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Nonlinear vibration of orthotropic circular plates—A comparison
Fibre Science and Technology 16 (1982) 111-119
NONLINEAR VIBRATION OF ORTHOTROPIC CIRCULAR PLATES--A COMPARISON#
M. SATHYAMOORTHY
Department of Mechanical and Industrial Engineering, C/arkson College of Technology, Potsdam, New York 13676 (USA) SUMMARY The purpose o f this paper is to make a comparison between two different approaches to the solution o f the same problem, namely the nonlinear vibration o f a circular plate. In this work the material o f the plate is taken to be rectilinearly orthotropic and the boundary o f the plate is clamped and immovable. The two different methods given here lead independently to the stress function, F, and in-plane displacements u and v which are exact solutions to the corresponding governing equations. It is possible to obtain u and v f r o m F and vice versa. The ultimate results are independent of the method chosen to solve the problem. It is shown that the Stress Function Approach has certain advantages over the Displacement Equations Approach. In particular, the Stress Function Approach will result in a substantial saving in analytical and computational efforts f o r nonlinear problems involving multiple modes in the assumed lateral displacement w or in cases where transverse shear and rotatory inertia effects are to be considered.
1.
INTRODUCTION
It is well known that the problem of nonlinear vibration of plates can be studied by means of nonlinear governing equations of the von Karm~in type 1 or by means of those written purely in terms of the displacement components u, v, w of the p l a t e ) The first step in solving a nonlinear problem is to determine the functional form for the stress function, F, in the von K~rman type approach, and for the in-plane displacements u and v in the so-called Displacement Equations Approach, for any given mode shape w. It is, however, very difficult to determine exactly the nature of t Presented at the 22nd ASME/AIAA/ASCE/AHSStructures, Structural Dynamics and Materials Conference, Atlanta, Georgia, April 1981. 111 Fibre Science and Technology 0015-0568/82/0016-0111/$02.75 f') Applied Science Publishers Ltd, England, 1982 Printed in Great Britain
112
M. SATHYAMOORTHY
the stress function, F, or the displacements u and v to satisfy the corresponding governing equations and in-plane boundary conditions for some given geometry of the plate. ~,2 In such cases approximate expressions for F o r u and v are obtained and Galerkin's method is used to satisfy the equation of transverse motion. Using Galerkin's method Yamaki 3 studied the nonlinear vibration behaviour of simply supported and clamped isotropic circular plates with stress-free and immovable edges by means of the Stress Function Approach. A similar approach was adopted by Nowinski* to study rectilinearly orthotropic circular plates. Sathyamoorthy and Chia 5 made use of the Displacement Equations Approach to investigate the effects of the transverse shear deformation and rotatory inertia on the nonlinear vibration of orthotropic circular plates. Using the semi-inverse method exact solutions to the in-plane equilibrium equations are reported in Reference 5. When the effects of transverse shear and rotatory inertia are neglected, the governing equations in Reference 5 will yield the necessary displacement equations for the present analysis. This paper is analytically concerned with nonlinear flexural vibrations of a circular plate with a clamped immovable boundary. The governing dynamic equations in terms of stress function F and lateral displacement w on the one hand and those in terms of the three displacement components u, v and w on the other hand form the two sets of equations corresponding to the Stress Function Approach and Displacement Equations Approach, respectively. These two sets of equations are presented here. For a lateral displacement w assumed to satisfy the boundary conditions, these two sets of equations are solved exactly to obtain the stress function F a n d the in-plane displacements u and v. At this stage each coefficient in F can be related to each coefficient in the expressions for u and v. Thus it is possible to obtain u and v from F a n d vice versa. With these quantities known in an explicit form the equation of motion in the lateral direction can be solved independently using Galerkin's method. This procedure results in a modal equation the coefficients of which are the same irrespective of the method used to arrive at this equation. These coefficients are tabulated for certain chosen plate parameters. Exact solutions to the modal equation are presented in tabular form considering the static case first and then free undamped vibration. It can be seen that the Stress Function Approach has advantages over the Displacement Equations Approach. Present results are in exact agreement with those reported in the literature. 3
2.
GOVERNING EQUATIONS
For an orthotropic circular plate of radius a, with the origin at the centre and x and y as the cartesian coordinates, the governing equations in terms of stress function F and lateral displacement w are 4
NONLINEAR VIBRATION OF ORTHOTROPIC CIRCULAR PLATES
F . . . . + k2Fr~.ry + mlF:x~.r = Ey(WZxy - W~xW.rr)
I 13 (1)
W. . . . . + k2w,ryy~, + 2(q 2 + 2p2)W,xxyr
= o~
F.,,w~x + f.~xW,. - 2 f ~ , w ~ , - ~ w,, +
(2)
where k2
Ey
m2 = ~ 2 ( k2 --
= E-~
q4 _
2p2q2)
q2 =
vyx
p2
=
Gxy# Ex
(3)
Exh 3 la = 1 - VxyVy x
D1 = 12p
vxY = qZ/k2
E x and Er are the elastic moduli along the x and y directions, respectively, Gxy is the shear modulus, and Vxy and vy x are the Poisson's ratios of the orthotropic material. In the case o f plates made o f isotropic material k 2 = 1, q2 = v and p2 = (1 - v)/2. Alternatively, the governing equations can be written in terms o f the in-plane displacements u, v and lateral displacement w as 2 k21:,yy + p2V x x + S2blxy = - - W , y ( k 2 w y y + p2W,xx) - S 2 W x W , x y
(5)
1 w. . . . . + k 2 ~,yyyy + 2(p 2 + S2)W x:yy = -D1 , - [W:,xNx: , + w.yyNyy --[-2 w ,,rNxr - pw.t t + q]
(6) U x x +p2U,yy + S21),xy = -- W x ( W x x + p 2 W y y ) -- S2W,yW,xy
(4)
where s 2 = p Z + q2
N x x = E l h ( e a + q2e2 )
N:,r = E l h p 2 7
Nry = E x h ( q 2 s l + k2e2)
1 2 /32 ~---/),y -+- ~W,y
el = U,x _L - ~1W2,x
7 = U,y + l~.x + W,xW,y
E1-
(7)
E~ #
The circular plate is assumed to be i m m o v a b l y clamped along the b o u n d a r y and therefore the b o u n d a r y conditions are u = v = w = w,x = w y = 0
along 1
X2
y2
02
a2 -----0
(8)
In Section 3 exact solutions to eqns (1), (4) and (5) are presented for an assumed lateral displacement w and then eqns (2) and (6) are solved using Galerkin's method. During this process, care is taken to see that all the b o u n d a r y conditions in eqn (8) are fully satisfied.
114
M. SATHYAMOORTHY
3. METHODOF SOLUTION A l t h o u g h the m e t h o d used and described below is valid for multiple term solutions, a single term polynomial form o f w is chosen to keep the analytical work within limits. The form of w chosen as follows satisfies the out-of-plane b o u n d a r y conditions in eqn (8):
( w =hA(z)
X2 .}'2)2 1
a2
a2
(9)
where A(z) is an u n k n o w n function of the nondimensional time z = t(Ex/pa2) ~/2 which will be determined later. This solution for w is substituted in eqn (1) and following a semi-inverse procedure an exact solution to the stress function F is obtained as follows:
[aoXS + a l y 8 + aZ(a2 x6 F = E~.Azh2 aS
+
a3Y 6)
+ a4(a4x 4 + asy 4) + a6(a6x 2 + aTy 2) + a8x6Y 2 +agx2y 6 +a2(alox4Y 2 + a l l x 2 y 4) +a12x4y 4 +a~3a4x2y 2]
(lO)
where the coefficients a 0 to a~ 3 are functions of time and are obtained in conjunction with the following immovable conditions, i.e. u = v = 0 along 1 - x2/a 2 - yZ/aZ = O: u =
v=
u.xd x =
v,~.dy =
[tl(k2F, yy_
fi
q Z F x x• )
-
~1w x2]
dx
= 0
(11) [t/(F.~.,. - q 2 Fo'r) - -2 1,2 ~ 0'] d) , = 0
where # tl - - E x ( k 2
_
q4)
It must be restated that the stress function in eqn (10) is an exact solution to eqn (1) for the assumed mode given by eqn (9). The coefficients a o to a~3 are lengthy and hence are not defined here. Having solved eqn (1) exactly, it is now necessary to satisfy the equation of motion in the transverse direction, i.e. eqn (2). It is clear that this equation will not be satisfied for the F a n d w given by eqns (10) and (9), respectively. Substituting these in eqn (2) results in an error function which is integrated over the area of the plate. Such a procedure will yield a modal equation in A (r) as
d2A dz -~ +klA
+k2A3
--
k3q° E~
(12)
NONLINEAR VIBRATION OF ORTHOTROPIC CIRCULAR PLATES
l 15
where qo is the uniformly distributed load per unit area of the plate. Coefficients k i depend upon the coefficients a o to a~ 3 and therefore are not defined here. Note that the title problem has now been fully solved using the Stress Function Approach, i.e. with the help of eqns (1) and (2). Turning our attention now to the second set of equations, namely, eqns (4), (5) and (6) ( the Displacement Equations Approach) the in-plane displacements u and v can be readily obtained by substituting the solution for w from eqn (9) in eqns (4) and (5). Following the semi-inverse procedure as before, the displacements u and v are obtained as follows: A2h 2 u = ~ -
[Cox7 q- clxSy 2 + c2x3y 4 d- c3xY 6 q- a2(c4x 5 q- csxay 2 + c6xY 4)
(13)
+ a4(cTx 3 + c8xY 2) + Cga6X] A2h 2 v = ~ff-
CloY 7 + c 1lySx 2 + c12y3x 's + c13YX 6 + a2(c14Y 5 + c15y3x 2 + cl6YX 4)
q.-a4(clvy 3 + c18yx 2) + c19a6y]
(14)
Here again, the coefficients c o to c19 are functions of time and are obtained following the standard mathematical procedure. For this reason these coefficients will not be defined explicitly but the relationships between c o to c19 and a o to a13 in eqn (10) will be presented in Section 4. The expressions for u and v given above are exact solutions to the in-plane equilibrium equations, eqns (4) and (5), for the assumed w given by eqn (9). These displacements satisfy all the boundary conditions in eqn (8) exactly. The only remaining equation, eqn (6), which represents the equation of motion in the lateral direction of the plate is now solved with the help of eqns (13) and (14) and Galerkin's procedure. The time-differential equation in this case becomes dZA
s3q o
d r 2 + s1A -~- s2 A3-'~- Ex
(15)
No attempt has been made here to define the coefficients s~ to s3, which in turn depend upon the coefficients c o to c19, as this would be a very lengthy procedure. Either eqn (12) or eqn (15) can be used to study the period-amplitude behaviour of immovably clamped rectilinearly orthotropic circular plates. This Dulling-type equation can be solved exactly in terms of a complete elliptic integral of the first kind 3 for the nonlinear period T w h e n qo = 0. In the case of static problems, A is independent of time and hence the resulting nonlinear algebraic equation will give the load-deflection relationship in the large-deflection regime. Some numerical results for both static and dynamic cases are presented and discussed later.
116
M. SATHYAMOORTHY 4.
COMPARISON
As the same problem has been solved by following two different approaches it is possible to make a comparison between the stress function F in the Stress Function Approach and the in-plane displacements u and v in the Displacement Equations Approach. The 14 coefficients in the expression for F i n eqn (10) are related to the 20 coefficients in u and v in eqns (13) and (14) as follows: c o = + ( k 2 b o - 8)
c 4 =~(keb4 + 16) c8 = b8
c9 = b 9
c12 = ½ ( k 2 b 1 2 - 8) C16 = b16
C1 = ½ ( k 2 b l
-
16)
c 2 = ½ ( k 2 b 2 - 8)
c 5 =½(kZb5 + 16)
c6 = b6
C~o = + ( k Z b ~ o - 8) Cl 3 : bl 3
Cl,
c17 =½(k2bl7 - 8)
c3 = b 3
c7 = ½ ( k 2 b 7 - 8)
cll =~(k2bl~ - 16)
c15 = ½(kZbls + 16)
=~(k2bt4 + 16)
cl8 = b18
(16)
c19 = b19
where b o = 2(ascii + 28aoC12)
bl = 6(2ax2cll + 5asclz)
bE = 6 ( 5 a 9 c l l + 2a12c12)
b3 =
b4 = 2(aloCll + 15a2c12)
b5 = 1 2 ( a l l c x x + a l o C 1 2 )
b 6 = 2(15a3cll + allCl2)
b7 = 2(a13cll + 6a4clz)
b 8 = 2(6a5cll + a13c12)
69 =
2(28alcl1 + a 9 c 1 2 )
2(a~cll
+ a6c12)
blo = 2(a9c22 + 28alcxz)
bll = 6(2a12c22 + 5a9c12)
blz = 6(5asczz + 2a12c12)
b13 =
b14 = 2 ( a l l c 2 2 + 15a3clz)
bl5 = 12(aloCz2
b16 =
2(15azCz2 + aloC12)
(17)
2(28aoCz2 + asClz) + a11c12 )
b17 = 2(al3czz + 6a5c12)
bib = 2(6a4c22 + a13c12)
b19 = 2(a6c22 + a7c12)
and 1 Cll = 1
C22 -- k2
c12 ~ -- q2c22
By using eqns (16) and (17) it is thus possible to obtain u and v from F and vice versa. The numerical values for coefficients Co to c 19 appearing in eqns (13) and (14) are given in Table I for isotropic and orthotropic circular plates using eqns (16) and (17). These coefficients are identical to those obtained directly by following the Displacement Equations Approach. Since the stress function F in eqn (10) and the displacements in eqns (13) and (14) are exact solutions, it is possible to relate Fwith u
NONLINEAR
VIBRATION
OF ORTHOTROPIC
TABLE
CIRCULAR
117
PLATES
1
VALUES FOR COEFFICIENTS CO TO C19 FOR ISOTROPIC AND ORTHOTROPIC CIRCULAR PLATES
lsotropic Co ct c2 c3 c4 cs C6 C7 Ca Co
= = = = = =
ct0 ctt cl 2 ct3 c1,, cts Ct6 = Ct 7 = Cta = Cl9 =
° K 2 = ~,
Orthotropic a
-
1.11667 3.35 3.35 1.11667 3" 133 33 6.26667 3"133 33 -- 2 ' 7 --2"7 0"683 33
co cI c2 c3 c4 cs C6 C7 C8 C9
- 1-14193 - 3"213 4 4 - 2.738 27 - 0 " 6 6 6 76 3" 198 35 5.41937 1"958 19 -- 2"673 34 -- 1"908 36 0"616 92
ct0 cl 1 c12 el3 c1,, cls C16 Cl 7 Cla Ct9
- 1.158 9 4 - 3.555 25 - 3"505 77 1"10946 3"286 78 6 . 7 0 3 12 3'288 42 -- 2"871 0 7 --2"9222 0"743 23 - -
vxy = 0-25, G ~ / E x = 0"015 ( f r o m R e f e r e n c e 5).
and v uniquely in the manner given in eqns (16) and (17). Although this can be done for circular and elliptical plates, it appears difficult to do the same with plates of other geometries. This is merely due to the fact that exact solutions to eqn (1) or eqns (4) and (5) are nonexistent for other geometries. Since the stress function and the in-plane displacements can be obtained one from the other, it is clear that the modal equation coefficients in eqns (12) and (15) will be identical, i.e. k 1 = Sl ; k 2 = $2 and k 3 = S3. These coefficients are presented in Table 2 for isotropic as well as orthotropic circular plates. TABLE 2 COEFFICIENTS OF MODAL EQUATION FOR ISOTROPIC AND ORTHOTROPIC CIRCULAR PLATES (a/h = 20)
Isotropic Orthotropic
5.
k 1 = sI
k2 = s 2
k3 = s 3
0 . 0 2 4 42 0 . 0 0 8 76
0.011 51 0 . 0 0 4 08
6 6 6 . 6 6 6 67 6 6 6 . 6 6 6 67
CONCLUDING
REMARKS
Numerical results are presented for the static and dynamic cases in Tables 3 and 4. In the static case the nonlinear load-deflection relationship for isotropic plates with v--0-3 is given by 3 ( ? )
5"861
(~)3
+ 2'761
q°a4
(18)
In the dynamic case the modal equation becomes 3
Eh 4 ph2A., + ~ - (9-768A + 4.602A 3) = 5/3qo
(19)
I 18
M. SATHYAMOORTHY TABLE 3 VALUES OF NON-DIMENSIONALLOAD qoa4/Eyh4 w0/h
lsotropic
Orthotropic
0"5 1'0 1"5 2"0
3"2756 8"6227 18'1126 33'8169
46"9242 123"2085 258"2113 481"2926
TABLE 4 VALUESOF T/To( × 104) WITH % = 0
A 0 0'5 1"0 1"5 2"0
~otropic
Orthotropic
10000 9584 8586 7460 6510
10000 9589 8599 7479 6529
It can be shown that the modal equation, eqn (15) (or eqn (12)) reduces exactly to eqn (18) for the isotropic static case, i.e. when A is independent of time, and to eqn (19) in the dynamic case in the presence of lateral load q0- Therefore present numerical results agree exactly with those of Reference 3. However, no comparison is possible in the case of orthotropic circular plates. In the dynamic case the wellknown hardening type of nonlinearity, i.e. decreasing period ratio with increasing amplitude of vibration, is observed for both isotropic and orthotropic cases. An analysis similar to the one presented in the preceding sections can be carried out for orthotropic circular plates with clamped and stress-free edge conditions. Since the governing equations are the same, the expressions for the stress function F and the in-plane displacements u and v would remain the same as given before. However, the coefficients a 0 to al ~ and c o to c 19 are determined in conjunction with the following stress-free boundary conditions: x F ~.~.- y F . x y = 0
along x 2 + y2 = a 2 yFxx
(20)
- xF.xy = 0
Here again, it can be readily shown that the stress function and in-plane displacements can be obtained one from the other and that the modal equation similar to eqn (15) is identical if either method is used. In the single-mode analysis which is presented in this paper the number of unknown coefficients in the stress function in eqn (10) is 14. In a multiple-mode analysis, consisting of two terms in the lateral displacement w, which has been carried out recently by the author, it was observed that the number of unknown coefficients in F b e c a m e 27, i.e. an increase of 13 coefficients for only one additional term in w. This clearly shows that the problem size becomes enormous when
NONLINEAR VIBRATION OF ORTHOTROPIC CIRCULAR PLATES
I 19
multiple-mode analysis is attempted. It is obvious that under these conditions if the Displacement Equations Approach is used the problem size becomes even higher because of a very large number of coefficients in the in-plane displacements, i.e. 42 coefficients instead of 20. A very similar situation arises in the case of moderately thick plates where the transverse shear and rotatory inertia effects are important. Here, due to the complex nature of the equation of motion in the lateral direction of the plate, the problem size for the Displacement Equations Approach is far greater than that of the Stress Function Approach even for a single-mode analysis. Therefore it is possible to conclude that a substantial amount of analytical and computational effort can be saved by using the Stress Function Approach even for a single-mode analysis. The selection of this method will be of crucial importance in complex problems, i.e. in problems involving multiple-modes or in cases where transverse shear and rotatory inertia effects are to be considered. Finally, it must be mentioned that exact solutions to eqn (1) or eqns (4) and (5) can be readily obtained when multiple terms are assumed in the expression for the lateral displacement w. The equivalence between the Stress Function Approach and the Displacement Equations Approach can, in this case, be established by following the procedure adopted in this paper.
REFERENCES 1. M. SATHYAMOORTHY,Nonlinear vibration of rectangular plates, J. Sound and Vibration, 58 (1978) pp. 301 4. 2. M. SATHYAMOORTHY,Nonlinear vibration of rectangular plates, J. Applied Mechanics, 46 (1979) pp. 215-17. 3. N. YAMAKI,Influence of large amplitude on flexural vibrations of elastic plates, Z A M M , 41 (1961) pp. 501-10. 4. J. L. NOWINSKI, Nonlinear vibrations of elastic circular plates exhibiting rectilinear orthotropy, Z A M P , 14 (1963) pp. 113-24. 5. M. SATHYAMOORTHYand C. Y. CHIA, 'Nonlinear Vibration of Orthotropic Circular Plates Including Transverse Shear and Rotatory Inertia', Proceedings of ASME Winter Annual Meeting, New York, (1979) pp. 357 72.
NONLINEAR VIBRATION OF ORTHOTROPIC CIRCULAR PLATES--A COMPARISON#
M. SATHYAMOORTHY
Department of Mechanical and Industrial Engineering, C/arkson College of Technology, Potsdam, New York 13676 (USA) SUMMARY The purpose o f this paper is to make a comparison between two different approaches to the solution o f the same problem, namely the nonlinear vibration o f a circular plate. In this work the material o f the plate is taken to be rectilinearly orthotropic and the boundary o f the plate is clamped and immovable. The two different methods given here lead independently to the stress function, F, and in-plane displacements u and v which are exact solutions to the corresponding governing equations. It is possible to obtain u and v f r o m F and vice versa. The ultimate results are independent of the method chosen to solve the problem. It is shown that the Stress Function Approach has certain advantages over the Displacement Equations Approach. In particular, the Stress Function Approach will result in a substantial saving in analytical and computational efforts f o r nonlinear problems involving multiple modes in the assumed lateral displacement w or in cases where transverse shear and rotatory inertia effects are to be considered.
1.
INTRODUCTION
It is well known that the problem of nonlinear vibration of plates can be studied by means of nonlinear governing equations of the von Karm~in type 1 or by means of those written purely in terms of the displacement components u, v, w of the p l a t e ) The first step in solving a nonlinear problem is to determine the functional form for the stress function, F, in the von K~rman type approach, and for the in-plane displacements u and v in the so-called Displacement Equations Approach, for any given mode shape w. It is, however, very difficult to determine exactly the nature of t Presented at the 22nd ASME/AIAA/ASCE/AHSStructures, Structural Dynamics and Materials Conference, Atlanta, Georgia, April 1981. 111 Fibre Science and Technology 0015-0568/82/0016-0111/$02.75 f') Applied Science Publishers Ltd, England, 1982 Printed in Great Britain
112
M. SATHYAMOORTHY
the stress function, F, or the displacements u and v to satisfy the corresponding governing equations and in-plane boundary conditions for some given geometry of the plate. ~,2 In such cases approximate expressions for F o r u and v are obtained and Galerkin's method is used to satisfy the equation of transverse motion. Using Galerkin's method Yamaki 3 studied the nonlinear vibration behaviour of simply supported and clamped isotropic circular plates with stress-free and immovable edges by means of the Stress Function Approach. A similar approach was adopted by Nowinski* to study rectilinearly orthotropic circular plates. Sathyamoorthy and Chia 5 made use of the Displacement Equations Approach to investigate the effects of the transverse shear deformation and rotatory inertia on the nonlinear vibration of orthotropic circular plates. Using the semi-inverse method exact solutions to the in-plane equilibrium equations are reported in Reference 5. When the effects of transverse shear and rotatory inertia are neglected, the governing equations in Reference 5 will yield the necessary displacement equations for the present analysis. This paper is analytically concerned with nonlinear flexural vibrations of a circular plate with a clamped immovable boundary. The governing dynamic equations in terms of stress function F and lateral displacement w on the one hand and those in terms of the three displacement components u, v and w on the other hand form the two sets of equations corresponding to the Stress Function Approach and Displacement Equations Approach, respectively. These two sets of equations are presented here. For a lateral displacement w assumed to satisfy the boundary conditions, these two sets of equations are solved exactly to obtain the stress function F a n d the in-plane displacements u and v. At this stage each coefficient in F can be related to each coefficient in the expressions for u and v. Thus it is possible to obtain u and v from F a n d vice versa. With these quantities known in an explicit form the equation of motion in the lateral direction can be solved independently using Galerkin's method. This procedure results in a modal equation the coefficients of which are the same irrespective of the method used to arrive at this equation. These coefficients are tabulated for certain chosen plate parameters. Exact solutions to the modal equation are presented in tabular form considering the static case first and then free undamped vibration. It can be seen that the Stress Function Approach has advantages over the Displacement Equations Approach. Present results are in exact agreement with those reported in the literature. 3
2.
GOVERNING EQUATIONS
For an orthotropic circular plate of radius a, with the origin at the centre and x and y as the cartesian coordinates, the governing equations in terms of stress function F and lateral displacement w are 4
NONLINEAR VIBRATION OF ORTHOTROPIC CIRCULAR PLATES
F . . . . + k2Fr~.ry + mlF:x~.r = Ey(WZxy - W~xW.rr)
I 13 (1)
W. . . . . + k2w,ryy~, + 2(q 2 + 2p2)W,xxyr
= o~
F.,,w~x + f.~xW,. - 2 f ~ , w ~ , - ~ w,, +
(2)
where k2
Ey
m2 = ~ 2 ( k2 --
= E-~
q4 _
2p2q2)
q2 =
vyx
p2
=
Gxy# Ex
(3)
Exh 3 la = 1 - VxyVy x
D1 = 12p
vxY = qZ/k2
E x and Er are the elastic moduli along the x and y directions, respectively, Gxy is the shear modulus, and Vxy and vy x are the Poisson's ratios of the orthotropic material. In the case o f plates made o f isotropic material k 2 = 1, q2 = v and p2 = (1 - v)/2. Alternatively, the governing equations can be written in terms o f the in-plane displacements u, v and lateral displacement w as 2 k21:,yy + p2V x x + S2blxy = - - W , y ( k 2 w y y + p2W,xx) - S 2 W x W , x y
(5)
1 w. . . . . + k 2 ~,yyyy + 2(p 2 + S2)W x:yy = -D1 , - [W:,xNx: , + w.yyNyy --[-2 w ,,rNxr - pw.t t + q]
(6) U x x +p2U,yy + S21),xy = -- W x ( W x x + p 2 W y y ) -- S2W,yW,xy
(4)
where s 2 = p Z + q2
N x x = E l h ( e a + q2e2 )
N:,r = E l h p 2 7
Nry = E x h ( q 2 s l + k2e2)
1 2 /32 ~---/),y -+- ~W,y
el = U,x _L - ~1W2,x
7 = U,y + l~.x + W,xW,y
E1-
(7)
E~ #
The circular plate is assumed to be i m m o v a b l y clamped along the b o u n d a r y and therefore the b o u n d a r y conditions are u = v = w = w,x = w y = 0
along 1
X2
y2
02
a2 -----0
(8)
In Section 3 exact solutions to eqns (1), (4) and (5) are presented for an assumed lateral displacement w and then eqns (2) and (6) are solved using Galerkin's method. During this process, care is taken to see that all the b o u n d a r y conditions in eqn (8) are fully satisfied.
114
M. SATHYAMOORTHY
3. METHODOF SOLUTION A l t h o u g h the m e t h o d used and described below is valid for multiple term solutions, a single term polynomial form o f w is chosen to keep the analytical work within limits. The form of w chosen as follows satisfies the out-of-plane b o u n d a r y conditions in eqn (8):
( w =hA(z)
X2 .}'2)2 1
a2
a2
(9)
where A(z) is an u n k n o w n function of the nondimensional time z = t(Ex/pa2) ~/2 which will be determined later. This solution for w is substituted in eqn (1) and following a semi-inverse procedure an exact solution to the stress function F is obtained as follows:
[aoXS + a l y 8 + aZ(a2 x6 F = E~.Azh2 aS
+
a3Y 6)
+ a4(a4x 4 + asy 4) + a6(a6x 2 + aTy 2) + a8x6Y 2 +agx2y 6 +a2(alox4Y 2 + a l l x 2 y 4) +a12x4y 4 +a~3a4x2y 2]
(lO)
where the coefficients a 0 to a~ 3 are functions of time and are obtained in conjunction with the following immovable conditions, i.e. u = v = 0 along 1 - x2/a 2 - yZ/aZ = O: u =
v=
u.xd x =
v,~.dy =
[tl(k2F, yy_
fi
q Z F x x• )
-
~1w x2]
dx
= 0
(11) [t/(F.~.,. - q 2 Fo'r) - -2 1,2 ~ 0'] d) , = 0
where # tl - - E x ( k 2
_
q4)
It must be restated that the stress function in eqn (10) is an exact solution to eqn (1) for the assumed mode given by eqn (9). The coefficients a o to a~3 are lengthy and hence are not defined here. Having solved eqn (1) exactly, it is now necessary to satisfy the equation of motion in the transverse direction, i.e. eqn (2). It is clear that this equation will not be satisfied for the F a n d w given by eqns (10) and (9), respectively. Substituting these in eqn (2) results in an error function which is integrated over the area of the plate. Such a procedure will yield a modal equation in A (r) as
d2A dz -~ +klA
+k2A3
--
k3q° E~
(12)
NONLINEAR VIBRATION OF ORTHOTROPIC CIRCULAR PLATES
l 15
where qo is the uniformly distributed load per unit area of the plate. Coefficients k i depend upon the coefficients a o to a~ 3 and therefore are not defined here. Note that the title problem has now been fully solved using the Stress Function Approach, i.e. with the help of eqns (1) and (2). Turning our attention now to the second set of equations, namely, eqns (4), (5) and (6) ( the Displacement Equations Approach) the in-plane displacements u and v can be readily obtained by substituting the solution for w from eqn (9) in eqns (4) and (5). Following the semi-inverse procedure as before, the displacements u and v are obtained as follows: A2h 2 u = ~ -
[Cox7 q- clxSy 2 + c2x3y 4 d- c3xY 6 q- a2(c4x 5 q- csxay 2 + c6xY 4)
(13)
+ a4(cTx 3 + c8xY 2) + Cga6X] A2h 2 v = ~ff-
CloY 7 + c 1lySx 2 + c12y3x 's + c13YX 6 + a2(c14Y 5 + c15y3x 2 + cl6YX 4)
q.-a4(clvy 3 + c18yx 2) + c19a6y]
(14)
Here again, the coefficients c o to c19 are functions of time and are obtained following the standard mathematical procedure. For this reason these coefficients will not be defined explicitly but the relationships between c o to c19 and a o to a13 in eqn (10) will be presented in Section 4. The expressions for u and v given above are exact solutions to the in-plane equilibrium equations, eqns (4) and (5), for the assumed w given by eqn (9). These displacements satisfy all the boundary conditions in eqn (8) exactly. The only remaining equation, eqn (6), which represents the equation of motion in the lateral direction of the plate is now solved with the help of eqns (13) and (14) and Galerkin's procedure. The time-differential equation in this case becomes dZA
s3q o
d r 2 + s1A -~- s2 A3-'~- Ex
(15)
No attempt has been made here to define the coefficients s~ to s3, which in turn depend upon the coefficients c o to c19, as this would be a very lengthy procedure. Either eqn (12) or eqn (15) can be used to study the period-amplitude behaviour of immovably clamped rectilinearly orthotropic circular plates. This Dulling-type equation can be solved exactly in terms of a complete elliptic integral of the first kind 3 for the nonlinear period T w h e n qo = 0. In the case of static problems, A is independent of time and hence the resulting nonlinear algebraic equation will give the load-deflection relationship in the large-deflection regime. Some numerical results for both static and dynamic cases are presented and discussed later.
116
M. SATHYAMOORTHY 4.
COMPARISON
As the same problem has been solved by following two different approaches it is possible to make a comparison between the stress function F in the Stress Function Approach and the in-plane displacements u and v in the Displacement Equations Approach. The 14 coefficients in the expression for F i n eqn (10) are related to the 20 coefficients in u and v in eqns (13) and (14) as follows: c o = + ( k 2 b o - 8)
c 4 =~(keb4 + 16) c8 = b8
c9 = b 9
c12 = ½ ( k 2 b 1 2 - 8) C16 = b16
C1 = ½ ( k 2 b l
-
16)
c 2 = ½ ( k 2 b 2 - 8)
c 5 =½(kZb5 + 16)
c6 = b6
C~o = + ( k Z b ~ o - 8) Cl 3 : bl 3
Cl,
c17 =½(k2bl7 - 8)
c3 = b 3
c7 = ½ ( k 2 b 7 - 8)
cll =~(k2bl~ - 16)
c15 = ½(kZbls + 16)
=~(k2bt4 + 16)
cl8 = b18
(16)
c19 = b19
where b o = 2(ascii + 28aoC12)
bl = 6(2ax2cll + 5asclz)
bE = 6 ( 5 a 9 c l l + 2a12c12)
b3 =
b4 = 2(aloCll + 15a2c12)
b5 = 1 2 ( a l l c x x + a l o C 1 2 )
b 6 = 2(15a3cll + allCl2)
b7 = 2(a13cll + 6a4clz)
b 8 = 2(6a5cll + a13c12)
69 =
2(28alcl1 + a 9 c 1 2 )
2(a~cll
+ a6c12)
blo = 2(a9c22 + 28alcxz)
bll = 6(2a12c22 + 5a9c12)
blz = 6(5asczz + 2a12c12)
b13 =
b14 = 2 ( a l l c 2 2 + 15a3clz)
bl5 = 12(aloCz2
b16 =
2(15azCz2 + aloC12)
(17)
2(28aoCz2 + asClz) + a11c12 )
b17 = 2(al3czz + 6a5c12)
bib = 2(6a4c22 + a13c12)
b19 = 2(a6c22 + a7c12)
and 1 Cll = 1
C22 -- k2
c12 ~ -- q2c22
By using eqns (16) and (17) it is thus possible to obtain u and v from F and vice versa. The numerical values for coefficients Co to c 19 appearing in eqns (13) and (14) are given in Table I for isotropic and orthotropic circular plates using eqns (16) and (17). These coefficients are identical to those obtained directly by following the Displacement Equations Approach. Since the stress function F in eqn (10) and the displacements in eqns (13) and (14) are exact solutions, it is possible to relate Fwith u
NONLINEAR
VIBRATION
OF ORTHOTROPIC
TABLE
CIRCULAR
117
PLATES
1
VALUES FOR COEFFICIENTS CO TO C19 FOR ISOTROPIC AND ORTHOTROPIC CIRCULAR PLATES
lsotropic Co ct c2 c3 c4 cs C6 C7 Ca Co
= = = = = =
ct0 ctt cl 2 ct3 c1,, cts Ct6 = Ct 7 = Cta = Cl9 =
° K 2 = ~,
Orthotropic a
-
1.11667 3.35 3.35 1.11667 3" 133 33 6.26667 3"133 33 -- 2 ' 7 --2"7 0"683 33
co cI c2 c3 c4 cs C6 C7 C8 C9
- 1-14193 - 3"213 4 4 - 2.738 27 - 0 " 6 6 6 76 3" 198 35 5.41937 1"958 19 -- 2"673 34 -- 1"908 36 0"616 92
ct0 cl 1 c12 el3 c1,, cls C16 Cl 7 Cla Ct9
- 1.158 9 4 - 3.555 25 - 3"505 77 1"10946 3"286 78 6 . 7 0 3 12 3'288 42 -- 2"871 0 7 --2"9222 0"743 23 - -
vxy = 0-25, G ~ / E x = 0"015 ( f r o m R e f e r e n c e 5).
and v uniquely in the manner given in eqns (16) and (17). Although this can be done for circular and elliptical plates, it appears difficult to do the same with plates of other geometries. This is merely due to the fact that exact solutions to eqn (1) or eqns (4) and (5) are nonexistent for other geometries. Since the stress function and the in-plane displacements can be obtained one from the other, it is clear that the modal equation coefficients in eqns (12) and (15) will be identical, i.e. k 1 = Sl ; k 2 = $2 and k 3 = S3. These coefficients are presented in Table 2 for isotropic as well as orthotropic circular plates. TABLE 2 COEFFICIENTS OF MODAL EQUATION FOR ISOTROPIC AND ORTHOTROPIC CIRCULAR PLATES (a/h = 20)
Isotropic Orthotropic
5.
k 1 = sI
k2 = s 2
k3 = s 3
0 . 0 2 4 42 0 . 0 0 8 76
0.011 51 0 . 0 0 4 08
6 6 6 . 6 6 6 67 6 6 6 . 6 6 6 67
CONCLUDING
REMARKS
Numerical results are presented for the static and dynamic cases in Tables 3 and 4. In the static case the nonlinear load-deflection relationship for isotropic plates with v--0-3 is given by 3 ( ? )
5"861
(~)3
+ 2'761
q°a4
(18)
In the dynamic case the modal equation becomes 3
Eh 4 ph2A., + ~ - (9-768A + 4.602A 3) = 5/3qo
(19)
I 18
M. SATHYAMOORTHY TABLE 3 VALUES OF NON-DIMENSIONALLOAD qoa4/Eyh4 w0/h
lsotropic
Orthotropic
0"5 1'0 1"5 2"0
3"2756 8"6227 18'1126 33'8169
46"9242 123"2085 258"2113 481"2926
TABLE 4 VALUESOF T/To( × 104) WITH % = 0
A 0 0'5 1"0 1"5 2"0
~otropic
Orthotropic
10000 9584 8586 7460 6510
10000 9589 8599 7479 6529
It can be shown that the modal equation, eqn (15) (or eqn (12)) reduces exactly to eqn (18) for the isotropic static case, i.e. when A is independent of time, and to eqn (19) in the dynamic case in the presence of lateral load q0- Therefore present numerical results agree exactly with those of Reference 3. However, no comparison is possible in the case of orthotropic circular plates. In the dynamic case the wellknown hardening type of nonlinearity, i.e. decreasing period ratio with increasing amplitude of vibration, is observed for both isotropic and orthotropic cases. An analysis similar to the one presented in the preceding sections can be carried out for orthotropic circular plates with clamped and stress-free edge conditions. Since the governing equations are the same, the expressions for the stress function F and the in-plane displacements u and v would remain the same as given before. However, the coefficients a 0 to al ~ and c o to c 19 are determined in conjunction with the following stress-free boundary conditions: x F ~.~.- y F . x y = 0
along x 2 + y2 = a 2 yFxx
(20)
- xF.xy = 0
Here again, it can be readily shown that the stress function and in-plane displacements can be obtained one from the other and that the modal equation similar to eqn (15) is identical if either method is used. In the single-mode analysis which is presented in this paper the number of unknown coefficients in the stress function in eqn (10) is 14. In a multiple-mode analysis, consisting of two terms in the lateral displacement w, which has been carried out recently by the author, it was observed that the number of unknown coefficients in F b e c a m e 27, i.e. an increase of 13 coefficients for only one additional term in w. This clearly shows that the problem size becomes enormous when
NONLINEAR VIBRATION OF ORTHOTROPIC CIRCULAR PLATES
I 19
multiple-mode analysis is attempted. It is obvious that under these conditions if the Displacement Equations Approach is used the problem size becomes even higher because of a very large number of coefficients in the in-plane displacements, i.e. 42 coefficients instead of 20. A very similar situation arises in the case of moderately thick plates where the transverse shear and rotatory inertia effects are important. Here, due to the complex nature of the equation of motion in the lateral direction of the plate, the problem size for the Displacement Equations Approach is far greater than that of the Stress Function Approach even for a single-mode analysis. Therefore it is possible to conclude that a substantial amount of analytical and computational effort can be saved by using the Stress Function Approach even for a single-mode analysis. The selection of this method will be of crucial importance in complex problems, i.e. in problems involving multiple-modes or in cases where transverse shear and rotatory inertia effects are to be considered. Finally, it must be mentioned that exact solutions to eqn (1) or eqns (4) and (5) can be readily obtained when multiple terms are assumed in the expression for the lateral displacement w. The equivalence between the Stress Function Approach and the Displacement Equations Approach can, in this case, be established by following the procedure adopted in this paper.
REFERENCES 1. M. SATHYAMOORTHY,Nonlinear vibration of rectangular plates, J. Sound and Vibration, 58 (1978) pp. 301 4. 2. M. SATHYAMOORTHY,Nonlinear vibration of rectangular plates, J. Applied Mechanics, 46 (1979) pp. 215-17. 3. N. YAMAKI,Influence of large amplitude on flexural vibrations of elastic plates, Z A M M , 41 (1961) pp. 501-10. 4. J. L. NOWINSKI, Nonlinear vibrations of elastic circular plates exhibiting rectilinear orthotropy, Z A M P , 14 (1963) pp. 113-24. 5. M. SATHYAMOORTHYand C. Y. CHIA, 'Nonlinear Vibration of Orthotropic Circular Plates Including Transverse Shear and Rotatory Inertia', Proceedings of ASME Winter Annual Meeting, New York, (1979) pp. 357 72.