- Email: [email protected]
On the dynamic response of rectangular plate, with moving mass
Thin-Walled Structures 39 (2001) 797–806 www.elsevier.com/locate/tws
On the dynamic response of rectangular plate, with moving mass M.R. Shadnam a, M. Mofid b
c
b,*
, J.E. Akin
c
a Department of Civil Engineering, Sharif University of Technology, Tehran, Iran Department of Civil Engineering, University of Kansas, 103 G, Learned Hall, Lawrence, KS 66045, USA Department of Mechanical Engineering and Material Sciences, Rice University, Houston, Texas, USA
Received 18 December 2000; received in revised form 23 April 2001; accepted 1 May 2001
Abstract In this article, the dynamics of plates under influence of relatively large masses, moving along an arbitrary trajectory on the plate surface is considered. The method consists of transformation of the governing equation into a series of eigenfunctions, which satisfy the boundary conditions of the plate. The method presented in this investigation is general and can be applied to general moving mass and moving force systems as well. 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. 2001 Published by Elsevier Science Ltd. Keywords: Rectangular plate with moving mass; Eigenfunctions; Boundary conditions; Response of structures
1. Introduction The dynamic behavior of structures under influence of moving loads is a subject of considerable engineering importance, and much attention had been given to the corresponding mathematical problem. Such studies have mostly been done for simpler structures, such as beams and strings, for which analytical complications are rather less significant [1]. The dynamic behavior of a beam subjected to moving
* Corresponding author. Tel.: +1-785-864-3807; fax: +1-785-864-5631. E-mail address: [email protected] (M. Mofid). 0263-8231/01/$ - see front matter 2001 Published by Elsevier Science Ltd. PII: S 0 2 6 3 - 8 2 3 1 ( 0 1 ) 0 0 0 2 5 - 8
798
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
loads or moving masses has been extensively studied in connection with machining processes, guide-way systems and design of railway bridges [2]. The problem is also a topic of interest in other engineering applications such as modern high-speed precision machinery processes [3]. The response of the plate subjected to moving force has been treated previously [4–7], however, there are clearly many problems of great physical significance in which load inertia is not negligible and alters the dynamic behavior of the system significantly [8,9]. The plate excited by an orbiting mass is of considerable interest in mechanical engineering, specifically in the analysis of rotating machinery. Additional applications are the study of the dynamic behavior of disk file memory units in the computer industry, or the study of guided circular saws commonly used in the wood products industry. The two above-mentioned systems can be modeled as circular plates excited by a moving mass [10]. The slab type bridges on which vehicles or trains travel and the decks of ships on which aircrafts land may also be modeled as moving masses on plates [11]. The fundamental mathematical complexity encountered in this problem lies in the fact that one of the coefficients of linear operator describing the motion is a function of space and time. This is caused by the presence of a Dirac-delta function as a coefficient necessary for a proper description of the motion. Physically, this term represents the interplay of inertial forces due to moving mass inertia. The present paper uses the theory proposed by Akin and Mofid [8], which is similar to that of Saigal [12]. However, in this paper it is clearly shown that the Akin formulation [8] has been extended extensively into 3D case. Also Saigal’s work [12] is significantly extended with more mathematical involvement, for “general trajectory of the moving mass”. The object of this paper is: 앫 To present a very simple and practical analytical–numerical technique for determining the response of plates, with various boundary conditions, carrying a moving mass. 앫 To develop a result with the option of investigating the contribution of each mode of vibration in the response of the system. 앫 To extend the procedure for unconditional general movement of the mass.
2. Problem definition A rectangular plate with a moving mass and different boundary conditions is considered. The mass is relatively large, i.e. its inertia cannot be neglected, and is moving along an arbitrary trajectory specified via parametric equations xo(t), yo(t) on the surface of the plate, as shown in Fig. 1. 2.1. Assumptions D(x, y) = D = Constant bending stiffness; r(x, y) = r = Constant mass per unit volume of the plate;
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
Fig. 1.
799
A moving mass traveling along an arbitrary trajectory on the surface of a rectangular plate.
No damping in the system; Uniform gravitational field, g; and M(t) = M = Constant mass moving on the plate 2.2. Initial conditions w(x, y, 0) = g1 (x, y) dw(x, y, 0)/dt = g2 (x, y)
3. Problem solution The equation of motion of a rectangular plate under moving mass M may be written as [9,12]: ⵜ4w⫹
r.h ∂2w P(x,y,t) ⫽ D ∂t2 D
where
冉
(1)
冊
d 2w P(x,y,t)⫽q(x,y,t)⫹M g⫺ 2 d(x⫺xo(t))d(y⫺yo(t)) dt
(2)
q(x, y, t) = general loading (zero for this case); xo(t), yo(t) = parametric equations of the trajectory of the moving mass; and d = Dirac Delta. The deflection can be written, using separation of variable technique, in the following form;
冘 ⬁
w(x,y,t)⫽
fn(x,y).T(t).
n⫽1
(3)
800
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
where, fn are the known eigenfunctions of the plate with the same boundary conditions. They have the form of ⵜ4fn⫺w4nfn⫽0
(4)
⍀2n.r.h w4n⫽ D
(5)
and
⍀n, n =1, 2,… are natural frequencies and Tn(t) are amplitude functions, which have to be calculated. Rewriting Eq. (2) in the form of a series;
冉
冊
d 2w M g⫺ 2 d(x⫺xo(t))d(y⫺yo(t))⫽ dt
冘
fn(x,y).Bn(t)
(6)
n
Substituting Eq. (3) into Eq. (6) and using a simplified subscript for differentiation results in
冉 冘
M g⫺
冊
冘
fn(x,y).Tn,tt(t) d(x⫺xo(t))d(y⫺yo(t))⫽
n
fn(x,y)Bn(t)
(7)
n
Multiplying both sides of Eq. (7) by fp(x, y); and integrating on area of the plate;
冉 冘冉冕
冊
冊
fn(x,y)fp(x,y)dA)Ttt(t) d(x⫺xo(t))d(y⫺yo(t))
M g⫺
n
A
冘冉冕
⫽
(8)
冊
fn(x,y).fp(x,y).dA Bn(t)
n
A
Considering Dirac delta properties and orthogonality of fn,
冉 冘
M Bp(t)⫽ g⫺ Vp
冊
fn(xo(t), yo(t)).Tn,tt(t) fp(xo(t), yo(t))
n
(9)
where
冕
Vp⫽ fp(x,y)2.dA
(10)
A
Substituting Eqs. (7) and (9) into Eqs. (1) and (2), results in an equation for Tn(t)
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
D
冉冘 冊 冘 冋冘 ⵜ4fn.Tn(t) ⫹r.h.
n
n
⫺
n
册
Applying Eq. (4) into Eq. (11),
n
f n.
M · Vn
(fq(xo(t), yo(t)).Tq,tt(t))⫹g .fn(xo(t), yo(t))
q
冘
冘再 冉 冊
(fn,Tn,tt(t))⫽
再
fn(x,y) Dw4n.Tn(t)⫹r.h.Tn,tt(t)⫺.
册
冎
(11)
冎
冉 冊冋 冘 M · ⫺ Vn
801
(fq(xo(t), yo(t)).Tq,tt(t))
(12)
q
⫹g fn(xo(t), yo(t)) ⫽0
This equation must be satisfied for arbitrary x, y (i.e. each point of the plate), and this is possible only when
冉 冊冋 冘
Dw4n.Tn(t)⫹r.h.Tn,tt(t)⫺·
册
M · ⫺ Vn
(fq(xo(t), yo(t))·Tq,tt(t))
(13)
q
⫹g fn(xo(t), yo(t))⫽0 n⫽1,2,3,…
The system in Eq. (13) is a set of coupled ordinary differential equations and a numerical procedure can be used to solve it. Rearranging it in matrix form results in; [H(t)]·{T,tt}⫽⫺{a.T}⫹{C(t)}
(14)
{T,tt}⫽[H(t)]·−1(⫺{a.T}⫹{C(t)})
(15)
or where, letting pi = M/Vi,
冤
m+p1f21 p1f2f1 .. .. p1fnf1 p2f1f2 m+p2f22 .. .. p2fnf2
[H(t)]·⫽ :
:
:
:
·
: · :
冥
(16)
2 n n
pnf1fn pnf2fn .. .. m+p f
an⫽Dw4n n⫽1,2,3,…
(17)
{aT} ⫽{a1T1, a2T2,…, anTn}
(18)
{T,tt}T⫽{T1,tt, T1,tt,..., Tn,tt}
(19)
T
802
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
{C(t)}T⫽g{p1f1, p2f2,..., pnfn}
(20)
in which, m = rh, is the mass per unit length of the beam. The eigenfunctions of the different types of plates are listed in [13].
4. Example problems To check the results of Eq. (18), the following simply supported plate shown in Fig. 1 is considered. The mass is traveling parallel to the X-axis at constant speed. Other data are listed below: r =2.77 e+5 kg/m3, D =3.52 e+9 kgm, v =18.791 m/s, b =30 m, a =60 m and Moving mass = M =49 860 kg. The eigenfunctions and eigenvalues are [13,14]:
冉 冊 冉 冊 冑 冋冉 冊 冉 冊 册冪
fi(x,y)⫽wmn(x,y)⫽
⍀i⫽⍀mn⫽p2
2
rab
m 2 n ⫹ a b
sin 2
mpx npy ·sin , i,m,n苸N a b D m
In each case the example problem has been solved by the above procedure employing a seventh order Runge–Kutta scheme. Fig. 2 provides information on participation of individual modes in deflection, right under the moving mass. The considered modes are (m, n) = (1,1), (1,2), (2,1), (2,2), (1,3), (3,1), respectively. In Fig. 3, the maximum deflections are compared with that of the classic Galerkin method. Figs. 4 and 5 show the total deflected geometry of the plate, when the moving
Fig. 2.
Scheme showing the participation of different modes in deflection under the moving mass.
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
Fig. 3.
Fig. 4.
803
Comparison of the maximum deflections.
Deflected geometry of the plate when the moving mass is in one forth of the span.
mass is in one fourth and one half of the span, respectively. The deflections are presented in SI units i.e. meters. Also, Fig. 6 compares the moving mass model with the moving force model. The deflection under the moving mass and the moving force are considered, accordingly. As the second example, consider a square plate simply supported at corners. It is excited by a mass rotating with a constant angular velocity equal to 10.46 rad/S on a circle of radius 1.27 m centered at the center of the plate. It is desired to determine the transient response for the first 0.6 S, the time necessary to complete one orbiting
804
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
Fig. 5.
Deflected geometry of the plate when the moving mass is in one half of the span.
Fig. 6. Comparison of the deflection under the moving load for moving mass and the moving force model.
cycle. It is assumed that the plate is originally at rest. Other data are listed below: r =2133 kg/m3, D =16557 kgm, a=b =5.08 m and Moving mass = M =4 530 kg. Similar formulas for eigenfunctions and frequencies are employed and the example problem has been solved by the above mentioned procedure, employing a seventh order Runge–Kutta scheme. Fig. 7 shows the displacement of the center point of the plate as a function of time.
5. Remarks on theory and results The problem is formulated for general movement of the mass on the plate and there is no restriction on the trajectory and the speed and/or acceleration of the
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
Fig. 7.
805
Transient response for the second example at the center of the plate.
moving mass. It is evident from Eq. (14) and Fig. 2 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. From Fig. 3, it could be concluded that the present method produces the same results of the Galerkin method. A comparison of the moving mass and moving force results in at least 30% difference between the two results and thus shows the importance of including mass in real design conditions, where the velocity and/or mass is large.
6. Conclusion A simple, yet general method has been introduced to compute the transient response of a plate excited by a traveling mass moving along an arbitrary general trajectory on the surface of a rectangular plate. The method is general so that having the free vibration eigenfunctions, one could extend it for nonrectangular plates. As examples, the dynamic response of a rectangular plate, simply supported on all its edges, under a mass moving parallel to one of its sides and also traveling along a circular trajectory is presented by means of operational calculus. From the analysis and results presented in the paper, it could be concluded that the method of eigenfunction expansion, presented in this paper, is able to capture the dynamic response of the moving mass problems. 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 personal computers.
806
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
References [1] Knowles JK. On the dynamic response of a beam to a randomly moving load. Journal of Applied Mechanics 1968;35. [2] Rao GV. Linear dynamics of an elastic beam under moving loads. ASME, Journal of Vibrations & Acoustics 2000;122. [3] Katz R, Lee CW, Ulsoy AG, Scott RA. Dynamic stability and response of a beam subjected to a deflection dependent moving load. Trans ASME, Journal of Vibrations & Acoustics 1987;109. [4] Livesly RK. Some notes on the mathematical theory of a loaded elastic plate resting on an elastic foundation. Quarterly Journal of Applied Mathematics 1953;6. [5] Gbadeyan JA, Oni ST. Dynamic behavior of beams and rectangular plates under moving loads. Journal of Sound and Vibration 1995;182(5). [6] Mindlin RD. Influence of rotary to inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics 1957;18. [7] Fryba L. Vibration of solids and structures under moving loads. The Netherlands: Noordhoff International, 1971. [8] Akin JE, Mofid M. Numerical Solution for Response of Beams with Moving Mass. ASCE Journal of Structural Engineering 1989;115(1). [9] Iwan WD, Stah KJ. The response of an elastic disk with a moving mass system. Journal of Applied Mechanics 1973;40. [10] Cifuentes A, Lalapet S. A general method to determine the dynamic response of a plate to a moving mass. Computers & Structures 1992;42(1). [11] Humar JL, Kashif AM. Dynamic response analysis of slab-type bridges. ASCE Journal of Structural Engineering 1995;121(1). [12] Saigal S, Agrawal OP, Stanisic MM. Influence of moving masses on rectangular plate dynamics. Ingenieur — Archiv 1987;57. [13] Soedel W. Vibrations of shells and plates. NY: Marcel Dekker, 1993. [14] Meirovitch L. Principles and techniques of vibrations. NJ: Prentice-Hall, 1997.
On the dynamic response of rectangular plate, with moving mass M.R. Shadnam a, M. Mofid b
c
b,*
, J.E. Akin
c
a Department of Civil Engineering, Sharif University of Technology, Tehran, Iran Department of Civil Engineering, University of Kansas, 103 G, Learned Hall, Lawrence, KS 66045, USA Department of Mechanical Engineering and Material Sciences, Rice University, Houston, Texas, USA
Received 18 December 2000; received in revised form 23 April 2001; accepted 1 May 2001
Abstract In this article, the dynamics of plates under influence of relatively large masses, moving along an arbitrary trajectory on the plate surface is considered. The method consists of transformation of the governing equation into a series of eigenfunctions, which satisfy the boundary conditions of the plate. The method presented in this investigation is general and can be applied to general moving mass and moving force systems as well. 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. 2001 Published by Elsevier Science Ltd. Keywords: Rectangular plate with moving mass; Eigenfunctions; Boundary conditions; Response of structures
1. Introduction The dynamic behavior of structures under influence of moving loads is a subject of considerable engineering importance, and much attention had been given to the corresponding mathematical problem. Such studies have mostly been done for simpler structures, such as beams and strings, for which analytical complications are rather less significant [1]. The dynamic behavior of a beam subjected to moving
* Corresponding author. Tel.: +1-785-864-3807; fax: +1-785-864-5631. E-mail address: [email protected] (M. Mofid). 0263-8231/01/$ - see front matter 2001 Published by Elsevier Science Ltd. PII: S 0 2 6 3 - 8 2 3 1 ( 0 1 ) 0 0 0 2 5 - 8
798
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
loads or moving masses has been extensively studied in connection with machining processes, guide-way systems and design of railway bridges [2]. The problem is also a topic of interest in other engineering applications such as modern high-speed precision machinery processes [3]. The response of the plate subjected to moving force has been treated previously [4–7], however, there are clearly many problems of great physical significance in which load inertia is not negligible and alters the dynamic behavior of the system significantly [8,9]. The plate excited by an orbiting mass is of considerable interest in mechanical engineering, specifically in the analysis of rotating machinery. Additional applications are the study of the dynamic behavior of disk file memory units in the computer industry, or the study of guided circular saws commonly used in the wood products industry. The two above-mentioned systems can be modeled as circular plates excited by a moving mass [10]. The slab type bridges on which vehicles or trains travel and the decks of ships on which aircrafts land may also be modeled as moving masses on plates [11]. The fundamental mathematical complexity encountered in this problem lies in the fact that one of the coefficients of linear operator describing the motion is a function of space and time. This is caused by the presence of a Dirac-delta function as a coefficient necessary for a proper description of the motion. Physically, this term represents the interplay of inertial forces due to moving mass inertia. The present paper uses the theory proposed by Akin and Mofid [8], which is similar to that of Saigal [12]. However, in this paper it is clearly shown that the Akin formulation [8] has been extended extensively into 3D case. Also Saigal’s work [12] is significantly extended with more mathematical involvement, for “general trajectory of the moving mass”. The object of this paper is: 앫 To present a very simple and practical analytical–numerical technique for determining the response of plates, with various boundary conditions, carrying a moving mass. 앫 To develop a result with the option of investigating the contribution of each mode of vibration in the response of the system. 앫 To extend the procedure for unconditional general movement of the mass.
2. Problem definition A rectangular plate with a moving mass and different boundary conditions is considered. The mass is relatively large, i.e. its inertia cannot be neglected, and is moving along an arbitrary trajectory specified via parametric equations xo(t), yo(t) on the surface of the plate, as shown in Fig. 1. 2.1. Assumptions D(x, y) = D = Constant bending stiffness; r(x, y) = r = Constant mass per unit volume of the plate;
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
Fig. 1.
799
A moving mass traveling along an arbitrary trajectory on the surface of a rectangular plate.
No damping in the system; Uniform gravitational field, g; and M(t) = M = Constant mass moving on the plate 2.2. Initial conditions w(x, y, 0) = g1 (x, y) dw(x, y, 0)/dt = g2 (x, y)
3. Problem solution The equation of motion of a rectangular plate under moving mass M may be written as [9,12]: ⵜ4w⫹
r.h ∂2w P(x,y,t) ⫽ D ∂t2 D
where
冉
(1)
冊
d 2w P(x,y,t)⫽q(x,y,t)⫹M g⫺ 2 d(x⫺xo(t))d(y⫺yo(t)) dt
(2)
q(x, y, t) = general loading (zero for this case); xo(t), yo(t) = parametric equations of the trajectory of the moving mass; and d = Dirac Delta. The deflection can be written, using separation of variable technique, in the following form;
冘 ⬁
w(x,y,t)⫽
fn(x,y).T(t).
n⫽1
(3)
800
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
where, fn are the known eigenfunctions of the plate with the same boundary conditions. They have the form of ⵜ4fn⫺w4nfn⫽0
(4)
⍀2n.r.h w4n⫽ D
(5)
and
⍀n, n =1, 2,… are natural frequencies and Tn(t) are amplitude functions, which have to be calculated. Rewriting Eq. (2) in the form of a series;
冉
冊
d 2w M g⫺ 2 d(x⫺xo(t))d(y⫺yo(t))⫽ dt
冘
fn(x,y).Bn(t)
(6)
n
Substituting Eq. (3) into Eq. (6) and using a simplified subscript for differentiation results in
冉 冘
M g⫺
冊
冘
fn(x,y).Tn,tt(t) d(x⫺xo(t))d(y⫺yo(t))⫽
n
fn(x,y)Bn(t)
(7)
n
Multiplying both sides of Eq. (7) by fp(x, y); and integrating on area of the plate;
冉 冘冉冕
冊
冊
fn(x,y)fp(x,y)dA)Ttt(t) d(x⫺xo(t))d(y⫺yo(t))
M g⫺
n
A
冘冉冕
⫽
(8)
冊
fn(x,y).fp(x,y).dA Bn(t)
n
A
Considering Dirac delta properties and orthogonality of fn,
冉 冘
M Bp(t)⫽ g⫺ Vp
冊
fn(xo(t), yo(t)).Tn,tt(t) fp(xo(t), yo(t))
n
(9)
where
冕
Vp⫽ fp(x,y)2.dA
(10)
A
Substituting Eqs. (7) and (9) into Eqs. (1) and (2), results in an equation for Tn(t)
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
D
冉冘 冊 冘 冋冘 ⵜ4fn.Tn(t) ⫹r.h.
n
n
⫺
n
册
Applying Eq. (4) into Eq. (11),
n
f n.
M · Vn
(fq(xo(t), yo(t)).Tq,tt(t))⫹g .fn(xo(t), yo(t))
q
冘
冘再 冉 冊
(fn,Tn,tt(t))⫽
再
fn(x,y) Dw4n.Tn(t)⫹r.h.Tn,tt(t)⫺.
册
冎
(11)
冎
冉 冊冋 冘 M · ⫺ Vn
801
(fq(xo(t), yo(t)).Tq,tt(t))
(12)
q
⫹g fn(xo(t), yo(t)) ⫽0
This equation must be satisfied for arbitrary x, y (i.e. each point of the plate), and this is possible only when
冉 冊冋 冘
Dw4n.Tn(t)⫹r.h.Tn,tt(t)⫺·
册
M · ⫺ Vn
(fq(xo(t), yo(t))·Tq,tt(t))
(13)
q
⫹g fn(xo(t), yo(t))⫽0 n⫽1,2,3,…
The system in Eq. (13) is a set of coupled ordinary differential equations and a numerical procedure can be used to solve it. Rearranging it in matrix form results in; [H(t)]·{T,tt}⫽⫺{a.T}⫹{C(t)}
(14)
{T,tt}⫽[H(t)]·−1(⫺{a.T}⫹{C(t)})
(15)
or where, letting pi = M/Vi,
冤
m+p1f21 p1f2f1 .. .. p1fnf1 p2f1f2 m+p2f22 .. .. p2fnf2
[H(t)]·⫽ :
:
:
:
·
: · :
冥
(16)
2 n n
pnf1fn pnf2fn .. .. m+p f
an⫽Dw4n n⫽1,2,3,…
(17)
{aT} ⫽{a1T1, a2T2,…, anTn}
(18)
{T,tt}T⫽{T1,tt, T1,tt,..., Tn,tt}
(19)
T
802
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
{C(t)}T⫽g{p1f1, p2f2,..., pnfn}
(20)
in which, m = rh, is the mass per unit length of the beam. The eigenfunctions of the different types of plates are listed in [13].
4. Example problems To check the results of Eq. (18), the following simply supported plate shown in Fig. 1 is considered. The mass is traveling parallel to the X-axis at constant speed. Other data are listed below: r =2.77 e+5 kg/m3, D =3.52 e+9 kgm, v =18.791 m/s, b =30 m, a =60 m and Moving mass = M =49 860 kg. The eigenfunctions and eigenvalues are [13,14]:
冉 冊 冉 冊 冑 冋冉 冊 冉 冊 册冪
fi(x,y)⫽wmn(x,y)⫽
⍀i⫽⍀mn⫽p2
2
rab
m 2 n ⫹ a b
sin 2
mpx npy ·sin , i,m,n苸N a b D m
In each case the example problem has been solved by the above procedure employing a seventh order Runge–Kutta scheme. Fig. 2 provides information on participation of individual modes in deflection, right under the moving mass. The considered modes are (m, n) = (1,1), (1,2), (2,1), (2,2), (1,3), (3,1), respectively. In Fig. 3, the maximum deflections are compared with that of the classic Galerkin method. Figs. 4 and 5 show the total deflected geometry of the plate, when the moving
Fig. 2.
Scheme showing the participation of different modes in deflection under the moving mass.
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
Fig. 3.
Fig. 4.
803
Comparison of the maximum deflections.
Deflected geometry of the plate when the moving mass is in one forth of the span.
mass is in one fourth and one half of the span, respectively. The deflections are presented in SI units i.e. meters. Also, Fig. 6 compares the moving mass model with the moving force model. The deflection under the moving mass and the moving force are considered, accordingly. As the second example, consider a square plate simply supported at corners. It is excited by a mass rotating with a constant angular velocity equal to 10.46 rad/S on a circle of radius 1.27 m centered at the center of the plate. It is desired to determine the transient response for the first 0.6 S, the time necessary to complete one orbiting
804
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
Fig. 5.
Deflected geometry of the plate when the moving mass is in one half of the span.
Fig. 6. Comparison of the deflection under the moving load for moving mass and the moving force model.
cycle. It is assumed that the plate is originally at rest. Other data are listed below: r =2133 kg/m3, D =16557 kgm, a=b =5.08 m and Moving mass = M =4 530 kg. Similar formulas for eigenfunctions and frequencies are employed and the example problem has been solved by the above mentioned procedure, employing a seventh order Runge–Kutta scheme. Fig. 7 shows the displacement of the center point of the plate as a function of time.
5. Remarks on theory and results The problem is formulated for general movement of the mass on the plate and there is no restriction on the trajectory and the speed and/or acceleration of the
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
Fig. 7.
805
Transient response for the second example at the center of the plate.
moving mass. It is evident from Eq. (14) and Fig. 2 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. From Fig. 3, it could be concluded that the present method produces the same results of the Galerkin method. A comparison of the moving mass and moving force results in at least 30% difference between the two results and thus shows the importance of including mass in real design conditions, where the velocity and/or mass is large.
6. Conclusion A simple, yet general method has been introduced to compute the transient response of a plate excited by a traveling mass moving along an arbitrary general trajectory on the surface of a rectangular plate. The method is general so that having the free vibration eigenfunctions, one could extend it for nonrectangular plates. As examples, the dynamic response of a rectangular plate, simply supported on all its edges, under a mass moving parallel to one of its sides and also traveling along a circular trajectory is presented by means of operational calculus. From the analysis and results presented in the paper, it could be concluded that the method of eigenfunction expansion, presented in this paper, is able to capture the dynamic response of the moving mass problems. 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 personal computers.
806
M.R. Shadnam et al. / Thin-Walled Structures 39 (2001) 797–806
References [1] Knowles JK. On the dynamic response of a beam to a randomly moving load. Journal of Applied Mechanics 1968;35. [2] Rao GV. Linear dynamics of an elastic beam under moving loads. ASME, Journal of Vibrations & Acoustics 2000;122. [3] Katz R, Lee CW, Ulsoy AG, Scott RA. Dynamic stability and response of a beam subjected to a deflection dependent moving load. Trans ASME, Journal of Vibrations & Acoustics 1987;109. [4] Livesly RK. Some notes on the mathematical theory of a loaded elastic plate resting on an elastic foundation. Quarterly Journal of Applied Mathematics 1953;6. [5] Gbadeyan JA, Oni ST. Dynamic behavior of beams and rectangular plates under moving loads. Journal of Sound and Vibration 1995;182(5). [6] Mindlin RD. Influence of rotary to inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics 1957;18. [7] Fryba L. Vibration of solids and structures under moving loads. The Netherlands: Noordhoff International, 1971. [8] Akin JE, Mofid M. Numerical Solution for Response of Beams with Moving Mass. ASCE Journal of Structural Engineering 1989;115(1). [9] Iwan WD, Stah KJ. The response of an elastic disk with a moving mass system. Journal of Applied Mechanics 1973;40. [10] Cifuentes A, Lalapet S. A general method to determine the dynamic response of a plate to a moving mass. Computers & Structures 1992;42(1). [11] Humar JL, Kashif AM. Dynamic response analysis of slab-type bridges. ASCE Journal of Structural Engineering 1995;121(1). [12] Saigal S, Agrawal OP, Stanisic MM. Influence of moving masses on rectangular plate dynamics. Ingenieur — Archiv 1987;57. [13] Soedel W. Vibrations of shells and plates. NY: Marcel Dekker, 1993. [14] Meirovitch L. Principles and techniques of vibrations. NJ: Prentice-Hall, 1997.