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Nonlinear dynamic damped response of an orthotropic circular plate
Compurers di Srrucrures Vol. 33. No. 5. pp. 1163-l 165. 1989 Printed in Great Britain.
0
NONLINEAR DYNAMIC ORTHOTROPIC
DAMPED RESPONSE CIRCULAR PLATE
0045-7949189 13.00 + 0.00 1989 Pergamon Press plc
OF AN
G. CHANDRASEKHARAPPA and H. R. SRIRANGARAJAN Department of Mechanical Engineering, Indian Institute of Technology, Powai, Bombay 400 076, India (Received
6 December
1988)
Abstract-Nonlinear dynamic damped response of a clamped-immovable orthotropic circular plate subjected to step pressure pulse excitation has been presented using the dynamic von Karman equations expressed in terms of the displacement component and solved by a one-term solution applying Galerkin technique to the deflection equation. This yields an ordinary nonlinear differential equation in time. The nonlinear dynamic damped response is obtained by applying the ultraspherical polynomial approximation (UPA) technique. The influences of orthotropic and damping parameters on the dynamic response are presented.
INTRODUCTION The nonlinear transverse vibration of elastic isotropic plates has attracted many researchers. In fact the amplitude of vibration depends on the relationship between the elastic moduli of an anisotropic structure. The resistances to mechanical actions in different directions are different for thin plates, and these have wide applications in modern technology. Nowinski [I] studied the nonlinear vibration of an elastic circular plate based on the von Karman dynamic equations generalized to the rectilinearly orthotropic case. Chaudhuri [2] presented the nonlinear dynamic behaviour of a clamped orthotropic circular plate to pulse excitations using the von Karman equations expressed in terms of displacement components and, by applying the Galerkin technique, reduced it to the nonlinear differential equation which was solved by the perturbation method, Ruei et al. [3] obtained an approximate solution to the dynamic von Karman equations for nonlinear forced vibration of a cylindrically orthotropic circular plate with a sinusoidally restrained edge. In this paper, the dynamic von Karman equations expressed in terms of displacement components are used for the nonlinear dynamic behaviour of a clamped-immovable orthotropic circular plate with damping and subjected to step-pressure-pulse excitation. By using the Galerkin technique to the deflection equation, the coupled nonlinear partial differential equations are reduced to the nonlinear differential equation in time. The ultraspherical polynomial approximation (UPA) technique introduced by Anderson [4,5] and extended by Srirangarajan and Srinivasan [6] has been applied to get the dynamic damped response and is compared with the digital solution computed on the CYBER 180/840A computing system using fourth-order the
Runge-Kutta method. The influences of orthotropic and damping parameters of material for the considered plate are presented.
GOVERNING
DIFFERENTIAL
EQUATIONS
Consider the potential energy of a circular orthotropic plate, taking damping into account, and acted on by a transverse pressure pulse load PO. Forming the Lagrangian function and applying Hamilton’s principle and Euler’s variational equations to the Lagrangian function, yields the following partial coupled differential equations [2].
ah
iau
k2
ar’+;jjy-y=-jy-
ph ah pch au1 adW 2 ah ~Jp+~~+g+;dr)-;ip+uJ~ .\ ,
awaJw -zF
(1)
k2a2w k2 aw
(2) where p is the density of the plate material, u is the displacement along the radius a, w is the deflection normal to the middle plane, h is the plate thickness, k2 = (E,i./E:), p2 = (E”iE;.), E: and EJ. are Young’s modulus along the x- and y-axes respectively, D., = E:h3/12 and E” is the elastic constant. For isotropy, E,:. = E,;. = E/(1 - v’), E” = vE/ (1 - v2), k2 = I, p2 = v, where v is Poisson’s ratio and E is Young’s modulus. 1163
G. CHANDRASEKHARAPPA
1164
and H. R.
SRIRANGARAJAN
NONLINEAR DYNAMIC DAMPED RESPONSE
The deflection function which satisfies the clamped boundary conditions is assumed as
01
018-
I-
r a
[
(3)
Here f(t) is a function of time t, to be determined. Substituting eqn (3) in (I), solving for immovable edge conditions (U = 0 at r = a), then substituting the expression for u in eqn (2) and applying Galerkin’s technique yields a nonlinear differential equation in nondimensional form f”(r)
+ Ef(?) + w:f(z) + qP(T.) = P:
I
I
I
I 0.8
I IO
I 12
, 14
I 16
T
Fig. 2. Orthotropic and isotropic dynamic damped response of the circular plate.
(4) Substituting
where nondimensional
2-2
;: ;:
2 2
w=hf(t)
Isotropic; Ym0.3 -----0rthotrc~ic K’. 0.3333
(5) in (4) yields
time (r) = t&@ G”(~)=~G’(~)+w~G(~)+~,[G(~)+s(~)]~=o
(6)
0: = 8(9 - k2) subject to the initial conditions f ‘2,
PF=12
C(T)
+
0
nondimensional damping factor (C) = c,/mi and prime (‘) denotes differentiation with respect to ‘c. 3 -p2k2
32(p2k2 + k) c’=46o8
(k +3)(k
5 -p2k2
-___
25-k2
+5)(k
+7)
7 -p2k2 +98-2k2
I +
= -S(T)
and
G’(r) = -S’(z)
at r
=o.
(7)
The solution of (6) is assumed as G(T) = A(t) cos $(T) where $(r) = Qr +
(8)
e(T)
18-2k2
(23 + 9p2k2)(P2k2 - 3) 240(9 - k 2, Following Anderson [5] yields
+ (19 + 5p2k2)(5 -p2k2)
120(25 - k2)
1.
+ (107 + 21p2k2)(p2k’ - 7) + _l_
1680(49 - k2)
A’(r)+:A(r)=z[G(T)fS(r)]‘sin$(r)
105
Now, make the transformation
B’(t) = &[G(Q+S(r)]‘cos+(r).
[7]
f(r) = G(T) + S(T)
-_-
(5)
--_-
_--__-.-.-
S”(r)
+ Lw(7)
+ of&s(t)
=
K2 = 0.3333 PZ = 0.2
E.2
E.4
where S(s) can be chosen as the particular solution of
E.6 E.
0
c.
0
(9)
fW
isotropic case
p:,
022 0.20 016 0 I6
;: a14 T a12 aI0 008 006 004 0.W
0.16 0.14 012
0
0.2
04
0.6
08
I.0
12
14
16
T
Fig. 3. Dynamic Fig. I. Comparison
of UPA with digital technique
results.
damped
response lar plate.
of an orthotropic
circu-
Nonlinear dynamic damped response of an orthotropic circular plate
The associated initial conditions A,cos8,=
are
-B(z)I,=,
and f A0 cos 0, + A, sin 0, = where A, =
and
A(r)(,,,
0,,=
S(T)[,
=o
(10)
O(r)(,=,.
Now, the right hand sides of (9) are expressed in ultraspherical polynomials in [0,2n], the expansion being restricted to the first term only, yielding
1165
tion computed on the CYBER 180/840A digital computing system using the classical fourth-order Runge-Kutta algorithm and this shown in Fig. 1. It is observed that the UPA technique results are in excellent agreement with the digital solution. Figure 2 shows the dynamic damped response of orthotropic and isotropic cases. It is seen that the dynamic behaviour of both are similar but the maximum deflection of the orthotropic plate is lower than that of the isotropic plate. The same behaviour is also seen in Fig. 3 plotted for E = 0.0. Figure 3 also shows the dynamic damped response for various values of E. It is seen that when ? increases, the maximum deflection decreases as in the case of the isotropic plate. REFERENCES
O’(7)
=
-nn.7, t’
A(7),
A)
(11)
Q.4(7)
where m and n are functions of 7, A (7) and 1, and 1 is the ultraspherical polynomial index. Equations (1 l), (10) and (8) determine G(t), which together with S(7) determines f(7), the nonlinear dynamic damped response.
1. J. L. Nowinski, Nonlinear vibration of elastic circular plates exhibiting rectilinear orthotropy. ZAMP 14, 112-124 (1963). 2. S. K. Chaudhuri, Note on nonlinear dynamic response
of a clamped orthotropic circular plate to pulse excitations. J. Sound Vibr. l%(3), 439441 (1983). 3. K. H. Ruei, C. Jiang and C. Y. Chia, Dynamic and static nonlinear analysis of cylindrically orthotropic circular plates with nonuniform edge constraints. ZAMP
NUMERICAL
In numerical evaluation,
[(;>’$I=0.5
the nondimensional and
35, 387400
(1984).
4. G. L. Anderson, An approximate analysis of nonlinear, nonconservative systems using orthogonal polynomials.
EVALUATION AND CONCLUSIONS
load
1 = 0.0
have been used. The dynamic damped response of the orthotropic plate for E = 2.0 obtained by the UPA technique has been compared with the digital solu-
J. Sound Vibr. 29(4), 463474
(1973).
5. G. L. Anderson, Application of ultraspherical polynomials to nonlinear, nonconservative systems subjected to step function excitation. J Sound Vibr. 32, 101-108 (1974).
6. H. R. Srirangarajan and P. Srinivasan, The pulse response of nonlinear systems. J. Sound Vibr. &l(3), 369-377 (1976).
7. S. T. Ariaratnam, Response of a nonlinear system to pulse excitation. J. h4ech. Engng Sci. 6, 2631 (1964).
0
NONLINEAR DYNAMIC ORTHOTROPIC
DAMPED RESPONSE CIRCULAR PLATE
0045-7949189 13.00 + 0.00 1989 Pergamon Press plc
OF AN
G. CHANDRASEKHARAPPA and H. R. SRIRANGARAJAN Department of Mechanical Engineering, Indian Institute of Technology, Powai, Bombay 400 076, India (Received
6 December
1988)
Abstract-Nonlinear dynamic damped response of a clamped-immovable orthotropic circular plate subjected to step pressure pulse excitation has been presented using the dynamic von Karman equations expressed in terms of the displacement component and solved by a one-term solution applying Galerkin technique to the deflection equation. This yields an ordinary nonlinear differential equation in time. The nonlinear dynamic damped response is obtained by applying the ultraspherical polynomial approximation (UPA) technique. The influences of orthotropic and damping parameters on the dynamic response are presented.
INTRODUCTION The nonlinear transverse vibration of elastic isotropic plates has attracted many researchers. In fact the amplitude of vibration depends on the relationship between the elastic moduli of an anisotropic structure. The resistances to mechanical actions in different directions are different for thin plates, and these have wide applications in modern technology. Nowinski [I] studied the nonlinear vibration of an elastic circular plate based on the von Karman dynamic equations generalized to the rectilinearly orthotropic case. Chaudhuri [2] presented the nonlinear dynamic behaviour of a clamped orthotropic circular plate to pulse excitations using the von Karman equations expressed in terms of displacement components and, by applying the Galerkin technique, reduced it to the nonlinear differential equation which was solved by the perturbation method, Ruei et al. [3] obtained an approximate solution to the dynamic von Karman equations for nonlinear forced vibration of a cylindrically orthotropic circular plate with a sinusoidally restrained edge. In this paper, the dynamic von Karman equations expressed in terms of displacement components are used for the nonlinear dynamic behaviour of a clamped-immovable orthotropic circular plate with damping and subjected to step-pressure-pulse excitation. By using the Galerkin technique to the deflection equation, the coupled nonlinear partial differential equations are reduced to the nonlinear differential equation in time. The ultraspherical polynomial approximation (UPA) technique introduced by Anderson [4,5] and extended by Srirangarajan and Srinivasan [6] has been applied to get the dynamic damped response and is compared with the digital solution computed on the CYBER 180/840A computing system using fourth-order the
Runge-Kutta method. The influences of orthotropic and damping parameters of material for the considered plate are presented.
GOVERNING
DIFFERENTIAL
EQUATIONS
Consider the potential energy of a circular orthotropic plate, taking damping into account, and acted on by a transverse pressure pulse load PO. Forming the Lagrangian function and applying Hamilton’s principle and Euler’s variational equations to the Lagrangian function, yields the following partial coupled differential equations [2].
ah
iau
k2
ar’+;jjy-y=-jy-
ph ah pch au1 adW 2 ah ~Jp+~~+g+;dr)-;ip+uJ~ .\ ,
awaJw -zF
(1)
k2a2w k2 aw
(2) where p is the density of the plate material, u is the displacement along the radius a, w is the deflection normal to the middle plane, h is the plate thickness, k2 = (E,i./E:), p2 = (E”iE;.), E: and EJ. are Young’s modulus along the x- and y-axes respectively, D., = E:h3/12 and E” is the elastic constant. For isotropy, E,:. = E,;. = E/(1 - v’), E” = vE/ (1 - v2), k2 = I, p2 = v, where v is Poisson’s ratio and E is Young’s modulus. 1163
G. CHANDRASEKHARAPPA
1164
and H. R.
SRIRANGARAJAN
NONLINEAR DYNAMIC DAMPED RESPONSE
The deflection function which satisfies the clamped boundary conditions is assumed as
01
018-
I-
r a
[
(3)
Here f(t) is a function of time t, to be determined. Substituting eqn (3) in (I), solving for immovable edge conditions (U = 0 at r = a), then substituting the expression for u in eqn (2) and applying Galerkin’s technique yields a nonlinear differential equation in nondimensional form f”(r)
+ Ef(?) + w:f(z) + qP(T.) = P:
I
I
I
I 0.8
I IO
I 12
, 14
I 16
T
Fig. 2. Orthotropic and isotropic dynamic damped response of the circular plate.
(4) Substituting
where nondimensional
2-2
;: ;:
2 2
w=hf(t)
Isotropic; Ym0.3 -----0rthotrc~ic K’. 0.3333
(5) in (4) yields
time (r) = t&@ G”(~)=~G’(~)+w~G(~)+~,[G(~)+s(~)]~=o
(6)
0: = 8(9 - k2) subject to the initial conditions f ‘2,
PF=12
C(T)
+
0
nondimensional damping factor (C) = c,/mi and prime (‘) denotes differentiation with respect to ‘c. 3 -p2k2
32(p2k2 + k) c’=46o8
(k +3)(k
5 -p2k2
-___
25-k2
+5)(k
+7)
7 -p2k2 +98-2k2
I +
= -S(T)
and
G’(r) = -S’(z)
at r
=o.
(7)
The solution of (6) is assumed as G(T) = A(t) cos $(T) where $(r) = Qr +
(8)
e(T)
18-2k2
(23 + 9p2k2)(P2k2 - 3) 240(9 - k 2, Following Anderson [5] yields
+ (19 + 5p2k2)(5 -p2k2)
120(25 - k2)
1.
+ (107 + 21p2k2)(p2k’ - 7) + _l_
1680(49 - k2)
A’(r)+:A(r)=z[G(T)fS(r)]‘sin$(r)
105
Now, make the transformation
B’(t) = &[G(Q+S(r)]‘cos+(r).
[7]
f(r) = G(T) + S(T)
-_-
(5)
--_-
_--__-.-.-
S”(r)
+ Lw(7)
+ of&s(t)
=
K2 = 0.3333 PZ = 0.2
E.2
E.4
where S(s) can be chosen as the particular solution of
E.6 E.
0
c.
0
(9)
fW
isotropic case
p:,
022 0.20 016 0 I6
;: a14 T a12 aI0 008 006 004 0.W
0.16 0.14 012
0
0.2
04
0.6
08
I.0
12
14
16
T
Fig. 3. Dynamic Fig. I. Comparison
of UPA with digital technique
results.
damped
response lar plate.
of an orthotropic
circu-
Nonlinear dynamic damped response of an orthotropic circular plate
The associated initial conditions A,cos8,=
are
-B(z)I,=,
and f A0 cos 0, + A, sin 0, = where A, =
and
A(r)(,,,
0,,=
S(T)[,
=o
(10)
O(r)(,=,.
Now, the right hand sides of (9) are expressed in ultraspherical polynomials in [0,2n], the expansion being restricted to the first term only, yielding
1165
tion computed on the CYBER 180/840A digital computing system using the classical fourth-order Runge-Kutta algorithm and this shown in Fig. 1. It is observed that the UPA technique results are in excellent agreement with the digital solution. Figure 2 shows the dynamic damped response of orthotropic and isotropic cases. It is seen that the dynamic behaviour of both are similar but the maximum deflection of the orthotropic plate is lower than that of the isotropic plate. The same behaviour is also seen in Fig. 3 plotted for E = 0.0. Figure 3 also shows the dynamic damped response for various values of E. It is seen that when ? increases, the maximum deflection decreases as in the case of the isotropic plate. REFERENCES
O’(7)
=
-nn.7, t’
A(7),
A)
(11)
Q.4(7)
where m and n are functions of 7, A (7) and 1, and 1 is the ultraspherical polynomial index. Equations (1 l), (10) and (8) determine G(t), which together with S(7) determines f(7), the nonlinear dynamic damped response.
1. J. L. Nowinski, Nonlinear vibration of elastic circular plates exhibiting rectilinear orthotropy. ZAMP 14, 112-124 (1963). 2. S. K. Chaudhuri, Note on nonlinear dynamic response
of a clamped orthotropic circular plate to pulse excitations. J. Sound Vibr. l%(3), 439441 (1983). 3. K. H. Ruei, C. Jiang and C. Y. Chia, Dynamic and static nonlinear analysis of cylindrically orthotropic circular plates with nonuniform edge constraints. ZAMP
NUMERICAL
In numerical evaluation,
[(;>’$I=0.5
the nondimensional and
35, 387400
(1984).
4. G. L. Anderson, An approximate analysis of nonlinear, nonconservative systems using orthogonal polynomials.
EVALUATION AND CONCLUSIONS
load
1 = 0.0
have been used. The dynamic damped response of the orthotropic plate for E = 2.0 obtained by the UPA technique has been compared with the digital solu-
J. Sound Vibr. 29(4), 463474
(1973).
5. G. L. Anderson, Application of ultraspherical polynomials to nonlinear, nonconservative systems subjected to step function excitation. J Sound Vibr. 32, 101-108 (1974).
6. H. R. Srirangarajan and P. Srinivasan, The pulse response of nonlinear systems. J. Sound Vibr. &l(3), 369-377 (1976).
7. S. T. Ariaratnam, Response of a nonlinear system to pulse excitation. J. h4ech. Engng Sci. 6, 2631 (1964).