THE DYNAMIC RESPONSE OF A TIMOSHENKO BEAM

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Journal of Sound and Vibration (1996) 198(2), 249256

LETTER TO THE EDITOR

THE DYNAMIC RESPONSE OF A TIMOSHENKO BEAM SUBJECTEDTO A MOVING MASS

H. P. L

Department of Mechanical and Production Engineering, National University of Singapore ,10 Kent Ridge Crescent, Singapore 0511

(Received6 September 1995, and in final form 6 March 1996)

1.

The dynamic behavior of a beam subjected to moving loads or moving masses has beenstudied extensively in connection with machining processes and the design of railway tracksand bridges. A comprehensive survey of the methods for analyzing the problem and thecorresponding numerical results using infinite series were presented in the book by Fryba[1]. Nelson and Conover [2], Benedetti [3], Steele [4], Florence [5] and Katz et al. [6]included in their studies the effects of elastic foundation, deflection dependent movingloads and moving masses in the form of a sequence. Timoshenkoet al. [7] simplified the

problem of a beam subjected to a moving mass, the original moving-mass moving-forceproblem, to the equivalent moving-force problem, by neglecting the inertial effects ofthe moving mass. Sadiku and Leipholz [8] compared the solutions for an Euler beam actedupon by a moving mass, and concluded that the approximate solution for themoving-force problem was not always an upper-bound solution in terms of deflectionunder the moving mass for the related moving-mass moving-force problem. In otherwords, the inertial effect of the moving mass cannot be neglected in comparison with thegravitational effect, even if the speed of the moving mass is relatively small.

The related problem of a moving load acting on a rotating shaft has also been studied

extensively in relation to machining processes. Katz et al. [9] examined the dynamicresponse of a constant-velocity moving load on a rotating shaft, on the basis of Euler,Rayleigh and Timoshenko beam theories. Huang and Chen [10] reported the dynamicresponse of a rotating orthotropic beam subjected to a moving harmonic load, on the basis

of Euler beam theory. The energy formulations were based on a co-ordinate system fixedto the rotating shaft. Han and Wu [11], by expressing the energy in terms of a co-ordinatesystem fixed to the rotating shaft, derived the equations of motion using Hamiltonsprinciple. The dynamic response of a rotating Timoshenko beam subjected to axial forcesand moving loads was also reported by Lee [12]. Other related studies on the dynamicresponse of a beam subjected to accelerating forces have been analyzed by Fryba [1] andSchlack [13]. The dynamic behaviors of rotating Euler beams and rotating Rayleigh beamssubjected to accelerating distributed surface forces were analyzed by Argento and Scott[14] and Argento et al. [15]. The problem of a viscoelastic Timoshenko beam subjected

to a transversely applied step-loading was solved using the Laplace transform method byRoss [16]. Zu and Han [17] dealt with the more difficult problem of a spinning Timoshenkobeam with general boundary conditions subjected to a moving load.

In the present study, the equation of motion in matrix form for a Timoshenko beam

acted upon by a concentrated mass moving with a constant speed is formulated using theLagrangian approach and the assumed mode method. Examples for a simply supportedbeam acted upon by a moving mass with various prescribed combinations of constant

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speed, mass and slenderness ratio of the beam are presented in the present study. Thepossibility of the mass separating from the beam during the course of the motion, whichviolates the assumption in the reported studies that the beam and the mass remain incontact at all times, is examined by monitoring the contact force between the mass andthe beam during the motion. The discrepancy between this contact force and the constant

gravitational force used in the equivalent moving-force formulation also gives anindication of the discrepancy between the moving-mass moving-force and thecorresponding moving force solutions by neglecting the inertial effect of the moving

mass.

2.

In Figure 1 is shown a simply supported Timoshenko beam acted upon by a movingconcentrated mass of mass M with a constant speed v along the beam. A co-ordinatesystem is assumed to be fixed in the inertial frame, with the X-axis parallel to theundeformed longitudinal axis of the beam and the Z-axis pointing vertically downwardin the same direction as the gravitational acceleration g.

On the basis of Timoshenko beam theory, the deformed beam can be described by the

transverse translations w(x,t) in the Z-direction and small rotations (x,t) about theY-axis. The variable x is the position of a point along the beam on the X-axis and t isthe time. The translationw consists of contributions due to bendingwband contributionsdue to transverse shear deformationws . The relationships between these variables are asfollows [18]:

w(x,t) = wb(x,t) + ws (x,t), (x,t) = wb(x,t)/x. (1, 2)

Using the assumed mode method, the quantities w and can be expressed as

w(x,t) =n

i= 1

p(t)i(x), (x,t) =n

i= 1

qi(t)i(x), (3, 4)

where iand iare spatial functions that satisfy the boundary conditions at the two endsof the beam. For a beam simply supported at both ends, these assumed functions are

i(x) =2 sinix

L , i(x) =2 cos

ix

L . (5, 6)

The resulting EulerLagrange equations, in matrix form, are

(AM) + MBp+ 2MvAp+ (kGAH+ Mv2C)p+ kGAETq= Mga, (7)

IdSq+ (EIK+ kGAS)q+ kGAEp= 0. (8)

Figure 1. A Timoshenko beam acted upon by a moving mass.


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The two equations are coupled second order differential equations. The matrices M,S,K, H, E, A, B and C are defined as follows:

(M)ij=

L

0

ijdx, (S)ij=

L

0

ijdx, (K)ij=

L

0

'i'j dx, (911)

(H)ij=L

0

'i'j dx, (E)ij=L

0

i'j dx, (12, 13)

(B)ij=i(x = s)j(x = s), (14)

(A)ij=12(i(x = s)'j(x = s)'i(x = s)j(x = s)), (15)

(C)ij=i(x = s)j (x = s). (16)

The vectors p, p andp are n 1 column vectors consisting ofpi, p iand p irespectively.The other vectors are also defined in a similar manner. The vectora is defined as

(a)i=i(x = s). (17)

The possibility of the separation of the mass from the beam during the course of motioncan be detected by the change of sign of the contact force between the mass and the beam.This contact force is defined to be positive if the force acting on the beam is pointingdownward in the Z-direction. A change of sign from positive to negative indicates thatthe mass has separated from the beam and the equation of motion is no longer valid fordescribing the ensuing motion. The expression for the contact force is given by

Fc = Mg M2wt2 + 2v 2wxt + v22wx2(x = s). (18)All of the terms within the pair of brackets are to be evaluated forx = s. The deflection

under the moving mass is also to be compared with the deflection under an equivalentmoving load given by Mg, neglecting the inertial effects of the mass. The equations ofmotion for a Timoshenko beam subjected to a moving force of magnitude Mg can bederived from the equations of motion for a Timoshenko beam subjected to a moving

mass by neglecting the inertial effect of the mass. These resulting equations of motion aregiven by

AMp+ kGAHp+ kGAETq= Mga, (19)

IdSq+ (EIK+ kGAS)q+ kGAEp= 0. (20)

The present assumed mode method requires that the assumed functions for w and

satisfy the respective geometric boundary conditions for a Timoshenko beam. The presentmethod is not efficient for a Timoshenko beam with other combinations of boundaryconditions, as the corresponding assumed functions tend to be more complex. Theseassumed functions for other combinations of boundary conditions for a Timoshenko beam

can be found in the work by Huang [19]. However, the simply supported boundarycondition analyzed in the present work is the most frequently encountered boundarycondition for the present problem.


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3.

The equations of motion generated in the preceding sections are included in numerical

simulation programs using the fourth order RungeKutta method for investigating the

response of a Timoshenko beam acted upon by a moving mass or an equivalent moving

load by neglecting the inertial effects of the mass. The deflection in theZ-direction underthis moving mass,wevaluated atx = s, is normalized by the static deflectionwstatx = L/2

when a loadMg is applied at the same point for an Euler beam. From the static analysis,

this deflection wst = MgL3/48EI. The reason for performing this normalization is to

facilitate the comparisons of the present results with the corresponding reported work [9].

This normalized deflection under the moving mass is denoted by U, defined by

U= w(x = s)/wst .

For the present study, the same Timoshenko beam parameters and material properties

as defined in Katz et al. [9] are used in the numerical simulations. These properties are

L = 1 m, E=207109 N/m2, G =776109 N/m2, k =09 and = 7700 kg/m3. The

cross-sectional area of the beam, A, is computed from the radius of gyration r0 defined

by a non-dimensional parameter (Rayleighs coefficient) =r0/L. The prescribed axial

Figure 2. Normalized deflections under the moving mass () and the equivalent constant moving loads() for 003. (a) a = 011; (b) a =05; (c) a =11.


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velocity of the moving mass,v , is similarly defined by a non-dimensional parameter givenby a = v/vcr , where vcr is given by vcr = (/L)EI/A.

It should be noted that a small value of Rayleighs coefficient implies that the beam isvery slender and that the behavior of the beam could be described by the Euler beamtheory. In Figure 2 is shown the normalized deflection under the moving mass and the

corresponding normalized deflection under the equivalent moving load for such a slenderbeam with = 003 and acted upon a moving mass with M=02AL. The numericalresults for the deflection under the equivalent moving load are in excellent agreement with

reported results by Katz et al. [9]. These results are generated using ten-term assumedfunctions forw and (n = 10), which have been found to give numerical results that haveconverged. It can also be seen from Figure 2 that there are significant discrepanciesbetween the deflection under the moving mass and the deflection under the equivalentconstant moving force, especially towards the later part of the prescribed motion.Moreover, as reported by Sadiku and Leipholz [8], the deflection under the equivalentconstant moving force is not an upper bound solution for the deflection under the movingmass.

In Figure 3, the same behavior is shown for a Timoshenko beam with = 015.Moreover, the shapes of the curves in Figure 3 are found to be slightly different from their

Figure 3. Normalized deflections under the moving mass () and the equivalent constant moving loads() for = 015. (a) a = 011; (b) a =05; (c) a =11.


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counterparts in Figure 2. The contact force between the mass and the beam for thenumerical results shown in Figures 2 and 3 is presented in Figure 4. It can be seen that,for a slender beam with = 003, separation of the mass from the beam is found to occurfor the case of a = 11 for s/L close to 097 when the sign of the contact force changesfrom positive to negative. For a Timoshenko beam with = 015, separation of the mass

from the beam is found to occur for both the two cases witha = 005 anda = 11. It shouldbe emphasized here that when the mass separates from the beam, the present equationsof motion are no longer valid for describing the ensuing motions. Besides reducing the axial

speed of the mass, an alternative way of avoiding the separation is to reduce the magnitudeof the mass, as reflected in the numerical results for a moving mass with M=01ALshown in Figure 5. It can be seen from Figure 5(a) that although there are slight differencesamong the deflections under the moving mass for various prescribed axial speeds of themass, the magnitude of the contact force increases drastically when the axial speed ischanged from a =1 to a = 19 (Figure 5(b)). Separation of the mass from the beam isfound to occur fora = 19 for such a mass of smaller magnitude, and the present equationsof motion are no longer valid for describing the ensuing motions.

4.

The equation of motion in matrix form has been formulated for the dynamic responseof a Timoshenko beam acted upon by a mass moving with a constant speed using theLagrangian approach and the assumed mode method. Numerical results are presented inthe present paper for a simply supported beam with a mass travelling with variousprescribed constant speeds. The deflections under the equivalent moving force byneglecting the inertial effects of the mass are found to be in excellent agreement with thereported works. It is found that separation of the mass from the beam may occur for ahigh axial speed of the mass. The present numerical results also confirm the finding by

Figure 4. The contact force between the mass and the beam. , a=011, ----, a = 005, , a =11.(a) =003; (b) = 015.


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U

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Figure 5. Normalized deflections under the moving mass and the contact force between the mass and the beamfor a beam with = 015 and a mass withM=01AL. , a=1, ----, a = 15, , a = 18, , a=19.(a) normalized deflection; (b) contact force.

Sadiku and Leipholz [8] that the deflection under the moving load computed using themoving-force formulation is not always an upper bound solution for the corresponding

moving-force moving-mass formulation for both the Euler beams and Timoshenkobeams. Other complicating effects, such as elastic foundations, axial forces and a movingmass of varying speed, can also easily be incorporated in the present formulation.

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THE DYNAMIC RESPONSE OF A TIMOSHENKO BEAM

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