RAILWAY VEHICLES
A THEORETICAL AND EXPERIMENTAL IN-VESTIGATION OF THE INTERACTION
BE-TWEEN TRACK AND VEHICLE
P R O E F S C H R I F T
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, TE VERDEDIGEN OP
WOENSDAG 6 DECEMBER 1967 TE 16 UUR
DOOR
HENDRIK GERARD STASSEN WERKTUIGKUNDIG INGENIEUR
GEBOREN TE GOES
Dit proefschrift is goedgekeurd door de promotoren prof.ir. R.G. Boiten
J e t i e n s , en premier lieu a a d r e s s e r , mes remerciements a l'Office de R e c h e r c h e s et d ' E s s a i s , organisme i n t e r n a t i o n a l , grace auquel il a é t é p o s s i b l e d'effectuer l e s travaux d é c r i t s dans c e t t e t h e s e .
En particulier, il m'est a g r é a b l e de remero'^r iwonsieur M. Mauzin, ingénieur en chef de la S.N.C.F. pour l ' a s s i s t a n c e qui il a bien voulu me fournir dans l ' e x é c u t i o n d e s m e s u r e s p r a t i q u e s . Monsieur Ie Dr.Ir. P . van Bommel, collaborateur d e s N.S. pour sa cooperation dans l'organisation du programme d e s mesures effectuées d a n s Ie cadre du comité d ' e x p e r t s C9 et e n s u i t e Monsieur F . C a s e de l ' O . R . E . , qui s ' e s t mis a ma d i s p o s i t i o n pour l e s travaux de correction du texte a n g l a i s ,
Il m'est éqalement a g r é a b l e d'exprimer mes remerciements a Monsieur Ie Professeur Ir. H.C.A. van Eldik Thieme de l ' U n i v e r s i t é Technologique de Delft, qui m'a t r e s aimablement autorisé a utiliser la c a l c u l a t r i c e analogique de son l a b o r a t o i r e . J e d e s i r e enfin remercier tous l e s c o l l a b o r a t e u r s du dit Laboratoire pour v e h i c u l e s et du Laboratoire de réqlage et de mesure de c e t t e université pour leur p r é c i e u s e a s s i s t a n c e d a n s l ' e x é c u t i o n d e s mesures et l ' é t a b l i s s e m e n t d e s d e s s i n s .
CONTENTS
C H A P T E R I I N T R O D U C T I O N
page
1.1. Preliminary remarks 9 1.2. Outline of the literature 10 1.3. Outline of the thesis 13 1.4. Some fundamental s t a t i s t i c a l observations 19
1.5. Definition of the model of the bogie observed 20
C H A P T E R II S T A T I S T I C A L P R O P E R T I E S O F T H E L A T E R A L D E V I A T I O N S O F T H E T R A C K
2 . 1 . Introduction 24 2.2. The rail-fault detection coach 24
2.3. Influence of the measuring system on the correlation
functions of the rail deviations 29 2.4. Instrumentation for measurements 32
2.5. Measurements 34 2 . 5 . 1 . Introduction 34 2.5.2. Checking measurements 35
2.5.3. Distribution functions, correlation functions, and power spectra of the lateral deviations
of the track 40 2.5.4. Some rough measurements to describe the profile
of the rail and the gauge of the track 46
2.6. Conclusions 49
C H A P T E R III T H E O R Y O F T H E L A T E R A L MOTION OF A F O U R - W H E E L E D BOGIE
3 . 1 . Introduction 52 3.2. Derivation of differential equations 55
3.2.1. Kinetic energy, potential energy and generalised
force components 55 3.2.2. Relations between the forces at the contact
6
3.2.3. Differential equations of the generalised
co-ordinates v and ip 58 3.3. Contact between wheel and rail 59
3.4. Stability of the system 65 3.5. Transfer functions between the lateral deviations
of the track and the lateral motions of the bogie
on a track with constant gauge 71 3.6. Transfer functions between the lateral deviations
of the track and the lateral motions of the bogie
on a track with variable gauge 76
C H A P T E R IV I N V E S T I G A T I O N S WITH T H E AID OF AN A N A L O G U E C O M P U T E R
4 . 1 . Introduction 80 4.2. Machine equations 81 4.3. Circuit diagrams 84 4.4. Response to a lateral displacement of the bogie 84
4.5. Description of the analogue model, based on the
describing function method 88 4.6. Influence of the track parameters on the behaviour
of the bogie 90 4.7. Comparison between the describing function method
and the method of s t a t i s t i c a l linearisation 101
C H A P T E R V E X P E R I M E N T A L I N V E S T I G A T I O N S
5.1. The experimental bogie 105 5.2. Instrumentation for measurements 108
5.3. Measurements 110 5.3.1. Introduction 110 5.3.2. Influence on the behaviour of the bogie of the
gauge of the track, the length of the rail and
the velocity in the direction of motion 112
5.3.3. Checking measurements 120 5.4. Comparison 'of measurements with theoretical
6 . 1 . R e s u l t s achieved 133 6.2. Further research 136 N O T A T I O N S 140 SUMMARY 148 SAMENVATTING 149 SOMMAIRE 150
The numbers in brackets refer to the references at the end of each chapter.
Chapter I INTRODUCTION
1.1. Preliminary remarks
Many studies and investigations have been made to describe the behaviour of railway vehicles concerning their movements. In particular the problem of acceleration and deceleration of railway vehicles has been studied in much detail [ l ] .
The stationary behaviour of railway vehicles in curves h a s been investigated by Heumann [ 2 ; 3 ] , Uebelacker [ 4 ] , Borgeaud [ 5 ] , Muller [ 6 ] and Levi [ 7 ] . Since the problem can be described in terms of algebraic equations, this phenomenon i s excluded from this t h e s i s .
Furthermore, much work has been done to investigate the para-sitic motions superposed on the movement in the direction of running. Play between the flanges of the wheels and the rail heads and between the wheelsets and the frame of a railway vehicle bogie or between a bogie and the body of the vehicle gives rise to very pronounced forces. As regards the safety and the comfort of p a s s e n g e r s the lateral move-ments of the bogie are of supreme importance. By numerical integration or in an analytic way many investigators have studied this phenomenon by supposing that the bogie runs over a straight track only. In some studies the problem i s extended to sinusoidal tracks; but in practice the variations in the position of the track, and in particular the pos-itions of the two r a i l s , have to be considered a s a random p r o c e s s . So, to get a better idea about the behaviour of the movements of railway vehicles it i s n e c e s s a r y to enter the field of random vibrations.
The interaction between track and railway vehicle can be indi-cated by the following greatly simplified block diagram.
VERTICAL MOVEMENTS TRACK PARAMETERS PLASTIC DEFORMATION OF THE TRACK STATIC ÜNLQAOEÖi TRACK J ELASTIC DEFORMATION OF THE TRACK H0RI2 DYNAMIC fLOAOEDl ITRACK J ONTAL MOV( RAILWAY VEHICLE :MENTS GENERALISED CO-ORDINATES
F i g . 1. Block diagram of the interaction between track and railway vehicle.
To construct a permanent way, a suitable construction of track and railway embankment has to be selected; a number of parameters such as gauge and length of the rails have to be defined. Due to the fact that in a given space of time many trains have p a s s e d over the section, p l a s t i c deformation a r i s e s . Hence, the deviations in the position of the unloaded track are dependent on the initial conditions of the construction of the railway and also the types of trains which have p a s s e d . At the moment a train i s passing, the position of the track i s seriously affected by e l a s t i c deformation. Hence, to consider the state of the track as an input variable of the railway vehicle, it is necessary to measure the track in a loaded s t a t e . This t h e s i s will report the study of a simplified model of a bogie with two degrees of freedom, having the lateral deviations, when the bogie i s passing, as input, and having the generalised co-ordinates which describe the movements of the bogie, as output. It should be noted that the problem mentioned here i s a non-linear problem with random inputs. From a non-linearised simplification the stability can be ascertained; however, by taking into account the e s s e n t i a l non-linearities the system behaviour can be described in a more satisfactory way.
1.2. Outline of the literature
Ever since trains have been running on tracks there have been made many investigations into the phenomenon of the lateral and parasitic movements of a train and, in particular, the hunting limit-cycles. These motions are due to the coned tyre profiles. These linear profiles, how-ever, change considerably after about 2 000 miles, and after 10 000 miles a non-linear profile is stabilised.
In 1887 Klingel [ 8 ] gave his well-known formula for the hunting wavelength \ of a single wheelset with conic profiles:
X* = 2 7 7 y i - ^ , (1)
' Jo
where r is the radius of the running circle of the wheel, b half of the wheelgauge and y^ the conicity of the profile. This result i s based on kinematic observations for pure rolling.
To describe the behaviour of a rigid bogie with two play-free wheelsets creep has to be taken into account; a dynamic approach has to be made. These investigations, carried out as early as 1887 by Bödecker [9] and later by others including Carter [ 1 0 ; l l ] , Cain [12; 13], Mauzin [ 1 4 ] , Rocard [ 1 5 ; 1 6 ] , Davies [17] and Royer [ 1 8 ] ,
I 11
resulted in a multiplicative correction term E dependent on half the wheelgauge b and half the wheelbase a:
^1^-
(2)This formula is based on a linear friction law between the forces at the contact point and without any play between wheelset and bogie frame. Later non-linear profiles were observed by Heumann [ 1 9 ] , who defined an effective conicity y^:
*
y n - - ^ ^ ' (3) P o - Pn
where p^ and p^ are respectively the radii of the curves of the rail and wheel profile at the contact point. It should be noted that the radius
p^ of a conic profile i s <», whereas the radius p^ of worn profiles is of
the same order a s p^. That is to say, the effective conicity y^ of a worn profile is much greater than y^; hence the natural frequency of the system i s much higher. The behaviour of the system deteriorates. On the other hand, the gravitational stiffness ' of conic wheels is equal to zero, while that of worn profiles has a certain value; this r e s u l t s in a stabilising effect.
In many papers, for example those of Heumann [ 20;2l] and Mauzin [ 14] the s t a t i c s of a system using the Coulomb friction law are des-cribed; this results in a more complicated form of E.
The studies of Rocard [ 16; 17] are based on a linear friction law to describe the dynamics of the system. So far, the investigations mentioned here are linearised observations of the non-linear problem, which only indicate the stability boundaries. The investigations of Langer [22] and Chartet [23] are a first approach in the field of non-linear vibrations. Langer has introduced a non-non-linear force of the rail acting on the wheel, while Chartet has also introduced a non-linear friction law.
In 1953 under the auspices of the committee of experts C9 of the Office de Recherches et d ' E s s a i s (ORE) of the Union Internationale
*)
T h e g r a v i t a t i o n a l s t i f f n e s s i s d e f i n e d a s t h e d i f f e r e n c e b e t w e e n t h e h o r i z o n t a l c o m p o n e n t s of t h e n o r m a l f o r c e s of o n e w h e e l s e t a c t i n g on b o t h w h e e l s p e r u n i t of d i s p l a c e m e n t , a s s u m i n g t h a t t h e v e r t i c a l l o a d i s c o n s t a n t .
des Chemins de Fer (UlC) an investigation on the hunting movements of railway vehicles was started. As a result of this De Pater [24] has derived the non-linear differential equations of a four-wheeled railway vehicle. Moreover, it was De Pater [25;26] who solved these non-linear differential equations applying the method of Krylov and Bogoljubov. Stimulated by his work and because of the advent of the digital com-puter, Van Bommel [27;28] was able to execute the numerical inte-gration of these equations.
The need for higher speeds of running trains demands a better idea of the behaviour of this complex system with many degrees of freedom. Two ways can be distinguished in the further development of the theory of lateral motions of railway vehicles.
1. The well-known matrix theory, often used for flutter calculations in aircraft design, makes it possible by linearisation to describe rail-way vehicle systems with many degrees of freedom.
The investigations of Bishop [ 2 9 ] , Braun [ 3 0 ] , Wickens [ 3 1 ; 32; 33] and Van Bommel [34] have given a considerable amount of information about the stability of the systems with different para-meters such a s the mass, the position of the mass centre, the mo-ments of inertia, the total load, the ratio of wheelgauge and wheel-base and the types of profiles of wheel and rail. The influence of lateral and longitudinal stiffnesses has been determined by Wickens [ 3 1 ; 32; 3 3 ] , Van Bommel [34] and Matsudaira [ 3 5 ] ; the first two investigators also introduced the theory of rolling contact including spin according to Kalker [36] . One of the first approaches describ-ing the behaviour in a dynamic s e n s e below the critical speed has been given by Matsudaira [35] ; later on similar calculations were made by Keizer [ 3 7 ] . Many of the investigators have, with
con-siderable s u c c e s s , made great use of digital computers.
2. However, a lot of non-linearities, such as the profiles of wheel and rail, the play between wheelset and bogie frame or between bogie frame and coach, and the non-linear friction of the forces arising between rail and wheel, play an important role in the behaviour of a vehicle. The best way to obtain an idea about the effects of these non-linearities is to start by studying these one by one. In this field can be placed the investigations of De Pater [24; 25; 26] and Van Bommel [ 2 7 ; 2 8 ] . The influence of non-linear features of the suspension and of the forces arising between rail and wheel are described by Gilchrist, Hobbs, King and Wasby [ 3 8 ] .
I 13
described only for straight tracks. In practice, however, the lateral and vertical irregularities of the track have an enormous influence on the dynamics of the railway vehicle. The contribution of Van Bommel [28] is the first investigation in this field. He observed the r e s p o n s e s of a sinusoidal track on a four-wheeled vehicle. More r e a l i s t i c is the appli-cation of random p r o c e s s e s to describe the irregularities of the track and the random motions of the vehicle. In particular in curves Birmann [39] shows the effect of the track geometry on the running properties of railway vehicles in a static and dynamic way. Nakamura [ 4 0 ; 4 l ] applied covariance functions to describe the vertical irregularities of the track and the r e s p o n s e s of the system on t h e s e . Some preliminary investigations to describe the lateral irregularities of the track in a statistical s e n s e , based on a special correlation technique were exe-cuted by the author [ 4 2 ; 4 3 ] . The t h e s i s to be presented will point out in more details the influence of lateral track irregularities on the dy-namic behaviour of a four-wheeled bogie [ 4 4 ] .
1.3. Outline of the t h e s i s
The t h e s i s is mainly concerned with the problem of lateral para-sitic motions, in particular the hunting movement of a four-wheeled bogie of a.railway vehicle.
F i r s t of all in this chapter an outline of the literature has been given; furthermore some fundamental remarks about s t a t i s t i c s and a definition of the bogie observed will be given.
The problem in hand is a non-linear problem; thus the system dynamics can only be defined in reference to specific input s i g n a l s . Therefore in chapter 11 some measurements of the probability density functions and the correlation functions of the inputs are described.
Based on the theory of De P a t e r [ 2 4 ; 2 5 ] , in chapter III the fund-amental differential equations of a bogie with two degrees of freedom are derived. From the non-linearity introduced by the contact between wheel flange and rail head, the effective conicity and the gravitational stiffness are calculated. The first one d e s t a b i l i s e s the system; the second one s t a b i l i s e s the motions of the bogie. The spin effect i s introduced, and is shown to have a destabilising effect; in fact it off-sets the gravitational stiffness. Applying the method of s t a t i s t i c a l linearisation, the influence of track parameters, such as gauge of the track and lateral deviations of both rails on the dynamics i s given. Based on the results of the determination of the inputs with the aid of correlation techniques an analogue simulation of the bogie system has been made. In chapter IV the influence of track parameters on the
cor-relation functions has been given. The influence of the spin effect is also illustrated.
In order to substantiate theoretical results, experiments have been carried out with a test bogie; they are described in chapter V. T h e s e experiments effected in co-operation with the Netherlands Rail-ways (Nederlandse Spoorwegen=NS) and the Société Nationale de Che-min de Fer(SNCF) and co-ordinated by ORE have indicated close agree-ment between theoretical and experiagree-mental results concerning the hunting wavelength. Qualitatively the theoretical and experimental values of the critical velocity agree very well; however, quantitatively there are some discrepancies. Some concluding remarks are made in chapter VI; this chapter also gives an outline of further research work to be performed.
Some of the relevant r e s u l t s of this study have already been pu-blished in condensed form [42;43;44].
1.4. Some fundamental statistical observations
The measurement of power spectra can be executed in two dif-ferent ways. A measurement in the frequency domain requires narrow-band filters. By measuring track irregularities the frequencies of the signals obtained are dependent on the speed of the inspection coach. Due to the fact that the speed has variations of at l e a s t 5% it i s n e c e s s -ary to have narrow-band filters controlled by the speed of the inspection coach. The experimental difficulties are nearly insurmountable.
By determining the correlation functions in the time domain a good correction of speed variations is obtained by sampling at regular intervals along the centre-line of the track. Only negligible errors ap-pear in the neighbourhood of the cut-off frequency. Later on from the correlation functions the power spectra can be calculated by taking the Fourier transformation. The determination of correlation functions is not a simple thing; it needs for a sufficient s t a t i s t i c a l certainty an enormous amount of multiplications and integrations; so only with fast and large digital computers it can be executed.
Thanks to Veltman and other investigators a simplified correl-ation technique called the polarity coincidence correlcorrel-ation method, can be used in many c a s e s [ 4 5 ; 4 6 ; 4 7 ] . The covariance function C JT) can be defined a s :
+ T +00 +«>
C ^ ^ ( T ) = lim - 1 ƒ u(t) v(t+T) dt = f [ U V f(U,V)dUdV, (4)
I 15
and the correlation function K (r) a s :
C (r)
K u v ( ^ ' = ^ ' ' ^ ' (5)
where u(t) and v(t) are realisations of the stationary and ergodic sto-chastic p r o c e s s e s U and V, a^ and a^ are the variances of u(t) and v(t), and f(U,V) is the joint probability density function for U and V. Furthermore the polarity coincidence correlation function of the same processes U and V is defined as:
+T P^^(r) = lirn^ - 1 r sgn{u(t)}sgn{v(t+T)}dt - T f f sgn{U}sgn{V}f(U,V)dUdV, (6) with sgn{x(t)} = +1 if x(t)>0 or x(t) = 0*, and s g n { x ( t ) } = - l if x(t)<0 or x(t) = 0 ~ .
Under the assumption that the p r o c e s s e s U and V p o s s e s s a joint prob-ability density function, having an elliptical transverse with a plane of constant probability and having zero means, and according to Van Vleck's [48] result on the auto-correlation function of the output of a hard limiter (clipper) Veltman has derived a relationship between
C (r) and P (r) : u v * u v '
C ir) =a a sin {^ P (r) }. (7) u v ' u V -J u v * '
Moreover, for a finite observation time Veltman [49] has shown that the estimator p (r) of P (r) p o s s e s s e s a s t a t i s t i c a l uncertainty of the same order as the estimator k (r) of K ( r ) . When th^ p r o c e s s e s
u v u v ~f
r-U and V have any arbitrary probability density function, by adding two auxiliary signals •w,(t) and w„(t) the polarity coincidence correlation method can be used too. T h e s e auxiliary signals have to p o s s e s s a uniform probability density function between the values —A and +A and they have to be non-correlated with each other and with u(t) and v(t).
In this case Veltman and other investigators [45;46;50] have proved that:
C ^ . (r) = C (r) = A 2 p ( r ) . (8)
u + w , , v + w „ u v ' ' u v ' '
The applicability of this simple correlation technique changes a multi-plication into a coincidence detection and changes an integration into a counting. In this manner the correlation functions can be calculated with a small digital computer.
Due to the fact that only a sampled correlation function over a finite time i s obtained, the Fourier transformation i s calculated by replacing the infinite Fourier integrals by finite sums. The consequence of this is elucidated by Blackman and Tukey [ 5 1 ] . The power spectrum E [v) related to the auto-covariance function C (T) is defined by the
uu u u
equations (9), where o) = 2TTV.
E iv) = f C (r) e-J'"'^dT= f C (r) COSOJT d r , (9a)
uu 7 "" J " " _ 0 0 —CO and +cn +00 C (r) = /" E {v)e*''^dv= f E {v) COSCOT dv. (9b) UU / UU J u u ' — 00 ^ 0 0
By defining the finite Dirac " c o m b " ^^{T; A T ) [ 5 1 , p. 7 1 - 7 2 ]
N - 1
V ^ , ( T ; A T ) ='/2ArS(T+NAr) + A r X S(T-qAT) + '/2ArS(T-NAr), (10)
N q = _ N + l
where N A r means the maximum value of the correlation variable r and 8(T) is called the Dirac function, it follows that the transformation of an infinite Fourier integral into a finite sum results in the calculation of the power spectrum E (z^):
I 17 By defining the " s p e c t r a l window" Q ( , ( V ; A T ) as the Fourier trans-formation of the finite Dirac comb V ( T ; A T ) :
Q „ ( I ^ ; A T ) = f V J ^ ( T ; A T ) e"-''^'^ dT = cot ( i i i ^ ) sin (Na)Ar)AT, (12) — 00
equation (11) leads to:
+00
Eljv) = ƒ E^Jy*) Q„
( Ï . - V * ; A T )dv*. (13)
Hence, the power spectrum E {vj can be considered as the power 1 2 spectrum E (v) averaged over the frequencies u, v +—i—# v +——i
"" \ A T . A T
etc., and weighted by the spectral window Q^ [v — v ; A T ) . T h i s implies the use of a " l a g window" D (T) defined a s follows:
D (T) = 1 if \T\ < N A T ,
and (14) D* (T) = 0 if | T | > N A T .
Now, by choosing another lag window it is possible to concen-trate the main lobe of the spectral window by keeping the side lobes as low as feasible. A useful method i s to multiply the covariance func-tion, before applying the Fourier transformafunc-tion, with the lag window D(T), known as the " H a n n i n g " window.
D ( T ) = 1/2(1 + C O S ^ ^ ^ ) if | T | < N A T , N A T
and (15)
D ( T ) = 0 if | T | > N A T .
It can be shown th-at if the power spectrum E {v) has been calculated uu
for V = ^ with m = 0(1)M, and if the quantities E = E^ (-25—)
2 N A T ""'™ " " 2 N A T
are introduced, the application of the Hanning window implies an oper-ation in the frequency domain given by the equoper-ations
E* ='/2 [ E ' + E ' , ] , u u , o u u , o u u , 1 ' E* =% [E^ , + 2 E ' + u u , m u u , m — 1 u u , m E* , , ='/2 [ E ^ M + E^ ,, , ] u u , M u u , M u u , M — 1 EL,m+ J ' ^"^™=1 (l)^^-l' (16)
The frequency separation between adjacent estimates is 1 / ( 2 N A T ) .
The spectrum E^^(v) obtained in this manner is called the smoothed spectrum. In the event of observing a cross-spectrum $ [v] defined in a similar way as (9a) and (9b), the smoothing operation has to be applied to both the real and imaginary part.
It should be noted that the smoothing procedure i s executed at regular intervals in the frequency domain; hence when calculating spec-tra as a function of wavelengths, these specspec-tra should first of all be smoothed in the frequency domain. Afterwards they can be transformed as a function of wavelengths.
From the input and output spectra, the transfer function of a linear system can be determined in an easy way according to figure 2.
ult), h(t) u(t)*h(t)
ii(t)
•€> jdli Euu(v) H(v) EUUOHHMI' ^ E J v )
M^u,!")
F i g . 2. B l o c k diagram of a l i n e a r s y s t e m with input and output in the time and frequency domain.
For a linear system characterised by an impulse response h(t) or a transfer function H(v) with an input u(t), a disturbance signal 'T){\.) non-correlated with u(t) and added to the output of the system and a total output v(t), the following well-known relations [52] are obtained:
E [v]
V V
E . . ( - ) + \\{{v)?E{v). (17a)
éiv) = H ( v ) E „ (z.). :i7b)
Similarly to the method just-mentioned, Booton [53;54] gave, in 1953, a method covering non-linear systems without memory. The method can be considered as being the s t a t i s t i c a l equivalent of the describing
I 19
u(t)
N[u(t)] v(t) u(t). Ku(t) l(t)
NON-LINEAR SYSTEM EQUIVALENT LINEAR SYSTEM
F i g . 3. Block diagram of the equivalent linear system for a non-linear system without memory.
function technique [ 5 5 ; 5 6 ] . Let it be assumed that the non-linear system i s given by:
v(t) = N [ u ( t ) ] , (18)
where v(t) is the output and u(t) is the input signal. By replacing the non-linear element by an equivalent gain factor K together with a noise source T7(t), the identity
v(t) = K u(t) + 77(t) (19)
is achieved. The cross-correlation between the noise rjd) and the input u(t) for zero-time difference is given by:
+T
^u7,(°) = i ! ! ^ 2 f i u(t)77(t)dt, (20)
and by taking into account the equations (18) and (19), it follows that:
+T +T C ^(o) = lim - ^ "'' T-oo 2T ƒ N [ u ( t ) ] u ( t ) d t - K ƒ [u(t)]2 dt L _ T - T (21a) or: +00
C^^(c>) = ƒ N(U) U f(U) dU - K ƒ u2f(U) dU, (21b)
The optimal value of K i s that value which makes the cross-correlation C (o) = 0, and furthermore, the second term of the right member of equation (21) is the mean square a of the input signal u(t); hence it follows that:
+ 00
K = - 1 / N(U) U f(U) dU. (22)
U — CO
Now, take a non-linear component with an input having a normal probability density function. If this component is not linked up in a closed loop, the equivalent gain can be calculated in an easy manner. However, by incorporating this component into a system with feedback having a normal input, a close approximation can be obtained only if the system contains a linear part which filters the higher har-monics out. In that c a s e the probability density function of the input of the non-linear component can be fitted to a normal probability density function.
1.5. Definition of the model of the bogie observed
The behaviour of a four-wheeled bogie on a track with random lateral deviations is very complicated. Taking into account the non-linear profiles of wheel and rail simplifications need to be made.
The bogie observed has two degrees of freedom, that i s to say the bogie is assumed to consist of a rigid frame, supported on two iden-tical and rigid wheelsets without any play. Likewise it is assumed that the mass centre is placed in the plane of the two w h e e l s e t s ; there are two vertical planes of symmetry with respect to the inertial and geome-trical properties. It is further assumed that at all times a one-point contact between wheel and rail is maintained. The most important non-linearity i s introduced by the contact between wheel flange and rail head. A linear friction law of the forces at the contact point between wheel and rail (Carter; Rocard) has been taken into account.
The bogie is considered to move along the track at a constant speed; no interaction between the vehicles in a train is taken into account; likewise the interaction between the bogie and the coach is disregarded. The lateral deviations of the track a s a train is passing, are assumed to be known. No interaction between the vertical and horizontal irregularities is taken into account.
I 21
Specialists Committee of ORE. Unless specifically mentioned the follow-ing numerical values are used, both for the theoretical and the ex-perimental investigations:
a. The total weight of the bogie G is 7.63 tonnes.
b. The total load of the bogie G , including the weight G, is 20 tonnes. c. The wheelbase 2a is 2.50 metres.
d. The wheelgauge 2b is 1.522 metres.
e. The moment of inertia 1 of the bogie aoout the vertical through the mass centre is equal to 1112 kgm/sec .
f. The radius of the running circle when the bogie is centrally posit-ioned on the track, is 0.448 metres.
g. The non-linear profiles are worn profiles.
The following variables are considered as the most important parameters of track and bogie:
a. The gauge of the track.
b. The mean square of the lateral deviations of the r a i l s . c. The length of the r a i l s .
d. The mean square of the deviations of the gauge of the track. e. The velocity in the direction of running of the bogie.
References 1. N O R D M A N N , H.; G e n a u i g k e i t s f r a g e n der Z u g f ö r d e r u n g s m e c h a n i k , G l a s . A n n . 8 0 ( 1 9 5 6 ) 1 0 , p . 3 1 8 - 3 3 5 . ' 2 . H E U M A N N , H . , D a s V e r h a l t e n von E i s e n b a h n f a h r z f e u g e n in G l e i s b o g e n , O r g a n . F o r t s c h r . E i s e n b a h n W e s e n 6 8 ( 1 9 5 3 ) , p . 1 0 4 - 1 0 8 , 1 1 8 - 1 2 1 , 1 3 6 - 1 4 0 , 1 5 8 - 1 6 1 . 3 . H E U M A N N , H . , G r u n d z ü g e d e r F ü h r u n g d e r S c h i e n e n f a h r z e u g e n , E l e k t r . B o h n e n 2 1 ( 1 9 5 0 ) , p . 81 e t c ; 2 4 ( 1 9 5 3 ) , p . 3 1 3 . 4 . U E B E L A C K E R , H., ü b e r d i e M a s s e w i r k u n g e n b e l p l ö t z l l c h e n R i c h t u n g s -a n d e r u n g e n im L-auf von E i s e n b -a h n f -a h r z e u g e n , O r g -a n . F o r t s c h r . E i s e n b a h n W e s e n 8 5 ( 1 9 3 0 ) , p . 271 — 2 8 4 . 5. B O R G E A U D , G., L e p a s s a g e e n c o u r b e s d e v e h i c u l e s de c h e m i n d e fer, d o n t l e s e s s i e u x f o u r n i s s e n t un effort de t r a c t i o n c o n t i n u e , t h e s i s Z u r i c h ( 1 9 3 7 ) , 167 p p . 6. M U L L E R , C . T h . , D e r E i s e n b a h n r a d s a t z . K i n e m a t i k , S p u r f ü h r u n g s g e o m e t r i e und F ü h r u n g s v e r m ö g e n , G l a s . A n n . 7 7 ( 1 9 5 3 ) , p . 2 6 4 — 2 8 1 . 7. L E V I , R., E t u d e r e l a t i v e au c o n t a c t d e s r o u e s s u r le r a i l . R e v . G é n . C h e m . d e fer 5 4 ( 1 9 3 5 ) , p . 8 1 - 1 0 9 . 8. K L I N G E L , Ü b e r d e n L a u f d e r E i s e n b a h n w a g e n , O r g a n . F o r t s c h r . E i s e n -,. b a h n W e s e n 3 8 ( 1 8 8 3 ) , p . 1 1 3 - 1 2 3 . 9. B Ó D E C K E R , D i e Wirkung z w i s c h e n R a d und S c h i e n e n , H a n n o v e r ( 1 8 8 7 ) , 113 p p . . 10. C A R T E R , F.W., T h e r u n n i n g of l o c o m o t i v e s w i t h r e f e r e n c e t o t h e i r t e n -d e n c y to -d e r a i l , I n s t . C i v . E n g . , S e l . p a p e r s 9 1 ( 1 9 3 0 ) , 25 p p . 1 1 . C A R T E R , F.W., On t h e s t a b i l i t y of r u n n i n g of l o c o m o t i v e s , P r o c . R o y . S o c . A 121 ( 1 9 2 8 ) , p . 5 8 5 - 6 1 1 . 12. C A I N , B . S . , N o t e s on t h e d y n a m i c s of e l e c t r i c l o c o m o t i v e s , J . A p p l . M e c h . 6 3 ( 1 9 4 1 ) , p . A 3 0 - A 3 6 , A 1 8 5 - A 1 8 7 .
13. C A I N , B . S . , V i b r a t i o n of r o a d a n d r a i l v e h i c l e s . N e w York a n d C h i c a g o ( 1 9 4 0 ) , 258 p p . 14. M A U Z I N , M., E t u d e s u r le l a c e t d e v e h i c u l e s . R e v . G é n . C h e m . de fer 5 2 ( 1 9 3 3 ) , p . 2 5 - 4 9 . 15. R O C A R D , Y., L a s t a b i l i t é de r o u t e d e s l o c o m o t i v e s 1, A c t u a l i t é s s c i . e t i n d u s t r . , P a r i s ( 1 9 3 5 ) . 16. R O C A R D , Y . ; J U L I E N , M . , L a s t a b i l i t é d e r o u t e d e s l o c o m o t i v e s 2, A c -t u a l i -t é s s c i . e-t i n d u s -t r . , P a r i s ( 1 9 3 5 ) . 17. D A V I E S , R . D . , Some e x p e r i m e n t s on t h e l a t e r a l o s c i l l a t i o n of r a i l w a y v e h i c l e s , J . I n s t . C i v . E n g . 11(1939), p . 2 2 4 - 2 6 1 . 18. R O Y E R , M., R e c h e r c h e s s u r l e s p r o p r i é t ê s d y n a m i q u e s d ' u n d i c ó n e s e d é p l a 9 a n t en a l i g n e m e n t d r o i t . B u l l . A s s . I n t . C o n g r . C h e m . d e fer 2 6 ( 1 9 4 9 ) , p . 8 0 1 - 8 1 6 . 19. H E U M A N N , H., L a u f der D r e h g e s t e l l - R a d s ö t z e in d e r G e r a d e n , O r g a n . F o r t s c h r . E i s e n b a h n W e s e n 9 2 ( 1 9 3 7 ) , p . 1 4 9 - 1 7 3 . 2 0 . H E U M A N N , H . , L a u f von E i s e n b a h n f a h r z e u g e mit z w e i o h n e S p i e l g e l a g e r -t e n R a d s ö -t z e n , G l a s . A n n . 6 2 ( 1 9 3 8 ) , p . 25—30, 43—48. 2 1 . H E U M A N N , H., L a u f von E i s e n b a h n f a h r z e u g e mit z w e i o h n e S p i e l g e l a g e r -t e n R a d s ö -t z e n , O r g a n . F o r -t s c h r . E i s e n b a h n W e s e n 9 5 ( 1 9 4 0 ) , p . 43—54. 2 2 . L A N G E R , B . F . ; S H A M B E R G E R , J . P . , D y n a m i c s t a b i l i t y of r a i l w a y s t r u c k s . T r a n s . A m e r . S o c . M e c h . E n q . 5 7 ( 1 9 3 5 ) , p . 4 8 1 - 4 9 3 . 2 3 . C H A R T E T , A., L a t h e o r i e s t a t i q u e du d e r a i U e m e n t d ' u n e s s i e u , R e v . G é n . C h e m . d e fer 6 9 ( 1 9 5 0 ) , p . 3 6 5 - 3 8 6 ; 7 1 ( 1 9 5 2 ) , p . 4 4 2 - 4 5 3 . 2 4 . P A T E R , A . D . d e , E x p o s é de l a t h e o r i e d e I ' i n t e r a c t i o n e n t r e la v o t e e t le v é h i c u l e de c h e m i n d e fer. M o u v e m e n t s u r u n e v o i e en a l i g n e m e n t d r o i t , r e p o r t p u b l i s h e d by O R E , U t r e c h t ( 1 9 6 3 ) , 111 p p . 2 5 . P A T E R , A . D . d e . E t u d e du m o u v e m e n t d e l a c e t d ' u n v é h i c u l e d e c h e m i n de fer, A p p l . S c i . R e s . 6 ( 1 9 5 7 ) , p . 2 6 3 - 3 1 6 . 2 6 . P A T E R , A . D . d e . T h e approximative d e t e r m i n a t i o n of t h e h u n t i n g m o v e m e n t of a r a i l w a y v e h i c l e by a i d of t h e m e t h o d of K r y l o v a n d B o g o l j u b o v , A p p l . S c i . R e s . 1 0 ( 1 9 6 1 ) , p . 2 0 5 - 2 2 8 . 2 7 . B O M M E L , P . v a n , M o u v e m e n t d e l a c e t d ' u n v é h i c u l e f e r r o v i a i r e c o n s i d é r e c o m m e un p h e n o m è n e n o n - l i n é a i r e , G e n e v a , C i v l c o I n s t i t u t e C o l o m b o ( 1 9 6 2 ) , 57 p p . 2 8 . B O M M E L , P . v a n , A p p l i c a t i o n de l a t h e o r i e d e s v i b r a t i o n s n o n - l i n é a i r e s s u r l e p r o b l è m e du m o u v e m e n t d e l a c e t d ' u n v é h i c u l e d e c h e m i n d e fer, t h e s i s D e l f t ( 1 9 6 4 ) , 305 p p . 2 9 . B I S H O P , R . E . D . , Some o b s e r v a t i o n s on l i n e a r t h e o r y of r a i l w a y v e h i c l e i n s t a b i l i t y , P r o c . C o n g r e s s on i n t e r a c t i o n b e t w e e n v e h i c l e a n d t r a c k , I n s t , of M e c h . E n g . ( 1 9 6 5 ) , p . 9 3 - 9 9 . 30. B R A N N , R . P . ; B I S H O P , R . E . D . , On t h e y a w i n g o s c i l l a t i o n s of a s i m p l e t r o l l e y h a v i n g four c o n e d w h e e l s , U n i v . C o l l . L o n d . , M e c h . E n g . D e p . , r e p o r t 6 2 / 1 ( 1 9 6 2 ) . 3 1 . W I C K E N S , A . H . , T h e e q u a t i o n s of m o t i o n of a four-wheeled r a i l w a y v e h i c l e , B r i t i s h R a i l w a y s B o a r d , r e p o r t E 4 6 8 ( 1 9 6 3 ) , 12 p p . 32. W I C K E N S , A . H . , A r e f i n e d t h e o r y of t h e l a t e r a l s t a b i l i t y of a f o u r - w h e e l e d r a i l w a y v e h i c l e h a v i n g a f l e x i b l e , u n d a m p e d s u s p e n s i o n , B r i t i s h R a i l w a y s B o a r d , r e p o r t 1 3 1 9 ( 1 9 6 6 ) , 34 p p . 3 3 . W I C K E N S , A . H . , T h e d y n a m i c s of r a i l w a y v e h i c l e s on s t r a i g h t t r a c k : F u n d a m e n t a l c o n s i d e r a t i o n s of l a t e r a l s t a b i l i t y , P r o c . C o n g r e s s on i n t e r a c t i o n b e t w e e n v e h i c l e a n d t r a c k , I n s t , of M e c h . E n g . ( 1 9 6 5 ) , p . 1 - 1 7 . 3 4 . B O M M E L , P . v a n . C o n s i d e r a t i o n s l i n é a i r e s c o n c e r n a n t l e m o u v e m e n t d e l a c e t d ' u n v é h i c u l e f e r r o v i a i r e . P a r t i e I: V é h i c u l e & d e u x e s s i e u x s a n s r o u l i s , r e p o r t p u b l i s h e d by O R E , U t r e c h t ( 1 9 6 7 ) , 71 p p . 3 5 . M A T S U D A I R A , T . , . On t h e m e t h o d of p r e v e n t i n g t h e h u n t i n g of r a i l w a y v e h i c l e s , p a r t i c u l a r l y of t w o - a x l e c a r s , r e p o r t p u b l i s h e d by O R E , U t r e c h t ( 1 9 6 0 ) , p . 9 9 - 1 7 1 . 3 6 . K A L K E R , J . J . , On t h e r o l l i n g c o n t a c t of t w o e l a s t i c b o d i e s in t h e p r e -s e n t -s of dry f r i c t i o n , t h e -s i -s D e l f t ( 1 9 6 7 ) , 160 p p . 37. K E I Z E R , C . P . , A c h s f ü h r u n g und G l e i s l a u f d e r S c h i e n e n f a h r z e u g e , G l a s . A n n . 9 0 ( 1 9 6 6 ) 7 , p . 2 4 4 - 2 4 8 . 38. G I L C H R I S T , A . O . ; H O B B S , A . E . N . ; KING, B . L . ; WASBY, V . , T h e r i d i n g of t w o p a r t i c u l a r d e s i g n s of f o u r - w h e e l e d r a i l w a y v e h i c l e , P r o c .
I 23 C o n g r e s s on i n t e r a c t i o n b e t w e e n v e h i c l e a n d t r a c k , I n s t , of M e c h . E n g . ( 1 9 6 5 ) , p . 1 7 - 3 2 . 39. B I R M A N N , F . , T r a c k p a r a m e t e r s , s t a t i c a n d d y n a m i c , P r o c . C o n g r e s s on i n t e r a c t i o n b e t w e e n v e h i c l e a n d t r a c k , I n s t , of M e c h . E n g . (196 5), p . 5 8 - 7 0 . 4 0 . N A K A M U R A , I., On t h e r e l a t i o n b e t w e e n s u p e r e l e v a t i o n a n d c a r r o l l i n g , P e r m a n e n t Way 5 ( 1 9 6 2 ) 1 4 , p . 10 — 17. 4 1 . N A K A M U R A , I . , D e s i g n of t r a c k i n s p e c t i o n c a r , q u a t e r l y r e p o r t J a p a n e e s R a i l w a y s 3 ( 1 9 6 2 ) 4 , p . 5 6 - 6 1 . 4 2 . B O M M E L , P . v a n ; S T A S S E N , H . G . , D e t e r m i n a t i o n d e q u e l q u e s c a r a c t é -r i s t i q u e s s t o c h a s t i q u e s d e s d e v i a t i o n s d e s f i l e s d e -r a i l s , e n v u e d e l ' é t u d e du m o u v e m e n t de l a c e t . R e v u e F r a n q a i s e d e M é c a n i q u e ( 1 9 6 5 ) 14, p . 6 5 - 7 0 . 4 3 . S T A S S E N , H.G., B O M M E L , P . v a n . D e t e r m i n a t i o n d e q u e l q u e s c a r a c t é -r i s t i q u e s s t o c h a s t i q u e s d e s d e v i a t i o n s d e s f i l e s de -r a i l s , en v u e de l ' é t u d e du m o u v e m e n t d e l a c e t , r e p o r t p u b l i s h e d by O R E , U t r e c h t ( 1 9 6 6 ) , 117 p p . 44. S T A S S E N . H . G . , On t h e i n t e r a c t i o n b e t w e e n t r a c k a n d r a i l w a y v e h i c l e , in p a r t i c u l a r w i t h r e s p e c t to t h e h u n t i n g p r o b l e m . P a p e r to be p r e -s e n t e d at t h e F o u r t h Conf. on N o n - l i n e a r O -s c i l l a t i o n -s , P r a a g ( 1 9 6 7 ) , 10 p p . 4 5 . V E L T M A N , B . P . T h . ; K W A K E R N A A K , H . , T h e o r i e und T e c h n i k d e r P o l a r i t a t s K o r r e l a t i o n fur d i e d y n a m i s c h e A n a l y s e N i e d e r f r e q u e n t e r S i g -n a l e -n u-nd S y s t e m e -n , R è g e l u -n g s t e c h -n i k 9(196 1), p . 357 — 3 6 4 . 4 6 . V E L T M A N , B . P . T h . ; B O S , A. v a n d e n . T h e a p p l i c a b i l i t y of t h e r e l a y c o r r e l a t o r a n d t h e p o l a r i t y c o i n c i d e n c e c o r r e l a t o r in a u t o m a t i c c o n -t r o l , I F A C C o n g r e s s , B a s l e ( 1 9 6 3 ) , p a p e r 2 6 4 , p . 1—7. 4 7 . V E L T M A N , B . P . T h . ; BOS, A. v a n d e n ; H E I N S , W., On t h e p o l a r i t y c o r r e l -a t i o n of n o n z e r o m e -a n G -a u s s i -a n v -a r i -a b l e s , r e p o r t 65—2, D e p . of A p p l . P h y s i c s , T e c h n . U n i v . D e l f t ( 1 9 6 5 ) , 33 p p . 4 8 . VAN V L E C K , J . H . , T h e s p e c t r u m of c l i p p e d n o i s e , H a r v . R a d i o R e s . L a b . , r e p o r t 5 1 ( 1 9 4 3 ) . 4 9 . V E L T M A N , B . P . T h . , Q u a n t i s a t i o n , s a m p l i n g f r e q u e n c y a n d d i s p e r s i o n w i t h c o r r e l a t i o n m e a s u r e m e n t s , R e g e l u n g s t e c h n i k 14(1966) 4, p . 1 5 1 -1 5 8 . 50. V E L T M A N , B . P . T h . , D e m e t i n g v a n c o r r e l a t i e f u n c t i e s , t h e s i s to b e p u b l i s h e d D e l f t ( 1 9 6 8 ) , 110 p p . 5 1 . B L A C K M A N , R . B , ; T U K E Y , J.W., T h e m e a s u r e m e n t of p o w e r s p e c t r a from t h e p o i n t of v i e w of c o m m u n i c a t i o n s e n g i n e e r i n g , D o v e r p u b l . . N e w York ( 1 9 5 8 ) , p . 14, 3 3 - 3 7 and 9 5 - 1 0 0 . 5 2 . S O L O D O V N I K O V , V . V . , I n t r o d u c t i o n to t h e s t a t i s t i c a l d y n a m i c s of a u t o -m a t i c c o n t r o l s y s t e -m s , D o v e r p u b l . , N e w York ( 1 9 6 0 ) , p . 123—136. 5 3 . B O O T O N , R . C . , T h e a n a l y s i s of n o n - l i n e a r c o n t r o l s y s t e m s w i t h r a n d o m i n p u t s , S y m p . on n o n - l i n e a r c i r c u i t a n a l y s i s , P o l y t e c h n . I n s t , of B r o o k l y n ( 1 9 5 3 ) , p . 3 6 9 - 3 9 1 . 54. B O O T O N , R . C . , N o n l i n e a r c o n t r o l s y s t e m s w i t h s t a t i s t i c a l i n p u t s . D y -n a m i c A -n a l y s i s a -n d C o -n t r o l L a b . , r e p o r t 6 1, M . I . T . ( 1 9 5 4 ) . 55. M a c M l L L A N , R . H . , N o n - l i n e a r c o n t r o l s y s t e m s a n a l y s i s , P e r g a m o n P r e s s , L o n d o n ( 1 9 6 2 ) , p . 7 6 - 7 9 . 56. P E R V O Z V A N S K I I , A.A., R a n d o m p r o c e s s e s in n o n - l i n e a r c o n t r o l s y s t e m s , A c a d e m i c P r e s s , N e w York ( 1 9 6 5 ) , p . 77 — 1 0 6 .
Chapter II
STATISTICAL PROPERTIES OF THE LATERAL DEVUTIONS OF THE TRACK
2.1. Introduction
The measurements of the lateral deviations in the rails had to be made using a moving railfault detection coach. All detection s y s -tems are constructed in such a way that a point of reference independent of the movements of the inspection coach is created [ l ; 2 ; 3 ; 4 ; 5 ] . In Europe there are only two inspection coaches suitable for the measure-ment of lateral deviations, both based on the principle of versed sine measurements. Because of its better transfer function for the measuring of large wavelengths, the SNCF inspection coach was chosen, rather than the inspection coach of the Deutsche Bundesbahn (DB).
To describe the random p r o c e s s e s , only the distribution functions and the correlation functions of the signals have been investigated. The correlation functions are calculated by means of the polarity coinci-dence correlation method. The required power spectra as a function of the wavelength are obtained by using the Fourier transformation.
The rail profiles of the tracks examined are measured manually by means of a copying device. From these measurements the average profiles are calculated.
2.2. The rail-fault detection coach
The measurements were made using the SNCF rail-fault detection coach [6;7] . With this detection coach three quantities can be meas-ured in the horizontal plane - the versed s i n e s of the left and right hand rails and the gauge of the track - a s well a s five recordings of the irreg-ularities in the vertical plane, for example the cross levels of the left "and right hand r a i l s .
Both versed s i n e s are measured on a base of 10 metres. Three probes are placed against the inner side of the rails, one exactly mid-way between the other two. Figure 1 shows a simplified outline of the versed sine measuring system. The versed sine measurements are re-corded on paper which is wound at a speed proportional with that of the coach. The movements of the inspection coach do not affect the quan-tities to be measured. It should be noted that the inspection coach described was designed specifically for checking the condition of
main-II 25
F i g . 1. D i a g r a m m a t i c v i e w of t h e d e v i c e for r e c o r d i n g t h e v e r s e d s i n e .
tenance of the track, and not as a measuring system for s t a t i s t i c a l analysis of the characteristics of a track.
By defining the following quantities: a. the ordinate ^ along the track,
b. the variable d ( ^ ) , which i s the variation of the position of the rail j in the horizontal plane (j=l for the right hand rail, j=2 for the left hand one),
c. the variable f.(^), which is the versed sine of the rail j in the plane just-mentioned,
d. the quantity 2a, which is the base of the two exterior probes, one can deduce for the versed sine f.:
J
fj(^) = d,(<f) - V2 [ d . ( ^ + a) + d . ( ^ - a ) ] , (la)
or written as a recurrent equation:
<li(«-a:
F i g . 2 . D e f i n i t i o n of t h e v e r s e d s i n e .
The characteristic equation of equation (lb) follows by substi-tuting p for d.
p2 _ 2p + 1 = 0 . (2)
Thus the roots p, and p^ are equal to unity and so the equation cannot have a stable solution. An exact calculation of the variable d is not possible.
Introduce an operator L [ u ( t ) ] applied to the random variable u(t) so that [ 8 ] :
L [ u ( t ) ] =-'/2 { u ( t + T ) + u ( t - T ) } ; (3)
then, by applying this operation k times to this variable, the following result i s obtained:
L'' [u(t)] = [-'AS^ 2 (^) u { t + (-k + 2q)T},
q = o ^ (4)
with k = 0. Consider now the random variable u(t) with zero means; this gives:
II 27
Referring back to equation ( l a ) , one can deduce, by taking into account equation (3), the following relation:
fj(^) = d . ( ^ ) + L [ d j ( ^ ) ] . (6)
So, it follows that:
d,(^) = 2 {-irL^iiAi)].
J k=o J
(7)
since, by substituting equation (6) into (7) and by taking equation (5) into account, the left and right hand members of relation (7) become identical. By taking into account the formula (4), equation (7) can be transformed as follows:
d,(^) = 1 i'A)^ 2 (^) f , { ^ + ( 2 q - k ) a } .
' k=0 q=0 ^ J (8)
When the c a s e f.(^) = sin {^^) is considered, the transfer function
H{k), being the quotient of the Fourier transformations of d and f,, of
the rail-fault detection coach will be:
Hik) 1 k = 0 COS ( 1 ^ ) k J (9) In other words:
if cos ^ ^ ;^ 1, then H(\) = 1 - cos i l l i ;
k k
(10) if cos 277-a 1, then H{k) = 0.
By observing this transfer function it will be realised that for wavelengths of ^ metres with k = 0 [ 1 ] oo the gain of the measuring system is equal to zero. Therefore, for a = 5 metres, it will be possible
F i g . 3 . T r a n s f e r f u n c t i o n s of r a i l - f a u l t d e t e c t i o n c o a c h e s , b a s e d on t h e p r i n c i p l e of v e r s e d s i n e m e a s u r e m e n t .
to obtain satisfactory and useful information only when 6<X.<50 metres. Assuming that the central probe i s b metres out of centre, it is possible, in the same way, to derive for the modulus of the transfer function | H (^) |:
| H ' ( M | = [(cos 2Zlb_ cos 2Z!H)2 . (sin ^Z!^- k sin 2zia^2
k k k k )']
'A
(11)
This formula shows that small deviations in the position of the central probe do not affect the modulus of the transfer function for wavelengths within the interval just-mentioned.
F i g . 4, I n f l u e n c e of t h e p o s i t i o n of t h e c e n t r a l p r o b e on t h e m o d u l u s of t h e t r a n s f e r f u n c t i o n of t h e S N C F r a i l - f a u l t d e t e c t i o n c o a c h .
II 29
2.3. Influence of the measuring system on the correlation functions of the rail deviations
The main concern is the detection of the correlation function of the lateral deviations of the r a i l s ; thus it i s n e c e s s a r y to compensate for the influence of the measuring system. From equation (8) a close approximation can be deduced of the inversed transfer function ui\\ when 6<k<50 metres, by replacing the infinite summation over k by a finite summation up to M and M—1 and then by taking the average of these two approximations. Hence, it follows that:
M - 1 k d,(f) = 2 ('/2)'^ Z (k)f [^ + (2q - k)a] + ' k=0 q=0 ^ ' M ^(^^)M+i ^ (M) f [^ + ( 2 q - M ) a ] . (12) q = 0 ^ '
By substituting equation (12) into the definition of the correlation function according to (1.4) and (1.5) one obtains, with ^ as correlation variable:
2 / o f M - 1 M - 1
'I, . 1 / ^ ^ . k + i
i.n
j j CTj 2 ^^-.CD 2i^ J L k = o m=o q=o n=o "^ " O .fj { ^ + ( 2 q - k ) a } f . { ^ + ( 2 n - m ) a + ^ * } + 2 M +2 ^ ^ (V,) T. T (M)(M)f / ^ + ( 2 q _ M ) a } . q=o n=o ^ •" . f j { ^ M 2 n - M ) a + ^*} + ^-^ k + M + l ^ ^ + S (/2) S 2 ( ^ ) ( M ) h {^ + ( 2 q - k ) a } k=o q = o n=o ^ I f j { ^ + ( 2 n - M) a + ^ * } + + f. { ^ M 2 q - k ) a + ^ * } f . { f + ( 2 n - M ) a } d^. (13)
By changing the order of integration and summation and by taking into account the definition of the correlation function, the result will be:
M - 1 M - 1 , . k m k + m
2 2
(>/2)
E T O O
k\ /m> k =o m =o q = o n =o q' ^n Kj J { [ 2 ( n - q ) + k - m ] a + ^ * } + j j 2M + 2 ^ ^ M'/2) 2 2 (M) (M)K. J { 2 a ( n - q ) . ^ * } . q = o n = o ^ j J M - 1 k + M + l "^ "^ + 2 C/ï) 2 -k=o q^ ^ (S^ (lï*) Kfx{t2(n-q)
=o n=o ^ I ' ' + k - r a ] a + ^ * } + Kj J { - [ 2 ( n - q ) + k - M ] a + ^*}} . (14) F i g . 5. M o d u l u s of t h e a p p r o x i m a t i o n of t h e i n v e r s e d t r a n s f e r f u n c t i o n of t h e r a i l - f a u l t d e t e c t i o n c o a c h , by d i g i t a l f i l t e r i n g a c c o r d i n g to e q u a t i o n (12) w i t h M = 10.Should a cross-correlation function between the two random pro-c e s s e s f (a versed sine) and v (the lateral movements of a bogie) be observed, one obtains, in the same way:
II 31 CT r M - l , k K . ( ^ * ) = - ^ 2 (/2) 2 (J) Kj J ( a ( k - 2 q ) + ^ * } + J er [ k = o q = o ^ J M+ 1 M + (/2) 2 (M) K j ^ { a ( M - 2 q ) + ^ * } q = o ^ J (15) MJ*) APO-AMF
F i g . 6. Attto-correlation functions of the versed s i n e f. and the J
lateral deviation d and cross-correlation functions between those variables and the displacement v* of the centre of any arbitrary bogie.
function
Figure 5 gives the modulus of the approximated inversed transfer 1
Hik) in the frequency domain. Figure 6 shows in the time
lateral deviation d,. In the same figure are given the cross-correlation functions between the versed sine f. and the displacement v of the centre of a bogie, and between the lateral deviation d, and the same displacement v respectively. In view of the symmetric property of auto-correlation functions k ( ^ ), they are only drawn for positive values
oir.
2.4. Instrumentation for measurements
The versed s i n e s of the left and right hand r a i l s , the gauge, and a signal to indicate the speed of the inspection coach and to mark the place on the tape were recorded simultaneously; so were the vertical irregularities and the accelerations of the measured coach.
RAIL-FAULT DETECTION COACH SPEED SIGNAL AND PLACE INDICATOR 1 — INDUCTIVE DISPLACEMENT METER ACCELERO-METER AMPLIFIER AMPLIFIER ~1 FREQUENCY MODULATOR TAPE RECORDER
F i g . 7. Block diagram of the instrumentation a s used when making the recordings.
The displacements were transformed into electrical quantities with the aid of inductive displacement meters, the accelerations were measured by force balanced accelerometers, and the speed was recorded by means of a tachometer. All these quantities were adapted to the frequency modulator by amplifiers and recorded on a 14-track tape recorder.
From these data the correlation functions and distribution functions were calculated on a small digital computer — a P . D . P . 8 —. Figure 8 illustrates the lay-out for the calculation of the distribution function. The P . D . P . 8 is controlled by pulses derived from the recorded speed s i g n a l s . Gate A is controlled by the place indicating signal. The signals to be analysed p a s s a band-pass filter and a D.C. operational amplifier; with the aid of an a / d converter the data are read into the digital com-puter, by which the data are classified in proportion to their v a l u e s . From the distribution function the mean square of the signal was cal-culated.
II 33 TAPE RECORDER BAND-PASS FILTER SPEED PULSE CONVERTER PLACE INDICATOR CONVERTER AMPLIFIER A / D CONVERTER GATE A DIGITAL COMPUTER
F i g . 8. Block diagram of the lay-out to determine distribution functions.
were determined using the polarity coincidence correlation method. F i r s t the signals were filtered by two band-pass filters; with the aid of two operational amplifiers the auxiliary signals were added. The sign of the signals, detected by the two polarity detectors, were read into the computer, which was controlled by the speed signal in the same way as described for the determination of the distribution function.
TAPE RECORDER
r
n
NOISE GENERATOR BAND-PASS FILTER AMPLIFIER POLARITY DETECTOR BAND-PASS FILTER NOISE GENERATOR AMPLIFIER SPEED PULSE CONVERTER PLACE INDICATOR CONVERTER POLARITY DETECTOR GATE A ~ DIGITAL COMPUTER JF i g . 9. Block diagram of the lay-out to determine correlation functions using the polarity coincidence correlation method.
I RULER '
TRACK GAUGE
F i g . 10. Definition of the gauge of the track.
The rail profiles were measured with a mechanical copying device. A ruler was placed on the track in a horizontal position. The ruler it-self had ends with slopes of 1 in 20. The profiles were measured from the reference line obtained in this way. The gauge was measured exact-ly 14 millimeters below the contact point A., between the rail and the ruler.
2.5. Measurements
2.5.1. Introduction
At the end of 1963 the first measurements were taken [ 9 ; 1 0 ] ; 14 sections were selected, so that:
a. The length of the track is at least five kilometres, so a s to ensure that the s t a t i s t i c a l uncertainty because of the finite length of ob-servation will be acceptable. No interruptions by switch and crossing work, etc. can be tolerated.
b. The track itself p o s s e s s e s different parameters, such as profile and length of the rail, gauge and average lateral deviation.
In the summer of 1966 five sections were measured again; in the following table the principal data of these five sections are given. The measurements were split up into three parts. First the rail-fault detection coach was inspected regarding its speed, a possible inter-action between the quantities measured and the natural frequencies of the coach and with regard to the direction of running. It was proved that the polarity coincidence correlation technique will be very useful in calculating the correlation functions. Some investigations were made in order to estimate the s t a t i s t i c a l deviation of the calculated correl-ation functions, because of the finite time of observcorrel-ation and the
non-RAIL NP<I -SLOPE 1:2
II section Assen — Hoogeveen Apeldoorn — Amersfoort Amersfoort -Apeldoorn Arnhem — Ede Zutphen — Lochem abbreviation A S N - H G V A P D - A M F A M F - A P D A H - E D Z P - L C length in km 16 10 6 11 9 rail length in m 14 k m - 3 0 m 2 km - 24 m 30 M welded 30 M 5 km — 30 m 6 k m - 2 4 m 24 m gauge in mm 1433 1434 1434 1438 1436 type of rail [11] NP 46 NP 46 NP 46 NS 63 BS 95 R T a b l e 1. Most s i g n i f i c a n t p r o p e r t i e s of the s e c t i o n s m e a s u r e d .
stationary behaviour of the quantities measured. Secondly the distri-bution functions and the correlation functions of the tracks were deter-mined, the length of the rails and the average values of the lateral deviation being considered a s the most significant parameters. Finally, the measurement of the profiles were made.
2.5.2. Checking measurements
The section ASN—H3V was measured five times at a constant speed of 19.4 m / s e c . From t h e s e records it was found that the corre-lation functions of the versed sines calculated repeated within 5%. With correlation techniques no interaction could be found between the movements of the inspection coach and the quantities describing the track. The five sections mentioned in table 1 were generally measured at speeds V of 8.3, 19.4 and 33.3 m / s e c . No influence of the speed could be shown. L i k e w i s e the running direction was not found to have any effect.
As already mentioned in chapter 1.4 Veltman has shown that for p r o c e s s e s with zero mean and a symmetrical joint probability density function with an elliptical transverse, the correlation function can be calculated from the polarity coincidence correlation function by means of the sine relation (1.7) [ 12; 13; 14; 15]. Moreover, by separately adding two auxiliary signals to the signals to be determined the correlation
function can immediately be calculated. The method with auxiliary signals h a s been used to check whether the correlation function could be calculated by means of the sine relation (1.7).
300 400 , , , 5 0 0
— • 5 [>"]
Fig. 11. Typical recordings of the versed s i n e s and the gauge of the track with the velocity V in the direction of running as parameter.
",.,.«*) • WITH AUXILIARY SIGNALS ASN-HGV
X WITHOUT AUXILIARY SIGNALS
Fig. 12, Correlation functions measured with and without auxil-iary s i g n a l s .
Figure 12 shows the correlation functions determined without and with the two auxiliary signals. The function on the left shows the agreement between the two correlation functions of the versed sine of a track without curves calculated by means of the two methods mentioned above. The right hand one shows the difference between any two correlation functions of section ASN-HGV determined with or without auxiliary
11 37
signals; the deviations being caused by errors in the measuring device and not a s a result of the peculiar behaviour of the probability distri-bution of the original s i g n a l s .
The variance of the polarity correlation function i s difficult to deal with. Note that the polarity procedure involves a priori normali-sation commonly used in measuring correlation functions. In calculating the degrees of accuracy after normalisation a quotient with dependent errors appears from which no general result can be derived. For very large shifts the correlation function approaches zero for normal pro-c e s s e s with zero means; whipro-ch only means that the signals are inde-pendent. In these conditions it is possible to derive the variance in a measured polarity coincidence correlation function [16; 17]:
L
lim {var [P(s^*)]}2 = _ L ( ^ - s^) P 2(^") d^^. (16)
= o o
For normal, often called " g a u s s i a n " , processes Van den Bos has given a variance calculation which holds good for all values of ^ and for a given observation length ^ [ 1 8 ] . Using the theorem of Price [ 19] the influence of finite clipping on the fourth moment of zero mean normal p r o c e s s e s is determined. Through partial integration a rather intricate form i s obtained in which the correlation function measured and its derivative to ^ appear, so an analytical approximation k (^ ) of the correlation function measured k(if ) is necessary for the numerical solution of the problem. A fairly close approximation can be found by means of the formula:
N - B < 5 * ^
k* (^*) = 2 A , e " - cos C q ^ ' \ (17)
q = l
By minimising the integral of the squared error {k(^ ) —k (^ )} the coefficients A„, Bq and Cq can be calculated; however, this includes the solution of a set of non-linear equations, which is rather difficult to handle. Therefore, an investigation has been made, calculating the Fourier transformation of the output of a hard limiter [ 2 0 ; 2 l ] and intro-ducing the quadrivariate normal characteristic function [ 2 2 ] . Now, the moment of the fourth order can be expressed in a four-dimensional sum-mation of powers of the correlation function measured, multiplied with correcting F-functions, faculties and sine functions.
/* 00 00 CO CO {var [p(f *)] }2 = - 1 - 5 ( ^ - e ' ) 2 2 2 2 {-HC'*)V. QTI^^^ J i = 0 q=0 J =0 k = 0 o . {-ki^'-^*)}i { _ k ( ^ ' + i*)}^ { - k ( ^ ' ) } ^ • H ( i , j , k , q ) d ^ ' - [ P ( ^ * ) ] 2 , (18) where H(i,j k q) = 2 2 ' 2 i ( i - m , k , j , m , q - n , n ) . G2(i-m,k,j,m,q-n,n) ^^ m=0 n=0 (i-m)!j!k!(q-n)!n!m! with Gi(i-m,k,j,m,q-n,n) = n ^ ^ ^ + q - m - n ) r ( L l i ^ i l = E ) . . r ( m ^ i +q-n) p^m+k + n) (ig^) 2 2 and
'2 (i-m,k,j,m,q—n,n) =[sin — (i+k +q—m—n] [sin — (i + j +n—m] .
. [sin 21 (m +j + q - n ] 2 [ s i n Z I ( m + k + n]2. (19c)
Owing to the good convergence the infinite power series can be changed into a finite power s e r i e s of 4 terras which are predominantly zero; the method derived i s very suitable for numerical "calculation for the'corre-lation functions observed in this t h e s i s .
This variance of the polarity coincidence correlation function can be transformed by the well-known sine relation (1.7) into the variance of the correlation function. For example figure 13 indicates the v a r [ k , , (^*)] for the section AMF-APD; the section has an observation length of 1000 metres. The small lines represent the values of twice the var [k^ ^ (^ ) ] , which means that the probability that the true
II 39
(f) 1.0
" d l dl A M F - A P D
F i q . 1 3 . V a r i a n c e of t h e c o r r e l a t i o n f u n c t i o n d u e t o t h e f i n i t e l e n g t h of o b s e r v a t i o n .
value is lying between these boundaries is 69% for normal distributions. Now, this section was split up into 6 parts of 1000 metres. From each part the correlation function was determined and afterwards the var[k , (^ )]was calculated. With regard to figure 13, figure 14 shows
1 1
that, due to the non-stationary behaviour of the signal, the var [ k , , (^ )] 1 1
measured is about twice as large as the var [ k , , (^ ) expected which
d j O j
might be having regard to the finite length of observation.
•«didi'i') ^''^ AMF-APD
"^['<d,d,<«*)] 0.8
F i g . 14, V a r i a n c e of t h e c o r r e l a t i o n f u n c t i o n d u e t o t h e f i n i t e l e n g t h of o b s e r v a t i o n and t h e n o n - s t a t i o n a r y b e h a v i o u r of t h e s i g n a l ,
2.5.3. Distribution functions, correlation functions, and power spectra
of the lateral deviations of the track
From the records of the versed sines of the five sections mentioned in table 1 the distribution functions are calculated for a band width of between 6 < \ < 5 0 metres. Figure 15 shows the distribution functions F(f ) of the versed sines of the left and rixjht hand rails of the sections ASN-HGV and ZP—LC; figure 16 illustrates the distribution functions of the versed s i n e s of the right hand rails of the five s e c t i o n s . Note
1. 2 0 - 2 - t - 6 - 8 -1 / -, 1 " ^ 1 1 1 ^^.^•^ 1 1 1 X ASN-HGV 1
A
, LEFT RIGHT 001 01 [mm] fj(i) 001 ai 10 16 50 84 90 99 999 9a99 8 6 1, 2 0 2 i 6 8 -- ^ y.y^
1 ^v
y
y
1 1 1 ^ Z P - L C ^ ' — F(fj) [%] ^ 1 LEFT RIGHT 1 1 50 84 90 99 999 9999 F(fj) [•/.] F i g . 1 5 . D i s t r i b u t i o n f u n c t i o n s o f t h e v e r s e d s i n e s of l e f t a n d r i g h t h a n d r a i l s .11 41
001 ai
Fig. 16. Distribution functions of the versed s i n e s of the right hand rails of the five s e c t i o n s measured.
that the distribution functions are drawn on the well-known proba-bility paper, on which a normal distribution function will be trans-formed into a straight line. From the figures 15 and 16 it follows that the distribution function will fairly closely approximate these curves within —2cr, <f.(/^)< + 2cr . Outside t h e s e limits an exponential function
j ^ j
seems to be more accurate [ 2 3 ] . From the distribution function of the
2
versed sine the mean square cr, is calculated :
J
w-+ f.
ƒ
-f, d F ( f J d f. f^df, (20)Assuming that a normal approximation will agree it should be noted that the mean square cr, can be obtained directly from the probability
j density function d F(fj) d f. , since [24] + 1
/T^
]
J 2 d f j = 0 . 6 8 2 6 . - 1 (21)Table 2 shows the values calculated as well as the values ob-tained by approximation of the versed sines of the left and right hand rails. section ASN-HGV A P D - A M F A M F - A P D A H - E D Z P - L C left l^ii) [mm] calculated 1,16 1.09 1.30 1.26 2.29 approximated 0.90 1.05 1.30 1.10 1.99 right fj(^) [mm] calculated 1,09 0.99 1.23 1.16 2.15 approximated
0.99 1
0.94 1.05 1.09 2.08 T a b l e 2. Mean s q u a r e s Cr of the v e r s e d s i n e s of left and r i g h thand r a i l s . •'
The correlation functions of the lateral deviations d. are deter-mined for a band with between 6<k<b0 metres. With the aid of the digital filter equation (14) the correlation functions of the lateral devi-ations d are calculated from the correlation functions of the versed sines f , and afterwards by using the Fourier transformation the power spectra of the lateral deviations d, are determined a s a function of the
SMOOTHED
NON-SMOOTHED
6 8 10 40 60 80100 X[m]
