Bowe (2008) Unsprung Wheel-beam Interactions Using Modal and Finite Element Models

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Unsprung wheel-beam interactions using modal and finiteelement models

C.J. Bowe *, T.P. Mullarkey

Department of Civil Engineering, National University of Ireland, Galway, Ireland

Received 7 September 2007; accepted 18 January 2008Available online 11 March 2008

Abstract

The aim of this study is to simulate the dynamic vertical response of an unsprung mass, i.e. wheel, traversing a beam, i.e. bridge or rail.This is achieved by using the authors modal or finite element method to model the dynamic interaction that exists between the unsprungmass and the beam. In this paper, the authors compare the final form of the modal method with the final form of the finite elementmethod, and they clearly demonstrate that if the mode shapes of the modal method are replaced by the finite element weighting andshape functions, the finite element method is derived, apart from the stiffness matrix. The results for the moving load compare very wellwith Akin and Mofids [Akin JE, Mofid M. Numerical solution for response of beams with moving mass. J Struct Eng 1989;115(1):12031] solution; however, the results for the moving mass are very different. Other authors who are getting agreement with Akin and Mofids(1989) moving mass are ignoring the issue of convective acceleration. 2008 Elsevier Ltd. All rights reserved.

Keywords: Unsprung moving mass; Convective acceleration; Modal superposition; Finite element; Similarity of forms

1. Introduction

Research on the dynamic response of vehicles traversingbridges has long been an interesting topic of civil engineer-ing. Depending on the complexity of the model and resultssought, one can simulate a vehicle as a moving constantforce as done by Biggs [2], Akin and Mofid [1], Wu et al.[3], and Bowe and Mullarkey [4]; a moving sprung massas done by Biggs [2], Yang and Wu [5], and Bowe and Mul-larkey [6]; or as a moving unsprung mass as done by Biggs

[2], Akin and Mofid [1] and Yang et al. [7]. In this paper,the authors focus on the latter vehicular model by develop-ing an analytical numerical model as well as a finite elementmodel for a moving unsprung mass traversing a beamwithin the ANSYS finite element program.

When one deals with a moving unsprung mass its verti-cal position must be the same as the vertical position of the

point of the beam directly underneath. However, since themoving unsprung mass is moving horizontally, its verticalvelocity is not the same as the vertical velocity of the pointof the beam directly underneath, (which we will call thelocal vertical velocity of the mass). In fact the verticalvelocity of the moving unsprung mass is equal to the verti-cal velocity of the beam plus a convective term. Similarlythe vertical acceleration of the moving unsprung mass isequal to the vertical acceleration of the beam plus addi-tional convective terms. In Akin and Mofids [1] moving

unsprung model the convective velocity and accelerationare omitted from the model; thus their solution is inaccu-rate. This inaccuracy is overcome in the authors paperby including both the local and convective terms for themoving unsprung mass and is clearly shown in the develop-ment and result sections.

The paper begins by developing the modal characteristicshape and natural frequencies for any beam support type,which is later particularized for a cantilever beam. Fromthe differential equation governing the vibration of a beamsubjected to the moving unsprung wheel the authors derive

0965-9978/$ - see front matter 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advengsoft.2008.01.002

* Corresponding author. Tel.: +353 91 524411x3086; fax: +353 91750507.

E-mail address: [email protected] (C.J. Bowe).

www.elsevier.com/locate/advengsoft

Available online at www.sciencedirect.com

Advances in Engineering Software 39 (2008) 911922
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a modal analytical solution with both the local and convec-tive acceleration of the unsprung mass taken into account.During the development, by normalising the beam modes,one can further reduce the complexity of the solutions.

Next, the finite element method for the movingunsprung mass is developed, which is somewhat different

from the development of the wheelrail contact (WRC) ele-ment in Bowe and Mullarkey [6] for a moving sprung mass.In the present paper additional mass, stiffness and dampingmatrices are added to the beam matrices, whereas onlyadditional stiffness matrices were required in the WRC ele-ment. This paper highlights the similarities and differencesbetween the modal and finite element model as far as theform of the final matrices is concerned. The weightingand shape functions of the finite element method play asimilar role to that played by the mode shapes in the ana-lytical numerical solution. The difference between the twomethods occurs in the representation of the stiffness matrix.Integration by parts gives the finite element stiffness matrix

a different form from that of the modal stiffness matrix.Additionally, the second derivatives of the beam elementshape functions are discontinuous in the finite elementmethod.

In order to validate the modal and finite elementmethod, the authors compare their results with those ofthe Bowe and Mullarkey [6] moving sprung model, wherethe Hertzian spring is given a reasonably large stiffnessand the moving sprung wheel is prevented from losing con-tact with the beam, i.e. no separation. In addition, thispaper highlights the drastic effects of omitting the convec-tive acceleration terms from an unsprung moving mass

problem as the results obtained are very different fromthe Akin and Mofids [1] solution. Nevertheless, authorssuch as Yang and Wu [5] are still comparing their devel-oped models with Akin and Mofids [1] inaccurate solution,ignoring the issue of convective acceleration.

2. Natural frequencies and modal shapes for a beam

In this section, the authors develop the natural frequen-cies and modal shapes of single-span beams as shown inFig. 1. The differential equation governing the free vibra-tion of a beam is expressed according to Biggs [2] as:

EIo

4vx; tox4

m o2vx; tot2

0 1

where E is Youngs modulus of elasticity, I is the secondmoment of area and m is the mass per unit length of thebeam. It should also be noted that beam deflection v(x,t)is expressed as a function of both time t and position xalong the span.

Since the right-hand side of Eq. (1) is set equal to zero,

the separation of variables method allows one to define thebeam deflection of the nth mode by:

vnx; t fnt/nx 2where fn(t) is a time function and /n(x) is the characteristicshape. The substitution of Eq. (2) into Eq. (1) gives rise totwo ordinary differential equation (ODE) governing fn(t)and /n(x) as follows:

d2

dt2fnt Hfnt 0 3a

d4

dx4/nx mH

EI/nx 0 3b

In order that Eq. (3) has a periodic solution, Hmust be po-sitive, and then the solution for fn(t) is as follows:

fnt C1 sinffiffiffiffiH

pt C2 cos

ffiffiffiffiffiHt

p4

whereffiffiffiffiH

pis the natural frequency of the beam, xn.

Equally, the ODE governing the characteristic shape/n(x) has the following solution:

/nx An sin anx Bn cos anx Cn sinh anx Dn cosh anx 5a

where

an ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimx2n=EI4q 5bA;Bn;Cn;Dn are constant values evaluated by the boundaryconditions of the beam. It should also be noted that Eq.(5a) is valid for any type of end restraint. In this paper,the dynamic effects caused by an unsprung wheel traversinga cantilever beam are investigated; hence, one must evalu-ate the beams natural frequencies and mode shapes, whichare found in the Appendix A.

3. Analytical numerical (modal superposition) solution of a

moving unsprung wheel

The authors are now in the position to develop theiranalytical numerical solution for a moving unsprung wheel

z

x

y, v

0m, EI

L

Fig. 1. Single-span beam and coordinate system.

z, k

x, i

y, v,j

0

m, EI

( )dX tc

dt=

L

Mw

Fig. 2. Moving unsprung wheel traversing a beam.

912 C.J. Bowe, T.P. Mullarkey / Advances in Engineering Software 39 (2008) 911922
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Mw traversing a beam at a constant speed as shown inFig. 2. The differential equation governing the vibrationof a beam subjected to the moving unsprung wheel masscan be expressed, repeating Eq. (54) in Appendix B, asfollows:

EI

o4v

x; t

ox4 mo

2v

x; t

ot2 px 6awhere

px Ftdx Xt 6bwhere F(t) is the force imparted to the moving unsprungmass by the beam, positive if acting in the positive y-direc-tion. The Dirac delta function is represented by d. The po-sition vector p(t) of the moving point mass is as follows:

pt Xti Ytj 7where i and j are the unit vectors in the x and y directions,respectively. The horizontal speed c of the moving mass

along the beam in Eq. (7) is

dXtdt

c 8

as shown in Fig. 2. The equation of motion for the movingmass in the y-direction is

Mwd2Yt

dt2 Mwg Ft 9

where g is the acceleration due to gravity. Eliminating F(t)in Eq. (6) using (9) gives the following:

EIo4vx; tox4

m o2vx; tot2

Mw d2Ytdt2

Mwg

dx Xt

10

Since the moving mass is always in contact with the beam,the constraint equation is

Yt vx; tjxXt 11

Therefore the vertical velocity of the moving mass is asfollows:

dYtdt

ovx; totxXt ovx; t

oxxXt dXt

dt|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}convective velocity

12

Furthermore the vertical acceleration of the moving mass isSubstituting Eq. (13) into (10) yields

EIo

4vx; tox4

m o2vx; tot2

Mw o2vx; tot2

xXt 2 o2vx; t

oxotxXt dXt

dt

o2vx; tox

2 xXtdXt

dt

dXt

dt ovx; t

oxxXt d2Xt

dt2 g

d x Xt 14

In order to solve Eq. (14), we use the method of modalsuperposition; whereby v(x,t) can be represented accordingto Biggs [2] as follows:

vx; t Xn

rnt/nx 15

where /n(x) is the nth characteristic shape discussed aboveand rn(t) is the nth function of time which has to be calcu-lated. Substituting Eq. (15) into (14) gives the followingequation:

EIXNn1rnt/ivn x m

XNn1

rnt/nx

MwXNn1

rnt/n Xt 2dXt

dt

XNn1

_rnt/0nXt"

dXtdt

2XNn1rnt/00n Xt

d2Xtdt2

XNn1rnt/0n Xt g

#d x Xt 16

where a dot over rn(t) represents a derivative with respect totime, and a dash over /n(x) represents a derivative with re-spect to x. Substituting Eqs. (3b) and (5b) into Eq. (16) toeliminate the fourth derivative of/n(x) gives the followingequation:

EIXNn1rnta4n/nx m

XNn1

rnt/nx

MwXNn1

rnt/n Xt 2dXt

dt

XNn1

_rnt/0nXt"

dXt

dt 2

XN

n1rn

t

/

0n

X

t

d

2Xtdt2

XNn1rnt/0nXt g

#dx Xt 17

As the reader will see, this replacement of/ivn x, using Eqs.(3b) and (5b), is an important distinction between the mod-

d2Ytdt2

o2vx; tot2

xXt

2 o2vx; toxot

xXt

dXtdt

o2vx; tox2

xXt

dXtdt

dXtdt

ovx; tox

xXt

d2Xtdt2

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}convective accelaration13

C.J. Bowe, T.P. Mullarkey / Advances in Engineering Software 39 (2008) 911922 913
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al method and the finite element system. Multiplying bothsides of Eq. (17) by /i(x), i= 1,2,3, . . . N, integrating alongthe beam length, and using Eq. (43) from the Appendix A,gives the following set of ordinary differential equationsgoverning rn(t), n = 1, 2,3 . . . N.

EIXNn1rnta4nLdin m

XNn1

rntLdin

MwXNn1

rnt/nXt 2dXt

dt

XNn1

_rnt/0nXt"

dXtdt

2XNn1rnt/00n Xt

d2Xtdt2

XNn1rnt/0nXt

g#/iXt; i 1; 2; 3; . . .N 18

where din = 1 when i= n, and din = 0 when i6 n. Rear-ranging Eq. (18) in the order of the mass, damping andstiffness terms with the forcing term on the right-hand sidegives the following:

XNn1mLdin Mw/iXt/nXtf grnt

XNn1

2MwdXt

dt/iXt/0nXt

_rnt

XNn1EILdina

4n Mw

dXt

dt 2

/iXt/00nXt(Mw d

2Xtdt2

/iXt/0nXtrnt Mwg/iXt;

i 1; 2; 3; . . .N 19

Since Akin and Mofid [1] only include the local accelera-tions of the moving unsprung mass, terms containingdX(t)/dt and d2X(t)/dt2 are omitted by Akin and Mofid[1]. Expanding Eq. (19) into matrix form gives:

mL

1 0 0 . . . 0

0 1 0 . . . 0

0 0 1 . . . 0

. . . . . . . . . . . . . . .

0 0 0 . . . 1

26666664

37777775r1

r2

r3

. . .

rN

8>>>>>>>>>>>:

9>>>>>>=>>>>>>;

Mw

/1/1 /1/2 /1/3 . . . /1/N

/2/1 /2/2 /2/3 . . . /2/N

/3/1 /3/2 /3/3 . . . /3/N

. . . . . . . . . . . . . . .

/N/1 /N/2 /N/3 . . . /N/N

26666664

37777775

r1

r2

r3

. . .

rN

8>>>>>>>>>>>:

9>>>>>>=

>>>>>>;

2Mw dXtdt

/1/01 /1/

02 /1/

03 . . . /1/

0N

/2/01 /2/

02 /2/

03 . . . /2/

0N

/3/01 /3/

02 /3/

03 . . . /3/

0N

. . . . . . . . . . . . . . .

/N/01 /N/

02 /N/

03 . . . /N/

0N

26666664

37777775_r1

_r2

_r3

. . .

_rN

8>>>>>>>>>>>:

9>>>>>>=>>>>>>;

EIL

a41 0 0 . . . 0

0 a42 0 . . . 0

0 0 a43 . . . 0

. . . . . . . . . . . . . . .

0 0 0 . . . a4N

2666666437777775r1

r2

r3

. . .

rN

8>>>>>>>>>>>:9>>>>>>=>>>>>>;

Mw dXtdt

2 /1/001 /1/

002 /1/

003 . . . /1/

00N

/2/001 /2/

002 /2/

003 . . . /2/

00N

/3/001 /3/

002 /3/

003 . . . /3/

00N

. . . . . . . . . . . . . . .

/N/001 /N/

002 /N/

003 . . . /N/

00N

26666664

37777775r1

r2

r3

. . .

rN

8>>>>>>>>>>>:

9>>>>>>=>>>>>>;

Mw d2

Xtdt2

/1/01 /1/

02 /1/

03 . . . /1/

0N

/2/01 /2/

02 /2/

03 . . . /2/

0N

/3/01 /3/02 /3/03 . . . /3/0N. . . . . . . . . . . . . . .

/N/01 /N/

02 /N/

03 . . . /N/

0N

26666664

37777775r1

r2

r3

. . .

rN

8>>>>>>>>>>>:

9>>>>>>=>>>>>>;

Mwg

/1

/2

/3

. . .

/N

8>>>>>>>>>>>:

9>>>>>>=>>>>>>;20

Rewriting Eq. (9), the reaction force between the unsprungwheel and the rail can be computed as;

Ft MwgMw d2Ytdt2

21

where F(t) is the force imparted tothe moving unsprungmassby the beam, positive if acting in the positive y-direction.

4. Finite element system of a moving unsprung wheel

In Figs. 3 and 4, the reader can see an unsprung wheeltraversing a beam finite element. The diagram presentsthe coordinate system i.e. x positive along the beam ele-ment, y positive upward and z positive outwards. The ori-

gin of the coordinate system is at local node 1 of the beam.For two-dimensional problems, the deflection in the y-direction and rotation about the z-axis are defined by R1and R2, respectively, at local node 1 of the beam, while

R1 R3

R2

R4z

x

y

0

m, EI

dX(t)c

dt=

Mw1 2

Fig. 3. Moving unsprung wheel traversing a beam finite element.

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R3 and R4 are the deflection in the y-direction and rotationabout the z-axis, respectively, at local node 2 of the beam.

From Bowe and Mullarkey [6] with a change of nota-tion, the vertical displacement at any point along the beamcan be calculated using Eq. (22), where U1 and U3 are thetransverse displacement shape functions and U2 and U4are the rotational shape functions.

vx; t

X4

n1Rn

tUn

x

22

where

U1x 1 2x=l3 3x=l2; implying U10 1;U010 0; U1l 0 and U01l 0

U2x xf1 2x=l x=l2g; implying U20 0;U020 1; U2l 0 and U02l 0

U3x 3x=l2 2x=l3; implying U30 0;U030 0; U3l 1 and U03l 0

U4

x

x

fx=l

2

x=l

g; implying U4

0

0;

U040 0; U4l 0 and U01l 123

Substituting Eq. (22) into (14) gives the following equation:

EIX4n1RntUivn x m

X4n1

RntUnx

MwX4n1

RntUnXt 2 dXtdt

X4n1

_RntU0nXt"

dXt

dt 2

X4

n1Rn

t

U00n

X

t

d

2Xtdt2

X4n1RntU0nXt g

#dx Xt 24

In order to apply Galerkins method [8] of weighted resid-uals, both sides of Eq. (24) are multiplied by Ui(x), i= 1, 2,3 and 4, and integrated along the element length;

EIX4n1

Zl0

UixUivn xdxRnt mX4n1

Zl0

UixUnxdxRnt

Mw Z

l

0

Ui

x

X4

n1

Un

X

t

Rn

t

2

dXtdt"

X4n1

U0nXt _Rnt dXt

dt

2X4n1

U00nXtRnt

d2Xtdt2

X4n1

U0nXtRnt g#dx Xtdx 25

Rearranging Eq. (25) in the order of the mass, dampingand stiffness terms with the forcing term on the right-handside gives the following:X4n1m

Zl0

UixUnxdxMwUiXtUnXt

Rnt

X4n1

2MwdXt

dtUiXtU0nXt

_Rnt

X4n1EI

Zl0

UixUivn xdx

MwdX

t

dt 2

UiXtU00nXtMw d

2Xtdt2

UiXtU0nXtRnt

MwgUiXt; i 1; 2; 3; and 4 26The term

Rl0UixUivn xdx will now be integrated by parts

twice using the following general expression:Zl0

udv uvjl0 Zl

0

vdu 27

resulting in the following:

Zl0

UixUivn xdx UixU000n xjl0 Zl0

U0ixU000n xdx

UixU000n xjl0 U0ixU00nxjl0Zl

0

U00i xU00nxdx

UixU000n xjl UixU000n xj0 U0ixU00nxjl U0ixU00nxj0Zl

0

U00i xU00nxdx 28

Eq. (28) implies the following:X4n1EI

Zl0

UixUivn xdxRnt

X4n1EIfUixU000n xjl UixU000n xj0

U0ixU00nxjl U0ixU00nxj0Zl

0

U00i xU00nxdxgRnt 29

By means of Eqs. (50) and (51) from the Appendix B, Eq.(29) becomes:

z

y

0

L

x

c

Mw

Fig. 4. Moving unsprung load traversing a fixed-free cantilever beam.

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X4n1EI

Zl0

UixUivn xdxRnt

UixQjl UixQj0 U0ixMjl U0ixMj0

X

4

n

1

EI

Zl

0

U00i xU00nxdxRnt 30

Substituting Eq. (55) from the Appendix B into Eq. (30)gives:

X4n1EI

Zl0

UixUivn xdxRnt

UilbQ2 Ui0bQ1 U0il ^M2 U0i0bM1X4n1EI

Zl0

U00i xU00nxdxRnt 31

Next, one substitutes Eq. (31) into (26) giving the followingequation as:

X4n1m

Zl0

UixUnxdxMwUiXtUnXt

Rnt

X4n1

2MwdXt

dtUiXtU0nXt

_Rnt

X4n1EI

Zl0

U00i xU00nxdx

MwdX

t

dt 2

UiXtU00nXtMw d

2Xtdt2

UiXtU0nXtRnt

MwgUiXt Ui0bQ1 UilbQ2 U0i0bM1 U0ilbM2; i 1; 2; 3; and 4 32

Rewriting Eq. (32) in matrix form, as well as substitutingEq. (23) into Eq. (32), gives the following:

m Zl0

U1U1 U1U2 U1U3 U1U4

U2U1 U2U2 U2U3 U2U4U3U1 U3U2 U3U3 U3U4

U4U1 U4U2 U4U3 U4U4

26664 37775dxR1R2R3R4

8>>>>>: 9>>>=>>>;Mw

U1U1 U1U2 U1U3 U1U4

U2U1 U2U2 U2U3 U2U4

U3U1 U3U2 U3U3 U3U4

U4U1 U4U2 U4U3 U4U4

2666437775

R1R2R3R4

8>>>>>:9>>>=>>>;

2Mw dXtdt

U1U01 U1U

02 U1U

03 U1U

04

U2U01 U2U

02 U2U

03 U2U

04

U3U01 U3U

02 U3U

03 U3U

04

U4U01 U4U02 U4U03 U4U04

26664

37775

_R1_R2_R3

_R4

8>>>>>:

9>>>=>>>;

EIZl

0

U001U001 U

001U

002 U

001U

003 U

001U

004

U002U001 U

002U

002 U

002U

003 U

002U

004

U003U001 U

003U

002 U

003U

003 U

003U

004

U004U001 U

004U

002 U

004U

003 U

004U

004

2666437775dxR1

R2

R3

R4

8>>>>>:9>>>=>>>;

Mw dXtdt

2U1U

001 U1U

002 U1U

003 U1U

004

U2U001 U2U002 U2U003 U2U004U3U

001 U3U

002 U3U

003 U3U

004

U4U001 U4U

002 U4U

003 U4U

004

26664 37775R1

R2

R3

R4

8>>>>>: 9>>>=>>>;Mw d

2Xtdt2

U1U01 U1U

02 U1U

03 U1U

04

U2U01 U2U

02 U2U

03 U2U

04

U3U01 U3U

02 U3U

03 U3U

04

U4U01 U4U

02 U4U

03 U4U

04

2666437775R1

R2

R3

R4

8>>>>>:9>>>=>>>;

Mwg

U1

U2

U3

U4

8>>>>>:

9>>>=>>>;

bQ1

1

0

0

0

8>>>>>:

9>>>=>>>;

bQ2

0

0

1

0

8>>>>>:

9>>>=>>>;

bM10

1

0

0

8>>>>>:9>>>=>>>; bM2

0

0

0

1

8>>>>>:9>>>=>>>; 33

Comparing Eq. (20) with (33), one can clearly see that themodal method is remarkably similar to the finite elementmethod apart from the representation of the stiffness ma-trix and the nodal forces, bQ1; bQ2; bM1 and bM2; moreover,during the finite element assembly Newtons Third Laweliminates these forces at internal nodes.

5. Results

In Sections 5.1 and 5.2, the authors analyse the dynamicresponse of a cantilever beam subjected to a movingunsprung wheel, while Section 5.3 examines Olssons [10]unsprung wheel system comprising an unsprung wheelmass and sprung mass body supported by a spring-dampertraversing a simply supported beam at a wide range ofspeeds. It should be noted that the gravitational and damp-ing effects of the beam are ignored in all cases. In eachexample, the authors use the Newmark-b time integrationmethod [9] with 500 equal time steps to solve the transientanalysis. In all cases, time t is arranged in such a mannerthat the wheel is at the left support at t = 0 s and the initialdisplacement and velocity of the unsprung wheel and beamare equal to zero. In Sections 5.1, 5.2 and 5.3 the authorsdescribe the deflection of the beam using 6 modes, whilein the finite element system the beam is discretized into10 beam elements.

5.1. Wheel as a moving unsprung load traversing a cantilever

beam

In this first example, the authors examine the effects of a

moving unsprung mass traversing a fixed-free cantilever

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7/12

beam at constant speed c = 50.8 m/s from left to right,where the inertia effects of the wheel mass are omitted; thusonly the gravitational effects of the mass are taken intoaccount i.e. the weight of the wheel Mwg is equal to25.79 kN. The authors adopt the same cantilever beam asAkin and Mofid [1], such that the beam has a length

L = 7.62 m, Young modulus of elasticity E= 2.07 1011

N/m2, moment of inertia I= 4.58 105 m4, and massper unit length m = 46 kg/m; therefore, using Eq. (48) thefirst natural frequency of this particular beam isx1 = 27.49 rad/s.

The vertical displacement at the free-end of the cantile-ver beam is plotted as a function of time in Fig. 5. Theresults obtained in Fig. 5a relate to the moving unsprungload traversing a fixed-free cantilever beam, while Fig. 5bshows the results of the moving unsprung load traversinga free-fixed cantilever. One can observe from the results astriking similarity between the authors system and theresults obtained from Akin and Mofid [1]. The reader

should be aware that in Akin and Mofids [1] paper, gravityacts in the upwards direction.

5.2. Wheel as a moving unsprung mass traversing a cantilever

beam

Using the same beam properties as Section 5.1, theauthors now investigate the effects of a moving unsprung

mass traversing a cantilever beam, where on this occasionthe inertia effects of the wheel mass as well as its gravita-tional effects are taken into account. The wheel is travers-ing at constant speed c = 50.8 m/s from left to rightacross the beam. In this particular model, the unsprungwheel has a mass Mw = 2629 kg, i.e. Mw/mL = 7.5.

In Fig. 6, the authors plot the vertical displacement atthe free-end of the fixed-free cantilever beam as a functionof time. Only the local acceleration of the moving unsprungwheel is taken into account in Fig. 6a, while Fig. 6b pre-sents the total (local plus convective) acceleration of themoving unsprung wheel. In addition, the authors plot theresults of a moving sprung wheel traversing the cantileverwhere the Hertzian spring is given a reasonable large stiff-ness (see Bowe and Mullarkey [6]). In one example, themoving sprung wheel is not allowed to separate from thebeam; however, it can be seen in Fig. 7 that wheel separa-tion actually occurs. Thus, the free-end deflection of thebeam due to the moving sprung wheel with separation is

also plotted in Fig. 6b. Examining the results of Fig. 6a,it can be seen that the results of the authors model arequite similar to the results of Akin and Mofids [1] modal

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.00 0.03 0.06 0.09 0.12 0.15

Time (sec)

VerticalDisplacement(m)

Akin & Mofid [1]

Modal (6 Modes)

Finite Element Model

(a) Fixed-free cantilever beam (fixed atx = 0)

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.00 0.03 0.06 0.09 0.12 0.15

Time (sec)

VerticalDisplacement(m) Akin & Mofid [1]

Modal (6 Modes)


(b) Free-fixed cantilever beam (fixed atx =L)

Fig. 5. Vertical displacement at the free-end of the cantilever beam due to

a moving unsprung load travelling at 50.8 m/s.

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.00 0.03 0.06 0.09 0.12 0.15

Time (sec)

VerticalD

isplacement(m)

Akin& Mofid [1]

Modal (6 Modes) - Local acceleration only

Finite Element Model - Local acceleration only

(a) Local acceleration of the moving unsprung wheel

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.00 0.03 0.06 0.09 0.12 0.15

Time (sec)

VerticalDisplacementm)

Modal (6 Modes) - Full Solution

Finite Element Model - Full Solution

Sprung Wheel - Without seperation

Sprung Wheel - With seperation

(b) Total (local plus convective) acceleration

of the moving unsprung wheel

Fig. 6. Vertical displacement at the free-end of a fixed-free cantilever

beam due to a moving unsprung mass travelling at 50.8 m/s.



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system, while the results for the authors unsprung modelin Fig. 6b all show a striking similarity with the resultsfrom the sprung model [6] with no separation. Fig. 6bshows that the results from the sprung model with separa-tion are slightly different.

The reader can see by comparing Fig. 6a with b thatconvective acceleration should not be omitted. It can beseen that the free-end deflection in Fig. 6b is a quarterthe deflection in Fig. 6a at time t = 0.15 s. Moreover, thegraphs in Fig. 6a are different in form from the graphs ofFig. 6b.

In Fig. 7, one can see that the reaction force between themoving unsprung wheel and the beam goes negative at timet = 0.085 s. In the case of the moving sprung wheel, separa-tion occurs at this time; contact is re-established at time

t = 0.114 s. For the moving unsprung systems, it is shownthat the reaction forces becomes positive at a much earliertime t = 0.099 s.

Next the authors examine the dynamic effects of themoving unsprung wheel traversing a free-fixed cantileverbeam as shown in Fig. 8. The vertical displacement at thefree-end of the cantilever beam as a function of time is plot-ted in Fig. 9. As before, only the local acceleration of themoving unsprung wheel is taken into account in Fig. 9a,while Fig. 9b presents the total (local plus convective)acceleration of the moving unsprung wheel. Examiningthe results of Fig. 9a, it can be clearly seen that the results

of the authors model are very similar to the results of Akinand Mofids [1] modal system, while the results for theauthors unsprung model in Fig. 9b are comparable withthe results from the sprung model [6] with no separation.

Comparing the free-end deflection of the beam in Fig. 9awith b, it can be seen that the results vary by a factor of4, which is a similar ratio that is observed in Fig. 6.

The authors now examine the effects of convective accl-eration at lower speeds. Fig. 10 presents the vertical dis-placement at the free-end of a cantilever beam as afunction of time subjected to a moving unsprung mass tra-versing at a speed c = 27.78 m/s as well as at a very slowspeed c = 1 m/s using the authors models. It should benoted that in Fig. 10 that the modal and finite elementmodel (FEM) have the same solution. The results obtainedfrom the fixed-free cantilever beam are shown in Fig. 10a,while Fig. 10b and c plots the results from the free-fixed can-tilever beam due to the moving unsprung mass. ExaminingFig. 10a and b, the reader can see that the free-end deflec-tion of the cantilever beam without convective accelerationvaries by a factor of 3 with the results for the total (localplus convective) acceleration at the speed c = 27.78 m/s.Whereas, at the very slow speed c = 1 m/s, Fig. 10a showsthat there are no significant difference between the convec-tive acceleration and the total (local plus convective) accel-eration, while Fig. 10c shows a noticeable difference in

results at lower speeds. Examining Fig. 10c, one can see that

z

y

0

L

Mw

x

c

Fig. 8. Moving unsprung mass traversing a free-fixed cantilever beam.

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.00 0.03 0.06 0.09 0.12 0.15

Time (sec)

VerticalDisplacement(m)

Akin & Mofid (1989)

Modal (6 Modes) - Local acceleration only

Finite Element Model - Local acceleration only

(a) Local acceleration of the moving unsprung wheel

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.00 0.03 0.06 0.09 0.12 0.15

Time (sec)

VerticalDisplacement

(m)




(b) Total (local plus convective) acceleration

of the moving unsprung wheel

Fig. 9. Vertical displacement at the free-end of a free-fixed cantileverbeam due to a moving unsprung mass travelling at 50.8 m/s.

-0.5

0.0

0.5

1.0

1.5

2.0

0 0.03 0.06 0.09 0.12 0.15

Time (sec)

ContactFo

rce/StaticWeight




Sprung Wheel - With seperation

Fig. 7. Dimensionless force between the moving mass and beam as afunction of time.



9/12

as the moving unsprung mass traverses the free-fixed canti-lever beam, the free-end undergoes an oscillatory motion,where the amplitude decreases to a constant value withoutconvective acceleration and increases to a constant withconvective acceleration.

5.3. Moving unsprung vehicle traversing a simply supported

beam

In this section, the authors examine the dynamicresponse of a simply supported beam subjected to a movingunsprung vehicle by means of the authors modal and finite

element models, at a wide range of speeds. The vehicle sys-

tem comprises an unsprung mass Mw in contact with beam,which supports a sprung vehicle mass Mv by means of aspring-damper as show in Fig. 11. The dimensionlessparameters adopted are similar to those of Olsson [6,10],such that the vehicle system to bridge mass ratio is 0.5;the unsprung wheel mass to sprung vehicle mass ratio is

0.25; the bridge to vehicle frequency ratio is 3; and the vehi-cle damping ratio is 0.125. Other dimensionless parametersused are a speed ratio (that is the vehicle speed divided bythe critical speed) and a dynamic amplification factor (ratiobetween the maximum dynamic deflection and static deflec-tion of the mid-point of the bridge). The critical speed ofthe vehicle is the speed such that the vehicle travels a dis-tance of twice the length of the bridge in a time equal tothe natural period of the bridge.

The simply supported beam used has a length L = 25 m,Youngs modulus of elasticity E= 2.87 106 kN/m2,moment of inertia I= 2.9 m4, mass per unit lengthm = 2.303 t/m and a Poissons ratio m = 0.2, as adopted

in Yang and Wu [7]. In Fig. 12, the author plots thedynamic amplification factor at mid-span of the beam fora wide range of speeds for the moving unsprung vehicle.From the results, one can observe that the authors modaland finite element model for the unsprung moving vehicleis remarkably similar to Olsson (1985) solution at allspeeds.

6. Conclusion

In this paper, the authors developed their own modaland finite element model of a moving unsprung mass tra-

versing a beam. It has been shown that the final form of

z

y

0

L

x

c

Mw

Mv

k1 c1

Fig. 11. Moving unsprung vehicle traversing a simply supported beam.

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0.0 0.2 0.4 0.6 0.8 1.0

Vehicle Speed/ Critical Speed

Dynamic/StaticDeflection

Olsson [10] Unsprung System

Modal (6 Modes)


Fig. 12. Moving unsprung vehicle traversing a simply supported beam ata wide range of speeds.

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.

0

Dimensionless timect/L

Vertical

Displacement(m)

Modal & FEM - Local acceleration (1m/s)

Modal & FEM - Local acceleration (27.78 m/s)

Modal & FEM - Local plus convective acceleration (1m/s)

Modal & FEM - Local plus convective acceleration (27.78 m/s)

(a) Fixed-free cantilever beam at both speeds

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0

Dimension less timect /L

VerticalD

isplacement(m)

Modal & FEM - Local acceleration (27.78 m/s)

Modal & FEM - Local plusconvective acceleration (27.78 m/s)

(b) Free-fixed cantilever beam due to

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Dimensionless timect /L

VerticalDispla

cement(m)

Modal & FEM - Local acceleration (1 m/s)

Modal & FEM - Local plus convective acceleration (1 m/s)

(c) Free-fixed cantilever beam due to

the moving mass travelling at 27.78 m/s

the moving mass travelling at 1 m/s

Fig. 10. Vertical displacement at the free-end of the cantilever beam dueto a moving unsprung mass travelling at lower speeds.



10/12

the modal method is remarkable similar to the final form ofthe finite element method, and that if the mode shapesof the modal method are replaced by the finite elementweighting and shape functions, the finite element methodis derived, apart from the stiffness matrix.

This paper also highlights the danger of omitting the

convective acceleration from moving unsprung wheel sys-tems. It is shown using both a modal and finite elementmodel that excluding these convective terms can have a sig-nificant effect on the results by as much a factor of 4. Earlydynamic studies by Biggs [2] and Akin and Mofid [1]neglected these terms; however more recent studies [7] arestill comparing their developed models with these inaccu-rate solution, while ignoring the issue of convectiveacceleration.

Appendix A. Cantilever beam natural frequencies and

mode shapes

In the following section, the authors derive the naturalfrequencies and mode shapes for a cantilever beam that isencastre at the origin; hence, the boundary conditions areas follows:

v ovox

0 at x 0 34a

EIo

2v

ox2 EIo

3v

ox3 0 at x L 34b

where v is the deflection of the beam, ov/o x is the slope,EIo2v/ox2 is the bending moment, and EIo3v/ox3 is theshear force. This follows the work of Biggs [2] and Fryba

[11]. The function /n(x) satisfies the same boundary condi-tion as v(x,t). This can be seen by substituting Eq. (2) intoEq. (34). Evaluating /n(x) at x = 0 using Eq. (5a) and thefirst derivative of that equation, gives:

/n0 0 Bn Dn Dn Bnd

dx/n0 0 Anan Cnan Cn An

35

Substituting Eq. (35) back into (5a) gives the following:

/nx Ansin anx sinh anx Bncos anx cosh anx36

Equally, one evaluates the second and third derivative ofEq. (5a) for x = L giving:

d2

dx2/nL Ana2n sin anL sinh anL

Bna2n cos anL cosh anL 0 37ad3

dx3/nL Ana3n cos anL cosh anL

Bna3nsin anL sinh anL 0 37bSince An,Bn cannot both be equal to zero for a non-trivialvalue of/n, then the determinant of the coefficients must be

zero, resulting in

sin anL sinh anL cos anL cosh anL cos anL cosh anL sin anL sinh anL

0 38which reduces to

cos anL cosh anL 1 0 39From Eq. (37a), one gets

An

Bn cos anL cosh anL

sin anL sinh anL 40

Rewriting Eq. (36) for /n(x) to within an undeterminedamplitude Bn as follows:

/nx BnAn

Bnsinh anx sin anx cosh anx cos anx

41

The undetermined amplitude Bn is evaluated by normaliz-ing /n (x) as follows:

ZL0

/2nxdx L 42

Eq. (42) gives a value of unity for Bn. In addition, themodes are orthogonal:ZL

0

/nx/ixdx Ldin 43

The evaluation of the natural frequencies, xn, of the canti-lever beam are related to the roots an of Eq. (39) throughEq. (5b). In order to compute the roots an of Eq. (39), itis rewritten as follows:

cos bn cosh bn 1 0 44awhere

bn anL 44bSubstituting Eq. (44b) into (5b) and rearranging in terms ofthe natural frequencies xn, one gets the following relation-ship (measured in rad/s):

xn b2n

L2

ffiffiffiffiffiEI

m

r45

The roots bn of Eq. (44a) are got by plotting the followingfunction:

Wb cosb coshb 1 46in Fig. 13. The roots occur when W crosses the b-axis.These roots are approximated by the points where cos bcrosses the b-axis as can be seen in Fig. 13. The roots ofcos b are given by the following equation:

bn 2n 1

2

p; n 1; 2; 3; . . . 47

In Table 1, the authors present bn, n = 1, 6 evaluated bothexactly and approximately for the cantilever beam. Forn = 1, the approximate value is unsuitable as the error is

over 16%. For n > 1, the approximate roots are more suit-



11/12

able because the differences between the evaluated andapproximate roots are less than 0.5%.

Using Table 1, one can write the first natural frequencyx1 using Eq. (45) as follows:

x1 1:875192

L2ffiffiffiffiffiEIm

r 48Alternatively, substituting Eq. (47) into (45) gives approx-imate natural frequencies for the cantilever beam, whichare most suitable when n > 1

xn n 1=22p2

L2

ffiffiffiffiffiEI

m

r49

Appendix B. Convention for shear force and bending moment

and equation of vibration for a beam

In the following section, the statement of convention forshear and bending moment as well as derivation of the gov-erning equation of vibration are presented. In Fig. 14, theauthor presents a differential beam segment dx subjectedto a load p(x) acting in the positive y-direction. The shearforces Q and bending moment M acting on this particularbeam segment obey the right-hand system; thus, the shearforce acts positively upwards on the positive face of thebeam segment and positively downwards on the negativeface of the beam segment, while the bending moment actspositively in a counter-clockwise direction about the z-axis

on the positive face of the beam segment and acts positively

in a clockwise direction about the z-axis on the negativeface of the beam segment as shown in Fig. 14.

Using the above convention, the constitutive law relat-ing bending moment to curvature states:

M EIo2v

ox250

Satisfying equilibrium, the sum of moments about the z-axis on the positive face of the free-body diagram mustbe equal to zero (ignoring rotary inertia); thus, the shearforce is equal to:

Q EIo3v

ox351

Next, satisfying Newtons Second Law in the verticaldirection:

Q oQox

dx pxdx Q mdx o2v

ot252

Simplifying Eq. (52) as follows:

oQ

ox px m o

2v

ot253

Substituting Eq. (51) into (53):

EIo

4v

ox4 m o

2v

ot2 px 54

The convention for the shear force and bending moment atlocal nodes 1 and 2 of the finite element are shown inFig. 15. The conventions for bending moment and shearforce of Figs. 14 and 15 are as follows:bM1 M0; bM2 MlbQ1 Q0; bQ2 Ql 55

z

x

y, v

0

M

xdx

dxxQQ +

p(x)

dx

M

Q

M +

Fig. 14. Free-body diagram of a load beam segment.

1Q

2Q

1M

2M

z

x

y

0m, EI

1 2

Fig. 15. Convention for bending moment and shear force at element

nodes.

-20

-15

-10

-5

0

5

10

0 2 4 6 8 10 12

True

Cos

()

cos

3

2

5

2

7

2

2

Fig. 13. Comparing the roots ofW and cos b graphically.

Table 1First 6 roots of a cantilever beam

n W (b) = 0 cos b = 0 % Differential

Roots of Eq. (46) Eq. (47)

1 1.875190 1.570796 16.232702 4.694091 4.712389 0.389813 7.854757 7.853982 0.009874 10.995541 10.995574 0.000305 14.137168 14.137167 0.000016 17.278760 17.278760 0.00000



12/12

References

[1] Akin JE, Mofid M. Numerical solution for response of beams withmoving mass. J Struct Eng 1989;115(1):12031.

[2] Biggs JM. Introduction to Structural Dynamics. 1st ed. McGraw-Hill; 1964.

[3] Wu JJ, Whittaker AR, Cartmell MP. The use of finite element

techniques for calculating the dynamic response of structures tomoving loads. Comput Struct 2000;78:78999.[4] Bowe CB, Mullarkey TP. Verification of contact elements by

triangular force technique. In: Symposium on Trends in the Appli-cation of Mathematics to Mechanics. Elsevier; 2000. p. 429.

[5] Yang YB, Wu YS. A versatile element for analyzing vehicle-bridgeinteraction response. Eng Struct 2001;23:45269.

[6] Bowe CJ, Mullarkey TP. Wheel-rail contact elements incorporatingirregularities. Adv Eng Software 2005;36:82737.

[7] Yang YB, Yau JD, Wu YS. Vehicle-bridge interaction dynamics withapplication to high speed railways. 1st ed. World Scientific PublishingCo. Pte Ltd.; 2004.

[8] Brebbia CA, Walker S. Dynamic analysis of offshore structures. 1sted. Newnes-Butterworths; 1979.

[9] Bathe KJ. Finite element procedures. 1st ed. Englewood Cliffs,NJ: Prentice-Hall; 1996.

[10] Olsson M. Finite element, modal co-ordinate analysis of structuresubjected to moving loads. J Sound Vib 1985;99:112.

[11] Fryba L. Dynamics of railway bridges. 1st ed. Thomas-Telford; 1996.


Documents

Bowe (2008) Unsprung Wheel-beam Interactions Using Modal and Finite Element Models