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Transverse vibration of a plate moving over multiple point supports
Applied Acousfics, Vol. 47, No. 4, pp. 291-301,
1996
Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0003-682X/96/$1 5.00 + 0.00 0003-682X(95)00058-5
Transverse Vibration of a Plate Moving over Multiple Point Supports H. P. Lee & T. Y. Ng Department
of Mechanical & Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 (Received 28 April 1995; revised version received 25 August 1995; accepted 29 September 1995)
ABSTRACT The equation of motion of a rectangular plate moving over multiple point supports is derived based on the Lagrangian approach and the assumed mode method. The point supports are assumed to be frictionless and are modelled by linear spring supports of large st@ness with the plate being pulled or pushed longitudinally over them. The transverse vibration of the plate is examined for various prescribed longitudinal sinusoidal motions of the plate. Keywords: Multiple supports, plate, vibration.
INTRODUCTION Many studies have been reported for the transverse vibration of a plate on point supports. These point-supported plates can be found frequently in space structures, electrical device supporting structures and many other engineering applications. The free vibration of an elastically pointsupported plate was analysed by Leuner,’ and Laura and Gutierrez2 for isotropic plates, and by Srinivasan and Munaswamy3 and Narita4 for composite plates. The steady-state response of elastically point-supported composite rectangular plates was reported by Ichinomiya et aL5 using the Ritz approach. Yamada et ~1.~ reported the forced response of an isotropic plate on viscoelastic point supports. The related studies for the dynamics of a plate subjected to a moving load were treated in the book by Fryba.7 The dynamics of a plate executing small motions relative to a reference frame undergoing large overall rigid-body motion was presented 291
292
H. P. Lee, T. Y. Ng
by Banerjee and Kane. 8 Yound and Lioug analysed the vibration of cantilever plates which were often used as models for rotating blades in turbomachinery. The present problem of a plate being pushed or pulled over multiple point support has not been analysed. The present study is an extension of an earlier work by Lee and Ng’O on the parametric excitation of a plate moving over multiple line supports.
THEORY
AND FORMULATIONS
The plate considered is assumed to be a thin, uniform plate of dimension ax,0 moving horizontally over multiple point supports, as shown in Fig. 1. Horizontal forces, not shown in the diagram, are applied at the left edge of the plate to pull or push the plate over these point supports. The assumptions made in the following formulation are that transverse deflections are small so that the dynamic behaviour of the plate is governed by classical thin plate theory. A set of mutually perpendicular unit vectors i, j, and k is assumed to be fixed in the plate. Flexibility of the plate in the two in-plane directions i and j is assumed to be negligible compared to the lateral direction k. With w, a function of X, y and t defined as the transverse deflection of a general point P with coordinates (x, y) on the plate, the velocity of the general point P is given by vp = ui + @k
(1)
where ti = dw/dt and 13 is the prescribed velocity of the plate at x = 0 as a result of the external applied forces at the left edge of the plate. Due to the assumption of in-plane rigidity, U is also the i component of the velocity
Fig. 1. A plate moving over multiple point supports.
293
Transverse vibration of a plate
for every point on the plate. With m denoting the mass of the plate per unit area, the kinetic energy T of the plate is T+n
(d+ti2)dxdy.
]j
(2)
0 0 Using classical thin plate theory, and defining D and p to be, respectively, the flexural rigidity and the Poisson’s ratio of the plate, the elastic strain energy of the plate due to bending is
0
0
Each point support is regarded to be a very stiff linear spring of stiffness k. The potential energy due to all the four point supports shown in Fig. 1 is vs =
;k
W2(%h)
+ w2(u,,b2)
+
w2(a2,14
.
+ w2(a*,b2)
(4)
>
(
The quantities within each pair of brackets after the term w2 are the coordinates of the point supports relative to the moving plate. For example w(al,bl) is the deflection of the plate at x = al and y = bi. The other terms are defined in the same manner. These relative locations of the point supports change with the motion of the plate. It can be seen from the above expression that increasing the number of point supports will only increase the number of terms in eqn (4). It will not directly affect the expressions for kinetic energy, strain energy and the following expression for the potential energy due to the axial forces of the plate. The potential energy due to the in-plane forces per unit width,f,, in the i direction caused by the motion of the plate is
+Ij j
va
0
fx( ~)2dxdy.
(5)
0
For a plate moving over the point supports shown in Fig. 1, the right edge of the plate is stress-free as the plate is being pulled or pushed over
294
H. P. Lee, T. Y. Ng
the line supports by external forces applied at the left edge of the plate. The in-plane forces per unit width in the i direction for the remaining part of the plate are
fx=
-m/tidr
(6)
X =
-mir(a-X)
where Y is a dummy variable for integration. method, the quantity w can be expressed as
Using the assumed mode
4nwon (X,Y)
w = i:
(7)
iZ=l
where c#+,are shape functions that satisfy the geometric boundary conditions at the four edges of the plate. For the present study, 4, are assumed to be products of the normalized modal functions for the vibration of a uniform unrestrained beam. The assumed functions are
(8) The shape functions $+(x) and cp,O) are defined as !A(x> = 1
?h(x) = & ‘@i(X)
=
COS y
+
(
(9)
1- 7
W-9
>
cash y
- yi_2
(11)
Xi_2X Xi_2.X sin - 1 + sinh -j-
(i = 3, .. ..I) I
$92(Y)
(pi(Y) = COS
(12)
1
cpl(Y>
=
=
( >
Jj-
1 -;
2Y
xi-2ycosh--Xi-2Y
w+
W
xj-2Y
sinw
xj-2Y
+ sinh 7
1
(13) Yj-2
o’= 3, . ....I)
(14)
295
Transverse vibration of a plate
where cos A, - cash A,
(15)
‘f~ = sin A, - sinh A,
for p=i orj and X1,...,XI or XJ are the consecutive roots of the transcendental equation 1 -cosXcoshX=O.
(16)
The functions $~t and cpl correspond to the rigid-body translation of an unrestrained beam, whereas & and cp2are the shape functions for rigidbody rotation. The total number of terms in the assumed form of u is N = ZxJ. In the following formulation, Z is made to be equal to J. c& are defined as (5&y) dh(x,y> =
= $~(x)(PI(Y) for n= 7h-1Wf2(y)
for . . = .
Mx,v>
=
&-(I-&)w(Y)
n =
l,...,Z I+
4 ..A
(17) (18)
. . .
for n = (Z- LIZ+ 1, .. ..Z2.
(19)
The Lagrangian of the plate involving u can be expressed as L=T=VE-Va-Vs.
(20)
The Euler-Lagrange equation for a plate moving over multiple point supports, shown in Fig. 1, is therefore given by mH~+(DM+mb!+kfD)q=O
(21)
where H, M, Y, Q and Q are matrices defined as b
a
(22)
(23)
H. P. Lee, T. Y. Ng
296
(Y),=
ji-(Lx)g$dy 0
P)q
=
m1>ij
(24)
0
+
Pl2)ij
+
(@21)ij
+
(@22&
(25)
with (%“)ii
= q&(x = @?I,b&j)j(x = a,, bn).
(26)
The matrices M, H, Y and @ are symmetric matrices. All the matrices except @ are independent of time. The matrix @ need to be updated as the plate is moving over the point supports. The vectors q and 4 are nx 1 column vectors consisting of qi and ii, respectively. The equations of motion generated can be included in numerical simulation programs for investigating the response of a plate undergoing various prescribed motions. The numerical integrations are performed using the variable time-step fourth-order Runge-Kutta method to ensure the stability of the numerical integrations.
RESULTS
AND SIMULATIONS
For the present numerical simulations, a = b = 1 m, m = 1 kg/m2 and D = 1 Nm. The initial configuration of the plate is such that the four point supports are located symmetrically at ai = 0.375 m, a2 = 0.625 m, bl = 0.25 m and b2=0.75 m. The initial shape of the plate is described by
(27) The quantities q1 and q3 are determined from the conditions that w = 0 m at all of the four point supports at aI = 0.375 m and a2 = 0.625 m, and w = 0.01 m at x= 0 m and at x = 1 m. Since 4, corresponds to the rigid-body translation for a beam, the initial shape of the plate is the first symmetric flexural mode shape of an unrestrained beam in the x direction passing through the four point supports. For a prescribed longitudinal sinusoidal motion of the plate in the i direction, the variation of x=ai is assumed to be ai = 0.375 - A sinS2t m.
(28)
For a plate moving over supports, shown in Fig. 1, the transverse displacements at various points on the plate with x=0 and x=0.375 m for
297
Transverse vibration of a plate
I
0
0.5
1
1.5
2
2.5
time (set) ,j x10-3 I
I
3 z
2
II:
0 3
0
0.5
1.5
1
2
2.5
time (set)
Fig. 2. Point displacements at several points on the plate. Upper diagram: -, y=O m; .... .. x=O,y=0.25m;---,x=O,y=0.5m.Lowerdiagram:-,x=0.375,y=Om; ...... x=O.375,y=O.25 m; ---, x=O.375,y=O.S m.
x=0,
R = 20 rad/s and A = 0.05 are shown in Fig. 2 using 25term approximation for w (N= 25, Z= J= 5), which has been found to result in converged results for the transverse displacements. It can be seen that the point displacements indicated by dotted lines for the point at x=0.375 m and y = 0.25 m near one of the point supports for k= lo4 N/m are almost negligible compared with the point displacements for the remaining points. Moreover, numerical results for k= lo5 N/m, not presented here, are found to be almost identical with that of k = IO4 N/m. The execution time for the numerical integration is found to increase with increased value of k. For the following numerical simulations, k is therefore chosen to be lo4 N/m for all the spring supports. The point displacements for three different points on the left edge of the plate for R = 20, 30, 32 and 34 rad/s and A = 0.05 are shown sequentially from the top diagram to the bottom diagram in Fig. 3. The point displacements for the two corners at the left edge are found to be identical, as the initial shape of the plate is assumed to consist of only the first rigidbody translation mode in the y direction (cpi (y)) as shown in eqn (27). The plate shows stable behaviour for R= 20, 30 and 32 rad/s. However, at R = 34 rad/s, an unstable behaviour in the form of an exponential increase
298
H. P. Lee, T.
R = 20
Y.Ng
rad/sec
I
I
0
1
2
3
5
4
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
time (set) R = 30 rad/sec
I
I
5
4
time (set)
R = 32 rad/sec
0.02 E R x z b
0
-0.02 J
I
0
1
2
3
5
4
time (set) 0 = 34 rad/sec
1
0.2,
-0.2 ’ 0
1 0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time (set) Fig. 3.
Point displacements
at the left edge of the plate. .‘..., x = 0, y = 0 m; --, y=0.5m;---,x=O,y=l m.
x = 0,
299
Transverse vibration of a plate
R = 20 rad/sec
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
IO
6
7
8
9
10
time (set) R = 30 rad/sx
x10-3 I
4
5 time (set)
II = 32 rad/sec
0
1
2
3
4
5 time (set)
R = 34 rad/sec 0.1 rl
0
0.5
1
1.5
2
2.5
3
time (set)
Fig. 4. Point displacements
at the left edge of the plate. ....., x = 0, y = 0 m; _ y=OSm;---,x=O,y=lm.
_
H. P. Lee, T. Y. Ng
300
of the amplitude of vibration is observed. Other regions of instability are also observed for different values of R. A general stability analysis can be performed by the method presented by Buffinton and Kane” using Floquet’s theory. Such a laborious analysis will not be undertaken in the present study. However, this paper will attempt to show the different behaviour of the plate if the initial shape of the plate is different. For the numerical results shown in Fig. 4, the initial shape of the plate is changed to u=
!h(P3q11
+ $3(P3q13.
(29)
The assumed function in the y direction is the first flexural mode for a beam. Once again, the point displacements at the two corners at the left edge of the plate are found to be identical, as the prescribed initial shape of the plate is symmetrical with respect to the locations of the point supports. Numerical results shown in Fig. 4 show that the behaviour of the plate at R =20, 30 and 32 rad/s, the first three diagrams presented sequentially from the top, are all stable. The behaviour of the plate is once again unstable at R = 34 rad/s, as shown in the bottom diagram of Fig. 4. These numerical results are consistent with the fact that the stability of a linear system is independent of the initial conditions. The behaviour of a plate with other combinations of point supports and prescribed longitudinal motions of the plate can also be easily verified using the present method.
CONCLUSION Approximate equations of motion in matrix form are derived for the motion of a plate moving over multiple point supports using the Lagrangian approach and the assumed mode method. The transverse vibration of the plate is caused by external forces applied at one end of the plate. For in-plane sinusoidal motion of the plate with a prescribed amplitude of excitation over the point supports, the stability of the plate is found to be dependent on the frequency of excitation, and independent of the initial prescribed shape of the plate.
REFERENCES 1. Leuner, T.R., An experimental-theoretical study of free vibrations of plates on elastic point supports. Journal of Sound and Vibration, 32 (1974) 481490. 2. Laura, P.A.A. & Gutierrez, R.H., Transverse vibration of thin, elastic plates with concentrated masses and internal elastic supports. Journal of Sound and Vibration,
75 (1981)
135-143.
Transversevibrationof a plate
301
3. Srinivasan, R.S. & Munaswamy, K., Frequency analysis of skew orthotropic point supported plates. Journal of Sound and Vibration, 39 (1975) 207-216. 4. Narita, Y., Note on vibrations of point supported rectangular plates. Journal of Sound and Vibration, 93 (1984) 593-597. 5. Ichinomiya, O., Narita, Y. & Maruyama, K., Steady state response analysis of elastically point supported composite rectangular plates. In Dynamics of Plates and Shells. ASME Publication PVP-178. ASME, USA, 1989, 65-70. 6. Yamada, F.G., Irie, T. & Takahashi, M., Determination of the steady state response of a viscoelastically point-supported rectangular plate. Journal of Sound and Vibration, 102 (1985) 285-295. 7. Fjba, L., Vibration of Solids and Structures Under Moving Loads. Noordhoff, Groningen, 1972. 8. Banerjee, A.K. & Kane, T.R., Dynamics of a plate in large overall motion. Transactions of the ASME, Journal of Applied Mechanics, 56 (1989) 887-891. 9. Young, T.H. & Liou, G.T., Coriolis effect on the vibration of a cantilever plate with time-dependent rotating speed. Transactions of the ASME, Journal of Vibration and Acoustics, 114 (1992) 232-241. 10. Lee, H.P. & Ng, T.Y., Effects of support configuration on the parametric excitation of a plate. International Journal of Solids and Structures, 31 (1994) 835-848. 11. Buffinton, K.W. & Kane, T.R., Dynamics of a beam moving over supports. International Journal of Solids and Structures, 21 (1985) 617-643.
1996
Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0003-682X/96/$1 5.00 + 0.00 0003-682X(95)00058-5
Transverse Vibration of a Plate Moving over Multiple Point Supports H. P. Lee & T. Y. Ng Department
of Mechanical & Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 (Received 28 April 1995; revised version received 25 August 1995; accepted 29 September 1995)
ABSTRACT The equation of motion of a rectangular plate moving over multiple point supports is derived based on the Lagrangian approach and the assumed mode method. The point supports are assumed to be frictionless and are modelled by linear spring supports of large st@ness with the plate being pulled or pushed longitudinally over them. The transverse vibration of the plate is examined for various prescribed longitudinal sinusoidal motions of the plate. Keywords: Multiple supports, plate, vibration.
INTRODUCTION Many studies have been reported for the transverse vibration of a plate on point supports. These point-supported plates can be found frequently in space structures, electrical device supporting structures and many other engineering applications. The free vibration of an elastically pointsupported plate was analysed by Leuner,’ and Laura and Gutierrez2 for isotropic plates, and by Srinivasan and Munaswamy3 and Narita4 for composite plates. The steady-state response of elastically point-supported composite rectangular plates was reported by Ichinomiya et aL5 using the Ritz approach. Yamada et ~1.~ reported the forced response of an isotropic plate on viscoelastic point supports. The related studies for the dynamics of a plate subjected to a moving load were treated in the book by Fryba.7 The dynamics of a plate executing small motions relative to a reference frame undergoing large overall rigid-body motion was presented 291
292
H. P. Lee, T. Y. Ng
by Banerjee and Kane. 8 Yound and Lioug analysed the vibration of cantilever plates which were often used as models for rotating blades in turbomachinery. The present problem of a plate being pushed or pulled over multiple point support has not been analysed. The present study is an extension of an earlier work by Lee and Ng’O on the parametric excitation of a plate moving over multiple line supports.
THEORY
AND FORMULATIONS
The plate considered is assumed to be a thin, uniform plate of dimension ax,0 moving horizontally over multiple point supports, as shown in Fig. 1. Horizontal forces, not shown in the diagram, are applied at the left edge of the plate to pull or push the plate over these point supports. The assumptions made in the following formulation are that transverse deflections are small so that the dynamic behaviour of the plate is governed by classical thin plate theory. A set of mutually perpendicular unit vectors i, j, and k is assumed to be fixed in the plate. Flexibility of the plate in the two in-plane directions i and j is assumed to be negligible compared to the lateral direction k. With w, a function of X, y and t defined as the transverse deflection of a general point P with coordinates (x, y) on the plate, the velocity of the general point P is given by vp = ui + @k
(1)
where ti = dw/dt and 13 is the prescribed velocity of the plate at x = 0 as a result of the external applied forces at the left edge of the plate. Due to the assumption of in-plane rigidity, U is also the i component of the velocity
Fig. 1. A plate moving over multiple point supports.
293
Transverse vibration of a plate
for every point on the plate. With m denoting the mass of the plate per unit area, the kinetic energy T of the plate is T+n
(d+ti2)dxdy.
]j
(2)
0 0 Using classical thin plate theory, and defining D and p to be, respectively, the flexural rigidity and the Poisson’s ratio of the plate, the elastic strain energy of the plate due to bending is
0
0
Each point support is regarded to be a very stiff linear spring of stiffness k. The potential energy due to all the four point supports shown in Fig. 1 is vs =
;k
W2(%h)
+ w2(u,,b2)
+
w2(a2,14
.
+ w2(a*,b2)
(4)
>
(
The quantities within each pair of brackets after the term w2 are the coordinates of the point supports relative to the moving plate. For example w(al,bl) is the deflection of the plate at x = al and y = bi. The other terms are defined in the same manner. These relative locations of the point supports change with the motion of the plate. It can be seen from the above expression that increasing the number of point supports will only increase the number of terms in eqn (4). It will not directly affect the expressions for kinetic energy, strain energy and the following expression for the potential energy due to the axial forces of the plate. The potential energy due to the in-plane forces per unit width,f,, in the i direction caused by the motion of the plate is
+Ij j
va
0
fx( ~)2dxdy.
(5)
0
For a plate moving over the point supports shown in Fig. 1, the right edge of the plate is stress-free as the plate is being pulled or pushed over
294
H. P. Lee, T. Y. Ng
the line supports by external forces applied at the left edge of the plate. The in-plane forces per unit width in the i direction for the remaining part of the plate are
fx=
-m/tidr
(6)
X =
-mir(a-X)
where Y is a dummy variable for integration. method, the quantity w can be expressed as
Using the assumed mode
4nwon (X,Y)
w = i:
(7)
iZ=l
where c#+,are shape functions that satisfy the geometric boundary conditions at the four edges of the plate. For the present study, 4, are assumed to be products of the normalized modal functions for the vibration of a uniform unrestrained beam. The assumed functions are
(8) The shape functions $+(x) and cp,O) are defined as !A(x> = 1
?h(x) = & ‘@i(X)
=
COS y
+
(
(9)
1- 7
W-9
>
cash y
- yi_2
(11)
Xi_2X Xi_2.X sin - 1 + sinh -j-
(i = 3, .. ..I) I
$92(Y)
(pi(Y) = COS
(12)
1
cpl(Y>
=
=
( >
Jj-
1 -;
2Y
xi-2ycosh--Xi-2Y
w+
W
xj-2Y
sinw
xj-2Y
+ sinh 7
1
(13) Yj-2
o’= 3, . ....I)
(14)
295
Transverse vibration of a plate
where cos A, - cash A,
(15)
‘f~ = sin A, - sinh A,
for p=i orj and X1,...,XI or XJ are the consecutive roots of the transcendental equation 1 -cosXcoshX=O.
(16)
The functions $~t and cpl correspond to the rigid-body translation of an unrestrained beam, whereas & and cp2are the shape functions for rigidbody rotation. The total number of terms in the assumed form of u is N = ZxJ. In the following formulation, Z is made to be equal to J. c& are defined as (5&y) dh(x,y> =
= $~(x)(PI(Y) for n= 7h-1Wf2(y)
for . . = .
Mx,v>
=
&-(I-&)w(Y)
n =
l,...,Z I+
4 ..A
(17) (18)
. . .
for n = (Z- LIZ+ 1, .. ..Z2.
(19)
The Lagrangian of the plate involving u can be expressed as L=T=VE-Va-Vs.
(20)
The Euler-Lagrange equation for a plate moving over multiple point supports, shown in Fig. 1, is therefore given by mH~+(DM+mb!+kfD)q=O
(21)
where H, M, Y, Q and Q are matrices defined as b
a
(22)
(23)
H. P. Lee, T. Y. Ng
296
(Y),=
ji-(Lx)g$dy 0
P)q
=
m1>ij
(24)
0
+
Pl2)ij
+
(@21)ij
+
(@22&
(25)
with (%“)ii
= q&(x = @?I,b&j)j(x = a,, bn).
(26)
The matrices M, H, Y and @ are symmetric matrices. All the matrices except @ are independent of time. The matrix @ need to be updated as the plate is moving over the point supports. The vectors q and 4 are nx 1 column vectors consisting of qi and ii, respectively. The equations of motion generated can be included in numerical simulation programs for investigating the response of a plate undergoing various prescribed motions. The numerical integrations are performed using the variable time-step fourth-order Runge-Kutta method to ensure the stability of the numerical integrations.
RESULTS
AND SIMULATIONS
For the present numerical simulations, a = b = 1 m, m = 1 kg/m2 and D = 1 Nm. The initial configuration of the plate is such that the four point supports are located symmetrically at ai = 0.375 m, a2 = 0.625 m, bl = 0.25 m and b2=0.75 m. The initial shape of the plate is described by
(27) The quantities q1 and q3 are determined from the conditions that w = 0 m at all of the four point supports at aI = 0.375 m and a2 = 0.625 m, and w = 0.01 m at x= 0 m and at x = 1 m. Since 4, corresponds to the rigid-body translation for a beam, the initial shape of the plate is the first symmetric flexural mode shape of an unrestrained beam in the x direction passing through the four point supports. For a prescribed longitudinal sinusoidal motion of the plate in the i direction, the variation of x=ai is assumed to be ai = 0.375 - A sinS2t m.
(28)
For a plate moving over supports, shown in Fig. 1, the transverse displacements at various points on the plate with x=0 and x=0.375 m for
297
Transverse vibration of a plate
I
0
0.5
1
1.5
2
2.5
time (set) ,j x10-3 I
I
3 z
2
II:
0 3
0
0.5
1.5
1
2
2.5
time (set)
Fig. 2. Point displacements at several points on the plate. Upper diagram: -, y=O m; .... .. x=O,y=0.25m;---,x=O,y=0.5m.Lowerdiagram:-,x=0.375,y=Om; ...... x=O.375,y=O.25 m; ---, x=O.375,y=O.S m.
x=0,
R = 20 rad/s and A = 0.05 are shown in Fig. 2 using 25term approximation for w (N= 25, Z= J= 5), which has been found to result in converged results for the transverse displacements. It can be seen that the point displacements indicated by dotted lines for the point at x=0.375 m and y = 0.25 m near one of the point supports for k= lo4 N/m are almost negligible compared with the point displacements for the remaining points. Moreover, numerical results for k= lo5 N/m, not presented here, are found to be almost identical with that of k = IO4 N/m. The execution time for the numerical integration is found to increase with increased value of k. For the following numerical simulations, k is therefore chosen to be lo4 N/m for all the spring supports. The point displacements for three different points on the left edge of the plate for R = 20, 30, 32 and 34 rad/s and A = 0.05 are shown sequentially from the top diagram to the bottom diagram in Fig. 3. The point displacements for the two corners at the left edge are found to be identical, as the initial shape of the plate is assumed to consist of only the first rigidbody translation mode in the y direction (cpi (y)) as shown in eqn (27). The plate shows stable behaviour for R= 20, 30 and 32 rad/s. However, at R = 34 rad/s, an unstable behaviour in the form of an exponential increase
298
H. P. Lee, T.
R = 20
Y.Ng
rad/sec
I
I
0
1
2
3
5
4
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
time (set) R = 30 rad/sec
I
I
5
4
time (set)
R = 32 rad/sec
0.02 E R x z b
0
-0.02 J
I
0
1
2
3
5
4
time (set) 0 = 34 rad/sec
1
0.2,
-0.2 ’ 0
1 0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time (set) Fig. 3.
Point displacements
at the left edge of the plate. .‘..., x = 0, y = 0 m; --, y=0.5m;---,x=O,y=l m.
x = 0,
299
Transverse vibration of a plate
R = 20 rad/sec
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
IO
6
7
8
9
10
time (set) R = 30 rad/sx
x10-3 I
4
5 time (set)
II = 32 rad/sec
0
1
2
3
4
5 time (set)
R = 34 rad/sec 0.1 rl
0
0.5
1
1.5
2
2.5
3
time (set)
Fig. 4. Point displacements
at the left edge of the plate. ....., x = 0, y = 0 m; _ y=OSm;---,x=O,y=lm.
_
H. P. Lee, T. Y. Ng
300
of the amplitude of vibration is observed. Other regions of instability are also observed for different values of R. A general stability analysis can be performed by the method presented by Buffinton and Kane” using Floquet’s theory. Such a laborious analysis will not be undertaken in the present study. However, this paper will attempt to show the different behaviour of the plate if the initial shape of the plate is different. For the numerical results shown in Fig. 4, the initial shape of the plate is changed to u=
!h(P3q11
+ $3(P3q13.
(29)
The assumed function in the y direction is the first flexural mode for a beam. Once again, the point displacements at the two corners at the left edge of the plate are found to be identical, as the prescribed initial shape of the plate is symmetrical with respect to the locations of the point supports. Numerical results shown in Fig. 4 show that the behaviour of the plate at R =20, 30 and 32 rad/s, the first three diagrams presented sequentially from the top, are all stable. The behaviour of the plate is once again unstable at R = 34 rad/s, as shown in the bottom diagram of Fig. 4. These numerical results are consistent with the fact that the stability of a linear system is independent of the initial conditions. The behaviour of a plate with other combinations of point supports and prescribed longitudinal motions of the plate can also be easily verified using the present method.
CONCLUSION Approximate equations of motion in matrix form are derived for the motion of a plate moving over multiple point supports using the Lagrangian approach and the assumed mode method. The transverse vibration of the plate is caused by external forces applied at one end of the plate. For in-plane sinusoidal motion of the plate with a prescribed amplitude of excitation over the point supports, the stability of the plate is found to be dependent on the frequency of excitation, and independent of the initial prescribed shape of the plate.
REFERENCES 1. Leuner, T.R., An experimental-theoretical study of free vibrations of plates on elastic point supports. Journal of Sound and Vibration, 32 (1974) 481490. 2. Laura, P.A.A. & Gutierrez, R.H., Transverse vibration of thin, elastic plates with concentrated masses and internal elastic supports. Journal of Sound and Vibration,
75 (1981)
135-143.
Transversevibrationof a plate
301
3. Srinivasan, R.S. & Munaswamy, K., Frequency analysis of skew orthotropic point supported plates. Journal of Sound and Vibration, 39 (1975) 207-216. 4. Narita, Y., Note on vibrations of point supported rectangular plates. Journal of Sound and Vibration, 93 (1984) 593-597. 5. Ichinomiya, O., Narita, Y. & Maruyama, K., Steady state response analysis of elastically point supported composite rectangular plates. In Dynamics of Plates and Shells. ASME Publication PVP-178. ASME, USA, 1989, 65-70. 6. Yamada, F.G., Irie, T. & Takahashi, M., Determination of the steady state response of a viscoelastically point-supported rectangular plate. Journal of Sound and Vibration, 102 (1985) 285-295. 7. Fjba, L., Vibration of Solids and Structures Under Moving Loads. Noordhoff, Groningen, 1972. 8. Banerjee, A.K. & Kane, T.R., Dynamics of a plate in large overall motion. Transactions of the ASME, Journal of Applied Mechanics, 56 (1989) 887-891. 9. Young, T.H. & Liou, G.T., Coriolis effect on the vibration of a cantilever plate with time-dependent rotating speed. Transactions of the ASME, Journal of Vibration and Acoustics, 114 (1992) 232-241. 10. Lee, H.P. & Ng, T.Y., Effects of support configuration on the parametric excitation of a plate. International Journal of Solids and Structures, 31 (1994) 835-848. 11. Buffinton, K.W. & Kane, T.R., Dynamics of a beam moving over supports. International Journal of Solids and Structures, 21 (1985) 617-643.