Multiple mode large amplitude vibration of thick orthotropic circular plates

Multiple mode large amplitude vibration of thick orthotropic circular plates

lm J. Non-Linear Mechanics, Vol. 19, No. 4, pp. 341-M8, 1984 Pnvled in Greal Britain. 0020-7462/84 $3.00 + IX) Pergamon Press Ltd. MULTIPLE MODE LAR...

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lm J. Non-Linear Mechanics, Vol. 19, No. 4, pp. 341-M8, 1984 Pnvled in Greal Britain.

0020-7462/84 $3.00 + IX) Pergamon Press Ltd.

MULTIPLE MODE LARGE AMPLITUDE VIBRATION THICK ORTHOTROPIC CIRCULAR PLATES

OF

M. SATHYAMOORTHY Department of Mechanicaland Industrial Engineering,Clarkson University,Potsdam, NY 13676, U.S.A. (Received 13 July 1983) Abstract--This paper is analytically concerned with the large amplitude vibration of thick

orthotropic circular plates incorporating the effectsof transverse shear and rotatory inertia. Von K,'irmhn-type fieldequations written in terms of the three displacement components of the plate are utilized to obtain solutions to clamped stress-freeand immovableplates. By means of Galerkin's techniqueand a numericalRunge-Kutta procedure a multiple-modeanalysis is carried out in both cases.Exactsolutionsare reportedfor two ofthe three governingequations.Effectsof transverseshear deformation and modal interaction are found to be significantfor orthotropic thick plates. The method given here could be extended to the multiple-modeanalysisof circular plates with other boundary conditions.

INTRODUCTION Large amplitude flexural vibrations of circular thin plates made ofisotropic and orthotropic materials have been extensively treated in the literature [I ]. Classical non-linear thin plate theory was used by Yamaki [2 ] to investigate the dynamic behavior of clamped and simply supported isotropic circular plates. Nowinski [3] derived yon K~rm,~n-type non-linear governing equations for rectilinearly orthotropic circular plates and utilized these equations in investigating the period-amplitude behavior of thin circular plates. In-plane boundary conditions corresponding to stress=free edges have been treated in [3 ] whereas immovable boundary conditions are considered in [2 ]. Several other investigations concerned with thin circular plates which make use of yon K~rm~.n-type theory can be found in [4] and [5 ]. A rather simple Berger-type theory which is based on the neglect of the second invariant of the middle surface strains in the expression for the extensional strain energy was used in the analysis of plates of various geometries with immovable in=plane boundary conditions. In the case of circular plates, Nash and Modeer [6 ], Wah [7 ], Pal [8 ] and several others. [4 ] made use of the Berger-type approximate non=linear theory and obtained reasonably acceptable results for various non-linear problems. It must be noted, however, that all these investigations are based on non-linear theories which do not account for the effects of transverse shear and rotatory inertia. Recent investigations indicate that these effects are very important for thick orthotropic plates [I ]. Several non-linear thick plate theories have been presented in the literature [I ] which account for the effects of transverse shear and rotatory inertia. Taking into consideration the difficulties involved in using these theories, Sathyamoorthy and Chia [9] presented a yon K~rm~n-type theory for thick rectangular and skew plates and demonstrated the applicability of the proposed theory to the solution of various non-linear static and dynamic plate problems [I0]. A similar theory based on the Berger-type approximation was also presented for application to thick plates with immovable in-plane boundary conditions [I I ]. While these two theories can be conveniently used for thick plates, a considerable simplification results when the Berger=type theory could be used. In this paper large amplitude free flexural vibrations of stress-free and immovable clamped rectilinearly orthotropic thick circular plates are considered. The effects of transverse shear and rotatory inertia are incorporated into the governing yon K~rm~n-typ¢ non=linear equations by means of suitable tracing constants. These equations are presented in terms of the three displacement components of the plate. Solutions to these three dynamic equations which satisfy all the in-plane as well as out-of=plane boundary conditions are obtained with the aid of a polynomial multiple=mode function for the lateral displacement w. Exact solutions to the in-plane displacements u° and v° are generated whilesatisfying the in-plane 341

342

M. SATHYAMOORTHY

boundary conditions simultaneously. The expressions for the lateral displacement w and inplane displacements u° and v° are finally used to solve the equation of motion in the lateral direction of the vibrating plate. This procedure results in a set of time-differential equations in terms of the amplitude functions found in the lateral displacement w. Numerical solutions to these modal equations are obtained by means of the Runge-Kutta integration procedure. Numerical results are presented for certain high modulus composite as well as isotropic plates. Effects of large amplitude, transverse shear, rotatory inertia, geometric and material plate parameters and modal interaction are discussed. Results corresponding to thin plates are presented for easy comparison. Present results are in close agreement with all available results for special cases. GOVERNING EQUATIONS For a thick circular plate of uniform thickness, h, and radius, a, made of rectilinearly orthotropic material the following equations apply when the effects of transverse shear and rotatory inertia are considered [9]. U,Ox + p 2 U,yy o o 4- S 2 V,xy = --w,x(W,xx 4- p2w,,r) -- S2W,yW,xr

(1)

k2vO,,, + p2U,xxO 4- S2U,xy°= --W,y(k2w, yy "}- p:W, xx) -- S2W, xW, xy

(2)

al (l),xxxx + a2(l),xxyy + aa(1),rrry + a4(l),m + as(l),x~at + ar(I),yr,t + a~(l),~x + as(l),rr + a g ( I ) , , - I + alo[(W),~ . + (w),r;,] 4- all (W),xxxx 4- al2(W),xxyy 4- al3(W),ryrr 4- ala(W),xx,yt, 4- al 5 (w),xxx,at 4- al 6(w),x~m 4- al 7(w),yyrrt, 4- al a(w),r~m 4- alg(w) . . . . xxx 4- a20(w),xxxxyr 4" a21(w),yyyyxx 4- a22(w),yyyyyr =- 0

(3)

where, k: = -~x,q Er 2 = Vxy, p2 = ~/aG~r 2 = p2 4- q2 ,s E~ /a = 1 - vxyvrx, I = q(x, y) - p h w , , + N ~ w , x ~ + Nyyw, yy + 2N~yw,~r.

(4)

In equations (1-4), u°, v° and w are the three displacement components of the middle surface of the plate. Also, a comma denotes partial differentiation with respect to the given coordinate. E~, E r, vxr, vy~, G~y are the orthotropic material constants of the plate and p is the mass density, q ( x , y ) is the applied load per unit area of the plate and N O are the stress resultants defined as given below. Nx~ = E~h (e~ + q2e~), Nry = E,,h (q2e, ax 4- k2eO) (5)

Nxy = hG~re°y

° are the middle surface strains given by where e~, e°r and exr 12

.

1 2

o

e ° = u,° + ~w,~,ey = v°,y + -~w,r,e~r = u°,r + V°,x + W, xW, r

(6)

In case of plates made of isotropic material the ratios of material constants reduce to k 2 = 1, q2 = v, p2 = (1 - V),s2 = (1 4- v)

2

2

(7)

where v is the Poisson's ratio of the isotropic plate material. Equations (1-3) are the governing equations applicable for thick circular plates made of orthotropic material. The coefficients a~are functions of the orthotropic parameters k 2, q2,p2, s 2 and tracing constants T~and Ri which are used to identify the effect of transverse shear and rotatory inertia, respectively. The expressions for a~ can be readily obtained from those given in [9]. In the case of thick plates T~ = R~ = 1 whereas these are zero for thin plates. It is a

Vibration of thick orthotr0pic circular plates

343

matter of simple exercise to reduce equations (1-3) to correspond to the classical non-linear thin plate theory in terms of u*, v° and w by equating T~ and Ri to zero. In what follows, solutions to the system of equations corresponding to thick plates are presented. Reductions are made to obtain further results for orthotropic and isotropic thin circular plates. MULTIPLE MODE SOLUTIONS The following boundary conditions apply for circular plates with stress-free and immovable clamped boundaries. Stress-free: w = w,x = w,y = 0 xNxx + yNxy = 0 yNyy + xN~r = 0

along x 2 + y2 = a 2.

(8)

Immovable: w = w,x = w,y = 0 u° = v° = 0

along x 2 + y2 = a 2.

(9)

A choice for w is made in the following form so that all the out-of-plane boundary conditions (those in terms of w and its derivatives) are satisfied. w ( x , y , T ) = h(x2a"

+ y 2 a2)2[A(z)+~_~B(r)(x2 + 1

y2

_

a2)]

(10)

where A(z) and B(z) are unknown functions of nondimensional time z such that z 2 = t 2 E J p a 2. Equation (10) is substituted into equations (1-2) and expressions for u° and v° are assumed in the following polynomial form. u o = b l x 11 + b2x 9 + b3x 7 + b , x S + b~x 3 + b6x + bTX9y2 + baxTy a + b9xSy 6 + bloxay a + b l l x y 1° + b12xTy 2 + b13xSy 4 + b14x3y 6 + b l s x y 8 + bx6xSy 2 + blvx3y 4 + b l s x y 6 + b19xay 2 + b20xy 4 + b21xy 2

(11)

v ° = b22y 11 + b23y9 + b24y 7 + b2sy 5 + b26Y3 + b27y + b28ygx 2 + b29yVx 4 + baoySx 6 + balyax s + b32yx 1° + b33yTx 2 + ba4ySx 4 + basy3x 6 + b36YX 8 + baTySx 2 + b3ay3x 't + b39yx 6 + b4oy3x 2 + b41yx 4 + ba2yx 2.

(12)

In equations (11) and (12) the coefficients bl-b42 are functions of orthotropic plate parameters and time. To determine these 42 coefficients explicitly, equations (11) and (12) are substituted in equations (1) and (2) along with equation (10) for w. The coefficients of 30 like terms are compared in these two equations to come up with 30 equations in terms of bi. The remaining 12 equations are generated by imposing the in-plane boundary conditions (either stress-free or immovable) given by equation (8) or (9). Thus, for each in-plane boundary condition 42 simultaneous equations are obtained in terms of the unknown coefficients bl-b42 and solved. It should be pointed out that u° and v° thus obtained are exact solutions of equations ( 1) and (2) satisfying the given boundary conditions completely. These coefficients are very lengthy and therefore not defined here explicitly. Since the in-plane displacements u* and v° are known, these are substituted along with w from equation (10) into equation (3) which represents the motion of the plate in the lateral direction. The resulting equation is multiplied by two functions ( x 2 + y 2 a2)2 and (x 2 + y2 a2)a and integrated over the area of the plate. This procedure will result in a set of two ordinary time-differential equations in A(r) and B(T) which will be coupled and nonlinear. These time-differential equations will have the same form for stress-free and immovable circular plates except that the coefficients A; and B~ will be different. The two modal equations in terms of the nondimensional time ~ and amplitude functions A(z) and B(z) are AI(A 3)..... + A2(A2B),rm + A3(AB2),,m + A,(Ba),,~, + As(A),,,,~,, _

A6(B) ....... + AT(A3),,, + As(A2B),,, + A9(AB2),~, + Alo(B3),,~ + Axl(A),~,,, + AI2(B) ..... + A13 Aa + A14A2B + A l s A B 2 + A16 B3

344

M. SATHYAMOORTHY

+ AIr(A),,, + Als(O),,, + AI9A + a2oO = q*

(13)

BI(A 3)..... + B2(A2B),,,,, + B3(AB2),,,. + B,,(B 3) ....

+ Bs(A) ....... + B6(B),.,.~ + BT(A3),,, + Bs(A2B),,, + B9(AB2),r, + Blo(B3),,, + BIl(A) ..... + B12(B),,,, + BI3A 3 + BI4A2B + B15AB 2 + B16B 3 + BIT(A),~, + BIs(B),,, + BI9A + B2oB = q*.

(14)

The nondimensional load q* in equations (13) and (14) is given by q* = qo [34/Ex where 13is the thickness parameter, a/h. The coefficients Ai and Bi in equations (13) and (14) will depend upon bl appearing in equations (11 ) and (12) and are far too long to be defined in this paper. However, some numerical values for Ai and B~ are tabulated later for certain chosen parameters. In addition to being functions of geometric and orthotropic plate parameters, A; and B~ are also functions of the tracing constants T~and R~. By taking these values as unity, modal equations applicable for thick plates are obtained. Similarly, when these tracing constants are taken as zero the modal equations will correspond to those of the classical nonlinear thin plate theory. In this case, the coefficients A ~-A 12 and BI-B12 vanish and therefore equations (13) and (14) are considerably simplified. These simplified equations are applicable for thin circular plates where the effects of transverse shear and rotatory inertia have been omitted. Equations (13) and (14) have been solved using an IMSL subroutine which is based on a Runge-Kutta-Verner numerical integration procedure. Static problems are also solved by treating A and B as independent of time and solving the resulting set of non-linear coupled algebraic equations by Newton's method. NUMERICAL RESULTS AND DISCUSSION Numerical values of period ratios T/To are presented for isotropic and composite plates. Effects of transverse shear, rotatory inertia, modal interaction, material and plate parameters, boundary conditions as well as amplitudes of vibration on period ratios are studied. The effects of transverse shear and rotatory inertia have been included in the nonlinear period T, whereas these effects have been omitted when calculating the linear period To. The orthotropic elastic constants of the plate material corresponding to glass-epoxy (GE), boron-epoxy (BE) and graphite.epoxy (GRE) composites are the same as given in [11]. When the effects of transverse shear deformation and rotatory inertia are both omitted by taking T~ = Ri = 0, the governing equations and all the numerical results obtained here are in close agreement with those presented in [2, 3 ]. Numerical results presented in [2, 3 ~ are based on a single-mode solution whereas a multiple-mode approach is used in this paper. Fundamental linear frequencies obtained on the basis of equation (10) are also in close agreement with those found in the literature. In the ease of static problems, the non-linear load-deflection values obtained by solving a set of non-linear algebraic equations are in good agreement with those given in [2, 3 ]. Variations of period ratios with nondimensional amplitude w* are given in Tables 2-5 for isotropic and composite plates with stress-free and immovable boundaries. Both thin (T~ = R~ = 0) and thick (T~ = Ri = 1) plates are considered. For all the cases investigated here period-amplitude behavior is of the hard-spring type i.e. period ratio decreases with increasing amplitude. The nondimensional amplitude w* corresponds to the ratio of the maximum deflection to the thickness of the plate. These results are obtained by solving equations (13) and (14) retaining only those coefficients which are significant in Table 1. Numerical values for the period ratios increase when the effect of transverse shear and rotatory inertia are taken into account. This type of increase is observed both in the single and multiple-mode analyses. The increase, however, becomes less pronounced at large amplitudes of vibration. It is clear from Tables 2-5 that the effect of coupling of modes is seen by a reduction in the period ratio in the case of thin plates. In the case of thick plates the period ratios are slightly higher at lower amplitudes and lower at large amplitudes of vibration when compared with the corresponding results of the single-mode solution. Results based on multiple-mode solutions are presented in Figs 1-4 for certain static and dynamic problems. Non-linear load-deflection curves in Fig. 1 indicate a behavior of the

Vibration of thick orthotropic circular plates

345

Table 1. Numericalvaluesofcoefficientsin equations (13) and (14) for boron-epoxy immovable circular plate

i

Ai

Bi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Thickness parameter, fl = I 0 0.019983 0.025064 - 0.056393 - 0.073464 0.050275 0.072312 -0.018557 -0.025898 1.101922 1.224364 -0.918265 - 1.049455 - 1.217783 1.494156 -3.399213 -4.427319 3.188393 4.481956 - 1.095748 - 1.590357 58.999050 65.950710 - 49.462660 - 57.292770 14.993610 18.275080 -41.854850 -54.681560 41.305250 56.662090 - 13.473200 - 19.940500 687.074700 770.227700 -577.664500 -675.922300 23.281750 23.281750 - 19.952040 - 30.429100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Thickness parameter, ~ = 30 0.000247 0.000309 - 0.000696 - 9.000907 0.000621 0.000893 - 0.000229 - 0.000320 0.122436 0.136040 - 0.102029 - 0.116606 0.112730 0.140981 -0.317665 -0.413816 0.285151 0.408859 -0.104253 -0.146246 55.039970 61.199840 -45.899390 -52.541770 11.330330 14.151340 -31.903710 -41.574930 28.921810 41.242260 - 10.455210 - 14.715050 5482.901000 6099.414000 -4574.519000 -5242.375000 23.281730 23.281750 - 17.737740 -28.214840

Table 2. Values of period ratios (T/To) 10" for thin circular plates with stress-free edges (T, = Ri = 0). S M S - - s i n g l e - m o d e solution; M M S - - m u l t i p l e - m o d e solution

wo* 0 0.5 1.0 ! .5

Isotropic SMS MMS

SMS

MMS

SMS

MMS

SMS

MMS

10000 9866 9495 8963

10000 9884 9560 9087

10000 9820 9277 8089

10000 9955 9823 9614

10000 9911 9666 9286

10000 9982 9930 9845

10000 9954 9825 9633

10000 9819 9211 7776

Table 3. Values of period ratios

w.*

0 0.5 1.0 1.5

GE

(T/To) 104

BE

GRE

for thick circular plates with stress-free edges (T~ = Rt = 1, /~-- lO)

lsotropic SMS MMS

SMS

MMS

SMS

MMS

SMS

MMS

I0159 10017 9623 9062

10225 10100 9746 9233

10657 10388 9454 7793

10704 10643 10467 10189

I 1705 11396 10298 8690

11242 10983 10032 9983

11552 11283 10138 8892

10468 10230 9337 7602

GE

BE

GRE

346

M, SATHYAMOORTHY Table 4. Values of period ratios (T/T.) 10" for thin circular plates with immovable edges (T~ = R, = 0) w**

Isotropic SMS MMS

SMS

GE MMS

SMS

MMS

SMS

MMS

0 0.5 1.0 1.5

100(30 9589 8608 7494

10000 9601 8648 7553

10000 9559 7911 6110

10000 9590 8617 7508

10000 9578 7928 6089

10000 9591 8621 7514

10000 9591 7951 6087

10000 9529 7860 6075

BE

GRE

Table 5. Values of period r a t i o s ( T / ~ ) 1 0 4 ~ r t h i c k circular plates with immovable edges ( ~ = R~ = 1,

# = 10) w~

Isotropic SMS MMS

SMS

GE MMS

SMS

MMS

SMS

MMS

0 ~5 1.0 1.5

10159 9717 8681 7520

10225 9784 8745 7579

10657 10035 7634 6213

10704 10113 8809 7451

11705 10017 7146 6224

11242 10460 8850 7304

11552 7760 6850 5985

10468 9899 7690 6144

400

BE

--- sMs '

_ qo /

GRE

///

MMS

/ /

300

//

k2

GRE

200

~

//

/

100 j 0

I 0.5

S

1.0

O

T

1.5

~ 2.0

Fig. 1. Load-deflection c u r v e s f o r i s o t r o p i c a n d g r a p h i t e - e p o x y c i r c u l a r p l a t e s w i t h i m m o v a b l e e d g e s (T, = R~ = 0). M M S - - multiple-mode solution; S M S - - single-mode solution.

hard-spring type which is also observed in dynamic problems as mentioned before. The same type of behavior is seen in Fig. 2 where the variations of period ratios are plotted for isotropic as well as composite circular plates. Variation of period ratios with thickness parameter are plotted in Figs 3 and 4 considering isotropic and boron-epoxy circular plates with stress-free and immovable boundaries. As the value of the thickness parameter increases the period ratios approach the values corresponding to thin plate theory. It is also observed that the static and dynamic behavior is very sensitive to the in-plane boundary conditions and the non-linearity is less pronounced in plates with stress-free edges. REFERENCES 1. M. Sathyamoorthy, Non-linear vibrations of plates-a review, The Shock and Vibration Digest 15, 3-16 (1983). 2. N. Yamaki, Influence of large amplitude on flexural vibrations of elastic plates, Z. angew. Math. Mech. 41, 501-510 (1961). 3. J.L. Nowinski, Non-linear vibrations of elastic circular plates exhibiting rectilinear orthotropy, Z. angew. Math. Phys. 14, 113-124 (1963). 4. M. Sathyamoorthy and K. A. V. Pandalai, Large amplitude vibrations of certain deformable bodies: Part II-plates and shells, d. aeronaut. Soc. India 25, 1-10 (1973). 5. C. Y. Chia, Non-linear Analysis of Plates. McGraw-Hill, New York (1980).

1.10 ~

34? GRE

1.00

Ge

~//I$OTROPI¢

0.90

_T

To 0.~0

0.70

0.60 0

n

n

~

J

0.5

1.0

1.5

2.0

w;

Fig. 2. Relation between period ratio and amplitude for isotropic and composite plates with immovable edge and B = 20, T, = R~ = 1. 1.10 'BE Ts"Ri'l

T

1.00

To !

~

0.g0

,

~'SOTROPIC

O.BO 5

, 10

~

T," Ri-]

- -

~,- ero

i 15

/3

, 20

I 25

, 30

Fig. 3. Variation ofperiod rado with thickness parameter for isotropic and boron.epoxy circular plates with stress-free edges and w~ = 1.0.

1.10 BE jTs~Ri-'I

_T i.oo

To 0.90

0.80

.. 5

J 10

~

ts'Ri'1

- - T

s" Ri'O

, 15

n 20

IC

n 2S

j 30

P Fig. 4. Relation between period ratio and thickness parameter for isotropic and boron-epoxy circular plates with immovable edges and w~ = 0.5.

348

M, SATHYAMOORTHY

6. W. A. Nash and J. R. Modeer, Certain approximate analysis of the non-linear behavior of plates and shells, Proceedings of the Symposium on the Theory of Thin Shells, pp. 331-354 (1960). 7. T. Wah, Vibration of circular plates at large amplitudes, J. Engng. Mech. Dir. Am. Soc. Cir. Engrs 89, 1-15 (1963). 8. M. C. Pal, Large amplitude vibration of circular plates subjected to aerodynamic heating, lnt. J. Solids Structures 6, 301-313 (1970). 9. M. Sathyamoorthy and C. Y. Chia, Effect of transverse shear and rotatory inertia on large amplitude vibration of anisotropic skew plates: Part I--Theory, J. appl. Mech. 47, 128-132 (1980). 10. M. Sathyamoorthy and C. Y. Chia, Effect of transverse shear and rotatory inertia on large amplitude vibration of anisotropic skew plates: Part I1, J. appl. Mech. 47, 133-138 (1980). I1. M. Sathyamoorthy, Transverse shear and rotatory inertia effects on non-linear vibration of orthotropic circular plates, Comput. Structures 14, 129-134 (I 981 ).