Numerical Solution for Response of Beams with Moving Mass

NUMERICAL SOLUTION FOR RESPONSE OF BEAMS WITH MOVING M A S S

By John E. Akin1 and Massood Mofld2

ABSTRACT: An analytical-numerical method is presented that can be used to determine the dynamic behavior of beams, with different boundary conditions, carrying a moving mass. This article demonstrates the transformation of a familiar governing equation into a new, solvable series of ordinary differential equations. The correctness of the results has been ascertained by a comparison, using finite element models, and very good agreement has been obtained. Furthermore, the article shows that the response of structures due to moving mass, which has often been neglected in the past, must be properly taken into account because it often differs significantly from the moving force model.

INTRODUCTION

In the early twentieth century, structural engineers realized the potential hazard produced when a mass traveled along a supported beam. They managed to calculate the dynamic response of this effect for only the simplest cases and their solutions were highly impractical (Jeffcott 1929). Recently, engineers designing highway and railroad bridges and space station facilities likely to be affected by sudden changes of mass have renewed the investigation of this problem (Stanisic and Hardin 1969; Stanisic et al. 1974). This article presents a numerical-analytical method of determining the behavior of beams with various boundary conditions and carrying a moving mass.

Calculating the response of beams affected by moving mass involves solving partial differential equations sufficiently complex that simplifying assumptions are usually adopted. These assumptions usually include fixed boundary conditions or neglect of the moving mass or beam mass inertia. In some cases, the inertia of a moving mass may be small and may be safely neglected. However, in cases involving a large moving mass, the neglect of inertia may cause an error of 20 to 80%, depending on such factors as the flexibility of the system, mass velocity, and other factors. These variations may be discussed in future papers.

This paper solves the system of ordinary differential equations describing a continuous beam with a moving mass and with various boundary conditions. The study used the Continuous System Modeling Programming (CSMP) to solve the system of ordinary differential equations (Speckheart 1976) and the Programs for Automatic Finite Element Calculations (PAFEC 1978; Akin 1982) to verify the analytical-numerical results independently.

LITERATURE REVIEW

T h e p rob lem was first a t tacked by Jeffcott in 1929. H e was fol lowed in 1934 by Steuding, and in 1951 by O d m a n . In these papers , the solution

'Chmn., Dept of Mech. Engrg. and Materials Sci. Dept., Rice Univ., Houston, TX 77251-1892.

2Grad. Student, Dept. of Mech. Engrg. and Materials Sci Dept., Rice Univ. Note. Discussion open until June 1, 1989. To extend the closing date one month,

a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on July 2, 1987. This paper is part of the Journal of Structural Engineering, Vol. 115, No. 1, January, 1989. ©ASCE, ISSN 0733-9445/89/0001-0120/S1.00 + $.15 per page. Paper No. 23105.

J 20

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involved many approximations and was rather impractical. Tn 1.951; Pestpi tried to solve this problem by means of the Rayleigh-Ritz technique, but no numerical results were presented. In 1969, Stanisic and Hardin presented a solution for a simply supported beam which was interested but not very easy to apply to different boundary conditions. What was needed was a reliable method that could be easily applied to many situations and a variety of boundary conditions.

The object of this paper is:

1. To present a very simple and practical analytical-numerical technique for determining the response of beams, with various boundary conditions, carrying a moving mass.

2. To develop a result with the option of investigating the contribution of each mode of vibration in the response of the system.

PROBLEM DEFINITION

A beam with a moving mass and different boundary conditions is shown in Fig. 1. Assumptions:

I(x) = I = constant moment of inertia; m(x) = m = constant mass per unit length of the beam, no damping in

the system; Uniform gravitational field, g; and M = constant mass moving on beam.

Initial conditions: y(x,0) = g,(*) dy(x,0)/dt = g2(x).

PROBLEM FORMULATION

Based on the assumption of a constant inertia, the equation governing motion of a beam under the moving mass, M, neglecting the damping, the rotary inertia, and shearing force effects can be written as

g

I(x), m(x)

FIG. 1. Beam with Moving Mass

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d.

d\(x, t) md2y(x,t) EI + ^ r - ^

dt2

= p(x,t). dxA

where

p(x,t) = F(t)b(x- vt) + Q(t)

F(t) = Mg - M d2y(vt,t)

dt2

(1)

(2)

(3)

Q(t) = general loading (zero for this case); v = speed of moving mass; 8 = Dirac Delta; and F(t) = resultant concentrated force caused by moving mass. Recall the Dirac delta integral properties:

J Jo

f(x)b(x — x0)dx = f(x0) 0 < x, x0 < L

f(x)h(x — x0)dx = 0 x < 0, x > L

(4a)

(4b)

Substituting Eqs. 3 and 2 into Eq. 1, the governing differential equation of a beam under moving mas can be written as

dAy(x,t) d2y(x,t) EI 7— + pA ;—

dt2 -M d2y(r\,t)

dt2 dx4

where

T] = vt m = pA

Assume a solution in the form of a series:

?(*. ') = X yn(x) Tn(t) n = 1, 2, 3, . . . N

+ Mg 8(X - T|) . (5)

(6)

(7)

where, Yn(x) are the known eigenfunctions of the beam (without M) with the same boundary conditions and n is the number of contributed modes. It has the form of

YZ(x) - ti,Yn(x) = 0

where

(8)

B! = EI

(9)

co„, n = 1, 2, 3, . . . are natural frequencies; Yn(x) = a sin (B„x) + b cos (B„x) + c sinh (B„x) + d cosh (B„x); a, b, c, d = constant coefficients; and T„(t) are functions of time which have to be calculated. Rewriting the right-hand side of Eq. 5 in form:

-M d2y(f\,t)

dt2 + Mg § ( - ~ •n) = X Y»W B„(t) • (10)

and substituting Eq. 7 into Eq. 10 and using a simplified subscript for differentiation results in

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r • I -M 2 Yn(r\) r„,„(0 + Mg |s(* - -n) = 2 Y«(.x) *»(') (11) L n J n

Multiplying both sides of Eq. 11 by Yp(x), and integrating along the beam length yields

M2y„(ti)r„, f^dx r„,„(0 Yp(xMx - rj)dx + Mg Yp(x)b(x Jo Jo

= E B»W r«(*)W«fc (12)

Considering of Eqs. 4a and Z? and applying the orthogonality of function Y„(x) with Eq. 12, it becomes

BP(t) = (~~) - 2 Tn,„YM + g W (13)

where

f Jo

Y2p(x)dx (14)

To get the equation for T„(t), we substitute Eqs. 13 and 11 into the equation of motion Eq. 5. This results in

EI 2 Y':{x)Tn{t) + pA 2 Yn(x)Tnil!(t)

= 2 {Y"(X\VJ " 2 Tq,„{t)YM) + g

Considering Eq. 8, Eq. 15 becomes

J, Yn(x) \EI&Tn(t) + pATn,„(t) n L

- E r„,„(0W + g

r.On) (15)

Y„(.i\) = 0 (16)

This equation must be satisfied for arbitrary x (i.e., each point of the beam), and this is possible only when the expression in the bracket is equal to zero for arbitrary n. So

EI&TM + PAr„,„(0

for n = 1, 2, 3, ...N.

M - 2 Tq,„{t)Yq{vt) + g Y„(vt) = 0

(17)

The system in the Eq. 17 is a set of coupled ordinary differential equations and a numerical procedure is used to solve it. Rearranging this equation in matrix form results in:

mOKT,,} = -WT} + {C(0} (18)

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{T,,} = [Hit)]-1 (-{OLT} + {C(t)}) (19)

where, letting P, = M/V^:

[H(t)]

~m + PXY\ PzY1Y2

PsYiYi

p,y 2 y, m + P2Y\

P3Y2Y3

PMYi . . . PJJ, P2Y3Y2 . . . P2YnY2

m+P3Y23...P3Y„Y3

PjYiYj

PnYJn

PjY2Yj...m + PjY}. PnY2Yn . . . PnYjYn. .

a„ = £ / p * n = 1, 2, 3

{aTf = {aiTi a2T2 a3T3 . . .}

{Tit} = {Tit„ T2t„ T3jt . . .}

{c{t)f = g{p,y, p2y2 P s ^ . . .}

p . y y.

(20)

(21)

(22)

(23)

(24)

It is very important to note that there is no need to calculate Eq. 14 in each mode because it can be normalized to a constant (length of beam):

Y2p(x)dx=Vp=L p= 1 , 2 , 3 ,

o

and

Pi=P2=P3 = ...Pn

(25)

M

L

Normalizing makes possible a symmetric matrix H(t), which saves computation time. This matrix H{t) shows that the contribution of each mode can be studied individually, and investigation of these modes offers a new area for potentially valuable research.

TEST PROBLEM

To check the results of Eq. 18, the following cantilever beam shown in Fig. 2 is considered:

•4" T | = V t

- L = 300'

• X - H

FIG. 2. Cantilever Beam

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FIXED-FREE BEAM WITH MOVING MASS

S

M = 3Lb.Sec2/In V = 2000 In.Sec L = 300 In

M/m =1.5 EI = 3.3e+09 Lb.In2

(1) analytical-numerica] solution (moving mass) (2) F. E. solution (moving mass) (3) analytical solution (moving force)

0.025 0.050 0.075 0.100

Time

0.125 0.150

(Sec.)

FIG. 3. Fixed-Free Beam with Moving Mass (M/m = 1.5; EI = 3.3 e+09 Ib • in.2)

_£


M = 15 Lb.Sec2/In V = 2000 In/Sec L = 300 In

20.0

15.0

• 1 T)

s 1 s

10.0 -

5.0-

• A

0.0 -f

-5.0

M/m = 7.5 EI = 3.3 e+09 Lb.In2

(1) analytical-numerical solution (moving mass)

(2) F.E solution (moving mass)

(3) analytical solution (moving force)

0.050 0.075

Time

0.100 0.125

(Sec.)

FIG. 4. Fixed-Free Beam with Moving Mass (M/m = 7.5; EI = 3.3 e+09 Ib • in.2)

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20.0

15.0 -

FREE-FIXED BEAM WITH MOVING MASS M=15 Lb.Sec2/In V = 2000 In/Sec L = 300 In

M/m = 7.5 EI = 3.3 e+09 Lb.In2

1 > 1 1 1 1 1 1 1

0.000 0.025 0.050 0.075 0.100 0.125 0.150

Time (Sec.)

Fixed Beam with Moving Mass (M/m = 7.5; EI = 3.3 e+09 Ib • in.2)

FIXED-HINGED BEAM WITH MOVING MASS M = 1 5 Lb.Sec2/In V = 2000 In/Sec L = 300 In

M/m = 7.5 EI = 3.3 e+09 Lb.In2 1.0-

0 . 8 -

0 . 6 -

0 .4 -

0 .2-

o.o-l

0 . 2 -

> ^ D p—=sSr ^

A

(2) yjT >k

(i) Js \

/ H / ^ ^ ^ \ U

(3) *V V

(1) analytical-numerical solution (moving mass) N .

(2)F.E. solution (moving mass)

(3) analytical solution (moving force)

T _ _ 1 — ™ _ ^ j , , (

0.000 0.025 0.050 0.075 0.100 0.125 0.150

Time (Sec.)

FIG. 6. Fixed-Hinged Beam with Moving Mass

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S O.S

FIXED-FIXED BEAM WITH MOVING_MASS___ M = 1 5 Lb.Sec2/In V = 2000 In/Sec L = 300 In

M/m = 7.5 EI = 3.3 e+09 Lb.In2

• (1) analytical-numerical solution (moving mass)

^ (2)F. E. solution (moving mass)

^ (3) analytical analytical solution (moving force)

0.025 0.050 0.075 0.100

Time 0.125 0.150

(Sec.)

FIG. 7. Fixed-Fixed Beam with Moving Mass

© a.

i

FREE-FIXED BEAM WITH MOVING MASS M = 1 5 Lb.Sec2fln V = 2000 In/Sec L = 300 In

1 0 - , M/m = 7.5 EI = 1 1 . 2 5 e+09 Lb.In2


^ (2)F. E. solution (moving mass) ^ (3) analytical solution (moving force)

0.000 0.025 0.075 0.100

Time

0.125

(Sec.)

FIG. 8. Free-Fixed Beam with Moving Mass (M/m = 7.5; EI = 11.25 e+09 lb • in.2

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10.0-1

M = 1 5 Lb.Sec2/In

M/m = 7.5

V = 2000 In/Sec L = 300 In

EI= 11.25 e+09 Lb.In2

2.0

£

-2.0

• (1) analytical-numerical solution (moving mass) ^ (2) F. E. solution (moving mass) A (3) analytical solution (moving force)

O.o-S—-a B-

yj) airatyucat solution vmuvuig wtixi • j

(3) ^--«G

0.000 0.025 0.050 0.075 0.100 0.125 0.150

Time (Sec.)

FIG. 9. Fixed-Free Beam with Moving Mass (M/m =7.5; EI =11.25 e+09 Ib • in.2)


M = 1 5 Lb.Sec2/In V = 2000 In/Sec L = 300 In

M/m = 7.5 EI = 6.81 e+09 Lb.In2

10.0

4.0-

2.0-

0.0 »

-2.0


^ (2)F.E. solution (moving mass)

^ (3) analytical solution (moving force)

0.025 0.050 0.075 0.100

Time

0.125 0.150

(Sec.)

FIQ. 10. Fixed-Free Beam with Moving Mass (M/m = 7.5; EI = 6.81 e+09 Ib • in.2)

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IV11> y jU^<*Ur IVJ t /1&a

(9

o

I

M = 15 Lb.Sec2/In

M/m = 7.5

V = 2000 In/Sec L = 300 In

EI = 6 .81e+09 Lb.In2


^ (2)F.E. solution (moving mass)

^ (3) analytical solution (moving force)

0.000 0.025 0.050 0.075

Time

0.100 0.125

(Sec.)

0.150

FIG. 11. Free-Fixed Beam with Moving Mass (M/m = 7.5; EI = 6.81 £+09 lb • in.2)

m = 0.0066666 lb s2/in.2, E = 3 X 107 psi, v = 2000 in./s

/ = 110.0 in.4 Vl = V2= ... =Vj = L = 300 in.

The eigenfunction and eigenvalues are

Y„(x) = cosh (P„x) - cos (P„.x) - X„[sinh (P„x) - sin (P,,*)] (26)

where (Young and Felgar 1949)

P,L = 1.8751041, p2L = 4.69409113, p 3 i = 7.85475743, . . .

\ i = 0.7340955, X2 = 1.01846644, \ 3 = 0.99922450, . . . .

Case No. 1 Moving mass = M = 3.0 lb sec2/in., and M/m = 1.5.

Case No. 2 Moving mass = M = 15.0 lb sec2/in., and M/m = 7.5. In each case, the test problems have been solved by the above procedure,

employing a fifth-order Runge-Kutta scheme, and by two additional ways: using CSMP to integrate Eq. 18, and by using a finite element model with Newmark Beta direct integration of the time varying mass matrix in PAFEC. The results are shown in Figs. 3 and 4. Note that a comparison of the results shows very good agreement between the analytical-numerical solution and

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the finite element time history approach. Different boundary conditions and stiffness were applied to the beam, re

sulting, in seven more cases. The results for these cases are shown in Figs. 5 - 1 1 .

REMARKS ON THEORY AND RESULTS

1. It is evident from Eq. 18 that the contribution of each mode of vibration has been taken into account, and It is possible to investigate the contribution of each mode of vibration individually.

2. Investigation of the results indicates that a higher mode of vibration in some time intervals, can be dominant.

3. A comparison of the moving mass and moving force (Fryba 1971) results in test problem Case 2, indicates an at least 80% different between the two results and thus shows the importance of including mass in real design conditions where the velocity is high.

CONCLUSION

The article has presented a theory to determine the response of beams, with different boundary conditions, carrying a moving mass. The theory is based on orthogonal functions, and the results indicate that the governing differential equation can be transformed into a series of coupled ordinary differential equations. Some special test problems were solved both by the move method and by a finite element model, and a comparison of the results indicate very good agreement between the two solutions and the important role of the moving mass of the system.

This numerical-analytical method can be applied to a wide range of problems in this area. Further, it is simple enough to carry out on most personal computers. Finally, the writers believe that this method will efficienty serve design engineers in real design conditions.

APPENDIX I. REFERENCES

Akin, J. E. (1982). Application and implementation of finite element methods. Academic Press, New York, N.Y.

Fryba, L. (1971). Vibration of solids and structures under moving loads Noordhoff International Publishing, Groningen, The Netherlands.

Jeffcott, H. H. (1929). "On the vibration of beams under the action of moving loads." Phil. Mag. 7(8), 66.

Mofid, M. (1989). "Response of structures with moving mass," thesis presented to Rice University in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Odman, S. T. (1948). "Differential equation for calculation of vibrations produced in load-bearing structures by moving loads." Preliminary Publication, Int. Assn. for Bridge and Struct. Engrg., 3d Congress, Liege.

PAFEC Data Preparation. (1978). Pafec Ltd., Nottingham, U.K. Pestel, E. (1951). "Tragweksauslenkung unter bewegter Last." Ingen.-Arch., 19,

378. Steuding, H. (1934). "Die Schwingung von Tragern bei bewegten Lasten." Ingen.-

Arch., 5, 275. Speckheart, F. H. (1976). A guide to using CSMP—The continuous system modeling

program. Prentice-Hall Inc., Englewood Cliffs, N.J.

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Stanisic. M. M., and Hardin, J. C. (1969). "On response of beams to an arhitry number of moving masses." / . Franklin Inst., 287, 115.

Stanisic, M. M., Euler, J. A., and Montgomery, S. T. (1974). "On a theory concerning the dynamic behavior of structures carrying moving masses." Ingen. Arch., 43, 295.

Young, D., and Flegar, R., Jr. (1949). "Tables of characteristic functions representing normal modes of vibration of a beam." University of Texas Austin, Tex.

APPENDIX II. NOTATION

The following symbols are used in this paper:

A E F g / L

M m P Q T„

t V X

Y„ y P» 8 P

wn

= = = = = = = = = = = = = = = = = = = =

beam cross-sectioned area; modulus of elasticity; concentrated load; acceleration of gravity; beam moment of inertia; beam length; lumped moving mass; beam mass per unit length; general loading; general loading; general function of time; time; constant horizontal velocity of the beam; horizontal coordinate for the beam; eigenfunctions of beam; transverse dynamic reflection of beam; eigenvalues; Dirac Delta; mass density of beam; and natural frequencies.

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Documents

Numerical Solution for Response of Beams with Moving Mass