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) k≥n h:=(y0−x0)/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
s
knk
→
→∞s
nk=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
3Fig. 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
P0(λi)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))/P0(λi)−
h
n−1
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 Recurrence formulae
ForReλi <0we haveσi =1and recursion ui(xk)= G
P0(λi)H(xk−vt)−hX
j≤k
ci(xj)ui(xj)
with solution
ui(xk)= G
P0(λi)H(xk−vt) 5
j≤k
1 1+ci(xj)h. Ifn→ +∞holds, then we obtain
n→+∞lim 5
xj≤x(1+ci(xj)h)=
n→+∞lim 5
xj≤x((1+ci(xj)h)1/(ci(xj)h))ci(xj)h=
exp lim
n→+∞
X
xj≤x
ci(xj)h=exp
x
Z
x0
ci(y)dy. In the caseReλi >0we have recurrence formula
ui(xk)= −G
P0(λi)H(vt−xk)+hX
j>k
ci(xj)ui(xj)
with solution
ui(x)= −G
P0(λi)H(vt−x) 5
xj>x(1+ci(xj)h)→
−G
P0(λi)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
P0(λi)
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))πx−1)·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( (
( )
ii 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
00
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
4N
m/s
= 40
v . The parameters of the subgrade are given by constants s
0= 9 ⋅ 10
7N/m
2, k
0= 4 . 6 ⋅ 10
4Ns/m
2and 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
240 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