Leissa_the Free Vibration of Rectangular Plates

Journal ofSound 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 freevibration of rectangular plates. Twenty-one cases exist which involve the possible combinations of clamped, simply-supported, and free edge conditions. Exact characteristicequations are given for the six cases having two opposite sides simply-supported. Theexistence of solutions to the various characteristic equations is carefully delineated. TheRitz method is employed with 36 terms containing the products of beam functions toanalyze the remaining 15 cases. Accurate frequency parameters are presented for a range ofaspect 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 changingPoisson's ratio are studied.

1. INTRODUCTION

A vast literature exists for the free vibrations of rectangular plates. Consider only the classicaltheory 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/oy2 in rectangular co-ordinates); D = Eh3/12(1 - v2), the flexural rigidity;E is Young's modulus; h is plate thickness; v is Poisson's ratio; p is mass density per unitarea of plate surface; and t is time. Exclude such complicating effects as orthotropy, in-planeforces, 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 alsofound that the majority of this voluminous literature dealt with a few specific problems, thatfor most of the problems the scope of treatment was limited or essentially non-existent, thatthe exactness of numerical results was ordinarily deficient, and that the effects of changingPoisson's ratio were generally ignored.

For rectangular plates there exist 21 distinct cases which involve all possible combinationsof classical boundary conditions (i.e., clamped, simply-supported, or free). For the six caseshaving two opposite edges simply-supported, well-known exact solutions exist which are theextensions ofVoigt's [2] early work. For the remaining 15 cases, three problems have receiveda great deal of attention. The completely clamped case is used frequently as a test problem foranalytical 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, particularlyfor simulatinglifting and stabilizing surfaces in the aerospace industry. The completely free case has a richhistory. 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 subsequentexperimental work and analytical discussion in the literature. Ritz [7] in 1909 used thecompletely free problem to demonstrate his now-famous direct method for extending theRayleigh principle for obtaining upper bounds on vibration frequencies.

The remaining 12 problems have received little coverage in the literature; indeed, for six ofthem virtually nothing at all can be found. An important step to remedy this situation wasmade by Warburton [8]. In this useful piece of work he presented frequency formulas for all21 types of problems derived by using the Rayleigh method with assumed mode shapeswhich are the products of vibrating beam eigenfunctions. Later, another set of formulas waspublished by Janich [9] for 18 cases (for fundamental modes only). Again, the Rayleightechnique was utilized, but simple trigonometric functions were chosen to represent the platedeflections. However, these functions do not represent the mode shapes nearly as well as thebeam 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, certainquestions concerning them still remain to be answered. Of particular interest is the generalquestion of accuracy. The beam functions do satisfy the geometric boundary conditions ofzero deflection and slope where required (although not the free edge or free corner conditionsof 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-termrepresentation of the deflection mode shape is used? And what happens to the relativeaccuracy as boundary conditions are changed or higher frequencies are sought?

The primary purpose ofthis work is to present in one place reasonably accurate results forfree vibration frequencies of all 21 combinations of classical boundary conditions forrectangular plates. Secondary purposes are (1) to evaluate the accuracy of Warburton'sformulas by direct comparison, (2) to point out some of the mathematical nuances of theVoigt and Ritz methods and of the resulting solutions, (3) to study the effects of changingedge conditions upon the frequencies and upon their accuracies, and (4) to study the effects ofchanging Poisson's ratio upon the vibration frequencies.

The first part ofthe paper deals with the exact solutions for the six cases having two oppositesides simply-supported. Extensive numerical results are given in Appendix A. The last partdeals with the remaining fifteen problems. Accurate frequencies are obtained by using theRitz method, 36-term mode shapes composed of beam functions, and the capabilities ofmodern, digital computers. These frequencies are tabulated in Appendix C. Comparison is

I II II II II II II 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 theyare 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 bemade. First, consider the plate having length dimensions aand b. For purposes ofdescription,a notation will be adopted as follows. The symbolism SS-C-SS-F, for example, will identify arectangular 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, itshould be remembered that the non-dimensional frequency parameter wa2 VPil5' (where OJ isfrequency, and a is a characteristic length) does not explicitly depend upon Poisson's ratio forthe rectangular shapes considered here unless at least one edge is free. However, the frequencyitselfdepends upon v in every case due to its inclusion in D. Unless otherwise stated, for the 15cases having a free edge, vwill be taken as 0'3, a widely-used practical value. (A list ofnotationis given in Appendix D.)

2. TWO OPPOSITE EDGES SIMPLY-SUPPORTED

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

for a simply-supported edge,

and for a free edge

oww=-=Oax ' (2)

(3)

02 W iFw oJ w oJ wox2 + 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 oftwo free edges the additional condition

02 W- =0 (5)oxoy

must be satisfied at the corner, although this condition will not be encountered when twoopposite edges are simply-supported.On the assumption of a sinusoidal time response for free vibration,

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

the classical Voigt [2] solution

Wm= [A msinVk2- (1.2 y + BmcosVk2- (/.2 Y + Cmsinh-ylk2+ (/.2 y +

+ Dm coshV k2 + (/.2 y] sin (Xx (7)

will be taken, where k 4= pOJ2/D, and (/. = mn/a, m = 1, 2, ... , and where k 2 is assumed to begreater 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 acharacteristic determinant of the fourth order for each m. Expanding the determinant andcollecting terms yields a characteristic equation. The characteristic equations for the six casesare listed below.

260

Case 1. SS-SS-SS-SS

A. W. LEISSA

(8)

Case 2. SS-C-SS-C

4>1 4>"{COS 4>1 cosh4>z - 1) - m2 n2 (~r sin 4>1 sinh 4>2 = O.

Case 3. SS-C-SS-SS

Case 4. SS-C-SS-F

4>14>2[A,z - m4n4(1 - V)2] + 4>1 4>2['1,2 +m4n4(1- V)2] cos 4>1 cosh 4>2 +

(9)

(10)

(11)

Case 5. SS-SS-SS-F

4>1[..1. + m2n2(1 - v)]2 tanh 4>2 - 4>2[..1. - m2n2(1 - v)]2 tan 4>1 = O. (12)

Case 6. SS-F-SS-F

24>14>2[..1.2 - m4n4(1 - V)2]2 (cos 4>1 cosh 4>2 - 1) + {4>HA. + m21l:2(1 - vW -- 4>HA. - m2 n2(1 - v)]4} sin 4>1 sinh 4>2: = O. (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

b4>1 =-VA.-m2 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 m2 n2). When this occurs it is necessary to replace sinv'P - (X2Y andcosv'k2- (X2Y in equation (7) by sinhY(X2 - k 2 y and coshy(X2 - P y, respectively. Then thecharacteristic equations become the following.

Case 1. SS-SS-SS-SS

sinh 111 sinh '12 = O.

Case 2. SS-C-SS-C

1'f11'f2(cosh I71 cosh '12 -1) - m2 n2 (~r sinh 111 sinh 112 = O.

Case 3. SS-C-SS-SS

. (16)

(17)

(18)


Case 4. SS-C-SS-F

111 I12P.2 - m4 n4(1 - V)2] + 1/1112[..1,2 + m4 n4 (1 - v?] cosh 111 cosh 1'/2 +

+m2n2(~r [,12(l-2v)-m4n4(1-v)2]sinhl'/lsinh1'/2=0. (19)

Case 5. SS-SS-SS-F

1'/1[1l + m2n2(1 - V)]2 tanh 1'/2 - 112[1l - m2n2(1 - V)]2 tanh 171 = 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+ m2 n2(1- vw--I'/~[Il- m2 n2 (l - V)]4} sinh III sinh 112 = O. (21)

It is seen that equations (16)-(21) are of the same form as equations (8)-(13), the former beingobtained from the latter by simply replacing sin, cos, and tan by sinh, cosh, and tanh, and<P1' cP2 by 1'/1,1'/2' respectively, where

1'/1 = %vm2 n2 -Il,

b1'/2 = - vm2 n2 + Il. (22)

aBecause 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 eithery-symmetric or y-antisymmetric. The characteristic equations corresponding to these modescan be obtained from equations (8), (9), (13), (16), (17) and (21) by factoring, or by newderivations 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 k2 > a2 (1l > m2 n2

). The resulting characteristic equations arethe following.

Case 1. SS-SS-SS-SS

Case 2. SS-C-SS-C

{

symmetric:

Il> m2 n2

antisymmetric :

{

symmetric:

Il < m2 n2

antisymmetric:

cP1 cP2cos - cosh-= 0

2 2 '. <P1 . h cP2

SIll- SIll -=02 2 .

1]1 h 1'/2cosh-cos - =02 2 '

. h III . h 1'/2SIn -SIll -=022'

(23a)

(23b)

(23c)

(23d)

{

symmetric:

Il> m2 n2

antisymmetric :

{

symmetric :

Il < m2 n2

antisymmetric:

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

rl. cP1 rl. h cP2'1'2 tan - - '1'1 tan -= 0,2 2

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

171 1121'/2 tanh - - '11 tanh - =O.

2 2

(24a)

(24b)

(24c)

(24d)

A. W. LEISSA262

Case 6. SS-P-SS-F

symmetric: ¢ d). -I- m 2 n2(l - vWtan ~1-1-

-I- 4>2[.:l. -m2 n2(l- v)]2tanh ~2 = 0,

antisymmetric: ¢2[). - m2 n2(1 - v)]2tan ¢1_2

-4>d). -I- m2 n2(l - v)]2 tanh ~2 = 0,

(2Sa)

(2Sb)

1'/1symmetric: 111[). -I- m2 n2(l - v)]2 tanh 2" -'12

- 1J2[.:l. - m2n2(1 - V)]2 tanh2 = 0,

antisymmetric: 1J2[). - m2n2(1- v)]2tanh 111 -2

1J2-1J1[.:l. + m2n2(l - v)]2 tanh 2" = 0.

(25c)

(25d)

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 thecorresponding frequencies) of a SS-C-SS-C plate of width b are the same as those of aSS-C-SS-SS plate of width b12. This is because the conditions along the antisymmetry axis ofa SS-C-SS-C plate are the same as conditions of a simple support. The same correspondenceexists 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 OPPOSITEEDGES SIMPLY-SUPPORTED

The characteristic equations presented in the preceding section were programmed androots of the equations were found by using Newton's method. Numerical results for thenondimensional frequency parameter). = ma2 y piD were obtained for each of the six casesover a range of aspect ratios and their reciprocals (alb = 2'5, 1'5, 1'0, 2/3, 0'4). Poisson'sratio, 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 valueof alb, the nine lowest values of ma2 '\!pID are displayed in increasing sequence. The resultsare exhibited in considerable accuracy simply because they were easily obtained to theaccuracy given, and because they may be of worth to someone desiring to investigate theaccuracy of an approximate method on some of these problems. In addition, for eacheigenvalue presented, the corresponding mode shape is described by the number of halfwaves in each direction. Thus, for example, a 32-mode has three half-waves in the x-directionand two in the y-direction. For all six cases the wave forms are, ofcourse, sine functions in thex-direction, according to equation (7). Furthermore, the wave forms in the y-direction arefound 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 ofthis result is that the node lines lying in the y-direction (two for a 32-mode) will be exactlystraight, parallel to the y-axis, and evenly spaced. On the other hand, those lying in thex-direction (one for a 32-mode), except for the SS-SS-8S-SS case, and except for an axis ofsymmetry, will not be exactly straight, parallel to the x-axis, or evenly spaced.


For purposes of subsequent discussion in this section, it will also be useful to clarify someterminology with respect to symmetry of modes. As already used in section 2, y-symmetricmodes 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 symmetryis present (e.g., SS-SS-SS-SS, SS-C-SS-C, SS-F-SS-F). Of course, all the six cases discussedin 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 gleanedfrom Tables Al to A6. For example, consider the SS-C-SS-C plate (Table A2) havingalb = 1·5. The 21 mode has a nodal line along x = al2 and has J, = 78·9836. Considering onlyone-half of the plate in this mode, one then has a SS-C-SS-C plate with alb = 0'75 withA= (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 inthe 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 informationfor plates having other boundary conditions as well. Returning to alb = 1·5 in Table A2, onesees that the 12 mode corresponds to a II mode for a SS-C-SS-SS plate having alb = 3 andthe same value of frequency parameter, A= 146·2677. (To accomplish these transformationssimply, one keeps the length a fixed and varies b to arrive at the desired alb ratio. Thus J, forthe 12 mode of a plate having b = (2/3) a is the same as that for the 11 mode of a plate havingb = (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 modeof a SS-C-SS-SS plate having alb = I and J, = (/3)2212'8169 = 23'6463 (already given inTable A3).

A word ofcaution should be mentioned regarding the procedure described in the precedingtwo paragraphs for obtaining additional information from the higher modes. In order for thecorrespondences 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 thex-direction satisfy these conditions in general only for Case 1, and in particular for the othercases only when the line is one of geometrical symmetry (y = bl2 in Cases 2 and 6). Thus, forthe SS-C-SS-SS plate (Table A3) having alb = 1'0, A for the 22 mode is less than four timesthat 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-directednodal line is noticeably curved, thereby supplying additional circumferential stiffness to theplate 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 timelater in this section. Specifically, when uniformly distributed, compressive force resultants Nx

(force per unit edge length of plate) act in the plane of the plate and perpendicular to thesimply-supported boundaries x = 0 and x = a, then the following correspondence existsbetween the frequency parameter (in the absence of N x) and the static buckling parameter(due to Nx):

2 {P fNxwa ...; 15 ~ mna ,J Ii' (26)

264 A. W. LEISSA

Some specific comments will be given below for each of the six cases having two oppositesides 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 bedetermined explicitly; i.e.,

wa2A= rc2[m2+ n2(~r] (m,n = 1,2, ...) (27)

and the mode shapes are the same as those of vibrating rectangular membranes. Numericalvalues for this case are listed in Table AI. From equation (27) it is clear that as alb -? 0,OJa2V piD -? m2n 2 and that as bla -? 0, wb2V piD -? n2 rc2.

In Appendix B it is shown that equation (16) for wa2VpID < m2 rc2 has no roots.

3.2. S8-C-SS-C

The eigenvalues listed in Table A2 are calculated from equations (24a) and (24b). InAppendix B it is shown that equations (24c) and (24d) for wa2vpiD < m2 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 beeasily obtained from the data presented for SS-C-SS-SS plates (see section 3.3), as discussedin section 3.

3.3. S8-C-SS-S8

The frequency parameters listed in TabIe A3 are calculated from equation (10). In AppendixB it is shown that equation (18) for wa2VpID < m2 rc 2 has no roots.

The lowest six frequencies of the square, as well as the fundamental frequencies for the fiveaspect ratios of Table A3, were also obtained to less accuracy in reference [10]. A few otherreferences dealing with this problem are described in reference [1].

Additional results for vibration frequencies ofSS-C-SS-SS plates can be found quite simplyfrom the y-antisymmetric results for SS-C-SS-C plates given in Table A2, as discussedpreviously in section 3. Specifically, fundamental frequencies for SS-C-SS-SS plates havingalb = 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 valuev = 0·3 being used. It is shown in Appendix B that roots to equation (19) can also exist forv = 0'3, provided thatmbla > 7·353. Thus,for alb = 1, the lowest frequency parameter of thistype 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] andduplicate 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 thatmbla> 7'228. Thus, for the square plate, the lowest frequency parameter which would be


encountered such that A< m2 n2 would be ASi' Furthermore, as shown in section 3.6, thisfrequency parameter would be close to m2 n2 (~ 630), and would be far beyond the range ofthe table. Similarly, for alb = 0-4, the lowest frequency parameter for the case of A. < m2 n2

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 datafor SS-P-SS-P

plates given in Table AG. In particular, fundamental frequencies for SS-SS-SS-P plateshaving 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 (seesection 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. Mostof the results are for A> m2 n2

, equations (25a) and (25b) being used. However, as is shown inAppendix B, there always exists one y-symmetric mode for each value ofm such that A< m2 n2

,

and, under rather restricted circumstances, y-antisymmetric modes arising from roots ofequation (25d) can also be found such that A< m2 n2 • In particular, for v = 0'3, it is seen inAppendix B that mbla must be greater than 14-455 in the latter case, and thus no frequenciesof this type appear among the first nine for the values of alb used in Table A6.

TABLE 1Representativefrequency parameters Afor the antisymmetric(n = 2) modes ofSS-F-SS-Fplates, with A< m2 n2 (v = 0'3)

alb m mbla m2 n2 A= wa2 VjjJD

1·0 15 15·0 2220·66 2220·221·0 14 14·0 1934·44 non-existent1'0 20 20·0 3947'84 3943·000'5 15 37'5 2220·66 2216'750·4 15 37·5 2220·66 2216·550·1 5 50 246·74 246·270'1 10 100 986·96 985·090·01 1 100 9·8696 9·85090·0690 1 14'49 9·86960 9·86946

The lack of importance of the antisymmetric frequencies for A< m2 n2 is seen further fromTable 1. All the values, although less than m2 n2 , are also quite close to m2n2 , the largestpossible deviation occurring for extremely large mbla: i.e., for mbla = 100, A. differs fromm 2 n2 by approximately 0·2 %. It was shown in reference [12] that for these modes a limitingvalue of A= 0·99810 m2 n2 is reached for mbla = ro (when v = 0'3). The nearness to m2 n2 towhich 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 squareplate 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 thisone for alb = 1!

It is interesting to note in Table A6 that the frequency parameters for the y-symmetricmodes (11, 21, 31, etc.) having A< m2 n2 are the only ones that decrease as alb is increased.Nor do the values of Adecrease with increasing alb for any fixed mode number in the fivepreceding Tables Al to AS. With increasing alb, the symmetric eigenvalues having.A. < m2 n2

266 A. W. LEISSA

decrease to a limiting value of m2 n2 '\1f="V2. The reason for this is that the frequencyparameter Aitself contains~within D, and for large alb the plate behaves as a beam oflength a simply-supported at both ends and undergoing anticlastic bending, the beambending frequency being independent of Poisson's ratio. Thus the limiting case of large albgives the beam frequency parameter of wa2 v'12p1Eh3 = m2 n2

• On the other end of the albrange for these modes, the eigenvalues for the 11 modes are A= 9·8351 and 9·8509 foralb = 0'1 and 0'01, respectively, approaching once more the limiting value of0'99810 m2 n2 asalb -+ O.

The SS-F-SS-F case has received reasonable attention in the literature. The solutionfunction (7) was used by Voigt [2] on this plate vibration problem in 1893, six years before themore widely recognized paper by Levy [16] proposing the same type of solution for staticproblems of plate bending. Another excellent piece of early work on this problem was byZeissig [17] who plotted extensive results. In reference [12] the first six frequencies for thesquare are given for v = 0'3 and agree with those of Table A6. Extensive tabular data forv=0'16 and 0·3 were also presented by Jankovic [18]; however, the numerical resultspresented are somewhat lacking in accuracy. A more serious fault of the latter paper is thelack of recognition ofthe existence of solutions such that.:l. < m2 n2 • This error gave values ofA= 11:2, (2n)2, (311:)2, ." for the 11,21, 31, ... modes/or all values o/a/b. Indeed, these valuesof .: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 wasseen above that the frequency parameter .:I. = OJa

2v'piD does not depend upon v unless oneor more of the edges of the plate is free. Thus, for example, among the six cases discussed insection 3, in only three was A. a function of v. For these three cases, v was fixed at a value of0·3.

However, it must not be forgotten that D =Eh3 /12(l- v2), and thus depends upon v;

therefore, in every case the frequency itself depends upon v. For purposes of quantitativecomparison, a frequency parameter Q is now defined which does not contain v, viz.

Q == wa2 JT1J=~. (28)

It is seen that Q is such that it equals Awhen v = O. Furthermore, in those cases where Adqesnot 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 forthe SS-F-SS-F plate when other values of v were used. This case was chosen because (1) thepresence of two free edges causes marked changes with v, (2) the y-antisymmetric modescorrespond to the SS-SS-SS-F case which is itself y-asymmetric, and (3) the existence ofeigenvalues is complicated, depending upon whether A. is greater or less than m2 n2

, asdiscussed 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, and2'5 and for those modes corresponding to the first nine frequencies which exist for the squareplate having v = 0·3 (as given in Table A6). As vis varied, the ordering ofthe modes can alsochange. For example, for alb = 0'4 it is seen that for v = 0'3, OJ16 < OJ23 , but for v = 0'5,W23 < OJ16' However, this is the only instance of mode reordering found for the range ofaspect ratios used in the table.

The modes are also separated into categories which are either y-symmetric or y-antisymmetric and having either A> m2 n2 or A< m2 n2• For the range of m and alb of the table,


only one y-antisymmetric mode was found for which A< m2 n2; 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 ifmbla> 4·051 for v = 0'5, and in this instance mbla = 5. The likelihood of these modesexisting increases with increasing v. Another example of this type which occurs beyond therange of the table is fo'r v = 0'5 and alb = 0·1. Then Al2 = 9·6635 (mbla = 10, Au/n2 = 0-9791)and A22 = 38·6228 (mbla = 20, A22I(2n)2 = 0'9783).

TABLE 2

Frequency parameter A = wa2 Yp/D as afunction ofv for SS-F-SS-F plates

J. > m2 rr2 vor a A

Type of mode A< m2 n2 b mil 0 0'3 0'5

y-symmetric0'4

11 9·8696 9·7600 9·450621 39·4784 39-2387 38·3771

11 9'8696 9·6314 9·079321 39'4784 38'9450 37·5192

<m2 n2 31 88·8264 87-9867 85·4899

11 9·8696 9·4841 8·7042

2·521 39·4784 38·3629 35-879931 88·8264 86'9684 82-509341 157·9137 155'3211 148·7256

13 15·7531 15·0626 14·13600·4 15 32'2511 31·1771 30·3335

23 45·7156 44·9416 41-0576

>m2 n2 13 39·2281 36·7256 34·778323 74·7963 70·7401 66·8020

2·5 13 160·3527 156'1248 153·197923 211·9354 199·8452 190·8194

y-antisymmetric 12 11·4098 11'0368 10·3901

0·414 22-6610 21-7064 20'865022 41·0521 40·5035 39-0826t16 44·7469 43-6698 42·8705

12 17·8821 16-1348 14·3516>ml n2 22 48·9147 46·7381 43·4768

14 77'5775 75·2834 73'661032 98,54·81 96'0405 91-1600

12 39·4631 33-6228 28·76282·5 22 84·9399 75'2037 66·0192

32 142'1102 130·3576 117'3358

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 alla/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 literaturesurvey of reference [1], only one mode ofone case (F-F-F-F) can be found wherein A increaseswith 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, itmust be noted that although 15 of the 21 cases ofsimple boundary conditions for rectangularplates have one or more free edges, yielding A.-dependence upon v, results in the literature forvalues of v other than 0·3 are quite sparse.

268 A. W. LEISSA

TABLE 3Variation of.A. with vfor the 22 mode

anda/b=O'4

v A. Difference

0·0 41·0521-0'0683

0-1 40·9838-{H683

0'2 40·8155-0,3120

0·3 40·5035-0,5322

0·4 39'9713-0,8870

0'5 39·0826

TABLE 4Frequency parameter Q = coa2V12p/Eh3 as a/unction ofv/or SS-F-SS-Fplates

A> m27C

2 vor a F

A

-Type of mode A. < m2

7C2 b mn 0 0·3 0'5

y-symmetric0·4 11 9·8696 10'2313 10·9126

21 39·4784 41·1333 44·3141

11 9·8696 10'0965 10·4839

121 39·4784 40·8255 43·3234

<m2 n2 31 88·8264 92'2351 98'7152

11 9·8696 9·9420 10'0507

2'521 39'4784 40·2152 41·430531 88'8264 91-1677 95·273741 157'9137 162·8208 171·7335

13 15·7531 15·7899 16'53070·4 15 32·2511 32'6825 35'0261

23 45·7156 47'1116 47'4092

>m27C

2 13 39·2281 38·4989 40·158523 74·7963 74·1558 77·1363

2'513 160·3527 163'6633 176·897723 211·9354 209·4947 220·3393

y-antisymmetric 12 11·4098 11'5697 11·9975

0·4 14 22·6610 22'7545 24·092822 41·0521 42·4592 45'1287i16 44·7469 45·7784 49·5026

12 17·8821 16·9139 16·5718>m2 n2

122 48·9147 48·9948 50·202714 77·5775 78·9184 85'056432 98·5481 100'6778 105·2625

12 39·4631 35'6228 33·21242'5 22 84·9399 78·8349 76·2324

32 142·1102 136·6519 135·4877

t Obtained by using solution for)" < m2 rc2•


Not only does A. decrease with increasing v for all modes in Table 2, but the decrease is atan increasing rate. That is, curves of A us. v all have negative curvature everywhere. Thisbehavior is demonstrated in Table 3 for the 22 mode and alb = 0·4. Therein, A is given forchanges in v having equal increment size 0'1. It is seen that the negative change in A. betweenv = 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 Aisprecisely m2 n2 , which corresponds to the frequency parameter of a beam simply-supportedat both ends. Thus the plate behaves according to Euler-Bernoulli beam theory in thisspecial case.

The variation of the frequency parameter Q not containing v is displayed in Table 4 for thesame alb ratios and mode shapes as in Table 2. Particularly interesting to note is that Q usuallyincreases with increasing v for the y-symmetric modes, but either increases or decreases forthe y-antisymmetric modes, and the tendency for Q to decrease with increasing v increases asalb 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 13mode 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. Frequencydecrease with increasing v is a strange phenomenon, but it can be found to occur in at leastone 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 thefundamental mode.

Upon considering the fundamental (i.e., 11) mode, it is seen from Table 4 that Q decreaseswith increasing alb for all non-zero values of v, and approaches the simply-supported beamfrequency Q = n2 as alb -+ oj. The curvature of the plate in the fundamental mode in they-direction is very small for small alb, thereby generating a bending moment My whichapproximately equals vMx everywhere. For large alb, the free edges are relatively closetogether, and because My is zero on these edges, only a small value ofit can be generated overthe small distance. Thus, the presence oflarge My for small alb causes considerable stiffeningand corresponding frequency increase for the plate. This effect increases, of course, withincreasing v. The same effects occur for the 21, 31, 41, etc., modes; however, for the othermodes one or more nodal lines lie in the x-direction, and the picture is then complicated bythe 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 numericalresults 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 usesfunctions 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 ofmotion of afreely vibrating, uniform beam. In addition, Xp and Yq satisfy desired clamped, simplysupported, or free edge conditions at the ends of the beam. The coefficients are determined bythe 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, butfree edge conditions are only approximated, making the approach usually less accurate when

270 A. W. LEISSA

a free edge is involved. Finally, in this short summary of the method, the orthogonality of thebeam functions and the consequent saving in numerical computational labor should bepointed out.

For each of the 15 problems considered here, the first six beam functions were used in eachco-ordinate direction, yielding 36 terms on the right-hand side ofequation (29). In the generalcase 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 antisymmetricwith respect to the geometric symmetry axis, and the determinant is uncoupled into two 18thorder 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 symmetricmodes 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 befound 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 thesix lowest values of the non-dimensional frequency parameter A. = C1Ja2V pID are given foralb = 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, theC-SS-F-F plate has the edge x = 0 clamped. The other edges in counterclockwise order arethen 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 thefrequency. In many cases the resulting nodal patterns have m - 1 and n - 1 nodal linesrunning parallel to, or approximately parallel to, the y- and x-axes, respectively, but in othercases this geometrical correspondence is at best vague (cf. reference [1]). Where symmetricaland antisymmetrical modes do exist, they can be identified by the mode numbers. Also, insome special cases of square plates having diagonal symmetry in their edge conditions, it isfound that A pq = Aqp 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 eigenfrequencyfrom the single-term Rayleigh solution is also given. The single-term solution is based uponusing the one dominant beam function for the mode and determining the frequency bymeans of Rayleigh's Quotient, inasmuch as no minimization is possible with a single beamfunction. This procedure is the basis of Warburton's [8] useful formulas. In the more generalRitz procedure, the single-term Rayleigh solution conveniently appears as the dominantelement on the diagonal of the characteristic determinant generated.

In the tables, values of A. are given to five significant figures. However, it should not bethought that these values are exact to this degree of accuracy. Indeed, in some scatteredinstances, particularly for alb = 1, more exact numerical results can be found elsewhere inthe published literature, as will be pointed out in detail in the discussion of the individualcases later in this section. From a rigorous mathematical standpoint the only statement thatcan be made concerning the relation of the Ato the exact values is that the present values areupper bounds for the exactvalues. However, from rate ofconvergence studies and comparisonwith known lower bounds from the literature, it is suggested that the Agiven in the tables areordinarily exact to three significant figures. It is also known that the exactness decreasesordinarily with increasing mode number and with the presence offree edges, as will be shownlater.

FREE VIBRATION OF RECTANGULAR PLATES 271A 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 differenceappears 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 afree 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 beenprepared, which gives the average per cent differences for the 30 values of A, available for eachof the 15 cases. Negative differences in the tables were added algebraically so as to diminishthe average differences. Table 5 shows that the average difference for the e-e-e-e case is thesmallest and that, as the constraints are relaxed, ordinarily (1) changing a clamped edge to asimply-supported one increases the difference slightly, (2) changing a simply-supported edgeto 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 all15 cases is seen to be somewhat less than 2%, and the mean difference is a little less than theaverage.

TABLE 5Average differences in A

between one-term and 36-termsolutions

Case

c-c-c-cc-c-c-SSc-c-c-pc-c-SS-SSc-c-p-SSc-c-p-pC-SS-C-PC-SS-SS-FC-SS-P-Fc-p-c-pc-p-ss-pc-p-p-pSS-Ss-p-pSs-p-p-pp-p-p-p

Average ofaverages

Per centdifference

0·380·410·910·461·104·270'540·613-800'580·774-862·93HI3·51

1·90

The significance of a negative error in the tables of Appendix C is particularly interestinginasmuch as it denotes that the one-term Rayleigh eigenvalue is smaller than the 36-termeigenvalue by the indicated per cent. And because both solutions are upper bounds for theexact eigenvalues, a negative error would appear to indicate that the one-term solution ismore 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 candecrease the accuracy of some of the eigenvalues. As a simple numerical illustration considerthe first two symmetric modes of the C-C-C-F square plate. Ifonly the first two x-symmetric

272 A. W. LEISSA

(30)

terms ofequation (29) are retained (i.e., All and A 12 being the only non-zero coefficients), thefollowing characteristic determinant is generated:

1

586,64 - ,1,2 -90,68 1=0.-90,68 1642·56 -.P

It has the roots A.ll = 24'07 and ..1 12 = 40·61. The one-term Rayleigh solutions are obtaineddirectly from the diagonal elements of equation (30), yielding All = 24,22 and ..112 = 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 TableC3 to be 24'020 and 40'039, respectively.

When the principle described above is applied to the ij mode, the terms preceding Al} inequation (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 ofterms ofequation (29) such that p > i and q > j will then decrease (i.e., improve the exactnessof) AiJ. Because the modes of each symmetry class become uncoupled in the solution, thelowest frequency of each symmetry class will never be increased by the addition of terms inequation (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 theco-ordinate system do not recognize this added symmetry. Thus, for example, the modeslabeled 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 representedby 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 upperbound. The inadequacy of the single-term solutions when diagonal symmetry is present waspointed out by Warburton [8].

TABLE 6Boundary condition identities

Antisymmetricmodes of General modes of

C-C-C-CC-C-C-SSC-C-C-FC-F-C-FC-SS-C-FC-F-F-FSS-F-F-FF-F-F-F

C-C-C-SS and C-C-SS-SSC-C-SS-SSC-C-SS-FC-F-SS-F, C-SS-C-F and C-SS-SS-FC-SS-SS-FC-SS-F-FSS-SS-F-FSS-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 obtainedby considering the antisymmetric modes of other cases. These correspondences are summarized in Table 6. Thus, for example, some values of Afor C-C-C-SS and C-C-SS-SS platescan be found from the frequencies of C-C-C-C plates, and conversely. This procedure isdescribed in detail in section 3.

For either very large or very small values ofalb, one set ofopposite edges is widely separatedand the other set is relatively close. In such cases the plate frequencies are related to the beamfrequencies. The frequency parameters for beams having length 1are given in Table 7, whereEIis the beam stiffness. For a beam strip ofthickness It and unit width, 1= h3112 and the beamparameter contains the same quantities as the plate parameter !J [see equation (28)], with a


substituted for I. However. in most cases the values ojTable 7 are the limiting values ojtheplateparameter A, rather than Q. This is due to the added stiffening in a plate due to Poisson ratioeffects, as discussed in section 4.

Aithough the plate parameters Aapproach the beam parameters of Table 7 for large andsmall values of alb, they do so at different rates for the various edge conditions, as can be seenin 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 theplate is 2·5 times as wide as it is long, the edge conditions at y = 0 and b are still quite sufficientto raise All significantly above the beam parameter of 22·373 in every case except the last. Inall six cases (with the possible exception of the last) Au -+ 22'373 as alb -+ O. In the table it isalso seen that the removal of the side constraints of deflection and slope reduces Au in everycase, and that the constraints of deflection are more significant than those of slope.

TABLE 7Frequency parameters w}2 Vp/Eljor beams

m C-C c-ss C-F Ss-ss SS-F F-F

1 22·373 15·418 3·5160 9,8696 same same2 61·673 49·965 22·034 39·478 as as3 120·903 104·248 61-697 88'826 C-SS C-C4 199·859 178'270 120·902 157-9145 298·556 272'031 199'860 246·740

>5 (2m + l?nz/4 (4m + 1)ZnZj16 (2m - l)2nZj4 m2 nZ

TABLE 8211 jor alb = 0·4

Edgeconditions

C-C-C-CC-C-C-SSC-SS-C-SSC-C-C-FC-SS-C-FC-F-C-F

23·64823044023·27722'57722·544220346

In Table 8 it is also seen that 211 for the C-F-C-F case is less than the fundamental beamparameter 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 and120'903, respectively, for all alb, and decreasing with increasing alb. This behavior was seenpreviously for another case when two opposite edges are free: the SS-F-SS-F case, as discussedin section 3.6. Upon looking further it is found that this behavior occurs for these modes in allcases 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 thesemodes approach a limiting value somewhat less than the corresponding beam parameters asalb -+ 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 forthese 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 thepublished 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 literaturefor the C-C-C-C plate is extensive. Reference [1] identifies 37 sources which deal explicitlywith this problem.

Relevant to the present paper, references giving lower bounds or more accurate upperbounds for the frequency parameters are particularly worth noting. Lower bound referencesinclude [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 TableCl. Upper bounds of A. = 35'9866, 131'58, 132·21 for alb = 1 which are closer than those ofTable Cl can be found in reference [29]. Reference [29] is particularly outstanding in thisrespect, giving close upper and lower bounds for the first 15 doubly symmetric modes for arange of0·125 < alb < 1. The extensive, accurate numerical results of reference [33], althoughnot 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 themas distinct, others do not. A proof of whether the modes are distinct, or whether they simplyreflect numerical approximations, is not yet available. Fortunately, this troublesome pointonly 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 isdevoted solely to this problem and therefore deals thoroughly with it. The eigenvaluespresented in reference [24] are also obtained by using the Ritz method with beam functionsfor v = 0·3. Results are given for alb = 0'5,0'75, 1·0, 1'5, 2·0 rather than those ofTable C3, sodirect comparisons can only be made for two of these values. Using ten beam functions inboth the x- and y-directions, thereby giving 100 terms in equation (29) was found to yieldA = 24·00, 40'03, 63·41, 76·73 and 80·70 for alb = 1which are, ofcourse, closer upper boundsthan those of Table C3. Similarly, using eight beam functions in the x-direction and seven inthe 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 withrespect to one diagonal. For example, the first five frequencies are symmetrical, antisymmetrical (A 12 = -A21), symmetrical (A 12 = A21), 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 andv = 0'3. Upon comparing with Table C6 it is then seen thatA for the 12 modefrom using onlynine terms is even further removed (i.e., a larger negative difference) from the one-termeigenvalue than the 36-term result. This is another example of the debilitating effect oflowerterms 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 usedalong 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 TableCIO but, unfortunately, no claim for bounds can be made for the former.


5.5. C-F-F-F (Table C12)

Considerable analytical and experimental results are available for this case from manyliterature sources, as summarized in reference [1]. Eighteen-term Ritz solutions with beamfunctions were used in references [20, 21, 34] with v = 0·3. For comparison purposes, valuesof A= 3'494,8'547,21'44,27'46 and 31·17 were given for alb = 1, which are only slightly lessexact than those of Table C12.

Bazley, Fox and Stadter [35] used 50-term Ritz solutions with beam functions, as well as amethod giving lower bounds to obtain extensive results for the symmetric (only) modes andv = 0'3. Direct comparison with Table C12 is only possible for alb = 1'0, for which the lowerbounds given in reference [35] are 3'4305, 20'874, 26·501 and 51'502. Sigillito [19] used thiscase to demonstrate the improved upper bound convergence possible when using functionsin equation (29) which are the products of beam functions and Legendre functions. Hethereby showed that a 36-term solution gave A= 3'4729,21'304, 27·291 and 54'262 for thesymmetric 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 allij except for j = 1, and that All decreases monotonically with increasing alb as for theSS-P-SS-P case (see the discussion in section 3.6), that the behavior of A21 is erratic. Thiserratic behavior of the 21 frequency compared with the 11 frequency is also seenin the preciseresults of references [19] and [35] for both lower and upper bounds, although the 31 modebehaves monotonically like the 11 frequency.

The inexactitude of the one-term solution for the first antisymmetric mode (112) in TableCl2 (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 representsexcessive stiffness in the system. However, because the error in the one-term solution appearsto be increasing without bound as alb increases, it would seem that gross lack of satisfactionof the free edge boundary conditions is the primary source of the error. This difficulty is seento occur for the A22 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 twostraight 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 insection 5.S also applies here. These two limitations combine to yield relatively poor accuracyfor the single-term Au.

5.7. SS-F-F-F (Table C14)

The erratic (i.e., non-monotonic) behavior of A21 and A31 with increasing alb is of someinterest in Table C14.

5.8. F-F-F-F (Table CIS)

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

276 A. W. LEISSA

The confusion in the literature concerning the existence of modes for this problem (and theshapes of the corresponding nodal patterns) is readily seen in the summarization laid out inreference [I]. Frequently certain vibration modes are not discovered in the analyses.

Particularly noteworthy is the work of references [28] and [19] where accurate upper andlower bounds for the doubly antisymmetric modes (corresponding to m and n both even inTable 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 Aof 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 (A22),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 AdelKadi, Who approximately three years before the author could find the time to write thispaper, wrote the computer programs and obtained the numerical results presented inAppendixes A and C. Without their efforts this paper would have been impossible tocontemplate.

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

transversalen Schwingungen rechteckiger Platten.3. E. F. F. CHLADNI 1787 Entdeckungen aber die Theorie des Klanges. Leipzig.4. E. F. F. CHLADNI 1802 Die Akustik. Leipzig.5. E. F. F. CHLADNI1825 Annalen der Physik, Leipzig 5,345.6. E. F. F. CHLADNI 1817 Neue Beitriige zur Akustik. Leipzig.7. W. RITZ 1909 Annalen der Physik 28, 737-786. Theorie der Transversalschwingungen einer

quadratischen Platte mit freien Randern.8. G. B. WARBURTON 1954Proceedings o/the Institute 0/Mechanical Engineers, ser. A, 168, 371-384.

The vibration of rectangular plates.9. R. JANICH 1962 Die Bautechnik 3,93-99. Die naherungsweise Berechnung der Eigenfrequenzen

von rechteckigen Platten bei verschiedenen Randbedingungen.10. S. IGUCHI 1938 Memorandum o/the Faculty 0/Engineering, Hokkaido University, 305-372. Die

Eigenwertprobleme fUr die elastische rechteckige Platte.11. H. LURIE 1951 Journal ofAeronautical Sciences 18, 139-140. Vibrations of rectangular plates.12. 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 oppositeedges supported.

13. S. T. A. ODMAN 1955 Proceedings NR 24, Swedish Cement and Concrete Research Institute, RoyalInstitute of Technology (Stockholm), 7-62. Studies of boundary value problems. Part II.Characteristic functions of rectangular plates.

14. S. TIMOSHENKO and J. M. GERE 1961 Theory 0/Elastic Stability. New York: McGraw-Hill BookCo., Inc.

15. A. S. VOLMIR 1963 Stability a/Elastic Systems. Moscow: Gos. Izd. Phys.-Mat. Lit. (In Russian.)16. M. LEVY 1899 Comptes rendues 129,535-539.17. C. ZEISSIG 1898 Annalen der Physik 64,361-397. Ein einfacher Fall der transversalen Schwin

gungen einer rechteckigen elastischen Platte.18. V. JANKOVIC 1964 Stavebnicky Casopis 12, 360-365. The solution of the frequency equation of

plates using digital computers. (In Czech.)19. V. G. SIGILLITO 1965 Applied Physics Laboratory, The Johns Hopkins University, Engineering

Memorandum EM-40l2. Improved upper bounds for frequencies of rectangular free andcantilever plates.

20. D. YOUNG 1950 Journal 0/Applied Mechanics 17, 448-453. Vibration ofrectangular plates by theRitz method.


21. M. V. BARTON 1951 Journal ofAppliedMechanics 18, 129-134. Vibration of rectangular and skewcantilever plates.

22. R. P. FELGAR 1950 University of Texas Circular No. 14. Formulas for integrals containingcharacteristic 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 ofMechanical Engineering Science 2,325330. Transient vibration of rectangular plates.

26. A. LEMKE 1928 Annalen der Physik 4, ser. 86, 717-750. Experimentelle Untersuchungen zurW. Ritzschen Theorie der Transversalschwingungen quadratischer Platten.

27. D. A. SIMONS and A. W. LEISSA 1971 Journal of Sound and Vibration 17,407--422. Vibrations ofrectangular cantilever plates subjected to in-plane acceleration loads.

28. N. W. BAZLEY, D. W. Fox and J. T. STADTER 1965 AppliedPhysicsLaboratory, The Johns HopkinsUniversity, Technical Memorandum TG·707. Upper and lower bounds for the frequencies ofrectangular free plates.

29. N. W. BAZLEY, D. W. Fox and J. T. STADTER 1965 AppliedPhysicsLaboratory, The Johns HopkinsUniversity, Technical Memorandum TG-626. Upper and lower bounds for the frequencies ofrectangular 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 ofeigenvalues-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 squareplate clamped at four edges.

32. S. TOMOTIKA 1935 Aeronautical Research Institute Report, Tokyo University 10, 301. On thetransverse 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 thinrectangular 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 HopkinsUniversity, Technical Memorandum TO-70S. Upper and lower bounds for frequencies ofrectangular 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. Vibrationsof a rectangular cantilever plate.

38. A. W. LEISSA and F. W. NIEDENFUHR 1963 American Institute of Aeronautics and AstronauticsJournal 1, 116-120. Bending of a square plate with two adjacent edges free and the othersclamped or simply supported.

278 A. W. LEISSA

APPENDIX ATABULATED DATA FOR PLATES HAVING TWO OPPOSITE SIDES SIMPLY-SUPPORTED

TABLE Al

Frequency parameters A. = wa2 .ypjD for SS-SS-SS-SS plates

albMode A

sequence 0·4 2/3 1'0 1'5 2-5

1 11 11 11 11 1111·4487 14-2561 19·7392 32·0762 71·5564

2 12 12 21 21 2116·1862 27·4156 49-3480 61-6850 101·1634

313 21 12 12 31

24·0818 43-8649 49·3480 98·6960 150'5115

4 14 13 22 31 4135-1358 49·3480 78·9568 111·0330 219·5987

5 21 22 31 22 1241·0576 57·0244 98'6960 128·3049 256·6097

622 23 13 32 22

45-7950 78'9568 98-6960 177-6529 286-2185

7 15 14 32 41 5149'3480 80-0535 128·3049 180·1203 308·4251

823 31 23 13 32

53'6906 93'2129 128·3049 209·7291 335-5665

9 16 32 41 23 6166'7185 106·3724 167·7833 239·3379 416·9908

TABLEA2

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

albMode A.

sequence 0·4 2/3 1·0 1'5 2-5

111 11 11 11 11

12-1347 17-3730 28-9509 56-3481 145-4839

212 12 21 21 21

18·3647 35·3445 54-7431 78·9836 164·7387

3 13 21 12 31 3127-9657 45·4294 69·3270 123-1719 202·2271

414 13 22 12 41

40-7500 62·0544 94·5853 146·2677 261·1053

521 22 31 22 51

41·3782 62'3131 102'2162 170·1112 342·1442

622 23 13 41 12

47·0009 88'8047 129·0955 189·1219 392·8746

723 31 32 32 22

56'1782 94'2131 140'2045 212-8169 415-6906

815 14 23 51 61

56'6756 97·4254 154'7757 276·0012 444-9682

924 32 41 42 32

68·7486 101·0788 170·3465 276'0125 455·3054


TABLEA3

Frequency parameters A = OJa2 y'pfD for SS-C-SS-SS plates

albMode A

sequence 0·4 2/3 1'0 1·5 2·5

111 11 11 11 11

11·7502 15'5783 23·6463 42·5278 103-9227

212 12 21 21 21

17·1872 31·0724 51-6743 69'0031 128'3382

313 21 12 31 31

25·9171 44·5644 58·6464 116·2671 172-3804

414 13 22 12 41

37-8317 55-3926 86·1345 120·9956 237·2502

521 22 31 22 12

41·2070 59-4627 100·2698 147-6353 320·7921

622 23 13 41 51

46'3620 83-6060 113·2281 184'1006 322-9642

715 14 32 32 22

52·9007 88·4384 133·7910 193'8025 346·7382

823 31 23 13 32

54'8720 93-6758 140·8456 243-4964 391·0659

924 32 41 42 61

66·6637 108'1069 168·9585 260·2020 429'2420

TABLEA4

Frequency parameters A= OJa2 y'pfDfor SS-C-SS-Fplates

albMode A

sequence 0·4 2/3 1·0 1'5 2·5

11 11 11 11 1110·1888 10·9752 12·6874 16·8225 30·6277

212 12 12 21 21

13-6036 20·3355 33·0651 45·3024 58·0804

3 13 13 21 12 3120·0971 37-9552 41·7019 61·0178 105·5470

414 21 22 22 12

29'6219 40'2717 63·0148 92'3073 149-4569

21 22 13 31 415 3%382 49·7317 72-3976 93·8293 173·1060

615 14 31 32 22

42'2425 64'1889 90·6114 141·7834 182-8110

722 23 23 13 32

42-9993 67-8993 103·1617 149·6055 235·0155

823 31 32 41 51

49·5740 89·3571 111'8964 162'2413 260·6371

916 24 14 23 42

58·0019 94·5150 131·4287 181·1868 305'2218

280 A. W. LEISSA

TABLEA5

Frequency parameters A. = wa2 y pJD for SS-SS-SS-Fplates

albMode A.

sequence 0·4 2/3 1·0 1·5 2'5

111 11 11 11 11

10·1259 10·6712 11'6845 13-7111 18'8009

212 12 12 21 21

13·0570 18·2995 27'7563 43·5723 50'5405

313 13 21 12 31

18·8390 33·6974 41·1967 47-8571 100·2321

414 21 22 22 12

27·5580 40·1307 59·0655 81·4789 110·2259

515 22 13 31 22

39-3377 48·4082 61·8606 92'6925 147'6317

621 14 31 13 41

39'6118 57·5929 90·2941 124'5635 169·1026

722 23 23 32 32

42·6964 64'7281 94·4837 132-8974 203-7304

823 24 32 23 51

48·7745 89·1859 108·9185 158·9180 257'4791

916 31 14 41 42

54'2497 89·2725 115'6857 161·4205 277·4280

TABLEA6

Frequency parameters A. = wa2 y'pJD for SS-F-SS-F plates

albMode r A

sequence 0·4 2/3 1·0 1'5 2·5

1 11 11 11 11 119·7600 9'6983 9'6314 9'5582 9·4841

212 12 12 12 12

11·0368 12·9813 16·1348 21·6192 33-6228

3 13 13 13 21 2115'0626 22·9535 36·7256 38'7214 38·3629

4 14 21 21 22 2221·7064 39'1052 38·9450 54·8443 75·2037

515 14 22 13 31

31·1771 40·3560 46·7381 65·7922 86·9684

621 22 23 31 32

39·2387 42·6847 70·7401 87-6262 130'3576

722 23 14 23 41

40·5035 54·2400 75·2834 103-9665 155'3211

8 16 15 31 32 1343·6698 66·2301 87-9867 105'1608 156·1248

923 24 32 14 23

44·9416 73·1982 96-0405 152·7784 199·8452


APPENDIX B

ON THE EXISTENCE OF EIGENVALUES SUCH THAT A< m2 n2 (k2 < ( 2)

An excellent attempt to determine the existence ofeigenvalues such that A < m2 n 2 was madein the unpublished work of Fletcher, Woodfield, and Larsen [12]. Therein, proofs similar tothose 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= m2 n2

• Thus,f(A) > 0 for all Ain the range 0 < A< mn, and no eigenvaluescan 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)

(B3)f(A)

For A= 0,j(0) = g(O) , but this is a trivial root. As Aincreases, 112/2 and tanh 112/2 both monotonically increase, and 111/2 and tanhl11/2 both monotonically decrease. At }., = m2 n2

,

171/2 = tanh171/2 = O. Thus,f(A) > g(A) for all Ain the range 0 < A< mn, and no y-symmetriceigenvalues can exist.

Equation (24d) for the y-antisymmetric modes is rewritten as

tanh I1d2 tanh 112/2I1d2 = 112/2 = g(A).

But tane/e < 1 for all e, and decreases monotonically as eincreases. 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-antisymmetriceigenvalues can exist.

CASE 3. ss-c-ss-ss

No eigenvalues can exist. The proof is the same as for the y-antisynunetric modes ofCase 2.

CASE 4. sS-C-SS-F

Equation (19) can be rewritten asf(A) = g(...t), where

f(A) = A2- m4 n4(1- V)2 + [.F + m4 n4(l- V)2] cosh 111 cosh 172' (B4)

g(A) = -m2 n2(~)2 [A2(1 _ 2v) _ m4n4(I _ V)2] sinh 111 sinh 112. (B5)a 111 112

Now

282

and

A. W. LEISSA

Therefore,J(O) = g(O). Furthermore, letting A= m2 n2 [1 - v] gives

(mnb ) (mnb )f(m2 n;2[1 - v]) = 2m4 n4 [1 - V]2 cosh -a- VV cosh -a-V2=V ,

;;- (mnb ) (mnb )g(m2 n2 [1 - v]) = 2m4 n4 [1 - V]2 ,J 2="""; sinh -a-vv sinh -a- vT=V .

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

,'

{V2 (mnb) (mnb .r;:;)}

= m4n;4 V2 -a- sinh -a- Y 2 ,

and then settingf(m2 n2) < g(m2 n2

) yields

(mnb ) v2 (mnb) (mnb )[1 - (1- V)2] + [1 + (1 - V)2] cosh -a-V2' < V2 a sinh -a-V"2 (B6)

as the condition for the existence of eigenvalues. Furthermore, for large values ofmnb V2/a,

(mnb )[1 - (1 - )/)2]« (1 + (1 - V)2] cosh --;-v'2

and

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

and condition (B6) can be approximated by

mb 1 + (1 - v)l V2->-~--

a v2 n(B7)


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

TABLE BlRoots ofequation (B6)

mb>v a

0·0 00

0·1 255·9730·2 57·9830·3 23·4130·4 12'0210'5 7·072

CASE 5. sS-SS-SS-F

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

f(A) = [A + m2 n2(I _ v)]2 tanh 112 ,112

tanh 111g(A) = fA - m2 n2(I - V)]2 --.

111

It is obvious thatf(O) =g(O) > O. Furthermore, letting A= m2 n2 [1 - v] gives

(B8)

(B9)

(BlO)

Therefore, a root of equation (20) will exist in the interval m2 n2(l- v) < A< m2 n2 iff(m2 n2) < g(m2 n2). Upon letting A= m2 n2

,

tanh(V2~)«_v)2• r;:; mnb 2 - v'v2-

a

as the condition for the existence ofeigenvalues. 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 )2-;; > V2n 2 - v

(Bll)

284 A. W. LEISSA

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

TABLEB2

Roots ofequation (BID)

mb>v a

0'0 00

0·1 81·2540·2 18'2310'3 7'2280'4 3'6010'5 2·026

CASE 6. SS~F-SS-F

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

111f(A.) = tanh T' (BI2)

(B13)

For A. = 0,111(0) = 112(0), thereforef(O) = g(O) = tanh(mrcb/2a). Furthermore,f(A.) is positiveand monotonically decreasing for all A. in the interval 0 ~ A. < m2 rc2

• But, g(m2 rc2 [l- v]) = O.Therefore, a root of equation (25c) will exist in the interval m2 rc2(1 - v) < A. < m2 rc2 iff(m 2 rc2 ) <g(m2 rc2

). Upon letting A. = m2 rc2 ,

( V)2 ( mrcb) (mrcb). 1g(m2 rc2) = m4 n4

-- vz-- tanh • r,:; hm -= 00,2-v a av21/1->01'/1

provided that v # O. If v = 0,f(m2 rc2) remains zero, and

( a) 111= -- lim 111 tanh -= O.mnb 1/ 1->0 2

Therefore, provided that v ~ 0, y-symmetric eigenvalues will always exist, and they will exist inthe interval m 2 rc2(l- v) < A. < m2 rc2

•

For the y-antisymmetrie modes, because equation (25d) is the same as equation (20) if 111and 112 are replaced by 111/2 and 112/2, respectively, the results for Case 5 apply, with theheading mb/a in Table B2 being replaced by mb/2a. For example, for v = O'3, y-antisymmetriceigenvalues will exist if mb/a > 2(7'228) = 14'455 (to five significant figure precision).


APPENDIX CTABULATED DATA FOR PLATES NOT HAVING TWO OPPOSITE SIDES SIMPLY-SUPPORTED

TABLECl

Frequency parameters A. = roa2 VpJD for C-C-C-C plates

Modealb

A

sequence 0·4 2/3 1·0 1'5 2·5

11 11 11 11 111 23-648 27·010 35-992 60-772 147'80

0-23% 0-31 % 0'33% 0'31% 0'23%

12 12 21 21 212 27-817 41'716 73·413 93'860 173·85

0'35% 0·44% 0·44% 0'44% 0'35%

13 21 12 12 313 35-446 66'143 73·413 148'82 221'54

0'31% 0'35% 0-44% 0'35% 0'31%

14 13 22 31 414 46'702 66·552 108·27 149'74 291·89

0'40% 0·45% 0·53% 0·45% 0'40%

15 22 31 22 515 61·554 79·850 131·64 179·66 384·71

0'43% 0'50% 0-64% 0'50% 0-43%

21 14 13 41 126 63-100 100'85 132·24 226·92 394'37

0·20% 0'47% 0'18% 0'47% 0'20%

TABLEC2

Frequency parameters A. = roazVpJD for C-C-C-SS plates

albMode r A

sequence 0'4 2/3 1·0 1·5 2'5

11 11 11 11 111 23-440 25-861 31-829 48·167 107·07

0'23% 0'34% 0·42% 0'46% 0'40%

12 12 12 21 212 27'022 38·102 63-347 85·507 13%6

0'22% 0'38% 0'49% 0'55% 0'54%13 13 21 12 31

3 33·799 60'325 71·084 123-99 194·410'29% 0'50% 0·48% 0'45% 0'45%

14 21 22 31 414 44'131 65·516 100·83 143'99 270·48

0·41% 0'34% 0'41% 0'51% 0'52%15 22 13 22 12

5 58·034 77·563 116·40 158·36 322·550'49% 0'30% 0·49% 0·50% 0'27%

21 14 31 32 22

6 62-971 92'154 130·37 214·78 353·430·18% 0'52% 0'40% 0'23% 0·44%

286 A. W. LEISSA

TABLEC3

Frequency parameters A. = a>a2 -yfpjD for C-C-C-Fplates (v = 0'3)

albMode r A

sequence 0·4 2/3 1·0 1·5 2·5

11 11 11 11 1122'577 23-015 24'020 26-731 37-6560·29% 0'57% 0'84% 1-01 % 0·92%

12 12 12 12 212 24·623 29-427 40'039 65-916 76·407

0'96% 1'28% 1'22% 1'04% 1-47%

13 13 21 21 313 29·244 44·363 63·493 66·219 135'15

0·58% 0'68% 0'75% 1-16% 1-47%

14 21 13 22 124 37-059 62·417 76·761 106·80 152·47

0·44% 0'42% 0-72% 1-67% 0'69%15 14 22 31 22

5 48·283 68·887 80·713 125·40 193·010'50% 0·64% 1'76% 0·91 % 1·16%

21 22 23 13 416 61-922 69·696 116-80 152·48 213-74

0'18% 1·37% 0'80% 0'60% 1'27%

TABLE C4

Frequency parameters A. = wa2 -yfp/D for C-C-SS-SS plates

albMode A

sequence 0-4 2/3 1-0 1-5 2·5

11 11 11 11 111 16-849 19·952 27·056 44·893 105-31

0'41% 0-55% 0-58% 0'55% 0·41 %

12 12 21 21 212 21-363 34'024 60'544 76-554 133-52

0·36% 0'52% 0'76% 0'52% 0·36%

13 21 12 12 313 29·236 54·370 60·791 122-33 182·73

0-44% 0'44% 0'35% 0-44% 0'44%

14 13 22 31 414 40-509 57'517 92-865 129-41 253-18

0-55% 0'58% 0'28% 0'58% 0'55%

21 22 13 22 125 51·457 67-815 114·57 152·58 321-60

0'24% 0-28% 0'54% 0·28% 0-24%

15 14 31 41 516 55-117 90'069 114-72 202·66 344-48

0-59% 0'55% 0-42% 0-55% 0'59%


TABLE C5Frequency parameters A= wa2 y'pfDfor C-C-SS-Fplates (v = 0'3)

albMode A

sequence 0·4 2/3 1·0 1'5 2·5

11 11 11 11 111 15-696 16·287 17·615 21'035 33-578

0'55% 1'04% 1·40% 1·46% 1-07%

12 12 12 21 212 18·373 24-201 36-046 55·184 66-612

1'60% 1'77% 1'46% 1·50% 1'66%

13 13 21 12 313 23'987 40·701 52-065 63'178 119-90

0-85% 0-83% 1'01 % 1'08% 1·71 %

14 21 22 22 124 32-810 50·822 71·194 99·007 150'83

0·61 % 0'58% 1·98% 1-53% 0'64%

15 22 13 31 225 44-862 59·071 74·349 109·22 187-61

0'63% 1'70% 0·72% 1·12% 0'80%21 14 31 13 41

6 50·251 66'262 106·28 150-90 193-230'25% 0'70% 0'65% 0·56% 1·46%

TABLE C6Frequency parameters A= wa2 VP7D for C-C-P-Pplates (v = 0'3)

albMode A.

sequence 0·4 2/3 1·0 1'5 2·5

11 11 11 11 111 3'9857 4·9848 6-9421 11·216 24·911

5'l2% 6'99% 7-23% 6'99% 5'22%

12 12 21 21 212 7'1551 13'289 24'034 29'901 44'719

5-93% 6·14% 10'24% 6'14% 5·93%

13 21 12 12 313 13'101 23-384 26'681 52'615 81·879

4'24% 2-87% -0'70% 2'87% 4-24%

14 13 22 31 414 21·844 30-262 47'785 68'090 136'52

5011% 3'30% 5-64% 3·30% 5'11 %

21 22 13 22 125 22·896 34·240 63·039 77-041 143·10

-0'55% 4'59% 4'00% 4'59% -0·55%

22 23 31 32 226 26-501 52·398 65·833 117·90 165-63

3-36% 3-82% -0,41 % 3'82% 3-36%

288 A. W. LEISSA

TABLEC7

Frequency parameters A. = maZyp/D for C-SS-C-Fplates (v = 0'3)

albMode A.

sequence 0·4 2/3 1'0 1·5 2·5

11 11 11 11 111 22·544 22-855 23-460 24·775 28'564

0'06%. 0'11% 0'17% 0·25% 0'46%12 12 12 12 21

2 24'296 27-971 35·612 53·731 70'5610'42% 0·73% 0'94% 1'09% 0·34%

13 13 21 21 123 28·341 40·683 63·126 64·959 114'00

0'50% 0'73% 0·15% 0'23% 0'97%14 21 13 22 31

4 35-345 62'310 66·808 97·257 130·840·49% 0·09% 0·85% 1-12% 0'27%

15 14 22 31 225 45·710 62·695 77'502 124·48 159·54

0·50% 0·73% 0-90% 0·18% 1'17%16 22 23 13 41

6 59'562 68'683 108·99 127·92 210-320'49% 0'58% 0'81% 0'74% 0-24%

TABLEC8

Frequency parameters A. = waz.vp/D for C-SS-SS-Fplates (v = 0'3)

albMode A

sequence 0·4 2/3 1'0 1·5 2'5

11 11 11 11 1115'649 16'067 16·865 18·540 23·0670-12% 0·20% 0'29% 0-40% 0·73%

12 12 12 12 212 17-946 22·449 31-138 50·442 59·969

0'73% HO% 1-25% 1'24% 0·36%

13 13 21 21 123 22-902 36·703 51·631 53·715 111·95

0·77% 0-93% 0'20% 0'28% 0·92%

14 21 13 22 314 30·892 50-696 64·043 88'802 115-11

0'70%. 0-12% 0'92% 1'01 % 0'32%

15 22 22 31 225 42·108 57·908 67·646 108·19 153·24

0'66% 0-72% 0'98% 0'22% 0·79%

21 14 23 13 416 50·222 59·840 101·21 126'09 189·49

0·06% 0'81 % 0'62% 0'70% 0'28%


TABLE C9Frequency parameters A= wazVpJD for C-SS-F-Fplates (v = 0'3)

albMode A.

sequence 0'4 2/3 1·0 1'5 2·5

11 11 11 11 113'8542 4-4247 5'3639 6'9309 10-1002'10% 3·66% 5-27% 8'24% 14'77%

12 12 12 21 212 6·4198 10·912 19-171 27·289 35-157

5-48% 6·65% 7'07% 3'12% 3·98%

13 21 21 12 313 11'576 22-958 24·768 38'586 74·990

4'68% 1-78% 0'67% 3·70% 1-87%

14 13 22 22 124 19·767 25-698 43-191 64·254 99'928

4'08% 3-06% 4-85% 6·43% 1'90%

21 22 13 31 225 22'521 32'425 53-000 67-467 127-69

0'01% 3-33% 2·48% -0'16% 5'28%

22 23 31 32 416 26-024 48·467 64·050 108·02 135-45

1-96% 4·52% 0-35% 2'89% 0'06%

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

albMode A

sequence 0·4 2/3 1'0 1-5 2-5

11 11 11 11 1122'346 22-314 22·272 22·215 22-130

0-12% 0-27% 0-46% 0·71% 1'10%

12 12 12 12 122 23'086 24·309 26'529 30·901 41·689

0'06% 0·13% 0'25% 0'56% 1·55%

13 13 13 21 213 25-666 31-700 43·664 61·303 61·002

0'42% 0'68% 0'98% 0'60% 1·10%

14 14 21 13 224 30'633 46·820 61·466 70'960 92-384

0'75% 1·01 % 0'34% 1-20% 0'61%

15 21 22 22 315 38-M7 61·566 67·549 74·259 119·88

0'33% 0'19% 0'16% 0'27% 0'85%

16 22 14 23 326 49·858 64'343 79·904 118'33 157·76

0'49% 0-09% 1-03% 0'99% 0'23%

290 A. W. LEISSA

TABLE CllFrequency parameters A = roa2 VpjD for C-P-SS-F plates (v = 0'3)

albMode .A

rsequence 0·4 2/3 1·0 1'5 2'5

11 11 11 11 111 15'382 15-340 15'285 15·217 15-128

0·24% 0·51 % 0'87% 1'32% 1·92%

12 12 12 12 122 16'371 17'949 20'673 25·711 37·294

0'12% 0-23% 0'44% 0'89% 1·98%13 13 13 21 21

3 19-656 26·734 39·775 49·550 49·2260·69% 0'98% 1'22% 0·84% 1'50%

14 14 21 22 224 25·549 43·190 49·730 64-012 83-325

1-08% 1·21 % 0·47% 0-25% 0'41%

15 21 22 13 315 34·507 49·840 56'617 68'126 103014

0'52% 0·25% 0'18% 1'25% 1'07%16 22 14 31 32

6 46·435 53·013 77·368 103·70 143·680'64% 0·11% 1'07% 0'53% 0·23%

TABLE C12Frequency parameters A = roa2 VpjD for C-P-F-F plates (v = 0'3)

albMode "-

sequence 0·4 2/3 1·0 1'5 2·5

11 11 11 11 111 3·5107 3·5024 3·4917 3·4772 3-4562

1·74% 0'39% 0·70% 1'12% 1'74%

12 12 12 12 122 4·7861 6-4062 8·5246 11'676 17'988

4-17% 7·09% 11'56% 17'45% 24-36%

13 13 21 21 213 8·1146 14·538 21·429 21·618 21·563

7'20% 9-09% 2'83% 1'93% 2'19%

14 21 13 22 224 13·882 22·038 27'331 39·492 57'458

6'69% -0'02% 5032% 4'74% 8'54%

21 22 22 13 315 21-638 26·073 31·111 53·876 60·581

1'83% 3·46% 3'17% 5'92% 1'84%

22 14 23 31 326 23·731 31-618 54·443 61-994 106'54

0-84% 3'37% 5'99% -0-48% 2-49%


TABLE C13Frequency parameters A= wa2 y'pjD for SS-SS-F-Fplates ("\I = 0'3)

albMode A

\

sequence 0·4 2/3 1·0 1·5 2'5

11 11 11 11 111·3201 2·2339 3·3687 5'0263 8·25067'56% 5-93% 5·37% 5'93% 7'56%

12 12 12 21 212 4·7433 9·5749 17-407 21·544 2%46

4'11 % 3'28% 7'80% 3·28% 4'11 %

13 21 21 12 313 10·362 16·764 19·367 37·718 64·760

2·02% 1'33% -3,11 % 1-33% 2'02%

21 13 22 31 124 15-873 24·662 38·291 55·490 99'206

0'80% 1·45% 4'24% 1·45% 0'80%

14 22 13 22 415 18'930 27·058 51·324 60·882 118'31

1·46% 3·31 % 3·01 % 3-31 % 1·46%

22 23 31 32 226 20·171 44·172 53·738 99·388 126'07

1'92% 2-96% -1-61 % 2·96% 1'92%

TABLE C14Frequency parameters A= wa2 y'pjD for SS-P-P-Fplates (v = 0'3)

albMode A

sequence 0·4 2/3 1·0 1·5 2'5

12 12 12 12 21It 2-6922 4·481 6'6480 9·8498 14·939

5-48% 5'63% 6'79% 8'11% 3·21 %

13 13 21 21 122 6·5029 13·009 15·023 15·013 16·242

4'37% 6·30% 2'63% 2·70% 9·27%

14 21 22 22 313 12-637 15-674 25·492 34·027 48·844

3-35% -1,63% 3-43% 4'61% 2'29%

21 22 13 31 224 15-337 20·373 26·126 48·332 52·089

0·53% 3·08% 1-87% 3'38% 6'84%

22 14 31 13 325 17'510 30'548 48·711 55·066 97·225

0'70% 1'14% 2'57% -0'50% 2'69%

15 23 23 32 416 21-699 33'411 50·849 70·695 102·34

3'03% 4'23% 2'37% 1'87% 1'87%

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

292 A. W. LEISSA

TABLE CI5Frequency parameters A= roa2 vpjD for F-F-F-Fplates (v = 0'3)

albMode .A

sequence 0·4 2(3 1'0 1·5 2'5

13 22 22 22 31It 3·4629 8·9459 13·489 20·128 21'643

3'37% 5·81 % 5·26% 5-81% 3-37%

22 13 13 31 222 5·2881 9·6015 19·789 21-603 33·050

7·40% 3'56% 13·06% 3'56% 7'40%

14 23 31 32 413 9·6220 20·735 24·432 46·654 60~137

2'55% 4·37% -8·43% 4'37% 2'55%

23 31 32 13 324 11·437 22-353 35'024 50'293 71·484

5'58% 0'09% 4'20% 0'09% 5·58%

15 14 23 41 515 18·793 25·867 35·024 58-201 117·45

2'94% 5-96% 4·20% 5'96% 2'94%

24 32 41 23 426 19·100 29·973 61·526 67·494 119·38

3·41% -1-67% 0'24% -1'67% 3·41%

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

APPENDIX DNOTATION

Am,Bm, Cm, Dm constants of integration in equation (7)Apq amplitudes of trial functions in equation (29)a, b length dimensions of a rectangular plate in the x and y directions, respectively

C clamped edge indicatorD plate flexural rigidity; equals Eh 3 /12(1-v2 )

e 2·71828 ...E Young's modulusF free edge indicatorh plate thicknessI second moment of the area ("moment of inertia") for a beamk [pw2 /D]1/4I length of a beam

M",My plate bending moments in the x and y directions, respectivelym, 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 inthe x direction

p,q beam function summation indices, according to equation (29)SS simply-supported edge indicator

t timew transverse plate deflection, W = w(x,y, t)W transverse plate deflection, W = W(x,y)

Xp , Yq beam functions in the x and y directions, respectively


x,y co·ordinates in the middle plane of a plateoc mnla

rPl' rP2 defined by equations (15)A. non·dimensional frequency parameter, equals wa2 yfplDv Poisson's ration 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 Laplacianoperator

Leissa_the Free Vibration of Rectangular Plates

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