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VIBRATION OF RECTANGULAR PLATES USING PLATE CHARACTERISTIC FUNCTIONS AS SHAPE FUNCTIONS IN THE RAYLEIGH–RITZ METHOD C. R, R. B. B G. D. X Department of Mechanical Engineering, Concordia University, Montreal, Quebec, Canada (Received 16 November 1994 and in final form 4 October 1995) The plate characteristic functions are used to express the deflection shapes in the Rayleigh–Ritz method to study rectangular plate vibrations. Since the plate characteristic functions are reasonable approximations to the vibration modes, they are found to improve the convergence of vibration frequencies. These plate frequencies are used to check the accuracy of the Rayleigh frequencies associated with the plate characteristic function modes. Computations are carried out for each of the four mode categories of a clamped square plate. In each mode category, the first 30 natural frequencies are tabulated and the first nine mode diagrams are drawn. The results show that the choice of specific shape functions enhances the effectiveness of the Rayleigh–Ritz method. 7 1996 Academic Press Limited

1. INTRODUCTION

Closed form solutions for rectangular plate vibration modes are known only for certain special cases in which at least a pair of opposite sides is simply supported. For other types of support conditions, approximate methods such as Rayleigh–Ritz are used to study rectangular plate vibrations, [1–3]. In the Rayleigh–Ritz method, the vibration mode is expressed as a linear combination of a set of assumed shape functions, which satisfy at least the geometrical boundary conditions. The coefficients in this modal expression are then chosen to minimize the Rayleigh quotient. This procedure culminates in a matrix eigenvalue problem, which is solved numerically using EISPACK subroutines [4]. It is well known that the accuracy of the Rayleigh–Ritz method depends on the choice of the shape functions used to describe the vibration mode. Reasonable approximations to the plate vibration modes are typically used as the shape functions. Since the plate characteristic functions are the optimum separable solutions to the plate vibration equation subjected to the plate boundary conditions, they are used as shape functions in the present investigation. These functions are obtained using an improved form of the variational method proposed by Kantorovich and Krylov [5]. In the Kantorovich–Krylov approach, the plate vibration mode is expressed as a product of an unknown function of one spatial variable and an assumed function of both variables. The variational technique is then used to reduce the partial differential equation to an ordinary differential equation in the unknown function. Kerr [6] observed that the presence of both variables in the assumed function results in an ordinary differential equation with variable coefficients. He further indicated that these coefficients can be made constant by restricting the independent variables of the assumed function to the other spatial variable only. Jones and Milne [7] used such an extension of the Kantorovich–Krylov 497 0022–460X/96/220497 + 13 $18.00/0

7 1996 Academic Press Limited

. .

498

approach alternatively in the principal directions to obtain convergent separable solutions to the vibration modes of clamped rectangular plates. Bhat, Singh and Mundkur [8] used Galerkin’s reduction to obtain expressions for averaged natural boundary conditions and applied the extended Kantorovich–Krylov approach successively in alternative principal directions until convergence, to obtain the approximate vibration modes and the corresponding Rayleigh frequencies for plates with different boundary conditions. Bhat et al. [8] named these functions as the plate characteristic functions. The authors further modified the extended Kantorovich–Krylov approach to obtain the optimum separable solutions to the partial differential equation governing plate vibration [9]. In this modification, the variational method is used to reduce the partial differential equation to two simultaneous ordinary differential equations with constant coefficients. Imposition of the averaged boundary conditions on the solutions of the simultaneous ordinary differential equation simplifies to four non-linear algebraic equations in four unknown modal parameters, which are solved numerically. This procedure can be adapted to generate accurate expressions for a large number of plate characteristic functions. Since the plate characteristic function is a separable function of the spatial variables, the zeroes of this function lie along straight lines parallel to the plate edges. Thus, the plate characteristic function cannot represent vibration modes with curved nodal lines. However, such a vibration mode can be represented as a linear combination of the plate characteristic functions. Thus, the plate characteristic functions are the natural choice for the shape functions in the Rayleigh–Ritz analysis of rectangular plate vibration. Plate functions obtained by a single application of the extended Kantorovich–Krylov reduction procedure once along either direction of a rectangular plate were used by Bhat and Mundkur [10] in the Rayleigh–Ritz method to study plate vibrations. In the present investigation, a large number of fully converged plate characteristic functions are generated, and are used as the shape functions for rectangular plate vibrations, in order to compute results with significantly improved accuracy. 2. THEORY

Rectangular plates with clamped or simply supported or free edges are frequently encountered in structural dynamics. Under such support conditions, the exact vibration mode of a uniform rectangular plate shown in Figure 1 can be obtained by minimizing the energy integral I=

gg$ 60 1 2

D

1 2w 1 2w + 1x 2 1y 2

1

2

− 2(1 − n)

0

0 1 17

1 2 w 1 2w 1 2w 2 2 − 1x 1y 1x1y

Figure 1. The rectangular plate.

2

%

− 12 mv 2w 2 dx dy.

499

The above energy integral can be rewritten in terms of the non-dimensional co-ordinates (x¯ , y¯ ) as I=

gg $60

1 2w 1 2w + 1x¯ 2 1y¯ 2

D 2ab

1

2

− 2(1 − n)

0

0 1 17

1 2w 1 2w 1 2w 2 2 − 1x¯ 1y¯ 1x¯1y¯

%

2

− V 2w 2 dx¯ dy¯ .

(1)

The function w(x¯ , y¯ ) that minimizes the energy integral I satisfies the exact plate vibration equation 4 1 1 4w 1 4w 2 1 w − V 2w = 0 2 4 +2 2 2+a a 1x¯ 1x¯ 1y¯ 1y¯ 4

and the corresponding boundary conditions. The exact solution of the plate vibration equation subjected to the plate boundary conditions is difficult to obtain except for the special case, when at least a pair of opposite sides is simply supported. However, approximate solutions for plate vibration modes can be obtained from equation (1) using variational methods. In particular, variational calculus can be used to obtain the best separable solution w(x¯ , y¯ ) = X(x¯ )Y(y¯ ) that minimizes the energy integral I in equation (1). These optimum separable solutions for the plate vibration modes are called the plate characteristic functions. Using equation (1), the variational energy equation dI = 0 can be simplified to

g$ 1

0

%

1 (00) iv B X + 2{B (02) − (1 − n)b (01)}X0 − {V 2B (00) − a 2B (04) − a 2(b (12) − b (03))}X dX dx¯ a2

g$

6

1

+

a 2A (00)Y iv +2{A (02) −(1−n)a (01)}Y0− V 2A (00) −

0

7%

1 (04) 1 (12) A − 2 (a −a (03)) Y dY dy¯ a2 a

1 1 1 1 −[V x dX]xx¯¯ = y dY]yy¯¯ = x dX']xx¯¯ = y dY']yy¯¯ = = 0 − [V = 0 + [M = 0 + [M = 0 = 0,

(2)

where A (00) =

g

1

g

1

X 2 dx¯ ,

A (02) =

0

B (00) =

g

1

g

1

XX0 dx¯ ,

A (04) =

0

Y 2 dx¯ ,

0

B (02) =

g

1

g

1

XX iv dx¯ ,

(3–5)

YY iv dy¯ ,

(6–8)

0

YY0 dy¯ ,

0

B (04) =

0

1 a (01) = [XX']xx¯¯ = = 0,

1 a (03) = [XX1]xx¯¯ = = 0,

1 a (12) = [X'X0]xx¯¯ = = 0,

(9–11)

1 b (01) = [YY']yy¯¯ = = 0,

1 b (03) = [YY1]yy¯¯ = = 0,

1 b (12) = [Y'Y0]yy¯¯ = = 0,

(12–14)

1 (00) B X1 + ((2 − n)B (02) − 2(1 − n)b (01))X' , a2

7

(15)

V y (y¯ ) = {a 2A (00)Y1 + ((2 − n)A (02) − 2(1 − n)a (01))Y'},

(16)

V x (x¯ ) =

M x (x¯ ) =

6

6

7

1 (00) B X0 + nB (02)X , a2

M y (y¯ ) = {a 2A (00)Y0 + nA (02)Y}.

(17, 18)

. .

500

Here [f(x¯ )] denotes f(1) − f(0). Since dX and dY are arbitrary variations satisfying the boundary constraints, equation (2) reduces to x¯ = 1 x¯ = 0

1 (00) iv B X + 2{B (02) − (1 − n)b (01)}X0 − {V 2B (00) − a 2B (04) − a 2(b (12) − b (03))}X = 0, a2

6

a 2A (00)Y iv + 2{A (02) − (1 − n)a (01)}Y0 − V 2A (00) −

7

1 (04) 1 (12) A − 2 (a − a (03)) Y = 0. a2 a

(19) (20)

Here, equation (19) is a fourth order differential equation in X with constant coefficients, and these coefficients are related to the variable Y through equations (6–8) and (12–14). A similar observation can also be made for equation (20). Furthermore, the averaged expressions for the boundary conditions of equations (19) and (20) can also be obtained from equation (2). For example, the boundary conditions at the edge x¯ = 0 can be written as (i) clamped:

X(0) = 0,

X'(0) = 0.

(ii) simply supported:

X(0) = 0,

X0(0) = 0,

(iii) free:

M x (0) = 0,

V x (0) = 0.

(21)

The boundary conditions at the other edges can be expressed in a similar manner. These eight homogeneous boundary conditions can be used to solve for the unknown functions X(x¯ ) and Y(y¯ ) and the unknown frequency parameter V from equations (19) and (20). In the context of rectangular plate vibration, the solutions of equations (19) and (20) can be expressed as X(x¯ ) = Cx1 cos p1 x¯ + Cx2 sin p1 x¯ + Cx3 cosh p2 x¯ + Cx4 sinh p2 x¯ ,

(22)

Y(y¯ ) = Cy1 cos q1 y¯ + Cy2 sin q1 y¯ + Cy3 cosh q2 y¯ + Cy4 sinh q2 y¯ ,

(23)

p22 − p12 = −2a 2{B (02) − (1 − n)b (01)}/B (00),

(24)

where

2 2

2 1

q − q = −2{A

(02)

− (1 − n)a }/(a A ), (01)

2

(00)

(25)

p22 p12 = a 2{V 2B (00) − a 2B (04) − a 2(b (12) − b (03))}/B (00), q q = {V A 2 2

2 1

2

(00)

(04)

2

− A /a − (a

(12)

− a )/a }/(a A ). (03)

2

2

(00)

(26) (27)

For certain plate boundary conditions, some of the modal parameters p1 , p2 , q1 and q2 need not take real values. Consequently, these parameters are treated as complex numbers in this analysis. The four homogeneous boundary conditions at x¯ = 0 and x¯ = 1 can be used to determine the four unknown coefficients in equation (22) within an arbitrary non-zero multiplication factor. Furthermore, these boundary conditions impose a relationship between the parameters p1 and p2 in the form P(p1 , p2 ) = 0.

(28)

Equation (28) is similar to the frequency equation of an analogous strut in the x direction. Similarly, the four homogeneous boundary conditions at y¯ = 0 and y¯ = 1 of equation (23) yield the frequency equation in the y-direction as Q(q1 , q2 ) = 0.

(29)

It can be observed that the four unknown modal parameters p1 , p2 , q1 and q2 can be determined from equations (24), (25), (28) and (29). From equations (6–8), it can be seen

501

that the right-hand side of equation (24) is a function of q1 and q2 . Similarly, the right-hand side of equation (25) is a function of p1 and p2 . The numerical method for the determination of parameters p1 , p2 , q1 and q2 from equations (24), (25), (28) and (29), for the ith mode, is discussed in detail in reference [9]. After determining p1(i) , p2(i) , q1(i) and q2(i), the ith frequency parameter corresponding to the plate characteristic function can be deduced from equations (19), (21) and (24–27) as V2 =

1 (i)2 (i)2 2n(1 − n)a (01)b (01) 1 (i)2 (i)2 (i)2 (i)2 2 (i)2 (i)2 . 2 p1 p2 + a q1 q2 − 2 (p2 − p1 )(q2 − q1 ) − A (00)B (00) a

(30)

These frequencies associated with the plate characteristic functions fi (x¯ , y¯ ) = Xi (x¯ )Yi (y¯ ) are good approximations for the plate frequencies [7–9]. Furthermore, these frequencies can be determined without solving the higher order matrix eigenvalue problem, which is prone to computational limitations. Since the zeroes of the plate characteristic function lie on straight lines parallel to the principal directions, these functions cannot accurately represent vibration modes with prominently curved nodal lines. However, a linear combination of the plate characteristic functions can be used to express a vibration mode with curved nodal lines. Since the plate characteristic functions are the best approximations to the plate vibration modes, these functions are assumed to form a set of linearly independent functions. Thus the Rayleigh–Ritz analysis with plate characteristic functions as assumed shape functions can be used to generate accurate results for the vibration frequencies and modes of rectangular plates under various support conditions. For such an analysis, it is desirable to arrange the plate characteristic functions so that the associated Rayleigh frequencies are in ascending order.

3. RAYLEIGH–RITZ ANALYSIS

For the Rayleigh–Ritz analysis, the vibration mode is expressed in terms of the plate characteristic functions as n

n

i=1

i=1

w(x¯ , y¯ ) = s ci fi (x¯ , y¯ ) = s ci Xi (x¯ )Yi (y¯ ).

(31)

Using equation (1), the expressions for the maximum kinetic and potential energy can be expressed as Tmax = 12 V 2

U max = 12

gg$6

1 1 2w 1 2w 2 +1 a 1x¯ 1y¯ 2

7

gg

w 2 dx¯ dy¯ ,

2

− 2(1 − n)

6

(32)

0 1 7%

1 2w 1 2w 1 2w 2 2 − 1x¯ 1y¯ 1x¯1y¯

2

dx¯ dy¯ .

(33)

Substitution of equation (31) into equations (32, 33) gives n

n

Tmax = 12 V 2 s s mij ci cj , i=1 j=1

n

n

U max = 12 s s kij ci cj , i=1 j=1

(34, 35)

. .

502 where

kij =

mij = Aij(00)Bij(00) ,

(36)

1 (22) (00) A B + a 2Aij(00)Bij(22) + n(Aij(02)Bij(20) + Aij(20)Bij(02) ) + 2(1 − n)Aij(11)Bij(11) , a 2 ij ij

(37)

Aij(mn) =

g

1

Xi(m)Xj(n) dx¯ ,

Bij(mn) =

0

g

1

Yi(m)Yj(n) dy¯ .

(38, 39)

0

In equations (38) and (39), the notation Xi(m) (or Yi(m) ) represents the mth derivative of the function Xi (or Yi ) with respect to x¯ (or y¯ ). Using integration by parts, the expression for kij in equation (37) can be further simplified to kij =

1 (22) (00) A B + 2Aij(11)Bij(11) + a 2Aij(00)Bij(22) + n(aij(01)bij(10) + aij(10)bij(01) ) a 2 ij ij − (aij(01) + aij(10) )Bij(11) − (bij(01) + bij(10) )Aij(11) ,

(40)

where 1 aij(mn) = [Xi(m)Xj(n) ]xx¯¯ = = 0,

1 bij(mn) = [Yi(m)Yj(n) ]yy¯¯ = = 0.

(41, 42)

Substitution of equations (34) and (35) into the energy equation U max = Tmax gives V2 =

{C}T[K]{C} . {C}T[M]{C}

(43)

In the Rayleigh–Ritz method, the coefficient vector {C} must be chosen to minimize the Rayleigh quotient on the right-hand sie of equation (43). This minimization procedure culminates in the matrix eigenvalue problem [K]{C} = V 2[M]{C}.

(44)

4. CLAMPED RECTANGULAR PLATES

For clamped rectangular plates, the vibration modes can be subdivided into four categories depending on their symmetry or antisymmetry about the axes x¯ = 1/2 and y¯ = 1/2. The abbreviated names SS, SA, AS and AA can be used to identify these mode categories. In this nomenclature, the first character represents the symmetric (S) or antisymmetric (A) property of the mode about x¯ = 1/2 and the second denotes a similar property about y¯ = 1/2. Since Xi (x¯ ) is either symmetric (S) or antisymmetric (A) about x¯ = 1/2, it can be expressed as

6 6

7 7

F cosh p2(i)(x¯ − 1/2) − cos p1(i)(x¯ − 1/2) G cosh 12 p2(i) cos 12 p1(i) Xi (x¯ )=g (i) (i) G sinh p2 (x¯1 −(i) 1/2) − sin p1 (x¯1 −(i) 1/2) sinh 2 p2 sin 2 p1 f

(S), (45) (A).

The frequency equation in the x-direction becomes P(p (i)1, p2(i)) = 0, where P(p1(i), p2(i)) =

6

p2(i) tanh 12 p2(i) + p1(i) tan 12 p1(i) (i) 2

1 2

(i) 2

(i) 1

1 2

p coth p − p cot p

(i) 1

(S), (A).

(46)

Using equations (45) and (46), the expressions for A from equations (38) as

(00) ij

503 (11) ij

,A

(22) ij

and A

can be determined

F 1 1 1 1 G2 p2(i)2 − p2( j)2 − p1(i)2 − p1( j)2 − p2(i)2 + p1( j)2 − p1(i)2 + p2( j)2 (li − lj ), Aij(00) =g G (p2( j)2 − p1( j)2) 2 f1 − 2p2( j)2p1( j)2 {1 − (1 − lj ) },

$6

7 6

7%

i$j i = j, (47)

F p1( j)2 p2( j)2 p2( j)2 p1( j)2 − 2 + + + li (i)2 ( j)2 (i)2 ( j)2 (i)2 ( j)2 (i)2 G p2 − p 2 p1 − p 1 p2 + p 1 p1 + p2( j)2 G Aij(11) =g p2(i)2 p1(i)2 p2(i)2 p1(i)2 +2 lj , (i)2 ( j)2 + (i)2 ( j)2 − (i)2 ( j)2 + (i)2 p2 − p 2 p1 − p 1 p2 + p 1 p1 + p2( j)2 G G 1 ( j)2 ( j)2 2 f−[ 2 (p2 − p1 ) + {1 − (1 − lj ) }],

$6 $6

7 6 7 6

7% 7%

F p2(i)2p2( j)2 p2(i)2p1( j)2 p1(i)2p1( j)2 p1(i)2p2( j)2 G2 (i)2 (li − lj ), ( j)2 − (i)2 ( j)2 + (i)2 ( j)2 − (i)2 p2 − p 2 p1 − p 1 p2 + p1 p1 + p2( j)2 Aij(22) =g G f12 (p2( j)4 + p1( j)4) + 12 (p2( j)2 − p1( j)2){1 − (1 − lj )2},

$6

7 6

7%

i $ j, i = j, (48) i $ j, i = j, (49)

where li =

6

p2(i) tanh 12 p2(i) (i) 2

1 2

p coth p

(i) 2

(S), (A).

(50)

The expressions for Aij(00) , Aij(11) and Aij(22) in equations (47–50) are useful in the determination of the mass and stiffness matrices associated with clamped rectangular plates from equations (36) and (40). For clamped plates, the expressions for Bij(00) , Bij(11) and Bij(22) can also be obtained from equations (47–50) by replacing the parameters p1 and p2 with q1 and q2 , respectively. 5. RESULTS AND DISCUSSION

Even though equations (47–50) are valid for any a, the special case of a clamped square plate (a = 1) is taken for numerical illustration. Thirty plate characteristic functions, reported in reference [9], for each of the four mode categories of a clamped square plate are used as shape functions for the Rayleigh–Ritz analysis. Computations of mij using these functions in equations (36) and (47) show that the non-diagonal elements of the mass matrix are several orders smaller than the diagonal elements. Since these non-diagonal elements are not exactly zero, the plate characteristic functions are not the exact solutions for the plate vibration mode. Even though these chosen shape functions are not orthogonal, they are assumed to form a complete linearly independent function basis for the expression of the vibration mode. Since the computation of the plate characteristic functions does not involve the numerical solution of a larger order matrix eigenvalue problem, several hundreds of these

. .

504

T 1 Rayleigh frequencies associated with plate characteristic functions of a clamped square plate i

SS mode

SA mode

AS mode

AA mode

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

35·99896 131·90213 131·90213 220·05865 309·03784 309·03784 393·35577 393·35577 562·17816 565·45203 565·45203 648·02057 648·02057 813·74701 813·74701 900·91730 900·91730 982·56171 982·56171 1062·22656 1146·34802 1146·34802 1315·37891 1315·37891 1392·38538 1392·38538 1396·46777 1396·46777 1559·01111 1559·01111 1720·19373

73·40536 165·02304 210·52634 296·36633 340·59042 427·35699 467·29092 510·64716 596·36670 677·74500 720·48639 723·30811 805·35010 927·70618 931·50360 969·99420 1054·16382 1098·27515 1179·61035 1217·16357 1259·10535 1342·71008 1345·76782 1467·53381 1546·11096 1552·22681 1587·78076 1633·11487 1670·92175 1795·20190 1839·07544

73·40535 165·02303 210·52634 296·36632 340·59043 427·35700 467·29093 510·64717 596·36670 677·74498 720·48641 723·30810 805·35007 927·70618 931·50358 969·99420 1054·16376 1098·27516 1179·61041 1217·16353 1259·10539 1342·71013 1345·76786 1467·53377 1546·11098 1552·22678 1587·78082 1633·11492 1670·92170 1795·20185 1839·07538

108·23591 242·66710 242·66710 371·37587 458·53113 458·53113 583·74861 583·74861 754·03538 754·03538 792·46216 877·32942 877·32942 1083·30164 1083·30164 1128·75174 1128·75175 1250·91502 1250·91502 1371·47051 1455·00663 1455·00663 1582·54414 1582·54414 1703·98651 1703·98651 1740·98236 1740·98236 1906·78810 1906·78810 2108·39589

functions can be generated accurately. For the Rayleigh–Ritz analysis, these plate characteristic functions must be arranged to keep their associated frequency parameters in ascending order, as shown in Table 1. Using the chosen shape functions, the mass and stiffness matrices are computed from equations (36), (40) and (47–50), and the EISPACK subroutines TRED2 and IMTQL2 are used to solve the resulting eigenvalue problem. In order to validate the convergence, the computations are repeated with an additional shape function, and the results are presented in Table 2. The convergence of all thirty frequency values in Table 2 indicates the suitability of the plate characteristic functions as shape functions in the Rayleigh–Ritz analysis. The same convergence trend has been observed even when smaller order matrices are used in the computation. This procedure can be used to generate very accurate results for the natural frequencies of rectangular plates under various types of support conditions. A comparison of Tables 1 and 2 shows that the frequencies associated with the plate characteristic functions are themselves reasonable approximations of the plate frequencies. An inspection of the coefficient vectors shows that the exact vibration modes can be categorized into two groups. In the first group, the coefficient vector has a single dominant coefficient which is several orders larger than the others. This dominant coefficient in the modal expression is found

505

Figure 2. The SS modes of the clamped square plate.

to be that of a plate characteristic function with a distinct frequency. Thus, every plate characteristic function with its associated distinct frequency sufficiently separated from that of the others is a reasonable approximation to a vibration mode. The second group contains pairs of coefficient vectors which have two numerically equal dominant coefficients corresponding to the plate characteristic function pairs with coincident frequency parameters. Here, the in-phase and out-of-phase superpositions of the plate characteristic function pair are reasonable approximations of the exact vibration mode pair. Thus, the plate characteristic functions the frequencies of which are sufficiently close interact strongly to form the exact vibration mode. Furthermore, although the plate characteristic functions for SS modes 2 and 3 have identical frequency parameters, the corresponding Rayleigh–Ritz frequencies differ slightly and the corresponding mode shapes shown in Figure 2 are also different. Hence, the Rayleigh–Ritz results are much more accurate than the simple frequency results obtained through the reduction procedure.

Figure 3. The SA modes of the clamped square plate.

. .

506

T 2 Rayleigh–Ritz frequencies of clamped square plate i

n

SS mode

SA mode

AS mode

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24

30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31

35·98538 35·98538 131·58085 131·58085 132·20665 132·20653 220·03892 220·03850 308·90243 308·90243 309·16700 309·16680 392·76638 392·76638 393·92083 393·91951 562·13657 562·13559 565·37982 565·37982 565·54680 565·54652 647·60129 647·60129 648·43214 648·43078 813·10605 813·10605 814·37293 814·37053 900·87286 900·87286 900·96142 900·96142 982·26217 982·26217 982·85409 982·85409 1062·21868 1062·21720 1145·81086 1145·81086 1146·88741 1146·88740 1315·34900 1315·34900 1315·40811 1315·40799

73·39431 73·39431 165·00328 165·00327 210·52298 210·52295 296·34347 296·34335 340·584543 340·58543 427·35404 427·35402 467·27017 467·27006 510·64316 510·64305 596·36261 596·36260 677·72603 677·72589 720·47382 720·47374 723·31846 723·31844 805·34707 805·34699 927·66660 927·66648 931·52312 931·52312 969·99691 969·99678 1054·16230 1054·16223 1098·27534 1098·27533 1179·60443 1179·60437 1217·15240 1217·15225 1259·10934 1259·10922 1342·69466 1342·69454 1345·78562 1345·78562 1467·53110 1467·53103

73·39431 73·39431 165·00327 165·00327 210·52298 210·52295 296·34349 296·34337 340·58543 340·58543 427·35403 427·35400 467·27018 467·27007 510·64320 510·64309 596·36261 596·36261 677·72600 677·72586 720·47385 720·47376 723·31849 723·31847 805·34705 805·34697 927·66659 927·66647 931·52314 931·52314 969·99692 969·99679 1054·16230 1054·16223 1098·27522 1098·27521 1179·60435 1179·60429 1217·15240 1217·15225 1259·10935 1259·10923 1342·69463 1342·69450 1345·78565 1345·78565 1467·53122 1467·53115

AA mode 108·21781 108·21775 242·15424 242·15424 243·15093 243·15057 371·35416 371·35365 458·22572 458·22572 458·82440 458·82392 583·12409 583·12409 584·34841 584·34707 753·84173 753·84173 754·21727 754·21683 792·45097 792·45004 876·83787 876·83787 877·81397 877·81260 1082·65036 1082·65036 1083·93154 1083·92952 1128·62025 1128·62025 1128·88672 1128·88672 1250·53452 1250·53452 1251·29003 1251·29003 1371·46114 1371·45991 1454·43817 1454·43817 1455·57701 1455·57701 1582·44989 1582·44989 1582·63837 1582·63808 [Continued

507

T 2—Continued i

n

SS mode

SA mode

AS mode

AA mode

25 25 26 26 27 27 28 28 29 29 30 30 31

30 31 30 31 30 31 30 31 30 31 30 31 31

1391·73065 1391·73065 1392·89920 1392·89918 1396·24815 1396·24815 1396·82404 1396·82300 1558·57325 1558·57325 1559·45742 1559·45340 1720·19808

1546·09717 1546·09697 1552·22884 1552·22883 1587·78620 1587·78596 1633·11686 1633·11677 1670·93012 1670·92989 1795·20616 1795·20539 1839·07681

1546·09713 1546·09692 1552·22895 1552·22894 1587·78620 1587·78596 1633·11673 1633·11664 1670·93018 1670·92995 1795·20622 1795·20545 1839·07684

1703·69018 1703·69018 1704·26567 1704·26451 1740·32351 1740·32351 1741·65551 1741·65551 1906·30580 1906·30580 1907·28043 1907·27685 2108·40018

Since the plate characteristic function is separable in spatial variables, it cannot represent a vibration mode with prominently curved nodal lines. However, the Rayleigh–Ritz results can be used to obtain accurate plots of the vibration modes. The node diagram of the first nine modes in each of the four mode categories of the clamped square plate are shown in Figures 2–4. Even though the second and the third SS modes have different shapes, their associated frequencies are reasonably close. An inspection of their modal coefficients shows that they are mainly the in-phase and the out-phase combinations of the second and third plate characteristic functions. A similar observation can be made about the other pairs of vibration modes with close natural frequencies. Except for the small loops in SS mode 7, its nodal pattern is similar to that of SS mode 2. The SS mode 2 is mainly an out-of-phase combination of the characteristic function pair (f2 , f3 ), whereas the SS mode 7 is similarly formed from another pair (f7 , f8 ). Since the characteristic functions are linearly independent, these two SS modes are different. The

Figure 4. The AA modes of the clamped square plate.

508

. .

presence of the three loops in SS mode 7 seems to be responsible for the threefold increase in the frequency. Within the accuracy of the mode plot computation, the nodal patterns of the second and the seventh AA mode appear to be identical. These modes are the out-of-phase superpositions of different characteristic function pairs of the linearly independent function set. Thus, these modes are different, even though their nodal patterns appear alike. Since the clamped square plate is a special case of a clamped rectangular plate corresponding to a/b = 1, its mode can be regarded as basic, and it changes continuously to rectangular plate modes with a gradual increase in the plate aspect ratio from unity. A similar relationship has been observed between the vibration modes of a circular plate and its elliptical extension [11].

6. CONCLUSIONS

The plate characteristic functions are used as shape functions for the Rayleigh–Ritz analysis of vibrating rectangular plates. Numerical results for the vibration frequencies and modes of a clamped square plate are presented for each of the four mode categories using 30 shape functions. The convergence of the results is validated by repeating the computations with 31 shape functions. The Rayleigh–Ritz coefficients are used to plot the first nine modes in each of the four mode categories. The Rayleigh frequencies associated with the plate characteristic functions are also found to be reasonably close to the plate frequencies. The plate characteristic functions with distinct frequency parameters are reasonable approximations to the plate vibration modes. For the plate characteristic function pairs with coincident frequencies, the in-phase and out-of-phase superpositions of the pair components represent the plate vibration mode pairs. The Rayleigh–Ritz frequencies are found to be very accurate and the present approach can be used to generate natural vibration frequencies of rectangular plates under various types of support conditions. These results of the Rayleigh–Ritz analysis are very useful in the study of the forced vibration of rectangular plates under various support conditions.

REFERENCES 1. A. W. L 1969 Vibration of Plates. NASA SP 160. Washington, D.C.: U.S. Government Printing Office. 2. A. W. L 1973 Journal of Sound and Vibration 31, 257–293. The free vibration of rectangular plates. 3. R. B. B, P. A. A. L, R. G. G, V. H. C and H. C. S 1990 Journal of Sound and Vibration 138, 205–219. Numerical experiments on the determination of natural frequencies of transverse vibrations of rectangular plate of non-uniform thickness. 4. B. T. S, J. M. B, J. J. D, B. S. G, Y. I, V. C. K and C. B. M 1976 in EISPACK Guide. New York: Springer-Verlag. Eigensystem routines. 5. L. V. K and V. I. K 1964 Approximate Methods in Higher Analysis. Groningen, The Netherlands: Noordhoff. 6. A. D. K 1968 Quarterly of Applied Mathematics 26, 219–229. An extension of Kantorovich method. 7. R. J and B. J. M 1976 Journal of Sound and Vibration 45, 309–316. Application of extended Kantorovich method to the vibration of clamped rectangular plates. 8. R. B. B, J. S and G. M 1993 Transactions of the American Society of Mechanical Engineers, Journal of Vibration and Acoustics 115, 177–181. Plate characteristic functions and natural frequencies of vibration of plates by iterative reduction of partial differential equation.

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9. C. R, R. B. B and G. D. X 1994 Journal of Sound and Vibration (in press). Vibration of rectangular plates by reduction of the plate partial differential equation into simultaneous ordinary differential equations. 10. R. B. B and G. M 1993 Journal of Sound and Vibration 161, 157–171. Vibration of plates using plate characteristic functions obtained by reduction of partial differential equation. 11. C. R, R. B. B and G. D. X 1995 International Journal of Mechanical Sciences 37(1), 61–75. On elliptical plate vibration modes as a bifurcation from circular plate modes. APPENDIX: NOTATION a a (01), a (03), a (12) A (00), A (02), A (04) b b (01), b (03), b (12) B (00), B (02), B (04) aij(mn) , bij(mn) Aij(mn) , Bij(mn) ci {C} D kij [K] m mij [M] M x , M y p1 , p2 q1 , q2 p1(i), q1(i), · · · P, Q

plate length in x-direction defined in equations (9)–(11) defined in equations (3)–(5) plate width in y-direction defined in equations (12)–(14) defined in equations (6)–(8) defined in equations (41) and (42) defined in equations (38) and (39) coefficient of ith mode coefficient vector flexural rigidity of plate stiffness coefficients stiffness matrix mass per unit area of plate inertia coefficients mass matrix averaged bending moment components, equations (17) and (18) modal parameters in X(x¯ ) modal parameters in Y(y¯ ) modal parameters in Xi Yi defined in equations (28) and (29)

Tmax U max V x , V Y w (x, y) (x¯ , y¯ ) X, Y Xi , Yi Xi(m) , etc. a dw, etc. fi (x¯ , y¯ ) n v V

non-dimensional maximum kinetic energy non-dimensional maximum strain energy averaged shear force components, equations (15) and (16) plate displacement Cartesian co-ordinates non-dimensional co-cordinates (x/a, y/b) components of plate characteristic function ith plate characteristic function components mth derivative of Xi , etc. =a/b, plate aspect ratio variation in w, etc. =Xi (x¯ )Yi (y¯ ), ith plate characteristic function Poisson ratio plate frequency =vab(m/D)1/2, nondimensional frequency