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Journal of Sound and

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the absence of tensile forces may be accompanied by the formation of &&no contact regions''.For an in"nite beam resting on a #at rigid foundation and subjected to an &&upward''concentrated moving force, Adams and Bogy [10] computed and illustrated graphically the

non-contact lengths, mode shapes and foundation reactions. To replace the &&upward''concentrated moving force by the &&downward'' one, the same problem was studied byChoros and Adams [11].

All the foregoing problems were solved with the analytical methods except that ofreference [2]. However, for most of the engineering problems, the analytical methods are

usually not available and must rely on the numerical methods. Although the "nite elementmethod (FEM) has already been used for the dynamic analysis of structures such as bridge

[12], beam [13], plate [14], etc., application of the FEM to the dynamic analysis of the

&&in"nite'' railway tracks is rare because the &&in"nite'' degrees of freedom (d.o.f ) ofthe &&in"nite'' railway track prevents the availability of the FEM. For this reason, based onthe equivalence between the resonant frequency, the critical moving speed, the frequency

spectrum and the maximum forced vibration responses of the "nite railway and those of thein"nite one, the present authors replaced the in"nite railway by a "nite one so that the FEMis available for the dynamic analysis of the "nite railway [15]. Hence, the dynamicbehaviors of the "nite railway studied in this paper are very close to those of the in"nite one.

The research on the inertial e!ects of the moving loads is rare. In addition to references[5] and [6], Michaltsos et al. [16] studied the inertial e!ect of a moving mass on thedynamic response of a simply supported beam, Esmailzadeh et al. [17] studied a similar

problem by replacing the point mass with the uniform partially distributed moving masses.

In both references [16, 17], the problem was solved by using the series solution method

incorporated with the conventional analytical technique. For a simply supported beam on

an elastic foundation and subjected to a moving load, Thambiratnam and Zhuge [18]

studied the inertial e!ect of the moving load with FEM. In reference [19], Krylovinvestigated the vibration characteristics of the track and the ground for the in"nite railwayanalytically by considering the e!ect of the sleepers and neglecting that of the inertial forceof the single axle load.

In most of the existing literature [1}11, 16, 17, 19], the total number of moving loads issingle and the dynamic response of the load itself was neglected. To accommodate the

practical applications, the dynamic responses of the railway together with the carriage

subjected to the action of a railway carriage with 2}4 suspension systems are studied in thispaper. E!ects such as the inertial forces due to carriage and axle loads, and the elastic forcesand the damping forces due to the springs and dampers of the suspension systems, were all

taken into account in the formulation of the problem.

2. PROPERTY MATRICES FOR THE RAILWAY AND ITS FOUNDATION

Figure 1 shows the mathematical model for a "nite railway track resting on the Winklerfoundation with linear spring constant k

U, where oxyz is the global co-ordinate system, is

the total length of the railway, l is the length of each beam element and the solid circles (E)

denote the nodes for the beam elements. The nth element was enlarged and shown in Fig. 2,

where x L and x L denote the distances from the left end of the railway (point o ) to nodeQ and nodeR of the nth beam element respectively. For convenience, a local co-ordinate

systemQxyz was introduced. The distance from the left end of the beam element (nodeQ)

to any cross-section of the beam element is represented by x, and the distance from node

Q to the ith linear spring is denoted by xG

respectively.

62 J.-S. WU AND P.-Y. SHIH

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In the last expressions, uG(t) (i"1}4) are the nodal displacements shown in Figure 2 and

aXG

and aWG

are the associated shape functions [20].

The substitution of equations (2a) and (2b) into equation (1) results in

;"

+u,2[k]+u, , (4)where

[k]"[k]

#[k]

, (5)

[k]

"l

EIW+a

WV,+a

WV,2 dx, (6)

[k]

"LCG

kU

+aX(

G),+a

X(

G),2. (7)

In the last equations, [k]

is the sti!ness matrix of the beam element alone, [k]

is the

sti!ness matrix due to the nC

linear springs supporting the beam element, and the

combination of [k]

and [k]

de"nes the e!ective sti!ness matrix of a beam element for therailway supported by the linear springs as shown in Figure 1. The notations + , and [ ]appearing in the last equations denote the column vector and the rectangular matrix

respectively.

2.2. ELEMENT MASS MATRIX

If the consistent mass model is employed, the element mass matrix of the beam element

shown in Figure 2 is given by [20]

[m]"[a]2[a] d

+aX(),+a

W(),

. (8b)

3. PROPERTY MATRICES FOR A RAILWAY CARRIAGE

For a multi-roller carriage resting on the railway, the mathematical model is shown in

Figure 3, where G is the center of gravity (c.g.) of the carriage, mCJ and JCJ denote the

e!ective mass and mass moment of inertia of the th carriage, FJ and MJ are the externalforce and moment, u

Jand

Jare the vertical displacement and rotation angle of the th

carriage at the c.g. of carriage, G respectively. The carriage is supported by n suspensionsystems and n rollers. The damping coe$cient and spring constant of the ith suspensionsystem are denoted by c

Gand k

Grespectively, while the mass of the ith roller is denoted by

mG

and the horizontal distance between the ith roller (or ith suspension system) and G is

denoted by rG

as shown in Figure 3.


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Figure 3. The mathematical model for a moving multi-roller carriage on the railway.

3.1. PROPERTY MATRICES

If the un-sprung masses of roller are assumed to be always in contact with the railway,

then the force on the axle of the ith roller is

FG"k

G(uG!uJ#rGJ)#cG (u G!uRJ#rG

QJ), (9)

where rG

is positive if the ith spring/damper system is located on the left side of G and is

negative on the right side of G as shown in Figure 3.

Dynamic equilibrium for the ith roller (mG) requires that

FXN

"FG#m

GuKG"0, (10a)

while the requirement for the dynamic equilibrium of the rigid carriage (mCJ and JCJ) is

FX

" L

G

FG#FJ!mCJuKJ"0, (10b)

M%

"L

G

!FGrG#MJ!JCJ$J"0. (10c)

Substituting equation (9) into equations (10a), (10b) and (10c), one obtains

kG(u

G!uJ#rGJ)#cG (u

G!uRJ#rG

QJ)#mGu

KG"0, (11a)

FJ"mCJuK J!L

G

FG"! L

G

kG(u

G!uJ#rGJ)!

LG

cG(u

G!u J#rG

QJ)#mCJuK J (11b)

MJ"JCJ$J#

L

G

FGrG"

L

G

kGrG(u

G!uJ#rGJ)#

L

G

cGrG(u

G!u J#rG

QJ)#JCJ

$J . (11c)

DYNAMIC RESPONSES OF RAILWAY AND CARRIAGE 65

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Writing equations (11a)}(11c) in the matrix form gives

+FA,"[k

A]+u,#[c

A]+u ,#[m

A]+uK, , (12)

where

+FA,"+0 FJ MJ, , (13a)

+u,"+uG uJ J, , (13b)+u ,"+u

Gu J

QJ, , (13c)

+uK,"+uKG

uKJ $J, . (13d)

From equations (12) and (13) one obtains the element sti!ness, damping and massmatrices for a multi-roller carriage (see Figure 3) to be

[kA]"

kG

!kG

!kGrG

!kG

LG

kG

!LG

kGrG

!kGr !

L

G

kGrG

L

G

kGrG

, (14a)

"

"

""

"""

"

""

"

""

"

"

} } } } } } } } } } } } } } } } } } }

[cA]"

cG

!cG

!cGrG

!cG

L

G

cG !

L

G

cGrG

!cGr !

L

G

cGrG

L

G

cGrG

, (14b)

"

"

""

"

""

"""

"

""

"

"

} } } } } } } } } } } } } } } } } } }

[mA]"

mG

0 0

0 mCT

0

0 0 JCT

. (14c)

"

"

""

"

""

"

"

} } } } } } } } }

For the case ofn "4, r

"a

, r

"a

, r

"!a

and r

"!a

, as shown in Figure 3, one

has

+FA,"+0 0 0 0 FJ MJ, , (15a)

+u,"+u

u

u

u

uJ J, , (15b)

+uJR,"+uJR

uJR

uR

uJR

uRJ QJ,, (15c)

+uJK,"+uJK

uJK

uJK

uJK

uKJ $J, , (15d)

[kA]"k[], (16a)

[cA]"c[], (16b)

[mA]"Um

m

m

m

mCJ JCJX , (16c)


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Figure 4. The static de#ection and rotational angle of the carriage, QR

and QR

, and the reactive forces on theaxles of the rollers, F

G(i"1}n ).

where UX is a diagonal matrix and

[]"

1 0 0 0 !1 a

0 1 0 0 !1 a

0 0 1 0 !1 !a

0 0 0 1 !1 !a

!1 !1 !1 !1 4 !(a

#a

!a

!a

)

a

a

!a

!a

!(a

#a

!a

!a

) (a

#a

#a

#a

)

.

"

"

""

"

""

"

"""

""

"

""

"

"

} } } } } } } }} } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } }

(17)

3.2. EXTERNAL LOADS ON THE RAILWAY

If the reactive forces on the axles of the rollers mG

(i"1}n) are represented by FG, while

the static de#ection and rotational angle of the carriage are represented by QRand

QRrespectively, as shown in Figure 4, then static equilibrium of the carriage requires

that

QR

"mCJg

LG

kG

, (18)

M%

"L

G

!F GrG"0. (19)

Since

F G

"kG(

QR#r

GQR

) (20)

the substitution of equation (20) into equation (19) gives

QR

"!L

GkGrG

LG

kGrG

QR

. (21)


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From equations (18), (20) and (21) one obtains

F G

"kG1!

LG

kGrG

LG

kGrGrG

mCJg

LG

kG (22)

and the total load on the railway at r"

rG is

FMXG"F

G#m

Gg"k

G1!L

GkGrG

LG

kGrGrG

mCJg

LG

kG#mGg (i"1, 2,2, n). (23)

For the special case of kG"k (i"1}n ), equation (23) reduces to

FMXG"1!

LG

rG

LG

rGrG

mCJgn #mGg (i"1, 2,2, n ). (24)

4. EQUATIONS OF MOTIONS FOR THE RELEVANT SYSTEMS

4.1. EQUATION OF MOTION FOR THE RAILWAY

Since the element property matrices for the railway together with the foundation have

been determined by equations (5)}(8), the assembly of these element property matrices willgive the following equation of motion for the whole railway

[M],;,+uK,,;#[K],;,+u,,;"+F,,;, (25)where

[K],;,

"K 2

K,

$ \ $

K, 2

K,,

,;,

, (26a)

[M],;,"

M

2M

,

$ \ $

M, 2

M,,,;,

, (26b)

+u,"+u

u

2 u,

,,;

"+w

w

2 w,M

,M,

,;, (27a)

+uK,"+uK

uK

2 uK,

,,;

"+wK

$

wK

$

2 wK,M

$,M

,,;

. (27b)

In equations (25)}(27), the notations NM and N denote the total number of nodes and that ofd.o.f. for the whole railway. Since there are two d.o.f. for each node, it is evident that

N"2NM.For convenience of formulation, the dimension of equation (25) is extended from N to

N"N#2, i.e.,

[MM ],Y;,Y

+uK,,Y;

#[KM],Y;,Y

+u,,Y;

"+FM,,Y;

, (28)


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where

[MM ],Y;,Y

"

M

2 M,

0 0

$ \ $ $ $

M,

2 M,,

$ $

0 2 2 0 0

0 2 2 0 0

, (29a)

"

"

""

"

""

"

""

"""

"

"

} } } } } } } } } } } } } } } }

[KM],Y;,Y

"

K

2 K,

0 0

$ \ $ $ $

K,

2 K,,

$ $

0 2 2 0 0

0 2 2 0 0

, (29b)

"

"

""

"

""

"

""

"

""

"

"

} } } } } } } } } } } } } } } }

+u,,Y;

"+u

u

2 2 u,\

u,

uJ J,,Y; (30a)"+w

w

2 w,M

,M

uJ J,,Y;,

+uK,,Y;

"+uK

uK

2 2 uK,\

uK,

uKJ $J,,Y; (30b)

"+wK

$

wK

$

2 wK,M

$,M

uKJ $J,,Y;,

+FM,,Y;

"+0 0 2 2 0 0 0 0,,Y;

. (30c)

Since all the external loads move with the carriage, they are considered as the coe$cients ofthe external force vector for the carriage, +FM

A,,Y;

[see equation (37)], rather than those for

the railway, for convenience. This is the reason why all the coe$cients of+FM,,Y;

as shown

in equation (30c) are equal to zero.

4.2. EQUATION OF MOTION FOR THE CARRIAGE

If the carriage moves on the railway with speed < from left of the railway to right, then

the position of the c.g. of the carriage, G, at any time t is given by

x%

(t)"x

#r

#

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Figure 5. De"nitions for the distance x%

(t) and the relevant notations.

where rG

is the distance between the ith roller and G, while n is the total number of rollers forone carriage. For the case of x

"0 at t"0, from equations (31) and (33) one obtains

N

"1 as it should be.

For convenience, we set

NL>

"N#1, NL>

"N#2"N (34a, b)

and extend the dimension of the equation of motion for the carriage (cf. equations (14)}(16))

from n#2 to N, i.e.,

[mA(t)]

,Y;,Y+uK (t),

,Y;#[c

A(t)]

,Y;,Y+u (t),

,Y;#[kM

A(t)]

,Y;,Y+u (t),

,Y;"+FM

A(t),

,Y;. (35)

All the coe$cients for the property matrices of equation (35) are equal to zero except thefollowing ones:

XM,G,H

"XGH

(i, j"1, 2,2, n , n#1, n#2), (36)

where XM"mA, c

Aor kM

A; X"m

A, c

Aor k

A; and the values of N

Gand N

Hare determined by

equations (33) and (34), while XGH appearing in equation (36) refer to the coe$cients of theproperty matrices as shown in equation (14) or equation (16).

Likewise, all the coe$cients of the external force vector +FMA(t),

,Y;are also equal to zero

except the ones below

FMA,G

"FMXG

(i"1, 2,2, n ), (37a)

FMA,L>

"FJ , FMA,L>"MJ , (37b, c)

where FMXG

(i"1}n ) are the external loads on the railway through the rollers of the carriage(see equations (23) and (24)).

In general, the nodal forces +FM GA

, induced by a concentrated force FMXG

on a beam element

are given by [21]

+FMGA

,"+aX(),2FM

XG, (38)


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in which +aX

(), is the shape function given by equation (3a) and +FMGA

, is to take the form

+FMG,"+ fG

m G

fG

mG

,, (39)

where the subscripts 1 and 2 for fG and mG refer to the nodal force and the nodal moment at

nodes Q and R of the beam element on which FMXG

is located, respectively (see Figure 2).

It is evident that one has

+FMG,"+FMXG

0 0 0, (40a)

if FMX G

is located at node Q. Similarly, if FMXG

is located at node R, one has

+FMG,"+0 0 FMXG

0,. (40b)

For simplicity, the locations for the rollers (r

, r

,2, rL), the length of each beam element

(l), the moving speed of the carriage (

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The given data for the railway are: Young's modulus E"20)6;10 N/m, moment ofinertia for the cross-sectional area I

W"2)037;10\ m, mass density "7850 kg/m,

cross-sectional area AM"6)37;10\ m, mass per unit length m "AM"50 kg/m, radius ofgyration

E"(I

W/AM"0)032 m, spring constant for the Winkler foundation

kU

"3)73;10 N/m, and length of each beam element l"1)0 m. While the given dataassociated with the carriage are: total mass of the carriage m

CT"2)0;10 kg, mass moment

of inertia JCT

"1)0;10 kg m, mass of each axle roller (or wheel) mG"100)0 kg

(i"1, 2,2, n ), spring constant for each suspension system kG"3)0;10 N/m, and

damping coe$cient for each damper cG"2)0;10 N s/m. For convenience, the external

loads on the c.g. of carriage (G) are assumed to be zero, i.e., FJ(t)"MJ(t)"0.

5.1. COMPARISON WITH SOME ANALYTICAL SOLUTIONS

For a "nite uniform beam (with length ) resting on the elastic foundation and simply

supported, the natural frequencies H and the vertical (transverse) displacements wX(M

, t) atposition M"x / due to a stationary pulsating concentrated force P (t)"PM sin Ct applying

at M

"x

/ are respectively given by [22]

H"a

(j#, j"1, 2,2, (43)

wX

(M, t)"2PM

m

H

sin(jM)sin(jM

)sin

Ct

a ( j#)!C

!

C

sin Ht

H(

H!

C) (44)

where

a"EI

Wm

, "

k

UEI

W

. (45a, b)

For the railway and elastic foundation studied in this paper with length "100 m, the

lowest 12 natural frequencies H

( j"1}12) obtained from equation (43) are shown incolumn 2 ofTable 1, while those obtained from the FEM introduced in this paper are listed

in column 3 for the lumped-mass model and in column 4 for the consistent-mass model,

respectively. From Table 1 one "nds that, among the two FEM models, the naturalfrequencies obtained from the lumped-mass model are more close to those obtained from

equation (43). This is because the assumptions made by the lumped-mass model are more

close to those made by equation (43). However, the di!erences between the values ofH

( j"1}12) obtained from the lumped-mass model and the corresponding ones obtainedfrom the consistent-mass model are negligible.

If the last railway is subjected to a stationary pulsating force P (t)"98000 sin 3t

N applied at the central point of the railway (i.e. M

"x

/ "0)5), the time histories for

vertical central displacement wX

(0)5, t) of the railway are shown in Figure 6, where the solid,

the dashed and the dotted lines denote the time histories obtained from equation (44) (with

j"1}1000), the FEM based on the lumped-mass model and the FEM based on theconsistent-mass model (with element length l"1)0 m, time interval t"0)01 s)

respectively. It is evident that the dynamic responses of the railway obtained from the FEM

(either based on the lumped-mass model or the consistent-mass model) are very close to

those obtained from the analytical solution given by equation (44).


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TABLE 1

he lowest 12 natural frequencies of a simply supported beam on elastic foundationR

Natural frequencies H

(rad/s)

Mode By equation (43) By FEM

numbers(j ) umped-mass model Consistent-mass model

1 863)712962 863)712962 863)7027602 863)713672 863)713672 863)7041483 863)716748 863)716748 863)7079434 863)725030 863)725030 863)7080435 863)742492 863)742492 863)7115426 863)774245 863)774245 863)7249257 863)826534 863)826534 863)7600918 863)906735 863)906734 863)8209879 864)023357 864)023354 863)916315

10 864)186033 864

)186026 864

)055930

11 864)405519 864)405505 864)25083512 864)693684 864)693655 864)513174

R Total beam length "100 m.

Figure 6. The time histories of a simply supported beam ("100 m) on elastic foundation subjected toa pulsating force P(t)"PM sin

Ct at mid-span (PM"98 000 N,

C"3 ) 0 rad/s): **, by equation (44); } } } } }, by

FEM (lumped-mass model); , by FEM (consistent-mass model).

5.2. NATURAL FREQUENCIES AND MODE SHAPES FOR THE RAILWAY

For the free-free railway with length "500 m (and total number of "nite elementsN

C"/l"500) studied in all the follwoing subsections, the natural frequencies and mode

shapes for the 1st, 4th and 10th modes are shown in Figure 7. From the "gure, one sees that

+

+

+863)712 rad/s, this is because the natural frequencies for a railway will

approach a continuous frequency spectrum when the ratio /g of the railway exceedsa certain value (e.g., /g"500/0)032'15000 for the present example) as shown inreference [15]. Besides, Figure 7 shows that all the higher mode shapes of the railway look


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Figure 7. The mode shapes for the free}free railway with length "500 m: (a) the 1st mode with

"863)712913 r/s; (b) the 4th mode with

"863)712915 r/s; (c) the 10th mode with

"863)713030 r/s.

like the travelling waves with wave length

I

"

(k/2)"

2

k, k"1, 2,2, (46)

where k refers to the kth mode shape. It is evident that if the resonance occurs at the kth

mode, then the associated exciting period will be

CI

"I

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