The free vibration of rectangular plates

Journal of Sound and Vibration (1973) 31(3), 257-293

THE FREE VIBRATION OF RECTANGULAR PLATES A. W. LEISSAt Federal Institute o/Technology, Zurich, Switzerland

(Received 10 April 1973)

This work attempts to present comprehensive and accurate analytical results for the free vibration of rectangular plates. Twenty-one cases exist which involve the possible combinations of clamped, simply-supported, and free edge conditions. Exact characteristic equations are given for the six cases having two opposite sides simply-supported. The existence of solutions to the various characteristic equations is carefully delineated. The Ritz method is employed with 36 terms containing the products of beam functions to analyze the remaining 15 cases. Accurate frequency parameters are presented for a range of aspect ratios (alb = 0'4, 2/3, 1'0, 1'5, and 2'5) for each case. For the last 15 cases, comparisons are made with Warburton's useful, approximate formulas. The effects of changing Poisson's ratio are studied.

1. INTRODUCTION

A vast literature exists for the free vibrations of rectangular plates. Consider only the classical theory governed by the differential equation

(1) where w is transverse deflection; '\74 is the biharmonic differential operator (Le., '\74 = \12 \12, '\72 = 02/0X2 + 02/ oy 2 in rectangular co-ordinates); D = Eh 3 /12(1 - v2), the flexural rigidity; E is Young's modulus; h is plate thickness; v is Poisson's ratio; p is mass density per unit area of plate surface; and t is time. Exclude such complicating effects as orthotropy, in-plane forces, variable thickness, the effects of surrounding media, large deflections, shear deformation and rotary inertia, and nonhomogeneity. Even with these restrictions a survey [1] made by the writer a few years ago uncovered 164 pertinent references. However, it was also found that the majority of this voluminous literature dealt with a few specific problems, that for most of the problems the scope of treatment was limited or essentially non-existent, that the exactness of numerical results was ordinarily deficient, and that the effects of changing Poisson's ratio were generally ignored. For rectangular plates there exist 21 distinct cases which involve all possible combinations of classical boundary conditions (i.e., clamped, simply-supported, or free). For the six cases having two opposite edges simply-supported, well-known exact solutions exist which are the extensions of Voigt's [2] early work. For the remaining 15 cases, three problems have received a great deal of attention. The completely clamped case is used frequently as a test problem for analytical methods because of the simplicity of the boundary conditions. The cantilever plate

t On leave from Ohio State University, Columbus, Ohio 43210, U.S.A. 257

258

A. W. LEISSA

has received extensive coverage because ofits practical importance, particularly for simulating lifting and stabilizing surfaces in the aerospace industry. The completely free case has a rich history. The first known observations of nodal patterns on plates were reported by Chladni [3-6J beginning in 1787 for completely free square plates, which inspired much subsequent experimental work and analytical discussion in the literature. Ritz [7] in 1909 used the completely free problem to demonstrate his now-famous direct method for extending the Rayleigh principle for obtaining upper bounds on vibration frequencies. The remaining 12 problems have received little coverage in the literature; indeed, for six of them virtually nothing at all can be found. An important step to remedy this situation was made by Warburton [8]. In this useful piece of work he presented frequency formulas for all 21 types of problems derived by using the Rayleigh method with assumed mode shapes which are the products of vibrating beam eigenfunctions. Later, another set of formulas was published by Janich [9] for 18 cases (for fundamental modes only). Again, the Rayleigh technique was utilized, but simple trigonometric functions were chosen to represent the plate deflections. However, these functions do not represent the mode shapes nearly as well as the beam functions; consequently, this latter work is of less practical value than Warburton's. Although Warburton's formulas are of considerable value to the design engineer, certain questions concerning them still remain to be answered. Of particular interest is the general question of accuracy. The beam functions do satisfy the geometric boundary conditions of zero deflection and slope where required (although not the free edge or free corner conditions of a plate); consequently, they are mathematically admissible functions for the Ritz variational procedure, and yield upper bounds for the fundamental (lowest) frequencies. However, to what accuracy can a free vibration frequency be obtained when only a single-term representation of the deflection mode shape is used? And what happens to the relative accuracy as boundary conditions are changed or higher frequencies are sought? The primary purpose ofthis work is to present in one place reasonably accurate results for free vibration frequencies of all 21 combinations of classical boundary conditions for rectangular plates. Secondary purposes are (1) to evaluate the accuracy of Warburton's formulas by direct comparison, (2) to point out some of the mathematical nuances of the Voigt and Ritz methods and of the resulting solutions, (3) to study the effects of changing edge conditions upon the frequencies and upon their accuracies, and (4) to study the effects of changing Poisson's ratio upon the vibration frequencies. The first part ofthe paper deals with the exact solutions for the six cases having two opposite sides simply-supported. Extensive numerical results are given in Appendix A. The last part deals with the remaining fifteen problems. Accurate frequencies are obtained by using the Ritz method, 36-term mode shapes composed of beam functions, and the capabilities of modern, digital computers. These frequencies are tabulated in Appendix C. Comparison is

I

I I

I

I I I

I

I

I

I I I I

x

Figure 1. A SS-C-SS-F rectangular plate with co-ordinate convention.

FREE VIBRATION OF RECTANGULAR PLATES

259

made with the results of the Warburton formulas and with other published results when they are available. The effects of changing Poisson's ratio are studied in a middle part. Before beginning, some explanatory comments which pertain to the entire paper should be made. First, consider the plate having length dimensions a and b. For purposes of description, a notation will be adopted as follows. The symbolism SS-C-SS-F, for example, will identify a rectangular plate with the edges x = 0, y = 0, x = a, Y = b having simply-supported, clamped, simply-supported, and free boundary conditions, respectively (see Figure 1). Secondly, it should be remembered that the non-dimensional frequency parameter wa 2 VPil5' (where OJ is frequency, and a is a characteristic length) does not explicitly depend upon Poisson's ratio for the rectangular shapes considered here unless at least one edge is free. However, the frequency itself depends upon v in every case due to its inclusion in D. Unless otherwise stated, for the 15 cases having a free edge, vwill be taken as 0'3, a widely-used practical value. (A list ofnotation is given in Appendix D.) 2. TWO OPPOSITE EDGES SIMPLY-SUPPORTED

For the sake of definiteness, the well-known classical boundary conditions will be repeated below for an edge parallel to the y-axis (for example, the boundaries x = 0 or x = a). For a clamped edge,

ow

w=-=O

ax

(2)

'

for a simply-supported edge,

(3) and for a free edge 02 W

oJ w

iFw

oJ w

ox2 + v oy2 = ox3 + (2- v) oxoy2

=

O.

(4)

Corresponding boundary conditions for the edges y = 0 and y = b are obtained by interchanging x and y in equations (2), (3), and (4). For a free corner formed by the intersection of two free edges the additional condition 02 W

=0 (5) oxoy must be satisfied at the corner, although this condition will not be encountered when two opposite edges are simply-supported. On the assumption of a sinusoidal time response for free vibration,

w(x,y, t) = W(x,y) el
(6)

the classical Voigt [2] solution

W m= [A m sinVk2 - (1.2 y + BmcosVk 2 -

+ D m coshV k2 +

(/.2

y] sin (Xx

(/.2

Y

+ Cmsinh-ylk2 + (/.2 y + (7)

will be taken, where k 4 = pOJ2/D, and (/. = mn/a, m = 1, 2, ... , and where k 2 is assumed to be greater than (/.2. The deflection function (6) exactly satisfies the governing field equation (1) and the simply-supported boundary conditions (3) along x = 0 and x = a. Substituting (7) into the four appropriate boundary conditions along the edges y = 0 and y = b leads to a characteristic determinant of the fourth order for each m. Expanding the determinant and collecting terms yields a characteristic equation. The characteristic equations for the six cases are listed below.

260

A. W. LEISSA

Case 1. SS-SS-SS-SS (8)

Case 2. SS-C-SS-C 4>1 4>"{COS 4>1 cosh4>z - 1) - m 2 n 2

(~r sin 4>1 sinh 4>2

=

O.

(9)

Case 3. SS-C-SS-SS (10)

Case 4. SS-C-SS-F 4>14>2[A,z - m4 n4(1 - V)2] + 4>1 4>2['1,2 +m4 n 4(1- V)2] cos 4>1 cosh 4>2 +

(11) Case 5. SS-SS-SS-F 4>1[..1. + m 2 n 2(1 - v)]2 tanh 4>2 - 4>2[..1. - m 2 n2(1 - v)]2 tan 4>1

=

O.

(12)

Case 6. SS-F-SS-F 24>14>2[..1. 2 - m4 n4(1 - V)2]2 (cos 4>1 cosh 4>2 - 1) - 4>HA. - m2 n2 (1 - v)]4} sin 4>1 sinh 4>2: = O.

+ {4>HA. + m21l:2(1 - vW (13)

In equations (8) through (13), A. is the nondimensional frequency parameter defined by (14)

and 4>1 and 4>2 are functions of A. given by b 4>1 =-VA.-m 2 n2 , a

(15) A point frequently overlooked in the literature is that it is possible for k2 to be less than a,z (that is, A less than m 2 n 2 ). When this occurs it is necessary to replace sin v'P - (X2 Y and cosv'k 2 - (X2 Y in equation (7) by sinh Y(X2 - k 2 y and coshy(X2 - P y, respectively. Then the characteristic equations become the following. Case 1. SS-SS-SS-SS sinh 111 sinh '12 = O.

. (16)

Case 2. SS-C-SS-C 1'f11'f2(cosh I71 cosh '12 -1) - m2 n2

(~r sinh 111 sinh 112

=

O.

(17)

Case 3. SS-C-SS-SS (18)

261

FREE VIBRATION OF RECTANGULAR PLATES

Case 4. SS-C-SS-F 111 I12P.2 - m 4 n 4 (1 - V)2] + 1/1112[..1,2 + m 4 n 4 (1 - v?] cosh 111 cosh 1'/2 +

+m2n2(~r [,12(l-2v)-m4n4(1-v)2]sinhl'/lsinh1'/2=0. Case 5. SS-SS-SS-F

22

22

1'/1[1l + m n (1 - V)]2 tanh 1'/2 - 112[1l - m n (1 - V)]2 tanh 171

(19)

=

O.

(20)

Case 6. SS-F-SS-F 21'/11'/2[,12 - m4 n4 (l- V)2]2(coshI'/1 cosh 1'/2 - 1) + {I7HIl+ m 2 n 2 (1-I'/~[Il-

m2 n 2 (l - V)]4} sinh III sinh 112 = O.

vw(21)

It is seen that equations (16)-(21) are of the same form as equations (8)-(13), the former being obtained from the latter by simply replacing sin, cos, and tan by sinh, cosh, and tanh, and
%vm

1'/2 =

-

b

a

vm

2

n2 -Il,

2

n2 + Il.

(22)

Because of the geometric symmetry which exists about the axis y = bl2 (i.e., "y-symmetry") in Cases 1,2, and 6, vibration modes in these cases will separate into ones which are either y-symmetric or y-antisymmetric. The characteristic equations corresponding to these modes can be obtained from equations (8), (9), (13), (16), (17) and (21) by factoring, or by new derivations in terms of a y' co-ordinate system having its origin in the middle of the plate (i.e., y' = y - b12) and, for example, retaining only the even functions of y in equation (7) for the symmetric modes having k 2 > a2 (1l > m 2 n 2 ). The resulting characteristic equations are the following. Case 1. SS-SS-SS-SS

cP1 cosh-= cP2 0 2 2 ' Il> m 2 n2 .
(23b)

1]1 h 1'/2 cosh-cos - =0

(23c)

symmetric:

cos -

symmetric: 2

Il < m n

2

2

{ antisymmetric:

2

'

. h -SIll III . h 1'/2 -=0

(23a)

22'

(23d)

cP1 + '1'2 rl. h cP2 2" tan 2"",0,

(24a)

rl. rl. cP2= 0, { antisymmetric : '1'2 tan -cP1 - '1'1 tanh -

(24b)

171 h 1'/2 III tanh "2 -1'/2 tan 2'= 0,

(24c)

SIn

Case 2. SS-C-SS-C symmetric:

Il> m2 n 2

rl.

'l'l tan

2

symmetric : 2

Il < m n

2

2

{ antisymmetric: 1'/2 tanh -171 - '11 tanh -112

2

2

=O.

(24d)

262

A. W. LEISSA

Case 6. SS-P-SS-F symmetric:

¢ d). -I- m 2 n 2(l - vWtan

~1-1-

-I- 4>2[.:l. -m 2 n 2 (l- v)]2tanh

antisymmetric: ¢2[). - m 2 n2 (1 - v)]2tan ¢1_ 2 -4>d). -I- m2 n 2 (l - v)]2 tanh symmetric:

1'/1

111[). -I- m 2 n2 (l - v)]2 tanh 2"

1J2[). - m 2 n2(1-

0,

(2Sa)

~2 =

0,

(2Sb)

=

0,

(25c)

2" =

0.

(25d)

-

- 1J2[.:l. - m 2 n 2(1 - V)]2 tanh

antisymmetric:

~2 =

'12

2

v)]2tanh 111 2 1J2 -1J1[.:l. + m2 n 2(l - v)]2 tanh

It is also seen that, for example, equations (24b) and (24d) are the same as equations (10) and

(18), respectively, except that 4>1 and 4>2 have been replaced by 4>1/2 and 4>2/2, respectively. The physical significance of this is that the y-antisymmetric modes of vibration (and the corresponding frequencies) of a SS-C-SS-C plate of width b are the same as those of a SS-C-SS-SS plate of width b12. This is because the conditions along the antisymmetry axis of a SS-C-SS-C plate are the same as conditions of a simple support. The same correspondence exists between equations (25b) and (25d) for thc SS·P-SS-F plate and equations (12) and (20) for the SS-SS-SS-P plate. 3. NATURAL FREQUENCIES OF PLATES HAVING TWO OPPOSITE EDGES SIMPLY-SUPPORTED

The characteristic equations presented in the preceding section were programmed and roots of the equations were found by using Newton's method. Numerical results for the nondimensional frequency parameter). = ma 2 y pi D were obtained for each of the six cases over a range of aspect ratios and their reciprocals (alb = 2'5, 1'5, 1'0, 2/3, 0'4). Poisson's ratio, where relevant (cases 4, 5, and 6), was taken uniformly to be 0·3. Numerical data for the six cases are presented in Appendix A. In each table, for each value of alb, the nine lowest values of ma 2 '\! pI D are displayed in increasing sequence. The results are exhibited in considerable accuracy simply because they were easily obtained to the accuracy given, and because they may be of worth to someone desiring to investigate the accuracy of an approximate method on some of these problems. In addition, for each eigenvalue presented, the corresponding mode shape is described by the number of half waves in each direction. Thus, for example, a 32-mode has three half-waves in the x-direction and two in the y-direction. For all six cases the wave forms are, of course, sine functions in the x-direction, according to equation (7). Furthermore, the wave forms in the y-direction are found to be sine functions exactly (i.e., Bm = em = Dm = 0) for the SS-SS-SS-SS case, whereas for the other cases the forms are only approximately sinusoidal. A consequence of this result is that the node lines lying in the y-direction (two for a 32-mode) will be exactly straight, parallel to the y-axis, and evenly spaced. On the other hand, those lying in the x-direction (one for a 32-mode), except for the SS-SS-8S-SS case, and except for an axis of symmetry, will not be exactly straight, parallel to the x-axis, or evenly spaced.

FREE VIBRATION OF RECTANGULAR PLATES

263

For purposes of subsequent discussion in this section, it will also be useful to clarify some terminology with respect to symmetry of modes. As already used in section 2, y-symmetric modes are those modes having an axis of symmetry with respect to the y co-ordinate (e.g., 11, 21, 13 modes are y-symmetric). Similarly, for example, the 12,22, and 14 modes are y-antisymmetric modes. Accordingly, these modes can exist only where double geometric symmetry is present (e.g., SS-SS-SS-SS, SS-C-SS-C, SS-F-SS-F). Of course, all the six cases discussed in this section can have vibration modes which are either x-symmetric or x-antisymmetric. Because the conditions along a straight nodal line are the same as those of a simplysupported straight edge, considerable additional results for other alb ratios can be gleaned from Tables Al to A6. For example, consider the SS-C-SS-C plate (Table A2) having alb = 1·5. The 21 mode has a nodal line along x = al2 and has J, = 78·9836. Considering only one-half of the plate in this mode, one then has a SS-C-SS-C plate with alb = 0'75 with A = (1/2)278'9836 = 19'7459 vibrating in the 11 mode. Similarly, the 31 mode for alb = 1·5 (with A = 123'1719) can be interpreted as the 11 mode for alb = 0'5, with J. = (/3)2123'1719 = 13'6858, or as the 21 mode for alb = 1·0 with A = (2/3)2123'1719 = 54'7431 (already given in the table). And the 41 and 51 modes for alb = 1·5 also yield the fundamental (11 mode) frequencies for plates having alb = 0·375 and 0·300. Considering the y-antisymmetric modes, one observes that the tables provide information for plates having other boundary conditions as well. Returning to alb = 1·5 in Table A2, one sees that the 12 mode corresponds to a II mode for a SS-C-SS-SS plate having alb = 3 and the same value of frequency parameter, A = 146·2677. (To accomplish these transformations simply, one keeps the length a fixed and varies b to arrive at the desired alb ratio. Thus J, for the 12 mode of a plate having b = (2/3) a is the same as that for the 11 mode of a plate having b = (lj3)a. Furthermore, for fixed values of a, p and D, J, is a direct measure of the frequency, w. Similarly, for example, the 32 mode for alb = 1·5 in Table A2 corresponds to the 11 mode of a SS-C-SS-SS plate having alb = I and J, = (/3)2212'8169 = 23'6463 (already given in Table A3). A word of caution should be mentioned regarding the procedure described in the preceding two paragraphs for obtaining additional information from the higher modes. In order for the correspondences described above to exist, the nodal lines must be straight, parallel to the xand y-axes, and evenly spaced. As discussed earlier in this section, the nodal lines lying in the x-direction satisfy these conditions in general only for Case 1, and in particular for the other cases only when the line is one of geometrical symmetry (y = bl2 in Cases 2 and 6). Thus, for the SS-C-SS-SS plate (Table A3) having alb = 1'0, A for the 22 mode is less than four times that of the II mode because the nodal line lying in the x-direction, while nearly straight, occurs at y > O· 5b. On the other hand, the frequency for the 22 mode of the SS-SS-SS-F plate (Table AS) for alb = 1 i!; more than four times that of the II mode because the x-directed nodal line is noticeably curved, thereby supplying additional circumferential stiffness to the plate in the vicinity of the nodal line. A useful analogy that exists (cf. [I, 10, 11]) between the vibration and buckling ofrectangular plates having two opposite edges simply-supported will be referred to from time to time later in this section. Specifically, when uniformly distributed, compressive force resultants N x (force per unit edge length of plate) act in the plane of the plate and perpendicular to the simply-supported boundaries x = 0 and x = a, then the following correspondence exists between the frequency parameter (in the absence of N x ) and the static buckling parameter (due to N x ):

2

{P

fNx

wa ...; 15 ~ mna ,J Ii'

(26)

264

A. W. LEISSA

Some specific comments will be given below for each of the six cases having two opposite sides simply-supported. 3.1. ss-SS-SS-SS In order for equation (8) to be satisfied it is necessary that cP! = mc, with integer values of n. Thus for this case (and only this case), the nondimensional frequency parameter can be determined explicitly; i.e.,

wa

2

A

= rc 2[m 2 +

n2(~r] (m,n = 1,2, ...)

(27)

and the mode shapes are the same as those of vibrating rectangular membranes. Numerical values for this case are listed in Table AI. From equation (27) it is clear that as alb - ? 0, OJa 2V pi D - ? m2n 2 and that as bla - ? 0, wb 2V pi D - ? n2 rc 2. In Appendix B it is shown that equation (16) for wa 2 VpID < m 2 rc 2 has no roots. 3.2. S8-C-SS-C The eigenvalues listed in Table A2 are calculated from equations (24a) and (24b). In Appendix B it is shown that equations (24c) and (24d) for wa 2 pi D < m 2 rc 2 have no roots. For the square plate (alb = 1) the same results were essentially also obtained by Iguchi [10], although he overlooked the 41 mode. Other extensive results are presented in references [12] and [13], and several other authors have solved this problem, some by approximate methods, as summarized in reference [l]. Additional numerical results for the y-antisymmetric modes of SS-C-SS-C plates may be easily obtained from the data presented for SS-C-SS-SS plates (see section 3.3), as discussed in section 3.

v

3.3.

S8-C-SS-S8

The frequency parameters listed in TabIe A3 are calculated from equation (10). In Appendix B it is shown that equation (18) for wa 2 VpID < m2 rc 2 has no roots. The lowest six frequencies of the square, as well as the fundamental frequencies for the five aspect ratios of Table A3, were also obtained to less accuracy in reference [10]. A few other references dealing with this problem are described in reference [1]. Additional results for vibration frequencies of SS-C-SS-SS plates can be found quite simply from the y-antisymmetric results for SS-C-SS-C plates given in Table A2, as discussed previously in section 3. Specifically, fundamental frequencies for SS-C-SS-SS plates having alb = 5,0, 3,0, 2'0, 5/3, 4/3, 0'80, 0'75, and 4/9 are thus obtained. 3.4. sS-C-SS-F The frequency parameters listed in Table A4 are calculated from equation (11), the value v = 0·3 being used. It is shown in Appendix B that roots to equation (19) can also exist for v = 0'3, provided thatmbla > 7·353. Thus,for alb = 1, the lowest frequency parameter of this type would be A8!' having a value of approximately 630 (see the discussion in section 3.6). The lowest six frequencies of the square were previously obtainr.:d in reference [12] and duplicate the corresponding values of Table A4. Additional numerical results are available for v = 0·25 from the plate buckling analogy (see section 3) and the data given in references [14] and [15], as well as in reference [1]. 3.5. sS-SS-SS-F Equation (12) is used with v = 0'3 to obtain the frequency parameters listed in Table A5. It is shown in Appendix B that roots to equation (20) can also exist for v = 0,3, provided that mbla> 7'228. Thus, for the square plate, the lowest frequency parameter which would be

FREE VIBRATION OF RECTANGULAR PLATES

265

encountered such that A < m2 n 2 would be ASi' Furthermore, as shown in section 3.6, this frequency parameter would be close to m 2 n 2 (~ 630), and would be far beyond the range of the table. Similarly, for alb = 0-4, the lowest frequency parameter for the case of A. < m 2 n 2 would be A3I ~ 88. The lowest six frequencies for the square were previously presented in reference [12] and duplicate the corresponding values in Table AS. Results which supplement those ofTable AS are easily obtained from the data for SS-P-SS-P plates given in Table AG. In particular, fundamental frequencies for SS-SS-SS-P plates having alb = 5'0, 3'0, 2'0, 5/3, 4/3, and 0·8 are obtained from the frequencies of the y-antisymmetric modes listed in Table A6. Numerical results are also available for v = 0·25 from the plate buckling analogy (see section 3) and the data given in references [14] and [15], as well as in reference [1]. 3.6. SS-F-SS-F The frequency parameters listed in Table A6 are calculated with v = 0'3 being used. Most of the results are for A > m 2 n 2 , equations (25a) and (25b) being used. However, as is shown in Appendix B, there always exists one y-symmetric mode for each value of m such that A < m 2 n2 , and, under rather restricted circumstances, y-antisymmetric modes arising from roots of equation (25d) can also be found such that A < m2 n 2 • In particular, for v = 0'3, it is seen in Appendix B that mbla must be greater than 14-455 in the latter case, and thus no frequencies of this type appear among the first nine for the values of alb used in Table A6. TABLE 1 Representativefrequency parameters A for the antisymmetric (n = 2) modes of SS-F-SS-F plates, with A < m 2 n2 (v = 0'3) alb

1·0 1·0 1'0 0'5 0·4 0·1 0'1 0·01

0·0690

m

15 14 20 15 15 5 10 1 1

mbla

15·0 14·0 20·0 37'5 37·5 50 100 100

14'49

m2 n 2

2220·66 1934·44 3947'84 2220·66 2220·66 246·74 986·96 9·8696 9·86960

A= wa2 VjjJD

2220·22 non-existent 3943·00 2216'75 2216·55 246·27 985·09 9·8509 9·86946

The lack of importance of the antisymmetric frequencies for A < m 2 n2 is seen further from Table 1. All the values, although less than m 2 n 2 , are also quite close to m 2 n 2 , the largest possible deviation occurring for extremely large mbla: i.e., for mbla = 100, A. differs from m 2 n 2 by approximately 0·2 %. It was shown in reference [12] that for these modes a limiting value of A= 0·99810 m2 n2 is reached for mbla = ro (when v = 0'3). The nearness to m 2 n 2 to which the eigenvalue can be forced is demonstrated by the value of mbla = 14·49 (>14'455) in the table. From the table it is also seen that the first eigenvalue of this type for the square plate would be reached when m = 15 (A = 2220'22). Prom Table A6 it is evident that approximately 200 free vibration frequencies of the other three types of modes would precede this one for alb = 1 ! It is interesting to note in Table A6 that the frequency parameters for the y-symmetric modes (11, 21, 31, etc.) having A < m 2 n 2 are the only ones that decrease as alb is increased. Nor do the values of A decrease with increasing alb for any fixed mode number in the five preceding Tables Al to AS. With increasing alb, the symmetric eigenvalues having.A. < m 2 n 2

266

A. W. LEISSA

decrease to a limiting value of m 2 n 2 '\1f="V2. The reason for this is that the frequency parameter Aitself contains ~ within D, and for large alb the plate behaves as a beam of length a simply-supported at both ends and undergoing anticlastic bending, the beam bending frequency being independent of Poisson's ratio. Thus the limiting case of large alb gives the beam frequency parameter of wa 2 v'12p1Eh 3 = m 2 n 2 • On the other end of the alb range for these modes, the eigenvalues for the 11 modes are A= 9·8351 and 9·8509 for alb = 0'1 and 0'01, respectively, approaching once more the limiting value of0'99810 m 2 n 2 as alb -+ O. The SS-F-SS-F case has received reasonable attention in the literature. The solution function (7) was used by Voigt [2] on this plate vibration problem in 1893, six years before the more widely recognized paper by Levy [16] proposing the same type of solution for static problems of plate bending. Another excellent piece of early work on this problem was by Zeissig [17] who plotted extensive results. In reference [12] the first six frequencies for the square are given for v = 0'3 and agree with those of Table A6. Extensive tabular data for v=0'16 and 0·3 were also presented by Jankovic [18]; however, the numerical results presented are somewhat lacking in accuracy. A more serious fault of the latter paper is the lack of recognition of the existence of solutions such that.:l. < m 2 n 2 • This error gave values of A = 11:2, (2n)2, (311:)2, ." for the 11,21, 31, ... modes/or all values o/a/b. Indeed, these values of .:I. are poles of equation (25a) rather than roots. 4. ON THE EFFECTS OF POISSON'S RATIO

For isotropic materials, Poisson's ratio (v) can vary between 0 and 0'5. However, it was seen above that the frequency parameter .:I. = OJa 2 v'pi D does not depend upon v unless one or more of the edges of the plate is free. Thus, for example, among the six cases discussed in section 3, in only three was A. a function of v. For these three cases, v was fixed at a value of 0·3. However, it must not be forgotten that D =Eh 3 /12(l- v2 ), and thus depends upon v; therefore, in every case the frequency itself depends upon v. For purposes of quantitative comparison, a frequency parameter Q is now defined which does not contain v, viz. Q

== wa 2

JT1J=~.

(28)

It is seen that Q is such that it equals A when v = O. Furthermore, in those cases where A dqes not depend upon v, then Q (and consequently the frequency) for v = 0'5 is 1'15 (i.e., v'4/3) times as great as Q for v = O. To determine further the effects ofv upon.:l. and Q, numerical results were also obtained for the SS-F-SS-F plate when other values of v were used. This case was chosen because (1) the presence of two free edges causes marked changes with v, (2) the y-antisymmetric modes correspond to the SS-SS-SS-F case which is itself y-asymmetric, and (3) the existence of eigenvalues is complicated, depending upon whether A. is greater or less than m 2 n 2 , as discussed in section 3. Table 2 gives values of A. for v = 0,0'3, and 0'5. Results are exhibited for alb = 0'4, 1, and 2'5 and for those modes corresponding to the first nine frequencies which exist for the square plate having v = 0·3 (as given in Table A6). As vis varied, the ordering ofthe modes can also change. For example, for alb = 0'4 it is seen that for v = 0'3, OJ16 < OJ 23 , but for v = 0'5, W23 < OJ16' However, this is the only instance of mode reordering found for the range of aspect ratios used in the table. The modes are also separated into categories which are either y-symmetric or y-antisymmetric and having either A> m 2 n 2 or A < m2 n2 • For the range of m and alb of the table,

267

FREE VIBRATION OF RECTANGULAR PLATES

only one y-antisymmetric mode was found for which A < m 2 n 2 ; it occurs where indicated for the 22 mode when alb = 0'4 and v = 0·5. As proven in Appendix B, these modes will exist if mbla> 4·051 for v = 0'5, and in this instance mbla = 5. The likelihood of these modes existing increases with increasing v. Another example of this type which occurs beyond the range of the table is fo'r v = 0'5 and alb = 0·1. Then Al2 = 9·6635 (mbla = 10, Au/n 2 = 0-9791) and A22 = 38·6228 (mbla = 20, A22I(2n)2 = 0'9783). TABLE 2

Frequency parameter A = wa 2 Yp/D as afunction ofv for SS-F-SS-F plates J. > m 2 rr 2

or Type of mode

A < m 2 n2

y-symmetric

v

a

b

A

mil 11

0

0'3

0'5

9·8696 39·4784

9·7600 39-2387

9·4506 38·3771

9'8696 39'4784 88·8264

9·6314 38'9450 87-9867

9·0793 37·5192 85·4899

21 31 41

9·8696 39·4784 88·8264 157·9137

9·4841 38·3629 86'9684 155'3211

8·7042 35-8799 82-5093 148·7256

13 15 23

15·7531 32'2511 45·7156

15·0626 31·1771 44·9416

14·1360 30·3335 41-0576

13 23

39·2281 74·7963

36·7256 70·7401

34·7783 66·8020

2·5

13 23

160·3527 211·9354

156'1248 199·8452

153·1979 190·8194

0·4

12 14 22 16

11·4098 22-6610 41·0521 44·7469

11'0368 21-7064 40·5035 43-6698

10·3901 20'8650 39-0826t 42·8705

12 14 32

17·8821 48·9147 77'5775 98,54·81

16-1348 46·7381 75·2834 96'0405

14·3516 43·4768 73'6610 91-1600

12 22 32

39·4631 84·9399 142'1102

33-6228 75'2037 130·3576

28·7628 66·0192 117'3358

0'4

21

11 21 31


11 2·5

0·4

>m2 n2

y-antisymmetric

>ml n2

22

2·5

t Obtained by using solution for A< m2 n2 • In Table 2 it is observed that A. decreases with increasing v for the SS-F-SS-F case for all a/b. Because the y-antisymmetric modes also contain all the modes of the SS-SS-SS-F case, the statement applies to the latter case as well. Indeed, reviewing the comprehensive literature survey of reference [1], only one mode ofone case (F-F-F-F) can be found wherein A increases with increasing v. In that case a nearly cirCUlar internal nodal line gives rise to large circumferential stiffening in the vicinity of the circle, thereby increasing the effect ofv. However, it must be noted that although 15 of the 21 cases ofsimple boundary conditions for rectangular plates have one or more free edges, yielding A.-dependence upon v, results in the literature for values of v other than 0·3 are quite sparse.

268

A. W. LEISSA TABLE 3 Variation of.A. with v for the 22 mode anda/b=O'4

v

A.

0·0

41·0521

0-1

40·9838

0'2

40·8155

0·3

40·5035

0·4

39'9713

0'5

39·0826

Difference -0'0683 -{H683 -0,3120 -0,5322 -0,8870

Frequency parameter Q

Type of mode

=

A> m2 7C2 or A. < m2 7C2

y-symmetric


TABLE 4 coa2V12p/Eh3 as a/unction ofv/or SS-F-SS-Fplates

A

F

b

mn

0·4

11 21

9·8696 39·4784

10'2313 41·1333

10·9126 44·3141

11 21 31

9·8696 39·4784 88·8264

10'0965 40·8255 92'2351

10·4839 43·3234 98'7152

2'5

11 21 31 41

9·8696 39'4784 88'8264 157'9137

9·9420 40·2152 91-1677 162·8208

10'0507 41·4305 95·2737 171·7335

0·4

13 15 23

15·7531 32·2511 45·7156

15·7899 32'6825 47'1116

16'5307 35'0261 47'4092

13 23

39·2281 74·7963

38·4989 74·1558

40·1585 77·1363

2'5

13 23

160·3527 211·9354

163'6633 209·4947

176·8977 220·3393

0·4

12 14 22 16

11·4098 22·6610 41·0521 44·7469

11'5697 22'7545 42·4592 45·7784

11·9975 24·0928 45'1287i 49·5026

1

12 22 14 32

17·8821 48·9147 77·5775 98·5481

16·9139 48·9948 78·9184 100'6778

16·5718 50·2027 85'0564 105·2625

2'5

12 22 32

39·4631 84·9399 142·1102

35'6228 78·8349 136·6519

33·2124 76·2324 135·4877

1

>m2 7C2

y-antisymmetric

>m2 n2

v

a

0

t Obtained by using solution for)" < m2 rc 2•

0·3

0'5

FREE VIBRATION OF RECTANGULAR PLATES

269

Not only does A. decrease with increasing v for all modes in Table 2, but the decrease is at an increasing rate. That is, curves of A us. v all have negative curvature everywhere. This behavior is demonstrated in Table 3 for the 22 mode and alb = 0·4. Therein, A is given for changes in v having equal increment size 0'1. It is seen that the negative change in A. between v = 0'4 and 0·5 is more than 13 times as great as between v = 0 and v = 0·1. For the y-symmetric modes indicated such that A < m2 n2 and for v = 0, it is seen that Ais precisely m2 n 2 , which corresponds to the frequency parameter of a beam simply-supported at both ends. Thus the plate behaves according to Euler-Bernoulli beam theory in this special case. The variation of the frequency parameter Q not containing v is displayed in Table 4 for the same alb ratios and mode shapes as in Table 2. Particularly interesting to note is that Q usually increases with increasing v for the y-symmetric modes, but either increases or decreases for the y-antisymmetric modes, and the tendency for Q to decrease with increasing v increases as alb increases. For the 13 and 23 modes of the square plate, the behavior is even more compli~ cated, Q first decreasing and then increasing with increasing v. A closer inspection of the 13 mode shows the following sequence of values for Q: 39'2281, 38'6400, 38'3887, 38'4989, 39'0406, 40'1585, corresponding to v = 0, 0'1, 0'2, 0,3, 0'4, 0'5, respectively. Frequency decrease with increasing v is a strange phenomenon, but it can be found to occur in at least one other case of boundary conditions, Specifically, from the very precise work of Sigillito [19] on the completely free square plate, this phenomenon can be seen occurring for the fundamental mode. Upon considering the fundamental (i.e., 11) mode, it is seen from Table 4 that Q decreases with increasing alb for all non-zero values of v, and approaches the simply-supported beam frequency Q = n2 as alb -+ oj. The curvature of the plate in the fundamental mode in the y-direction is very small for small alb, thereby generating a bending moment My which approximately equals vMx everywhere. For large alb, the free edges are relatively close together, and because My is zero on these edges, only a small value ofit can be generated over the small distance. Thus, the presence oflarge My for small alb causes considerable stiffening and corresponding frequency increase for the plate. This effect increases, of course, with increasing v. The same effects occur for the 21, 31, 41, etc., modes; however, for the other modes one or more nodal lines lie in the x-direction, and the picture is then complicated by the presence of curved nodal lines (for which My '# 0) and internal shearing forces, Qy. 5. PLATES HAVING OTHER EDGE CONDITIONS For the other 15 cases of edge conditions not having two opposite sides simply-supported, the classical Rayleigh-Ritz method was used with beam functions to obtain numerical results for the frequency parameters, A.. This procedure is very well known (cf. references [1, 7, 20-28]) and will not be described in detail again here. Suffice it to say that the method uses functions W(x,y) in equation (6) in the variables separable form,

W(x,y) =

2: ApqXp(x) Yiy),

p,q

(29)

where X p and Yq are normalized eigenfunctions exactly satisfying the equation of motion of a freely vibrating, uniform beam. In addition, X p and Yq satisfy desired clamped, simplysupported, or free edge conditions at the ends of the beam. The coefficients are determined by the Ritz method so as to minimize an energy functional and thereby yield a best approximation to the satisfaction of the equation of motion (1) for the plate. Clamped and simplysupported plate boundary conditions are exactly satisfied by use of the beam functions, but free edge conditions are only approximated, making the approach usually less accurate when

A. W. LEISSA 270 a free edge is involved. Finally, in this short summary of the method, the orthogonality of the beam functions and the consequent saving in numerical computational labor should be pointed out. For each of the 15 problems considered here, the first six beam functions were used in each co-ordinate direction, yielding 36 terms on the right-hand side ofequation (29). In the general case the procedure then reduces to the evaluation of eigenvalues of a 36th order determinant. However, in cases where one geometric symmetry axis is present (e.g., C-C-C-SS, C-C-C-F, C-SS-C-F, C-F-SS-F, C-F-F-F, SS-F-F-F), all modes are either symmetric or antisymmetric with respect to the geometric symmetry axis, and the determinant is uncoupled into two 18th order determinants. Similarly, in cases having two geometric symmetry axes (C-C-C-C, C-F-C-F, F-F-F-F), four symmetry classes of modes exist, yielding four ninth-order eigenvalue determinants. However, no effort was made to separate further the fourfold symmetric modes of the square plate having C-C-C-C or F-F-F-F edges. For several of the cases, more than the first six frequencies, particularly for alb = 1, can be found elsewhere in the literature. On the other hand, for six ofthe cases (C-C.SS-F, C-SS-C-F, C-8S-SS-F, C-SS-F-F, S8-8S-F-F, S8-F-F-F) no explicit previous results have been found. Numerical results for the 15 cases are displayed in the tables of Appendix C. Therein the six lowest values of the non-dimensional frequency parameter A. = C1Ja 2 V pI D are given for alb = 0'4,2/3, 1'0, 1'5, and 2'5 in each case. Poisson's ratio, which is an independent parameter in the 12 cases involving a free edge, is taken uniformly as 0·3 in these tables. Again the edge condition notation defined in section I is stressed. That is, for example, the C-SS-F-F plate has the edge x = 0 clamped. The other edges in counterclockwise order are then simply-supported (y = 0), free (x = a) and free (y = b). In the tables the identifying mode number mn associated with each frequency is also given. The mode number is determined by the largest A pq in the eigenvector associated with the frequency. In many cases the resulting nodal patterns have m - 1 and n - 1 nodal lines running parallel to, or approximately parallel to, the y- and x-axes, respectively, but in other cases this geometrical correspondence is at best vague (cf. reference [1]). Where symmetrical and antisymmetrical modes do exist, they can be identified by the mode numbers. Also, in some special cases of square plates having diagonal symmetry in their edge conditions, it is found that A pq = A qp or A pq = -A qp for all p and q, indicating diagonal symmetry or antisymmetry in the mode shapes. For each entry in the tables the per cent difference of the 36-term solution eigenfrequency from the single-term Rayleigh solution is also given. The single-term solution is based upon using the one dominant beam function for the mode and determining the frequency by means of Rayleigh's Quotient, inasmuch as no minimization is possible with a single beam function. This procedure is the basis of Warburton's [8] useful formulas. In the more general Ritz procedure, the single-term Rayleigh solution conveniently appears as the dominant element on the diagonal of the characteristic determinant generated. In the tables, values of A. are given to five significant figures. However, it should not be thought that these values are exact to this degree of accuracy. Indeed, in some scattered instances, particularly for alb = 1, more exact numerical results can be found elsewhere in the published literature, as will be pointed out in detail in the discussion of the individual cases later in this section. From a rigorous mathematical standpoint the only statement that can be made concerning the relation of the Ato the exact values is that the present values are upper bounds for the exact values. However, from rate ofconvergence studies and comparison with known lower bounds from the literature, it is suggested that the Agiven in the tables are ordinarily exact to three significant figures. It is also known that the exactness decreases ordinarily with increasing mode number and with the presence offree edges, as will be shown later.

FREE VIBRATION OF RECTANGULAR PLATES

271

A detailed inspection of the tables of Appendix C reveals that as often as not the difference between the one-term and 36-term solutions is less than 1 %. Furthermore, the difference appears to be less for the stiffer cases (Le., those having the larger fundamental frequencies), and to increase as free edges are added. Isolated examples of large differences are 24,36, 17·45 and 11'56 %for the l2mode ofC-F-F-Fplates havinga/b = 2,5, 1'5, and 1, respectively; 14'77% for the 11 mode ofa C-SS-F-F plate havinga/b = 2'5; and n06%and minus 8·43 % for the 13 and 31 modes, respectively, of a F-F-F-F plate having a/b = 1. The presence of a free corner appears especially to detract from the exactness of the one-term solutions. To study the question of exactness somewhat more quantitatively, Table 5 has been prepared, which gives the average per cent differences for the 30 values of A, available for each of the 15 cases. Negative differences in the tables were added algebraically so as to diminish the average differences. Table 5 shows that the average difference for the e-e-e-e case is the smallest and that, as the constraints are relaxed, ordinarily (1) changing a clamped edge to a simply-supported one increases the difference slightly, (2) changing a simply-supported edge to a free one increases the difference considerably, and (3) the intersection of two free edges (Le., a free corner) causes large per cent differences. The overall average of differences for all 15 cases is seen to be somewhat less than 2 %, and the mean difference is a little less than the average. TABLE

5

Average differences in A between one-term and 36-term solutions Case

c-c-c-c c-c-c-SS c-c-c-p c-c-SS-SS c-c-p-SS c-c-p-p C-SS-C-P C-SS-SS-F C-SS-P-F c-p-c-p c-p-ss-p c-p-p-p SS-Ss-p-p Ss-p-p-p p-p-p-p

Average of averages

Per cent difference 0·38 0·41 0·91 0·46 1·10

4·27 0'54 0·61 3-80 0'58

0·77 4-86 2·93 HI 3·51 1·90

The significance of a negative error in the tables of Appendix C is particularly interesting inasmuch as it denotes that the one-term Rayleigh eigenvalue is smaller than the 36-term eigenvalue by the indicated per cent. And because both solutions are upper bounds for the exact eigenvalues, a negative error would appear to indicate that the one-term solution is more exact than the 36-term solution in such instances! It is not widely known that the a~dition of terms to a Rayleigh-Ritz formulation can decrease the accuracy of some of the eigenvalues. As a simple numerical illustration consider the first two symmetric modes of the C-C-C-F square plate. If only the first two x-symmetric

272

A. W. LEISSA

terms of equation (29) are retained (i.e., All and A 12 being the only non-zero coefficients), the following characteristic determinant is generated: 586,64 - ,1,2 -90,68 1-90,68 1642·56 -.P

1=0.

(30)

It has the roots A.ll = 24'07 and ..1 12 = 40·61. The one-term Rayleigh solutions are obtained directly from the diagonal elements of equation (30), yielding All = 24,22 and ..1 12 = 40,53. Thus, the two-term solution gives a more exact upper bound on the fundamental frequency., but also makes the A12 approximation worse. Accurate values of All and ..112 are seen in Table C3 to be 24'020 and 40'039, respectively. When the principle described above is applied to the ij mode, the terms preceding Al} in equation (29) cause Al} to increase above the value obtained from the Ai) term taken alone, although approximations to the other eigenvalues are then also obtained. The addition of terms of equation (29) such that p > i and q > j will then decrease (i.e., improve the exactness of) AiJ. Because the modes of each symmetry class become uncoupled in the solution, the lowest frequency of each symmetry class will never be increased by the addition of terms in equation (29). However, diagonal symmetry is also present for many cases when alb = 1 (i.e., C-C-C-C, C-C-SS-SS, C-C-F-F, SS-SS-F-F, P-F-P-P) and the one-term solutions in the co-ordinate system do not recognize this added symmetry. Thus, for example, the modes labeled 13 and 31 for the F-F-F-P square plate (see Table CIS) are actually doubly antisymmetric and doubly symmetric with respect to the diagonals and are both being represented by one-term eigenfunctions not possessing this symmetry. Hence, the frequency of the oneterm solution for the 31 mode (8'43 %less than the 36-term solution) need not be an upper bound. The inadequacy of the single-term solutions when diagonal symmetry is present was pointed out by Warburton [8]. TABLE 6 Boundary condition identities

Antisymmetric modes of

General modes of

C-C-C-C C-C-C-SS C-C-C-F C-F-C-F C-SS-C-F C-F-F-F SS-F-F-F F-F-F-F

C-C-C-SS and C-C-SS-SS C-C-SS-SS C-C-SS-F C-F-SS-F, C-SS-C-F and C-SS-SS-F C-SS-SS-F C-SS-F-F SS-SS-F-F SS-F-F-F and SS-SS-F-F

Straight nodal lines duplicate simply-supported boundary conditions; consequently, additional frequency parameters, particularly for other aspect ratios, can often be obtained by considering the antisymmetric modes of other cases. These correspondences are summarized in Table 6. Thus, for example, some values of A for C-C-C-SS and C-C-SS-SS plates can be found from the frequencies of C-C-C-C plates, and conversely. This procedure is described in detail in section 3. For either very large or very small values of alb, one set ofopposite edges is widely separated and the other set is relatively close. In such cases the plate frequencies are related to the beam frequencies. The frequency parameters for beams having length 1are given in Table 7, where EI is the beam stiffness. For a beam strip ofthickness It and unit width, 1= h3 112 and the beam parameter contains the same quantities as the plate parameter !J [see equation (28)], with a

FREE VIBRATION OF RECTANGULAR PLATES

273

substituted for I. However. in most cases the values ojTable 7 are the limiting values ojthe plate parameter A, rather than Q. This is due to the added stiffening in a plate due to Poisson ratio effects, as discussed in section 4. Aithough the plate parameters Aapproach the beam parameters of Table 7 for large and small values of alb, they do so at different rates for the various edge conditions, as can be seen in Table 8. Therein values ofthejundamental frequency parameter All are collated in descending order for the six cases where two opposite edges x = 0 and a are clamped. Although the plate is 2·5 times as wide as it is long, the edge conditions at y = 0 and b are still quite sufficient to raise All significantly above the beam parameter of 22·373 in every case except the last. In all six cases (with the possible exception of the last) Au -+ 22'373 as alb -+ O. In the table it is also seen that the removal of the side constraints of deflection and slope reduces Au in every case, and that the constraints of deflection are more significant than those of slope. TABLE 7 Frequency parameters w}2 Vp/Eljor beams C-C

c-ss

C-F

Ss-ss

SS-F

F-F

22·373 61·673 120·903 199·859 298·556

15·418 49·965 104·248 178'270 272'031

3·5160 22·034 61-697 120·902 199'860

9,8696 39·478 88'826 157-914 246·740

same as C-SS

same as C-C

m

1 2 3 4 5 >5

(2m

+ l?nz/4

(4m

+ 1)ZnZj16

(2m - l)2n Zj4

m2 nZ

TABLE 8 211 jor alb = 0·4 Edge conditions C-C-C-C C-C-C-SS C-SS-C-SS C-C-C-F C-SS-C-F C-F-C-F

23·648 230440 23·277 22'577 22·544 220346

In Table 8 it is also seen that 211 for the C-F-C-F case is less than the fundamental beam parameter value of 22·373. And in Table ClO All is found to be less than 22·373 for all alb, and decreases with increasing alb. Similarly, A.2l and A31 are found to be less than 61·673 and 120'903, respectively, for all alb, and decreasing with increasing alb. This behavior was seen previously for another case when two opposite edges are free: the SS-F-SS-F case, as discussed in section 3.6. Upon looking further it is found that this behavior occurs for these modes in all cases when two opposite sides are free: that is, C-F-SS-F (Table Cl1), C-F-F-F (Table e12), SS-F-F-F (Table C14) and F-F-F-F (Table CI5). Whether the plate parameters for these modes approach a limiting value somewhat less than the corresponding beam parameters as alb -+ 0, as occurred for the SS-F-SS-F case (see section 3.6) remains to be determined. However, it is clear that, as for the SS-F-SS-F case discussed earlier, as alb -+ 00 the A for these modes approach the beam frequencies multiplied by vT="V2 (= 0·95394 for v = 0'3), because My bending moments cannot effectively be generated for free edges and small b. Some observations and, where fruitful, comparisons with other numerical results in the published literature will be made in detail below for several of the 15 cases.

274

A. W. LEISSA

5.1. c-c-c-c (Table Cl) Because of the relative mathematical simplicity of its boundary conditions, the literature for the C-C-C-C plate is extensive. Reference [1] identifies 37 sources which deal explicitly with this problem. Relevant to the present paper, references giving lower bounds or more accurate upper bounds for the frequency parameters are particularly worth noting. Lower bound references include [29, 30, 31, 32]. From these four sources closest lower bounds for the square (alb = 1) of A = 35'986, 73'354, 108'12, 131·55 and 132'18 can be extracted for comparison with Table Cl. Upper bounds of A. = 35'9866, 131'58, 132·21 for alb = 1 which are closer than those of Table Cl can be found in reference [29]. Reference [29] is particularly outstanding in this respect, giving close upper and lower bounds for the first 15 doubly symmetric modes for a range of 0·125 < alb < 1. The extensive, accurate numerical results of reference [33], although not bounds, should also be mentioned here. The existence of distinct" 13" and "31" modes for the square having distinct eigenvalues (i.e., 131·64 and 132'24) is still a matter of speculation at this time. Some authors report them as distinct, others do not. A proof of whether the modes are distinct, or whether they simply reflect numerical approximations, is not yet available. Fortunately, this troublesome point only has mathematical, and not practical, significance. 5.2.

C-C-C-F

(Table C3)

Reference [24] is 'the only known previous work dealing with this case. However, it is devoted solely to this problem and therefore deals thoroughly with it. The eigenvalues presented in reference [24] are also obtained by using the Ritz method with beam functions for v = 0·3. Results are given for alb = 0'5,0'75, 1·0, 1'5, 2·0 rather than those of Table C3, so direct comparisons can only be made for two of these values. Using ten beam functions in both the x- and y-directions, thereby giving 100 terms in equation (29) was found to yield A = 24·00, 40'03, 63·41, 76·73 and 80·70 for alb = 1 which are, of course, closer upper bounds than those of Table C3. Similarly, using eight beam functions in the x-direction and seven in the y-direction gave A = 26'72,65'89,66'19, 106'77 and 125'34 for alb = 1·5. 5.3.

C-C-F-F

(Table C6)

All the modes for the square in this case are either symmetrical or antisymmetrical with respect to one diagonal. For example, the first five frequencies are symmetrical, antisymmetrical (A 12 = -A 21 ), symmetrical (A 12 = A 21 ), symmetrical, and antisymmetrical (A 13 = -A31 ), in that order. To demonstrate further the rate of convergence of the Ritz method when using beam functions, it can be mentioned that Young [20] used three beam functions in each direction, yielding ninth order determinants and A. = 6'958, 24'80, 26'80, 48·05 and 63·14 for alb = 1 and v = 0'3. Upon comparing with Table C6 it is then seen thatA for the 12 modefrom using only nine terms is even further removed (i.e., a larger negative difference) from the one-term eigenvalue than the 36-term result. This is another example of the debilitating effect oflower terms in a Ritz solution as discussed earlier in this section. 5.4.

C-F-C-F

(Table CI0)

Extensive work for this case was done by Claassen and Thorne [33] who used an interesting, direct approach to the problem. The double sine series for SS-SS-SS-SS plates was used along with additional edge and corner functions to obtain the necessary boundary periodicities. Accurate results for A are given in reference [33] which are lower than those of Table CIO but, unfortunately, no claim for bounds can be made for the former.

FREE VIBRATION OF RECTANGULAR PLATES

5.5.

275

(Table C12)

C-F-F-F

Considerable analytical and experimental results are available for this case from many literature sources, as summarized in reference [1]. Eighteen-term Ritz solutions with beam functions were used in references [20, 21, 34] with v = 0·3. For comparison purposes, values of A = 3'494,8'547,21'44,27'46 and 31·17 were given for alb = 1, which are only slightly less exact than those of Table C12. Bazley, Fox and Stadter [35] used 50-term Ritz solutions with beam functions, as well as a method giving lower bounds to obtain extensive results for the symmetric (only) modes and v = 0'3. Direct comparison with Table C12 is only possible for alb = 1'0, for which the lower bounds given in reference [35] are 3'4305, 20'874, 26·501 and 51'502. Sigillito [19] used this case to demonstrate the improved upper bound convergence possible when using functions in equation (29) which are the products of beam functions and Legendre functions. He thereby showed that a 36-term solution gave A= 3'4729,21'304, 27·291 and 54'262 for the symmetric modes of alb = 1. The abundant numerical results of references [36] and [37] should also be mentioned here. It is somewhat interesting to note that although the AIj increase with increasing alb for all ij except for j = 1, and that All decreases monotonically with increasing alb as for the SS-P-SS-P case (see the discussion in section 3.6), that the behavior of A21 is erratic. This erratic behavior of the 21 frequency compared with the 11 frequency is also seenin the precise results of references [19] and [35] for both lower and upper bounds, although the 31 mode behaves monotonically like the 11 frequency. The inexactitude of the one-term solution for the first antisymmetric mode (1 12 ) in Table Cl2 (e.g., 24·36 %for alb = 2'5) is partially the result of the rudimentary beam function Y2(Y) representing rigid rotation about the axis Y = b12. This simple mode shape by itself represents excessive stiffness in the system. However, because the error in the one-term solution appears to be increasing without bound as alb increases, it would seem that gross lack of satisfaction of the free edge boundary conditions is the primary source of the error. This difficulty is seen to occur for the A2 2 mode and also for the 11 mode of the C-SS-P-F plate (Table C9). 5.6.

SS-SS-F-F

(Table C13)

Because the mode shape of the fundamental single-term solution (i.e., the product of two straight lines) is the deflected shape of the plate with apoint load acting at the free corner [38], rather than a distributed inertia load, the equation ofmotion is poorly satisfied by the function. The lack of satisfaction of the free edge conditions by this simple function as discussed in section 5.S also applies here. These two limitations combine to yield relatively poor accuracy for the single-term Au. 5.7.

SS-F-F-F

(Table C14)

The erratic (i.e., non-monotonic) behavior of interest in Table C14. 5.8.

F-F-F-F

A21

and

A31

with increasing alb is of some

(Table CIS)

In terms of ease in obtaining accurate analytical solutions, this is no doubt the most poorly behaved of all 21 cases of rectangular plates. The difficulty is delineated partly by (1) the presence of free edges and free corners, (2) the presence of additional symmetry (and the increased confusion of identifying modes) for the square, and (3) the fact that the difficulties of the SS-P-P-P and SS-SS-F-P cases are also inherent as parts of the overall problem (see Table 6).

276

A. W. LEISSA

The confusion in the literature concerning the existence of modes for this problem (and the shapes of the corresponding nodal patterns) is readily seen in the summarization laid out in reference [I]. Frequently certain vibration modes are not discovered in the analyses. Particularly noteworthy is the work of references [28] and [19] where accurate upper and lower bounds for the doubly antisymmetric modes (corresponding to m and n both even in Table C15) for v = O' 3 are presented, although no effort is made to identify the mode shapes. For alb = 1·0 upper and lower bounds for A of 13'464, 69'576, 76'904 and 13'092, 66'508, 75·146 from references [28] and [19] can be compared with the upper bounds of 13·489 (A 22 ), 69'7620'24), and 77'825 (A42) of the present work. Similarly, for alb = 2/3, bounds of 8'9351, 38,294,66,965 and 8'6667, 36,651, 64·844 can be compared with 8'9459, 38'434, 67'287. ACKNOWLEDGMENTS

The author wishes to acknowledge the painstaking work of Donald Simons and Adel Kadi, Who approximately three years before the author could find the time to write this paper, wrote the computer programs and obtained the numerical results presented in Appendixes A and C. Without their efforts this paper would have been impossible to contemplate. REFERENCES 1. A. W. LEISSA 1969 NASA SP-160. Vibration of plates.

2. W. VOIGT 1893 Nachr. Ges. Wiss. (Gottingen), no. 6, 225-230. Bemerkungen zu dem Problem der 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

transversalen Schwingungen rechteckiger Platten. E. F. F. CHLADNI 1787 Entdeckungen aber die Theorie des Klanges. Leipzig. E. F. F. CHLADNI 1802 Die Akustik. Leipzig. E. F. F. CHLADNI1825 Annalen der Physik, Leipzig 5,345. E. F. F. CHLADNI 1817 Neue Beitriige zur Akustik. Leipzig. W. RITZ 1909 Annalen der Physik 28, 737-786. Theorie der Transversalschwingungen einer quadratischen Platte mit freien Randern. G. B. WARBURTON 1954Proceedings o/the Institute 0/Mechanical Engineers, ser. A, 168, 371-384. The vibration of rectangular plates. R. JANICH 1962 Die Bautechnik 3,93-99. Die naherungsweise Berechnung der Eigenfrequenzen von rechteckigen Platten bei verschiedenen Randbedingungen. S. IGUCHI 1938 Memorandum o/the Faculty 0/ Engineering, Hokkaido University, 305-372. Die Eigenwertprobleme fUr die elastische rechteckige Platte. H. LURIE 1951 Journal of Aeronautical Sciences 18, 139-140. Vibrations of rectangular plates. H. J. FLETCHER, N. WOODFIELD and K. LARSEN 1956 Brigham Young University. Contract DA-04-495-0RD-560 (CFSTI No. AD 107 224). Natural frequencies of plates with opposite edges supported. S. T. A. ODMAN 1955 Proceedings NR 24, Swedish Cement and Concrete Research Institute, Royal Institute of Technology (Stockholm), 7-62. Studies of boundary value problems. Part II. Characteristic functions of rectangular plates. S. TIMOSHENKO and J. M. GERE 1961 Theory 0/ Elastic Stability. New York: McGraw-Hill Book Co., Inc. A. S. VOLMIR 1963 Stability a/Elastic Systems. Moscow: Gos. Izd. Phys.-Mat. Lit. (In Russian.) M. LEVY 1899 Comptes rendues 129,535-539. C. ZEISSIG 1898 Annalen der Physik 64,361-397. Ein einfacher Fall der transversalen Schwingungen einer rechteckigen elastischen Platte. V. JANKOVIC 1964 Stavebnicky Casopis 12, 360-365. The solution of the frequency equation of plates using digital computers. (In Czech.) V. G. SIGILLITO 1965 Applied Physics Laboratory, The Johns Hopkins University, Engineering Memorandum EM-40l2. Improved upper bounds for frequencies of rectangular free and cantilever plates. D. YOUNG 1950 Journal 0/ Applied Mechanics 17, 448-453. Vibration of rectangular plates by the Ritz method.

FREE VIBRATION OF RECTANGULAR PLATES

277

21. M. V. BARTON 1951 Journal ofAppliedMechanics 18, 129-134. Vibration of rectangular and skew cantilever plates. 22. R. P. FELGAR 1950 University of Texas Circular No. 14. Formulas for integrals containing characteristic functions of a vibrating beam. 23. V. S. GONTKEVICH1964 in Natural Vibrations ofPlates and Shells (A. P. Filippov, ed.). Kiev: Nauk. Durnka. (Translated by Lockheed Missiles & Space Co., Sunnyvale, California.) 24. G. F. ELSBERND and A. W. LEISSA 1970 Developments in Theoretical and Applied Mechanics, 19-28. Free vibration of a rectangular plate clamped on three edges and free on a fourth edge. 25. E. M. FORSYTH and G. B. WARBURTON 1960 Journal of Mechanical Engineering Science 2,325330. Transient vibration of rectangular plates. 26. A. LEMKE 1928 Annalen der Physik 4, ser. 86, 717-750. Experimentelle Untersuchungen zur W. Ritzschen Theorie der Transversalschwingungen quadratischer Platten. 27. D. A. SIMONS and A. W. LEISSA 1971 Journal of Sound and Vibration 17,407--422. Vibrations of rectangular cantilever plates subjected to in-plane acceleration loads. 28. N. W. BAZLEY, D. W. Fox and J. T. STADTER 1965 AppliedPhysics Laboratory, The Johns Hopkins University, Technical Memorandum TG·707. Upper and lower bounds for the frequencies of rectangular free plates. 29. N. W. BAZLEY, D. W. Fox and J. T. STADTER 1965 AppliedPhysicsLaboratory, The Johns Hopkins University, Technical Memorandum TG-626. Upper and lower bounds for the frequencies of rectangular clamped plates. 30. N. ARONSZAJN 1950 Oklahoma A. and M. College, Stillwater, Oklahoma, Technical Report No.3, Project NR 041,090. The Rayleigh-Ritz and the Weinstein methods for approximation of eigenvalues-III: Application of Weinstein's method with an auxiliary problem of type 1. 31. S. TOMOTIKA 1936 Philosophical Magazine 21, 745-760. The transverse vibration of a square plate clamped at four edges. 32. S. TOMOTIKA 1935 Aeronautical Research Institute Report, Tokyo University 10, 301. On the transverse vibration of a square plate with clamped edges. 33. R. W. CLAASSEN and C. J. THORNE 1960 U.S. Naval Ordnance Test Station, China Lake, California, NOTS Tech. Pub. 2379, NAVWEPS Rept. 7016. Transverse vibrations of thin rectangular isotropic plates. (Errata available from CFSTI as AD 245 000.) 34. M. V. BARTON 1949 Defense Research Laboratory, University ofTexas Report DRL·222, CM 570. Free vibration characteristics of cantilever plates. 35. N. W. BAZLEY, D. W. Fox and J. T. STADTER 1965 AppliedPhysicsLaboratory, The Johns Hopkins University, Technical Memorandum TO-70S. Upper and lower bounds for frequencies of rectangular cantilever plates. 36. R. W. CLAASSEN and C. J. THORNE 1962 Pacific Missile Range, Technical ReportPMR-TR-61-1. Vibrations of a rectangular cantilever plate. 37. R. W. CLAASSEN and C. J. THORNE 1962 Journal ofAerospace Science 29, 1300-1305. Vibrations of a rectangular cantilever plate. 38. A. W. LEISSA and F. W. NIEDENFUHR 1963 American Institute of Aeronautics and Astronautics Journal 1, 116-120. Bending of a square plate with two adjacent edges free and the others clamped or simply supported.

278

A. W. LEISSA

APPENDIX A TABULATED DATA FOR PLATES HAVING TWO OPPOSITE SIDES SIMPLY-SUPPORTED TABLE Al Frequency parameters A. = wa2 .y pjD for SS-SS-SS-SS plates

alb Mode sequence

1 2 3

4

5 6

A

0·4

11 11·4487 12 16·1862 13 24·0818 14 35-1358 21 41·0576 22 45-7950 15 49'3480 23 53'6906 16 66'7185

7 8 9

2/3

1'0

1'5

2-5

11 14-2561 12 27·4156 21 43-8649 13 49·3480 22 57·0244

11 19·7392 21 49-3480 12 49·3480 22 78·9568 31 98'6960

11 32·0762 21 61-6850 12 98·6960 31 111·0330 22 128·3049

23 78'9568 14 80-0535 31 93'2129 32 106·3724

13 98-6960 32 128·3049 23 128·3049

32 177-6529 41 180·1203 13 209·7291

11 71·5564 21 101·1634 31 150'5115 41 219·5987 12 256·6097 22 286-2185 51 308·4251 32 335-5665

41 167·7833

23 239·3379

61 416·9908

TABLEA2

Frequency parameters A. = wa2 y pjD for SS-C-SS-C plates alb Mode sequence

A.

0·4

2/3

1·0

1'5

2-5

1

11 12-1347

11 17-3730

11 28-9509

11 56-3481

11 145-4839

2

12 18·3647

12 35·3445 21 45·4294 13 62·0544 22 62'3131 23 88'8047 31 94'2131 14 97·4254

21 54-7431 12 69·3270 22 94·5853 31 102'2162 13 129·0955

21 78·9836 31 123-1719 12 146·2677 22 170·1112 41 189·1219

21 164·7387 31 202·2271 41 261·1053 51 342·1442 12 392·8746

32 140'2045 23 154'7757

32 212-8169 51 276·0012

22 415-6906 61 444-9682

32 101·0788

41 170·3465

42 276'0125

32 455·3054

3 4 5 6 7 8 9

13 27-9657 14 40-7500 21 41·3782 22 47·0009 23 56'1782 15 56'6756 24 68·7486

279

FREE VIBRATION OF RECTANGULAR PLATES TABLEA3

Frequency parameters A =

OJa 2 y' pfD for

SS-C-SS-SS plates

alb

Mode sequence

0·4

2/3

1'0

1·5

2·5

1

11 11·7502

11 15'5783

11 23·6463

11 42·5278

103-9227

2

12 17·1872

12 31·0724

21 51-6743

21 69'0031

21 128'3382

3

13 25·9171

21 44·5644

12 58·6464

31 116·2671

31 172-3804

4

14 37-8317

13 55-3926

22 86·1345

12 120·9956

41 237·2502

5

21 41·2070

22 59-4627

31 100·2698

22 147-6353

12 320·7921

6

22 46'3620

23 83-6060

13 113·2281

41 184'1006

51 322-9642

7

15 52·9007

14 88·4384

32 133·7910

32 193'8025

22 346·7382

8

23 54'8720

31 93-6758

23 140·8456

13 243-4964

32 391·0659

9

24 66·6637

32 108'1069

41 168·9585

42 260·2020

61 429'2420

A

11

TABLEA4

Frequency parameters A = OJa2 y' pfDfor SS-C-SS-F plates alb Mode sequence

A

0·4

2/3

11 10·1888

11 10·9752

11 12·6874

16·8225

11 30·6277

2

12 13-6036

12 20·3355

12 33·0651

21 45·3024

21 58·0804

3

13 20·0971

13 37-9552

21 41·7019

12 61·0178

31 105·5470

4

14 29'6219

21 40'2717

22 63·0148

22 92'3073

12 149-4569

5

21 3%382

22 49·7317

13 72-3976

31 93·8293

41 173·1060

6

15 42'2425

14 64'1889

31 90·6114

32 141·7834

22 182-8110

7

22 42-9993

23 67-8993

23 103·1617

13 149·6055

32 235·0155

8

23 49·5740

31 89·3571

32 111'8964

41 162'2413

51 260·6371

9

16 58·0019

24 94·5150

14 131·4287

23 181·1868

42 305'2218

1·0

1'5

11

2·5

280

A. W. LEISSA TABLEA5

Frequency parameters A. = wa 2 y pJD for SS-SS-SS-F plates alb Mode sequence

0·4

2/3

1·0

1·5

2'5

1

11 10·1259

11 10·6712

11 11'6845

11 13-7111

11 18'8009

12

12 18·2995

12 27'7563

21 43·5723

21 50'5405

21 41·1967 22 59·0655

12 47-8571 22 81·4789

31 100·2321 12 110·2259

A.

2

13·0570

3

13 18·8390

4

14 27·5580

13 33·6974 21 40·1307

5

15 39-3377

22 48·4082

13 61·8606

31 92'6925

22 147'6317

6

21 39'6118

14 57·5929

31 90·2941

13 124'5635

41 169·1026

7

22 42·6964

23 64'7281

23 94·4837

32 132-8974

32 203-7304

8

23 48·7745

24 89·1859

32 108·9185

23 158·9180

51 257'4791

9

16 54'2497

31 89·2725

14 115'6857

41 161·4205

42 277·4280

TABLEA6

Frequency parameters A. = wa 2 y'pJD for SS-F-SS-F plates alb Mode sequence

A

r

0·4

2/3

1·0

1

11 9·7600

11 9'6983

11

11

9'6314

9'5582

11 9·4841

2

12 11·0368

12 12·9813

12 16·1348

12 21·6192

12 33-6228

13 15'0626 14 21·7064

13 22·9535

13 36·7256

21 38·3629

21 39'1052

21 38·9450

21 38'7214 22 54·8443

5

15 31·1771

14 40·3560

22 46·7381

13 65·7922

31 86·9684

6

21 39·2387

22 42·6847

23 70·7401

31 87-6262

32 130'3576

7

22 40·5035

23 54·2400

14 75·2834

23 103-9665

41 155'3211

16 43·6698 23 44·9416

15 66·2301 24 73·1982

31 87-9867 32 96-0405

32 105'1608 14 152·7784

13 156·1248 23 199·8452

3

4

8 9

1'5

2·5

22 75·2037

FREE VIBRATION OF RECTANGULAR PLATES

281

APPENDIX B

ON THE EXISTENCE OF EIGENVALUES SUCH THAT A < m2 n 2 (k 2 < ( 2 ) An excellent attempt to determine the existence ofeigenvalues such that A < m2 n 2 was made in the unpublished work of Fletcher, Woodfield, and Larsen [12]. Therein, proofs similar to those given below for Cases 3, 4, and 5 are correctly presented. CASE 1. ss-ss-ss-ss Equation (16) can be rewritten as f(A) = sinh 111 sinh 112 = 0,

(Bl)

where 111 = 111(A) and 112 = I1zCA) according to equation (22). For A= 0, sinhl11 = sinh 112 > O. As A increases, sinh172 increases monotonically and sinhl11 decreases monotonically, becoming zero at A = m 2 n2 • Thus,f(A) > 0 for all A in the range 0 < A < mn, and no eigenvalues

can exist. CASE 2. ss-c-ss-c Consider first the y-symmetric modes. Equation (24c) is rewritten as f(}.,) = 171 tanh!:= 112 tanh 172 =g(}.,) 2 2 2 2 .

(B2)

For A = 0,j(0) = g(O) , but this is a trivial root. As A increases, 112/2 and tanh 112/2 both monotonically increase, and 111/2 and tanhl11/2 both monotonically decrease. At }., = m 2 n 2 , 171/2 = tanh171/2 = O. Thus,f(A) > g(A) for all A in the range 0 < A< mn, and no y-symmetric

eigenvalues can exist. Equation (24d) for the y-antisymmetric modes is rewritten as f(A)

tanh I1d2 I1d2

=

tanh 112/2 112/ 2 = g(A).

(B3)

But tane/e < 1 for all e, and decreases monotonically as e increases. And 112> 111 for all ...t, except A = 0, which is a trivial root. Therefore,f(A) > g(}.,) for alI...t > 0, and no y-antisymmetric

eigenvalues can exist. CASE 3. ss-c-ss-ss No eigenvalues can exist. The proof is the same as for the y-antisynunetric modes of Case 2. CASE 4. sS-C-SS-F Equation (19) can be rewritten asf(A) = g(...t), where f(A)

=

A2 - m 4 n4 (1- V)2 + [.F + m 4 n4 (l- V)2] cosh 111 cosh 172'

g(A) = -m 2 n 2 (~)2 [A2(1 _ 2v) _ m4n4(I _ V)2] sinh 111 sinh 11 2 . a 111 112 Now

(B4) (B5)

282

A. W. LEISSA

and

Therefore,J(O) = g(O). Furthermore, letting A= m2 n 2 [1 - v] gives

f(m 2 n;2[1

-

v]) = 2m 4 n 4 [1 -

g(m 2 n 2 [1 - v]) = 2m 4 n4 [1 -

V]2

V]2

(mnb )

(mnb

cosh -a- VV cosh -a- V2=V

),

,J;;2="""; sinh (mnb -a- vv)sinh (mnb -a- vT=V ).

e,

But coshe > sinha for all and 1 > vv/(2 - v); thereforef(m 2 n 2 [1- v]) > g(m2 n2 [1 - v]). Therefore, a root of equation (19) will exist in the interval m 2 n 2 (1 - v) < A < m 2 n 2 if f(m 2 n Z )
=

(mnb)

(mnb .

V2 m4 n;4 { V2 -a- sinh -a- Y r;:;)} 2 ,

and then settingf(m2 n2 ) < g(m 2 n 2 ) yields

mnb )< V2 v (mnb) a sinh (mnb -a-V"2) (-a-V2' 2

[1 - (1- V)2] + [1 + (1 -

V)2]

cosh

as the condition for the existence of eigenvalues. Furthermore, for large values of [1 - (1 - )/)2]«

(1

+ (1 -

V)2]

cosh

(B6)

mnb V2/a,

mnb ) (--;-v'2

and sinh

mrcb) :::::!cosh (mnb) -a-V2 , (-a-0

and condition (B6) can be approximated by

mb

1 + (1 -

v)l V2

->-~--

a

2

v

n

(B7)

283

FREE VIBRATION OF RECTANGULAR PLATES

The roots of equation (B6) are given in Table Bl for 0 < v < 0·5. They are found to be the same as the roots of equation (B7) for the number of significant figures given. TABLE Bl Roots ofequation (B6)

CASE

5.

v

mb> a

0·0 0·1 0·2 0·3 0·4 0'5

255·973 57·983 23·413 12'021 7·072

00

sS-SS-SS-F

Equation (20) can be rewritten asf(A) = g(J.), where

f(A) = [A

g(A) It is obvious thatf(O)

=

+ m 2 n2(I

fA - m2 n2(I

_ v)]2 tanh 112 , 112 -

(B8)

tanh 111

(B9)

V)]2 - - .

111

= g(O) > O. Furthermore, letting A = m 2 n 2 [1 -

v] gives

Therefore, a root of equation (20) will exist in the interval m 2 n2(l- v) < A < m 2 n 2 if f(m 2 n 2 ) < g(m 2 n 2 ). Upon letting A = m 2 n 2 ,

tanh(V2~) « _ v)2 • r;:;

mnb

'v2-

2- v

(BlO)

a

as the condition for the existence of eigenvalues. Furthermore, for large values of 'V'2(mnb/a) , tanh(V2mnb/a) is approximately unity, and condition (BlO) can be approximated by

mb 1 ( v -;; > V2n 2 - v

)2

(Bll)

284

A. W. LEISSA

The roots of equation (BIO) are given in Table B2 for 0.;;; v ~ 0·5. TABLEB2

Roots ofequation (BID)

CASE

6.

v

mb> a

0'0 0·1 0·2 0'3 0'4 0'5

81·254 18'231 7'228 3'601 2·026

00

SS~F-SS-F

Consider first the y-symmetric modes. Equation (25e) is rewritten asf(A.) = g(A.), where

111 f(A.) = tanh T'

(BI2)

(B13)

For A. = 0,111(0) = 112(0), thereforef(O) = g(O) = tanh(mrcb/2a). Furthermore,f(A.) is positive and monotonically decreasing for all A. in the interval 0 ~ A. < m2 rc 2 • But, g(m2 rc 2 [l- v]) = O. Therefore, a root of equation (25c) will exist in the interval m 2 rc2 (1 - v) < A. < m2 rc 2 if f(m 2 rc 2 )
( g(m 2 rc 2 ) = m4 n4 (-V)2 2-v

mrcb) tanh (mrcb). vz-• hm a av21/ r,:;

1

- = 00,

1->01'/1

provided that v # O. If v = 0,f(m 2 rc 2 ) remains zero, and

a) 1/ (mnb

= - - lim 111 tanh -111= O. 1->0

2

Therefore, provided that v ~ 0, y-symmetric eigenvalues will always exist, and they will exist in the interval m 2 rc 2(l- v) < A. < m 2 rc 2 • For the y-antisymmetrie modes, because equation (25d) is the same as equation (20) if 111 and 112 are replaced by 111/2 and 112/2, respectively, the results for Case 5 apply, with the heading mb/a in Table B2 being replaced by mb/2a. For example, for v = O'3, y-antisymmetric eigenvalues will exist if mb/a > 2(7'228) = 14'455 (to five significant figure precision).

285

FREE VIBRATION OF RECTANGULAR PLATES

APPENDIX C TABULATED DATA FOR PLATES NOT HAVING TWO OPPOSITE SIDES SIMPLY-SUPPORTED TABLECl

Frequency parameters A. = roa2 V pJD for C-C-C-C plates alb

Mode sequence

A

0·4

2/3

1·0

1'5

2·5

1

11 23-648 0-23%

11 27·010 0-31 %

11 35-992 0'33%

2

12 27-817 0'35%

12 41'716 0·44%

21 73·413 0·44%

11 60-772 0'31% 21 93'860 0'44%

147'80 0'23% 21 173·85 0'35%

3

13 35-446 0'31%

12 73·413 0-44%

12 148'82 0'35%

31 221'54 0'31%

4

14 46'702 0'40%

21 66'143 0'35% 13 66·552 0·45%

22 108·27 0·53%

31 149'74 0·45%

15 61·554 0'43% 21 63-100 0·20%

22 79·850 0'50% 14 100'85 0'47%

31 131·64 0-64% 13 132·24 0'18%

22 179·66 0'50% 41 226·92 0'47%

41 291·89 0'40% 51 384·71 0-43%

5

6

11

12 394'37 0'20%

TABLEC2

Frequency parameters A. = roazV pJD for C-C-C-SS plates alb Mode sequence 1

2

3

4

5

6

r

A

0'4

2/3

1·0

1·5

2'5

11 23-440 0'23% 12 27'022 0'22% 13 33·799 0'29% 14 44'131 0·41% 15 58·034 0'49% 21

11 25-861 0'34% 12 38·102 0'38% 13 60'325 0'50% 21 65·516 0'34% 22 77·563 0'30% 14

11 31-829 0·42% 12 63-347 0'49% 21 71·084 0·48% 22 100·83 0'41% 13

11 48·167 0'46% 21 85·507 0'55% 12 123-99 0'45% 31 143'99 0'51% 22

107·07 0'40% 21 13%6 0'54% 31 194·41 0'45% 41 270·48 0'52% 12

116·40 0·49% 31

158·36 0·50% 32

322·55 0'27% 22

62-971 0·18%

92'154 0'52%

130·37 0'40%

214·78 0'23%

353·43 0·44%

11

286

A. W. LEISSA TABLEC3

Frequency parameters A. = a>a2 -yf pjD for C-C-C-Fplates (v = 0'3) alb Mode sequence

2

3

4

5

6

A

r

1·0

1·5

2·5

0·4

2/3

11 22'577 0·29% 12 24·623 0'96% 13 29·244 0·58%

11 23-015 0'57% 12 29-427 1'28% 13 44·363 0'68%

11 24'020 0'84% 12 40'039 1'22% 21 63·493 0'75%

11 26-731 1-01 % 12 65-916 1'04% 21 66·219 1-16%

31 135'15 1-47%

14 37-059 0·44% 15 48·283 0'50%

21 62·417 0'42% 14 68·887 0·64%

13 76·761 0-72% 22 80·713 1'76%

22 106·80 1-67% 31 125·40 0·91 %

12 152·47 0'69% 22 193·01 1·16%

21 61-922 0'18%

22 69·696 1·37%

23 116-80 0'80%

13 152·48 0'60%

41 213-74 1'27%

11 37-656 0·92% 21 76·407 1-47%

TABLE C4

Frequency parameters A. = wa2 -yf p/D for C-C-SS-SS plates Mode sequence

alb A

0-4

2/3

1-0

1

11 16-849 0'41%

11 19·952 0-55%

27·056 0-58%

2

12 21-363 0·36%

12 34'024 0'52%

3

13 29·236 0-44%

4

14 40-509 0-55%

5

6

1-5

2·5

21 60'544 0'76%

11 44·893 0'55% 21 76-554 0'52%

11 105-31 0·41 % 21 133-52 0·36%

21 54·370 0'44%

12 60·791 0'35%

12 122-33 0-44%

31 182·73 0'44%

13

57'517 0'58%

22 92-865 0'28%

31 129-41 0'58%

41 253-18 0'55%

21 51·457 0'24%

22 67-815 0-28%

13 114·57 0'54%

22 152·58 0·28%

12 321-60 0-24%

15 55-117 0-59%

14 90'069 0'55%

31 114-72 0-42%

41 202·66 0-55%

51 344-48 0'59%

11

287

FREE VIBRATION OF RECTANGULAR PLATES TABLE C5 Frequency parameters A = wa 2 y' pfDfor C-C-SS-Fplates (v = 0'3)

alb Mode sequence

A

0·4

2/3

1·0

1'5

2·5

1

11 15-696 0'55%

11 16·287 1'04%

11 17·615 1·40%

11 21'035 1·46%

11 33-578 1-07%

2

12 18·373 1'60%

12 36-046 1'46%

21 55·184 1·50%

21 66-612 1'66%

3

13 23'987 0-85%

12 24-201 1'77% 13 40·701 0-83%

21 52-065 1'01 %

12 63'178 1'08%

31 119-90 1·71 %

14 32-810 0·61 % 15 44-862 0'63% 21 50·251 0'25%

21 50·822 0'58% 22 59·071 1'70% 14 66'262 0'70%

22 71·194 1·98% 13 74·349 0·72% 31 106·28 0'65%

22 99·007 1-53%

12 150'83 0'64%

31 109·22 1·12% 13 150-90 0·56%

22 187-61 0'80% 41 193-23 1·46%

4

5

6

TABLE

Frequency parameters A = wa2

C6

VP7D for C-C-P-P plates (v = 0'3) alb

Mode sequence

A.

0·4

2/3

1·0

1'5

2·5

1

11 3'9857 5'l2%

11 4·9848 6'99%

11 6-9421 7-23%

11 11·216 6'99%

24·911 5'22%

2

12 7'1551 5-93%

12 13'289 6·14%

21 24'034 10'24%

21 29'901 6'14%

21 44'719 5·93%

3

13 13'101 4'24%

21 23-384 2-87%

12 26'681 -0'70%

12 52'615 2'87%

31 81·879 4-24%

4

14 21·844 5011%

13 30-262 3'30%

22 47'785 5-64%

31 68'090 3·30%

5

21 22·896 -0'55%

6

22 26-501 3-36%

22 34·240 4'59% 23 52·398 3-82%

13 63·039 4'00% 31 65·833 -0,41 %

22 77-041 4'59% 32 117·90 3'82%

41 136'52 5'11 % 12 143·10 -0·55%

11

22 165-63 3-36%

288

A. W. LEISSA TABLEC7

Frequency parameters A. = maZy p/D for C-SS-C-F plates (v = 0'3) Mode sequence 1

2

3

4

5

6

alb A.

0·4

2/3

1'0

1·5

2·5

11 22·544 0'06%. 12 24'296 0'42%

11 22-855 0'11% 12 27-971 0·73%

11 23-460 0'17% 12 35·612 0'94%

11 24·775 0·25% 12 53·731 1'09%

11 28'564 0'46% 21 70'561 0·34%

13 28·341 0'50% 14 35-345 0·49%

13 40·683 0'73% 21 62'310 0·09%

21 63·126 0·15% 13 66·808 0·85%

21 64·959 0'23% 22 97·257 1-12%

12 114'00 0'97% 31 130·84 0'27%

15 45·710 0·50% 16 59'562 0'49%

14 62·695 0·73% 22 68'683 0'58%

22 77'502 0-90% 23 108·99 0'81%

31 124·48 0·18% 13 127·92 0'74%

22 159·54 1'17% 41 210-32 0-24%

TABLEC8

Frequency parameters A. =

waz.vp/D for

C-SS-SS-F plates (v = 0'3)

alb Mode sequence

A

0·4

2/3

1'0

1·5

2'5

11 15'649 0-12% 17-946 0'73%

11 16'067 0·20% 12 22·449 HO%

11 16·865 0'29% 12 31-138 1-25%

13 22-902 0·77% 14 30·892 0'70%.

13

36·703 0-93% 21 50-696 0-12%

21 51·631 0'20%

11 23·067 0·73% 21 59·969 0·36% 12 111·95 0·92%

64·043 0'92%

11 18·540 0-40% 12 50·442 1'24% 21 53·715 0'28% 22 88'802 1'01 %

22 57·908 0-72%

22 67·646 0'98%

31 108·19 0'22%

22 153·24 0·79%

14 59·840 0'81 %

23 101·21 0'62%

13 126'09 0'70%

41 189·49 0'28%

12

2

3

4

5

6

15 42·108 0'66% 21 50·222 0·06%

13

31 115-11 0'32%

289

FREE VIBRATION OF RECTANGULAR PLATES TABLE

C9

Frequency parameters A = wa z VpJD for C-SS-F-F plates (v = 0'3) Mode sequence

2

3

4

5

6

alb A.

0'4

2/3

1·0

1'5

2·5

11 3'8542 2'10% 12 6·4198 5-48%

11 4-4247 3·66%

11 5'3639 5-27%

11 6'9309 8'24%

11 10-100 14'77%

12 10·912 6·65% 21 22-958 1-78%

12 19-171 7'07% 21 24·768 0'67%

21 27·289 3'12% 12 38'586 3·70%

21 35-157 3·98% 31 74·990 1-87%

13 25-698 3-06%

22 43-191 4-85%

22 64·254 6·43%

22 32'425 3-33% 23 48·467 4·52%

13 53-000 2·48% 31 64·050 0-35%

31 67-467 -0'16% 32 108·02 2'89%

12 99'928 1'90% 22 127-69 5'28% 41 135-45 0'06%

13 11'576 4'68% 14 19·767 4'08% 21 22'521 0'01%

22 26-024 1-96%

TABLE CIO Frequency parameters A = wazvpJD for C-F-C-F plates (v = 0'3)

alb Mode sequence

A

0·4

2/3

1'0

1-5

2-5 11 22-130 1'10% 12 41·689 1·55% 21 61·002 1·10%

11 22'346 0-12%

11 22-314 0-27%

11 22·272 0-46%

2

12 23'086 0'06%

3

13 25-666 0'42%

12 26'529 0'25% 13 43·664 0'98%

4

14 30'633 0'75%

12 24·309 0·13% 13 31-700 0'68% 14 46·820 1·01 %

11 22·215 0·71% 12 30·901 0'56% 21 61·303 0'60%

21 61·466 0'34%

13 70'960 1-20%

22 92-384 0'61%

5

15 38-M7 0'33%

21 61·566 0'19%

22 67·549 0'16%

22 74·259 0'27%

6

16 49·858 0'49%

22 64'343 0-09%

14 79·904 1-03%

23 118'33 0'99%

31 119·88 0'85% 32 157·76 0'23%

290

A. W. LEISSA TABLE Cll

Frequency parameters A = roa2 V pjD for C-P-SS-F plates (v = 0'3) alb Mode sequence

1

2

3

4

5

6

.A

r

0·4

2/3

1·0

1'5

11 15'382 0·24%

11 15-340 0·51 %

11 15'285 0'87%

11 15·217 1'32%

11 15-128 1·92%

12 16'371 0'12% 13 19-656 0·69% 14 25·549 1-08%

12 17'949 0-23% 13 26·734 0'98% 14 43·190 1·21 % 21 49·840 0·25% 22 53·013 0·11%

12 20'673 0'44% 13 39·775 1'22% 21 49·730 0·47%

12 25·711 0'89% 21 49·550 0·84% 22 64-012 0-25%

22 56'617 0'18% 14 77·368 1'07%

13 68'126 1'25% 31 103·70 0'53%

12 37·294 1·98% 21 49·226 1'50% 22 83-325 0'41% 31 103014 1'07% 32 143·68 0·23%

15 34·507 0'52% 16 46·435 0'64%

TABLE

2'5

C12

Frequency parameters A = roa2 V pjD for C-P-F-F plates (v = 0'3) alb Mode sequence

1

2

3

4

5

6

"-

0·4

2/3

1·0

1'5

2·5

11 3·5107 1·74% 12 4·7861 4-17% 13 8·1146 7'20%

11 3·5024 0'39% 12 6-4062 7·09% 13 14·538 9-09%

11 3·4772 1'12% 12 11'676 17'45% 21 21·618 1'93%

11 3-4562 1'74% 12 17'988 24-36% 21 21·563 2'19%

14 13·882 6'69% 21 21-638 1'83% 22 23·731 0-84%

21 22·038 -0'02% 22 26·073 3·46% 14 31-618 3'37%

11 3·4917 0·70% 12 8·5246 11'56% 21 21·429 2'83% 13 27'331 5032%

22 39·492 4'74%

22 31·111 3'17% 23 54·443 5'99%

13 53·876 5'92% 31 61-994 -0-48%

22 57'458 8'54% 31 60·581 1'84% 32 106'54 2-49%

FREE VIBRATION OF RECTANGULAR PLATES

291

TABLE C13 Frequency parameters A = wa 2 y' pjD for SS-SS-F-F plates ("\I = 0'3)

alb Mode sequence

A

\

0·4

2/3

1·0

1·5

2'5

11 1·3201 7'56%

11 2·2339 5-93%

11 3·3687 5·37%

11 5'0263 5'93%

11 8·2506 7'56%

12 4·7433 4'11 %

12 9·5749 3'28%

12 17-407 7'80%

21 21·544 3·28%

21 2%46 4'11 %

13 10·362 2·02% 21 15-873 0'80%

21 16·764 1'33% 13 24·662 1·45%

21 19·367 -3,11 % 22 38·291 4'24%

12 37·718 1-33% 31 55·490 1·45%

31 64·760 2'02% 12 99'206 0'80%

5

14 18'930 1·46%

22 27·058 3·31 %

13 51·324 3·01 %

22 60·882 3-31 %

41 118'31 1·46%

6

22 20·171 1'92%

23 44·172 2-96%

31 53·738 -1-61 %

32 99·388 2·96%

22 126'07 1'92%

2

3

4

TABLE C14 Frequency parameters A= wa 2 y' pjD for SS-P-P-F plates (v = 0'3)

alb Mode sequence

A

0·4

2/3

1·0

1·5

2'5

12 2-6922 5-48% 13 6·5029 4'37%

12 6'6480 6'79% 21 15·023 2'63% 22 25·492 3-43%

12 9·8498 8'11% 21 15·013 2·70% 22 34·027 4'61%

21 14·939 3·21 % 12 16·242 9·27% 31 48·844 2'29%

13 26·126 1-87% 31 48·711 2'57%

31 48·332 3'38%

3

14 12-637 3-35%

12 4·481 5'63% 13 13·009 6·30% 21 15-674 -1,63%

4

21 15-337 0·53%

22 20·373 3·08%

22 17'510 0'70% 15 21-699 3'03%

14 30'548 1'14% 23 33'411 4'23%

It 2

5

6

23 50·849 2'37%

13 55·066 -0'50% 32 70·695 1'87%

22 52·089 6'84% 32 97·225 2'69% 41 102·34 1'87%

t In a mathematical sense the first mode has a zero frequency, and corresponds to rigid body rotation about the simply-supported end.

292

A. W. LEISSA TABLE

Frequency parameters A= roa2

CI5

vpjD for F-F-F-F plates (v = 0'3) alb

Mode sequence

It 2

3

4

5

6

.A

0·4

2(3

13

22 8·9459 5·81 %

3·4629 3'37% 22 5·2881 7·40% 14 9·6220 2'55% 23 11·437 5'58% 15 18·793 2'94% 24 19·100 3·41%

1·5

2'5

22 13·489 5·26% 13 19·789 13·06% 31 24·432 -8·43%

22 20·128 5-81% 31 21-603 3'56% 32 46·654 4'37%

31 21'643 3-37% 22 33·050 7'40% 41

32 35'024 4'20% 23 35·024 4·20% 41 61·526 0'24%

13

1'0

13

9·6015 3'56% 23 20·735 4·37% 31 22-353 0'09% 14 25·867 5-96% 32 29·973 -1-67%

50'293 0'09% 41 58-201 5'96% 23 67·494 -1'67%

60~137

2'55% 32 71·484 5·58% 51 117·45 2'94% 42 119·38 3·41%

t In a mathematical sense the first three modes have zero frequencies and correspond to rigid body translation in the transverse direction and rigid body rotations about the symmetry axes.

APPENDIX D NOTATION

Am,Bm, Cm, D m A pq a, b C D e E F

constants of integration in equation (7) amplitudes of trial functions in equation (29) length dimensions of a rectangular plate in the x and y directions, respectively clamped edge indicator plate flexural rigidity; equals Eh 3 /12(1-v 2 ) 2·71828 ... Young's modulus free edge indicator h plate thickness I second moment of the area ("moment of inertia") for a beam k [pw 2 /D]1/4 I length of a beam M",My plate bending moments in the x and y directions, respectively m, n number of half waves in the x and y directions, respectively, of a mode shape N" compressive in-plane force resultant (force per unit edge length of plate) acting in the x direction p,q beam function summation indices, according to equation (29) SS simply-supported edge indicator t time w transverse plate deflection, W = w(x,y, t) W transverse plate deflection, W = W(x,y) X p , Yq beam functions in the x and y directions, respectively

FREE VIBRATION OF RECTANGULAR PLATES

293

x,y co·ordinates in the middle plane of a plate oc mnla rPl' rP2 defined by equations (15) A. non·dimensional frequency parameter, equals wa 2 yfplD v Poisson's ratio n 3'14159 ... p mass density per unit area of plate '11,1/2 defined by equations (22) Q non·dimensional frequency parameter, equals A./~ OJ circular frequency, radians per unit time '\74- biharmonic differential operator; equals '\72 '\72, where '\72 is the scalar Laplacian operator