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LETTER TO THE EDITOR: ON THE RELATIONSHIP BETWEEN THE FUNDAMENTAL FREQUENCY AND STATIC DEFLECTIONS OF THIN ELASTIC PLATES
Journal of Sound and Vibration (1996) 196(4), 523–528
LETTERS TO THE EDITOR ON THE RELATIONSHIP BETWEEN THE FUNDAMENTAL FREQUENCY AND STATIC DEFLECTIONS OF THIN ELASTIC PLATES G. R B. N R Structural Engineering Group M. S. S Applied Mathematics Division, Vikram Sarabai Space Centre, Trivandrum—695 022, India (Received 7 August 1995, and in final form 6 February 1996)
The precise determination of the frequencies of elastic plates of arbitrary geometry and boundary conditions involves considerable difficulties in the integration of the fourth order partial differential equation. From the point of view of practical application, approximate methods of determination of the fundamental mode could be helpful. Jones [1] examined the applicability of a frequency-static deflection relation, rhv 2(Wmax /q) = C
(21·630),
(1)
for plates of various geometry and boundary conditions. This relation is obtained from the expression for the fundamental frequency (v) of a clamped elliptical plate [2], and the maximum deflection (Wmax ) of the same plate under a uniformly distributed load, q [3]. Here r is the mass per unit area and h is the plate thickness. Jones cautioned that this relation may be inappropriate if any portion of the plate boundary is freely supported. Maurizi, Belles and Laura [4] have made a comparative study of various existing expressions [1,4,5] to obtain the fundamental frequency for the case of a clamped elliptical plate. Sundararajan [6] also presented a similar type of relation, which is based on Rayleigh’s method, where the fundamental mode of a rectangular plate is approximated by the deflection functions of beams subjected to uniform distributed loads. The constant C in equation (1) for rectangular plates having all edges clamped, all edges simply supported, two opposite edges simply supported and the others clamped, and two opposite edges simply supported with the third edge clamped and the fourth edge free, examined by him, was found to be 1·723, 1·613, 1·667 and 1·978, respectively. This variation in the values of the constant indicates that its value changes with the geometry of the plate and the boundary conditons. Although the simple approximate expressions suggested by Jones [1] and Sundararajan [6] are good estimates for various plate configurations, there is no formal derivation of the frequency-static deflection relation for a plate of arbitrary shape and complex boundary conditions. The purpose of this study is to examine the possibility of such frequency-static displacement relations, and to propose a methodology for estimating the fundamental frequency of a plate through its static deflections under a uniformly distributed load. Consider a linear elastic plate occupying an area A, inside the boundary S, undergoing free harmonic vibration. According to the Rayleigh–Ritz method, the fundamental frequency (v) can be obtained by selecting a function c(x,y) for the lateral deflection (w) 523 0022–460X/96/390523 + 6 $18.00/0
7 1996 Academic Press Limited
524
of the plate which would satisfy the boundary conditions and then minimizing the resultant potential and kinetic energies. As a first approximation, the function c can be selected as an expression which is proportional to a static deflection with the same conditions of edge fixing and subjected to a uniformly distributed load. This is equivalent to the assumption that the plate surface, which corresponds to the fundamental mode, is identical to that deflected by a uniformly distributed load. The expression for the possible deflection is taken in the form w = Wmax c(x,y)
(2)
where Wmax is the maximum deflection of the plate under a uniformly distributed load, q, and =c(x,y)= E 1,[(x,y)$S. By using expression (2), expressions for the potential energy (U) and the work done (W) can be written as U = 12 D
gg06 7 6 7 1 2w 1x 2
2
+
1 2w 1y 2
6 71
1 2w 1 2w 1 2w 2 2 + 2(1 − n) 1x 1y 1x1y
2
+ 2n
A
2
2 dx dy = 12 DWmax Uc ,
(3) W=
gg
qw dx dy = qWmax Wc ,
(4)
A
where Uc =
gg06 7 6 7 1 2c 1 2c 2 + 1x 1y 2
2
+ 2n
A
Wc =
gg
6 71
12c 1 2c2 1 2c 2 2 + 2(1 − n) 1x 1y 1x1y
2
dx dy,
cdx dy,
A
and D is the flextural rigidity. Minimizing the total potential energy (0U − W) with respect to Wmax , one obtains DUc Wmax = qWc ,
(5)
The lateral deflection of a plate undergoing free harmonic vibration having frequency v, can be expressed as w = f(t)c(x,y),
(6)
Then the expressions for the potential energy (U) and the kinetic energy (T) become
T = 12 rhv 2
U = 12 Df 2Uc ,
(7)
gg
(8)
A
w 2 dx dy = 12 rhw 2f 2Tc ,
525
where Tc =
gg
c 2 dx dy.
A
Minimizing the resultant potential and kinetic energies with respect to f, one obtains DUc = rhw 2Tc ,
(9)
From equations (5) and (9), a relation between v and Wmax is obtained as rhv2(Wmax /q) = af ,
(10)
where af = Wc /Tc =
gg
>gg
c dx dy
c 2 dx dy.
A
A
Equation (10) is helpful for determining the fundamental frequency (v) of a plate from its static deflection under a uniform distributed load, q. The method of solution for a simply supported elliptical plate is briefly explained below. For a simply supported elliptical plate under a uniformly distributed load, q, a three-term deflection function for w is chosen as w = Wmax {1 + d1 j 2 + d2 h 2}(j 2 + h 2 − 1),
(11)
where j = x/a, h = y/b, and a and b are the semi-major and semi-minor axes of the elliptical plate, respectively. Equation (11) satisfies the geometric boundary conditions of zero edge deflection and possesses two-fold symmetry. Substituting the expression for w given in equation (11) in equations (3) and (4) and minimizing the total potential energy (0U − W), one obtains [aij ]{Xi }=−(qb 4/D){bi },
(12)
where [aij ] is a 3 × 3 symmetric matrix having elements a11 = (2K 4 + 4nK 2 + 2),
a12 = (3K 4 + 4nK 2 + 1)/2,
a22 = (63K 4 + (18n + 8)K 2 + 3)/12,
a13 = (K 4 + 4nK 2 + 3)/2,
a23 = (3K 4 + (6n + 8)K 2 + 3)/12,
a33 = (3K 4 + (18n + 8)K 2 + 63)/12,
K = b/a.
{Xi } is a 3 × 1 column matrix having elements X1 = Wmax ,
X2 = Wmax d1 ,
X3 = Wmax d2 .
{bi } is a 3 × 1 column matrix having elements b1 = 1/4,
b2 = 1/24,
b3 = 1/24.
The maximum deflection, Wmax and the constants d1 and d2 are obtained by solving equation (12). The deflection function or mode shape, c(x,y) for the simply supported elliptical plate undergoing free harmonic vibration is assumed as c(x,y) = {1 + d1 j 2 + d2 h 2}(j 2 + h 2 − 1).
(13)
Then the constant af in equation (10) becomes af = 20(6 + d2 + d2 )/(80 + 20(d1 + d2 ) + 3(d12 + d22 ) + 2d1 d2 ).
(14)
526
The fundamental frequency (v) is obtained by using the values of Wmax and af in equation (10). In order to examine the adequacy of the present method of computing the fundamental frequency of a plate from its static deflections under a uniformly distributed load, elliptical and rectangular plates with different boundary conditions have been considered. The results are presented in non-dimensional form as Wmax = a(qb 4/D),
l = zrh/Dvb 2 = zaf /a.
(15,16)
For the case of a rectangular plate, a and b are its length and width, whereas in the case of a elliptical plate, they are the semi-major and semi-minor axes respectively. With the definitions j = x/a, h = y/b and K = b/a, the mode shape function c(x,y) utilized, and af and a obtained for the clamped elliptical plate are c(x,y) = (j 2 + h 2 − 1)2, af = 5/3 and a = 1/8(3K 4 + 2K 2 + 3). For the case of rectangular plates with different boundary conditions, the mode shape function c(x,y) utilized, and af and a obtained, are given below. Here the edges of the rectangular plate are designated by S, C and F for simply supported, clamped and free boundary conditions respectively. Case (a). All edges simply supported (SSSS): c(x,y) = sin(pj)sin(ph),
af = 16/p 2,
a = 16/p 6(K 4 + 2K 2 + 1).
Case (b). Two opposite long edges simply supported, others simply supported and free (SSSF): c(x,y) = j sin(ph),
af = 6/p,
a = 1/p 5(1/6 + (1 − n)K 2/p 2).
Case (c). Two opposite long edges simply supported, others clamped and simply supported (SSCS): c(x,y) = 12 (cos(3pj/2) − cos(pj/2))sin(ph),
af = 64/3p 2,
a = 1024/3p 6(41K 4 + 40K 2 + 16). Case (d). Two opposite long edges freely supported, others simply supported (FFSS): c(x,y) = sin(pj),
af = 4/p,
a = 4/p 5K 4.
Case (e). All edges clamped (CCCC): c(x,y) = 14(cos(2pj) − 1)(cos(2ph) − 1),
af = 16/9,
a = 1/p 4(3K 4 + 2K 2 + 3).
Case (f). Two opposite long edges simply supported, others clamped (SSCC): c(x,y) = 12(cos(2pj) − 1)sin(ph),
af = 16/3p,
a = 16/p 5(16K 4 + 8K 2 + 3).
From Table 1, the non-dimensional fundamental frequency parameter l for the simply supported elliptical plate obtained from equation (16) compares well with the existing values given in references [1,2,7,8]. As far as the clamped elliptical plate is concerned, the deflection function c(x,y) used for a uniformly distributed load satisfies exactly the fourth order differential equation. Hence, Wmax is also exact. However, for the case of a clamped elliptical vibrating plate, the exact solution is not available and so a three-term deflection function, as used in reference [7], is required to obtain a result close to the exact value. When the mode shape is assumed to be proportional to the static deflected shape of the clamped elliptical plate under a uniform distributed load, the value of af is equal to 1·667, which is close to 1·630, as given in references [1,4]. However, for the results presented in Table 2 for the case of a rectangular plate with different boundary conditions, the constant af in the frequency-static deflection relation varies. The maximum static deflections of
527
T 1 A comparison of the non-dimensional frequency parameter (l) for simply supported elliptic plates
n
Present study ZXXXXCXXXXV a af l
Leissa [8], a
Leissa [9], l
Jones [1], l
Majumdar [2], l
Prasad et al., [7], l
K = b/a = 1 0·0 0·0781 0·25 0·0656 0·30 0·0637 0·50 0·0573
1·548 1·558 1·559 1·566
4·452 4·872 4·947 5·227
0·0781 0·0656 — 0·0573
4·447 4·865 — 5·219
4·569 4·985 5·043 5·335
4·444 4·862 — 5·217
4·258 4·761 4·855 5·215
K = b/a = 1/2 0·00 0·1542 0·25 0·1439 0·30 0·1421 0·50 0·1356
1·563 1·572 1·573 1·579
3·183 3·305 3·327 3·412
0·1542 0·1441 — 0·1356
3·172 3·292 — 3·99
3·252 3·366 3·357 3·468
3·017 3·301 — 3·542
3·103 3·281 3·315 3·449
T 2 A comparison of the fundamental frequency parameter (l) for rectangular plates (K = b/a = 1/2; n = 0·3) Edge conditions SSSS SSSF SSCS FFSS CCCC SSCC
Present study ZXXXXXXXCXXXXXXXV a af l 0·0107 0·0177 0·0214 0·2091 0·0028 0·0087
1·6211 1·9099 2·1615 1·2732 1·7778 1·6977
12·3370 10·3813 13·1868 2·4674 25·2700 13·9577
Roark [10], a
Sundararajan [6], l
0·0102 0·0151 0·0092 0·2240 0·0025 0·0084
12·6198 11·4832 13·2943 2·3812 26·0428 14·0543
rectangular plates were obtained by using the deflection functions of Leissa [5]. The results are found to be in reasonably good agreement with existing results by different methods [10]. It is better to use the mode shape of the plate proportional to its static deflection under uniformly distributed load, for evaluating the constant af and determining the fundamental frequency (v) from equation (10).
1. R. J 1975 Journal of Sound and Vibration 38, 503–504. An approximate expression for the fundamental frequency of elastic plates. 2. J. M 1971 Journal of Sound and Vibration 18, 147–155. Transverse vibrations of elastic plates by the method of constant deflection. 3. S. P. T and S. W-K 1959 Theory of plates and shells. New York: McGraw-Hill. 4. M. J. M, P. B and P. A. A. L 1994 Journal of Sound and Vibration 171, 141–144. Free vibration of clamped elliptical plates. 5. A. W. L 1969 Vibration of Plates (NASA SP-160). Washington, DC: U.S. Government Printing Office.
528
6. C. S 1978 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 45, 936–938. An approximate solution for the fundamental frequency of plates. 7. K. L. P, A. V R and B. N R 1992 Journal of Sound and Vibration 158, 383–386. Free vibration of simply supported and clamped elliptical plates. 8. A. W. L, W. E. C, L. E. H and A. T. H 1969 American Institute of Aeronautics and Astronautics Journal 7, 920–928. A comparison of approximate methods for the solution of plate bending problems. 9. A. W. L 1967 Journal of Sound and Vibration 6, 145–148. Vibration of a simply supported elliptical plate. 10. R. J. R 1965 Formulas for Stress and Strain. New York: McGraw-Hill.
LETTERS TO THE EDITOR ON THE RELATIONSHIP BETWEEN THE FUNDAMENTAL FREQUENCY AND STATIC DEFLECTIONS OF THIN ELASTIC PLATES G. R B. N R Structural Engineering Group M. S. S Applied Mathematics Division, Vikram Sarabai Space Centre, Trivandrum—695 022, India (Received 7 August 1995, and in final form 6 February 1996)
The precise determination of the frequencies of elastic plates of arbitrary geometry and boundary conditions involves considerable difficulties in the integration of the fourth order partial differential equation. From the point of view of practical application, approximate methods of determination of the fundamental mode could be helpful. Jones [1] examined the applicability of a frequency-static deflection relation, rhv 2(Wmax /q) = C
(21·630),
(1)
for plates of various geometry and boundary conditions. This relation is obtained from the expression for the fundamental frequency (v) of a clamped elliptical plate [2], and the maximum deflection (Wmax ) of the same plate under a uniformly distributed load, q [3]. Here r is the mass per unit area and h is the plate thickness. Jones cautioned that this relation may be inappropriate if any portion of the plate boundary is freely supported. Maurizi, Belles and Laura [4] have made a comparative study of various existing expressions [1,4,5] to obtain the fundamental frequency for the case of a clamped elliptical plate. Sundararajan [6] also presented a similar type of relation, which is based on Rayleigh’s method, where the fundamental mode of a rectangular plate is approximated by the deflection functions of beams subjected to uniform distributed loads. The constant C in equation (1) for rectangular plates having all edges clamped, all edges simply supported, two opposite edges simply supported and the others clamped, and two opposite edges simply supported with the third edge clamped and the fourth edge free, examined by him, was found to be 1·723, 1·613, 1·667 and 1·978, respectively. This variation in the values of the constant indicates that its value changes with the geometry of the plate and the boundary conditons. Although the simple approximate expressions suggested by Jones [1] and Sundararajan [6] are good estimates for various plate configurations, there is no formal derivation of the frequency-static deflection relation for a plate of arbitrary shape and complex boundary conditions. The purpose of this study is to examine the possibility of such frequency-static displacement relations, and to propose a methodology for estimating the fundamental frequency of a plate through its static deflections under a uniformly distributed load. Consider a linear elastic plate occupying an area A, inside the boundary S, undergoing free harmonic vibration. According to the Rayleigh–Ritz method, the fundamental frequency (v) can be obtained by selecting a function c(x,y) for the lateral deflection (w) 523 0022–460X/96/390523 + 6 $18.00/0
7 1996 Academic Press Limited
524
of the plate which would satisfy the boundary conditions and then minimizing the resultant potential and kinetic energies. As a first approximation, the function c can be selected as an expression which is proportional to a static deflection with the same conditions of edge fixing and subjected to a uniformly distributed load. This is equivalent to the assumption that the plate surface, which corresponds to the fundamental mode, is identical to that deflected by a uniformly distributed load. The expression for the possible deflection is taken in the form w = Wmax c(x,y)
(2)
where Wmax is the maximum deflection of the plate under a uniformly distributed load, q, and =c(x,y)= E 1,[(x,y)$S. By using expression (2), expressions for the potential energy (U) and the work done (W) can be written as U = 12 D
gg06 7 6 7 1 2w 1x 2
2
+
1 2w 1y 2
6 71
1 2w 1 2w 1 2w 2 2 + 2(1 − n) 1x 1y 1x1y
2
+ 2n
A
2
2 dx dy = 12 DWmax Uc ,
(3) W=
gg
qw dx dy = qWmax Wc ,
(4)
A
where Uc =
gg06 7 6 7 1 2c 1 2c 2 + 1x 1y 2
2
+ 2n
A
Wc =
gg
6 71
12c 1 2c2 1 2c 2 2 + 2(1 − n) 1x 1y 1x1y
2
dx dy,
cdx dy,
A
and D is the flextural rigidity. Minimizing the total potential energy (0U − W) with respect to Wmax , one obtains DUc Wmax = qWc ,
(5)
The lateral deflection of a plate undergoing free harmonic vibration having frequency v, can be expressed as w = f(t)c(x,y),
(6)
Then the expressions for the potential energy (U) and the kinetic energy (T) become
T = 12 rhv 2
U = 12 Df 2Uc ,
(7)
gg
(8)
A
w 2 dx dy = 12 rhw 2f 2Tc ,
525
where Tc =
gg
c 2 dx dy.
A
Minimizing the resultant potential and kinetic energies with respect to f, one obtains DUc = rhw 2Tc ,
(9)
From equations (5) and (9), a relation between v and Wmax is obtained as rhv2(Wmax /q) = af ,
(10)
where af = Wc /Tc =
gg
>gg
c dx dy
c 2 dx dy.
A
A
Equation (10) is helpful for determining the fundamental frequency (v) of a plate from its static deflection under a uniform distributed load, q. The method of solution for a simply supported elliptical plate is briefly explained below. For a simply supported elliptical plate under a uniformly distributed load, q, a three-term deflection function for w is chosen as w = Wmax {1 + d1 j 2 + d2 h 2}(j 2 + h 2 − 1),
(11)
where j = x/a, h = y/b, and a and b are the semi-major and semi-minor axes of the elliptical plate, respectively. Equation (11) satisfies the geometric boundary conditions of zero edge deflection and possesses two-fold symmetry. Substituting the expression for w given in equation (11) in equations (3) and (4) and minimizing the total potential energy (0U − W), one obtains [aij ]{Xi }=−(qb 4/D){bi },
(12)
where [aij ] is a 3 × 3 symmetric matrix having elements a11 = (2K 4 + 4nK 2 + 2),
a12 = (3K 4 + 4nK 2 + 1)/2,
a22 = (63K 4 + (18n + 8)K 2 + 3)/12,
a13 = (K 4 + 4nK 2 + 3)/2,
a23 = (3K 4 + (6n + 8)K 2 + 3)/12,
a33 = (3K 4 + (18n + 8)K 2 + 63)/12,
K = b/a.
{Xi } is a 3 × 1 column matrix having elements X1 = Wmax ,
X2 = Wmax d1 ,
X3 = Wmax d2 .
{bi } is a 3 × 1 column matrix having elements b1 = 1/4,
b2 = 1/24,
b3 = 1/24.
The maximum deflection, Wmax and the constants d1 and d2 are obtained by solving equation (12). The deflection function or mode shape, c(x,y) for the simply supported elliptical plate undergoing free harmonic vibration is assumed as c(x,y) = {1 + d1 j 2 + d2 h 2}(j 2 + h 2 − 1).
(13)
Then the constant af in equation (10) becomes af = 20(6 + d2 + d2 )/(80 + 20(d1 + d2 ) + 3(d12 + d22 ) + 2d1 d2 ).
(14)
526
The fundamental frequency (v) is obtained by using the values of Wmax and af in equation (10). In order to examine the adequacy of the present method of computing the fundamental frequency of a plate from its static deflections under a uniformly distributed load, elliptical and rectangular plates with different boundary conditions have been considered. The results are presented in non-dimensional form as Wmax = a(qb 4/D),
l = zrh/Dvb 2 = zaf /a.
(15,16)
For the case of a rectangular plate, a and b are its length and width, whereas in the case of a elliptical plate, they are the semi-major and semi-minor axes respectively. With the definitions j = x/a, h = y/b and K = b/a, the mode shape function c(x,y) utilized, and af and a obtained for the clamped elliptical plate are c(x,y) = (j 2 + h 2 − 1)2, af = 5/3 and a = 1/8(3K 4 + 2K 2 + 3). For the case of rectangular plates with different boundary conditions, the mode shape function c(x,y) utilized, and af and a obtained, are given below. Here the edges of the rectangular plate are designated by S, C and F for simply supported, clamped and free boundary conditions respectively. Case (a). All edges simply supported (SSSS): c(x,y) = sin(pj)sin(ph),
af = 16/p 2,
a = 16/p 6(K 4 + 2K 2 + 1).
Case (b). Two opposite long edges simply supported, others simply supported and free (SSSF): c(x,y) = j sin(ph),
af = 6/p,
a = 1/p 5(1/6 + (1 − n)K 2/p 2).
Case (c). Two opposite long edges simply supported, others clamped and simply supported (SSCS): c(x,y) = 12 (cos(3pj/2) − cos(pj/2))sin(ph),
af = 64/3p 2,
a = 1024/3p 6(41K 4 + 40K 2 + 16). Case (d). Two opposite long edges freely supported, others simply supported (FFSS): c(x,y) = sin(pj),
af = 4/p,
a = 4/p 5K 4.
Case (e). All edges clamped (CCCC): c(x,y) = 14(cos(2pj) − 1)(cos(2ph) − 1),
af = 16/9,
a = 1/p 4(3K 4 + 2K 2 + 3).
Case (f). Two opposite long edges simply supported, others clamped (SSCC): c(x,y) = 12(cos(2pj) − 1)sin(ph),
af = 16/3p,
a = 16/p 5(16K 4 + 8K 2 + 3).
From Table 1, the non-dimensional fundamental frequency parameter l for the simply supported elliptical plate obtained from equation (16) compares well with the existing values given in references [1,2,7,8]. As far as the clamped elliptical plate is concerned, the deflection function c(x,y) used for a uniformly distributed load satisfies exactly the fourth order differential equation. Hence, Wmax is also exact. However, for the case of a clamped elliptical vibrating plate, the exact solution is not available and so a three-term deflection function, as used in reference [7], is required to obtain a result close to the exact value. When the mode shape is assumed to be proportional to the static deflected shape of the clamped elliptical plate under a uniform distributed load, the value of af is equal to 1·667, which is close to 1·630, as given in references [1,4]. However, for the results presented in Table 2 for the case of a rectangular plate with different boundary conditions, the constant af in the frequency-static deflection relation varies. The maximum static deflections of
527
T 1 A comparison of the non-dimensional frequency parameter (l) for simply supported elliptic plates
n
Present study ZXXXXCXXXXV a af l
Leissa [8], a
Leissa [9], l
Jones [1], l
Majumdar [2], l
Prasad et al., [7], l
K = b/a = 1 0·0 0·0781 0·25 0·0656 0·30 0·0637 0·50 0·0573
1·548 1·558 1·559 1·566
4·452 4·872 4·947 5·227
0·0781 0·0656 — 0·0573
4·447 4·865 — 5·219
4·569 4·985 5·043 5·335
4·444 4·862 — 5·217
4·258 4·761 4·855 5·215
K = b/a = 1/2 0·00 0·1542 0·25 0·1439 0·30 0·1421 0·50 0·1356
1·563 1·572 1·573 1·579
3·183 3·305 3·327 3·412
0·1542 0·1441 — 0·1356
3·172 3·292 — 3·99
3·252 3·366 3·357 3·468
3·017 3·301 — 3·542
3·103 3·281 3·315 3·449
T 2 A comparison of the fundamental frequency parameter (l) for rectangular plates (K = b/a = 1/2; n = 0·3) Edge conditions SSSS SSSF SSCS FFSS CCCC SSCC
Present study ZXXXXXXXCXXXXXXXV a af l 0·0107 0·0177 0·0214 0·2091 0·0028 0·0087
1·6211 1·9099 2·1615 1·2732 1·7778 1·6977
12·3370 10·3813 13·1868 2·4674 25·2700 13·9577
Roark [10], a
Sundararajan [6], l
0·0102 0·0151 0·0092 0·2240 0·0025 0·0084
12·6198 11·4832 13·2943 2·3812 26·0428 14·0543
rectangular plates were obtained by using the deflection functions of Leissa [5]. The results are found to be in reasonably good agreement with existing results by different methods [10]. It is better to use the mode shape of the plate proportional to its static deflection under uniformly distributed load, for evaluating the constant af and determining the fundamental frequency (v) from equation (10).
1. R. J 1975 Journal of Sound and Vibration 38, 503–504. An approximate expression for the fundamental frequency of elastic plates. 2. J. M 1971 Journal of Sound and Vibration 18, 147–155. Transverse vibrations of elastic plates by the method of constant deflection. 3. S. P. T and S. W-K 1959 Theory of plates and shells. New York: McGraw-Hill. 4. M. J. M, P. B and P. A. A. L 1994 Journal of Sound and Vibration 171, 141–144. Free vibration of clamped elliptical plates. 5. A. W. L 1969 Vibration of Plates (NASA SP-160). Washington, DC: U.S. Government Printing Office.
528
6. C. S 1978 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 45, 936–938. An approximate solution for the fundamental frequency of plates. 7. K. L. P, A. V R and B. N R 1992 Journal of Sound and Vibration 158, 383–386. Free vibration of simply supported and clamped elliptical plates. 8. A. W. L, W. E. C, L. E. H and A. T. H 1969 American Institute of Aeronautics and Astronautics Journal 7, 920–928. A comparison of approximate methods for the solution of plate bending problems. 9. A. W. L 1967 Journal of Sound and Vibration 6, 145–148. Vibration of a simply supported elliptical plate. 10. R. J. R 1965 Formulas for Stress and Strain. New York: McGraw-Hill.