EVALUATION OF
RAILWAY SYSTEMS DYNAMICS
BY MODEL ADJUSTMENT
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DYNAMICS BY MODEL ADJUSTMENT
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. H. VAN BEKKUM, HOOGLERAAR IN DE AFDELING DER SCHEIKUNDIGE
TECHNOLOGIE, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE
VAN DEKANEN TE VERDEDIGEN OP WOENSDAG 30 JUNI 1976
TE 16.00 UUR
DOOR
PETRUS MATHEUS THEODORUS BROERSEN
natuurkundig ingenieur
geboren ie Niedorp
T'fi'nTr^
BIBLIOTHEEK TU Delftr
I
Dit proefschrift is goedgekeurd door de promotoren
PROF. DR. IR. A. D. DE PATER
PROF. IR. B. P. TH. VELTMAN
ACKNOWLEDGMENT
I express my gratitude to Mr. R. Terrasse and his collaborators of the "Departement Essais de la Direction du Materiel de la Societe Nationale des Chemins de Fer Francais". Only with the experienced assistance of the teams of Mr. Line and later on of Mr. Moreau, it was possible to execute the test runs which are the basis of this thesis. This co-operation was realized by the effort of Mr. A. Pettelat of the Office for Research and Experiments of the International Union of Railways.
I am also indebted to Mr. M. Balda of the §koda Works, Plzen, &.S.S.R., for his assistance in programming the hybrid computer during his stay as research fellow on the Delft University of Technology.
-CONTENTS
Samenvatting Summary
List of symbols
Chapter 1. Introduction
Chapter 2. Stability and riding quality 2.1. Stability and critical speed 2.2. Stability in practice
2.3. Assessment of the riding quality 2.4. Maintenance of the track
Chapter 3. Derivation of the equations of motion of a bogie 3.1. Basic equations
3.2. Wheel-rail contact forces 3.3. Equations of motion
3.4. Discussion of the equations of motion
Chapter 4. Model adjustment with two parallel models 4.1. The choice of the error criterion
4.2. Selection of a parameter estimation procedure 4.3. Adjustment with two parallel models
Chapter 5. Description of the experiments and the bogie 5.1. Introduction
5.2. The experimental bogie 5.3. The measurements
5.4. Profiles of wheels and rails
5.5. Non-linear description of the wheel-rail contact Chapter 6. Realization on a hybrid computer
6.1. Reconstruction of rail position from recorded signals 6.2. Equivalent input signals for a non-linear system 6.3. Implementation
Chapter 7. Results
7.1. Power spectra of rail irregularities
7.2. Correspondence with theoretical and adjusted paramete 7.3. Sensitivity
7.4. Investigations with the optimum model 7.5. Non-linear investigations
7.6. Discussion of the results
7.7. Recommended model and parameter values Chapter 8. Conclusions
SAMENVATTING
Dit proefschrift beschrijft de toepassing van parameterschatting op de zijdelingse stochastische bewegingen van spoorwegvoertuigen. Er wordt een wiskundig model gegeven dat deze bewegingen kan beschrijven. Dit model heeft
onafhankelijke parameters voor alle termen die te maken kunnen hebben met de krachten tussen wiel en rail. Dit zijn onder andere de parameters voor kruip, equivalente coniciteit en gravitatiestijfheid. Voor alle parameters kunnen waarden berekend worden. Maar een experimenteel onderzoek moet uitwijzen of deze waarden ook geldig zijn voor versleten wielen en rails. Hiertoe worden de gemeten bewegingen van een draaistel vergeleken met de responsies van het model, dat gesimuleerd wordt met een hybriede rekenmachine. De gemeten
on-regelmatige ligging van het spoor is het ingangssignaal voor het model. Door aanpassing van de parameters in het model worden daarvoor geschatte waarden gevonden die zorgen voor een optimale overeenkomst tussen de gemeten bewegingen en de modelresponsie.
Voor de meeste parameters komen de zo geschatte waarden redelijk overeen met de berekende. Hierdoor wordt deze berekeningswijze experimenteel
onder-steund, evenals de structuur van het wiskundige model. Maar voor twee para-meters worden schattingen gevonden die niet theoretisch verklaard kunnen
worden. Deze parameters zijn echter onmisbaar voor een juiste beschrijving van de bewegingen omdat ze de stabiliteit beheersen.
De overeenkomst tussen gemeten bewegingen en de responsie van een aange-past model is bevredigend. Het is dus mogelijk om het model, met aangeaange-paste waarden van de parameters, te gebruiken bij het ontwerpen van nieuwe spoorweg-voertuigen.
-SUMMARY
In this thesis parameter estimation is applied to the random lateral motions of railway vehicles. A mathematical model describing these motions
is given, with independent parameters for all terms which can be related to wheel-rail forces. This includes creep coefficients, the equivalent conicity and the gravitational stiffness parameter, for which theoretical values can be derived. However, the validity of these values in the case of worn wheel and rail surfaces needs experimental investigation. This can be done by simu-lating the lateral motions with a hybrid computer model, using the measured rail position as input. Model adjustment, by a variation of creep coefficients and other uncertain parameters, yields the parameter values for which an
optimal correspondence between model output and measured vehicle response is reached.
Most of the resulting parameter estimates agree reasonably with their theoretical values. This validates the theory and the structure of the
mathematical model. However, two parameters have estimates whose values cannot be explained with theory. These parameters have a great impact on the stabil-ity and are indispensable for a proper description of the motions.
The satisfactory correspondence that was obtained between experiments and an adjusted model justifies the conclusion that a model with adjusted parameter values can be used reliably for the design of new vehicles.
LIST OF SYMBOLS
(The numbers in parentheses refer to the section where the quantity is defined or first introduced.)
a half wheelbase (3.1)
a, coefficient of kth order derivative (6.1) k
b half mean gauge (3.1)
(in 4.3) vector of process parameters with elements b b half distance between longitudinal springs (3.1)
c
b, coefficient of kth order derivative (6.1) k
c longitudinal suspension stiffness (3.1) c lateral suspension stiffness (3.1)
y
e error (4.2)
e* error of adjusted model (4.3) g acceleration of gravity (5.3)
(in 4.3) functional relationship for process and model g gauge (7.4)
w
e . gauge at wheelset i (5.3) wi
i suffix for wheelset, i = 1: leading i = 2: trailing j suffix for rail ,. j = 1 : right hand
j = 2: left hand k damping factor of hunting motion (7.3)
k lateral damping coefficient between bogie and body (3.1) k, rotational damping coefficient between bogie and body (3.1) m mass of wheelset (3.1)
a
m^ mass of bogie frame (3.1) n additive noise (4.2)
r mean rolling radius in wheel-rail contact point (3.1) r.. variable rolling radius (3.2)
s complex frequency t time
t time (4.2) o
u model input signal (4.2) u model input signal (6.2)
u additional model input signal (6.2) u, . additional model input signal (6.2)
ki
w model output signal (4.2) x measured signal (6.1)
x (in 6.2) output signal of non-linear model y lateral co-ordinate of bogie frame (3.1) y* reference for y (7.4)
y. lateral co-ordinate of wheelset i (3.1)
y. , lateral co-ordinate of wheelset i in model (6.2) i,mod
y lateral rail centre-line deviation (3.2) w
y . y at wheelset i (3.2) wi w
z process output signal (4.2)
E expectation operator (5.5) F non-linear function (7.5) F external lateral force (7.6) G{ } power spectrum (7.1)
G(s) transfer function (6.1)
G (s) transfer function for time delay (6.3)
G (s) transfer function for prediction (6.3) H(s) transfer function of filter (6.1)
I moment of inertia of wheelset around lateral axis (3.1) ya _
I moment of inertia of wheelset around vertical axis (3.1) za
1 , moment of inertia of bogie frame around vertical axis (3.1) zb
M external torque (7.6) z
N normal force between wheel and rail (3.2) Q external lateral force on bogie frame (3.1) Q . lateral wheel-rail force on wheelset i (3.1) Q external moment on bogie frame (3.1)
Q,. wheel-rail moment on wheelset i (3.1)
ill
T averaging time (4.2)
T .. longitudinal tangential wheel-rail force (3.2)
xij
T .. lateral tangential wheel-rail force (3.2)
yij
V speed (3.1)
X. . longitudinal wheel-rail force (3.2) Y. . lateral wheel-rail force (3.2) a coefficient with local meaning a parameter in simplified model (7.3) 2 damping factor in filter (6.1)
(in 4.3) vector of model parameters with elements 6, k (in 7.5) statistically linearized coefficient of n o n -linearity in contact angles
3, parameter in simplified model (7.3)
3* improved estimate of parameter (4.3) Y.. angle of wheel-rail contact (3.2)
Ö parameter for sum of left and right hand contact angles (7.3) 6. 6 at wheelset i (3.2)
1
e parameter for gravitational stiffness effect (7.3)
e. E at wheelset i (3.2)
1
e parameter in non-linearity with broken lines (5.5)
'-7
K spin creep coefficient (3.2)
K longitudinal creep coefficient (7.3) K . K at wheelset i (3.2)
XI X
K lateral creep coefficient (7.3) K . K at wheelset i (3.2)
yi y
X equivalent conicity (7.3)
X . X at wheelset i (3.2)
A coefficient of polynomial approximation (5.5)
X coefficient of gauge (5.5)
u friction coefficient (3.2)
\i, moment of order k (5.5)
k
p correlation coefficient (4.3)
a coefficient for coupling between the roll angle and the
lateral displacement of a wheelset (7.3)
a. a at wheelset i (3.1)
T time constant for delay (6.3) T time constant for prediction (6.3) u . . spin creep (3.2) nij u .. longitudinal creep (3.2) xij u .. lateral creep (3.2) yij
(J) rotation of bogie frame around longitudinal axis (5.3)
1
rotation of wheelset i around longitudinal axis (3.2) <(; c r o s s - l e v e l of r a i l s ( 3 . 2 )
w
d) . * at w h e e l s e t i ( 3 . 2 ) wi w
ij; rotational co-ordinate of bogie frame (3.1)
ip. rotational co-ordinate of wheelset i (3.1)
CO frequency in rad s~^
ÜJ eigenfrequency of filter (6.1) o
(in 7.3) eigenfrequency of simplified model
-r numbe-r fo-r damping (7.3)
Aw vector of differences between model outputs (4.3) Aw, element of Aw (4.3)
k
Ag vector of differences between parameters (4.3) Ag, element of A6 (4.3)
k
A6 difference in 6 between both wheelsets (7.3)
Ae difference in e between both wheelsets (7.3)
AK difference in ic between both wheelsets (7.3) x X
AK difference in tc between both wheelsets (7.3) y y
AA difference in A between both wheelsets (7.3) Ao difference in a between both wheelsets ( 7 . 3 ) .
Chapter 1
INTRODUCTION
The interaction between railway vehicles and track has interested many in-vestigators. A complete vehicle, with several degrees of freedom, both in the horizontal and in the vertical direction, was analyzed by de Pater (1963). However, most studies that have been made are limited to the vertical, the
lateral or the longitudinal parasitic motions.
Generally only a weak coupling exists between vertical and horizontal
vibrations. With some precautions the lateral motions can be studied separately. The main problems are related to these lateral motions: the adequate description of the wheel-rail forces, and the introduction of the flexibility of the rails in the theory. The problem of rail flexibility can be evaded by studying the motions of a vehicle in the centre of a train. Then it may be assumed that the
track is deformed by the leading vehicles of the train and will move hardly during the passage of the vehicle which is used for experiments. In a mathemat-ical model the track can thus be characterized as a stochastic input signal, without interaction with the responses of the vehicle.
The dynamic behaviour of a railway vehicle on a straight track is dominated by parasitic hunting motions in lateral direction. This oscillation is a major limitation to satisfactory running of railway vehicles. Hunting, a coupled translation and yaw rotation of wheelsets and bogie frame, is largely due to the combined action of creep forces and the mutual geometry of wheel and rail profiles. A proper description of this hunting motion corresponds to a satisfy-ing description of the motions of a vehicle, huntsatisfy-ing besatisfy-ing the dominant mode. However, comparisons between theoretical and experimental results have not been reported frequently, as far as lateral motions are concerned. This is not sur-prising, because quantitatively the correspondence is poor. Measured power
spectra of vehicle motions in vertical direction can be predicted very well with the theory available. But an ORE research (1972) of the power spectra of the motions in lateral direction showed important differences between measured and computed spectra.
This thesis reports a study of the lateral parasitic motions of a bogie with six degrees of freedom on a straight irregular track. The difference between theory and experiment is investigated. A clear comprehension of the
lateral motions is essential for a realistic use of the theory in designing advanced railway vehicles. The theoretical knowledge is still insufficient to rely upon it completely.
The motions of a wheelset in the lateral direction are limited by the
-wheel flanges. But for satisfactory behaviour, it is favourable if the guidance along the track in normal running conditions is not achieved by the flanges.
Wickens (1969) suggests that the guidance should be offered by the wheel conicity, so that the displacements of the wheelsets normally remain within the flangeway clearances, without flange contact. If a wheelset is rolling along the track and is disturbed from the central position, there is a difference in rolling radii between each wheel due to the conicity. As the wheels of one wheelset are fixed
to their axle, the wheels cover different distances for one rotation of that axle. This causes a lateral velocity of the wheelset in the direction of the central position on the rails. The wheelset returns, but the lateral velocity is not yet vanished in the central position. So the wheelset goes to the other side and the same mechanism causes a velocity in the opposite direction. Based on a kinematic description of this phenomenon with pure rolling. Klingel (1883) derived a well-known formula for the fundamental wavelength of this hunting motion. This wavelength is inversely proportional to the square root of the conicity. To describe the behaviour of a bogie with two wheelsets creep has to be taken into account. Creep arises in rolling contact and is intermediate between pure rolling and pure sliding. A description including creep gave rise
to multiplicative correction factors in the formula of Klingel. For not purely conical wheels an equivalent conicity has been defined. Further refinements have been made to the equations of motion, including inertial terms, spring and damper
forces and the gravitational stiffness effect. But the hunting motion remained the most important mode of vibration. Law and Cooperrider (1974) give a survey of recent developments in the research of railway vehicle dynamics.
For small displacements and low running speed the conicity offers the
guidance of the wheelset within its clearances. On the other hand, at high speeds the conicity of the wheels causes dynamic instability. See Wickens (1969). A satisfying response of the system to irregularities in the track requires a high conicity. But the conicity should be low to have satisfactory dynamic stability at high speeds. The conflicting requirements for guidance and stability are the fundamental problems in railway vehicle design. It is clear that the "equivalent" conicity has to be determined properly. Stassen (1967, p. 62) proposed statistical linearisation over the range of actual displacements to obtain a value for the equivalent conicity. The applicability of this method to non-linear wheel and rail profiles is investigated in this thesis.
Two different approaches are possible to study the motions of a railway vehicle. In the first approach the system of differential equations is derived for a vehicle on perfectly straight track; in the second approach the inevitable irregularities in the actual rail position are taken into account. In the case
without inperfections in the rail, the system of equations is linearised and the stability of the linearised differential equations is investigated. The stability of this set of equations can be characterized by the critical speed, which is defined as the highest speed where all roots of the set of equations still have a positive damping. For a linear system, this is a necessary and sufficient con-dition for stability. However, the set of linear differential equations will certainly have lost its validity when the actual displacements of a vehicle become equal to the clearances between wheels and rails. The critical speed assures only that an initial disturbance will die out finally below that speed, but gives no information about the amplitude of the transient response. Conse-quently the critical speed cannot be sufficient for stability in practice.
A more refined study has been made by Keizer (1968), who computed the logarithmic decrements as a function of the speed. But unfortunately the logar-ithmic decrements do not provide sufficient information about the maximum dis-placements if the system has more than one degree of freedom.
Even if the transfer functions are computed as functions of the wavelength, knowledge of the power spectral densities of actual track irregularities is necessary in order to determine the magnitude of actual displacements. All theor-etical studies with perfectly straight track can at best provide a means for mutual comparison of sets of differential equations with slightly different para-meter values. For larger parapara-meter variations a comparison must always include realistic estimations of the actual displacements. For sufficient conditions for stability the irregularities of the rail have to be taken into account.
The random nature of the irregularities of the track leads to a description with random processes for the displacements of a vehicle. For the lateral motions,
the contribution of Stassen (1967) was the first investigation in this field. He proved that the combination of track irregularities and wheel profiles is the dominant feature for a description of the actual displacements of a bogie. Qualitatively his description was correct, but nevertheless significant differ-ences remained between experimental and theoretical displacements. Stassen carried out his experiments in conjunction with the committee of experts B52 of the ORE
(Office for Research and Experiments) of the UIC (International Union of Railways). The experimental part of this thesis is a continuation of the co-operation with B52. The interaction between vehicles and track is also investigated by committee C116 of the ORE; see ORE (1972, 1974).
Much effort has been spent in the derivation of the differential equations describing the motions of railway vehicles, both in mathematically oriented studies and from a physical point of view. Independently numerous experiments have been carried out in parallel. Unfortunately, only a few quantitative
-parisons have been reported for lateral motions, mostly with unsatisfying results. The parameter values in the differential equations usually have been chosen according to theoretical derivations. But for several parameters the theoretical values have not yet been supported by experimental evidence for vehicles running along the track. This is certainly the case for the equivalent conicity and the gravitational stiffness; Hobbs (1967) gave a survey of experi-ments that have been carried out on creep, but these experiexperi-ments were not in normal operating conditions. The poor correspondence between theory and experi-ments may be due to incomplete terms in the differential equations or to wrong values of parameters in the hitherto used descriptions. Both possibilities are investigated in this thesis.
With a known structure of the mathematical model, a proper parameter esti-mation technique will yield parameter values for an optimal correspondence
be-tween theory and experiments. By adjusting the parameters in an analog model, the difference between the theoretically expected and the experimentally determined displacements of a bogie can be minimized. The estimates of the parameters can also be used as experimental evidence for a description of the creep forces, the equivalent conicity and the gravitational stiffness effect. The correspondence between theory and experiments is an indication for the accuracy of the model description of the motions.
The outline of this thesis is as follows. In the present introduction the importance of the study of the random lateral motions is indicated. For compari-sons between theory and experiments, the definition of a criterion of equivalence is required. In chapter 2 the stability of railway vehicles is investigated to find a correct basis for this criterion. The phylosophy behind some criticism about such a stability has been discussed by Broersen (1974). The differential equations for a bogie with six degrees of freedom are derived in chapter 3. Chapter 4 is devoted to a description of the chosen model adjustment technique.
In chapter 5 the bogie and the experiments are described. Then follows the measurement of the rail position in chapter 6. Here also the equivalence of a non-linear system and a linear system with additional inputs is discussed. Finally, details are given about the implementation on a hybrid computer.
A railway vehicle has a non-linear response to the track. Therefore the dynamics can only be given in reference to specific input signals. Some spectra of the rail input are given in chapter 7. Furthermore the results enclose the parameter values giving optimal correspondence between theory and experiments, as well as an indication about the goodness of fit. Some of these results have already been published; see Broersen (1973, 1975). Finally, a recommendation is given for both terms necessary for the mathematical model and for the parameter
Chapter 2
STABILITY AND RIDING QUALITY 2.1. Stability and critical speed
It is easily accepted that every railway vehicle should be stable. But this consensus keeps concealed the difference that may exist between the concepts of stability in theory and in practice.
In theoretical investigations, the stability is often determined as a prop-erty of a railway vehicle; it is then considered independent of the irregular position of the track. A set of linearised differential equations is derived for small displacements on a perfectly straight track. The masses of the vehicle represent an increasing destabilizing factor at higher speeds; the damping of the fundamental hunting motion decreases. The stability is determined by the roots of the characteristic equation of the set of differential equations, and it depends on the running speed. The critical speed is defined as the maximum value of the speed where all roots still have a negative real part. So at the critical speed the real parts of one or more roots become zero and above this speed at least one root has a positive real part. Generally, this root will belong to the hunting oscillation. Below the critical speed the solution of the set of autonomous dif-ferential equations is stable. Consequently the railway vehicle is considered to be stable, which means that the vehicle finally will return to its equilibrium position after an initial disturbance. This is a necessary and sufficient condi-tion for the stability of a linear system. But the differential equacondi-tions of a railway vehicle are certainly non-linear for displacements of the order of the clearances between wheel flanges and rails. These clearances impose an upper limit on the displacements. So it must be certain that the displacements remain smaller than their limits. The critical speed, however, gives no information about the magnitude of the displacements, although it may be expected that they will become extremely large in the neighbourhood of the critical speed. Due to
the lack of information about the amplitudes, the critical speed cannot provide a sufficient condition for stability. For the same reason investigations of the autonomous system with perfectly straight track can never provide a sufficient condition for stability in practice. This includes the computation of logarithmic decrements, transfer functions and non-linear extensions of the differential equations.
To obtain a sufficient condition for stability the irregularities of the rail must be taken into account. Only then it is possible to verify whether the displacements remain within the clearance. So it is impossible to attribute the qualification stability to a vehicle. Stability is determined by the interaction
-of a vehicle and a track. This follows also from the experience that certain vehicles ride well on some networks, but not on others. A vehicle can even have a stable behaviour on one part, and an unstable on another part of the same track.
Equations, linearised for perfectly straight track, are valid for small lateral displacements. But the actual displacements of a vehicle are never small because of the response to rail irregularities. Even for a well maintained track
the actual displacements will increase to critical values if the speed of a vehicle approaches the limit of stability. Then these linearised differential equations are no longer valid; consequently they are not suited for the purpose of analysis of stability.
The behaviour of a vehicle becomes essentially non-linear by wheel-rail con-tact near the flanges. To arrive at linearised equations in this case, the large displacements with flange contact must be taken into account. A basic difference exists between the stability of linear and non-linear systems. La Salle and Lefschetz (1961, p. 57) state that in determining stability in practice, linear approximations are definitely unsatisfactory. In other words, to determine
stability in practice one has to examine non-linear responses for the actual displacements. Only if these displacements can be described with an equivalent linear system, it is possible that the stability of that linear system is a reliable guide for the stability in practice of the real system. Generally, the stability of a non-linear system does not depend on the stability of the
linearised system, but on the nature of the non-linearity. If the non-linearity has a destabilizing effect, the stability of the system, linearised for small displacements, is a necessary condition for stability in practice, but it is never sufficient. In the case of railway vehicles, the non-linearity has a stabilizing effect, so that the stability of the linear system with small dis-placements can predict nothing but that flange contact probably will occur. In practice there will be flange contact for at least some points of a track, other-wise the track would be maintained in a too perfect state, what is very expens-ive. So flange contact will occur before the behaviour of a vehicle becomes unstable, and it has to be incorporated in an analysis of stability.
Stability of the linearised system for small displacements is not always a necessary condition for stability in practice. The desired state of a system may be mathematically unstable, but yet its performance may be acceptable for a criterion of practical stability if the system oscillates sufficiently near this state. An example for railway vehicles is a stable limit-cycle with displace-ments smaller than the clearances between rails and wheel flanges. Such a limit-cycle is possible if a free play exists in the lateral suspension between bogie frame and axle boxes.
The easy computation of the critical speed is an important advantage of this concept of stability. In theoretical studies the critical speed can be used for a mutual comparison of slightly different values of parameters in linearised
equa-tions. In this way it is possible to optimize suspension stiffnesses. But if the optimization leads to larger parameter variations, it becomes disputable whether the mutual comparisons via the critical speed are reliable, unless these vari-ations would have a small influence on actual displacements. In general there is no guarantee that a vehicle with a very high critical speed for small displace-ments would have an acceptable behaviour on a real track. On the other hand, a vehicle with a much lower critical speed can behave properly in practice, also above its critical speed.
In practice it is impossible to measure one single value for the critical speed of a vehicle, which can be compared with the theoretical value on perfectly straight track. A vehicle will produce different critical speeds on different tracks, and even have a partly stable and partly unstable behaviour on parts of the track with seemingly the same statistical properties. A practical criterion that a vehicle has reached its critical speed can be that the level of lateral accelerations on a wheelset suddenly changes to a level two or three times higher. The resulting motions must then have a pattern with a regular frequency in agreement with the hunting wavelength. But it is difficult to distinguish between unstable motions, and stable motions in responses to irregularities in the track for which the vehicle is sensitive. For instance, the resulting motions show dominantly the hunting wavelength if the rail irregularities have a pattern that excites the hunting oscillation. Nevertheless these motions can be described with a stable linear system.
We conclude that the critical speed and all other results of a study of the motions of the autonomous system on perfectly straight track provide neither a
sufficient nor a necessary condition for stability in practice. The interaction between vehicle and track must be taken into account.
2.2. Stability in practice
A theoretical concept of 'practical stability' exists. It is an extension of Liapunov's definition; see La Salle and Lefschetz (1961). This definition gives the maximum values of the initial disturbance and of the perturbing forces during all the time, which guarantee that the displacements will never exceed a given maximum. This maximum is given by the clearances between wheel flanges and rails for a railway vehicle. The track irregularities can be seen as the pertur-bing forces. The limits to be imposed on the displacements, however, are not
-absolute but relative to the irregular rail position. Therefore, this definition cannot be applied rigorously to the stability of railway vehicles; it will be difficult to obtain a useful description of stability from a mathematical point of view.
In practice, however, there is empirical know-how about the desired behav-iour of a railway vehicle and about its 'stability'. But these experiences are not often reported, unless in discussions during symposia, e.g. Wickens
(1965-1966, p. 29-44, 71 and 149-151). But the coupling is weak between the experi-ences in practice and the theoretical definition of stability. Yet the demands derived from a practical point of view can be quantified with a theoretical investigation. For the lateral motions the demands are distinguished into two aspects: passenger comfort and vehicle safety.
Firstly, the level of the accelerations to which the pay load is subjected may not exceed a maximum. This imposes limits on the accelerations on the place where they are transmitted to the pay load, e.g. on the seat of a passenger.
Secondly, demands exist for the area of contact between wheels and rails, related to forces and guidance along the track. This latter group of demands contains a number of effects:
- danger of derailment - damage to the track - large lateral forces
- fatigue failures of the vehicle structure - wear of rails and wheel flanges
- noise level from the contact area
- tractive power lost in these detrimental effects.
Regarding all these effects it is favourable if the lateral forces between wheels and rails are small. Peterson (1972) found these forces by measuring bending moments in a wheelset, accelerations, spring forces and so on. The lateral wheel-rail forces can be derived from the measured signals, but quite a lot of work must be done to have the forces available continuously. However, only high values of the forces are interesting here, because with small forces no problems with stability can arise from the area of contact. Large forces can only occur if the contact area between wheel and rail is situated near the flange of a wheel. Otherwise the wheel would immediately slide over the rail until there is flange contact. So flange contact can be used as an indication for undesirable behaviour of a vehicle. A definition of flange contact still must be given. If a wheelset is in its central position on a track, wheel-rail contact will generally take place on the flat, more or less conical part of the wheel profile, and on top of the rail. The area of contact shifts gradually with lateral wheelset
displace-ments. For one specific displacement, contact can take place in two points, one on the flat part of the wheel profile and one on the flange. The contact area will move discontinuously over the wheel and rail surfaces around this two-point contact, which separates flange contact from contact in the flat part. A combina-tion of a wheel and a rail profile sometimes does not give a discontinuity in the position of the contact area. Then, by an arbitrary definition, flange contact takes place if the angle of contact is greater than 45 . Finally, a third indica-tion for large wheel-rail forces can be derived from wheelset acceleraindica-tions. The lateral accelerations can be measured easily by mounting an accelerometer some-where on a wheelset. These accelerations, however, depend not only on wheel-rail
forces, but also on suspension forces. The suspension forces depend on transfer functions within the vehicle structure. In the frequency range of the eigenfre-quencies of the construction some filtering can be applied. The resulting filtered accelerations can contain sufficient information for the purpose of determining the stability. Recapitulating, we have three methods to find large lateral forces between wheel and rail:
- continuous measurement of wheel-rail forces - the occurrence of flange contact
- (filtered) wheelset accelerations.
As a result of practical considerations, we define a railway vehicle to be stable if:
- the level of the accelerations to which the pay load is subjected remains below some critical value
- the lateral forces between wheel and rail do not exceed an allowable maximum. So these accelerations together with the lateral forces determine the limit of stability. Both accelerations and wheel-rail forces will depend on the speed. The speed for which the limit of stability is reached will be the critical speed in practice. The level of accelerations and the level of wheel-rail forces deter-mine the riding quality. This riding quality will be our quantitative measure for stability, below the critical speed. The riding quality, depending on the inter-action between vehicle and track, determines which vehicle-track combination is very good, good, satisfactory, just admissible, not admissible or dangerous for a certain speed. On the other hand, given the vehicle-track combination, it is possible to determine the maximum speed for which the riding quality is just admissible: the critical speed in practice, as well as ranges for the speed where the riding quality is very good, satisfactory etc.
The riding quality is essentially based on two different parts: the wheel-rail forces and the accelerations transmitted to the pay load. These forces and accelerations have to be considered separately, because they act on different
-places. Moreover, every frequency is equally important for wheel-rail forces, but for accelerations filtering must be carried out according to human sensi-tivity.
One vehicle-track combination is better than another if its riding quality is better for a given speed. But the riding quality is not necessarily improved if the motions of a wheelset or the irregularities in the track are diminished. It is very well possible that the motion of a wheelset matches the irregularities. Generally, wear favours this situation. If a wheelset follows the rail irregular-ities, the resulting wheel-rail forces will be small, although both wheel motions and rail irregularities separately are not small. Therefore, the combination of vehicle and track determines the riding qualityl The riding quality can be used for a comparison of different vehicles on the same track. This conclusion was also drawn by Peterson (1972), as a result of experimental evidence for wheel-rail forces.
From a practical point of view, the riding quality contains all essential information about the stability of a railway vehicle on straight track. Also the desired quantitative aspects of stability can be included in the riding quality. Details about the assessment will be discussed in the following section. The
riding quality can be measured in practice and computed in theory with an adequate mathematical model. In contradistinction to the mathematically derived stability
concepts like the critical speed of a vehicle, the riding quality agrees with practical experiences. It provides a useful criterion in designing new vehicles which will satisfactory travel at high speed over existing tracks. This conclusion agrees with one of the design cases used by the British Railways. See Wickens, Gilchrist and Hobbs (1969-70).
2.3. Assessment of the riding quality
The definition of the riding quality was rather broad. It has been pointed out that different practical demands for the behaviour of a railway vehicle can be put together into a framework of the riding quality. However, at least two separate parts are necessary to determine the riding quality: wheel-rail forces and accelerations on the pay load. Of course the riding quality can be extended with more aspects. This might be useful if the forces or the accelerations in construction elements of a vehicle can become critical (e.g. automatic coupling of vehicles). Experience will show whether a definition of the riding quality guarantees a satisfactory behaviour of a vehicle. Otherwise improvements will be recommended by experience, because the practice remains the basis for the riding quality. In this thesis only proposals on possible methods for assessing the riding quality are given. A complete evaluation of these methods is beyond the
scope of this thesis.
It is clear that limits exist for the accelerations to which the pay load can be subjected. Goods may not be damaged, but generally no problems will arise because breakable goods will be well packed to withstand loading and unloading. A maximum shock value for the accelerations or a level for the variance of the accelerations can be determined, depending on the type of goods. In every case an upper limit to the accelerations is given by the fact that the vehicle itself may not be damaged. The accelerations must be measured or calculated on the place where they are transmitted to the goods, e.g. the floor of a vehicle.
The criteria are more complex for the transport of persons. It is important that the level of vibrations experienced by a traveller will not discourage him to buy tickets in future. So the acceptance of vibration effects by the passenger must be investigated. The International Organization for Standardization ISO (1972) gives limits for the duration of vibration, depending on frequency, amplitude and direction of accelerations. These limits are different for preserving passenger comfort, working efficiency and finally health. The evaluation of the acceler-ations between 1 and 80 Hz is straightforward with the ISO recommendacceler-ations. The greatest sensitivity of the human body is between 1 and 2 Hz for horizontal
vibrations, and between 4 and 8 Hz for vertical vibrations. It is emphasized that the accelerations shall be measured as close as possible to the point through which the vibration is transmitted to the body. For example, if the passenger is
standing on a floor, the measuring transducer can be fastened to the structure of the coach. Where some resilient element, such as a seat cushion, does exist be-tween the body and the vibrating structure, it is permissible to interpose a measuring transducer between passenger and cushion. For random vibrations the root-mean-square value is to be evaluated separately for each third-octave frequency band, or the accelerations can be passed through a filter, inversely proportional to the human sensitivity. The weighted vibration levels are to be compared to the frequency range with the highest human sensitivity . The ISO curves give a relation between the acceleration level and the duration until the reduced comfort boundary is reached; the lower the acceleration level the longer the duration. This duration is the quantity to be used for assessing the riding quality, with dimension: time.
The cushion is generally a non-linear element in the transfer of acceler-ations from vehicle structure to human body. Therefore it is required to measure accelerations on the cushion, unless one is prepared to apply a non-linear
analysis for the cushion. This part of the riding quality, related to passenger comfort, can be improved in two ways: a lower level of weighted accelerations in the vehicle structure or a lower transfer of vibrations through the cushions.
-Another part of the riding quality must be derived from the wheel-rail con-tacts, where the forces may not be too large. These forces are relatively small if there is no flange contact. But it is impossible to prevent flange contact at all times, because of the random nature of the motions. So during a fraction of
the time flange contact must be expected. The smaller this fraction is, the better the behaviour of a vehicle on that track. Broersen (1974) showed that determining a time fraction of flange contact can be replaced by the measurement of the stan-dard deviation of the relative displacements between wheel and rail.
The time fraction of flange contact determines the second part of the riding quality. Only the time fraction is known during which the wheel-rail forces can be large; the precise magnitude of the forces is unknown. If the time fraction of flange contact is small, however, the maximum values of the forces will also remain moderate. On the other hand, very large lateral forces will cause flange climbing: the wheel has to climb upwards if the wheel-rail distance diminishes when the contact is situated already on the wheel flange. If desired, very large forces can be detected separately by the occurrence of flange climbing.
Two other methods have been indicated in section 2.2, to find large lateral wheel-rail forces: continuous measurement of forces or filtered wheelset acceler-ations. It is then possible to derive the time fraction that the lateral wheel-rail forces exceed a prescribed value. This time fraction might replace the pre-vious fraction of flange contact in the second part of the riding quality.
The two parts of the riding quality have been discussed. In the first part, the vibrations on the human body are filtered according to the physiological sensitivity and yield a duration for passenger comfort. The second part contains information about the forces in the most critical points of the vehicle structure; the contacts between wheel and rail. This part is expressed as a time fraction of flange contact. So both parts have the dimension: time. The riding quality is a weighted sum of the two parts, but the weighting factors can depend on the use of a train. For instance, the duration for passenger comfort shall be more important for long distance non-stop trains than for local networks.
In experiments, the quantities required for the assessment of the riding quality can be measured. In theoretical investigations, the response to the input track irregularities can be computed with an adequate mathematical model of a vehicle. The accelerations in the vehicle structure follow directly from the computed response, just as the wheel-rail displacements and the fraction of flange contact. It is advisable to replace the non-linear cushion by some equiv-alent linear transfer function. Then filtering can be applied according to the human sensitivity to find the duration. So both parts of the riding quality can also be evaluated in theory.
A criterion in the design of new vehicles should be that they have a good riding quality. This can be computed with an adequate mathematical model. So the parameters in this model (e.g. suspension stiffness, dampers, cushion characteris-tics) can be optimized with respect to the riding quality. This optimization applies to the behaviour of a new railway vehicle on existing straight track, at a constant design speed.
An important conclusion can be drawn for the sequel of this thesis. A mathe-matical model for the interaction between vehicle and track is a good model if the riding quality can be predicted with that model. A model is not particularly good because it has many degrees of freedom or many parameters, or because it includes non-linearities; only its capability to compute the riding quality deter-mines the reliability of a mathematical model.
2.4. Maintenance of the track
We saw in section 2.2 that the riding quality depends on the vehicle, the track and the speed. It was concluded that the riding quality contains the essen-tial information about the behaviour of a vehicle. The implications for the track are investigated here.
The type of maintenance meant here is the intermediate removal of
misalignments in a track that not yet needed maintenance for reasons of aging. This maintenance is usually based on periodical inspections with a rail measuring coach. The existing measuring machines have a poor transfer function; see Stassen (1967, p. 28). Moreover, they measure only partially the relevant rail quantities; for instance, the rolling line offset (see section 5.4) is not measured. A scheme of maintenance is based on this partial information about the irregular track. However, even if all relevant rail quantities could be measured, conclusions about maintenance cannot be independent of vehicles. Given the fact that the behaviour of a railway vehicle depends on the combination track-vehicle, it is curious that the maintenance of a track is independent of the vehicles which normally use that track. This is a consequence of the view that the track should be as straight as (economically) possible and the speed should be below the critical speed, calculated for a linearised system on perfectly straight track. In this view, track and vehicle are unjustly considered separately.
From the riding quality it can be derived that the quality of a particular track depends on the wheel-rail forces and on the accelerations transmitted to the pay load for vehicles at normal operating speeds. So track is to be classi-fied by the responses of vehicles, possibly of standard vehicles. These measured responses also include the effects of parts of the track position which them-selves cannot yet be measured adequately (e.g. rolling line offset, track
-bility, influence of subsoil), as well as possibly unknown interactions between different track irregularities. Broersen (1974) indicated that a standard vehicle for track inspection can exist because of the similarity of the dominating hunting motion for different vehicles on the same track.
The aim of track maintenance should not be: putting the rails as straight as economically possible, but maximizing the riding quality of the vehicles which use the track. This does not necessarily coincide with a track as straight as possible.
Both a vehicle and a track are judged by the same riding quality. A vehicle design can be optimized by its response to a standard track; the condition of a track can be assessed by the response of a standard vehicle. This agrees with the knowledge that the interaction between vehicle and track is important.
Results based on investigations of either track or vehicle separately can easily lead to wrong conclusions. The interaction is investigated further on in this thesis.
Chapter 3
DERIVATION OF THE EQUATIONS OF MOTION OF A BOGIE 3.1. Basic equations
Much effort has been spent by many investigators in deriving the differential equations governing the motions of railway vehicles. A theoretical analysis of the kinematic oscillation of a slowly rolling single wheelset with conical wheels has already been given by Klingel (1883). Many contributions followed about the in-fluence of creep, profiled wheels and non-linear aspects and also discussions about stability appeared. De Pater (1963) has given a mathematically oriented study including all free co-ordinates that are possible for rigid bodies in a vehicle. Nowadays the investigations about a mathematical model for the inter-action between vehicle and track are coordinated by committee CI 16 of ORE. One report of this committee, ORE (1974), presents results which were mainly obtained by the Research Department of British Railways. Joly (1974) has published about stability studies with a mathematical model for the French Railways (SNCF).
The following derivation of the equations of motion of a bogie will assume small amplitude motion about the mean position of any mass element, so that most coefficients can be taken linear. We will assume that the structural elements of the bogie are rigid, and joined together by frictionless suspension components. This bogie, resting in contact with the rails on 4 points, has 14 degrees of freedom: 6 for the bogie frame, 6 for each of the two wheelsets minus 2 contact relations for each wheelset. Only the lateral vibrations are studied in this thesis. These motions are anti-symmetric with respect to the longitudinal plane of symmetry. The roll angle of the bogie frame around the longitudinal axis is not coupled with the other lateral motions by the special choice of the centre of gravity of the bogie frame on the level of the axles. ORE (1971) gives a description of this bogie. The bogie can be described laterally with 6 free co-ordinates: lateral displacement and rotation of frame and wheelsets, see figure 3.1.
-- • V
2b
2a
FIGURE 3.2.
Lateral motion ao-ordinates of a bogie.
The equations of motion for the bogie with constant speed are;
m^y + k ; + 2c (2y-yj-y2) = Q
I 'I) + k i + 2c b2(2i{;-i|; -i|;„) + 2c a(2a<J;-y +y„)
Z b i\) X c 1 2 y 1 z I ^Va V l ^ 2Cy(yj-y-a^) - - ^ ^ ^^ = Q^j I l^ + 2c b2(^ -^) + y^^ y, = Q, za 1 X c 1 br M ^I|J1 I Vo^ V 2 ^ 2Cy(y2-y-Ha^) - ^ ^ ^^ = %2 I Va I l^ + 2c b2(^ -^) + yf-^ y^ = Q^, za 2 X c 2 br '2 ^iil (3.1)
In the left-hand sides the inertia terms, the spring forces and the gyroscopic forces can be recognized; see ORE (1974). The wheel-rail forces constitute the right-hand sides of the wheelset equations. On the bogie frame also forces are acting because of the connection between frame and the coach above, although the construction has been designed to have a negligible interaction. These forces are non-linear functions of the differences in velocity between bogie frame and coach. But the equations of motion of the coach are left out of the eqs. (3.1).
So these forces appear as external forces: Q -k y and Q -k ]J. The separated
vis-y vis-y 'v V
cous damping terms are written on the left-hand side. They cause no loss in generality, but can give computational advantages.
3.2. Wheel-rail contact forces
The vertical motions are excluded from (3.1) to keep only a small number of degrees of freedom. Moreover, the coupling between vertical and lateral motions is weak. But the variations of the vertical forces cannot be calculated comple-tely with a model for only lateral motions. Therefore, the vertical force per wheel, or better: the force normal to the wheel-rail contact area, is taken constant as one quarter of the total vertical load. The lateral forces, acting on the wheelset in the contact area, are given by:
Q . = - Y., - Y.„ yi il i2 Q,. = b(X. -X. ) + b^.(Y -Y. ) 1^1 il i2 1 il i2 (3.2) X., =T .„ i2 xi2 FIGURE 3.2.
Contact forces acting on the wheels.
These forces can be expressed in the tangential forces with the relationships:
X. . T . .
XIJ
Y. . = (-l)-^'^'NsinY. . + T ..cosy
(3.3)
LJ ^J y i j ij
Kalker (1967) gives expressions for tangential forces in his theory of rolling contact. When one elastic body is rolling over another, small deviations from the pure rolling motion set up forces acting in the area of contact between the
-two bodies. Conversely, externally applied forces which are small in comparison with the limiting dry friction values uN, cause small deviations from the steady
rolling motion. The differences in the velocities of wheel and rail in the con-tact area, divided by the forward speed of the vehicle, are denoted as the creep u. Likewise, the relative angular velocity about an axis normal to the contact plane, divided by the forward speed, is denoted as the spin u . So creep is the
n
phenomenon intermediate between pure rolling and gross sliding. For small creep and spin, Kalker (1967) gives a linear relation between creep, spin, and the transmitted force. Experiments also indicate a linear relation, but the measured creep is always larger than predicted with the theory; see Hobbs (1967). This may be caused by a high surface pressure in the area of contact, by the roughness of the surfaces or by contamination or lubrication of the surfaces. Moreover, the physical background of pure elastic virginal bodies may be poor for wheel-rail contacts.
In this thesis linear relations will be assumed between creep and trans-mitted force. The coefficients of these relations are a subject of investigation.
Before embarking upon the details of the derivation, some general remarks must be made. Committee CI 16 of ORE (1974) has given a more detailed derivation along the same lines. Differences are discussed in section 3.4, as well as the reason why some apparently small terms will not be neglected. The equations are set up to give a maximum flexibility in the model to be investigated. For that reason different parameters will be introduced sometimes for the leading and the trailing wheelset. These differences can be motivated by slightly different wheel profiles and by different standard deviations of the motions of both wheelsets. It is investigated whether these differences yield an improvement.
Expressed in creep and spin, the tangential forces become:
T . . = K . Nu . .
xij XI xij
(3.4) T . . = K .Nu . . + < Nu ...
yij yi yij s nij
Creep and spin are determined completely by the velocity of the wheels in the contact area. The rails are assumed not to move in their deformed position. The deformation takes place by the leading vehicles, of the train before the test bogie arrives. Consequently the rail is fixed during the passage of this bogie and does not contribute to the creep. For small deviations from pure rolling, excluding flange contact or gross sliding, creep and spin are given by:
-r. .+r . bijj. u .. = ^ i — + (-l)J - i xij r V y r ^ u .. = -i - — 1 - ^. (3.5) yij V V 1 r^. U . . = (-l)-J Y; • - 't. + -TT- . nij Lj 1 V
where cosy.- is taken equal to 1, as will be done further. The quantities r.,,
(^. and Y-. can be expressed in the co-ordinates of the bogie and in the irregu-lar track position, which appears here in the equations of motion. The irreguirregu-lar track is characterized by y and i> ', y is the lateral centre-line deviation that
W W W
will be discussed in detail in section 5.4, and $ is the cross-level, the angle w
between the line, connecting the tops of both rails, and the horizontal plane. For simplicity of notation, the rail position at wheelset i will be denoted as y .. The rail position is only a function of the longitudinal co-ordinate, so
wi y^2(x^2a) = y^j(x) *w2(^^^^) = *wl<^>' ^'"'^ and dy . , dy . wi dx _ ^ wi wi dx dt dx d* . , d* . (3.7) wi dx dt dx
The mean value of the rolling radii of the wheel is r; r.. and Y-• are the
momentary values of rolling radius and contact angle for wheel ij. The formulae, required for a derivation of the wheel-rail forces are:
r. -r. = 2A. (y.-y .-rt}) .) il i2 1 1 wi wi
(3.8) Y-, + Y-T = 26.
il i2 1
Y. -Y.„ = -2<i) . + 2e.(y.-y .-rcj) . )/b il i2 wi 1 1 wi wi <i>. = 4) . - o . ( y . - y .-r(t! . ) / b .
1 Wl 1 1 wi wi
The parameters A., 6., e. and a. are defined by the linearised relations (3.8); A. is called the equivalent conicity and e. is the gravitational stiffness para-meter. The linear expressions (3.8) can only be valid for small displacements;
they are certainly not valid for displacements causing flange contact. Some non-linear models for A. and e. are given in section 5.5. The results in chapter 7 include an evaluation of both linear and non-linear models.
Combining (3.3), (3.4), (3.5) and (3.8) with (3.2), we arrive at the wheel rail forces: y. ra. 2 K Nr e. K (e.-a.) - 2N { ^ - ^ ?- M ( y - - y - - r * . ) + 2N()) . b b •^ 1 Wl Wl wi (3.9) 2K .Na.r 2 K .Nr ra. bV ^wi V ^ b '^ *wi A. Q = -2bNK {_L (y_-y _-j-4, .) + - ,J,.} + 2bN(l-< )6.J;.. i|ji XI r 1 Wl wi V 1-' s 1 1 3.3. Equations of motion
All terms required for the equations of motion of a bogie on an irregular track have been given. The external forces Q and Q on the bogie frame are
y V
caused by friction; the irregular rail position produces the exciting forces for the wheelsets. The normal forces N in the wheel-rail contact areas are assumed to be constant in the equations for the lateral motions. In a more extensive model, including vertical motions, constraints exist for the vertical displace-ment and the rotation of a wheelset around the longitudinal axis, because the wheelset must remain in contact with the rails. The normal forces can then be computed as the forces of constraint to maintain these contacts.
The equations of motion (3.1) become, with (3.9):
m^^y + k y + 2c (Zy-y^-y^) = Q (3.10)
I 'i, + k i + 2c b2(2i|j-i|; -*„) + 2c a(2a*-y +y-) = Q, zb p X c 1 / y 1 2 V
ra y.
V l ^ 2c^(yi-y-a^) + 2KyjN {(1+-Y-)
T " '^^
2 K Nr I o V e, K (e -a,) ^ , s va 1 V • J. OM r 1 s i l l £ K ( E -a ) 2N i-^ - ——'- —\ (y , + r* ,) + 2N(|) , ^ b b •' wl wl wl 2K ,Na.r 2K ,Nr ra, bV ^wl V ^ b '^ ^wl - ^za*l " 2c^b2(^j-^) . 2bNK^j (-1 y, - - ^ ) I Vo A . - ^ ^ Yl - 2bN(l-K^) 6,^j = 2bNK^^ - (y^j . r^^j)
ra y^
V 2 * 2c^(y2-y+a^) +
2^^^M(^\
^ ^ T " ^ 2 ^
2K Nr I a„V , G„ K {z-a^) + ( s _ _Y3.2_-^ ^ 2 N U - ^ 2 2 , ^ *- V br -^ ^2 ^ b b ^ ^2 2Nf-^ T — ^ (y Q-"^"* Q) ^ 2N<* b b ' w2 w2 w2 2K „Nö„r 2K „Nr ra„ + y2 2 . ^ _ j r 2 _ ^ ^ ^ _ • bV ^w2 V ^ b -^ *w2 - ^za*2 ^ 2c^b2(^^-^) . 2bNK^2(^ ^2 ^ ^ ^. ^
^ y^ - 2bN(l-K^)52^2 = 2bNK^2
-T
Kl^'Kl^-3.4. Discussion of the equations of motion
Several assumptions have been made during the derivation of (3.10). Some of these are required to derive equations at the present state of the knowledge about the wheel-rail interaction. Some are certainly not completely valid in practice because of the more than infinitesimal displacements caused by the
actual irregularities in the rail position. Most assumptions are discussed below. The coefficients in the equations of motion (3.10) are all taken to be
constant. This is reasonable for inertia terms and suspension stiffnesses. The creep coefficients, however, will vary with the shape of the contact area, which in turn depends on the lateral displacements. This causes variations along the track that are not included in the equations of motion. The geometrical contact parameters A., 5., c. and a. will also vary along the track. It is impossible to take these variations into account with (3.10). So for creep and geometrical parameters average values have to be used, that remain constant for at least a part of the track. The only modification is given by a difference of these para-meters between both wheelsets. Such a difference may be due to the different standard deviations of the displacements of both wheelsets.
The linear relation (3.4) between creep and force is only valid for small creep and spin rates. For larger creep this relation is non-linear. Moreover, the normal force N is taken constant in the derivation of the creep forces. But N will vary in practice both with vertical and with lateral motions. Finally,
the influence of the small creep moment is omitted in (3.4).
The only flexibility is assumed to be concentrated in the suspension spring elements. Torsion and bending of bogie frame and of wheelsets are neglected, so
the wheels are assumed to be rigidly connected by their axle. If the wheels were
-allowed to rotate slightly with respect to each other, the calculation of the kinematical creep quantities in (3.5) would be quite different. But axles are generally so stiff that this relative rotation can be omitted from the equations.
The rails are assumed not to move in their deformed position.during the passage of the bogie, although in practice some flexibility will be present in the track structure. In practice, the rail will therefore move somewhat due to the varying wheel-rail forces and the rail velocity will contribute to the creep. This effect has been omitted in the equations of motion. The other effect, a change in the rail position due to the rail flexibility, gives no problems be-cause the rail position can be measured near the wheels of both wheelsets during the passage of. the bogie.
The irregular rail position has been characterized by the lateral centre-line deviation of the track. At least two aspects have not been taken into account in (3.8): the variations of the gauge and the rolling line offset. Some attention is given to the introduction of gauge variations in section 7.5. The rolling line offset will be discussed in section 5.4.
Special consideration has been given to the derivation of terms containing (J), and (J) . in (3.8) and (3.9), because little unanimity exists among different
1 Wl
investigators. It looks suspicious that y . and <p .appear in the equations for the forces, as the track is assumed not to move. But these rail derivatives are used for an approximation for (j). in (3.5), which is found by using (3.8) for (j). . This is necessary because (J), itself is not included as one of the co-ordinates of the mathematical model.
Some small differences exist when the equations for the wheel-rail forces (3.9) are compared with the results of ORE (1974). In our equations, the creep moment has been omitted, giving less terms for Q,-» On the other hand, the
care-ful analysis of è., y. -y.^ and Y-I ^ Y - ^ yielded three extra terms: 2 K .Nr* ./V,
1 il i2 il I 2 yi wi
2Nó . and 2bN(l-K )<5.i^. respectively.
Wl s 1 1
Finally, it must be stressed that omission of terms remains dangerous at this stage. Omissions are only justified if they are based on their influence on the solution of the equations of motion. These solutions are still unknown; nevertheless some physical effects have been omitted and other small effects have been taken into account.
A motivation for omissions in this thesis is given by the fact that the mathematical model will be used in a model adjustment process. The solutions of
the differential equations, the model responses, will be compared with the actual bogie responses on an irregular track. Parameters as well as coefficient values will be determined for an optimal correspondence between measured and computed motions with model adjustment. Where can we expect that adjustment of parameters
may be required? The mathematical model describes a system with six degrees of freedom. The description of the bogie frame contains only inertia forces, spring forces, and external forces that have to be measured. This part of the model is completely known, and needs no experimental verification, just like the inertia and spring forces for both wheelsets. But the description of the wheel-rail forces is the part of the model that is not yet firmly based on experimental evidence, as far as the influence on the solutions is concerned. Apart from the rail input signal, a maximum of 4 independent parameters can describe all possible influ-ences in one wheelset equation of the model. The 4 parameters are the coefficients of y., y., il. and IJJ. in the equations for y. and ip.. Those coefficients may be unknown or incorrectly determined in the theoretical derivation of the wheel-rail forces. But in this thesis they are estimated with model adjustment. The demand for a model that is to be adjusted is that every influence which is not yet experimentally verified, be present in the model, with an independent parameter. For the 4 wheelset equations, 16 parameters are then required. According to the derivation in this chapter, based on the theoretical knowledge of the wheel-rail interaction, several coefficients in the equations of motion contain the same parameters. Table 3.1 gives a scheme of the distribution of the parameters among
the coefficients in the equations. In this scheme those parameters are given, which are related to the wheel-rail interaction, and also I , the parameter for
ya the gyroscopic effect.
e q u a t i o n f o r : ^1 •• ^2 c o e f f i c i e n t of: • ^1 Ky,, a j , N y a ' 1 ^ 2 %2' ° 2 ' ^
Sa'
"2
^1 e , , a j . K^, N^1' S ' ^
^2 ^ 2 ' ° 2 ' % ' ^ ^ 2 ' ^ 2 ' ^ ^ ^ 1 ' ^ ^2S' V '
'2'
^
^ ^ y l ' ^ 6 , K , N 1 s ^2S' ^' ^
Table 3.1. Distribution of the parameters over the coefficients in the equations of motion.
The 15 different parameters in table 3.1 are combined into 16 coefficients. The dependence in those coefficients is restricted to the coefficients containing I , viz. the gyroscopic coupling terms. This dependence could easily be removed
-by introducing one more parameter, but it will turn out later that these coef-ficients are not important. The influence of all parameters is investigated, so the mathematical model (3.10) is complete and cannot be made more general by considering more terms. If other physical effects with new parameters would be included in the derivation, the number of parameters would be greater than the maximum number of independent coefficients to be estimated. This would inevitably yield interdependent estimates. On the other hand, if an important physical effect has been omitted in the derivation, the estimates obtained with model adjustment will show this. In this case, one or more coefficients in the equation would have an estimate that is not in agreement with the derivation in this chapter. An indication is given then where a careful re-examination of the theoretical model is required.
Summarizing: in the derivation of the equations of motion only those physi-cal effects are taken into account, that are necessary to give independent coefficients for model adjustment. Other effects must be omitted to prevent an undesired interdepence of parameter estimates. The theoretical derivation of the equations of motion can be validated with experimental evidence when the optimal parameter estimates are in agreement with this derivation. On the other hand, if an estimate does not agree with the value given by the derivation, there is evidence that this derivation is not complete.
Chapter 4
MODEL ADJUSTMENT WITH TWO PARALLEL MODELS' 4.1. The choice of the error criterion
First, the background phylosophy of the objective of this thesis is developed. It is clear that it is not obvious from all work in the past what goals have to be achieved in railway vehicle analysis.
Much effort has been spent on the derivation of the equations of motion of railway vehicles. Many phenomena can be explained qualitatively with a mathemat-ical model, but this is not sufficient to justify the use of a model for the improved design of vehicles. Then, the quantitative aspects of the theory become important, and they must be ascertained by a comparison between theory and
experiment. A comparison becomes cumbersome and hard to interpret if the experi-mental vehicle is unnecessarily complicated. It is better to verify first the wheel-rail system, then the bogie-wheel-rail system and the bogie-body system, and finally the complete system of rail-wheel-bogie-body.
Verification of the wheel-rail system has been done already. The creep
forces have been measured (see Hobbs (1967)) and compared with theoretical values according to the results of Kalker (1967). From these measurements it follows clearly that the theory predicts too high forces for a given creep velocity in railway practice. Correct values for the creep coefficients depend heavily on the surface conditions, and may be 30 to 50% lower than the theoretical values. These results have been obtained with laboratory experiments, because it is extremely difficult to measure creep forces and creep velocities along an irregular track. Also measurements have been carried out on the difference in rolling radii and contact angles for both wheels of a wheelset. Linearisation of these quantities yields the equivalent conicity and the gravitational stiffness respectively.
The next step is the verification of the creep theory, the conicity and the gravitational stiffness in experiments along the track. It is impossible to do these experiments with one isolated wheelset, because the essential longitudinal and lateral suspensions of that wheelset introduce a coupling between the wheel-set dynamics and the motions of the rest of the vehicle. The simplest isolated vehicle can be a bogie or a 2-axled freight car. Both vehicles have' heen used in comparisons: Stassen (1967) used a bogie and ORE (1972) a 2-axled special vehicle of British Railways. The results of Stassen were mainly qualitative. BR obtained some correspondence in measured and computed spectra of lateral vehicle motions, but only with a rather fancy multiplication of the input rail spectra with a factor four.
It is easier to accomplish the verification of the bogie-body system. The
