw, c symmetric bending, d antisymmetric bending.

Though the application of direct structural connections between the bogies is not considered here, it is worth noting that the antisymmetric bending mode can be suppressed, without degrading the ability of the vehicle to steer properly, by providing a direct shear connection between the bogies. Such inter-couplers are used in practice with the objective of reducing forces in curves, [20].

6.6 Application to Specific Configurations Analytical studies of body-steered bogie vehicles were initiated by Bell and Hedrick [21] and Gilmore [22] who identified various instabilities which were promoted by low conicities and reduced creep coefficients. A considerable body of work by Anderson and Smith and colleagues is reported in [23] to [28], covering the analysis of a vehicle with bogies having separately steered wheelsets. Weeks [29] has described dynamic modelling and track testing of vehicles with steered bogies, noting the enhanced sensitivity of this type of configuration to constructional misalignments. The theoretical considerations of the previous Section are illustrated by the scheme shown in Figure 6.9. Two additional stiffnesses ka and kb , representing linear springs between the wheelsets and the car body, are added to the basic scheme

THE BOGIE VEHICLE

191

h7

ka

h9

kc

h3

k

kb

h

h

c

ϕb

yb dc7

ϕb

ka

db7

yb

ks = [ ka kb kc ] as =

1 - h7 0 0 0 0 0 -1 dc7 -(c + h - h7) 0 0 0 0 0 0 0 0 0 0 0 0 1 h9 -1 dc9 -(c - h +h9) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 - h3 0 0 0 - 1 -h4 0 0

Figure 6.9 Simplified generic bogie with additional stiffnesses: top, plan view, bottom, end view looking forward at the leading bogie and outer wheelset. (h7 = - ch/(c + h); h9 = - ch/(c - h); h4 = 2h - h3).

discussed above. Then the matrices ks and as for the leading bogie are also shown in Figure 6.9, and there will be similar matrices for the trailing bogie. The first two rows relate to the stiffnesses ka and kb, and satisfy equations (4.5), (4.6) and (4.10) as do the corresponding rows for the trailing bogie. In accordance with the above discussion only four additional stiffnesses are required to decrease the degeneracy of the stiffness matrix to the maximum of four that is required. The addition of interwheelset shear stiffness kc such as might be provided by cross-bracing is covered by the third row in as which relates to kc and is also shown in Figure 6.9 for the leading bogie. There is a corresponding row for the trailing bogie and both rows satisfy equations (4.5), (4.6) and (4.10). This element could replace either ka and kb and the criteria for stability and perfect steering would still be satisfied. Thus a


192

f

c

b

d

a

g

Figure 6.10 Wiesinger bogie: a and d pivots connecting wheelsets to bogie frame g (ky1 and ky2 very large), b pivot connecting bogie frame to car body (kyb very large), c pivot connecting steering arm e to car body (ka very large), f linkage providing shear connection between the wheelsets (kc very large).

e4

e3 e1

c5 c1

e2 c6 c4 c3

c2

ks = [ ka kb kc ] 0 - 1 0 0 1 + h/c 0 0 0 0 - h/c 0 0 0 0 0 0 0 as =

0 0 0 0 1 - h/c 0 -1 0 0 h/c 0 0 0 0 0 0 0 1 - h3 0 0 0 -1 -h4 0 0 0

0

0 0 0 0 0 0 0

ka = e1e3c22(c5 - c1)2/{e1(c2 - c1)2 + e3(c5 - c1)2} kb = e2e4c32(c6 - c4)2/{e2(c4 - c3)2 + e4(c6 - c4)2} c2(c5 - c1)/c1(c5 - c2) = c/(c + h) c3(c6 - c4)/c4(c6 - c3) = c/(c - h) Figure 6.11 Linkage steered bogie with separately steered wheelsets.

THE BOGIE VEHICLE

193

practical choice for design is to steer the outer wheelset from the car body and steer the inner wheelset by means of a shear connection from the outer wheelset. This is exemplified by designs of steered bogie due to Robinson (1881) (cited by Smith [12]) and Wiesinger (1932) [11], Figure 6.10. The need for elastic elements in order to achieve a margin of stability and satisfactory response was not then recognised; so in these schemes ka and kc are very large, leading to the use of the pivots a, b, c and d. However, the analysis above suggests that it is possible to dispense with pivots and accommodate the necessary relative displacements by the use of suitably disposed elastic elements. A serious drawback of this configuration is that the direct coupling of the wheelset motions to the car body degrades stability and ride quality as shown in [30]. In the configuration shown in Figure 6.11 each wheelset is separately steered through a linkage connected to the car body, and is the basis of the steered bogie developed by UTDC. Such arrangements have been analysed by Smith and Anderson [23-27] and Shen [31]. It can be seen that the effective stiffnesses ka and kb are functions of the linkage stiffnesses e1, e2 and e3. Expressions for ka and kb are derived by considering the equilibrium of the system in terms of the generalised coordinates and the angular position of the link, and then eliminating the latter as inertia effects in the links are negligible. As in the previous example, alternative configurations can be derived by providing inter-wheelset shear stiffness kc. Then any one of ka, kb and kc, or the corresponding element stiffnesses, could be set to zero and the condition for stability satisfied. Alternative configurations could therefore have either the outer or the inner wheelsets steered together with an interwheelset shear connection. This could represent a useful simplification, which does not appear to have been studied in modern times, though it is the equivalent of the schemes by Robinson and Weissinger mentioned above. Careful choice of the dimensions of the linkages is necessary in order to keep within practical sizes. For this reason schemes have been developed in which the linkage is in the vertical plane. A more common variant of body steered bogie in which the wheelsets are jointly steered by one lever is shown in Figure 6.12. To represent this configuration 3 additional rows of the compatibility matrix are required for the leading bogie, corresponding to stiffnesses ka, kb and kc which are functions of the element stiffnesses e1, e2 and e3 as shown in Figure 6.12. The matrices ks and as for the leading bogie are also shown in Figure 6.12, and there will be similar matrices for the trailing bogie. Elements of this design go back as far as 1841, and there are many modern examples. The third row of a is equal to the sum of the first and second rows and thus only two of the three rows in the compatibility matrix for the linkage are independent. kc is equivalent to an inter-wheelset shear stiffness, and thus the arrangement not only provides the steering action defining the yaw angles of the wheelsets relative to the body, but also imposes inter-wheelset shear restraint. Figure 6.13 shows a scheme which received intensive development by Liechty [11]. A further example is provided by the configuration shown in Figure 6.14 in which the wheelsets are steered by a linkage pivoted on a member joining the wheelsets. This is clearly derived from the scheme of Figure 6.11 and provides a useful model for the discussion of stability below.


194

c1

ks = diag[ ka kb kc ] as =

0 - 1 0 0 1 + h/c 0 0 0 0 - h/c 0 0 0 0 0 0 0 0 0 0 0 1 - h/c 0 -1 0 0 h/c 0 0 0 0 0 0 0 0 -1 0 0 2 0 -1 0 0 0 0 0 0 0 0 0 0

ka = e1e3c22(c5 - c1)2/ ∆

kb = e2e3c12(c5 - c3)2/ ∆

kc = e1e2c22(c3 - c1)2/ ∆

∆ = {e1(c2 - c1)2 + e2(c3 - c1)2+ e3(c5 - c1)2} c2(c5 - c1)/c1(c5 - c2) = c/(c + h) c3(c5 - c1)/c1(c5 - c3) = c/(c - h) Figure 6.12 Steered bogie with shared wheelset linkage.

Figure 6.15 shows the dynamic response of a vehicle with the configuration of Figure 6.14 and parameters given in Table 6.1 to entry into a curve consisting of a linearly increasing curvature over a distance of 20 m and the constant curvature of 225 m. The track is assumed to be canted so cant deficiency is zero. Figure 6.15 should be compared with Figure 6.6 which gives the corresponding results for the conventional bogie vehicle. When the vehicle is completely on the uniform section of the curve, the wheelsets move outwards and adopt a radial position with negligible longitudinal creep and creep forces. The spin on the outer wheels is relatively large and results in corresponding lateral creep forces but the lateral forces on the inner wheels are small. These results are consistent with equations (3.2.8-9). The bogie frames and car body also adopt radial positions consistent with the wheelset positions so that in the steady state the suspensions are not strained. However, during the transition quite large longitudinal and lateral creep forces are exerted, as on the transition the suspension is strained and the resulting forces must be reacted by inertia and creep forces. These forces reduce to zero as the vehicle

THE BOGIE VEHICLE

195

e3 c1

e1 e2 h c2 ks = diag[ ka kb kc ]

⎡1 − h − 1 0 c 2 a = ⎢⎢0 0 − 1 0 c2 ⎢⎣1 − h 0 0 0 s

0 0 − c2 1 h 0 0 − c2 0 −1 − h 0 0 0

0

kb = e2e3c12/∆

ka = e1e3c12/∆

kc = e1e2c22/∆

0 0 0 0 0 0 0⎤ 0 0 0 0 0 0 0⎥⎥ 0 0 0 0 0 0 0⎥⎦ 2 2 2 ∆ = e1c2 + e2c2 + e3c1

Figure 6.13 Liechty’s bogie. c2 = h2/c.

e3 c5 c1

e1

e4 a

e2

kc

c6 c4 c3

c2

ks = diag[ ka kb kc ] ⎡(1 + h / c) / 2h − 1 0 0 0 − (1 + h / c) / 2h 0 0 0 − h / c 0 0 0 0 0 0 0⎤ a s = ⎢⎢(1 − h / c) / 2h 0 0 0 0 − (1 − h / c) / 2h − 1 0 0 h / c 0 0 0 0 0 0 0⎥⎥ ⎢⎣ 1 0 0 0 0 0 0 0 0⎥⎦ −h 0 0 0 −1 − h 0 −1

Figure 6.14 Bogie with wheelsets separately steered from an inter-wheelset member. Expressions for ka and kb and the relationship between the ci are the same as the scheme of Figure 6.11.

196


leaves the transition. This is consistent with the results given in Figure 6.15. A compensating steering arrangement that would account for transition geometry has been proposed by Smith [32].

Figure 6.15 Dynamic response of steered bogie vehicle of Figure 6.14 to curve entry. V = 15 m/s, R0 = 225 m, length of transition L = 20 m.

The concept of the body-steered bogie has been applied to freight vehicles with three piece bogies by Scales [33], List [34] and Shen [31] shows that dynamic stability as well as curving performance is significantly improved in comparison with a conventional design. So far it has been shown that a general formulation of the equations of motion of a symmetric railway vehicle with two unsymmetric bogies, capable of perfect steering, makes it possible to consider a wide variety of possible configurations on a common basis. Though there is a large number of disposable parameters theoretically available, only a limited number are needed to derive practical configurations which are simple and yet are capable of perfect steering. The configurations discussed satisfy the necessary conditions for dynamic stability at low speeds, and so the actual range of values of parameters necessary to achieve both static and dynamic stability must be discussed next.

THE BOGIE VEHICLE

197

6.7 Stability of Bogie Vehicles with Steered Wheelsets In this section the dynamic behaviour, as exemplified by the root locus as speed and suspension stiffness is varied, of the body-steered four-axle bogie vehicles is considered. The configuration of steered bogie vehicle chosen for discussion is that of Figure 6.14. Consider firstly the behaviour at low speeds, and how the eigenvalues of the system vary with suspension stiffness. Figure 6.16 shows the variation of the eigenvalues with k, a factor applied to ky, ka, kb and kc. When k = 0 there are four undamped oscillations at wheelset kinematic frequency ω =V(λ/lr0)1/2. Introduction of stiffness results in damping of these oscillations, the real part of the eigenvalue being initially proportional to k. Two of these oscillations can be identified with the steering or bending oscillation A and D of the leading and the trailing bogies respectively. The other two oscillations can be identified with the shear oscillation B and C of the leading and the trailing bogies respectively. In both cases the steering oscillation is lightly damped. The shear oscillations are heavily damped, splitting into four subsidences B1, B2, C1 and C2 as k increases. It can be seen that if these results are compared with those for the isolated bogie discussed in Section 5.8, the eigenvalues corresponding to the steering and shear oscillations are significantly changed in character. In fact, both oscillatory instability and divergence occur for certain values of k, and the conditions for this are discussed below in detail. Not shown in Figure 6.16 are eigenvalues which are largely independent of k. These eigenvalues, of which there are 9 complex conjugate pairs, refer to modes

Figure 6.16 Variation of the eigenvalues for the body-steered vehicle at low speeds as a function of a factor on the stiffnesses ka and kc (eigenvalues of large modulus not shown).

198


Figure 6.17 Root locus as speed is varied for body-steered bogie vehicle of Figure 6.11. The left hand plot shows the root locus in the complex plane and the right hand plot is the variation of the imaginary parts with speed. Parameters of Table 6.1 with ka = 40 MNm, kb = 40 MNm, kc = 0, ky = 0.

which are substantially oscillations of the vehicle body and the bogie frames on the primary and secondary suspension as if the wheels were fixed. The influence of speed on the eigenvalues for the steered bogie vehicle is shown in Figure 6.17, for values of ky, ka, kb and kc which give stability at low speeds. Comparison with Figure 6.3 shows that the results are very similar to those for a conventional bogie vehicle, with the notable exception that the branches A1 and A2, initially proportional to speed, corresponding to the bogie steering modes, differ. Inspection of the eigenvectors shows that one branch refers to motions mainly involving the leading bogie and the other branch refers to the trailing bogie. The eigenvalues associated with the vehicle body and bogie frame modes, substantially independent of speed, are broadly similar to those of the conventional bogie vehicle, as are the 8 relatively large real negative eigenvalues corresponding to wheelset subsidences in lateral translation and yaw. As in the case of the conventional bogie vehicle, at speeds for which the steering frequencies approach the natural frequencies of the vehicle body on the suspension, there is the possibility of body instability. Such body instability can be eliminated by a suitable choice of secondary suspension parameters as discussed for the conventional bogie vehicle. As speed is increased, the combined effect of speed and inertia modifies the damping in the steering oscillations. For each steering oscillation which is stable for low speeds, the damping will vanish at a bifurcation speed VB. The instability corresponds to a form of bogie hunting, and unlike the conventional bogie vehicle where the two bifurcation speeds are approximately the same, as Figure 6.17 shows those for the body steered vehicle can be quite different.

THE BOGIE VEHICLE

199

6.8 Simple Bogie Model Examination of the reduced order stiffness matrix E* defined in equations (4.19) and (4.20) shows that, for practical vehicles, E* is of the form E*= E*0 + ∆E where ∆E is small compared with E0*, and

e11

e12

-e11

e14

0

0

0

0

e12

e22

-e12

e24

0

0

0

0

-e11

-e12

e11

-e14

0

0

0

0

e14

e24

-e14

e44

0

0

0

0

0

0

0

0

e11

e14

-e11

e12

0

0

0

0

e14

e44

-e14

e24

0

0

0

0

-e11

-e14

e11

-e12

0

0

0

0

e12

e24

-e12

e22

E0*=

where

(1)

e11 = kaρ2/4h2 + kbσ2/4h2 + kc e12 = - kaρ/2h - kch e14 = - kbσ/2h - kch e22 = ka + kch2 e24 = kch2 e44 = kb + kch2

where ρ = 1 + h/c, σ = 1 - h/c. E0* is the stiffness matrix derived by assuming that the car body is restrained from deviating from a uniform forward motion. As in the case of the conventional bogie vehicle, for the purposes of trend studies a good approximation to the eigenvalues associated with bogie motions can be obtained by treating the vehicle body as an inertial mass. Accordingly, referring to the equations of motion put yc, φc, ψc = 0. Furthermore, if it assumed that ky is large and remembering that kψ = 0, the bogie frame co-ordinates are given by yb = ( y1 + y2)/2 and ψb = ( y1 - y2)/2h. If d1 = 0 then the bogie frame roll coordinate is laterally uncoupled from the rest of the vehicle. The system is thus reduced to two uncoupled sets of four degrees of freedom each, involving y1, ψ1, y2, ψ2


200

and y3, ψ3, y4, ψ4. Finally the secondary lateral suspension stiffness kyb has only a small influence on the stability of the bogies and will be neglected. Introducing sum and difference coordinates ϕi defined by equation (4.2.3)

and neglecting all suspension damping terms, the equations of motion will be {(m/2 + mb/4)s2 + fs/V + e11}φ1 - f φ2 ± e12 φ2+ e14φ4 = 0 (fλl/r ± e12) φ1 + {(I/2)s2 + fl2s/V + e22}φ2 ± e24φ4 = 0

(3) (4)

{(m/2 + mb/4)s2 + fs/V }φ3 - f φ4 = 0

(5)

e14φ1 ± e24φ2 + (fλl/r)φ3 + {( I/2)s2 + fl2s/V + e44}φ4 = 0

(6)

where now e11 = kaρ2/4h2 + kbσ2/4h2 + kc e12 = - kaρ/4h + kbσ/4h e14 = - kaρ/4h - kbσ/4h - kch e22 = ka /4 + kb /4 e24 = ka /4 - kb /4 e44 = ka /4 + kb /4 + kch2 The upper sign refers to the leading bogie and the lower sign refers to the trailing bogie. A particularly simple case arises when the bogie frame mass mb = 0, corresponding to the configuration shown in Figure 6.14 with a light inter-wheelset member, for then if it is assumed that I = ml2 it is possible to write

D = ms2 /2f + s/V

(7)

then a trial solution q = aeD t leads to the characteristic equation p4D4 + p3D3 + p2D2 + p1D + p0 = 0 where p4 = f 4l 4

(8)

THE BOGIE VEHICLE

201

p3 = f 3l 2{ ka (1/2 +ρ2l2/4h2) + kb(1/2 +σ2l2/4h2) + kc( l2 + h2 )} p2 = 2f 4l 2α + f 2kakc{l2 + h2 + l2(1-ρ)2 }/4 + f 2kakb{1+ l2/h 2 + l2( ρ - σ )2/4h2}/4 + f 2kbkc{l2 + h2 + l2(1-σ)2 }/4 ± f 3l2(kbσ - kaρ )(1 - α )/4h p1 = f 3α{ ka(1/2 + ρ2 l2/4h2) + kb(1/2 + σ2 l2/4h2) + kc(l2 + h2)} ± f 2(1 - α){-kakc(ρ -1)h + kbkc(σ - 1)h + kakb( σ - ρ )/2h}/4 p0 = f 4α2 + f 2α (kakb/h2 + kakc + kbkc)/4 ± f3α(kbσ - kaρ)(1 - α )/4h and α = λl/r0. 6.9 Stability of Simple Bogie Model First consider stability at low speeds so that D becomes equal to s/V. In this case the results are applicable to the configurations shown in Figures 6.11 and 6.14. Routh’s criterion for oscillatory stability, equation (3.4.3), will be applied. It can be shown that if any two of the stiffnesses ki are zero, T3 = 0 and an undamped oscillation at wheelset kinematic frequency occurs in accordance with the discussion of Section 4. Consequently, it is of interest to discuss the three cases, each for the leading and trailing bogie, firstly where the outer wheelset is steered and there is shear stiffness between the wheelsets, secondly where the inner wheelsets are steered and there is shear stiffness between the wheelsets and thirdly where both outer and inner wheelsets are steered. Firstly consider the possibility of oscillatory instability at very low speeds. This can occur only for very large values of the stiffnesses ki and then the sign of T3 is determined by the sign of p1 so that on substitution for r and s the condition for oscillatory stability is ± (1 - α){kakc + kbkc + kakb/h2} < 0

(1)

where the upper sign refers to the leading bogie and the lower to the trailing bogie. Thus the stability depends on ± (1-α ).For the leading bogie, oscillatory instability occurs if α 1. For the trailing bogie, oscillatory instability occurs if α >1, but not if α 0 or 4f2α + kakb/h2 + kakc + kbkc ± f(kbσ - kaρ)(1 - α )/h > 0

(10)

Assuming the usual case where α 4fhα /ρ ( 1 - α ) and for ka very large if kc < fρ ( 1 - α )/h. if the inner wheelset is steered the leading bogie is statically stable for all • values of kb and kc and the trailing bogie is statically unstable for sufficiently large values of kb and sufficiently small values of kc. Instability occurs for kc = 0 if kb > 4fhα /σ ( 1 - α ) and for kb very large if kc < fσ ( 1 - α )/h. • if both wheelsets are steered, and kc = 0, the leading bogie is statically unstable for sufficiently large values of ka and for sufficiently small values of kb. Instability will occur for kb = 0 if ka > 4fhα /ρ ( 1 - α ) and for ka very large if kb < fρh( 1 - α ). The trailing bogie is statically unstable for sufficiently large values of kb and for sufficiently small values of ka. Instability occurs for ka = 0 if kb > 4fhα /σ ( 1 - α ) and for kb very large if ka < fσ h( 1 α ). In all cases low values of conicity promotes divergence and increasing conicity in-

204


creases static stability. The divergence boundaries shown in Figure 6.19 occur for values of the stiffnesses closely in accordance with the simple criteria discussed above. Note that these static instabilities all occur if ρ =1. They are the result of

Figure 6.20 Stability boundaries showing contours of the bifurcation speed VB for body steered bogies with wheelsets separately steered from an interwheelset member (Figure 6.14) with car body fixed: outer wheelset steered through stiffness ka with inter-wheelset shear connection kc, trailing bogie (a) and leading bogie (b); inner wheelset steered kb with inter-wheelset shear connection kc, trailing bogie (c) and leading bogie (d); both inner and outer wheelsets steered with no inter-wheelset shear connection, trailing bogie (e) and leading bogie (f). Coned wheels, λ = 0.05. D refers to divergence, O refers to low speed oscillatory instability.

THE BOGIE VEHICLE

205

assymmetry rather than steering as such. For if ρ =1 then the bogie behaves as an unsymmetric vehicle. The stability of unsymmetric vehicles is discussed in Chapter 9 but the salient feature of the static instability is revealed by considering a wheelset mounted on a pivoted arm as discussed in Chapter 3. Inspection of equation (10) shows that static instability can be eliminated completely if the linkage stiffnesses ka and kb are chosen to be different such that kb/ka = ρ/σ. The form of these results is very similar to those obtained by Bell and Hedrick [21] for a bogie in which the relative angular displacement between the wheelsets was proportional to the relative motion between the bogie frame and car body, a configuration similar in some respects to that of Figure 6.10. The results given here may also be compared with those given by Smith and Anderson [25] for a bogie in which there is a fixed relationship between ka and kb. Figure 6.19 also shows contours of the bifurcation speed VB. The instability corresponds to a form of bogie hunting which is an inertia driven instability of the steering oscillation. It is similar to the instability of a conventional bogie, but as the linkage restraining the bogie can be made stiff without degrading curving performance, quite high critical speeds can be achieved when parameters are chosen appropriately. Figure 6.20 indicates the stability as a function of the stiffnesses ki for the leading and trailing bogies for each of the cases where one of the three ki are zero. For a complete vehicle values of the ki must be chosen to give a margin of stability for both leading and trailing bogies. Moreover, as a range of conicities and creep coefficients must be considered in practice, values of the stiffnesses ki must be chosen accordingly. Figure 6.21 shows an example of the effect of varying equivalent conicity and creep coefficient on stability. In this case, the usual bogie instability is promoted by large conicities and the instability of the leading bogie discussed above is promoted by low conicities and creep coefficients. The result is a restricted range of speeds for stable operation unless the range of conicities is restricted.

Figure 6.21 Stability of the body-steered vehicle of Figure 6.11 showing the effect of varying conicity and creep coefficient on the bifurcation speed VB (m/s) as ka (107 MNm) and kb (107 MNm) are varied. f/2 indicates that f11, f22 and f23 are halved. ky = 6 MN/m.

206


In general, detailed optimisation of the parameters is necessary considering not only the linkage stiffnesses but the lateral primary suspension stiffness. Whilst it has been shown that arrangements providing only two of the stiffnesses ka, kb and kc are sufficient it is possible that schemes in which all three stiffnesses are implemented could provide better performance. More generally, the approach followed in this Chapter makes it possible to consider many possible configurations on a common basis. Though there are many disposable parameters theoretically available, only a limited number are needed to derive practical configurations which are relatively simple and which improve the resolution of the conflict between steering and stability. Full assessment of a given design must depend on design detail and performance in sharp curves, a situation in which these schemes are likely to be applied, obviously requires a nonlinear analysis. Furthermore, it has been shown that the introduction of more refined suspension arrangements results in more forms of potential static and dynamic stability.

References 1. Lewis, M.J.T.:Early Wooden Railways. Routledge and Kegan Paul, London, 1974, p. 291. 2. Stover, J.F.: American Railroads. University of Chicago Press, Chicago, 1961, p. 25. 3. White, J.H.: A History of the American Locomotive. The John Hopkins Press, Baltimore, 1968, pp. 169-174. 4. Ahrons, E.L.: The British Steam Railway Locomotive 1825-1925. The Locomotive Publishing Co., London, 1927, p. 62. 5. White, J.H.: The American Railroad Passenger Car. The John Hopkins Press, Baltimore, 1978, p. 497. 6. Ahrons, E.L. p. 157. 7. Matsudaira, T.: Hunting problem of high-speed railway vehicles with special reference to bogie design for the New Tokaido Line. Inst. Mech. Eng. Proc. 180, Part 3F (1965), pp. 58-66. 8. Sauvage, G.: Running quality at high speeds. In: O. Nordstrom (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 9th IAVSD Symposium, Linkoping, Sweden, June 1985. Swets and Zeitlinger Publishers, Lisse, 1986, pp. 496-508. 9. Evans, J.R.: The modelling of railway passenger vehicles. In: G. Sauvage (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 12th IAVSD Symposium, Linkoping, Sweden, August 1991. Swets and Zeitlinger Publishers, Lisse, 1992, pp. 144-156.

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10. Eickhoff, B.M., Evans, J.R. and Minnis, A.J.: A review of modelling methods for railway vehicle suspension components. Vehicle System Dynamics 24 (1995), pp. 469496. 11. Liechty, R.: Das Bogenlaufige Eisenbahn-Fahrzeug. Schulthess, Zurich, 1934. 12. Smith, R.E.: Steering rail vehicle axles − a historical review. Proc. CSME Canadian Engineering Centennial Convention, 1987. 13. Liechty, R.: Studie uber die Spurfuhrung von Eisenbahnfahrzeugen. Schweizer Archiv f. Angewandte Wissenschaft und Technik, 3 (1937), pp. 81-100. 14. Liechty, R.: Die Bewegungen der Eisenbahnfahrzeuge auf den schienen und die dabei auftretenden Kräfte. Elektrische Bahnen, 16 (1940), pp. 17-27. 15. Schwanck, U.: Wheelset steering for bogies of railway vehicles. Rail Engineering International 4 (1974), pp. 352-359. 16. Wickens, A.H.: Steering and dynamic stability of railway vehicles. Vehicle System Dynamics, 5 (1975), pp. 15-46. 17. Wickens, A.H.: Stability criteria for articulated railway vehicles possessing perfect steering. Vehicle System Dynamics, 7 (1979), pp. 33-48. 18. de Pater, A.D.: Optimal design of a railway vehicle with regard to cant deficiency forces and stability behaviour. Delft University of Technology Lab. for Eng. Mech. Report 751, 1984. 19. de Pater, A.D.: Optimal design of railway vehicles. Ingenieur-Archiv 57 (1987), pp. 25-38. 20. Topham, W.L.: Methods of reducing flange wear on diesel and electric locomotives. J. Inst. Loco. Engineers 49 (1959), pp. 771-825. 21. Bell, C.E. and Hedrick, J.K.: Forced steering of rail vehicles: stability and curving mechanics. Vehicle System Dynamics 10 (1981), pp. 357-385. 22. Gilmore, D.C.: The application of linear modelling to the development of a light steerable transit truck. In: A.H. Wickens (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 7th IAVSD Symposium, Cambridge, September 1981. Swets and Zeitlinger Publishers, Lisse, 1982, pp. 371-384. 23. Fortin, J.A. and Anderson, R.J.: Steady-state and dynamic predictions of the curving performance of forced-steering rail vehicles. In: J.K. Hedrick (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 8th IAVSD Symposium, Cambridge, Mass., August 1983. Swets and Zeitlinger Publishers, Lisse, 1984, pp. 179-192.

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24. Fortin, J.A.C., Anderson, R.J. and Gilmore, D.C.: Validation of a computer simulation forced-steering rail vehicles. In: O. Nordstrom (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 9th IAVSD Symposium, Linkoping, June 1985. Swets and Zeitlinger Publishers, Lisse, 1986, pp. 100-111. 25. Smith, R.E and Anderson, R.J.: Characteristics of guided-steering railway trucks. Vehicle System Dynamics, 17 (1988), pp. 1-36. 26. Anderson, R.J., Fortin, C.: Low conicity instabilities in forced-steering railway vehicles. In: M. Apetaur (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 10th IAVSD Symposium, Prague, August 1987. Swets and Zeitlinger Publishers, Lisse, 1988, pp. 17-28. 27. Smith, R.E.: Forced-steered truck and vehicle dynamic modes-resonance effects due to car geometry. In: A. Apetaur (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 10th IAVSD Symposium, Prague, August 1987. Swets and Zeitlinger Publishers, Lisse, 1988, pp. 423-424. 28. Smith, R.E.: Dynamic characteristics of steered railway vehicles and implications for design. Vehicle System Dynamics, 18 (1989), pp. 45-69. 29. Weeks, R.: The design and testing of a bogie with a mechanical steering linkage. In: M. Apetaur (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 10th IAVSD Symposium, Prague, August 1987. Swets and Zeitlinger Publishers, Lisse, 1988, pp. 497508. 30. Li, W.: The dynamics of perfect steering bogie vehicles and its improvement with a reconfigurable mechanism. Doctoral Dissertation, Loughborough University, 1995. 31. Shen, Z.Y., Yan, J.M., Zen, J., and Liu, J.X.: Dynamical behaviour of a forced-steering three-piece freight car truck. In: M. Apetaur (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 10th IAVSD Symposium, Prague, August 1987. Swets and Zeitlinger Publishers, Lisse, 1988, pp. 407-418. 32. Smith, R.E.: Multiaxle steered articulated railway vehicle with compensation for transitional spirals. U. S. Patent Application 157565, 1989. 33. Scales, B.T.: Behaviour of bogies on curves, Railway Engineering, December, 1972. 34. List, H.A.: Means for improving the steering behaviour of railway vehicles. Annual Meeting of the Transportation Research Board, Washington, D. C. January, 1976.

7 The Three-Axle Vehicle 7.1 Introduction Various forms of three-axle vehicle have been used widely in the past. In most of these designs the wheelsets were connected to the car body by a conventional suspension similar to that used in two-axle vehicles. Negotiation of curved track was catered for by allowing greater flexibility or clearances for the central wheelset. However, there is also a long history of inventions which attempt to ensure that wheelsets are steered so that they adopt a more or less radial position on curves. It was argued that three axles, connected by suitable linkages, would assume a radial position on curves and then re-align themselves correctly on straight track. As will be shown below, a wide range of new potential instabilities are introduced. In fact, the three-axle configuration is also important because it gives considerable insight into the dynamic behaviour of articulated vehicles discussed in Chapter 8. Three-axle vehicles were in use from an early date. According to Liechty [1] a three-axle vehicle in which the lateral displacement of the central axle steered the outer axles through a linkage was tried out on the Linz-Budweis railway in 1826. Germain patented a design in 1837 in which radial steering was provided [2], and in 1844 Themor built a similar vehicle which was operated for some time [1]. Fidler also patented a similar arrangement in 1868 [3], Figure 7.1(a). In these last three schemes the outer wheelsets were pivoted to the car body. In 1889, Robinson’s arrangement [4] introduced the refinements of guides for the central wheelset, the body pivots was placed slightly inboard of the outer wheelsets, and the central wheelset had a much smaller radius than the outer wheelsets. Faye’s 1898 patent [5] removed the guides for the central wheelset in order to avoid reported difficulties with Robinson’s design on reverse curves. There were, of course, many different ways of providing inter-wheelset steering, such as complex linkages, and this is exemplified by the variety of designs produced since. Fidler introduced direct shear connection between the outer wheelsets in 1868, [3]. The central wheelset was mounted without lateral freedom in the car body, Figure 7.1(b). A similar arrangement was invented by Grover in 1880 [6]. All these developments were based on very simple ideas about the mechanics of vehicles in curves. In this Chapter, the basic instabilities of three-axle vehicles with a single car body will be considered, and how they are related to the natural steering properties of the three-axle vehicle. Particular emphasis will be given to the various

210


Figure 7.1 Three-axle vehicles with steered wheelsets due to Fidler (a) outer wheelsets pivoted on car body (b) central wheelset pivoted on car body, direct shear connection between outer wheelsets.

possibilities for the connections between the wheelsets and the car body in order to meet the conflicting requirements of stability and curving. The three-axle vehicle was first examined in this context in a series of papers [7]-[14]. A similar approach, though with slightly different assumptions, has been followed by de Pater [15, 16] and Keizer [17].

7.2 Steering and Stability of Three Axle Vehicles Consider the idealised three-axle vehicle shown in Figure 7.2. As the rotation of the wheels cannot generate any forces in the suspension consider the set of nine generalised coordinates q = [ y1 ψ1 y2 ψ 2 yb φb ψb y3 ψ 3 ]T

(1)

as indicated in Figure 7.2. The conditions governing steering and stability of multi-axle vehicles have been discussed in Section 6.4 and it can be seen that the criterion expressed by equation (6.4.15) can be satisfied by a three-axle vehicle. As in the case of the bogie vehicle, the total number of disposable parameters is very large. Fully generic schemes could

THE THREE-AXLE VEHICLE

211

(a)

ψ3

y3

ψb

yb

ψ2

y2

ψ1

y1

2h*

(b)

ϕb

yb

Figure 7.2 Generalised coordinates for three-axle vehicle.

be the starting point for an analysis of the three-axle vehicle. However, it is more important to derive the simplest arrangements which are capable of both perfect steering and stability. The main thrust will be that the vehicle should be made as flexible as possible in plan view, consistent with the need for stability. Noting that fully generic arrangements may be useful in generating design variants, the further development of the argument is based on a simpler form of arrangement shown in Figure 7.3. Firstly, consider the case where all wheelsets are similar and each wheelset is connected to the car body in the conventional manner. Then, the element stiffness and compatibility matrices are k = diag [ ky1 kφ1 kψ1 ky2 kφ2 kψ2 ky1 ⎡1 ⎢ ⎢0 ⎢0 ⎢ ⎢0 a = ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢⎣0

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 − 1 d1 0 0 1 0 0 0 0 −1 d4 0 0 1 1 0 0 0 − 1 d1 0 0 1 0 0 0

− 2h * 0 −1 0 0 −1 2h * 0 1

kφ1 kψ1 ]

(2)

0 0⎤ ⎥ 0 0⎥ 0 0⎥ ⎥ 0 0⎥ 0 0⎥ ⎥ 0 0⎥ ⎥ 1 0⎥ 0 0⎥ ⎥ 0 − 1⎥⎦

(3)

212


2h* ky1/2

kψ2/2

ky2/2

kψ 1

ky1/2 k ψ1

kyd kt kyd

ky1/2

ky2/2

kψ2/2

ky1/2

h1 Figure 7.3 Schematic showing suspension stiffnesses for three-axle vehicle.

The solution of equations (6.4.10) will initially be discussed for this simplified generic arrangement. Equations (6.4.10) become ky1(y1 - yb + d1ϕb - 2h*ψb ) = 0

(4)

kϕ1 ϕb = 0

(5)

kψ1( ψ1 - ψb ) = 0

(6)

ky2(- yb + d4ϕc + y2 ) = 0

(7)

kϕ2 ϕb = 0

(8)

kψ2( ψb - ψ2 ) = 0

(9)

ky1( y3 - yb + d1ϕb + 2h*ψb ) = 0

(10)

kϕ1 ϕb = 0

(11)

kψ1(ψ3 - ψb) = 0

(12)

From equations (5), (8) and (11) ϕb = 0. From equations (6), (9), (12), either ψ1 = ψb or kψ1 = 0

(13)

either ψ2 = ψb or kψ2 = 0

(14)

either ψ3 = ψb or kψ1 = 0

(15)


213

Figure 7.4 Geometry of two alternative configurations on uniform curve: top, ky1 = 0 and yb = y2; bottom, ky2 = 0 and yb = y1.

For steering without lateral or longitudinal creep on a uniform curve, within the assumptions of Section 6.4, and relative to the unstrained vehicle y1 = y3 = - lr0/λR0 + 2h*2/R0

(16)

y2 = - lr0/λR0

(17)

ψ1 = -ψ3 = 2h*/R0

(18)

ψ2 =

(19)

so that from (19) and (14) ψb = 0 and the value of kψ2 can be arbitrarily chosen. From (4), (7) and (10), either ky2 = 0 and yb = y1 = y3 or ky1 = 0 and yb = y2. Figure 7.4 indicates the geometry for each of these configurations in which the wheelsets adopt a radial position with radial displacements appropriate to their conicity, and the car body takes up the positions connecting the wheelsets without strain of the suspension elements. These two configurations satisfy the conditions for steering so it is now necessary to consider stability. For the first configuration, in which ky2 = 0 and ky1, kφ1, kφ2, and kψ2 are non-zero, and the degeneracy of E is 5, and the stability criterion, equation (6.4.15), is not satisfied and there are two pairs of imaginary eigenvalues at low speeds corresponding to undamped kinematic wheelset oscillations. At least two additional stiffnesses are required to reduce the degeneracy to three as required by the stability criterion. For the second configuration, in which ky1 = 0 and ky2, kφ1, kφ2, and kψ2 are nonzero, the degeneracy of E is 6, and again the stability criterion, equation (6.4.15), is not satisfied and there are three pairs of imaginary eigenvalues at low speeds corresponding to three undamped kinematic wheelset oscillations. In this case at least

214


three additional stiffnesses are required to reduce the degeneracy to three as required by the stability criterion. As in the case of the bogie vehicle, for these additional stiffnesses, the corresponding rows in the compatibility matrix a must satisfy the rigid body conditions expressed by equations (6.4.5) and (6.4.6), and the steering condition expressed by equation (6.4.10). In addition they must be linearly independent from the existing rows of a. (Of course, the ki can be repeated as long as the corresponding row in a is linearly independent, thus increasing the rank of a ). The sub-matrices of k and a relating to the additional elastic elements will be denoted by ks and as for convenience. Figure 7.3 shows that further elastic connections corresponding to direct shear connections between adjacent wheelsets (of stiffness kyd) and the outer wheelsets (of stiffness kt) have been introduced. As in the case of the body-steered bogie, in schemes of the early period, pivots were used to provide lateral restraint and yaw freedom. For example, in Robinson’s scheme ky1 and kψ2 were very large though the principles discussed above remain valid. As mentioned above, such a vehicle would be unstable at low speeds. The sub-matrices of ks and as relating to these additional elastic elements are (20) ks = diag [ kyd kyd kt ] ⎡1 − h1 a = ⎢⎢0 0 ⎢⎣1 − 2h * s

− 1 − h2 1 − h2 0 0

0 0 0 0 0 ⎤ 0 0 0 − 1 − h1 ⎥⎥ 0 0 0 − 1 − 2h * ⎥⎦

(21)

where h2 = 2h* - h1. The condition for stability at low speeds has been shown, in Section 6.4, to depend on the stiffness matrix E* involving only the wheelset coordinates. In the present case the number of disposable parameters will be (R2 - 4R + 3)/4 = 4 and so E* may be expressed in terms of the given elements e13*, e15*, e16*, and e22* and then e11*= - e13*- e15* e12*= e13*h*+ e16* e14*= 4e15*h* - 2e16* + e13*h* e23*= - e13*h* e24*= - 2e22* - 3e13*h*2 - 4e16*h* e25*= - e16* e26*= e22* + e13*h*2 e33*= - 2e13*


215

e34*= 0 e35*= e13*

(22)

e36*= e13*h* e44*= 4e22* + 16e16*h* - 16e15*h*2 + 2e13*h*2 e45*= - e14* e46*= e24* e55*= e11* e56*= - e12* e66*= e22* For the enhanced configuration, E* has the following four independent elements e13* = - kyd e15* = - kt - ky1kψ2/2(8ky1h*2 + kψ2)

(23)

e16* = - 2kth* e22* = h12kyd + 4kth*2 From equation (6.4.15), for stability, the degeneracy of E* must be not more than three and this is satisfied if either (a) ky1 = 0 and kyd, kt, ky2 and kψ2 are non-zero, (b) ky2 = kt = 0 and ky1, kψ2 and kt are non-zero, or (c) ky2 = kψ2 = 0 and ky1, kyd and kt are non-zero.

7.3 Steering with Unequal Conicities The above discussion of curving requires modification to account for unequal conicities of the wheelsets. There is a geometric requirement that a bending displacement enabling all three wheelsets to take up radial alignment is compatible with the lateral displacements required, by their various equivalent conicities, for pure rolling. Figure 7.5 shows the geometry when such a vehicle negotiates a curve. For zero longitudinal and lateral creep, the wheelsets must adopt a radial position, and, assuming for simplicity purely coned wheels, move outwards lr0/λR0 (for the outer wheelsets) and lr1/λ1R0 (for the central wheelset). If the radial displacements are small compared with the radius of the curve, the geometry of Figure 7.5 then yields the position of an effective pivot, measured from the central points between the

216


lr1/R0λ1

h1 lr0/R0λ

Figure 7.5 Geometry of three-axle vehicle for zero lateral and longitudinal creep on uniform curve when conicities are not uniform.

wheelsets, necessary for perfect steering h*- h1= l( r0/λ - r1/λ1 )/2h*

(1)

The possible range of configurations which satisfy this equation is constrained by two practical considerations. Firstly, large values of conicity are not achievable in practice. Secondly, assuming that the difference in conicity between outer and central wheelsets is achieved by a difference in wheel diameters, it is unlikely that in a good mechanical design the ratio of diameters could exceed about two, or exceptionally three. Figure 7.6 then summarises the possible range of configurations which steer perfectly on curves for the case where h* = 1.25 m. Lines A and B give the relationship between λ1 and λ for the virtual pivot position being at the central and outer wheelsets respectively. For longer wheelbases the position of the effective pivot lies nearer the centre of the wheelbase. The lines C and D represent the limits of a 2:1 ratio in conicity between central and outer wheelsets. A smaller ratio in conicity leads to a pivot position between the wheelsets as shown at E, and large ratios in conicity could require pivot positions outside the wheelbase.

0.3

A

0.2

λ1l/r0

E C

0.1

B D

0

0

0.1

λ1l/r0

0.2

0.3

Figure 7.6 Relationship between conicities and virtual pivot position for negotiation of uniform curve with zero longitudinal and lateral creep. h* = 1.25 m.


217

7.4 Stability of Vehicle with Uniform Conicity In order to discuss stability attention will be concentrated on option (c) of Section 2 in which ky2 = kψ2 = 0 and ky1 and the inter-wheelset stiffnesses kyd and kt are nonzero. Moreover in this Section, the case where h1 = h2 = h* and λ1 = λ will be considered, deferring discussion of non-uniform conicity to the following Section. It is convenient to consider, firstly, the behaviour at low speeds. As already discussed, at low speeds, the car body degrees of freedom can be eliminated and the equations of motion will be of the form of (6.4.20). Neglecting the small terms associated with the contact stiffness, and assuming for simplicity that all the creep coefficients are equal, the non-zero terms of B and C are B11 = B33 = B55 = 2f C12 = C34 = C56 = -2f

B22 = B44 = B66 = 2fl2

C21 = C65 = 2flλ/r0

C43 = 2flλ1/r1

and the characteristic equation becomes p6D6 + p5D5 + p4D4 + p3D3 + p2D2 + p1D + p0 = 0

(1)

where D = d/Vdt, α = λl/r0 and α1 = λ1l/r0 and p6 = 1 p5 = (l2 + 4h*2)kt/fl2 + (2l2 + h12 + h22)kyd/fl2 p4 = (2α + α1 )/ l2 + {2(l2 + 4h*2)(2l2 + h12 + h22) - (l2 + 2h*h1)2}kydkt/2f 2l 4 + {(2l2 + h12 + h22)2 - (l2 - h22)2}kyd2/4f 2l 4 p3 = (l2 + 4h*2)(α + α1)kt/f l4 + (α1l2 + 3αl2 + h12α1 + h12α + 2h22α)kyd/f l4 + (3l2 + 8h*2)(3l2 + h12)kyd2 kth22/4f 3l 6 p2 = α(α + 2α1)/l4 + {α1 ( l2 + h12)2 + 8h22 l2α + 2α( l2 + h12)( l2 + h22) - h12 l2(α - 1)2 - 2h1h2 l2(α - 1)(1 - α1)}kyd2/4f 2l 6 + [(2h* - αh1)(2h*α - h1)l2 + 2(l2 + 4h*2){α1(l2 + h12 ) + α(l2 + h22)} - α1(l2 + 2h*h1)2]kydkt/2f 2l 6 p1 = α(α l2 + α1 l2 + h12α1 + h22α)kyd/f l6 + αα1(l2 + 4h*2)kt/f l 6 + {24h*2α + 4h*h1α(α1 - α) + (α1 + 2α)(3l2 + h12)}kyd2kth22/4f 3l 6 p0 = α2α1/ l6 + {4α2h22 - α1h12(α - 1)2 - 2h1h2α(α - 1)(1 - α1)}kyd2/4f 2l 6 + α1(2h* - h1α)(2h*α - h1)kydkt/2f 2l 6

218


Figure 7.7 Root locus for three-axle vehicle at low speeds as a factor k on the stiffnesses kt and kyd is varied, for the parameters of Table 7.1.

Figure 7.7 shows the variation of the eigenvalues with k, a factor applied to kyd and kt. When k = 0 there are three undamped oscillations A, B and C at wheelset kinematic frequency ω = V(λ/lr0)1/2 as discussed above. Introduction of stiffness results in damping of these oscillations, the real part of the eigenvalue being initially proportional to k. One of these oscillations can be identified with a steering or bending oscillation A of the complete vehicle, in which there is very little interwheelset shear. As k is varied from zero to very large values the steering oscillation remains lightly damped. As k increases, the frequency of the steering oscillation decreases, tending towards a constant value for large values of k. The other two oscillations B and C involve much more shear of the inter-wheelset structure and are heavily damped, splitting into four subsidences B1, B2, C1 and C2 as k increases. The damping of one of these subsidences, C2, eventually decreases as k is increased so that at large values of k it vanishes. An important feature of these results is the small damping in the steering oscill-


219

Table 7.1 Example parameters for three-axle vehicle. suspension ky1 = 0.23 MN/m cy1 = 100 kNs/m kyd = 0.70 MN/m creep coefficients f11 = 8.83 MN vehicle geometry h* = 4.125 m inertia m =1250 kg Izb = 269000 kgm2

kψ1 = kψ2 = 0 cψ1 = cψ2 = 0 kt = 0.20 MN/m

kφ1 = kφ2 = 1MNm cφ1 = cφ2 = 50 kNms ct = cyd = 0

f22 = 8.06 MN

f23 =17.7 kNm

W = 100.6 kN

d1 = d4 = 0.2 m

r0 = 0.45 m

l = 0.7452 m

I = 700 kgm2 Iy = 250 kgm2

mc=27000 kg

Ixc=32400 kgm2

ky2 = 0 dy2 = 0

ation A and in the subsidence C2 at large values of k. In fact, both oscillatory instabilities and divergence occur for certain values of the parameters, and the conditions for stability will now be discussed in detail. If k is very large, it is appropriate to neglect terms in k of order less than two in (1) which then reduces to p4D4 + p3D3 + p2D2 + p1D + p0 = 0

(2)

where p4 = {2(l2 + 4h*2)(2l2 + h12 + h22) - (l2 + 2h*h1)2}kydkt/2f 2l 4 + {(2l2 + h12 + h22)2 - (l2 - h22)2}kyd2/4f 2l 4 p3 = (3l2 + 8h*2)(3l2 + h12)kyd2 kth22/4f 3l 6 p2 = {α1 ( l2 + h12)2 + 8h22 l2α + 2α( l2 + h12)( l2 + h22) - h12 l2(α - 1)2 - 2h1h2 l2(α - 1)(1 - α1)}kyd2/4f 2l 6 + [(2h* - αh1)(2h*α - h1)l2 + 2(l2 + 4h*2){α1(l2 + h12 ) + α(l2 + h22)} - α1(l2 + 2h*h1)2]kydkt/2f 2l 6 p1 = {24h*2α + 4h*h1α(α1 - α) + (α1 + 2α)(3l2 + h12)}kyd2kth22/4f 3l 6 p0 = {4α2h22 - α1h12(α - 1)2 - 2h1h2α(α - 1)(1 - α1)}kyd2/4f 2l 6 + α1(2h* - h1α)(2h*α - h1)kydkt/2f 2l 6 If α = α1 and h1 = h2 and k is taken to the limit then this reduces to p 3D 3 + p 1D = 0

(3)

the solutions of which consist of a purely imaginary root corresponding to an un-

220


(a)

(b)

Figure 7.8 Mode shapes for three-axle vehicle, kyd = kt = ∞ (a) corresponding to zero root (b) one cycle corresponding to si = iωI where ωI is given by equation (4).

damped oscillation frequency given by

ωI2 = 9αV2(l2 + 3h*2)/(8h*2 + 3l2)( h*2 + 3l2)

(4)

and a zero root corresponding to a mode of neutral stability. This zero root implies that the vehicle is capable of quasi-static misalignment as discussed in connection with two-axle vehicles, Section 4.3, and the corresponding mode shape is shown in Figure 7.8(a). The oscillation at frequency ωI is one in which the creep forces are in overall equilibrium even though the motion of individual wheelsets is not one of pure rolling. This is analogous to the behaviour of a rigid two-axle vehicle, Section 5.3. Figure 7.8(b) shows that the mode shape of this oscillation resembles the steering oscillation mentioned above. Instability at low speeds can occur either by the real part of a complex conjugate pair of eigenvalues becoming positive (oscillatory instability) or by a real eigenvalue becoming positive (divergence). As the instabilities occur for large values of the shear stiffnesses it is most instructive to use the perturbation method introduced in Section 3.3. The terms in the expressions for the coefficients which are of order k2 can be considered as small perturbations δpi of the initial values pi* which are of order k3. The corresponding perturbation in the ith eigenvalue then follows from equation (3.3.12). Thus, from (2), the initial values pi* are


221

p 4* = 0 p3* = (3l2 + 8h*2)(3l2 + h*2)kyd2 kth*2/4f 3l 6 p 2* = 0

(5)

p1* = {24h*2α + 3α(3l2 + h*2)}kyd2kth*2/4f 3l 6 p 0* = 0 and

δp4 = (3l4 + 16h*2l2 + 12h*4)kydkt/2f 2l 4 + (3l2 + h*2)(l2 + 3h*2)kyd 2/4f 2l 4 δ p3 = 0 δp2 = {α (3l2 + h*2)(l2 + 3h*2) + h*2l2(1 + α)2}kyd2/4f 2l 6 + {(2 - α)(2α - 1)h*2l2 + α(3l4+ 16h*2l2 + 12h*4)}kydkt/2f 2l 6

(6)

δ p1 = 0 δp0 = (1 + α)2αh*2kyd2/4f2l6 + (2 - α)(2α - 1)αh*2kydkt/2f 2l 6 The approximations for the eigenvalues found using equation (3.3.12) are sI = µΙ ± ωI where ωI is given by equation (3) and

µΙ = - V(δp4 ωI4 - δp2 ωI2 + δp0 )/(- 3p3 ωI2 + p1 )

(7)

and sII = µΙΙ = - Vp0/p1. Equation (7) can be expressed in the form

µΙ = Vp4 (ωI2 - ω12)( ωI2 - ω22)/2p1

(8)

where ω1 and ω2 are the roots of

δp4s4 + δp2s2 + δp0 = 0

(9)

which are given by

ω12 = V2{(1 + α)2αh*2kyd + 2(2 - α)(2α - 1)αh*2kt}/ {2(3l4 + 16h*2l2 + 12h*4)kt + (3l2 + h*2)(l2 + 3h*2)kyd} (10)

ω22 = V2α

222


Equation (7) shows that for stability either

ω1 < ωI < ω2

or

ω1 > ωI > ω2

(11)

This is identical to Leonhard’s criterion [18] and is, of course fully equivalent to Routh’s criterion; it provides a simple way of discussing the stability of the system. From (4) and (10) it can be seen that ωI < ω2 for all values of h* and l, independent of α. Therefore for stability ω1 < ωI which for given kyd and kt is a quadratic in α. For practical values of the parameters this has solutions for small values of α, which can occur in practice, and very large values which are not of interest. It follows from (4) and (10) that for very small values of α stability cannot be achieved if kyd > 4kt. For large values of kyd and h* >> l the condition that ω1 < ωI implies that oscillatory instability will only occur for low values of conicity, such as α < 1/10. The mode shape of the unstable mode is similar to that shown in Figure 7.8(a) and thus instability of the steering oscillation which occurs for large values of kyd and kt is promoted by low conicity, inter-wheelset flexibility, and short wheelbase. Even where it occurs, it should be noted that the real part of the corresponding pair of eigenvalues, though positive, remains small. The criterion for static stability is p0 > 0 or

α3 + α(1+ α)2kyd2 h*2/4f2 + α(2 -α )(2α - 1)ktkyd h*2/2f 2 > 0

(12)

Divergence can only occur if α 2 which is beyond the practical range of values), and for kt > kyd/4. As divergence only occurs when α is small (12) reduces to kyd/ kt > 2(2 - α)(1 - 2α )/ (1 + α)2

(13)

independent of wheelbase and creep coefficient. The mode of neutral static stability which exists for very large values of kyd and kt is destabilised by the introduction of flexibility between the wheelsets. It is now possible to consider the influence of speed on the eigenvalues and this is illustrated by the root locus shown in Figure 7.9. As speed is increased the eigenvalues corresponding to branches A, B and C1and C2 are initially proportional to speed and are closely equal to the solutions of (1). For the chosen set of parameters there is stability at low speeds. Similar to the case of the two-axle vehicle, the eigenvalues associated with the vehicle body modes, D1, D2 and D3 are substantially independent of speed and are closely equal to the wheels-fixed eigenvalues as illustrated in Table 7.2. For the three-axle vehicle, like other configurations, at speeds for which the kinematic or steering frequencies approach the natural frequencies of the vehicle body on the suspension, there is the possibility of body instabilities. As in the case of the two-axle vehicle, stability can be obtained by choosing suitable values for ky1, ky2, dy1, and dy2. As a result, for the chosen set of parameters there is little interaction between the wheelset modes and the modes involving the car body oscillating on the suspension so body instabilities do not arise. In addition there are the usual 6 relatively large eigenvalues corresponding to the wheelset subsidences in lateral translation and yaw,


223

Figure 7.9 Root locus as speed is varied for three-axle vehicle with parameters of Table 7.1 except that kt = kyd = 0.1 MN/m. Table 7.2 Eigenvalues for three-axle vehicle with parameters of Table 7.1 except that kt = kyd = 0.1 MN/m.

1 2 3,4 5 6 7,8 9 10 11,12 13 14 15,16 17,18

At Speed, V = 10 m/s -1290 -1395 -1377 ± 6.066i -1377 -1397 -2.406 ± 9.254i -45.30 -2.415 -3.468 ± 2.019i -14.18 -2.500 -1.796 ± 5.722i -0.7354 ± 5.770i

Wheels - Fixed

-2.407 ± 9.244i -48.19 -2.415 -3.735 ± 1.828i

Label Wheelset subsidence Wheelset subsidence Wheelset subsidence Wheelset subsidence Wheelset subsidence Car body upper sway Car body yaw Car body yaw Car body lower sway Shear subsidence C1 Shear subsidence C2 Shear oscillation B Steering oscillation A

224


two of which appear at E and X in Figure 7.9. As the speed is increased, the steering oscillation A loses stability at a bifurcation speed VB = 48.7 m/s. Figure 7.10 summarises stability as a function of kyd and kt, for the extreme cases of λ = 0.5 and λ = 0.05 with the creep coefficients halved. Analogous to the case of the two-axle vehicle, the behaviour is different depending on the value of kyd/2flα1/2 and kt/2flα1/2. The ‘stiffness to ground’ provided by the yaw stiffness for the twoaxle vehicle is provided by kyd and kt for the three axle vehicle. As already discussed, stability depends on non-zero values of kyd and kt; if kyd = 0, there will be two undamped eigenvalues at wheelset kinematic frequency at low speed, one mode involving motion of the central wheelset, and the other mode involving motion of the outer wheelsets such that there is zero shear between the outer wheelsets. If kt = 0 there will be one undamped eigenvalue at wheelset kinematic frequency at low speed, the motion involving zero shear between adjacent wheelsets. As mentioned above, a suitable choice of parameters has been made so that the body instabilities do not occur. If both kyd and kt are non-zero, examination of the root locus has shown that it is the steering oscillation, branch A, that becomes unstable above the bifurcation speed VB. VB is increased if kyd and kt are increased. This region is labelled O1 in Figure 7.10. This instability is analogous to wheelset instability in which the inertia forces induced by the steering oscillation drive the oscilla-

λ = 0.5

λ = 0.05 f x 0.5

150

150 V (m/s)

V (m/s) 100

O1

50

0 150 V (m/s)

50

S 0.2

kyd (MN/m)

S 0.8

1

O1

0

0

100

S 0.2

kt (MN/m) 0.8

50

10

D

S

O1

150 V (m/s)

100 50

O2

100

0

0.2

kyd (MN/m) 0.8

1

O1 O2

S

D 0.2 kt (MN/m) 0.8

1

Figure 7.10 Joint effect of conicity and creep coefficient on stability as a function of kt and kyd for three-axle vehicle with the parameters of Table 7.1. D = divergence, O = oscillatory instability and S = stable.


225

tion, there being little involvement of the car body. Low conicity and creep coefficients, stabilise the instability O1, but a separate instability O2 is introduced where the critical speed reduces with increasing kyd and kt. The mode shape of this oscillation is similar to that of the low speed steering oscillation shown in Figure 7.8(b) and involves significant yaw of the car body. In addition, the combination of low conicity and creep coefficient introduces divergence over a range of small values of kyd and above a limiting value of kt in accordance with equation (13). For very large values of kyd and kt, beyond the range of values in Figure 7.10, there is a further region of oscillatory instability which occurs at low speeds in accordance with equation (11). In the light of these results, it might be thought that the selection of values for kyd and kt to obtain the maximum critical speed for a range of conicity and creep coefficient is a difficult compromise. However, it should be borne in mind that a number of the instabilities, both dynamic and static, discussed above are associated with eigenvalues which small in magnitude. This is particularly true when the system has low values of equivalent conicity. In these circumstances, judgement of parameter values should be informed by nonlinear response studies.

7.5 Stability with Unequal Conicities The preceding results apply to a configuration in which the wheelset conicities are uniform and the inter-wheelset shear structure is symmetric so that the shear spring kyd is located midway between the wheelsets. As discussed above, it is necessary to vary the spring position h1(measured from the leading wheelset) if the conicities are not uniform in order to achieve perfect steering. However, it is convenient to consider the influence of h1 and conicity on stability separately. Figure 7.11 shows the effects on stability of variations of h1, α and α1 for the vehicle with parameters of Table 7.1, based on the linearised equations of motion. Both oscillatory and static instabilities occur. From equation (4.1), for static stability p0 > 0 so in the case where α = α1 then

α2 + {4αh22 + h1(α - 1)2(2h2 - h1)}kyd2/4f 2 + (2h* - h1α)(2h*α - h1)kydkt/2f 2 > 0

(1)

Without discussing (1) in detail, it can be seen in general terms that movement of the pivot towards the central wheelset and increasing kt all promote divergence. This is illustrated by the results in Figure 7.11(c). Two regions of oscillatory instability occur as h1 is varied and are shown in Figure 7.11(c). Movement of the position of the spring kyd towards either the outer or inner wheelsets is destabilising and oscillatory instability occurs at all speeds when either h1 = 0 or h1 = 2h*. This is as expected as s = ± iV(λ/r0l)1/2 is then a solution of the characteristic equation for all values of λ, λ1, kyd and kt, an undamped kinematic oscillation occurring because of the lack of elastic restraint.

226

RAIL VEHICLE DYNAMICS (a)

300 V (m/s)

0

O

V (m/s)

O

200 100

(b)

300

O

200 D

S

0.1

100

0.3

λ 0.5

0

0.1

0.3 λ 0.5 1

(c)

300 V (m/s)

S

O

O

200 100

0

S 4

D

h1 (m)

8

Figure 7.11 The influence of unequal conicities and effective pivot point on stability for the three-axle vehicle with parameters of Table 7.1. D = divergence; O = oscillatory instability; S = stable.

As indicated in Figure 7.11(c) the steering oscillation of the example three-axle vehicle, with h3 = 3 m., loses its stability at small amplitudes at V =168.4 m/s. The eigenvalues are given in Table 7.3 and the eigenvector corresponding to the mode in which µ = 0 is given in Table 7.4. This solution is a sub-critical bifurcation leading to a limit cycle in which the lateral displacement of all three wheelsets have a similar amplitude of 5 mm at V= 178 m/s, the details of which are shown in Figure 7.12. This shows the numerical solution of the nonlinear equations of motion for the vehicle with the parameters of Table 7.1 and the wheel rail geometry of Figure 2.2. Figures 7.12(a) and (b) indicate that the limit cycle is in the form of a steering oscillation very similar to that predicted by the small amplitude solution and there are many points of similarity with the corresponding results for the unrestrained bogie shown in Figure 5.21. The displacements of the car body are small. The frequency of the oscillation is 7.46 Hz and is closely equal to that given by the small amplitude eigenvalues. Figure 7.12(c) shows the variation of the rate of rotation of the leading wheelset which is about 0.1% of V/r0. This fluctuates at twice the frequency of the oscillation as a major component of the torque applied to the wheelset is the product of the longitudinal creep force and the rolling radius both of which are varying at the frequency of the oscillation. Figures 7.12(d) and (e) show the longitudinal and lateral creepages for the right hand wheel of the leading wheelset. As the variation in the wheelset rotational speed is small, reference to equation 2.5.13 shows that the largest


227

Table 7.3 Eigenvalues for the three-axle vehicle of Table 7.1, h1 = 3 m, V = VB = 168.4 m/s. Eigenvalue(s-1)

No. 1,2 3.4 5 6 7,8 9 10,11 12 13,14 15,16 17 18

-41.04± 269.1i -38.29 ± 159.7i -176.7 -147.8 -36.03 ± 83.43i -79.52 ± 47.50i -46.98 -2.44 ± 9.32i -4.30 ± 3.68i -4.64 -2.42

Label Large wheelset roots Large wheelset roots Large wheelset root Large wheelset root Shear oscillation Shear subsidence Steering Car body yaw Car body upper sway Car body lower sway Shear subsidence Car body yaw

Table 7.4 Eigenvector for mode 10 of the isolated bogie of Table 7.1 , V = VB = 168.4 m/s.

y1 ψ1 y2 ψ2 yb φb ψb y2 ψ2

1

-0.2454 + 0.3839i -0.2901 - 0.8140i 0.1594 + 0.09210i -0.01508 + 0.001898i 0.002628 - 0.00006271i 0.07224 - 0.05877i -1.0462 - 0.19032i 0.4814 - 0.4148i

contribution to the longitudinal creep is made by the variation in rolling radius with lateral displacement and this is reflected in the waveform. Similarly, reference to equation 2.5.15 shows that the largest contribution to the lateral creep is made by the angle of yaw, as indicated by the waveform. Figure 7.12(f) shows the spin of the leading wheelset, and reference to equation 2.5.17 shows that the spin behaves like δr/r0 and so the waveform reflects the geometrical nonlinearity of the contact slope variation with lateral displacement. Figures 7.12(g) and (h) shows the longitudinal and lateral creep forces for the right hand wheel of the leading wheelset, and consideration of their resultant and the normal force indicate that slipping is occurring over part of the cycle of the oscillation. Figure 7.12(i) shows the variation of the normal force acting on the right hand wheel which is in phase with the lateral displacement and is relatively large in magnitude.

228

RAIL VEHICLE DYNAMICS 0.01

(a)

.005

yb

0

ψ1

(b)

0

-0.01 0 .005

y1 y2

y3

-.005 0.1 t (s) 0.2 0 .005

(d)

γ1r1

γ2r1

0

(c)

1

ψ3

-V/r0-Ω1

0

ψ2

ψb 0.1

0.2

(e)

0

-1 0 0.5 0

0.1

0.2

(f) ω3l1

ω3r1 -5 0 20

0.1

(g)

-.005 0.2 0 20

T1r1 (kN)

T2r1

0

0

-20

0

0.1

0.2

-20 0

0.1

(h)

0.2

-0.5 0 100

-T3r1

0.1

0.2

(i)

50

0.1

0.2

0

0

0.1

0.2

Figure 7.12 Displacements, creepages and forces in the limit cycle of three-axle vehicle of Table 7.1 (except that h3 = 3 m) at V = 178 m/s.

Figures 7.11(a) and (b) show the effect of variation of λ and λ1 on stability. For static stability p0 > 0 so if h = h1 then from equation (4.1)

α2α1 + (1 + α){α(1 + α1) + α - α1}kyd2h2/4f2 + α1(2 - α)(2α - 1)kydkth2/2f2 > 0

(2)

Reduction of λ promotes divergence as shown in Figure 7.11(a) In order to get a feel for the behaviour of the system, it is useful to consider the simple case of a vehicle in which kt = 0 and kyd is very large. Figure 7.13 shows the root locus as h1 and λ1 are varied. Solution of (4.1) for this case shows that there are two purely imaginary eigenvalues when α = α1 and h1 = h2. One of these, A in Figure 7.13, corresponds to kinematic oscillations of the wheelsets in which the phase between the wheelset motions is determined by the shear connection between the wheelsets so that the shear spring is unstressed. The other, B in Figure 7.13, refers to a steering oscillation in which creep occurs. The frequency of this steering oscillation, when α = α1 and h1 = h2, is given by

ω12 = (1 + α)2V2h*2/(3l2 + h*2)(l2 + 3h*2)

(3)

It can be seen that even if the conicity is zero, a steering oscillation occurs in which the wavelength is solely determined by the geometry of the vehicle. Consider the variation of the conicities, but maintaining h1 = h2. As α1 is reduced in


229

Figure 7.13 Root locus as h1 and α1 are varied for three-axle vehicle of Table 7.1 with α = 0.025 and kt =0.

value, the two eigenvalues approach each other and, at a critical value, frequency coalescence takes place, further reductions in α1 leading to eigenvalues of the form ±µ ± iω. This form of instability may be termed ‘flutter’ as it is closely related to instabilities experienced with other non-conservative systems such as those arising in Aeroelasticity [20]. It can be shown [11] that the equations of motion of the system with kt = 0 and kyd very large are of the same form as those of undamped aeroelastic systems which have been extensively studied [20]. Increasing α1 beyond the value α1 = α results in the natural frequencies diverging in value. The frequency of the steering oscillation, branch B, reduces in value until

α1 = 2α(1 + α )/(1 - 2α2 )

(4)

consistent with (3), when it becomes zero. Further reductions in the value of α1 result in eigenvalues of the form ±µ and divergence occurs. Now, in this simple case of a vehicle in which kt = 0 and kyd is very large, consider the variation of the position of the inter-wheelset shear spring kyd, but maintaining α = α1. Solution of (4.1) for this case shows that there are two purely imaginary eigenvalues. Irrespective of the position of the pivot, one of these corresponds to kinematic oscillations of the wheelsets, and the other refers to a steering oscillation. The frequency of this steering oscillation is given by

230

RAIL VEHICLE DYNAMICS ω2 =

V 2 {h2 2 (1 + α ) 2 − (1 − α ) 2 (h2 − h1 ) 2 } {(2l 2 + h1 2 + h2 2 ) 2 − (l 2 − h2 2 ) 2 }

(5)

When the effective pivots of the shear springs move towards the central axle, the frequency of the steering oscillation reduces in value until h1/2h = 2/(3 - α )

(6)

when it becomes zero. Further increases in h1 result in roots of the form ±µ and divergence occurs. As the pivots move toward the outer axles, the frequency of the steering oscillation reaches a maximum and then decreases in value until it vanishes at a point outside the wheelbase when h1/2h = -2α /(1 - 3α )

(7)

Further outward movement of the pivots result in roots of the form ±µ and divergence occurs. These results show that for the three-axle vehicle there are many potential modes of instability. However, the lateral stiffness and damping between the car body and wheelsets may be chosen appropriately to eliminate the body instabilities, in a similar way to that prescribed for the two-axle vehicle. Then stability may be obtained up to high speeds providing that extreme values of conicity and inter-wheelset stiffnesses are avoided. The historical difficulties with the stability of three-axle vehicles with inter-wheelset linkages can be ascribed to the linkages either being deficient in that the criteria of equation (6.4.15) are not satisfied or that linkages were too stiff.

7.6 Dynamic Response In addition to problems with stability, experience with three-axle vehicles has shown that whilst steady-state curving may be improved, the form of the inter-wheelset connections may give rise to large steering errors on non-uniform curves [5]. Accordingly, it is important to consider the dynamic response of three-axle vehicles to track geometry and the following examples illustrate the principles involved. Figures 7.14 and 7.15 show the dynamic response of the three-axle vehicle of Table 7.1 (except that kyd = kt = 0.1 MN) to entry into a curve of radius R0 = 225 m, with a cubic parabolic transition of length L0 = 20 m as computed from the full nonlinear equations of motion. The vehicle speed V = 15 m/s and the coefficient of friction µ = 0.3. The track is canted so that the cant deficiency is zero throughout. The results may be compared with the results for the elastically restrained wheelset in Section 3.7 and the results for the two-axle vehicle in Section 4.6. Figure 7.14 shows the motion of the wheelsets which after an initial transient on the transition take up a radial attitude with an outwards lateral displacement closely equal to that necessary to negate longitudinal creep as given by equation (3.2.23). In the transient, the wheelsets attempt to follow the curve, and the steering errors are quite small.


231

0.005 0

ψ1

ψ2

ψ3

y1

y2

y3

-0.005 0

2

t (s)

4

0

2

4

0

2

4

Figure 7.14 Dynamic response of the wheelsets of the three-axle vehicle with the parameters of Table 7.1 (except that kyd = kt = 0.1 MN) to curve entry with a cubic parabolic transition of length 20 m, V = 15 m/s and R0 = 225 m. 2

5

ψ1 (mr)

0

T2r1 (kN)

0

-2

y1 (mm)

-4 -6 0

2

t (s)

4

T1r1 (kN)

10 0

-5 -10

T2l1 (kN) 0

2

4

zyW

10 0

T1l1 (kN)

-10 0

2

4

T2r1+ T2l1 -10 0

2

4

Figure 7.15 Dynamic response of the leading axle of the three-axle vehicle with the parameters of Table 7.1 (except that kyd = kt = 0.1 MN) to curve entry with a cubic parabolic transition of length 20 m, V = 15 m/s and R0 = 225 m.

However, on very sharp curves flange contact readily takes place, and steering errors can be large. This effect can be pronounced on reverse curves as has been found in the past. Figure 7.15 shows the time history of the creep forces acting on the leading wheelset. As in the case of a single wheelset, once the steady-state motion is established, the lateral creep forces are mainly generated by spin and are largely cancelled out by the gravitational stiffness force, and the longitudinal creep forces are small. However, during the transition the deviations of the wheelsets from a radial attitude induce elastic forces generated by the inter-wheelset suspension which have to be

232


5 0

ψ1 (mr)

-5

-5

-10

y1 (mm) -10 0

T2r1 (kN)

0

2

t (s)

4

-15

T2l1 (kN) 0

2

4

20

T1r1 (kN)

10 0

zwW (kN)

0

-10 -20

2

T1l1 (kN) 0

2

4

-2 0

T2r1 + T2l1 2

4

Figure 7.16 Dynamic response of the leading axle of the three-axle vehicle with the parameters of Table 7.1 to curve entry with a cubic parabolic transition of length 20 m, V = 15 m/s and R0 = 225 m.

reacted by longitudinal creep forces, which therefore reach a peak and then subside to their steady-state value. Figure 7.16 presents the corresponding results for the same vehicle but with the nominal values of the stiffnesses kyd and kt ( kyd = 0.7 MN, kt = 0.2 MN). The increased stiffness of the inter-wheelset structure results in larger longitudinal creep forces both in the steady-state and the transition. It is clear from these indicative results that the three-axle vehicle offers the possibility of improved steady-state curving in which the wheelsets take up a closely radial position, but the dynamic response in transitions and reverse curves requires careful consideration. Thus, the trade-off between stability and curving is not completely eliminated.

References 1. Liechty, R.: Das Bogenlaufige Eisenbahn-Fahrzeug. Schulthess, Zurich, 1934, p. 23. 2. White, J.H.: The American Railroad Freight Car. The John Hopkins Press, Baltimore, 1993, p. 168. 3. Fidler, C.: British Patents 2399, 3825, 1868.


233

4. Elsner, H.: Three-Axle Streetcars. N.J. International, Hicksville, 1994, Vol. 1, Chapter 1. 5. Elsner, H.: Three-Axle Streetcars. N.J. International, Hicksville, 1994, Vol. 1, p. 60. 6. Liechty, R.: Das Bogenlaeufige Eisenbahn-Fahrzeug. Schulthess, Zurich, 1934, p. 31. 7 Wickens, A.H.: Steering and dynamic stability of railway vehicles. Vehicle System Dynamics 5, No. 1-2 (1975), pp. 15-46. 8. Wickens, A.H.: Stability criteria for articulated railway vehicles possessing perfect steering. Vehicle System Dynamics 7, No.1 (1979), pp. 33-48. 9. Wickens, A.H.: Static and dynamic stability of a class of three-axle railway vehicles possessing perfect steering. Vehicle System Dynamics 6, No.1 (1977), pp. 1-19. 10. Wickens, A.H.: Flutter and divergence instabilities in systems of railway vehicles with semi-rigid articulation. Vehicle System Dynamics 8, No.1 (1979), pp. 3348. 11. Wickens, A.H.: The stability of a class of multi-axle railway vehicles possessing perfect steering. In: K. Magnus (Ed.): Proc. IUTAM Symposium on Dynamics of Multibody Systems, Munich, August-September 1977, pp. 345-356. SpringerVerlag, Berlin, 1978. 12. Wickens, A.H.: Static and dynamic stability of unsymmetric two-axle railway vehicles possessing perfect steering. Vehicle System Dynamics 11, No. 2 (1982), pp. 89-106. 13. Wickens, A.H.: Stability optimisation of multi-axle railway vehicles possessing perfect steering. ASME Journal of Dynamic Systems Measurement and Control 110, No.1 (1988), pp. 1-7. 14. Wickens, A.H.: Static and dynamic stability of a generalised symmetric threeaxle railway vehicle possessing perfect steering. Archives of Transport Quarterly 1, No.2 (1989), pp. 139-160. 15. de Pater, A.D.: Optimal design of a railway vehicle with regard to cant deficiency forces and stability behaviour. Delft University of Technology, Laboratory for Engineering Mechanics, Report 751, 1984. 16. de Pater, A.D.: Optimal design of railway vehicles. Ingenieur-Archiv 57, No.1 (1987), pp. 25-38.

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17. Keizer, C.P.: A theory on multi-wheelset systems applied to three wheelsets. In O. Nordstrom (Ed.): The Dynamics of Vehicles on Roads and Tracks. Proc. 9th IAVSD Symposium, Linkoping, June 1985. Swets and Zeitlinger Publishers, Lisse, 1986, pp. 233-249. 18. Porter, B.: Stability criteria for linear dynamical systems. Oliver and Boyd, Edinburgh, 1967, p. 37. 19. Bolotin, V.V.: Non-conservative problems of the theory of elastic stability. Pergamon, Oxford, 1963. 20. Done, G.T.S.: The flutter and stability of undamped systems. British Aeronautical Research Council Reports and Memorandum No.3553, 1966.

8 Articulated Vehicles 8.1 Introduction The economics of railways in which expensive infrastructure is justified by large traffic flows requires the operation of either many vehicles with short headways or very large vehicles to provide the required capacity. Hence the concept of the train. In many cases the interaction between the vehicles in a train is minimised by the form of coupling between the vehicles, so that longitudinal forces can be transmitted between car bodies, but the coupler is capable of transmitting little or no lateral force or yaw couple. In this case it is a good approximation to treat each vehicle as if it were isolated and the lateral dynamics of each vehicle can be considered to be largely independent of that of the rest of the train. However, the need to improve curving performance, maximise the use of the clearance gauge, minimise axle loads, reduce mass, aerodynamic drag and cost has led to many designs in which there is articulation of the car bodies of a vehicle or train, so that the connections between vehicles form an essential part of the running gear. In this Chapter articulated vehicles, in which the relative motion between the car bodies is used to influence the stability and guidance of the vehicle, are considered. The first articulated locomotive was designed by Horatio Allen in 1832 [1]. Though this had a short career, it probably stimulated several of the articulated designs for the Semmering Contest in 1851. Thereafter, there was a succession of articulated locomotives the development of which is described by Weiner [2], and which sought to resolve the conflict between the long wheelbase made necessary by high power and the large curvature of many railway lines. Most of these had unsymmetric fore-and-aft configurations which are considered in Chapter 9. In the early days of the railways, it had become customary to link together two and three axle vehicles not only by couplings but also by side chains to provide yaw restraint between adjacent car bodies in order to stabilise lateral motions [3]. The need to lengthen vehicles stimulated measures which allowed bending in plan view. As an alternative to the use of the bogie, in 1837 W.B. Adams proposed an articulated two-axle carriage [4]. The first three-axle vehicle with articulated car body was proposed by Fidler in 1868 [5]. In Fidler’s patent the wheelset is mounted on the car body at the point where the car body is tangential to the curve, and the central wheelset is mounted on a steering beam, Figure 8.1(a). Machlachan [6] proposed a similar configuration in 1878 but made the significant addition of a shear connection

236


(a)

(b)

(c)

(d)

(e)

Figure 8.1 Historical articulated railway vehicle configurations. (a) Fidler, 1868. (b) Machlachan, 1878. (c) Barber, 1907. (d) Liechty, 1931. (e) Configuration using steering linkage.

or pivot between the car bodies, Figure 8.1(b). In Barber’s 1907 patent [7] the outer bodies are pivoted together but instead of the wheelset being mounted on the body at the point of tangency, the outer wheelsets are freely pivoted and are connected to the central wheelset by cross-bracing, Figure 8.1(c). The steering beam has disappeared. A similar objective is achieved by Liechty’s 1931 patent [8] in which the outer wheelset is mounted on an arm pivoted on the car body and actuated by the steering beam, Figure 8.1(d). A similar scheme has been used recently in the Boa design [9]. An alternative approach is to use a linkage, Figure 8.1(e), similar to the arrangement used in body-steered bogies, but driven by the angle between the car bodies. In trains of bogie vehicles, the use of the Jacob bogie which is shared by adjacent car bodies reduced mass (and cost and drag) of an articulated rake. However, the resulting fixed consist was disliked by some operators, and individual body lengths had to be shorter to maintain limits on the ‘throwover’ (lateral displacements of the car bodies) on curves. Nevertheless, articulation was successfully exploited by Gresley [10] in a number of carriage designs between 1900-1930. In 1939, Sta-

ARTICULATED VEHICLES

237

(a) (b) (c) (d) (e)

Figure 8.2 Examples of some of the many configurations of vehicles with articulated car bodies and single-axles. Equivalent bogie versions are common.

nier employed a double point articulation, one at each end of the bogie, which allowed longer car bodies [11]. Application to a new generation of passenger trains in the U.S. in 1930-50 was less successful, and generally articulated trains acquired a reputation for bad riding. More recently, articulation of bogie vehicles has been used, most successfully, in high speed trains such as the TGV [12]. Another important modern development is the use of articulation on vehicles with single axle running gear, as discussed in Chapters 4 and 5.The dynamic behaviour of the Talgo train is influenced by its unsymmetric configuration and the problems arising will be considered in Chapter 9. Current examples of trains which have single-axle running gear such as the Copenhagen S-Tog [13] embody forced steering of the wheelsets through mechanical linkages or hydraulic actuators driven by the angle between adjacent car bodies. Extensive design calculations were carried on this train [13] and its lateral stability is also discussed in [14]. Another recent example is provided by the Wien trams [15]. A wide range of articulated configurations have been used in practice, particularly for trams, and some of the variations are shown in Figure 8.2. This figure is

238


based in part on [16]. The simplest form of articulated vehicle is one with two car bodies supported on three or four axles. Not only has this been a common configuration, particularly for trams, but its study reveals many of the dynamic characteristics of articulated trains with many axles. For a vehicle with three or more axles it was shown in Chapter 7 that stability could be achieved without compromising steering on uniform curves, the stiffness necessary for stability being provided in a way that does not impede radial steering. This provides a theoretical basis for the discussion of articulated vehicles exploiting single-axles in this Chapter. As in the case of the bogie vehicle considered in Chapter 6, two broad approaches to configurations can be distinguished. The first may be termed forced steering (steering derived from relative motions between car bodies) and the second self steering (steering derived from wheelset motions only). Various possible future stages in the development of single-axle suspensions may be envisaged, in which mechanical linkages are progressively displaced by systems with actuators and sensors and either passive or active control. 8.2 Steering and Stability As a wide variety of vehicles with articulated car bodies exist, it will be necessary to consider a limited number of variants, so the behaviour of a number of representative configurations with three and four axles will be discussed. Figure 8.2(a) shows two two-axle vehicles coupled together; Figure 8.2(b) and 8.2(c) show similar con-

\3

yd y3 \d h

\c

yc

\2

y2

y3

\2

\b

h1 c yd \4

y4

h

\d

\3

h

yb

y2

\b \1 y 1

c

Figure 8.3 Generalised coordinates for articulated vehicles.

yb y1 \1


239

k\b (a)

ky2/2

k\1

kyb

k\2

ky1/2

ky2/2

ky1/2 kyb

k\b

ky2/2

k\1

ky1/2

(b) k\1

k\2

k\2 kyb

k\1

Figure 8.4 Simplified arrangement of suspension showing basic stiffnesses for articulated vehicles.

figurations but with one axle removed; Figure 8.2(d) and (e) show symmetric threeaxle vehicles with either two or three car bodies, including vehicles with two car bodies carrying a steering beam on which the central wheelset is mounted. Comparison of the behaviour of these various configurations will give insight into the dynamics of articulated vehicles in general. Consider the equations of motion of these configurations. It is clear that the configurations of Figure 8.2(b) and (c) are derivable from that of Figure 8.2(a) by the deletion of a wheelset, and such unsymmetric configurations will be discussed in Chapter 9. Moreover, the configurations of Figure 8.2(d) and (e) are equivalent in terms of their degrees of freedom. Consequently, making the same assumptions that have been made in the case of the two-axle vehicle and other configurations, the motions of these vehicles may be defined by the generalised coordinates shown in Figure 8.3. The form of the elastic stiffness matrix will be determined by applying the criteria for stability of a vehicle capable of perfect curving given in Chapter 6. The solution of equations (6.4.10) will initially be discussed for the three-axle vehicle with three car bodies of Figure 8.2(d) so that neglecting the variation of the rotational speed of the wheelsets q >y\ybMb\by\ycMc\cy\ydMd\d@T

(1)

so that yb, yd and yc refer to lateral translation of the vehicle bodies and \b ,\d , and \c refer to yaw of vehicle bodies. The other yi, \i are standard wheelset coordinates. The equations of motion will be of the form of (2.11.23) where

240


F = block diagonal[ F1 O33 F2 O33 F3 O33 ]

(2)

and O33 is the 3 u 3 null matrix, and Fi refers to the ith wheelset. The inertia matrix is A = diag [m I mb Ixb Izb m I mc Ixc Izc m I mb Ixb Izb ]

(3)

The component stiffness and compatibility matrices for the basic form of secondary suspension shown in Figure 8.4(a) are k = diag[ ky1 kI1 k\1 kyb kIb k\b ky2 kI2 k\2 kyb kIb k\b ky1 kI1 k\1]

a

ª1 «0 « «0 « «0 «0 « «0 «0 « «0 «0 « «0 « «0 «0 « «0 «0 « «¬0

0 1

d1

h

0 0

0

0

0

0 0

0

0

0

0

1

0

0 0

0

0

0

0 0

0

0

1 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0

0 d2 1 0 0 0 0 0 0 0 0 0 0

1 h1 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 1 d 3 h1 c 0 0 1 0 0 0 0 1 0 1 d4 0 0 0 1 0 1 0 0 1 0 1 d 3 c h1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 d2 0 0 1 0 0 0 0 1 d1 0 0 1 1 0 0

(4)

0º 0 »» 0» » 0» 0» » 0» 0» » 0 » (5) 0 »» h1 » » 0» 1» » h» 0» » 1»¼

Then equations (6.4.10) become ky1( y1 - yb + d1Mb - h\b ) = 0

(6)

kM1Mb = 0

(7)

k\1( \1 - \b ) = 0

(8)

kyb{ yb - d2Mb - h1\b - yc + d3Mc - (c - h1)\c} = 0

(9)

kIb( Mb - Mc ) = 0

(10)

k\b( \b - \c ) = 0

(11)


241

ky2(- yc + d4Mc + y2 ) = 0

(12)

kM2Mc = 0

(13)

k\2( \c - \2 ) = 0

(14)

kyb{ yd - d2Md + h1\d - yc + d3Mc + (c - h1)\c} = 0

(15)

kIb(Md - Mc ) = 0

(16)

k\b(\d - \c ) = 0

(17)

ky1(y3 - yd +d1Md + h\d) = 0

(18)

kM1Md = 0

(19)

k\1(\3 - \d) = 0

(20)

From equations (7), (13) and (19) Mb = Mc = Md = 0. From equations (8), (11), (14), (17), and (20) either \1 = \b or k\1 = 0

(21)

either \c = \b or k\b = 0

(22)

either \2 = \c or k\2 = 0

(23)

either \d = \c or k\b = 0

(24)

either \3 = \d or k\1 = 0

(25)

For the case where the outer wheelsets have radius r0 and conicityO and the inner wheelset has radius r1 and conicityO1, for steering on a uniform curve without lateral and longitudinal creep, the displacements measured from the unstrained position are y1 = y3 = - lr0/OR0 + (c + h )2/2R0

(26)

y2 = - lr1/O1R0

(27)

\1 = -\3 = (c + h )/R0

(28)

\2 = 0

(29)

Because of symmetry yd = yb, \d = -\b , and \c = 0 so that from (23) the value of

242


k\2 can be arbitrarily chosen. From (12), either ky2 = 0 or yc = y2. Choosing the latter case, from (6), (9), (26) and (27) yb = {- h1lr0/O- hlr1/O1+ h1(c + h )2/2}/R0( h + h1)

(30)

\b = {- lr0/O+ lr1/O1+ (c + h )2/2}/R0( h + h1)

(31)

If the condition (21) is satisfied by putting \1 = \b then, from (28) and (31), - lr0/O + lr1/O1 + (c + h )( c - h - 2h1)/2 = 0

(32)

or, if the inner and outer wheelsets have the same radius and conicity h1 = (c - h)/2

(33)

If this condition is satisfied the conditions for perfect steering can be satisfied with a nonzero yaw stiffness k\1. The remaining conditions (22) and (24) requires that k\b = 0. This implies that there are two main possibilities for symmetric articulated three-axle vehicles. The first has geometry which satisfies the geometric requirement (32) or (33) and the second does not satisfy this requirement so that k\1 = 0. In addition, there is, of course, the further possibility of the vehicle with a single car body which has been discussed in Chapter 7. An analogous analysis can be carried out for the configurations of Figure 8.2(a), where the basic stiffnesses are shown in Figure 8.4(b). As this vehicle consists of two-axle vehicles coupled together with simple suspension elements, it is not necessary to reiterate the equations of motion. The conditions for steering give the simple result y1 = y4 = - lr0/OR0 + (c + h )2/2R0

(34)

y2 = y3 = - lr1/O1R0 + (c - h )2/2R0

(35)

\1 = -\4 = (c + h )/R0

(36)

\2 = -\3 = (c - h )/R0

(37)

yb = yd = - lr0/2OR0 - lr1/2O1R0 + (c2 + h2)/2R0

(38)

\b = -\d = - lr0/2OhR0 + lr1/2O1hR0 + c/R0

(39)

Similar results, of course, apply to the configurations of Figures 8.2(b) and 8.2(c). Figure 8.5 indicates the geometry for each of these configurations in which the wheelsets adopt a radial position with radial displacements appropriate to their conicity, and the car bodies take up the positions connecting the wheelsets without


243

strain of the suspension elements. Thus, for small displacements and large radius curves, steering without lateral and longitudinal creep is obtained because of the combined action of creep and conicity which is the primary mechanism of guidance. It is evident that on curves with moderate and large curvature, a similar analysis may be carried out based on the full nonlinear equations of motion, cf. Section 3.2. These configurations satisfy the conditions for steering so it is now necessary to consider stability. For simplicity it will be assumed that all wheelsets have the same radius and conicity. The necessary condition for the vehicle to be able to steer without lateral and longitudinal creep and be dynamically stable at low speeds is given by (6.4.15). For the three-axle configuration of Figure 8.4(a) if the geometrical condition (33) is not satisfied and the yaw stiffness k\1 is zero, the degeneracy of the basic stiffness matrix E is 7, so that in this case three additional stiffnesses are required to

(a)

c h1

h

(b)

\b y b

(c + h)2/2R0

y1

Ol/R 0r0

c (c)

h

\b y

b

(c + h)2 /2R 0

y1

Figure 8.5 Attitude on curve of various articulated configurations.

Ol/R 0 r0

244


reduce the degeneracy to 3 so that the criterion may be satisfied. For configurations where the geometrical condition (33) is satisfied and there are nonzero yaw stiffnesses k\1, the degeneracy of E is 4, so that in this case only one additional stiffness is required to satisfy the criterion. The basic stiffness matrix E for the four-axle configuration of Figure 8.4(b) has degeneracy 7 and therefore requires 4 additional stiffnesses in order to satisfy the criterion. Thus additional stiffnesses must be provided, and this is considered next. 8.3 Application to Specific Configurations Based on the configurations discussed in Section 1, Figure 8.6 shows an extension of Figure 8.4 in which further elastic connections have been introduced. Figure 8.6(a) and (b) show the body-steered and self-steered versions of the three-axle vehicle and Figure 8.6(c) and (d) show the corresponding versions of the four-axle vehicle. Firstly, it is useful to consider the early schemes shown in Figure 8.1 in the light of the above criteria. As mentioned in Chapter 7, pivots were originally used to provide lateral restraint and yaw freedom, instead of elastic elements. For example, in Fidler’s patent for the three-axle vehicle, Figure 8.1(a), the geometric condition (2.33) is satisfied and ky1, ky2, kyb, k\1, and k\2, were very large though the principles discussed above remain valid. As mentioned above, E has degeneracy of 4 and the vehicle would be unstable at low speeds. The steering beam is equivalent to the central car body of Figure 8.2(d). In Machlachan’s scheme, Figure 8.1(b), there is a shear connection or pivot between the outer car bodies, which may be regarded as an additional stiffness kyc. From the above discussion, and assuming that the geometrical condition (2.33) is satisfied, the addition of an additional row and corresponding stiffness kyc would have the effect of stabilising the vehicle so that all eigenvalues had negative real parts at low speeds. In Barber’s scheme, Figure 8.1(c), the outer bodies are connected by a shear stiffness kyc but the geometrical condition (2.33) is not satisfied. Instead the outer wheelsets are freely pivoted and are connected to the central wheelset by cross-bracing, with shear stiffness kyd. In this case the three stiffnesses kyc and kyd (leading and trailing bay) would ensure that the degeneracy of E would be reduced to 3 as required by the stability criterion. A similar objective is achieved by Liechty’s scheme, Figure 8.1(d) in which the outer wheelset is controlled by a steering beam. Again, the steering beam is the central car body of Figure 8.2(d) steered through springs of stiffness kd and ke, but if the steering beam is small its independent coordinates can be eliminated and its effect represented as shown in Figure 8.7. It is therefore an equivalent arrangement to a steering linkage connecting the yaw angle of the leading wheelset with the angle between adjacent car bodies. Thus, these additional stiffnesses are provided in two distinct ways, consisting of the basic configuration as defined in Section 3 with, either body steering or self steering. In the former case the outer wheelsets are steered by a linkage, of stiffness kst, from the angle between the outer and central car body, the third additional


(a) e1

245

kyc

c1 c2

c3 e2 ks =diag [ kst

as

kst kyc ]

0 0 0 º ª0 c h 0 0 2(h h1 ) 0 0 0 0 c h 2h1 0 0 » «0 0 0 0 0 0 0 0 0 2 0 0 0 2 2 c h h c h h h 1 1 » « «¬0 0 1 d5 0 0 0 0 0 0 0 1 d5 c c »¼

kst = e1e2c22(c3 - c1)2/{e1(c2 - c1)2 + e2(c3 - c1)2}

(b)

c2(c3 - c1)/c3(c2 - c1) = (c + h)/(c - h -2h1)

kt

kyd

kyd

ks = diag[ kyd kyd kt]

as

0 0 0 1 c h h3 h3 ª0 «0 0 0 0 0 1 c h h3 « «¬1 c h 0 0 0 0 0

0 0 0 0 0 0 0 0º 0 0 0 1 h3 0 0 0»» 0 0 0 1 c h 0 0 0»¼

k st2 (c)

k st1

k st1 k st2

ks = [ kst1 kst2 kst2 kst1 ]

ª0 «0 « «0 « ¬0

as

1 0 0 1 h / 2c 0 0 0 0 0 0 0 0 0 1 h / 2c 0 1 0 0 0 0 0 0 0 0 0 0

h / 2c h / 2c

kyf

(d) kyd

0 0º 0 0»» 0 0 0 1 0 0 1 h / 2c 0 0» » 0 0 0 0 0 0 1 h / 2c 0 0¼

kt

h / 2c h / 2c

kyd

kyf

kye

ks = [ kyd kyd kt kye kyf kyf]

as

h ª1 «0 0 « «1 c h « 0 «0 «1 c « 0 «¬0

0 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0

0 0 0

1

0 0 0 0 0 º 0 0 0 1 h »» 0 0 0 1 c h» » 0 » c h 1 c h 0 0 0 0 0 0 0 0 0 0 » 1 c » c »¼ c 0 0 0 0 0 1 h 0 0

0

0

1 0

h 0

Figure 8.6 Additional stiffnesses for articulated vehicles.

246


c1 \d

O

c-h

c1

R P

Q

S

\b

\2

c2 h 2c

c1 c2 c h 2 c2

as = [ 0 0 0 0 -1+ h/2c 0 1 0 0 0 0 -h/2c 0 0 ]

Figure 8.7 Steering beam applied to rear wheelset of leading vehicle showing the equivalence to a steering linkage.

stiffness required for stability is provided by kc, as shown in Figure 8.6(a). In the latter case the adjacent wheelsets connected by a shear spring kyd and the outer wheelsets connected by a shear spring kt, as shown in Figure 8.6(b). Similarly, there are equivalent schemes for the four-axle vehicles with two car bodies as shown in Figures 8.6(c) and (d). For long articulated vehicles direct connection of the wheelsets may seem difficult to implement, but it is possible to utilise passive sensors and actuators replicating the stiffness and damping characteristics of mechanical linkages and arranged so as to be de-coupled from the motions of the car bodies. 8.4 Stability of an Articulated Three-Axle Vehicle As might be expected, there is considerable similarity between the behaviour of the articulated three-axle vehicle and the three-axle vehicle with single car body considered in Chapter 7. Firstly, the scheme in which the adjacent wheelsets connected by a shear spring kyd and the outer wheelsets connected by a shear spring kt, as shown in Figure 8.6(b) will be considered in detail. Consider the influence of speed on the eigenvalues of the configuration of Figure 8.6(b) with the nominal set of parameters given in Table 8.1. The root locus as speed is varied is shown in Figure 8.8, and values for the eigenvalues at V = 10 m/s are given in Table 8.2. It was shown in Section 6.4 that, at low speeds, the equations of motion can be separated into uncoupled sets involving either the wheelset coordinates or the car body coordinates. The resulting system of equations involving the wheelset coordinates is of exactly the same form, at low speeds, as those derived in Section 7.2 for the three-axle vehicle with single car body and, in particular, the form of the stiffness matrix E* is as given in (7.2.22-23). It follows that the system has six eigenvalues proportional to speed consisting of a steering oscillation A, a


247

Table 8.1 Example parameters for three-axle articulated vehicle. suspension of basic system ky1 = ky2 = 0.23 MN/m k\ = 0 kI = kI = 1MNm cy2 = 50 kNs/m cy1 = 200 kNs/m c\= c\= 0 cI = cI = 50 kNms kyb = 1 MN/m k\b = 0 kIb = 0 cyb = 0 cIb = 0 c\b = 0 additional suspension for body-steered system kst = 30 kNm kyc = 1 MN/m k\ = 2 MNm c\b = 0.1 MNms cst = cyc = c\2 = 0 additional suspension for self-steered system kyd = 0.1 MN/m kt = 0.1 MN/m ct = cyd = c\2 = c\b = 0 k\ = 5 MNm vehicle geometry r0 = 0.45 m l = 0.7452 m c = 6.25 m d = 0.2 m h=2m h1 = 3.75 m creep coefficients f11 = 8.83 MN f22 = 8.06 MN f23 =17.7 kNm inertia m =1250 kg I = 700 kgm2 mb=10000 kg Ixc=12000 kgm2 Izb = 130000 kgm2 mc=7000 kg Ixc=8400 kgm2 Izb = 9000 kgm2 2 Iy = 250 kgm

shear oscillation B and two shear subsidences C1 and C2 which are closely equal to the solutions of equation (7.4.1). For the chosen set of parameters there is stability at low speeds. A further set of eigenvalues is associated with wheelset motions and consist of six large subsidences (three roots are equal to -2f/mV and three roots are equal to -2fl2/IV) and these are not shown in Figure 8.8. The eigenvalues associated

Figure 8.8 Root locus of the three-axle configuration of Figure 8.6(b) with the nomi-

nal set of parameters given in Table 8.3.

248


Table 8.2 Eigenvalues for articulated three-axle vehicle with parameters of Table 8.1.

1 2 3 4 5 6 7,8 9,10 11 12,13 14,15 16,17 18,19 20,21 22,23 24 25,26 27,28 29,30

At Speed, V = 10 m/s -1398 -1392 -1383 -1331 -1331 -1331 -0.6423 ± 45.74i -2.368 ± 22.60i -14.18 -2.909 ± 10.40i -2.259 ± 8.833i -2.162 ± 8.801i -3.820 ± 7.333i 0.2710 ± 5.210i -0.8677 ± 5.601i -2.542 -3.199 ± 4.351i -1.303 ± 3.329i -3.162 ± 4.389i

Wheels - Fixed

-0.02164 ± 45.83i -2.445 ± 22.60i -2.909 ± 10.40i -2.181± 8.927i -2.165 ± 8.797i -0.2211± 7.246i

-3.266 ± 4.448i -1.317 ± 3.305i -3.012 ± 4.471i

Label Wheelset subsidence Wheelset subsidence Wheelset subsidence Wheelset subsidence Wheelset subsidence Wheelset subsidence D9 D8 Shear subsidence C2 D7 D6 D5 D4 Steering oscillation A Shear oscillation B Shear subsidence C1 D2 D1 D3

Figure 8.9 Natural modes of articulated vehicle with parameters of Table 8.3 with wheels fixed and zero suspension damping, D = 0.


249

V= 8.5

Z 4.829

73.0

37.36

78.9

42.26

Figure 8.10 Mode shapes of the critical modes at the bifurcation speeds of articulated vehicle with parameters of Table 8.3, showing one cycle of the oscillation.

with the vehicle body modes, D1-D9, are substantially independent of speed except at speeds for which the kinematic or steering frequencies approach the natural frequencies of the vehicle body on the suspension. They are closely equal to the wheels-fixed eigenvalues, as illustrated in Table 8.2. The mode shapes of these wheels-fixed modes are shown in Figure 8.9 for the case where the suspension damping is zero. It is noteworthy that many of the wheels-fixed modes are completely uncoupled from the wheelset motions but in other cases there is considerable interaction, leading to body instability. For the given parameters, body instability occurs at a bifurcation speed VB1 = 8.5 m/s and the associated mode shape is shown in Figure 8.10(a). As in the case of the two-axle vehicle, stability can be obtained by choosing suitable values for ky1, ky2, dy1, and dy2, but in addition inter-car body dampers may be effective. At higher speeds, the shear oscillation B loses stability at a bifurcation speed VB2 = 73.0 m/s. As shown in Figure 8.10(b), this is analogous to the wheelset instability, there is little motion of the car bodies and is predicted accurately if the car bodies are assumed fixed. A further instability at a bifurcation speed VB3 = 78.9 m/s involves yaw of the central car body and the mode shape is shown in Figure 8.10(c).

250


Figure 8.11 Variation of the imaginary part of the eigenvalues for the three-axle configuration of Figure 8.6(b) with the nominal set of parameters given in Table 8.1.

O = 0.5

80

O2

S

V (m/s)

80

O = 0.05, f x 0.5

O1 D S

O1 0 80

kyd (MN/m) 1 O2

O2 S

V (m/s)

O1 0 kt (MN/m) 0.5 80 V (m/s)

0 kyd (MN/m) 1 80

O1

S

0 k\2 (MNm) 0.5

S

D

0 kt (MN/m) 0.5 80 S

0 k\2 (MNm) 0.5

Figure 8.12 Joint influence of conicity, creep coefficient and stiffnesses kyd, kt and k\2 on stability for articulated three-axle vehicle of Fig. 8.6(b) with parameters of Table 8.1 except that cy1 = cy2 = cyb = 0.1 MNs/m, kyd = 0.7 MN/m, kt = 0.2 MN/m and k\2 = 5 MNm. S denotes stability, D divergence and O1 and O2 oscillatory instabilities.


251

O

80

O

V (m/s)

0 80

C

S

S

kst (MNm) 0.4 O

V (m/s)

O

V (m/s)

B A

O= 0.5 f/2

80

0 80

kst (MNm) 0.4

V (m/s)

S

S 0 80 V (m/s)

kyc (MN/m) 1

O

S O

S 0

ky2 (MNm)

0 kyc(MN/m) 1 80 O V (m/s)

1

0 k (MNm) 1 y2

Figure 8.13 Joint influence of conicity, creep coefficient and stiffnesses kst, kyc and k\2 on stability for articulated three-axle vehicle of Figure 8.6(a) with forced steering with parameters of Table 8.3. S denotes stability and O oscillatory instability.

Figure 8.11 shows the variation of the imaginary parts of the eigenvalues with speed for a range of low speeds. Frequency coincidence occurs between the steering oscillation A and shear oscillation B and the car body on suspension modes, but in only one case in the chosen speed range does instability occur. Clearly, coupling of the wheelset modes and the car body modes depends not only on a degree of frequency coincidence but also on compatibility of mode shape for the energy transfer that would sustain the amplitude of the oscillation. In this connection, the zig-zag mode shape of the car body mode D4 is particularly likely to couple with the steering oscillation, and this illustrates the tendency of articulated vehicles to suffer this form of body instability. The joint influence of the stiffnesses kyd, k\2 and kt, conicity and creep coefficient is shown in Figure 8.12. This may be compared with Figure 7.10 which shows a similar plot for the three-axle vehicle with a single car body. If any of the stiffnesses kyd, kt and k\2 are zero then the critical speed is zero in accordance with the discussion in the preceding Section. For large values of the conicity, instability is confined to smaller values of kyd, k\2 and kt. The instability, labelled O1 in Figure 8.12, is the body instability discussed above. The instability O2 is analogous to the wheelset instability. For low values of the conicity and creep coefficients, divergence occurs for a range of values of kyd and for larger values of kt. Moreover, the instability O2 occurs at lower speeds.

252


Turning to the scheme of Figure 8.6(a) in which the wheelsets are connected by a steering linkage to the car bodies, it is found that the root locus is qualitatively similar to the root locus for the variant with direct connections between the wheelsets, but there is a very lightly damped steering oscillation at low speeds. Three additional points arise. Firstly, as might be expected, in this case there is more interaction between the car body modes and the wheelset modes. Secondly, the addition of yaw damping, c\b, between the car bodies is effective in eliminating the body instability. Thirdly, very large values of c\b are effective in stabilising the system at low speeds, and increasing the damping in the steering mode, even when kyc = 0 and the stability criteria discussed in Section 3 are not satisfied. However, the bifurcation speeds associated with the wheelset modes are little affectedҠ. Hence Figure 8.13 shows the influence of the stiffnesses kst, kyc and k\2 on stability for extreme values of the conicity and creep coefficient for the scheme of Figure 8.6(a) and the parameters of Table 8.1. In accordance with the discussion above if the contact stiffnesses were neglected then for stability at low speeds either kyc or k\2 is needed in addition to kst. If both kyc and k\2 are zero, the critical speed is determined by the contact stiffnesses and would be low. Initially, increases in kst increase the critical speed, A in Figure 8.13(a). Beyond a certain value of kst the critical speed is not influenced by kst, B in Figure 8.13(a), because the mode of instability involves mainly kinematic motion of the central wheelset. For large values of kst, an instability involving interaction with the lateral bending mode of the car bodies occurs. This instability occurs for smaller values of kst in the case of small values of conicity and reduced creep coefficient as shown in Figure 8.13(b). Thus, in this case as in others that have already been discussed, there is a conflict between the requirement that kst must be sufficiently large in order to achieve a large margin of stability at high conicity, but this will tend to encourage instability for low values of conicity and creep coefficient. Figures 8.13(c)-(f) show that both k\2 and kyc are effective in stabilising the system for low conicities and creep coefficients, but not at high conicities. 8.5 Stability and Response of an Articulated Four-Axle Vehicle For the configuration of Figure 8.6(c) and the nominal set of parameters given in Table 8.3, the root locus as speed is varied is shown in Figure 8.14. As might be expected, in this case there are eight eigenvalues proportional to speed representing four oscillations at kinematic frequency O. These oscillations are stable at low speeds provided that elastic stiffness has been provided in accordance with the prescription of the Section 3. In addition there are the usual set of eigenvalues also associated with wheelset motions and consisting of eight large subsidences (at low speeds, four roots are equal to -2f/mV and four roots are equal to -2fl2/IV) which become heavily damped oscillations at higher speeds as shown at S in Figure 8.14. As usual, the eigenvalues associated with the vehicle body modes, D, are more or less independent of speed except at speeds for which the kinematic or steering frequencies approach the natural frequencies of the vehicle body on the suspension. They are closely equal to the wheels-fixed eigenvalues. As in the case of the threeaxle vehicle,


253

Table 8.3 Example parameters for four-axle articulated vehicle suspension of basic system ky1 = ky2 = 0.23 MN/m k\ = k\ = 0 kI = kI = 1MNm cy1 = 50 kNs/m c\= c\= 0 cI = cI = 50 kNms additional suspension for body-steered system kst1 = 2.5MN/m kst2 = 2.5MN/m dst1 = dst2 = 0 additional suspension for self-steered system kyd = 1MN/m kt = 0.1MN/m kyf = 0.1MN/m ct = cyd = cyf = 0 vehicle geometry l = 0.7452 m r0 = 0.45 m h = 3.5 m c=6m d = 0.2 m creep coefficients f11 = 7.44 MN f22 = 6.79 MN f23 =13.7 kNm inertia m =1250 kg I = 700 kgm2 mb=10000 kg Ixc=12000 kgm2 Izb = 130000 kgm2 2 mc=7000 kg Ixc=8400 kgm Izb = 9000 kgm2 2 Iy = 250 kgm

kyb = 40 MN/m cy2 = cyb = 0

kye = 1MN/m

Figure 8.14 Root locus of the four-axle configuration of Figure 8.6(c) with the nominal set of parameters given in Table 8.3.

254


it is noteworthy that many of the wheels-fixed modes are completely uncoupled from the wheelset motions but in other cases there is major interaction, potentially leading to body instability. One of the oscillations at kinematic frequency, shown at O1 in Figure 8.14, has very low damping and is associated with near coincidence between one of the wheels-fixed modes and a steering mode. At higher speeds, all four steering modes lose stability at bifurcation speeds between 85.5 and 89.5 m/s. This is analogous to the wheelset instability, there is little motion of the car bodies and is predicted accurately if the car bodies are assumed fixed. Then application of equation (3.3.15) shows that in this case it is the stiffness of the steering linkage kst1 which determines the bifurcation speed. A full survey of the stability boundaries will not be given here, but as an example Figure 8.15 shows stability as a function of speed and the stiffness in the steering linkage kst (= kst1 =kst2), for three values of the equivalent conicity and a case of reduced creep coefficient. If kst is zero then the system is unstable at low speed as already discussed. For small values of kst instability O1 occurs above a certain speed. For large values of kst another form of instability O2, which involves relatively large amounts of yaw of the car bodies, occurs for which low conicity and creep coefficient is destabilising. This again illustrates the difficult trade-off for many configurations of railway vehicle, for if a design has to cater for a wide range

100

O = 0.5

O = 0.2

100

O1 V (m/s)

V (m/s)

S

S

O2 0 100

k st (MNm)

20

O = 0.05

0 100

V (m/s)

V (m/s) S

0

20 kst (MNm) O = 0.05 f x 0.5

k st (MNm)

S 20

0

kst (MNm)

20

Figure 8.15 The effect of variations of conicity and creep coefficient on stability as a function of the stiffness of the steering linkage kst and speed for the four-axle vehicle with the parameters of Table 8.5. S = stability, O1 and O2 instability.


255

of conicities, comparison of the boundaries for high and low conicity in Figure 8.15 shows that the speed range for stability is severely reduced. These results can be compared with those for the configuration of Figure 8.6(d), for a set of nominal parameters, given in Table 8.3, comparable with those used in the discussion of the configuration of Figure 8.6(c), in which the wheelsets are connected directly and not through the car bodies. Figure 8.16 shows the root locus as speed is varied, and can be compared with Figure 8.14. In this case the eight eigenvalues which are proportional to speed, as suggested by the analysis of Section 6.4, consist of two conjugate complex pairs corresponding to steering oscillations, shown as O1 and O2 in Figure 8.16 and four real roots, shown at R. One of the latter is very small indicating a condition of marginal static stability, though for the chosen set of parameters there is stability at low speeds in accordance with the discussion of Section 3. The lightly damped kinematic oscillation of the body steered configuration is absent. The usual set of eigenvalues associated with wheelset motions consists of six large subsidences (three roots are equal to -2f/mV and three roots are equal to -2fl2/IV) and these are not shown in Figure 8.16. The six pairs of eigenvalues associated with the vehicle body modes are substantially independent of speed except at speeds for which the kinematic or steering frequencies approach the natural frequencies of the vehicle body on the suspension and they are closely equal to the wheels-fixed eigenvalues. These are shown as D1-D5 in Figure 8.16. (D6 is off the scale). It is noteworthy that in the present case the wheels-fixed modes are completely uncoupled from the wheelset motions. motions.re 8.17 shows stability as a function of speed and the stiffnesses kyd (= kye), kyf and kt for two extreme values of the equivalent conicity and creep coefficient. Note that in accordance with the discussion of Section 3, stability at low speeds is obtained if either kt or kyf are zero. However, an appropriate choice of stiffnesses

Figure 8.16 Root locus of the four-axle configuration of Figure 8.6(d) with the nominal set of parameters given in Table 8.3.

256


additional to the minimum required for stability at low speeds enhances stability. Compared with the body steered vehicle, the range of stability is much increased for larger values of conicity and there is considerable scope for optimisation. However, divergence occurs for smaller values of kyd and larger values of kyf and kt when the conicity is low and the creep coefficient is reduced. It was seen above that this behaviour is characteristic of some articulated vehicles having three axles. The conditions for static stability depend in a complicated way on the inter-wheelset stiffnesses, as indicated for three-axle vehicles. In order to assess the practical significance of these results it is necessary to consider a typical solution of the complete nonlinear equations of motion for the directly steered system of Table 8.3. Figures 8.18(a) and (b) show, for the set of parameters indicated, the dynamic response to an initial condition y1(0) = 4 mm applied to the lateral displacement of the leading wheelset. The response is dominated by a lightly damped oscillation, following this transient, by the vehicle taking up a steady-state attitude in which overall balance of the creep forces is achieved. Figure 8.18(d) shows the stability boundaries, consistent with those shown in Figure 8.17, in the kyd (= kye) - O plane. As O is a function of the wheelset lateral displacement, it can be

O =

O = 0.05 f /2 O

V

S

D S kyd

kyd

O V

S D

S kyf

V

S

kyf

O D S

kt

kt

Figure 8.17 Qualitative diagram showing the joint influence of conicity, creep coefficient and stiffnesses kyd (= kye), kyf and kt on stability for a directly steered four-axle articulated vehicle. S denotes stability, D divergence and O oscillatory instability.


257

(a)

4

1

2 y (mm)

4 3

0

-4 -6

2 0

2

4

6

b 3 1

0

b d

-2

(b)

1 \(mr) 0.5

-0.5

d

-1

2

-1.5 0

8

4

2

t (s)

4 t (s)

6

8

(c)

(d)

0.2

S

C

O A

D

0.1

B O1 0

O2 C1

A1 D 1

kyd = kye (MN)

2

Figure 8.18 Dynamic response and stability of vehicle with low conicity subjected to initial condition of 4 mm applied at leading wheelset. (a) and (b) lateral and yaw displacements (c) steady-state attitude of vehicle (d) stability chart. Parameters of Table 8.3 except that kyd = kye = 1 kN/m, kt = 2 MN/m, kf = 1 MN/m, creep coefficients halved and wheel-rail geometry consistent with O= 0.065.

258


0

2

\1 (mr)

0 y1 (mm) -10

3

t (s)

-5

10

5 T1l

T2r

(kN)

(kN) T1r

T2l

-10

-15

Figure 8.19 Dynamic response of linkage steered vehicle of Figure 8.6(d) to curve entry. Speed 15 m/s, length of transition 20 m., curve radius 225 m cant deficiency zero. y1 lateral displacement, \1 yaw angle, T1r and T1l longitudinal creep forces on right and left, T2r and T2l lateral creep forces, for leading wheelset.

2

0.5 0

0

\(mr)

y1 (mm) -10 0

t (s)

3

1.5 (kN)

2 T1l

0 T1r -1.5

-1.5

0

T2r

(kN)

T2l

-10

Figure 8.20 Dynamic response of self-steered vehicle of Figure 8.6(c) to curve entry. Speed 15 m/s, length of transition 20 m., curve radius 225 m cant deficiency zero.


259

seen that if A corresponded to the starting point of the transient, the motion would decay towards the stability boundary at B, and in this case this point corresponds to a zero real root. Reference to equation (3.7.5) shows that for the equivalent linear system with a zero real root the response to an impulse tends to a motion in the mode corresponding to the zero real root. In fact, Figure 8.18(c) shows that the attitude of the vehicle is closely similar to that indicated by the corresponding linear system. If C were the starting point, the motion would decay to point D and a limit cycle. If A1 and C1 were the starting points then the motions would grow until the steady state is established at B and D respectively. It is clear that because, in this case, the instabilities O2 and D are associated with low conicity, displacements and creep forces will be small. This is not the case with the instability O1. The self-steered configuration therefore allows considerable freedom in the selection of the parameters. While both configurations can be expected to perform well on curves of constant radius, the forces generated in a transition or reverse curve will be influenced by the suspension parameters and there is conflict between the requirements for stability and dynamic response. This can be illustrated by comparing the dynamic response in curve entry for examples of the two configurations. Figure 8.19 shows a typical dynamic response of the linkage-steered vehicle of Figure 8.6(d) on entry to a curve in which the transition has linearly increasing curvature. The track is canted and is being traversed at the speed for zero cant deficiency. On the transition, the varying angles between the wheelsets and the car bodies induces forces in the steering linkage so that the leading wheelset fails to steer and moves out beyond the rolling line, large lateral and longitudinal creep forces being induced. These forces are significantly reduced when running on the uniform part of the curve as the wheelsets take up a radial position and attempt to move out to the rolling line. For a much smaller radius curve, the wheelsets are still able to take up a radial position but because of the restricted flange-way clearance, cannot move out to the rolling line and hence significant creep forces are generated. If the cant deficiency is not zero then further yaw movements of the wheelsets are necessary and the creep forces are increased. In the case of the self-steered configuration of Figure 8.6(c) it is possible to select suitably low values of the stiffnesses which, whilst maintaining stability for a range of conicities, also give low forces in the transition to a uniform curve. This is illustrated in Figure 8.20. References 1. White, J.H.: American Locomotives, Revised Ed., The John Hopkins Press, Baltimore, 1997, p. 509. 2. Weiner, L.: Articulated Locomotives, Constable, London, 1930. 3. Fryer, C.E.J.: A History of Slipping and Slip Carriages, Oakwood, Oxford, 1997 p. 7.

260


4. White, J.H.: The American Railroad Passenger Car, The John Hopkins Press, Baltimore, 1978, p. 627. 5. Fidler, C.: British Patents 2399, 3825. 1868. 6. Liechty, R.: Das Bogenlaeufige Eisenbahn-Fahrzeug, Schulthess, Zurich, 1934, p. 31. 7. Barber, T.W.: British Patent 24632, 1907. 8. Liechty, R.: British Patent 390036, 1933. 9. Anon.: International Railway J. September, 1988. p. 26. 10. Jenkinson, D.: British Railway Carriages of the 20th Century, Volume 1: The End of an Era 1901-22. Patrick Stephens, Wellingborough, 1988, p. 155. 11. Jenkinson, D.: British Railway Carriages of the 20th Century, Volume 2: The Years of Consolidation 1923-53. Patrick Stephens, Wellingborough, 1990, p. 197. 12. Tachet, P. and Boutonnet, J.-C.: The Structure and Fitting Out of the TGV vehicle Bodies. French Railway Techniques 21, No. 1 (1978), pp. 91-99. 13. Rose, R.D.: Lenkung und Selbstlenkung von Einzelradsatzfahrwerken am Beispiel des KERF im S-Tog Kopenhagen. Proc. 4th International Conference on Railway Bogies and Running Gears, Budapest, 1998, pp. 123-132. 14. Slivsgaard, E. and Jensen, J.C.: On the dynamics of a railway vehicle with a single-axle bogie, Proc. 4th Mini Conference on Vehicle System Dynamics, Identification and Anomalies, Budapest, 1994, pp. 197-207. 15. Anon.: Wien Light Rail Vehicles by Duwag-Bombardier. Railway Gazette International, September, 1992, p. 586. 16. Charlton, E.H.: Articulated Cars of North America. Light Railway Transport League, London, 1966, Figure 34.

9 Unsymmetric Vehicles

1 Introduction Though early railway passenger and freight vehicles were generally symmetric, locomotives early adopted an unsymmetric configuration in order to maximise the axle-load on driving wheels and make use of the available adhesion. Additional smaller wheelsets were soon provided to improve the steadiness of running. It has already been mentioned, in Chapter 6, how the introduction of the leading swivelling bogie improved both stability and curving behaviour, and it was a matter of experience that the running behaviour of these unsymmetric configurations was strongly dependent on the direction of motion [1]. A later example of an unsymmetric vehicle design, used in trams, is provided by “maximum traction trucks” devised in the 1890’s and in which the driving wheels were followed or preceded by pony wheels which had a diameter about two-thirds of the driving wheels [2]. In this case each bogie was unsymmetric but the complete vehicle was usually symmetric. The classic work on the stability of unsymmetric railway vehicles is by Carter [3] which was directed toward the configurations then current in railway practice. Carter applied Routh's stability theory, not only to electric bogie locomotives, then exhibiting many problems of instability, but also to a variety of steam locomotives. In his mathematical models, a bogie consists of two wheelsets rigidly mounted in a frame, and locomotives comprise wheelsets rigidly mounted in one or more frames. Following Carter’s first paper of 1916 the theory was elaborated in a chapter of his 1922 book [4]. Carter’s next paper [5] gave a comprehensive analysis of stability within the assumptions mentioned above. As he was concerned with locomotives the emphasis of his analyses was on the lack of fore-and-aft symmetry characteristic of the configurations he was dealing with, and he derived both specific results and design criteria. Carter's work expressed, in scientific terms, what railway engineers had learnt by hard experience, that stability at speed required rigid-framed locomotives be unsymmetric and uni-directional. His analysis of the 0-6-0 locomotive found that such locomotives were unstable at all speeds if completely symmetric and he comments that this class of locomotive is “much used in working freight trains; but is not employed for high speed running on account of the proclivities indicated in the previous discussion.” Carter analysed the 4-6-0 locomotive both in forward and reverse motion and found that in forward motion beyond a sufficiently high speed or sufficiently stiff


262

(a)

(b) 60

60

O

V (m/s)

O

V (m/s) 40

40

D 20

20

S

S 0

1

2 3 ky (MN/m)

4

0

1

2 3 ky (MN/m)

4

Figure 9.1 Carter’s stability diagram for the 4-6-0 locomotive in (a) forward motion and (b) reverse motion. ky is the centring stiffness. (Recalculated in modern units from [5]). S = stable; O = oscillatory instability; D = divergence.

bogie centring spring (laterally connecting the bogie to the locomotive body) oscillatory instability occurs, but as the mass of the bogie is small compared with the main mass of the locomotive, the resulting oscillation was unlikely to be dangerous at ordinary speeds. Carter’s stability diagrams, the first of their kind in the railway field, are shown in Figure 9.1(a) and 9.1(b). In reverse motion, Figure 9.1(b), Carter found that beyond a certain value of the centring spring stiffness buckling of the wheelbase occurred which would tend to cause derailment at the leading wheelset. As this wheelset is incorporated in the main frame of the locomotive, the lateral force acting between wheel and rail would be proportional to the mass of the main frame and would be correspondingly large, and potentially dangerous. This was the explanation of a number of derailments at speed of tank engines such as the derailment of the Lincoln to Tamworth mail train at Swinderby on 6 June 1928, as discussed in his final paper [6]. Carter’s analysis of the 2-8-0 with a leading Bissel, or single-axle bogie, similarly explained the need for a very strong aligning couple for stability at high speed, whilst noting that in reverse motion a trailing Bissel has a stabilising effect for a large and useful range of values of aligning couple. Rocard [7,8] also considered unsymmetric configurations and proposed the use of different conicities fore-and-aft. A general theory for the stability of unsymmetric vehicles and the derivation of theorems relating the stability characteristics in forward motion with those in reverse motion was given in [9,10]. In the case of articulated two-axle vehicles at low speeds it was shown that a suitable choice of elastic restraint in the inter-wheelset connections results in static and dynamic stability in forward and reverse motion and which will steer perfectly, without any modification dependent on the direction of motion. However, the margin of stability is small. A further practical result of Carter’s work was a series of design measures, the subject of various patents [11], for the stabilisation of symmetric electric bogie locomotives, because the introduction of the symmetric electric locomotive had been accompanied by a more or less common experience of lateral instability at high speed. In particular, Carter suggested an arrangement of the running gear in which

UNSYMMETRIC VEHICLES

263

the configuration was altered depending on the direction of motion. This makes it easier to resolve the conflict of stability and steering with an adequate margin of stability. Such re-configurable systems have been studied by Li [12]. Chapter 3 discusses the use of the pony axle (a wheelset mounted on a leading or trailing arm) in conjunction with freely rotating wheels, and the example of the Talgo train was cited, in which the wheelsets are mounted on the car bodies in an unsymmetric way. A further source of lack of symmetry in railway vehicles can occur when a configuration which is intended to be symmetric suffers from badly distributed loading or unequal wheel wear, thus giving rise to asymmetry about a transverse plane. Asymmetry about a longitudinal plane can, of course, also occur for the same reasons; however, this tends to merely alter the equilibrium position of the vehicle, and, while it should be taken into account in detailed numerical calculations, no new phenomena are introduced. Moreover, there is evidence to suggest that asymmetry about a transverse plane due to unequal wheel wear is more important for vehicles which have inherently poor curving ability, such as the three-piece freight truck. In this case, calculations have been described by Tuten, Law, and Cooperrider [13]. Illingworth [14] suggested the use of unsymmetric stiffness in steering bogies. Elkins [15] showed both by calculation and experiment that a configuration of bogie, with the trailing axle having independently rotating wheels and the leading axle conventional, significantly improved stability and curving performance and reduced rolling resistance. Suda et al [16-17] have studied bogies with unsymmetric stiffnesses and symmetric conicity, and their development work has led to application in service [18]. The concept has been extended to include lack of symmetry of the wheelsets by equipping the trailing axle with freely rotating wheels. This provides a practical example of a re-configurable design as the wheelsets are provided with a lock which is released on the trailing wheelset (allowing free rotation of the wheels) and locked on the leading wheelset (providing a solid axle). The lock is switched depending on the direction of motion. In modern vehicles the provision of secondary suspension between bogies and car body has the result that lack of symmetry in the car body rarely introduces new phenenoma. However, lack of symmetry in the bogie itself can be exploited to improve performance and this is discussed below.

9.2 Stability Theorems for Rigid and Semi-Rigid Vehicles Important insights are obtained by considering, first of all, the static and dynamic stability of vehicles in which the wheelsets are incorporated rigidly in one or more frames, which are themselves connected by joints imposing non-elastic constraints. Thus, in these cases the elastic stiffness matrix E is null. A rigid vehicle is one in which all the interwheelset elastic stiffnesses are infinite, so that the number of degrees of freedom M = 2, corresponding to lateral translation and yaw. A semi-rigid vehicle is defined as one in which E is null but 2N > M > 2, where N is the number of wheelsets, so that in addition to possessing rigid body freedoms, the vehicle is able to articulate like a mechanism. Now referring to the simple form of the equa-


264

tions of motion of a single wheelset, because the wheelset is symmetrical about a transverse plane through its centre, for a motion in the reverse direction, while retaining the same definition of generalised coordinates, the equations of motion (2.10.47-48) become y + 2f22 y /V + kyy + 2f22ψ = Qy m (1) 2

-2f11λly/r0 + Iz ψ + 2f11l ψ /V + kψψ = Qψ

(2)

and in general the equations of motion for reversed motion of a complete vehicle are [ As2 + Bs/V - C + E] q

= Q

(3)

Thus, the equations of reversed motion are obtained by reversing the sign of the creep stiffness matrix. In the case of the semi-rigid or rigid vehicle the equations of motion reduce to [As2 + (B/V)s ± C ] q

= Q

(4)

where + refers to forward motion and - refers to reversed motion. The trial solution q α est leads to the determinantal equation

⏐As2 + (B/V)s ± C⏐ = 0

(5)

On making the substitution s = VD this becomes

⏐AV2 D2 + BD ± C⏐ = 0

(6)

and for low speeds, this reduces to

⏐BD ± C⏐ = 0

(7)

which expands to, for forwards motion, pMDM + pM-1DM-1 +..........+ p1D + p0 = 0

(8)

For reversed motion, since all the elements of C change sign, some of the pi change sign. In particular, p0 and p1 will have opposite signs. The condition for static stability is that p0 > 0 and a necessary (but not sufficient) condition for dynamic stability is that all the pi have the same sign. Hence, if an unsymmetric rigid or semi-rigid vehicle is statically and dynamically stable in forward motion, it must be dynamically unstable in reverse motion. If the number of degrees of freedom M is even, p0 does not change sign in reversed motion and if the vehicle is statically stable in both directions, it will be dynamically unstable in one direction. If M is odd, p0 does change sign in reversed motion and the vehicle will be statically unstable in one direction of motion.


265

Hence, an unsymmetric semi-rigid or rigid vehicle can possess a margin of stability in one direction only. This margin of stability is derived from the lack of symmetry, for if the vehicle were symmetric since (8) must be invariant with the direction of motion it follows that pM-1, pM-3,...... must all be zero. Then, in many cases (8) can be factorised in the form (for M even) pM(s2 + ω12) (s2 + ω22)........ (s2 + ωP2) = 0

(9)

where ω1, ω2,......... ωP are a set of P = M/2 steering frequencies, or (for M odd) pM(s2 + ω12) (s2 + ω22)........ (s2 + ωP2)s = 0

(10)

involving a set of P = (M-1)/2 steering frequencies, with the addition of a zero root. This zero root indicates that the vehicle is capable of quasi-static misalignment similar to that discussed for two-axle vehicles in Section (4.3). In other cases, roots coalesce to form a pair of roots ±µ ± iω indicating oscillatory instability or alternatively roots of the form ±µ occur indicating divergence. The assumption of low speeds has made it possible to neglect the inertia terms in the equations of motion. For semi-rigid vehicles in the more general case, (6) leads to a characteristic equation of order 2M. However, the coefficients p1 and p0 will remain unchanged, and it follows that irrespective of speed, an unsymmetric semirigid or rigid vehicle will be unstable in at least one direction of motion. This suggests that the examination of stability at low speeds is likely to yield useful information on the behaviour of various configurations, as is done in the following.

9.3 Unsymmetric Rigid Vehicle The general theory is exemplified by the unsymmetric rigid vehicle as considered by Carter [5]. Whereas Carter considered rigid assemblies of an arbitrary number of wheelsets, it will be sufficient for present purposes to consider only the case of a two-axle vehicle or bogie. If y and ψ represent the lateral displacement and yaw of the vehicle then the standard form of the simplified equations of motion of Section 2.10 applies where ⎡ m0 m 0 c ⎤ A=⎢ ⎥ ⎣ m0 c I 0 ⎦ ⎡4 f B=⎢ ⎣0

(1)

⎤ 4 f (h + l ) ⎥⎦ 0

2

2

(2)


266

−4f 0 ⎤ ⎡ C=⎢ ⎥ ⎣2 f (α1 + α 2 ) 2 fh(α1 − α 2 ) ⎦

(3)

where m0 = 2m + mb and I0 = Izb + 2mh2 and c is the distance of the centre of mass ahead of the mid point, and α1 = λ1l/r0, α2 = λ2 l/r0. The characteristic equation of this system is p4s2 + p3 s3 + p2 s2 + p1 s + p0 = 0

(4)

At low speeds, this reduces to p2 s2 + p1 s + p0 = 0

(5)

and then p2 = 2(h2 + l2)/V2 p1 = h(α1 − α2)/V p0 = (α1 + α2) In accordance with the general theory discussed in the previous section, since the number of degrees of freedom is even, p0 is invariant with change in direction of motion and is essentially positive. On the other hand, the sign of p1 changes when the direction of motion changes. Thus, the distribution of conicity can be arranged to give stability in one direction of motion, but not in both. This was suggested as a means of stabilising a vehicle by Rocard [7] who states that a successful experiment was made by French National Railways in 1936. Equation (5) yields an eigenvalue µ ± iω where, if p12 α1 there is dynamic stability. When α2 is sufficiently greater than α1 there is divergence. In reverse motion, the frequency of the steering oscillation remains unaltered but the real parts of the eigenvalues change sign. This is, of course, in accordance with the discussion in Section 2.


274

1

ω1

0.5

ω 0

ω2 0.2 0.4 0.6 α2

0

0.4 0.2

µ1

0

µ

-0.2 -0.4

µ2 0

0.2

0.4 α2

0.6

Figure 9.6 Behaviour of eigenvalues of semi-rigid unsymmetric vehicle at low speeds as conicity of rear wheelset is varied. Parameters of Table 4.1 except that α1 = 0.22, h1 = 0.75.

If µ1 is small, the characteristic polynomial can be factorised approximately, yielding a real root µ1

µ1 = −

p0 {α h (α − 1) + α1h2 (1 − α 2 )}V = − 2 1 21 p1 α 2 (l + h12 ) + α1 (l 2 + h22 )

(5)

and a complex pair λ = µ ± iω where

ω2 =

p1 {α 2 ( l 2 + h12 ) + α1 (l 2 + h22 )}V 2 = p3 ( 2l 2 + h12 + h22 )l 2

µ=−

=−

(6)

( p 2 p1 − p 3 p 0 ) 2 p 3 p1

(α 2 − α1 ){h1 ( l 2 + h22 )(1 − α1 ) + h2 ( l 2 + h12 )(1 − α 2 )}V 2l 2 ( 2l 2 + h12 + h22 ){α 2 ( l 2 + h22 ) + α1 ( l 2 + h12 )}

(7)


275

h1 = 0

1

1

h1 = h/3, h2 = 2h/3 D

S

S

α2

α2

O

O

0

0

1

α1

1

α1

h1 = h2 = h

1

D

α2 O

0

1

1

α1

h1 = 2h/3, h2 = h/3 D

α2

h2 = 0

1 D

α2

D,O

D,O

O 0

α1

1

0

α1

1

Figure 9.7 Regions of stability in the α1, α2 plane as a function of h1 and h2 for the semi-rigid vehicle at low speeds; S = stability, D = divergence and O = oscillatory instability.

Though these expressions are based on an approximate solution of the characteristic equation, they do describe accurately enough the behaviour of the system as the conicities and pivot position vary. The real root represents a subsidence or divergence depending on the relative magnitudes of h1 and h2 and α1 and α2. In fact, the exact condition for static stability is p0 > 0 or

α 2 h1 (α1 − 1) + α1 h2 (1 − α 2 ) > 0

(8)

α 2 < 1 / {1 + h1 (1 − α1 ) / α1 h2 }

(9)

or

The oscillatory root represents an oscillation with frequency roughly equal to the mean frequency of the kinematic frequencies of the two wheelsets, and is influenced


276

by joint position. The damping in this oscillation is strongly dependent on the difference in equivalent conicities and the pivot position exercises a less important influence. The condition for stability is simply

α 2 > α1

(10)

The regions of stability, as given by (9) and (10) are plotted in the α1, α2 plane for various spring positions in Figure 9.7. It is possible to select parameters which will ensure both static and oscillatory stability for one direction of motion, but these parameters would lead to both static and oscillatory instability in reverse motion. Since, in most practical cases, α1 a2h1. This criterion can be combined with (10) to give the necessary but not sufficient condition h2/h1 > α2/α1 >1

(11)

Thus, for stability in forward motion, the pivot must be nearer the front wheelset than the back wheelset and the rear wheelset must have larger conicity (e.g., smaller radius) than the front wheelset. These results contrast with those obtained for the rigid vehicle.

9.6 The Influence of Elastic Stiffness on Stability The influence of the stiffness on the stability of the two-axle vehicle shown in Figure 9.3(b) is now considered. The analysis will show that the configuration studied is not only stable for a range of parameters in both forward and reverse motion, but it is also capable of perfect steering. Consider the behaviour of the eigenvalues as a factor k on the stiffnesses ks, kb and ksb is varied, and the steering conditions (4.8) and (4.9) are applied. There are two regimes in which the behaviour is quite distinct. The first of these is for small values of kl/2f and the second is for all other values of kl/2f. As will be explained, the former regime is one of small interaction between the wheelsets and the latter regime is one of intensive interaction. When kl/2f = 0 the eigenvalues are purely imaginary and distinct, corresponding to undamped kinematic oscillations. As the eigenvalues are distinct it is possible to employ the perturbation analysis of Section (3.3), equation (3.3.12). Therefore, for small values of kl/2f , the eigenvalues are µ1 ± iω1 and µ2 ± iω2 where ω12 =

V 2 λ V 2 ( k sb − k s h)(1 − λ1l / r0 ) ± lr0 2 fl 2

(1) µ1 = −V ( k s l 2 + k s h 2 + k b − 2 k sb h) / 4 fl 2


277

ω 22 =

V 2 λ V 2 ( k sb + k s h)(1 − λ2 l / r0 ) ± lr0 2 fl 2

(2) µ2 = −V ( k s l + k s h + k b + 2 k sb h) / 4 fl 2

2

2

the ± signs referring to forward and reverse motion respectively. Both µ1 and µ2 are negative for all values of ks and kb and the system is stable. These results are consistent with a wheelset oscillating under asymmetric elastic restraint, the other wheelset being restrained from lateral motion. In fact, (1) can be derived by the solution of the binary set of equations of motion involving y1 and ψ1, whilst (2) similarly can be derived from the sub-system involving y2 and ψ2. The analysis is then a simple extension of that for a single symmetric wheelset given in Chapter 3, to which the present analysis is reducible. These two wheelset subsystems (leading and trailing wheelsets) are, for the small values of kl/2f under consideration here, weakly-coupled in the sense of Milne [19], and his analysis could be applied to give more complete rigour. The validity of the perturbation method applied here depends on the eigenvalues of the sub-systems being sufficiently separated. This requirement is, of course, not satisfied for the symmetric vehicle with equal conicity on both wheelsets, and a different approach is needed as employed in Chapter 4. As kl/2f is increased the interaction between the wheelsets dominates the behaviour of the vehicle. Some insight into the pattern of behaviour is given by the results for the symmetric vehicle discussed in Chapter 4, where it was shown that two modes of oscillation exist; a lightly damped steering oscillation which takes place at the kinematic frequency and a "shear" oscillation which for smaller values of kl/2f takes place at the kinematic frequency. As the real part of the eigenvalue increases rapidly as kl/2f increases this oscillation is replaced by two subsidences. Similar results may be expected for the unsymmetric vehicle, but there is the possibility of static and oscillatory instability. The condition for static stability is that p0 > 0 and if the steering conditions (4.8) and (4.9) are applied, this becomes 2fα1α2 ± ksh(α1 -α2) ± ksb(α1 +α2 - 2 α1α2) > 0

(3)

which reduces to (5.8) as kl/2f becomes large. Routh’s condition for oscillatory stability is that the test function of equation (3.4.3) should be positive or, on substitution ±(α1 -α2)[-A2{± f(α1 -α2) - ksh(2 −α1 - α2) - ksb(α1-α2)} + Α ksbh{2ksh(α1 -α2) - 2ksb(α1+α2) + 4ksb} ± 4ksb2h2f(α2 -α1)] > 0

(4)

where A = k s l 2 + k s h 2 + k b . Equation (4) reduces to (3.10) as kl/2f becomes large. For small values of kl/2f, for the vehicle shown in Figure 9.3(b), (4) is satisfied


278

for all values of the parameters consistent with (1) and (2), and the vehicle is oscillatory stable for both forward and reverse motion, except when α1 = α2 when it is of course marginally stable, irrespective of the position of the spring. Figure 9.8 summarises these results in the form of a stability diagram in the ks, h1 plane, for both forward and reverse motion at low speeds, where the position of the spring and the conicities are chosen so that (4.8) and (4.9) are satisfied. As the position of the spring moves aft from the central position, there is the possibility of both oscillatory and static instability. If the spring were placed at the trailing axle, then from (5) for oscillatory stability k

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