Geometrically nonlinear axisymmetric vibrations of polar orthotropic circular plates

Geometrically nonlinear axisymmetric vibrations of polar orthotropic circular plates

Pergamon Inr ,I Mech. Sci. Vol. 38, No. 3. pp. 325 333. 1996 Copyright %~. 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0...

409KB Sizes 0 Downloads 208 Views

Pergamon

Inr ,I Mech. Sci. Vol. 38, No. 3. pp. 325 333. 1996

Copyright %~. 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020 7403/96 $15.00 + 0.00

0020-7403 (95)00052-6 GEOMETRICALLY NONLINEAR AX1SYMMETRIC VIBRATIONS OF POLAR ORTHOTROPIC CIRCULAR PLATES CHORNG-FUH LIU* and GE-TZUNG CHEN Department of Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, R.O.C. (Received 5 August 1994; and in revi.sed [orm 20 March 1995) Abstraet--The geometrically nonlinear free vibrations of polar orthotropic circular plates are analyzed by the axisymmetric finite element. The formulation is based on the three-dimensional elasticity with all the nonlinear terms in the strains included and is considered to be more general and precise theoretically compared to the plate-theory-based nonlinear approaches. Numerical results with various geometries, orthotropies, amplitudes and boundary conditions by the present formulation are also shown and compared to those by other approaches.

INTRODUCTION

Circular plates are very common engineering structural components and their vibrations have attracted many researchers in the past. For large amplitude vibration analyses, most of the approaches, analytical or numerical, are based on plate theories, e.g. [1-10] and more can be found in [ 11-15-1. With such plate-theory approaches, approximations are imposed inevitably. In this paper, axisymmetric finite element, which has been used successfully in the linear vibration analyses of circular plates [,,163, is extended to obtain the nonlinear frequencies of polar orthotropic circular plates. The element is developed on the threedimensional elasticity theory and therefore has no simplifications or approximations. All the nonlinear terms of the Lagrangian strains are included in the analysis and boundary conditions are satisfied exactly, whereas it can be done only approximately in the platetheory-based methods [,16]. Ratios of linear to nonlinear fundamental frequencies by the present formulation are shown for different values of orthotropies, amplitudes and aspect ratios, and for different types of boundary conditions FORMULATION

Consider a circular plate of thickness h and radius a. The cylindrical coordinate system (r, 0, z) is chosen such that the r0-plane coincides with the bottom surface of the plate. The origin of the coordinate system is located at the center of the bottom surface with the z-axis upward in the thickness direction. For the axisymmetric vibration, the displacement field can be written as u(r, O, z, t) = u(r. z, ~}

v(r, 0, z, t) = 0

(1)

wtr, 0, z, t) = w(r, z, tl where u and w denote the displacements in the radial and the thickness directions, t is the time variable. The nonlinear strain-displacement relations are then: ~:r =

+

+

~:"=-~-z+2\?z/ + 2 t ?z)

' Author to whom correspondence should be addressed.

326

Chorng-l- uh l.iu and Ge-Tzung Chen

~;0 ~

F

(2)

+2~

+

+ ?-:N+NN

}'~o = 7:o = O

The polar orthotropic constitutive equations are as follows: O"r

Cl!

C12

('~3

0

O"z

I ('I:

C22

C2~

0

gz

GO

--} Ct,

C23

('.~

0

gO

0

0

C66

~rz

i

0

where

"t

(3)

(I - V~oVo~) ~'11 --

E~ E o A (Vzr + VOrVzO)

('~ 2 -

E~EoA

('~ 3 = (Vo~ + Vz~Voz) E~EoA

(1 (7"22 - -

V~o vo,)

(4)

E~EoA {YOz + ~'rzl'Or)

( ' 2 3 --

C33 -

E~EoA ( 1 .- Vrz V~) E,E_A

A = (l - vr~v~, - v~ovo~ - vo, v,o -- 2v~,Vo~Vro)/E~E~Eo and E , , Eo, E~ are Young's moduli in the radial, circumferential and thickness directions respectively, vu are Poisson's ratios with the first subscript denoting the stressed direction. Gr~ is shear modulus. The variational form associated with vibration states that

O = fi [ f,o. [(a,6e, + a~&= + aofeo + rr:@r~) - p(fi6fi + ~'6~)]dv]dt.

(5)

Substituting the strain-displacement (2) and the stress-strain (3) relations into the above variational equation, it can be seen that u and w are the primary dependent variables. Express displacements u and w in (5) as the products of their nodal values ui, wi and shape functions Ni just the same as in the usual finite element formulation, n

and integrate through the domain of an element, we obtain the following elemental equation of motion: [m]~t)} + ([k] + [k]y){U} = 0 where {U} v = [u~ u2 ... u , w~ " z . . ~ , ] , n is the number of nodes in an element. [m], [k] are the linear elemental mass and stiffness matrix which are the same as those in [16], and [ k i n is the nonlinear stiffness matrix (see Appendi× for details). All of them are of dimension

327

Geometricallynonlinear axisymmetnc vibrations

Ice /

/

_

Clamped

I

z

Z

u=o

a

z_-h,2

u__0

fi//

I

~ r

SS--2

SS-1

Z

~z J

dec u=O

atz=0

;a'zr

=0

u=0

m r

SS-4

SS-3

>'-1/////"

Fig. 1 Types of boundary conditions.

2n x 2n. After assembling all the element equations, the global equation of motion is obtained: [ M ] {:~} + ([K] + [ K ] , , ) : X l = 0, If synchronous, harmonic motion is assumed, the above equation then becomes an eigenvalue-value equation of the following form:

([K] + rK]N){_X} = ;-[~U {_X;. The eigenvalue 2 denotes the square of vibration frequency c,). Since the nonlinear stiffness matrix [kin contains the dependent variables u , w and their derivatives, the above equation is to be solved iteratively. NUMERICAL RESULTS AND DISCUSSION In this section, numerical results obtained by using the present axisymmetric finite element model with the complete Lagrangian nonlinear strains considered are presented. The type of finite element used is the 8-node isoparametnc element and is square in shape with side length h, the thickness of the plate. A clamped and four kinds of simply-supported conditions are considered in the present work, see Fig. I. Fhe materials of the plates analysed have the following properties: (a) isotropic E = 210 GPa v =0.3 p = 7810 kg,,m ~

328

Chorng-F uh Liu and Ge-Tzung Chen Table 1. (oL U)NI lor clamped isotropic circular plate a/h

5

10

20

0.2

0.9895 0.9921

0.9899 0.9927

0.9901 0.9928 0.9928

Present [3] 121

0.4

0.9599 0.9699

0.9616 0.9718

0.9621 0.9722 0.9724

Present [3l [2]

0.6

0.9162 0.9366

0.9196 0.9402

0.9205 0.9410 0.9413

Present [3] [2]

0.8

0.8641 0.8965

0.8688 0.9015

0.8699 0.9026 0.9029

Present [31 [21

1.0

0.8088 0.8533

0,8147 0.8591

0.8164 0.8603 0.8607

Present [3] [21

c/h

(b) polar orthotropic

E, = 70 GPa Eo/E, = 0.2, 1.8 E_ = Eo (;,: = 26.92 GPa V0r = Vrz = VzO = 0.3

t~ = 2710 kg/rn 3. Tables 1 and 2 show ratios of linear to nonlinear fundamental frequencies (O)L/(DNL)of clamped and simply-supported isotropic circular plates for different values of a/h and c/h where a is the radius of the plate and c is the maximum amplitude of the central point on the midplane of the plate. The results based on shear deformable plate theory [3] and classical plate theory [2] with von Karman nonlinear strains are also included to compare with the present solutions wherein the values obtained by the present formulation and shear deformable theory for a/h = 20 are considered to be those for a thin plate, Tables 3 and 4 give the ratios of linear to nonlinear frequencies of the fundamental mode for clamped and simply supported polar orthotropic circular plates. Two orthotropies with different values of a/h and c/h are considered. The present solutions are compared to those from classical plate theory [8], which are not changed, as before, with the change of a/h. From the results presented in this paper, the following conclusions can be drawn. (a) For isotropic circular plates (i) For clamped cases, values of (2)L/(!)NL of shear deformable theory are always higher than the present solutions, although they have the same trend as c/h and a/h change, i.e. the effect of geometric nonlinearity is more pronounced when complete nonlinear strains are used. The difference is greater for larger c/h. Classical plate solutions are independent of a/h and are close to those of shear deformation theory for thin plate cases. (ii) For simply supported cases, frequency ratios for conditions SS-1 and SS-2, for any particular c/h value, show the opposite trend compared to those of the shear deformable theory as a/h changes. However. the results of SS-4 change in the same way as the shear deformable theory. Values of ~L/CONLfor SS-3 condition increase first and then decrease

329

Geometrically nonlinear axisymmetric vibrations Table 2. COL/e)NLfor simply supported isotropic circular plate a/h

5

10

20

0.2

0.9673 0.9949 0.9120 0.9868 0.9733

0.9664 0.9946 0.9125 0.9925 0.9741

0.9660 0.9946 0.9122 0.9940 0.9743 0.9744

SS- I SS-2 SS-3 SS-4 [3] [2l

0.4

0.8854 0.9802 0.8309 0.9639 0.9058

0.8827 0.9790 0.8319 0.9748 0.9081

0.8815 0.9788 0.8315 0.9777 0.9087 0.9089

SS-2 SS-3 SS-4 [3] [2]

0.6

0.7847 0.9571 0.7572 0.9327 0.8216

0.7811 0.9551 0.7586 0.9488 0.8250

0.7793 0.9547 0.7584 0.9531 0.8258 0.8261

SS-I SS-2 SS-3 SS-4 [3] [21

0.8

0.6877 0.9269 0.691 ! 0.8947 0.7382

0.6848 0.9262 0.6927 0.9176 0.7420

0.6813 0.9232 0.6931 0.9210 0.7429 0.7432

SS- ! SS-2 SS-3 SS-4 [3] [2]

e/h

SS- 1

Table 3. COL/CONLfor clamped orthotropic circular plate a/h = 5

F-,0/F,r:

a/h = 10

a/h =

20

0.2

i.8

0.2

1.8

0.2

1.8

0.9888

0.9903

0.9882

0.9908

0.9881 0.9907

0.9909 0.9937

Present

0.9549 0.9644

0.9655 0.9756

Present

e/h 0.2

0.4

0.9571

0.9636

0.9554

0.9651

[8]

[8]

0.6

0.9111

0.9235

0.9074

0.9265

0.9067 0.9254

0.9274 0.9478

Present [8]

0.8

0.8547

0.8757

0.8515

0.8794

0.8508 0.8788

0.8803 0.9131

[8]

0.7922 0.8289

0.8301 0.8745

1.0

0.7967

0.8242

0.7928

0.8289

Present

Present

[8l

with the increase of a/h. Just like the cases in linear vibration analysis, different simply supported conditions lead to different results [16]. The difference is very difficult, or may be impossible, to be shown by conventional analyses. (iii) For larger c/h and a/h, results with SS-1 condition are close to those with SS-3 condition, and the same for SS-2 and SS-4. Their difference indicates that specifying zero radial displacement on boundary has a very significant effect on the frequency ratios. This agrees with the results in 1-17] where the effect of radial constraint on finite amplitude vibration was discussed. (iv) The simply supported condition used in I-2] and I-3] resembles SS-1 of the present study, however, it seems difficult to identify the results of the plate theories to those of the present solution with any of the four simply supported conditions.

330

Chorng-Fuh Liu and Ge-Tzung Chen Table 4. e~L/U~NLfor simply supported orthotropic circular plate a/h=5

c/h

0.2

0.4

0.6

0.8

Eo/Er:

a/h = 10

a/h = 20

0.2

1.8

0.2

1.8

0.2

1.8

0.9549 0.9987 0.8774

0.9724 0.9944 0.9264

0.9517 0.9976 0.8766

0.9716 0.9942 0.9264

0.9506 0.9975 0.8758

0.9712 0.9941 0.9261

SS-I SS-2 SS-3

0.9923

0.9852

0.9961

0.9917

SS..4 [8]

0.8482 0.9951 0.7741 0.9825 0.7267 0.9895 0,6865 0.9707 0.6164 0.9818 0.6130 0.9566

0.9020 0.9783 0.8558 0.9601

0.8396 0.9910 0.7728 0.9880

0.8129 0.9535 0.7895 0.9262

0.7155 0.9803 0.6856 0.9758

..... ..... 0.7283 .....

0.8995 0.9778 0.8561 0.9729 0.8087 0.9520 0.7900 0.9446

0.6055 0.9663 0.6122 0.9603

..... ..... 0.7290 .....

0.9971

0.9935

0.9414

0.9818

0.8366 0.9902 0.7718 0.9895 0.8138

0.8983 0.9776 0.8556 0.9764 0.9334

SS-i SS-2 SS-3 SS-4

0.7116 0.9786 0.6850 0.9775 0.6840

0.8068 0.9517 0.7896 0.9498 0.8683

SS-1 SS-2 SS-3 SS-4

0.6019 0.9633 0.6121 0.9618 0.5771

..... ..... 0.7289 ..... 0.7989

[8]

[8]

SS-I SS-2

SS-3 SS-,4 [81

..... Not converge

(b)

For polar orthotropic circular plates

(i) F o r clamped cases, different orthotropies lead to different trends of the frequency ratios as a/h increases. F o r Eo/E, = 0.2, ~OL/O~r~Ldecreases and for Eo/E, = 1.8, it increases for any particular c/h value, i.e. for Eo/E, = 0.2, a thinner plate is not necessarily a plate with smaller nonlinear effect for clamped cases. (ii) Nonlinear effect is greater for the present formulation than the classical plate theory for clamped orthotropic cases, and the difference is larger for higher c/h. (iii) F o r both orthotropies with conditions SS-1 and SS-2, the values of ~L/~ONL decrease with increase of a/h for any particular value of c/h and the trend is reversed for SS-4 condition. W h e n the b o u n d a r y condition is of SS-3 type, ~OL/~NL decreases for Eo/E, = 0.2 and increases first then decreases for Eo/E, = 1.8 as a/h increases. (iv) Values of COL/~NLfor SS-2 and SS-4 are close to each other for both orthotropies when a/h and c/h increase. These values are much greater than those with SS-1 and SS-3 conditions. The constraints of radial displacement on b o u n d a r y for simply supported cases again show quite a great influence on the nonlinear effect just the same as for isotropic plates. However, for all cases of both isotropic and polar orthotropic circular plates, the effect of geometric nonlinearity changes slightly as thickness ratio changes for any value of c/h and changes m u c h greatly as c/h changes. Also, the difference between the simply-supported conditions SS-1 and SS-3 is more obvio'us for smaller c/h and diminishes for larger vibrating amplitude. Nonlinear effect is much less for SS-2 and SS-4 where radial constraint is not imposed on the boundary. It is n o t e w o r t h y that conventional nonlinear vibration analyses of isotropic and polar orthotropic circular plates, based on either classical or shear deformable plate theories have shown that, I-2, 3, 8] 1. Thicker plate (smaller a/h) has more p r o n o u n c e d nonlinear effect for both clamped and simply supported conditions. 2. F o r the same material, geometry and value of c/h, the nonlinear effect with simply supported condition is always greater than with clamped condition.

Geometrically nonlinear axisymmetric vibrations

331

3. Changing c/h or Eo/E, individually, the corresponding change of frequency ratios is more pronounced with simply supported condition than clamped condition. 4. For all cases, nonlinear effect increases with the increase of the value of c/h. From the results of the present investigation, it is observed that, except for the last one of the above four conclusions derived previously by conventional plate theories, the other three are not necessarily true or valid. The frequency ratios, as shown by the present formulation, can have more complex change along with change of material, geometry boundary condition, and so on~ CONCLUDING

REMARKS

The axisymmetric finite element formulation with the complete Lagrangian nonlinear strains is employed to investigate the effect of the geometric nonlinearity on the axisymmetric vibration of circular plates. The formulation is a particular axisymmetric model of 3-D elasticity without any simplification or approximation, and therefore can be considered to be more accurate than plate theories. Also, with the present formulation, the boundary conditions can be specified more clearly and satisfied exactly. Results of the present study show that effect of geometric nonlinearity on axisymmetric vibration is a complicated combination of factors like the aspect ratio a/h, the amplitude ratio c/h, the orthotropy ratio Eo/E, and boundary conditions. It is therefore recommended that, to have a better understanding of the nonlinear axisymmetric vibration of any particular circular plate, the present approach, which provides one of the best analyses, should be considered. REFERENCES 1. Chuh Mei, Finite element displacement method for large amplitude free flexural vibrations of beams and plates. Comput. Struct. 3, 163 (1973). 2. G. Venkateswara Rao, K. Kanaka Raju and 1. S. Raju, Finite element formulation for the large amplitude free vibrations of beams and orthotropic circular plates. Comput. Struct. 6, 169 (1976). 3. K. Kanaka Raju and G. Venkateswara Rao, Axisymmetric vibrations of circular plates including the effects of geometric nonlinearity, shear deformation and rotary inertia, J. Sound Vib. 47, 179 (1976). 4. G. Venkateswara Rao and K. Kanaka Raju, Large amplitude axisymmetric vibrations of orthotropic circular plates elastically restrained against rotation. J. Sound Vib. 69, 175 (1980). 5. J. N. Reddy and C. L. H uang, Large amplitude free vibrations of annular plates of varying thickness. J. Sound. Vib. 79, 387 (1981). 6. A. V. Srinivasan, Large amplitude free oscillations of beams and plates. AIAA J. 3, 1951 (1965). 7. P. Z. Bulkelay, An axisymmetric nonlinear vibration of circular plates. Trans ASME, d. Appl. Mech. 4, 630 (1963). 8. K. Kanaka Raju, Large amplitude vibrations of circular plates with varying thickness. J. Sound Vib. 50, 399 (1977). 9. R. Bhattacharya and B. Banerjee, Influence of large amplitudes, shear deformation and rotary inertia on axisymmetric vibrations of moderately thick circular plates: a new approach. J. Sound Vib. 133, 185 (1989). 10. J, Ramachandran, Nonlinear vibrations of circular plates with linearly varying thickness, J. Sound Vib. 38, 225 (1975). 11. A. W. Leissa, Recent research in plate vibrations, 1973-1976: complicating effects. Shock Vib. Dig. 9, 21 (1977). 12. A. W. Leissa, Plate vibration research, 1976- 1980: complicating effects. Shock Vib. Dig. 13, 19 (1981). 13. A. W. Leissa, Recent studies in plate vibrations, 1981-1985: Part II. Complicating effects. Shock Vib, Dig. 19, 10 (1987). 14. M. Sathyamoorthy, Nonlinear vibrations of plates--a review~ Shock Vib. Dig. 15, 3 (1983). 15. M. Sathyamoorthy, Nonlinear vibration analysis of plates: a review and survey of current developments. Appl. Mech. Rev. 40, 1553 (1987). 16. C. F. Liu and G. T. Chen, A simple finite element analysis of axisymmetric vibration of annular and circular plates. Int. J. Mech. Sci. 37. 861 871 (1995). 17. C. L. D. Huang and I. M. Al-Khattat, Finite amplitude vibrations of a circular plate. Int. J. Non-linear Mech. 12, 297 (1977). APPENDIX The nonlinear stiffness matrix [k]n can be decomposed as

LEk~,~] [k~nl j

332

C h o r n g - F u h Liu and Ge-Tzung Chen

where

[k~ 1N]q = J {CI ~[(3:2t(du;drjNi ,Nj r~ + C~ a f( l/2t(gu/&)(N,.,Nj.~ + N~.~Nj.,) + (1/2)(Ou/Or)N~.,Ni.,]

+ C~ 3[(1/2)lu/r2)(N~.,N) * N~Nj.,) + (1/2rZ)(Ou/Or)N~Nj] + Cl3[(12rJ(c~u/drj(N~Nj., + N,.,Nj) + ([/2)(u/r)Ni.,Nj,, + C23 [1F2r)(Ou/dz)(N, Nj., + N,.~N~) + (1/2)(u/r)N,.~Nj.= + C3~[(3'2)(u/r3)NiNj] + C66 [(eu/i~r)N,.=Nj., + (gu/c~z)Ni.,Nj., + N,.,Nj.,)] + C12 [(l'2}(?w/caz)N,.,Nj.,] + C22[{l'21(?w/~z)NI.~N2.~] ± C23 [il '2r2)|gw/Oz)NiNj] + C~6 [II/2)[~w..'&)(N~.~Nj., + N~.,N2.~)] + C~ ~[( [, 2t(~u/&) 2N.., N~.,]

+ Czz[(12)teu/c~z)gNi.~Nj,~] + C33[(l/2){u2/r*)NiNi] • (', 2 [(1 2){Ou/gz)(gu/Or)(N..,Ni. ~ + N~.,Nj.,)] + Ct3 [(1. 211u,'r2)(,,qu/car)(Ni,,Nj + N/N.~.,)]

+ C23[ll,211u;r2)(du/c~z)(Ni.~Nj + N~Nj,,)] + C~6[(l/2Rgu/,arIZN.,N.~ + (1/2)(gu/Oz)2N~.,Ni.,] + C66[(12t(~u/&)(Ou/dr)iN~.~Nj., + N~.,N2.~)]} dv [kl:N]O = f { c l t [

1/21 ~w'dr)Ni ,Nj

r~

+ C22[il/2)(Ou/t~zlNi.zN).~] + C~3[[1/2r)(gw/Or)N,N~.,]

+ Cz3[( l2r)(t~w/~z)NiN~.~ + (1/2)(u/r2)N~Nj.~'[

+ C66[ll2)tgw~ar)Nc~N:., + (1/2)(8w/&)N,.~N~.,] • C6~ [ll2)t~u/&)N,.~N~., + (1/2)(Ou/Oz)N~.,N~.,] + C~ [t l'2){Ow/dr)(Ou/(~rjNi.,Nj,,] + C22 [(1/2)(c~w/Ozll~u/dz)Ni.,N~.,] + C~2[(l:21(tqw/dzl(~,u/grlNi.,Nj., + (1/2)(Ow/Or)(Ou/Oz)Ni.,Nz,] + C~ ~[(t.:2)(gw/Or)(u/r2)NiNj.,] + C~3 [( l:2)(gw/Oz)(u/r 2) N. N2.~] + C~ 6 [( 1<2)(~,w/gr)(Ou/&)N/. ~Nj., + ( 1/2)(0w/&)(Ou/Or)N~,, Ni.,]

+ C~6[ll'2)(c~w/c~r){du/~z)Ni.,Ns., + 11 2)t?wgz)(?u/Oz)Ni.,N~.,] } dv = [/<" ~ ] i, [~]~

f{ C,2 [ I 2)(~w/ar)(N,.,Ni.= + N,.,N~.,) + (I/2)(Ow/&)N,,,N~,,] + Ca;[13.2)t?w/&)Ni,~Ns,,] + Co6 [(?w/?rl{N,.,N~.~ + N,.,Ns,,) + (Ow/Oz)N~.,N~.,] + C~l [(l/2)(Ou/dr)N~.,Nj.,]

Geometrically nonlinear axlsymmetric vibrations + Ci 2 [1 l/2)(?,u/~r)N, ~N, z]

+ Ci3[(l/2)(u/r)N,,rNs, ] ~- C23[(I/2)(u/r)N,:Nj~] + C6~[(1/2)(c3u,,~z)INi..Ni~ ~ Ni,,Nj.~)] + C1~ [(l'2)(Ow./~ri:N,.,N,.,] ~- ('22 [( 1/2)(?'wOzl 2 N,: N; :]

+ C~a[(1/2)(?,w4z)({~t~ ?r)l~\~..:'~j,~ + N..~Nj.,)] * C.~ [I 1/2)i?w ?r): N,.: N , : - ( 1"2)(0w/Oz)2 Ni,,Nj.,)]

333