Acta Mech 210, 351–359 (2010) DOI 10.1007/s00707-009-0207-y
Erol Uzal · L. Emir Sakman
Dynamic response of a circular plate to a moving load
Received: 24 November 2008 / Revised: 23 June 2009 / Published online: 11 August 2009 © Springer-Verlag 2009
Abstract An analytical series solution for the dynamic response of a circular, thin, isotropic plate under a moving singular load is given. The mass of the agent applying the force is ignored. The response of the plate is computed in two example cases, to illustrate the solution, including the force moving on a straight line through the plate, and also for a load moving on a circular trajectory on the plate.
1 Introduction Plates are among the most studied structures in solid mechanics. This is due not only to the frequent occurrences of plate structures in several engineering disciplines, but also to the simplified mathematical structure of governing equations compared with full three-dimensional equations. Circular plates are one of the few cases for which analytical solutions can be worked out depending on the boundary conditions. This has long been done for the eigenproblem for the circular plate. But there seems to be no work in which the dynamic response of the circular plate is computed in detail. In this paper we give an exact analytical expression for the dynamic response of a thin, isotropic plate, simply supported throughout its sides, loaded by a discrete load moving on an arbitrary trajectory on the surface of the plate. The solution of this problem may have an importance in civil engineering structures; for example, one can think of a heavy load being moved on the circular-shaped floor of a building; or a vehicle moving on the floor of a parking lot. Other examples can be found in manufacturing in which some type of machining process is being carried out on a thin circular work piece. At any rate, the present problem may be considered as one of the canonical problems of elastic plate theory. There are a number of studies concerning moving loads on rectangular plates and beams. The book by Fryba [1] collects many such results. Ichikawa et al. [2] considered the general continuous Euler-Bernoulli beam subjected to forces applied by a moving mass and gave an analytical solution based on eigenfunction expansion. Henchi and Fafard [3] worked on a similar problem, but with several loads and ignoring the inertia of the load-producing agents, using a variational finite element approach. Michaltsos [4] gave analytical solutions for a vehicle modeled as two moving loads on a bridge approximated as a beam ignoring the dynamics of the vehicle itself. A similar problem taking the dynamics of the vehicle into account was considered by Esmailzadeh and Jalili [5]. These studies were concentrated on the Euler-Bernoulli beam model. More complicated beam models, for example the Timoshenko beam, were considered in [6–8]. Kargarnovin and Younesian [6] and Lee [7] considered the Timoshenko beam on an elastic foundation subject to moving loads, both by analytical approaches. Wang and Lin [8] considered random vibrations of the Timoshenko beam under a random moving load. Kim [9] studied axially loaded beams on elastic foundations subject to harmonic moving loads. E. Uzal (B) · L. E. Sakman Department of Mechanical Engineering, Istanbul University, Istanbul, Turkey E-mail:
[email protected] Tel.: +90-212-5931941
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F
r = rf (t) θ = θf (t)
Fig. 1 Vertical singular load moving on an arbitrary trajectory on a circular plate
Zibdeh and Juma [10] considered a rotating beam subjected to a random moving load for the Euler-Bernoulli, Rayleigh and Timoshenko beam models. Moving loads on plate structures were also studied quite extensively for rectangular plates. Shadnam et al. [11] gave the analytical solution for the dynamic response of a rectangular plate under the effect of a moving mass. The nonlinear plate version of the same problem was also considered by Shadnam et al. [12]. Takabatake [13], using a Galerkin approach, solves the same problem when the plate thickness changes discontinuously. Composite plates subject to multiple moving loads were considered by Lee and Yhim [14] using finite elements and third-order plate theory. Au and Wang [15] computed sound radiated from a composite plate due to moving loads. Finally, References [16–18] consider plate or plate-like structures under the effect of a moving vehicle model. Kim et al. [16] used a finite element model for a bridge. Li et al. [17] used a similar model but also took into account the wind loads on the vehicle, Yagiz and Sakman [18] used a Lagrangian approach for the bridge-vehicle interaction. The differential quadrature method proposed by Bert et al. [19] seems to be a good candidate for solving dynamic response problems, too. The studies cited above consider elastic materials. The dynamic response of plates with viscoplastic behavior can be computed by employing the Green’s functions computed by Fotiu [20]. Studies pertaining to the dynamic response of circular plates seem to be rare. In this paper, we give an analytical solution for the dynamic response of a circular plate to a singular load moving arbitrarily on the surface of the plate. Any details of the manner the force is applied is ignored, i.e., the dynamics of the agents producing the force are assumed to be much faster than the dynamics of the plate. In the next section, the formulation and solution of the problem is presented, and in Sect. 3 example computations are given for two cases. 2 Problem formulation and solution A circular plate made of a homogeneous, isotropic, linear material is rigidly held (simply supported) at its sides while a singular load F, always vertical to the natural plane of the plate, moves on a prescribed trajectory on the surface of the plate (Fig. 1). The governing equation for small displacements is ρh
∂ 2w F + D∇ 4 w = δ(r − r f (t))δ(θ − θ f (t)). 2 ∂t r
(1)
We use polar coordinates with the origin placed at the center of the plate; r = r f (t) and θ = θ f (t) describe the trajectory of the force; ρ, h and D are the plates’ density, thickness and bending rigidity, respectively. w(r,θ, t) is the displacement of the plate. Non-dimensionalizing lengths by r0 , the plates’ radius, and time by ρhr03 /F, the governing equation can be written as, using the same symbols for the non-dimensional variables, 1 ∂ 2w + α∇ 4 w = δ(r − r f (t))δ(θ − θ f (t)), 2 ∂t r
(2)
where α=
D r0 F
(3)
Dynamic response of a circular plate to a moving load
353
is a non-dimensional constant. The boundary conditions can be written as w=
∂ 2w = 0 at r = 1. ∂r 2
(4)
We assume that the plate is not loaded before the force starts moving, so the initial conditions are w=
∂w = 0 for t = 0. ∂t
(5)
To solve the problem, Eqs. (2), (4) and (5), we utilize the eigenmodes of the circular plate defined by ∇ 4 W − β 4 W = 0, ∂2W = 0 at r = 1, W = ∂r 2
(6) (7)
where β4 =
ω2 , α
(8)
and ω is the eigenvalue defined by w(r, θ, t) = W (r, θ )eiωt . Writing Eq. (6) in the form (∇ 2 − β 2 )(∇ 2 + β 2 )W = 0
(9)
and assuming a separated solution W (r, θ ) = R(r ) cos mθ
or
R(r ) sin mθ,
it is found that m has to be an integer due to periodicity and R(r ) satisfies, in both cases, 2 2 d d 1 d 1 d m2 m2 2 2 + − β + + β − − R = 0. dr 2 r dr r2 dr 2 r dr r2
(10)
(11)
A set of four linearly independent solutions of this equation is R = Jm (βr ), Ym (βr ),
Im (βr ),
K m (βr ),
(12)
of which the second and fourth are discarded since they become infinite at r = 0. Therefore, we can express the eigenfunction as a Fourier series ∞
A0 (r ) W (r, θ ) = + [Am (r ) cos mθ + Bm (r ) sin mθ], 2
(13)
m=1
where Am (r ) = cm Im (βr ) + dm Jm (βr ), Bm (r ) = em Im (βr ) + f m Jm (βr ). Applying the boundary conditions Eq. (7), it is found that β satisfies 2 2 d Jm (βr ) d Im (βr ) − J (β) = 0. Im (β) m dr 2 dr 2 r =1 r =1 Denoting the roots of this equation by βmn ; m = 0, 1, 2, . . . ; n = 1, 2, . . . ,
(14.1) (14.2)
(15)
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and superposing over n, the solution can be written as Eq. (13), in which Am (r ) =
∞
amn Rm (βmn r ),
(16.1)
bmn Rm (βmn r ),
(16.2)
n=1
Bm (r ) =
∞ n=1
where we have defined Rm (βmn r ) = Jm (βmn )Im (βmn r ) − Im (βmn )Jm (βmn r ),
(17)
which are orthogonal over n with weight r. Thus, eigenfunctions are each term in Eq. (13), with the above relevant definitions. To solve the time-dependent problem, Eqs. (2) and (4), we therefore assume that ∞
w(r, θ, t) =
∞
∞
1 a0n (t)R0 (β0n r ) + Rm (βmn r ) [amn (t) cos mθ + bmn (t) sin mθ ]. 2 n=1
(18)
m=1 n=1
Substituting in Eq. (2), and using the orthogonality relations, we find d 2 amn 2 + ωmn amn = a¯ mn , dt 2 d 2 bmn 2 + ωmn bmn = b¯mn , dt 2 where ωmn =
√ 2 αβmn
(19.1) (19.2)
(20)
and 1 Rm (βmn r f (t)) cos mθ f (t), π mn 1 Rm (βmn r f (t)) sin mθ f (t), b¯mn (t) = π mn 1 mn = r [Rm (βmn r )]2 dr ,
a¯ mn (t) =
(21.1) (21.2)
(21.3)
0
with the initial conditions damn dbmn = 0, bmn = = 0 for t = 0. dt dt The solution of Eqs. (19) and (22) is amn =
1 amn (t) = ωmn bmn (t) =
1 ωmn
(22)
t a¯ mn (τ ) sin ωmn (t − τ )dτ ,
(23.1)
b¯mn (τ ) sin ωmn (t − τ )dτ .
(23.2)
0
t 0
Substituting Eqs. (23.1, 2), in Eq. (18) completes the solution for the response to the moving load. 3 Results and discussion We demonstrate the solution expressed by Eqs. (18), (21.1) and (23.1) by considering two case studies. First, the load passes through the plate along a straight line at a distance p from the center with constant velocity U. In this case, the trajectory of the force is given by
Dynamic response of a circular plate to a moving load
355
t=1
t=2
t=3
t=4
t=5 Fig. 2 Response of the plate at various times for p = 0 and U = 0.4
r f (t) =
p 2 + [U t −
θ f (t) = arctan
1 − p 2 ]2 ,
p . U t − 1 − p2
(24.1) (24.2)
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0 -0.1
w ( 0,0,t )
-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8
0
1
2
3
4
5
3
4
5
3
4
5
t
Fig. 3 Response of the center of the plate for p = 0 and U = 0.4
0 -0.1
w ( 0,0,t )
-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8
0
1
2
t
Fig. 4 Response of the center of the plate for p = 0.1 and U = 0.4
0 -0.1
w ( 0,0,t )
-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8
0
1
2
t
Fig. 5 Response of the center of the plate for p = 0.5 and U = 0.4
Here the angle θ f is measured from the positive x-axis. It should be noted that in non-dimensional form the radius of the plate is 1 and all other physical parameters are collected in the non-dimensional α. We took α = 0.075 which roughly corresponds to a 2 mm thick steel plate with 1 m radius carrying 1,000 N. Figure 2 shows the response of the plate for p = 0 as a three-dimensional plot at various times, while Figs. 3, 4, 5 and 6 show the displacement at the center of the plate as a function of time for p = 0, 0.1, 0.5 and 0.8 (going through a diameter). The load passes (through) from left to right, on the upper part. The velocity U is 0.4 in all cases. The last picture in the three-dimensional figure is for the time when the load departs the plate’s surface. Similarly the two-dimensional time plots are for the time-period when the load is on the plate. The vertical scales on each figure are the same to facilitate comparison. When the load passes through the middle of the plate the average displacement of the plate is larger. It is seen that, when the load passes away
Dynamic response of a circular plate to a moving load
357
0 -0.1
w ( 0,0,t )
-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8
0
0.5
1
1.5
2
2.5
3
t
Fig. 6 Response of the center of the plate for p = 0.8 and U = 0.4
t=1
t=2
t=3
t=4
Fig. 7 Response of the plate at various times for a = 0.5 and = 0.5π
from the center, the half of the plate that is not carrying the load has a deformation that is opposite to the other half towards the end of the passage of the load. As the second case, the load rotates at a constant radius a with constant angular velocity . In this case the trajectory is r f (t) = a, θ f (t) = t.
(25.1) (25.2)
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0 -0.1
w ( 0,0,t )
-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8
0
1
2
3
4
5
3
4
5
3
4
5
t
Fig. 8 Response of the center of the plate for a = 0.1 and = 0.5π
0 -0.1
w ( 0,0,t )
-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8
0
1
2
t
Fig. 9 Response of the center of the plate for a = 0.5 and = 0.5π
0 -0.1
w ( 0,0,t )
-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8
0
1
2
t
Fig. 10 Response of the center of the plate for a = 0.8 and = 0.5π
Figure 7 shows the response of the plate for a = 0.5 as a three-dimensional plot at various times while Figs. 8, 9 and 10 show the response of the plate for a = 0.1, 0.5 and 0.8. The angular velocity is 0.5π for all cases. The shape of the plate is shown at times t = 1, 2, 3 and 4 in the three-dimensional plot while the two-dimensional plots are drawn up to time t = 5. Here again the average displacement is larger when the load is closer to the center. The peaks and valleys formed on the plate seem to be rotating in phase with the load. The convergence of the computation is demonstrated by a typical case in which the load passes through the center of the plate in Table 1.
Dynamic response of a circular plate to a moving load
359
Table 1 The convergence of the computation for load passing through the center of the plate Number of Mods
w(0.72, π/5, 3.2)
1 2 3 4
−0.256732000 −0.287076000 −0.289973270 −0.289592055
4 Conclusions An exact solution in the form of a Fourier-Bessel series was given for the dynamic response of a circular plate to an arbitrarily moving singular load on the surface of the plate. The solution was computed for two special cases: one in which the load passes through a straight line with constant velocity, and another one in which the load rotates on a concentric circle on the plate with constant angular velocity. The results seem to be in agreement with physical expectations; no comparison with other work was possible since this seems to be the first instance that such a computation was carried out. The main aim of the paper was to report the exact solution and point out that it can be applied for loads moving on any trajectory on the plate. References 1. Fryba, L.: Vibration of Solids and Structures Under Moving Loads. Noordhoff International, Groningen (1972) 2. Ichikawa, M., Miyakawa, Y., Matsuda, A.: Vibration analysis of the continuous beam subjected to a moving mass. J. Sound Vib. 230, 493–506 (2000) 3. Henchi, K., Fafard, M.: Dynamic behavior of multi-span beams under moving loads. J. Sound Vib. 199, 33–50 (1997) 4. Michaltsos, G.T.: Dynamic behavior of a single-span beam subjected to loads moving with variable speeds. J. Sound Vib. 258, 359–372 (2002) 5. Esmailzadeh, E., Jalili, N.: Vehicle-passenger-structure interaction of uniform bridges traversed by moving vehicles. J. Sound Vib. 260, 611–635 (2003) 6. Kargarnovin, M.H., Younesian, D.: Dynamics of Timoshenko beams on Pasternak foundation under moving load. Mech. Res. Commun. 31, 713–723 (2004) 7. Lee, H.P.: Dynamic response of a Timoshenko beam on a Winkler foundation subjected to a moving mass. Appl. Acoust. 55, 203–215 (1998) 8. Wang, R.T., Lin, T.Y.: Random vibration of Multi-Span Timoshenko beam due to a moving load. J. Sound Vib. 213, 127– 138 (1998) 9. Kim, S.M.: Vibration and stability of axial loaded beams on elastic foundation under moving harmonic loads. Eng. Struct. 26, 95–105 (2004) 10. Zibdeh, H.S., Juma, H.S.: Dynamic response of a rotating beam subjected to a random moving load. J. Sound Vib. 223, 741– 758 (1999) 11. Shadnam, M.R., Mofid, M., Akin, J.E.: On the dynamic response of rectangular plate with moving mass. Thin-Walled Struct. 39, 797–806 (2001) 12. Shadnam, M.R., Rofooei, F.R., Mehri, B.: Dynamics of nonlinear plates under moving loads. Mech. Res. Commun. 28, 453– 461 (2001) 13. Takabatake, H.: Dynamic analysis of rectangular plates with stepped thickness subjected to moving loads including additional mass. J. Sound Vib. 213, 829–842 (1998) 14. Lee, S.Y., Yhim, S.S.: Dynamic analysis of composite plates subjected to multi-moving loads based on a third order theory. Int. J. Solids Struct. 41, 4457–4472 (2004) 15. Au, F.T.K., Wang, M.F.: Sound radiation from forced vibration of rectangular orthotropic plates under moving loads. J. Sound Vib. 281, 1057–1075 (2005) 16. Kim, C.W., Kawatani, M., Kim, K.B.: Three-dimensional dynamic analysis for bridge–vehicle interaction with roadway roughness. Comput. Struct. 83, 1627–1645 (2005) 17. Li, Y., Qiang, S., Liao, H., Xu, Y.L.: Dynamics of wind–rail vehicle–bridge systems. J. Wind Eng. Ind. Aerodyn. 93, 483– 507 (2005) 18. Yagiz, N., Sakman, L.E.: Vibrations of a rectangular bridge as an isotropic plate under a traveling full vehicle model. J. Vib. Control 12, 83–98 (2006) 19. Bert, C.W., Wang, X., Striz, A.G.: Static and free vibrational analysis of beams and plates by differential quadrature method. Acta Mech. 102, 11–24 (1994) 20. Fotiu, P.A.: Elastodynamics of thin plates with internal dissipative processes. Part II. Computational aspects. Acta Mech. 98, 187–212 (1993)
