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Ŕ periodica polytechnica

Transportation Engineering 39/2 (2011) 83–85 doi: 10.3311/pp.tr.2011-2.06 web: http://www.pp.bme.hu/tr c Periodica Polytechnica 2011 RESEARCH ARTICLE

On dynamics of the track/vehicle

system in presence of inhomogeneous rail supporting parameters

VilmosZoller/IstvánZobory

Received 2010-09-06

Abstract

We study the dynamics of the train/track system in case of an inhomogeneous longitudinal subgrade stiffness/damping distri- bution. Our model consists of a Bernoulli-Euler beam, fixed at infinity, laying on a viscoelastic Winkler foundation of con- tinuously varying stiffness/damping parameters, and a damped oscillatory load moving along the beam at a constant velocity.

In order to obtain an approximate, semianalytical solution we build up a new discretization method based on the approxima- tion of the discretized stiffness/damping values by generalized functions. The approximate solutions tend to continuous func- tions represented in a closed-form, analytical fashion.

Keywords

railway track dynamics · beam equation · inhomogeneous supporting field

Vilmos Zoller

Dept. Railway Vehicles, BME, H-1521 Budapest, Hungary e-mail: zoller.vilmos@rkk.uni-obuda.hu

István Zobory

Dept. Railway Vehicles, BME, H-1521 Budapest, Hungary e-mail: railveh@rave.vjt.bme.hu

1 Introduction

In our simple model we consider a viscoelastic Winkler foun- dation of continuously varying stiffness/damping parameters given by functionss0+s(x), k0+k(x), a Bernoulli-Euler beam of parameters E I, ρA laying on the subgrade, and a load of weightG moving along the beam at a longitudinal velocity v, and vibrating dampedly at complex frequencyw=α+iω. In caseα =0 we have a harmonic load, whilew = 0stands for the case of a constant load.

The motion of the system is governed by the Bernoulli-Euler beam equation

E I∂4z

∂x4 +ρA∂2z

∂t2 +(k0+k(x))∂z

∂t +(s0+s(x))z=

Gexp(wt)δ(x−vt), (1)

whereδstands for Dirac’s unit impulse distribution, while con- tinuous functions s and k vanish outside the finite interval [x0,y0].

Eq. (1) satisfies boundary conditions lim

|x|→∞z(x,t)=0 (2) at±∞.

Fig. 1. System model

2. APPROXIMATE BOUNDARY PROBLEM

First we discretize parameter functions of the subgrade in the following way.

We build up step functions

otherwise.

0

, 1 , , 1 , 0 ), / , [ if / ) : ( )

( + =

= s x k n x x x hn k j n x

skn j j j K

by discretizing continuous function (or ), for , where and

)

s(x k(x) kn h:=(y0x0)/n n

j jh x

xj:= 0+ , =0,1,K, hold.

In this case is satisfied. For the limit, which is a generalized function, in case

) ( ) ( lim snn x sx

n =

+∞

k→ +∞

δ

=

= +∞ = n

j j j

k kn

n x s x s x x x h

s

1( ) ( )

) ( lim : ) ( holds, and for its limit

) ( d ) ( ) ( ) (

lim 0

0

x s y y x y s x s

y x

n n = δ =

+∞

is satisfied, hence we have the following commutative diagram:

2

Fig. 1.System model

On dynamics of the track/vehicle system inhomogeneous rail supporting parameters 2011 39 2 83

(2)

s

kn

k

s

n

k=n

Ì

s

n

The different limits of discretized function sequences are illustrated in Fig. 2.

Æ

Æ

Æ

Ì È

Fig.2. The limits of the discretized function sequences

The beam equation discretized to n parts in case

k → +∞

has the form

4 2

0 0

4 2

u u u

EI A k s

t u

x t

∂ + ρ ∂ + ∂ +

∂ ∂ ∂ =

1

exp( ) ( ) n ( ) ( , )j j ( )j ( , ) (j j)

j

G wt x vt s x u x t k x u x t h x x

= t

⎛ ∂ ⎞

δ − − ∑ ⎜⎝ + ∂ ⎟⎠ δ −

. (3)

Response to the first term is

3

Fig. 2. The limits of the discretized function sequences

2 Approximate boundary problem

First we discretize parameter functions of the subgrade in the following way.

We build up step functions skn(x):=

( s(xj)knifx∈[xj,xj +hnk ),j =0,1, . . . ,n−1, 0otherwise.

by discretizing continuous functions(x)(ork(x)), fork ≥ n, whereh :=(y0−x0)/nandxj :=x0+ j h, j =0,1, . . . ,n hold.

In this case lim

n→+∞snn(x)= s(x)is satisfied. For the limit, which is a generalized function, in casek→ +∞

sn(x):= lim

k→+∞skn(x)=

n

X

j=1

s(xj)δ(x−xj)h

holds, and for its limit

n→+∞lim sn(x)=

y0

Z

x0

s(y)δ(x−y)dy=s(x)

is satisfied, hence we have the following commutative diagram:

is satisfied, hence we have the following commutative diagram:

n

k

kn

s

s

k=n

Ì

s

n→∞

The different limits of discretized

1

The different limits of discretized function sequences are il- lustrated in Fig. 2.

The beam equation discretized ton parts in casek → +∞

has the form E I∂4u

∂x4+ρA∂2u

∂t2 +k0∂u

∂t +s0u=Gexp(wt)δ(x−vt)−

n

X

j=1

s(xj)u(xj,t)+k(xj)∂

∂tu(xj,t)

hδ(x−xj). (3) Response to the first term is

G

4

X

i=1

σi

P0i)exp(wt+λi(x−vt))H(σi(x−vt)) with characteristic polynomial

P(λ)=E Iλ4+ρAv2λ2−v(k0+2ρAw)λ+(s0+k0w+ρAw2) (4) and signsσi := −sgn Reλi, P(λi)=0, i =1, . . . ,4, see e.g.

[1],[3].

The full response has the formu(x,t)=

4

P

i=1

exp(wt+λi(x− vt))ui(x), and can be given in a recursive way as

4

X

i=1

σiexp(wt+λi(x−vt)){GH(σi(x−vt))/P0i)−

h

n1

X

j=1

ci(xj)ui(xj)H(σi(x−xj))}

with functions defined by

ci(x):=σiexp(λix)s(x)+(w−λiv)k(x) 4E Iλ3i ,

i =1,2,3,4, (5)

cf. [2],[4].

Per. Pol. Transp. Eng.

84 Vilmos Zoller/István Zobory

(3)

3 Recurrence formulae

ForReλi <0we haveσi =1and recursion ui(xk)= G

P0i)H(xk−vt)−hX

jk

ci(xj)ui(xj)

with solution

ui(xk)= G

P0i)H(xk−vt) 5

jk

1 1+ci(xj)h. Ifn→ +∞holds, then we obtain

n→+∞lim 5

xjx(1+ci(xj)h)=

n→+∞lim 5

xjx((1+ci(xj)h)1/(ci(xj)h))ci(xj)h=

exp lim

n→+∞

X

xjx

ci(xj)h=exp

x

Z

x0

ci(y)dy. In the caseReλi >0we have recurrence formula

ui(xk)= −G

P0i)H(vt−xk)+hX

j>k

ci(xj)ui(xj)

with solution

ui(x)= −G

P0i)H(vt−x) 5

xj>x(1+ci(xj)h)→

−G

P0i)H(vt−x)exp

−

x

Z

y0

ci

, n→ +∞.

Summarizing the results obtained above we get a finite closed- form integral formula for the continuously supported problem (1-2) in form

z(x,t)=G

4

X

i=1

σi

P0i)

exp

wt+λi(x−vt)−

x

Z

li

ci(y)dy

H(σi(x−vt)) (6)

withli :=

( x0if Reλi <0, y0if Reλi >0. 4 Numerical results

In the example, similar to that of [5], the parameters of the beam areE I =6·106Nm2,ρA=60kg/m. The weight of the constant load isG = 6.5·104N, while its horizontal velocity isv = 40m/s. The parameters of the subgrade are given by constantss0 = 9·107N/m2,k0 = 4.6·104Ns/m2and single sinusoidal waves

s(x)=

( (cos((20πm)x )−1)·107N/m2, if 0 m≤x≤40m, 0 otherwise,

k(x)=

( (cos((20m))πx1)·2500Ns/m2, if 0 m≤x≤40m, 0 otherwise.

Summarizing the results obtained above we get a finite closed-form integral formula for the continuously supported problem (1-2) in form

4 1

( , ) exp ( ) ( )d H( (

( )

i

i x

i i i

i i l

z x t G wt x vt c y y x vt

=

P

⎛ ⎞

= ∑ ′ λ σ ⎜ ⎜ ⎝ + λ − − ∫ ⎟ ⎟ ⎠ σ − )) (6)

with

0

0

if Re 0, : if Re 0.

i i

i

l x y

λ <

= ⎨ ⎧ ⎩ λ >

4. NUMERICAL RESULTS

In the example, similar to that of [5], the parameters of the beam are

, . The weight of the constant load is ,

while its horizontal velocity is

2 6

Nm 10 6 ⋅

EI = ρ A = 60 kg/m G = 6 . 5 ⋅ 10

4

N

m/s

= 40

v . The parameters of the subgrade are given by constants s

0

= 9 ⋅ 10

7

N/m

2

, k

0

= 4 . 6 ⋅ 10

4

Ns/m

2

and single sinusoidal waves

7 2

(cos( /(20 m)) 1) 10 N/m , if 0 m 40 m, ( ) 0 otherwise,

x x

s x = ⎨ ⎧ ⎪ π − ⋅ ≤ ≤

⎪⎩

(cos( /(20 m)) 1) 2500 Ns/m , if 0 m

2

40 m, ( ) 0 otherwise.

x x

k x ⎧ ⎪ π − ⋅ ≤ ≤

= ⎨ ⎪⎩

Fig.3. The vertical position z vt t ( , ) of the load

5

Fig. 3.The vertical positionz(vt,t)of the load

References

1 De Pater A D, Inleidend onderzoek naar het dynamisch gedrag van spoorstaven, 1948. Thesis: Waltman, Delft.

2 Zobory I, Zoller V,Dynamic response of a railway track in case of a moving complex phasor excitation., Progress in Industrial Mathematics at ECMI 96 (Brøns M, Bendsøe M P, Sørensen M P, eds.), Teubner/Stuttgart, 1997, 85- 92.

3 Zobory I, Zoller V, Zibolen E,Theoretical investigations into the dynam- ical properties of railway tracks using a continuous beam model on elastic foundation., Periodica Polytechnica Ser. Transp. Eng.,22(1), (1994), 35-54.

4 Zoller V, Zobory I,Relations between the motion-responses caused by fixed and moving loads acting on discretely supported strings and beams., Progress in Industrial Mathematics at ECMI 2000 (Anile M, Capasso V, Greco A, eds.), Springer/Berlin, 2002, 657-661.

5 Zoller V, Zobory I,Track dynamics with longitudinally varying track stiff- ness., Proc. 9th Mini Conf. Vehicle System Dynamics, Identification and Anomalies (I. Zobory, ed.), Budapest Univ. Technology Economics, 2004, 119-126.

On dynamics of the track/vehicle system inhomogeneous rail supporting parameters 2011 39 2 85

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