ARTICLE IN PRESS. International Journal of Mechanical Sciences

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1 International Journal of Mehanial Sienes 51 (009) 6 55 Contents lists available at SieneDiret International Journal of Mehanial Sienes journal homepage: Exat solutions for the free in-plane vibrations of retangular plates Y.F. Xing, B. Liu The Solid Mehanis Researh Center, Beijing University of Aeronautis and Astronautis, Beijing , China artile info Artile history: Reeived 3 August 008 Reeived in revised form 0 Deember 008 Aepted 3 Deember 008 Available online 0 January 009 Keywords: Retangular plate In-pane free vibration Frequeny Mode shape Exat solution abstrat All lassial boundary onditions inluding two distint types of simple support boundary onditions are formulated by using the Rayleigh quotient variational priniple for retangular plates undergoing inplane free vibrations. The diret separation of variables is employed to obtain the exat solutions for all possible ases. It is shown that the exat solutions of natural frequenies and mode shapes an be obtained when at least two opposite plate edges have either type of the simply-supported onditions, and some of the exat solutions were not available before. The present results agree well with FEM results, whih show that the present solutions are orret and the diret separation of variables is pratial. The exat solutions an be taen as the benhmars for the validation of approximate methods. & 009 Elsevier Ltd. All rights reserved. 1. Introdution Corresponding author. address: xingyf@buaa.edu.n (Y.F. Xing). There is no doubt that transverse vibrations of plates are of great pratial importane, sine their natural frequenies are prone to most of the external exitation, and as suh there is an extensive literature relevant to the free transverse vibrations of retangular plates. On the ontrary, only a few studies are dediated to the free in-plane vibrations (FIV) of plates over the years, sine the natural frequenies involved are muh higher and beyond the level of available exitations. However, it has been found that the in-plane vibrations an be exited in the strutures suh as the hulls of oean-going ships and the shells of flight vehiles, et. Hene it is also important to study the in-plane vibrations of plates. A signifiant ontribution to this subjet was made by Bardell et al. [1], who alulated the in-plane vibrational frequenies using the Rayleigh Ritz method, and provided a valuable review of the related literature available up to that time, inluding the pioneering wor of Lord Rayleigh [] dealing with what was referred to as simply-supported plates. Gorman [3] introdued the superposition method as a means to obtain the analytial-type FIV solutions of retangular plates with ompletely free boundaries, fully lamped boundaries [] and elasti supports normal to the boundaries [5]. Du et al. [6] also analyzed the FIV of retangular plates with elastially restrained edges by using an improved Fourier series method, in whih the in-plane displaements are expressed as the superposition of a double Fourier osine series and four supplementary funtions. Additionally, Seo et al. [7] performed an FIV analysis of a antilevered retangular plate by using a variational approximation proedure, wherein the differential equations and trationfree onditions on two opposite edges are satisfied exatly and the remaining onditions are satisfied variationally. Singh et al. [8] investigated the FIV of isotropi non-retangular plates aording to the variational method, wherein the displaement fields are represented by muh higher order polynomials than the ones used for the geometri representation. And Woodo et al. [9] studied the effets of the ply orientation on in-plane vibrations based on the Rayleigh Ritz formulation in onjuntion with Hamilton priniple. It is noteworthy that there have been some exat solutions for the FIV of plates. Par [10] derived the exat frequeny equations for the FIV of the lamped irular plate by using the separation of the variables. Gorman [11] obtained the exat solutions for the FIV of retangular plates with two opposite edges simply supported, the other opposite edges being both lamped or both free. In Gorman s elegant wor, only one quarter of the retangular plate was analyzed, and it was shown that by this approah, the interpretation of the omputed mode shapes with mode family separation beomes a muh more manageable tas, the probability of missing an eigenvalue an be greatly redued, and the problem of repeated eigenvalues an be avoided. In present study, the exat solutions for the FIV of retangular plates are attempted. There are several apparent differenes between the present wor and Gorman s [11] as follows. (1) Rayleigh quotient variational priniple is employed to derive the mathematial representations of all possible boundary /$ - see front matter & 009 Elsevier Ltd. All rights reserved. doi: /j.ijmesi

2 Y.F. Xing, B. Liu / International Journal of Mehanial Sienes 51 (009) onditions inluding lamped ondition, free ondition and two distint types of simple support boundary onditions that are denoted by the symbols SS1 and SS [11]. () The diret separation of variables is used to solve the governing equations. By this approah, the exat solutions an be obtained readily. The solution proedure shows that the exat solutions are available only when at least two opposite plate edges are simply supported. (3) The entire retangular plate is analyzed diretly, and there are no problems suh as the interpretation of the omputed mode shapes, the probability of missing an eigenvalue, and the repeated eigenvalues. () All possible exat solutions are obtained, inluding the solutions for the ases SS1 C and SS1 SS, et., whih were not available before. The paper is organized as follows. In Setion, the formulations of all boundary onditions are given by using the Rayleigh quotient variational priniple; then in Setion 3 the exat solutions are obtained through the separation of variables; finally in Setion the numerial experiments are onduted and the results are ompared with those by FEM.. Differential equations and boundary onditions Consider the harmoni normal vibrations of a retangular plate as shown in Fig. 1. The maximum strain energy P max and the maximum ineti energy T max an be represented, respetively, as P max ¼ 1 ZZ e T Ee dx dy A T max ¼ o T 0 ; T 0 ¼ 1 ZZ rðu þ v Þ dx dy (1) A where e and E are the strain vetor and elasti matrix, respetively. By means of Rayleigh quotient variational priniple dp max ¼ o dt 0, one an obtain @y þ ro u du @x þ ro v dv dx dy A Z ½ðs x l þ tmþdu þðs y m þ tlþdvš ds ¼ 0 () G where l ¼ osðn; xþ ¼os y, and m ¼ osðn; yþ ¼sin y, n is the outer normal diretion of the boundary. Due to the neessary and suffiient onditions of Rayleigh quotient priniple, the governing equations and the homogeneous boundary onditions an be obtained. The latter are given in Table 1, and the governing Table 1 The lassi boundary onditions of retangular plate. B.C. x ¼ 0ora y ¼ 0orb Clamped (C) u ¼ v ¼ 0 u ¼ v ¼ 0 Free (F) s x ¼ 0, t ¼ 0 s y ¼ 0, t ¼ 0 Simply supported u ¼ 0, t ¼ 0 (SS) u ¼ 0, s y ¼ 0 (SS1) v ¼ 0, s x ¼ 0 (SS1) v ¼ 0, t ¼ 0 (SS) equations @y þ ro u ¼ @x þ ro v ¼ 0 (3) þ u þ þ u o u 1 ¼ þ v þ þ u o v 1 ¼ 0 () where u and v are the in-plane p displaements in x and y diretions, respetively, ¼ ffiffiffiffiffiffiffiffiffi G=r is the shear wave veloity, and u 1 ¼ 1 u ; u ¼ 1 þ u (5) By solving Eq. (), one an obtain the modes and natural frequenies. Further, it is not diffiult to solve Eq. () when the four plate edges are simply supported through the inverse method. 3. The exat eigensolutions For the free transverse vibrations of retangular thin plate, the authors have obtained the exat results for the ases SSCC, SCCC and CCCC by means of the diret separation of variables [1]. The diret separation of variables is employed again to obtain the exat solutions for the FIV of plates. The separation-of-variable solution of Eq. () an be written as uðx; yþ ¼Ae mx e ly vðx; yþ ¼Be mx e ly (6) Substitution of Eq. (6) into Eq. () leads to m þ u 1 l o 3 þ u 1 lmu A 6 lmu l o 7 5 ¼ 0 þ u 1 m þ u B 0 1 The existene of nontrivial solutions of A and B requires that ðl þ m Þ þ 3 u or l þ m þ o o ðl o þ m Þþu 1 (7) ¼ 0 (8) l o þ m þ u 1 ¼ 0 (9) From Eq. (9), one an have m 1;3 ¼iO; m ; ¼iL (10) l 1;3 ¼iT; l ; ¼iZ (11) Fig. 1. Plate and oordinates. where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi O ¼ l þ o rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; L ¼ l o þ u 1 (1)

3 8 Y.F. Xing, B. Liu / International Journal of Mehanial Sienes 51 (009) 6 55 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ¼ m þ o rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o ; Z ¼ m þ u 1 Therefore, the solutions of Eq. () an be written as (13) uðx; yþ ¼f 1 ðxþf ðyþ vðx; yþ ¼ 1 ðxþ ðyþ (1) where f 1 ðxþ ¼A 1 os Ox þ A sin Ox þ A 3 os Lx þ A sin Lx 1 ðxþ ¼C 1 os Ox þ C sin Ox þ C 3 os Lx þ C sin Lx f ðyþ ¼B 1 os Ty þ B sin Ty þ B 3 os Zy þ B sin Zy ðyþ ¼D 1 os Ty þ D sin Ty þ D 3 os Zy þ D sin Zy (15) By substituting Eq. (1) into Eq. (), one an find that the solutions are meaningless exept for the following forms: uðx; yþ ¼f 1 ðxþe ly vðx; yþ ¼ 1 ðxþe ly or uðx; yþ ¼f ðyþe mx vðx; yþ ¼ ðyþe mx (16a,b) It is shown below that Eqs. (16) an be satisfied only when at least two opposite plate edges are simply supported. The relations of f 1 (x) and 1 (x) should be determined prior to the derivation of the exat solutions, whih an be obtained by inserting Eq. (16a) into Eq. () as C ¼ O l A ) 1 ¼ 1 A 1 1 ¼ O l C 1 ¼ O l A where ¼ 1 A l ¼ O o (17) C ¼ l L A 3 ¼ A 3 C 3 ¼ l L A ¼ A ) where ¼ L l l ¼ L u o (18) 1 Then the f 1 (x) and 1 (x) in Eq. (15) an be rewritten as f 1 ðxþ ¼A 1 os Ox þ A sin Ox þ A 3 os Lx þ A sin Lx 1 ðxþ ¼ A 1 os Ox þ A 1 1 sin Ox A os Lx þ A 3 sin Lx(19) Similarly, one an obtain the relations of f (x) and (x). It follows from Eq. (19) that, if f 1 (x) is a sine funtion, 1 (x) must be a osine funtion, and vie versa. Therefore, Eqs. (16) an only be satisfied when at least two opposite plate edges are simply supported. This paper assumes the opposite plate edges x ¼ 0 and a to be simply supported, i.e., Eq. (16b) will be used below The eigensolutions for the simply-supported edges x ¼ 0 and a There are four ombinations of the simply-supported onditions for the edges x ¼ 0 and a, as given in Table. The ase SS SS is solved here for the eigenfuntions and eigenvalue equations. The SS onditions are u ¼ 0; t @x or f 1 ð0þ ¼f 1 ðaþ ¼0 ¼ 0 (0) 0 1 ð0þ ¼0 1ðaÞ ¼0 (1) Substituting Eq. (19) into Eq. (1), one an obtain A 1 ¼ A 3 ¼ 0, and " #" # sin O a sin La A ¼ 0 () 1 O sin Oa L sin La 0 A Then the eigenvalue equation is ð L 1 OÞsin Oa sin La ¼ 0 (3) Sine sin Oa ¼ 0 and sin La ¼ 0 are equivalent, here only sin Oa ¼ 0 is onsidered, and A ¼ 0 from Eq. (). Thus the normal eigenfuntions in Eq. (19) an be obtained as f 1 ðxþ ¼sin Ox 1 ðxþ ¼ 1 os Ox () As the eigenfuntions f 1 (x) and 1 (x) are the fators of modes, whih are used below to derive the eigenfuntions f (y) and (y), Eq. () an be rewritten as f 1 ðxþ ¼sin Ox 1 ðxþ ¼os Ox (5) All possible eigenfuntions and eigenvalue equations for the simply-supported edges x ¼ 0 and a an be solved in the same way. These eigensolutions are listed in Table from whih one an see that, although there are four ombinations of SS1 and SS, there are only two distint types of eigenfuntions and eigenvalue equations. 3.. The eigenfuntions for arbitrary opposite edges y ¼ 0 and b Based on f 1 (x) and 1 (x) obtained by using the simplysupported onditions of the edges x ¼ 0 and a, the eigenfuntions f (y), (y) and orresponding eigenvalue equations an be derived by using the boundary onditions of the other opposite edges y ¼ 0 and b. Assume that the in-plane natural mode of the plate are given in separation of variables form as uðx; yþ ¼f ðyþf 1 ðxþ vðx; yþ ¼ ðyþ 1 ðxþ (6) where f 1 (x) and 1 (x) are given in Table. Let m ¼ io, and substitute it into Eq. (13) to obtain rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o T ¼ O rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o Z ¼ u 1 O (7) whih show that T and Z may be real or pure imaginary. And there are three ases as follows. Case 1: u 1 ðo=þ XO. The eigenvalues T and Z are real from Eq. (7). Thus the eigenfuntions in Eq. (15) an be used diretly, Table 3 The oeffiient relations of the funtions f (y) and (y). f 1 ðxþ ¼sin O x 1 ðxþ ¼os O x f 1 ðxþ ¼os O x 1 ðxþ ¼sin O x Table The four ombination of SS1 and SS for the edges x ¼ 0 and a and eigensolutions. Eigenvalue equations Eigenfuntions SS SS sin Oa ¼ 0 f 1 (x) ¼ sin Ox, 1 (x) ¼ os Ox SS1 SS1 sin Oa ¼ 0 f 1 (x) ¼ os Ox, 1 (x) ¼ sin Ox SS SS1 os Oa ¼ 0 f 1 (x) ¼ sin Ox, 1 (x) ¼ os Ox SS1 SS os Oa ¼ 0 f 1 (x) ¼ os Ox, 1 (x) ¼ sin Ox Case 1 B 1 ¼ 3 D ; B ¼ 3 D 1 B 3 ¼ D ; B ¼ D 3 B 1 ¼ 3 D ; B ¼ 3 D 1 B 3 ¼ D ; B ¼ D 3 Case B 1 ¼ 3 D ; B ¼ 3 D 1 B 3 ¼ D ; B ¼ D 3 B 1 ¼ 3 D ; B ¼ 3 D 1 B 3 ¼ D ; B ¼ D 3 Case 3 B 1 ¼ 3 D ; B ¼ 3 D 1 B 3 ¼ D ; B ¼ D 3 B 1 ¼ 3 D ; B ¼ 3 D 1 B 3 ¼ D ; B ¼ D 3

4 Y.F. Xing, B. Liu / International Journal of Mehanial Sienes 51 (009) given by f ðyþ ¼B 1 os Ty þ B sin Ty þ B 3 os Zy þ B sin Zy ðyþ ¼D 1 os Ty þ D sin Ty þ D 3 os Zy þ D sin Zy (8) Case : ðo=þ XO u 1 ðo=þ. The eigenvalue T is real, and Z is pure imaginary, from Eq. (7). For simpliity, the eigenvalue Z is hanged from imaginary to real, as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o T ¼ O rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o Z ¼ u 1 þ O (9) Table The eigensolutions for simply-supported edges y ¼ 0 and b. Eigenvalue equations Eigenfuntions SS SS sintb ¼ 0 f ðyþ ¼ 3 os Ty; ðyþ ¼sin Ty sin Zb ¼ 0 f ðyþ ¼ os Zy; ðyþ ¼sin Zy SS1 SS1 sintb ¼ 0 f ðyþ ¼ 3 sin Ty; ðyþ ¼os Ty and the eigenfuntions in Eq. (15) should be modified aordingly as f ðyþ ¼B 1 os Ty þ B sin Ty þ B 3 osh Zy þ B sinh Zy ðyþ ¼D 1 os Ty þ D sin Ty þ D 3 osh Zy þ D sinh Zy (30) Case 3: ðo=þ oo. Both T and Z are pure imaginary values, and they an also be hanged to be real as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ¼ O o rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ¼ O o u 1 (31) The orresponding eigenfuntions in Eq. (15) are rewritten as f ðyþ ¼B 1 osh Ty þ B sinh Ty þ B 3 osh Zy þ B sinh Zy ðyþ ¼D 1 osh Ty þ D sinh Ty þ D 3 osh Zy þ D sinh Zy (3) To substitute Eqs. (8), (30) and (3) into Eq. (), the oeffiient relations of f (y) and (y) an be determined for the above three ases that are given in Table 3, in whih 3 ¼ T/O, ¼ O/Z. sin Zb ¼ 0 f ðyþ ¼ sin Zy; ðyþ ¼os Zy. The eigenvalue equations and numerial results SS SS1 os Tb ¼ 0 f ðyþ ¼ 3 os Ty; ðyþ ¼sin Ty Table 5 The four ombinations of eigenvalue equations. 1 3 sin Oa ¼ 0 sin Tb sin Zb ¼ 0 os Zb ¼ 0 sin Oa ¼ 0 os Tb os Zb ¼ 0 f ðyþ ¼ os Zy; ðyþ ¼sin Zy SS1 SS os Tb ¼ 0 f ðyþ ¼ 3 sin Ty; ðyþ ¼os Ty os Zb ¼ 0 f ðyþ ¼ sin Zy; ðyþ ¼os Zy os Oa ¼ 0 sin Tb sin Zb ¼ 0 os Oa ¼ 0 os Tb os Zb ¼ 0 The eigenvalue O and the orresponding eigenfuntions f 1 (x) and 1 (x) for the simply-supported edges x ¼ 0 and a are given in Table. And the eigenfuntions for another two edges y ¼ 0 and b have also been derived for all three ases, see Eqs. (8), (30) and (3) and Table 3. The remaining problems are to derive the eigenvalue equations orresponding to the opposite edges y ¼ 0 and b aording to the relevant boundary onditions, and then to solve for the natural frequenies by using Eqs. (7), (9) and (31). Consider a retangular plate with in-plane dimension a b ¼ 1 m 1. m, the volume density r ¼ 800 g/m 3, Young s modulus E ¼ Pa, and Poisson s ratio u ¼ 0.3. The alulated frequenies are given in dimensionless frequeny parameter b ¼ oa/p, and the frequenies by FEM are denoted by b*. Exept the results for the simplest ases with four simply-supported edges, all frequenies and mode shapes are ompared with those by FEM, whih are obtained by MSC/NASTRAN with the mesh and using membrane elements. Table 6 The frequeny parameter b and the values of m, n and i. x ¼ 0, a SS1 SS1 SS SS y ¼ 0, b SS1 SS1 SS SS SS SS SS1 SS (m,n) ¼ (0,1) (m,n) ¼ (0,1) (m,n) ¼ (1,1) (m,n) ¼ (1,0) (m,n) ¼ (1,0) (m,n) ¼ (1,1) (m,i) ¼ (0,1) (m,n) ¼ (1,1) (m,n) ¼ (1,1) (m,n) ¼ (0,) (m,i) ¼ (1,0) (m,i) ¼ (0,1) (m,n) ¼ (0,) (m,i) ¼ (1,0) (m,n) ¼ (1,) (m,n) ¼ (1,) (m,n) ¼ (1,) (m,n) ¼ (1,).1667 (m,n) ¼ (,1).0000 (m,n) ¼ (,0) (m,n) ¼ (,0).1667 (m,n) ¼ (,1).003 (m,i) ¼ (1,1).1667 (m,n) ¼ (,1) (m,n) ¼ (,1).003 (m,i) ¼ (1,1).603 (m,n) ¼ (,).003 (m,i) ¼ (1,1) (m,i) ¼ (1,1).5000 (m,n) ¼ (0,3).696 (m,n) ¼ (1,3).603 (m,n) ¼ (,) (m,n) ¼ (0,3).603 (m,n) ¼ (,).817 (m,i) ¼ (0,).696 (m,n) ¼ (1,3) (m,n) ¼ (,).696 (m,n) ¼ (1,3) (m,n) ¼ (3,1).817 (m,i) ¼ (0,) T ¼ 0, yes T ¼ 0, no T ¼ 0, no T ¼ 0, yes Z ¼ 0, no Z ¼ 0, yes Z ¼ 0, yes Z ¼ 0, no O ¼ 0, Tb ¼ np, yes O ¼ 0, Tb ¼ np, yes O ¼ 0, Tb ¼ np, no O ¼ 0, Tb ¼ np, no O ¼ 0, Zb ¼ ip, no O ¼ 0, Zb ¼ ip, no O ¼ 0, Zb ¼ ip, yes O ¼ 0, Zb ¼ ip, yes

5 50 Y.F. Xing, B. Liu / International Journal of Mehanial Sienes 51 (009) The edges y ¼ 0 and b are simply supported For the simply-supported opposite edges y ¼ 0 and b, all possible eigensolutions an be obtained readily by using Eqs. (8), (30) and (3) and boundary onditions, whih are given in Table. It follows from Tables and that there are many different ombinations of SS1 and SS, whih is a signifiant differene from the free out-of-plane vibrations of plate when the four edges are simply supported. For the retangular plate with four simply-supported edges, the frequenies are available only for Cases 1 and, and the Case is involved in Case 1. It is noteworthy that the Case 3 is available only for the plate with at least one free edge. Using the obtained eigenvalues O, T and Z from the eigenvalue equations, the frequenies an be alulated from Eq. (7) as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o mn ¼ T n þ O m sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z o i þ O m mi ¼ (33) u 1 and o mn ao mi for any positive integers n and i. There are no rigid body modes, and hene the eigenvalues T and O annot be zero Fig.. Mode shapes of plate with four simply-supported edges, b ¼ (a) SS1 SS1 SS1 SS1, (b) SS1 SS SS1 SS, () SS SS SS SS and (d) SS-SS1-SS-SS1. Table 7 The eigenvalue equations when the edges y ¼ 0 and b are arbitrary. Case 1 C C 1 os Tb os Zb ¼ 1 3 þ sin Tb sin Zb 3 SS C 3 tan Zb ¼ tan Tb SS F a 1 tan Tb ¼ a tan Zb 1 os Tb os Zb ¼ 1 a 1 þ a sin Tb sin Zb a a 1 SS1 C 3 tan Tb ¼ tan Zb a tan Tb ¼ a 1 tan Zb a 3 1 a os Tb os Zb þ a 3 a 1 sin Tb sin Zb ¼ a 1 a 3 Case C C 1 os Tb osh Zb ¼ 3 sin Tb sinh Zb 3 SS C 3 tanh Zb ¼ tan Tb SS F a 1 tan Tb ¼ a 3 tanh Zb 1 os Tb osh Zb ¼ 1 a 3 a 1 sin Tb sinh Zb a 1 a 3 SS1 C 3 tan Tb ¼ tanh Zb a 3 tan Tb ¼ a 1 tanh Zb a 3 1 þ a 3 os Tb osh Zb þ a 3 3 a 1 sin Tb sinh Zb ¼ a 3 a 1 3 Case 3 1 osh Tb osh Zb ¼ 1 sinh Tb sinh Zb a 3 þ a a a 3 SS F a tanh Tb ¼ a 3 tanh Zb a 3 tanh Tb ¼ a tanh Zb a 3 þ a 3 osh Tb osh Zb a 3 3 þ a sinh Tb sinh Zb ¼ a 3 a 3

6 Y.F. Xing, B. Liu / International Journal of Mehanial Sienes 51 (009) simultaneously, as also Z and O. There are four ombinations, as shown in Table 5, for the eigenvalue equations in Tables and. Here only the first ombination (the first olumn of Table 5) is disussed for the analysis of frequeny properties. For this ombination, Eq. (33) an be written as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o mn ¼ ðnp=bþ þðmp=aþ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðip=bþ o þðmp=aþ mi ¼ u 1 (3) It an be seen from Tables and that the same frequenies an be alulated from Eq. (3) for the four ombinations SS SS SS SS, SS SS1 SS SS1, SS1 SS SS1 SS and SS1 SS1 SS1 SS1. The first 10 frequenies are given in Table 6 from whih it follows that, if O, T and Z are not equal to zero, there are four repeated frequenies; if one of O and T or one of O and Z equals zero, the boundary onditions should be heed, and there are two repeated frequenies. For the ase SS1 SS1 SS1 SS1, T an be zero, but not Z; and if O ¼ 0, Tb ¼ np is the root, but not Zb ¼ ip. Similar analysis an be performed for the other three ombinations. The mode shapes are drawn in Fig. for b ¼ , whih orresponds to the four repeated frequeny... The edges y ¼ 0 and b are arbitrary Aording to the oeffiient relations in Table 3 and Eqs. (8), (30) and (3), one an derive the eigenfuntions f (y), (y) and the orresponding eigenvalue equations by means of the arbitrary boundary onditions of the edges y ¼ 0 and b. But only for the seven ases C C,, SS1 C, SS C,, SS F and, the eigenvalue equations and the oeffiient relations of eigenfuntions are derived here, whih are given in Table 7 and Table A1, respetively. The simply-supported and lamped boundary onditions are separable, but the free boundary onditions are not separable for FIV of plate, as is the ase for the free transverse vibrations. For the sae of brevity, only the ase is onsidered below. The free boundary onditions are t ¼ @x ¼ 0 ) f 1f 0 þ 0 1 ¼ 0 s y ¼ 0 @y ¼ 0 ) uf0 1 f þ 1 0 ¼ 0 (35) Table 8 The frequeny parameters b for the ase y ¼ 0 and b are lamped (C C). 1(1) (1) 3() (1) 5(1) 6() 7(1) 8(1) 9(1) 10() Oa/p Tb/p Zb/p b b* x ¼ 0,a SS1 SS1 SS SS Both SS1 SS1 Both Both Both SS1 SS1 Both Both Table 9 The frequeny parameters b for the ase y ¼ 0 and b are free (). 1(3) (1) 3() (1) 5(1) 6(1) 7(1) 8(3) 9(3) 10(1) Oa/p Tb/p Zb/p b b* x ¼ 0,a Both SS1 SS1 Both SS SS SS1 SS1 Both Both Both Both Both Table 10 The frequeny parameters b for the ase y ¼ 0 is lamped, and y ¼ b is free (). 1(1) (1) 3(3) (1) 5() 6(3) 7(1) 8(1) 9(1) 10(1) Oa/p Tb/p Zb/p b b* x ¼ 0,a SS1 SS1 SS SS Both SS1 SS1 Both Both Both Both SS1 SS1 SS SS Table 11 The frequeny parameters b for the ase SS1 C SS F. 1() (1) 3(3) (1) 5(1) 6() 7(3) 8(1) 9(1) 10(1) Oa/p Tb/p Zb/p b b *

7 5 Y.F. Xing, B. Liu / International Journal of Mehanial Sienes 51 (009) 6 55 There are two distint types of eigenfuntions for the simplysupported edges x ¼ 0 and a as given in Table, whih are f 1 ðxþ ¼sin Ox 1 ðxþ ¼os Ox and f 1ðxÞ ¼os Ox 1 ðxþ ¼sin Ox (36a,b) for whih the signs of the oeffiients of f and are opposite. This means that the sign of the seond olumn in Table 3 is opposite to that of the third olumn for three ases. Thus, the substitution of Eq. (36a) or Eq. (36b) into Eq. (35) leads to the same result, as f 0 þ O ¼ 0 uof 0 ¼ 0 (37) Then one has f 0 ð0þþo ð0þ ¼0 uof ð0þ 0 ð0þ ¼0 ; f 0 ðbþþo ðbþ ¼0 uof ðbþ 0 (38a,b) ðbþ ¼0 By solving Eqs. (38), one an obtain the eigenvalue equations as shown in Table 7, and the eigenfuntion oeffiients as shown in Table A1. The parameters a i (i ¼ 1,,3,) in these two tables are given by a 1 ¼ 3T O Z þ O ¼ T O ; a O ¼ 3Ou T uþ ¼ ZTð1 Ou þ Z O u þ Z a 3 ¼ 3Ou T ZTðu 1Þ ¼ Z Ou Z O u ; a ¼ 3T þ O þ O Z þ O ¼ T (39) O Fig. 3. Mode shapes of plate with both lamped edges y ¼ 0 and b (C C). (a) SS1 C SS1 C, (b) SS C SS C, () SS1 C SS1 C and (d) SS C SS C. Fig.. Mode shapes of plate with both free edges y ¼ 0 and b (). (a), (b) SS F SS-F, () and (d) SS F SS F.

8 Y.F. Xing, B. Liu / International Journal of Mehanial Sienes 51 (009) The numerial results and omparisons are presented in Tables 8 11, wherein the numbers in parentheses (*) of the first rows indiate that the frequenies are omputed from the eigenvalue equation of Case*, see Table 7. The both in Tables 8 10 means that the two ases SS1 SS1 and SS SS (for x ¼ 0 and a) have the same frequenies. The boundary onditions for the edges x ¼ 0 and a are SS1 SS in Table 11. The mode shapes are shown and ompared with those by FEM in Figs There are several points pertaining to the numerial results as follows. (1) The omputed frequenies and mode shapes agree with those by FEM, see Tables 8 11 and Figs () For the ases when the edges y ¼ 0 and b are C C, and, the eigenvalues T and Z annot be zero, but O an be zero, see Tables IfO ¼ 0, Tb ¼ np/ (n ¼ 1,,3,y) orrespond to SS1 SS1 (for x ¼ 0 and a), but Zb ¼ ip/ (i ¼ 1,,3,y) orrespond to SS SS (for x ¼ 0 and a). (3) If the edges x ¼ 0 and a are SS SS1 or SS1 SS orresponding to the eigenvalue equation os O a ¼ 0, O, T and Z annot be zeros, and there are no repeated frequenies, see Table 11. Fig. 5. Mode shapes of plate with lamped edge (y ¼ 0) and free edge (y ¼ b) (). (a) SS1 C, (b) SS C SS F, () SS1 C and (d) SS C SS F. Fig. 6. Mode shapes of plate with SS1 C SS F.

9 5 Y.F. Xing, B. Liu / International Journal of Mehanial Sienes 51 (009) Conlusions Anowledgement All possible exat solutions for the free in-plane vibrations of retangular plate have been derived by using the diret separation of variables for the first time. The exat solutions for the ases in whih two opposite edges are simply supported and the other two opposite edges are asymmetrial suh as SS1 C, et. were not available before. One an see from present wor that the method of diret separation of variables is powerful. The present wor provides further insight into the overall subjet of the free in-plane vibrations of retangular plate. The authors gratefully anowledge the support from the National Natural Siene Foundation of China (Grant no ). Appendix A See Table A1 for details. Table A1 The oeffiients of the eigenfuntions f (y) and (y). Case 1 C C D 3 ¼ D 1 ; D ¼ 3 D ; D ¼ f 1 D 1 D 3 ¼ D 1 a 1 ; D ¼ D a ; D ¼ D 1 f 3 f 1 ¼ ðos Tb os ZbÞ sin Tb þ 3 sin Zb SS C sin Tb D 1 ¼ D 3 ¼ 0; D ¼ D sin Zb SS F sin Tb D 1 ¼ D 3 ¼ 0; D ¼ D a 1 sin Zb f 3 ¼ a 1ðos Tb os ZbÞ a 1 sin Tb a sin Zb SS1 C D ¼ D ¼ 0; D 3 ¼ D 1 os Tb os Zb D ¼ D ¼ 0; D 3 ¼ D 1 a 1 os Tb os Zb D 3 ¼ D 1 ; D ¼ 3 D ; D ¼ D 1 f 6 ; f 6 ¼ a 1 os Tb þ os Zb a 1 sin Tb þ 3 sin Zb Case C C D 3 ¼ D 1 ; D ¼ 3 D ; D ¼ f D 1 D 3 ¼ D 1 a 1 ; D ¼ D a 3 ; D ¼ D 1 f f ¼ ðos Tb osh ZbÞ sin Tb 3 sinh Zb SS C sin Tb D 1 ¼ D 3 ¼ 0; D ¼ D sinh Zb SS F sin Tb D 1 ¼ D 3 ¼ 0; D ¼ D a 1 sinh Zb f ¼ a 1ðos Tb osh ZbÞ a 1 sin Tb a 3 sinh Zb SS1 C os Tb D ¼ D ¼ 0; D 3 ¼ D 1 osh Zb os Tb D ¼ D ¼ 0; D 3 ¼ D 1 a 1 osh Zb D 3 ¼ D 1 ; D ¼ 3 a 1 os Tb þ osh Zb D ; D ¼ D 1 f 7 ; f 7 ¼ a 1 sin Tb þð 3 = Þsinh Zb Case3 D 3 ¼ D 1 a ; D ¼ D a 3 ; D ¼ f 5 D 1 D 3 ¼ D 1 ; D ¼ 3 D ; D ¼ D 1 f 8 f 5 ¼ a ðosh Tb osh ZbÞ a sinh Tb a 3 sinh Zb f 8 ¼ a osh Tb þ osh Zb a sinh Tb þ 3 sinh Zb SS F sinh Tb D 1 ¼ D 3 ¼ 0; D ¼ D a sinh Zb D ¼ D ¼ 0; D 3 ¼ D 1 a osh Tb osh Zb

10 Y.F. Xing, B. Liu / International Journal of Mehanial Sienes 51 (009) Referenes [1] Bardell NS, Langley RS, Dunsdon JM. On the free in-plane vibration of isotropi plates. J Sound Vib 1996;191(3): [] Lord Rayleigh. The theory of sound, vol. 1. New Yor: Dover; 189. p [3] Gorman DJ. Free in-plane vibration analysis of retangular plates by the method of superposition. J Sound Vib 00;7: [] Gorman DJ. Aurate analytial type solutions for the free in-plane vibration of lamped and simply supported retangular plates. J Sound Vib 00;76: [5] Gorman DJ. Free in-plane vibration analysis of retangular plates with elasti support normal to the boundaries. J Sound Vib 005;85: [6] Du JT, Li WL, Jin GY, Yang TJ, Liu ZG. An analytial method for the in-plane vibration analysis of retangular plates with elastially restrained edges. J Sound Vib 007;306: [7] Seo JW, Tiersten HF, Sarton HA. Free vibrations of retangular antilever plates. Part : In-plane motion. J Sound Vib 00;71: [8] Singh AV, Muhammad T. Free in-plane vibration of isotropi non-retangular plates. J Sound Vib 00;73: [9] Woodo RL, Bhat RB, Stiharu IG. Effet of ply orientation on the in-plane vibration of single-layer omposite plates. J Sound Vib (007), doi: / j.jsv [10] Par CI. Frequeny equation for the in-plane vibration of a lamped irular plate. J Sound Vib (008), doi: /j.jsv [11] Gorman DJ. Exat solutions for the free in-plane vibration of retangular plates with two opposite edges simply supported. J Sound Vib 006;9: [1] Xing YF, Liu B. New exat solutions for free vibrations of thin orthotropi retangular plates. Compos Strut (008), doi: /j.ompstrut