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Effects of large amplitude, shear and rotatory inertia on vibration of rectangular plates
Journal qf’Sound
and Vibration (1979) 63(2), 161-167
EFFECTS OF LARGE AMPLITUDE, SHEAR AND ROTATORY INERTIA ON VIBRATION OF RECTANGULAR PLATES M. SATHYAMOORTHY Department cf Civil Engineering, The University of Calgary, Calgary, Alberta, (Received
12 November
1977, bnd in revisedform
Canadu
16 June 19781
In this paper, the governing equations applicable for the large amplitude free, flexural vibration of orthotropic rectangular plates are formulated in terms of the displacement components u, v and w. The formulation and the solutions presented for the cases of isotropic and orthotropic simply supported rectangular plates incorporate the effects of the transverse shear deformation and the rotatory inertia on the large amplitude vibration behaviour. The influences of shear and rotatory inertia are significant in the case of moderately thick plates undergoing large amplitude vibration. The method suggested here does not require the use of the Berger approximation for plates with immovable in-plane edge conditions. A comparison with the results available in the literature indicates the inadequacy of the approach adopted earlier for the study of the large amplitude free flexural vibration of orthotropic rectangular plates.
1. INTRODUCTION
The large amplitude vibrations of plates and shells of various geometries have been investigated by several authors [ 11. In the area of large amplitude vibration of plates of various shapes, rectangular plates have been studied more frequently than others. The simplified Berger formulation has often been used to investigate the non-linear vibration behaviour and the applicability of this approximation to plates has been studied in reference [2]. Although the effect of the large amplitude on vibration of plates has been studied by several authors [l], the transverse shear and rotatory inertia effects have been neglected in many cases. These effects play a considerably important role in the large amplitude vibration of moderately thick plates. Wu and Vinson [3] investigated only the effect of the transverse shear deformation although their Berger type formulation does include the effect of rotatory inertia. Reissner [4] extended the classical plate theory to take into account the effect of the transverse shear deformation on the static behaviour of plates and a further improvement in plate theory was suggested in reference [S] in order to include the effects of the transverse shear deformation and rotatory inertia on the dynamic behaviour of plates. Recently, the author reported results on the combined effects of the transverse shear deformation and rotatory inertia on the large amplitude vibration behaviour of isotropic skew plates, obtained by making use of the Berger approximation applicable for skew plates [6]. It is worth pointing out here that the effect of assuming the Berger hypothesis amounts to overestimating the non-linear period and the error in doing so does seem to increase with the increase in the amplitude of vibration [7]. In the case of orthotropic rectangular plates, all the results so far available in the literature are obtained on the basis of the Berger approximation. No attempt has been made so far to improve the accuracy of these results. In this paper, the large amplitude free flexural vibration of a simply supported rectangular plate is considered, with attention given to the effects of the transverse shear deformation 161 0022-460X/79/060161
+ 07 802.00/O
0
1979 Academic
Press Inc. (London)
Limited
162
M. SATHYAMOORTHY
and rotatory inertia. A system of equations governing the large amplitude vibration of rectangular plates is presented in terms of the displacement components u, L’and w and the rotation components c( and j?. The nature of these equations is such that they can readily be used for non-linear problems with other in-plane edge conditions and are suitable for application of other known methods of non-linear analysis. The solutions to the governing equations are obtained on the basis of an assumed vibration mode and the application of Galerkin’s method. The formulation presented in this paper enables one to study the nonlinear vibration behaviour of plates with immovable in-plane edge conditions without using the Berger approximation. Thus, the formulation and the results presented here for the case of an orthotropic simply supported rectangular plate are considered to be better than those presented earlier [3, 81. The time differential equation is integrated by using the numerical Runge-Kutta method. The time period is plotted for different aspect ratios and thickness to length ratios of isotropic and orthotropic rectangular plates at various amplitudes of vibration. The results presented here show an interesting influence of transverse shear and rotatory inertia on the large amplitude vibration behaviour of rectangular plates.
2. GOVERNING
EQUATIONS
The geometry of the plate and the co-ordinate system are shown in Figure 1. The governing non-linear equations, applicable to the large amplitude free, flexural vibration of ty
26
B
X
20
Figure 1. Geometry
and co-ordinate
system.
and rotatory
orthotropic rectangular plates influenced by transverse shear deformation inertia effects, can be derived as [6]
ahyax + aivx,fay= 0, aN,,.iax + aN,,,iay = 0,.
(1, 2)
a + w,~- Kla,xx - KZa,,Y - K38,xv f K4a.,, = 0,
* (3)
B + w,,- K5P,yy - KGB.., , - K,a,.,).
+ K8P.rr = 0,
(4)
1 - Wa..,
+ dy,,
(5)
+ w. ,J - 4 ,(B,y + w.J
= 0,
where 1 = N&v.,,
N,, = E,&,
+ N,,w,,, + ~N,P,,~,
NY,,= E, = u,x + Lw2 2 .X’ K2
=
E,JE,,
q2
=
v*p,
F2
=
p2
v’ + .U =
$W’
GdJE,,
. Y’
+ q2sJ,
E,h(q2c, + K’E,), Y =
U.?
+
E, = EJv’,
u.,
+
W,P
N,, = E,hp2y, .V’
v’- (1 - \‘xrVJ’
LARGE VIBRATION OF RECTANGULAR
K,
K, = K,,E,,
= GK,,,
K, = GK,,, K,,
K, = GE,K,
,K, 3,
K,,
= h2/10G2,
K,,
163 K, = E,K,,.
K, = PK,,, K,
K, = PK,,,
K, = GE,Kj,K,,,
= - 5hGJ6,
PLATES
= h’/lOG,\,
K,,
= -5hGJ6. = {(W,)
+bl,.,/G)l.
of the plate, p is the mass per unit volume, a and p are the rotation components in the xand y directions, respectively, and Ex, Ey. vxy, vyx, G, G, and G, are theelastic constants of the plate material. Solving for CLand B from equations (3) and (4) and substituting for these in equation (5) gives
is the thickness
N(I,) + R(w) = 0,
(ht
where the operators N and R are defined as 4
a4
N=m,$+m,
axZayZ
+
a4
R = mlo7
+ ml,
ZX
a4 ____
mz = K,K,
- K,K,
m6 = K,K,
+ K,K,,
m, = K, + K,,
a2 K9zT
12 aX2ay*
aY4
m, = -KIK,, ms = K,K8
4 JL+m
-
a’+ ““(@ a’- %&2 !E_ 1. ml 2x’ c74 (?2 z4
Klllu
- K,K,, + K,K,,
ml0= K,K,,
m1.l = -K,K,,
2y:i, -_-if
m6
+
~-__ aX2at2
m3 = &K2KS, rn; = K, + K,,
ml1= K&,0>
+
ml4
pJ,2ir2
m4 = -K,K,, mA = K2 f Ki,
ml2= K,(K, - K,) + K,,(K, -
I, = I - K,w,,,
ml4 = -K,K,,.
m13
K-1,
- K, ow,,,,. - phw.u.
Equations (1) and (2) can be written in terms of the displacement components u, r and W,by use of appropriate substitutions for N,.,, Nyy and Nx,, as u ,xX+ P2U,,, + (P2 + 4*)u.,, = -w.dw.xx + PZWJ - (P2 + q2)w,Vw,xr. K’u.,.,~+ ~‘0. x,+ (P’ + q2)u,,
= -wJK~w,~,.
+ pzw,,.)
- (P’ + q2)~~,x~~~,x,..
(7) (81
Equations (6), (7) and (8) are then the governing equations for the large amplitude free flexural vibrations of orthotropic rectangular plates, with account taken of the effects of transverse shear deformation and rotatory inertia. These three equations are now all in terms of the displacement
components
u, v and w.
3. NUMERICAL EXAMPLE As an example, a simply supported rectangular plate undergoing free, flexural vibration is considered here. For a rectangular plate of dimensions 2a and 2b, the following mode shape w will satisfy all the boundary
conditions:
w/h = f(z) cos(7rx/2a) cos(rry/2b).
(9)
By substituting for w in equations (7) and (8) and solving for u and u, the following in-
164
M. SATHJ’AMOORTH).
plane displacements are obtained: u = A,[(1 - q2r2) + cos(ny/b)] sin(zx/a) + B.Y. U = A, [((K2r2
- q’)/K’r)
+ r cos(mz/a)] sin(zy/b) + Cy.
where A, = nS2h2/32a and r = a/b. If the edges of the plate are assumed to be immovable [6], the in-plane boundary conditions are u = 0 at x = f a and u = 0 at y = + b. It can be verified from equations (10) and (I 1) that these immovable edge conditions will be satisfied if B = C = 0. With all the /
/
(‘0)
1.212 -
I.14g
bo
Es
I.06
ai
co
ci
bi
doi
0.98 0.5
I.0
1. 5
(cl
0.8
0*74c
04
IT
k
0.8
bo
0.66
co do
0.8
0.62E 0.5
I.0
Figure 2. T/T, us. aspect ratio. (a) Amplitude 0; (b) amplitude 0.5; (c) amplitude b, h/2a = l/20; c, h/2a = l/30; d, no transverse shear or rotatory inertia; i, isotropic; tude.
1.: 5
1.0. Key: a, /1!2u = I :lO: o. orthotropic; A. ampli-
LARGE
VIBRATION
OF RECTANGULAR
165
PLATES
I
I
1 (b)
0.:
I.2
1.C
dio
do
do 0%
o
I.0 I.0
I
do 0.6
dl
I.0 I.0
m
0.6
Figure 3. T/7;, cs. h/2u.(a,
1.0. Key as for Figure 2
0.5; (b) I
r
boundary conditions satisfied, it is now required to solve equation (6). Application of Galerkin’s method to equation (6) with u and u taken from equations (10) and (11) yields the following modal equation: .::. (17) c,.t + C,f, + c,j; + c1.1, = 0,
.12 = f’+ (C,,‘3C’,).f .l.
.r; = f’+ (C,/3C,)f3,
+ C,,
C, = 16EA(C,,m,,
+ C,:C,,
+ 7aw,,
= B~~2~~o,or2 C,,
C 15 = -Cz2(3 Cl-
C,, C,,
C,,
C,,
+ mo6r2) + m,,,
+ C,,),
B, = h/2a, (ml; m,; m3) = h,(m,,,
mo2, m,,),
c,> K,;
+ mo,,r4
= -C,2r2(y2
m,,).
= h3Ex(mOlo,mol,.m,,,),
1,
+ K2r2)(3 - yZ/K2r2).
= (B,n)‘(m,,,
C,,
= m,,
+ m,,,r’),
+ mOhr2.
= -(/T&W. K,,)
+ m,),Ir’l.
+ mo8r2) -
C,,
C,,
= hE.yW,,;
m4 = Cf,m,,.
cm,, m,) = h’tm,,, (mlo~mll~ m,,)
= (B,z)“(m,,,
= (1 + CJCB,~)‘,
= 3C,:(C1,
C, = -64bTE,m,,,
+ mo2r2 + mo3r4) - (P,n)‘(m,,
+ 2y2r2 - q4r4),
= W,@“(m,,
I),
C, = 48/3TE,tC, i + C, Jmos,
+ K,,),
= (B,n)“(m,,
+ C,,,I.
+ n2fi:C1<, -
+ m<,‘J,
C, = -4C,,13:Ex, Cl,
Cl = E,C,,(C15
+ C,,C,,),
C, = -4E,P:(C,,
),f 3.
z = t, E,:pcr”.
$laT = (.), C, = E,(C,,
f,, = f + (C,/C,
= ph=!E\.
f&,).
(m,; me) = Czjh’(m,,,.
m, = Cz3m09, (m,,,m,,)
= ph3(m,,,,m,,,).
mnh).
166
M. SATHYAMOORTHk.
Equation (12) is the modal equation applicable for the large amplitude free, flexural vibration of simply supported orthotropic rectangular plates with immovable in-plane edge conditions. Solutions to this non-linear time differential equation are obtained by means of the numerical Runge-Kutta method, with the time interval VT taken as 0401 to obtain reasonably accurate results. The ratio of the non-linear period of vibration, 7; in which the effects of transverse shear deformation and rotatory inertia are included, to the corresponding linear period, T,, ofa classical plate, in which these effects are not included. has been computed for different non-dimensional amplitudes, plate aspect ratios and thickness length ratios of isotropic and orthotropic rectangular plates. The results are presented in graphical form in Figures 2 and 3. The orthotropic constants of the material of the plate are taken as in reference [8].
4. CONCLUSIONS
A system of equations governing the large amplitude free vibration of rectangular plates has been derived which includes the effects of transverse shear deformation and rotatory inertia. The method altogether eliminates the use of the Berger approximation for the study of large amplitude vibration of rectangular plates with immovable in-plane edge conditions. The results presented here are in exact agreement with those presented by Chu and Herrmann [9] for the case of vibration of rectangular plates when no transverse shear and rotatory inertia effects are considered. The effects of transverse shear deformation and rotatory inertia on the large amplitude vibration of rectangular plates are shown by an increase in the time period, the increase being the largest at small amplitudes of vibration and decreasing gradually as the amplitude TABLE 1
Values of T/To h/2a
r @5
l/20 1.0
0.5 l/30 1.0
0.5 l/40 1.0
0.5 l/50 1.0
t Values obtained $ Present results.
0.5
Amplitude 1.0
3.3364.f 0.8767f 1.0989 0.8842
0.9242 0.6478 08621 0.6563
0.7246 0.6910 0.6666 0.6552
I.0526 0.8570 1.0309 0.8642
0.8696 0.6363 0.8264 06449
0.7018 0.6334 0.6494 0.6128
1.0101 08498 09901 0.8569
0.8475 0.6322 0.8065 06409
06897 0.6168 0.6410 0.6000
09901 0.8464 09709 0.8535
0.8264 06303 0.7937 0.6390
0.6849 0.6096 0.6369 0.5944
from Figure
I of reference [8].
1.5
LARGE VIBRATION OF RECTANGULAR
PLATES
167
of vibration increases. It can be seen that the increase in the time period is quite significant even at moderately large amplitudes for plates made of orthotropic material. The combined effects of transverse shear and rotatory inertia decrease as the plate thickness is reduced and the results approach those of the classical plate theory for thin plates at small amplitudes of vibration. The period ratio decreases with increase in amplitude thereby exhibiting the well known hardening behaviour. At moderately large amplitudes of vibration the period ratio increases with an increase in the aspect ratio for a plate made of orthotropic material whereas in the case of an isotropic plate the period ratio increases with the aspect ratio. reaches a maximum around an aspect ratio of 1.1 and then starts decreasing. However. when the vibration is at small amplitudes, the transverse shear and rotatory inertia effects increase with the aspect ratio for isotropic plates and remain practically unaltered with change in the aspect ratio for orthotropic plates. In Table 1, some of the numerical results obtained here are compared with those of reference [8]. It is clear that the Berger approach as presented in reference [8] is inaccurate for the non-linear vibration analysis of orthotropic rectangular plates. The results in reference [8] overestimate the period ratio at all amplitudes greater than that applicable for the small amplitude linear vibration. This could be attributed to the nature of the assumed functional forms for u and /I and the assumption of the Berger hypothesis in reference [81.
REFERENCES 1. M. SATHYAM~RTHYand K. A. V. PANDALAI1973 Journal of the Aeronautical Soc,iet_vof India 25. l-10. Large amplitude vibrations of certain deformable bodies-Part II. 2. J. L. NOWINSKIand H. OHNABE1972 International Journal of Mechanical Science 14. 165-170. On certain inconsistencies in Berger equations for large deflections of elastic plates. 3. CHENG-IH Wu and J. R. VINSON1969 .Iournal of Applied Mechanics 36, 254-260. Influences of large amplitudes, transverse shear deformation and rotatory inertia on lateral vibrations of isotropic plates. 4. E. REISSNER1945 Journal of Applied Mechanics 12, A69-A77. The effect of transverse shear deformation on the bending of elastic plates. 5. R. D. MINDLIN 1951 Journal of Applied Mechanics 18, 31-38. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. 6. M. SATHYAMOORTHY 1977 Journal of Sound and Vibration 52, 155-163. Shear and rotatory inertia effects on large amplitude vibration of skew plates. 7. M. SATHYAMOORTHY 1978 Journal of Sound and Vibration 58, 301-304. Non-linear vibration of rectangular plates. 8. CHENG-IH WV and J. R. VINSON 1969 Journal of Composite Materials3,548-561. On the nonlinear oscillations of plates composed of composite materials. 9. H. N. CHU and G. HERRMANN1956 Journal of Applied Mechanics 23, 532-540. Influence of‘large amplitudes on free flexural vibrations of rectangular elastic plates.
and Vibration (1979) 63(2), 161-167
EFFECTS OF LARGE AMPLITUDE, SHEAR AND ROTATORY INERTIA ON VIBRATION OF RECTANGULAR PLATES M. SATHYAMOORTHY Department cf Civil Engineering, The University of Calgary, Calgary, Alberta, (Received
12 November
1977, bnd in revisedform
Canadu
16 June 19781
In this paper, the governing equations applicable for the large amplitude free, flexural vibration of orthotropic rectangular plates are formulated in terms of the displacement components u, v and w. The formulation and the solutions presented for the cases of isotropic and orthotropic simply supported rectangular plates incorporate the effects of the transverse shear deformation and the rotatory inertia on the large amplitude vibration behaviour. The influences of shear and rotatory inertia are significant in the case of moderately thick plates undergoing large amplitude vibration. The method suggested here does not require the use of the Berger approximation for plates with immovable in-plane edge conditions. A comparison with the results available in the literature indicates the inadequacy of the approach adopted earlier for the study of the large amplitude free flexural vibration of orthotropic rectangular plates.
1. INTRODUCTION
The large amplitude vibrations of plates and shells of various geometries have been investigated by several authors [ 11. In the area of large amplitude vibration of plates of various shapes, rectangular plates have been studied more frequently than others. The simplified Berger formulation has often been used to investigate the non-linear vibration behaviour and the applicability of this approximation to plates has been studied in reference [2]. Although the effect of the large amplitude on vibration of plates has been studied by several authors [l], the transverse shear and rotatory inertia effects have been neglected in many cases. These effects play a considerably important role in the large amplitude vibration of moderately thick plates. Wu and Vinson [3] investigated only the effect of the transverse shear deformation although their Berger type formulation does include the effect of rotatory inertia. Reissner [4] extended the classical plate theory to take into account the effect of the transverse shear deformation on the static behaviour of plates and a further improvement in plate theory was suggested in reference [S] in order to include the effects of the transverse shear deformation and rotatory inertia on the dynamic behaviour of plates. Recently, the author reported results on the combined effects of the transverse shear deformation and rotatory inertia on the large amplitude vibration behaviour of isotropic skew plates, obtained by making use of the Berger approximation applicable for skew plates [6]. It is worth pointing out here that the effect of assuming the Berger hypothesis amounts to overestimating the non-linear period and the error in doing so does seem to increase with the increase in the amplitude of vibration [7]. In the case of orthotropic rectangular plates, all the results so far available in the literature are obtained on the basis of the Berger approximation. No attempt has been made so far to improve the accuracy of these results. In this paper, the large amplitude free flexural vibration of a simply supported rectangular plate is considered, with attention given to the effects of the transverse shear deformation 161 0022-460X/79/060161
+ 07 802.00/O
0
1979 Academic
Press Inc. (London)
Limited
162
M. SATHYAMOORTHY
and rotatory inertia. A system of equations governing the large amplitude vibration of rectangular plates is presented in terms of the displacement components u, L’and w and the rotation components c( and j?. The nature of these equations is such that they can readily be used for non-linear problems with other in-plane edge conditions and are suitable for application of other known methods of non-linear analysis. The solutions to the governing equations are obtained on the basis of an assumed vibration mode and the application of Galerkin’s method. The formulation presented in this paper enables one to study the nonlinear vibration behaviour of plates with immovable in-plane edge conditions without using the Berger approximation. Thus, the formulation and the results presented here for the case of an orthotropic simply supported rectangular plate are considered to be better than those presented earlier [3, 81. The time differential equation is integrated by using the numerical Runge-Kutta method. The time period is plotted for different aspect ratios and thickness to length ratios of isotropic and orthotropic rectangular plates at various amplitudes of vibration. The results presented here show an interesting influence of transverse shear and rotatory inertia on the large amplitude vibration behaviour of rectangular plates.
2. GOVERNING
EQUATIONS
The geometry of the plate and the co-ordinate system are shown in Figure 1. The governing non-linear equations, applicable to the large amplitude free, flexural vibration of ty
26
B
X
20
Figure 1. Geometry
and co-ordinate
system.
and rotatory
orthotropic rectangular plates influenced by transverse shear deformation inertia effects, can be derived as [6]
ahyax + aivx,fay= 0, aN,,.iax + aN,,,iay = 0,.
(1, 2)
a + w,~- Kla,xx - KZa,,Y - K38,xv f K4a.,, = 0,
* (3)
B + w,,- K5P,yy - KGB.., , - K,a,.,).
+ K8P.rr = 0,
(4)
1 - Wa..,
+ dy,,
(5)
+ w. ,J - 4 ,(B,y + w.J
= 0,
where 1 = N&v.,,
N,, = E,&,
+ N,,w,,, + ~N,P,,~,
NY,,= E, = u,x + Lw2 2 .X’ K2
=
E,JE,,
q2
=
v*p,
F2
=
p2
v’ + .U =
$W’
GdJE,,
. Y’
+ q2sJ,
E,h(q2c, + K’E,), Y =
U.?
+
E, = EJv’,
u.,
+
W,P
N,, = E,hp2y, .V’
v’- (1 - \‘xrVJ’
LARGE VIBRATION OF RECTANGULAR
K,
K, = K,,E,,
= GK,,,
K, = GK,,, K,,
K, = GE,K,
,K, 3,
K,,
= h2/10G2,
K,,
163 K, = E,K,,.
K, = PK,,, K,
K, = PK,,,
K, = GE,Kj,K,,,
= - 5hGJ6,
PLATES
= h’/lOG,\,
K,,
= -5hGJ6. = {(W,)
+bl,.,/G)l.
of the plate, p is the mass per unit volume, a and p are the rotation components in the xand y directions, respectively, and Ex, Ey. vxy, vyx, G, G, and G, are theelastic constants of the plate material. Solving for CLand B from equations (3) and (4) and substituting for these in equation (5) gives
is the thickness
N(I,) + R(w) = 0,
(ht
where the operators N and R are defined as 4
a4
N=m,$+m,
axZayZ
+
a4
R = mlo7
+ ml,
ZX
a4 ____
mz = K,K,
- K,K,
m6 = K,K,
+ K,K,,
m, = K, + K,,
a2 K9zT
12 aX2ay*
aY4
m, = -KIK,, ms = K,K8
4 JL+m
-
a’+ ““(@ a’- %&2 !E_ 1. ml 2x’ c74 (?2 z4
Klllu
- K,K,, + K,K,,
ml0= K,K,,
m1.l = -K,K,,
2y:i, -_-if
m6
+
~-__ aX2at2
m3 = &K2KS, rn; = K, + K,,
ml1= K&,0>
+
ml4
pJ,2ir2
m4 = -K,K,, mA = K2 f Ki,
ml2= K,(K, - K,) + K,,(K, -
I, = I - K,w,,,
ml4 = -K,K,,.
m13
K-1,
- K, ow,,,,. - phw.u.
Equations (1) and (2) can be written in terms of the displacement components u, r and W,by use of appropriate substitutions for N,.,, Nyy and Nx,, as u ,xX+ P2U,,, + (P2 + 4*)u.,, = -w.dw.xx + PZWJ - (P2 + q2)w,Vw,xr. K’u.,.,~+ ~‘0. x,+ (P’ + q2)u,,
= -wJK~w,~,.
+ pzw,,.)
- (P’ + q2)~~,x~~~,x,..
(7) (81
Equations (6), (7) and (8) are then the governing equations for the large amplitude free flexural vibrations of orthotropic rectangular plates, with account taken of the effects of transverse shear deformation and rotatory inertia. These three equations are now all in terms of the displacement
components
u, v and w.
3. NUMERICAL EXAMPLE As an example, a simply supported rectangular plate undergoing free, flexural vibration is considered here. For a rectangular plate of dimensions 2a and 2b, the following mode shape w will satisfy all the boundary
conditions:
w/h = f(z) cos(7rx/2a) cos(rry/2b).
(9)
By substituting for w in equations (7) and (8) and solving for u and u, the following in-
164
M. SATHJ’AMOORTH).
plane displacements are obtained: u = A,[(1 - q2r2) + cos(ny/b)] sin(zx/a) + B.Y. U = A, [((K2r2
- q’)/K’r)
+ r cos(mz/a)] sin(zy/b) + Cy.
where A, = nS2h2/32a and r = a/b. If the edges of the plate are assumed to be immovable [6], the in-plane boundary conditions are u = 0 at x = f a and u = 0 at y = + b. It can be verified from equations (10) and (I 1) that these immovable edge conditions will be satisfied if B = C = 0. With all the /
/
(‘0)
1.212 -
I.14g
bo
Es
I.06
ai
co
ci
bi
doi
0.98 0.5
I.0
1. 5
(cl
0.8
0*74c
04
IT
k
0.8
bo
0.66
co do
0.8
0.62E 0.5
I.0
Figure 2. T/T, us. aspect ratio. (a) Amplitude 0; (b) amplitude 0.5; (c) amplitude b, h/2a = l/20; c, h/2a = l/30; d, no transverse shear or rotatory inertia; i, isotropic; tude.
1.: 5
1.0. Key: a, /1!2u = I :lO: o. orthotropic; A. ampli-
LARGE
VIBRATION
OF RECTANGULAR
165
PLATES
I
I
1 (b)
0.:
I.2
1.C
dio
do
do 0%
o
I.0 I.0
I
do 0.6
dl
I.0 I.0
m
0.6
Figure 3. T/7;, cs. h/2u.(a,
1.0. Key as for Figure 2
0.5; (b) I
r
boundary conditions satisfied, it is now required to solve equation (6). Application of Galerkin’s method to equation (6) with u and u taken from equations (10) and (11) yields the following modal equation: .::. (17) c,.t + C,f, + c,j; + c1.1, = 0,
.12 = f’+ (C,,‘3C’,).f .l.
.r; = f’+ (C,/3C,)f3,
+ C,,
C, = 16EA(C,,m,,
+ C,:C,,
+ 7aw,,
= B~~2~~o,or2 C,,
C 15 = -Cz2(3 Cl-
C,, C,,
C,,
C,,
+ mo6r2) + m,,,
+ C,,),
B, = h/2a, (ml; m,; m3) = h,(m,,,
mo2, m,,),
c,> K,;
+ mo,,r4
= -C,2r2(y2
m,,).
= h3Ex(mOlo,mol,.m,,,),
1,
+ K2r2)(3 - yZ/K2r2).
= (B,n)‘(m,,,
C,,
= m,,
+ m,,,r’),
+ mOhr2.
= -(/T&W. K,,)
+ m,),Ir’l.
+ mo8r2) -
C,,
C,,
= hE.yW,,;
m4 = Cf,m,,.
cm,, m,) = h’tm,,, (mlo~mll~ m,,)
= (B,z)“(m,,,
= (1 + CJCB,~)‘,
= 3C,:(C1,
C, = -64bTE,m,,,
+ mo2r2 + mo3r4) - (P,n)‘(m,,
+ 2y2r2 - q4r4),
= W,@“(m,,
I),
C, = 48/3TE,tC, i + C, Jmos,
+ K,,),
= (B,n)“(m,,
+ C,,,I.
+ n2fi:C1<, -
+ m<,‘J,
C, = -4C,,13:Ex, Cl,
Cl = E,C,,(C15
+ C,,C,,),
C, = -4E,P:(C,,
),f 3.
z = t, E,:pcr”.
$laT = (.), C, = E,(C,,
f,, = f + (C,/C,
= ph=!E\.
f&,).
(m,; me) = Czjh’(m,,,.
m, = Cz3m09, (m,,,m,,)
= ph3(m,,,,m,,,).
mnh).
166
M. SATHYAMOORTHk.
Equation (12) is the modal equation applicable for the large amplitude free, flexural vibration of simply supported orthotropic rectangular plates with immovable in-plane edge conditions. Solutions to this non-linear time differential equation are obtained by means of the numerical Runge-Kutta method, with the time interval VT taken as 0401 to obtain reasonably accurate results. The ratio of the non-linear period of vibration, 7; in which the effects of transverse shear deformation and rotatory inertia are included, to the corresponding linear period, T,, ofa classical plate, in which these effects are not included. has been computed for different non-dimensional amplitudes, plate aspect ratios and thickness length ratios of isotropic and orthotropic rectangular plates. The results are presented in graphical form in Figures 2 and 3. The orthotropic constants of the material of the plate are taken as in reference [8].
4. CONCLUSIONS
A system of equations governing the large amplitude free vibration of rectangular plates has been derived which includes the effects of transverse shear deformation and rotatory inertia. The method altogether eliminates the use of the Berger approximation for the study of large amplitude vibration of rectangular plates with immovable in-plane edge conditions. The results presented here are in exact agreement with those presented by Chu and Herrmann [9] for the case of vibration of rectangular plates when no transverse shear and rotatory inertia effects are considered. The effects of transverse shear deformation and rotatory inertia on the large amplitude vibration of rectangular plates are shown by an increase in the time period, the increase being the largest at small amplitudes of vibration and decreasing gradually as the amplitude TABLE 1
Values of T/To h/2a
r @5
l/20 1.0
0.5 l/30 1.0
0.5 l/40 1.0
0.5 l/50 1.0
t Values obtained $ Present results.
0.5
Amplitude 1.0
3.3364.f 0.8767f 1.0989 0.8842
0.9242 0.6478 08621 0.6563
0.7246 0.6910 0.6666 0.6552
I.0526 0.8570 1.0309 0.8642
0.8696 0.6363 0.8264 06449
0.7018 0.6334 0.6494 0.6128
1.0101 08498 09901 0.8569
0.8475 0.6322 0.8065 06409
06897 0.6168 0.6410 0.6000
09901 0.8464 09709 0.8535
0.8264 06303 0.7937 0.6390
0.6849 0.6096 0.6369 0.5944
from Figure
I of reference [8].
1.5
LARGE VIBRATION OF RECTANGULAR
PLATES
167
of vibration increases. It can be seen that the increase in the time period is quite significant even at moderately large amplitudes for plates made of orthotropic material. The combined effects of transverse shear and rotatory inertia decrease as the plate thickness is reduced and the results approach those of the classical plate theory for thin plates at small amplitudes of vibration. The period ratio decreases with increase in amplitude thereby exhibiting the well known hardening behaviour. At moderately large amplitudes of vibration the period ratio increases with an increase in the aspect ratio for a plate made of orthotropic material whereas in the case of an isotropic plate the period ratio increases with the aspect ratio. reaches a maximum around an aspect ratio of 1.1 and then starts decreasing. However. when the vibration is at small amplitudes, the transverse shear and rotatory inertia effects increase with the aspect ratio for isotropic plates and remain practically unaltered with change in the aspect ratio for orthotropic plates. In Table 1, some of the numerical results obtained here are compared with those of reference [8]. It is clear that the Berger approach as presented in reference [8] is inaccurate for the non-linear vibration analysis of orthotropic rectangular plates. The results in reference [8] overestimate the period ratio at all amplitudes greater than that applicable for the small amplitude linear vibration. This could be attributed to the nature of the assumed functional forms for u and /I and the assumption of the Berger hypothesis in reference [81.
REFERENCES 1. M. SATHYAM~RTHYand K. A. V. PANDALAI1973 Journal of the Aeronautical Soc,iet_vof India 25. l-10. Large amplitude vibrations of certain deformable bodies-Part II. 2. J. L. NOWINSKIand H. OHNABE1972 International Journal of Mechanical Science 14. 165-170. On certain inconsistencies in Berger equations for large deflections of elastic plates. 3. CHENG-IH Wu and J. R. VINSON1969 .Iournal of Applied Mechanics 36, 254-260. Influences of large amplitudes, transverse shear deformation and rotatory inertia on lateral vibrations of isotropic plates. 4. E. REISSNER1945 Journal of Applied Mechanics 12, A69-A77. The effect of transverse shear deformation on the bending of elastic plates. 5. R. D. MINDLIN 1951 Journal of Applied Mechanics 18, 31-38. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. 6. M. SATHYAMOORTHY 1977 Journal of Sound and Vibration 52, 155-163. Shear and rotatory inertia effects on large amplitude vibration of skew plates. 7. M. SATHYAMOORTHY 1978 Journal of Sound and Vibration 58, 301-304. Non-linear vibration of rectangular plates. 8. CHENG-IH WV and J. R. VINSON 1969 Journal of Composite Materials3,548-561. On the nonlinear oscillations of plates composed of composite materials. 9. H. N. CHU and G. HERRMANN1956 Journal of Applied Mechanics 23, 532-540. Influence of‘large amplitudes on free flexural vibrations of rectangular elastic plates.