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Large amplitude vibration of elastic plates
Compuwrs % Swucrures Vol. 32. No. I. pp. 243-244. 1989 Printed in Great Britain.
0
0045s7949/89 $3.00 + 0.00 1989 Pergamon Press plc
TECHNICAL NOTE LARGE
AMPLITUDE
VIBRATION
OF ELASTIC
PLATES
G. CHANDRASEKHARAPPAand H. R. SRIRANGARAJAN Department of Mechanical Engineering, Indian Institute of Technology, Powai, Bombay 400076, India (Received 23 March 1988)
Abstract-An approximate solution is obtained for the nonlinear time differential equation of the large amplitude vibration of elastic and isotropic square plates with all sides hinged, where the effects of transverse shear and rotary inertia are taken into account by applying the ultraspherical polynomial approximation technique and are compared with results of Singh et al. and the digital results.
INTRODUCTION In recent years, the large amplitude flexural vibrations of elastic plates have had many applications in modern thin plate structures. The classical plate theory is no more applicable when the deflection of the plate exceeds its thickness. Large amplitude vibrations of plates have been studied by many researchers. Singh er al. [l] presented a method for finding the fundamental frequency of large amplitude vibration of elastic and isotropic rectangular plates, where the effects of transverse shear and rotatory inertia are taken into account. Sathyamoorthy[2] studied the large amplitude free, undamped flexural vibration of elastic, isotropic skew plates taking into consideration the effects of transverse shear and rotatory inertia. Singh er al. [3] presented a method for obtaining the exact solution of the type of nonlinear equations that occur in large amplitude vibrations of structural elements. In this study, an approximate solution for the nonlinear ordinary differential equation has been obtained by applying the ultraspherical polynomial approximation (UPA) technique[4] and the results are compared with those of Singh et al. [3] and the digital results using the classical fourth-order Runge-Kutta method computed on the WIPRO:B-200 computing system.
Expanding (a3 + a4fz)-’ in binomial expansion, retaining only two terms and after simplification, eqn (1) becomes
where n8r6(3 - v)
5n6(3 - v)*r4
a’ = 2304(1 - v2)*+ 4608(1 - v*)*
25n4r2(3 - v) +4608(1+v)*(l-v) 25z4r2
5n6r4 a2 = 1728(1 + v)*(l -v*)+
n4r4(27 - 5v)
1728(1+ ~)~(l - v)* 5n2r2(23 - 1Iv)
a3 = 3456(1 + v)(l - v2) + 3456(1 + v)(l -v*) 25
ANALYSIS
+ 576(1 + v)*
The time differential equation for the large amplitude flexural vibrations of elastic plates given by Sathyamoorthy [2] is
-v*)+
nV(3 -v)* ar = 768(1 - v)(l - v*)
. .. .
w+=w+-
I
W a2f+
cf’
+
a&i)*
=
o
(1) (a3
+
ad*
1
subject to initial conditions f=A,andj‘=Oats=O.
768(1 -v*)
+ 5n4r4(3-v) 384(1 - v2)
where r = h/a, h = thickness, a = side length and v = Poisson’s ratio. Now applying the UPA technique [4] to eqn (3), we get the nonlinear frequency (w*) as
where (‘) = d/(dr ), r = rq ‘I* and q = E/pa*. Neglecting the coefficients a6, a,, as, a,, a,, and a,, as reported by Sathyamoorthy [2], the above equation becomes
P+
5n4r4(3 - v)
nV(3 - v)~ a4= 1536(1 -v)(l
(2) 243
I(>!!g(l_c,)+ 1
a3
y$4 (
A: >
244
Technical Note J,(m)
= Bessel function;
w2 = a2/a,. The frequency o,, given by the classical theory [2], is w,==
7r*r
[3( 1- v*)p ’
The ratio of the nonlinear period (r*) to that given by the classical theory (T,) is n2r
?=-
w*[3(1 - \J)]“Z’ 0
0.2 0.4 D6
0.8 1.0
0
0.2 0.4 0.6 O-8 1.0
Ao
Ao NUMERICAL
0
0.2 0.1 0.6 0,6 1.0
Ao Fig. 1. Time period versus amplitude. (a) r = l/IO; (b) r = l/20; (c) r = l/30. UPA technique; 0 digital results; @ results of Singh et al. [3].
Plots of f (the ratio of the nonlinear period obtained by the UPA technique for A = l/2 to that given by the classical theory) versus A, (the “amplitude” of vibration) are shown in Fig. la, b and c for r = l/10, r = l/20 and r = l/30 respectively, taking Poisson’s ratio (v) as 0.3, and are compared with those of Singh et ai. [3] and the digital solution using the classical fourth-order Runge-Kutta method, taking the time step AT as 0.001 computed on the WIPRO:B-200 computing system. By observing Fig. la-c, it is seen that the results of the UPA technique are in close agreement with the digital results rather than those of Singh ef al. [3], and by increasing the amplitude of vibration, the nonlinear period decreases. REFERENCES
P. N. Singh, Y. C. Das and V. Sundararajan, Large amplitude vibration of rectangular plates. J. Sound Vibr. 17, 235-240
where
c = ”
w + 1)
_ J,(m), (nx/2)”
n = 2, 4, 6;
E.= ultraspherical polynomial index; r ( ) = Gamma function;
RESULTS AND DISCUSSION
(1971).
M. ~thyamoor~y, amplitudes including Inr. J. Solids Struct. P. N. Singh, A. M. solution of nonlinear
Vibration of skew plates at large shear and rotatory inertia effects. 14, 8699880 (1978). Awin and A. R. Shouman, Exact oscillation problems. J. Sound Vibr.
75, 303-306 (1981). S. C. Sinha and P. Srinivasan,
Application of ultraspherical polynomials to nonlinear autonomous systems. J. Sound Vibr. 18, 55 (1971).
0
0045s7949/89 $3.00 + 0.00 1989 Pergamon Press plc
TECHNICAL NOTE LARGE
AMPLITUDE
VIBRATION
OF ELASTIC
PLATES
G. CHANDRASEKHARAPPAand H. R. SRIRANGARAJAN Department of Mechanical Engineering, Indian Institute of Technology, Powai, Bombay 400076, India (Received 23 March 1988)
Abstract-An approximate solution is obtained for the nonlinear time differential equation of the large amplitude vibration of elastic and isotropic square plates with all sides hinged, where the effects of transverse shear and rotary inertia are taken into account by applying the ultraspherical polynomial approximation technique and are compared with results of Singh et al. and the digital results.
INTRODUCTION In recent years, the large amplitude flexural vibrations of elastic plates have had many applications in modern thin plate structures. The classical plate theory is no more applicable when the deflection of the plate exceeds its thickness. Large amplitude vibrations of plates have been studied by many researchers. Singh er al. [l] presented a method for finding the fundamental frequency of large amplitude vibration of elastic and isotropic rectangular plates, where the effects of transverse shear and rotatory inertia are taken into account. Sathyamoorthy[2] studied the large amplitude free, undamped flexural vibration of elastic, isotropic skew plates taking into consideration the effects of transverse shear and rotatory inertia. Singh er al. [3] presented a method for obtaining the exact solution of the type of nonlinear equations that occur in large amplitude vibrations of structural elements. In this study, an approximate solution for the nonlinear ordinary differential equation has been obtained by applying the ultraspherical polynomial approximation (UPA) technique[4] and the results are compared with those of Singh et al. [3] and the digital results using the classical fourth-order Runge-Kutta method computed on the WIPRO:B-200 computing system.
Expanding (a3 + a4fz)-’ in binomial expansion, retaining only two terms and after simplification, eqn (1) becomes
where n8r6(3 - v)
5n6(3 - v)*r4
a’ = 2304(1 - v2)*+ 4608(1 - v*)*
25n4r2(3 - v) +4608(1+v)*(l-v) 25z4r2
5n6r4 a2 = 1728(1 + v)*(l -v*)+
n4r4(27 - 5v)
1728(1+ ~)~(l - v)* 5n2r2(23 - 1Iv)
a3 = 3456(1 + v)(l - v2) + 3456(1 + v)(l -v*) 25
ANALYSIS
+ 576(1 + v)*
The time differential equation for the large amplitude flexural vibrations of elastic plates given by Sathyamoorthy [2] is
-v*)+
nV(3 -v)* ar = 768(1 - v)(l - v*)
. .. .
w+=w+-
I
W a2f+
cf’
+
a&i)*
=
o
(1) (a3
+
ad*
1
subject to initial conditions f=A,andj‘=Oats=O.
768(1 -v*)
+ 5n4r4(3-v) 384(1 - v2)
where r = h/a, h = thickness, a = side length and v = Poisson’s ratio. Now applying the UPA technique [4] to eqn (3), we get the nonlinear frequency (w*) as
where (‘) = d/(dr ), r = rq ‘I* and q = E/pa*. Neglecting the coefficients a6, a,, as, a,, a,, and a,, as reported by Sathyamoorthy [2], the above equation becomes
P+
5n4r4(3 - v)
nV(3 - v)~ a4= 1536(1 -v)(l
(2) 243
I(>!!g(l_c,)+ 1
a3
y$4 (
A: >
244
Technical Note J,(m)
= Bessel function;
w2 = a2/a,. The frequency o,, given by the classical theory [2], is w,==
7r*r
[3( 1- v*)p ’
The ratio of the nonlinear period (r*) to that given by the classical theory (T,) is n2r
?=-
w*[3(1 - \J)]“Z’ 0
0.2 0.4 D6
0.8 1.0
0
0.2 0.4 0.6 O-8 1.0
Ao
Ao NUMERICAL
0
0.2 0.1 0.6 0,6 1.0
Ao Fig. 1. Time period versus amplitude. (a) r = l/IO; (b) r = l/20; (c) r = l/30. UPA technique; 0 digital results; @ results of Singh et al. [3].
Plots of f (the ratio of the nonlinear period obtained by the UPA technique for A = l/2 to that given by the classical theory) versus A, (the “amplitude” of vibration) are shown in Fig. la, b and c for r = l/10, r = l/20 and r = l/30 respectively, taking Poisson’s ratio (v) as 0.3, and are compared with those of Singh et ai. [3] and the digital solution using the classical fourth-order Runge-Kutta method, taking the time step AT as 0.001 computed on the WIPRO:B-200 computing system. By observing Fig. la-c, it is seen that the results of the UPA technique are in close agreement with the digital results rather than those of Singh ef al. [3], and by increasing the amplitude of vibration, the nonlinear period decreases. REFERENCES
P. N. Singh, Y. C. Das and V. Sundararajan, Large amplitude vibration of rectangular plates. J. Sound Vibr. 17, 235-240
where
c = ”
w + 1)
_ J,(m), (nx/2)”
n = 2, 4, 6;
E.= ultraspherical polynomial index; r ( ) = Gamma function;
RESULTS AND DISCUSSION
(1971).
M. ~thyamoor~y, amplitudes including Inr. J. Solids Struct. P. N. Singh, A. M. solution of nonlinear
Vibration of skew plates at large shear and rotatory inertia effects. 14, 8699880 (1978). Awin and A. R. Shouman, Exact oscillation problems. J. Sound Vibr.
75, 303-306 (1981). S. C. Sinha and P. Srinivasan,
Application of ultraspherical polynomials to nonlinear autonomous systems. J. Sound Vibr. 18, 55 (1971).