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Further results on the vibration and stability of a circular plate elastically restrained against rotation
Journal of Sound and Vibration (1975) 41(3), 388-390
FURTHER RESULTS ON THE VIBRATION AND STABILITY OF A CIRCULAR PLATE ELASTICALLY RESTRAINED AGAINST ROTATION This letter deals with the analysis of axially symmetric modes of vibrations of a thin, elastic plate elastically restrained against rotation along the edge arid subjected to a hydrostatic state of in-plane stress. This investigation constitutes an extension of a recent publication [1]. 1. INTRODUCTION
It already has been shown that use of the Galerkin method yields excellent accuracy when the plate displacement function is approximated by means of a polynomial [1]. A very simple frequency equation, ideal for design purposes, is obtained in that case. The procedure is generalized in the present study for a M-term polynomial solution. Excellent accuracy is achieved for a two-term, approximate mode expansion. 2. VARIATIONALSOLUTION
In the case of normal axisymmetric modes the classical equation of motion for the displacen~ent, w(r,t), of a thin, elastic plate is D[74 W - N V 2 W - phoa2 W = 0, (1) where w(r, t) = W ( r ) e ' % D is the flexural rigidity, p is the density of the plate material and h is the thickness of the plate. The boundary conditions are: W(a) = 0, (2a)
dW(a)
w
,,dW\l
dr =-(ibn~--~rZ + 7- 711,. ,/I ~ where r is the distributed spring constant of the elastic restraint. Following reference [1 ], one can approximate W(r) by a summation of polynomials:
(2b)
M
W ( r ) " W , ( r ) = ~ A~(ajr4 + fljr 2 + 1)r2<
(3)
J.O
The coefficients aj and flj are evaluated by making use of conditions (2): 1 (a/OD) + (l + 70 + 4j as = a4 (a/d?D) + (5 + lt) + 4j' 2 (a/q~D) + (3 + ,u) + 4j fls = a2(a/c~D) + (5 +/t) + 4j" Substituting expression (3) in equation (1) results in the error function, e(r):
= J=.~r ~ ,4j ([ - , h o : - - F - ajr'+2 _ [
388
p11c~
+ ~A'( 4 + 2 j )
a~t]r "+2~]
(4a) (4b)
389
LETTERS TO THE EDITOR
The orthogonality condition 2g a
-f f e(r) Wkrdr=O
(k=O, 1. . . . M),
(6)
0 0
where Wk = (akr 4 + flkr 2 + 1)r 2k, yields
JY- Ma{jp ~ -_ ~ % a l ~% j.o
pho92
10 + 2 j + 2 k
/--~
Loj ~k + flk a j) +
-DN
(4 + 2j) 2 ~j ~k} aS§+ 2j +
2k
N
(~k + #j/~, + c,j) + ~ [~, Fj(2 + 2j) 2 + ~j 8, (4 + 2jy] - (2 + 2j)~(4 + 2j) 2 ~ ~,
Iphco2 ~
a 6+2J+2k
N
' x 6 + 2j + 2k - ["D--- (pJ + ilk) + ~ [aft % + flj fl~(2 + 2j) 2 + %(4 + 2j) 2]
- (2 + 2 j y [(4 + 2 j y ~j L + 4 f ~j e~l 4 + 2j + 2k
+ g [4f F~ + (2 + 2j) ~/~A a2J+2k+2
- (2j - 2 y 4fl ~ - 4fl(2 + 2j)~.BjB. - (2 + 2j)'- (4 + 2j)" ~j 2 j + 2k + 2 -
2j--~-~+(2j-2)~4f2j--T~-
4j~-(2j-2)~4j'~.-4.F(2+2j)~
2 =0.
(7)
Equation (7) yields a system of linear, homogeneous equations in the A/s. From the nontriviality condition one obtains a determinantal equation in the frequency coefficients: P~h D COoj a 2 .
Qoj =
Tables 1 and 2 show numerical results obtained by making use of a two-term solution (M : 1 ) . The results are compared with values available in the literature in Table 1 and it can be concluded that the accuracy of the present approach is excellent. TABLE 1
Comparison of numerical results t f
Boundary donditions
Frequency coefficient = (ph/D)cooja 2 "~ I2ot 0o2 A,
1 term 2 terms
9
Other authors
9
2 terms
,
h
a ~
Other authors
Buckling coefficient =
N,,a2 / D N 9
A
9
Other 1 term 2 terms authors
Clamped 10.33 10.2170 10.2158 [2] 43.058 39.771 [2] 16.00 14.70 14.68 [I] Simply /t=0.30 4.947 4.935 4.977[2] 30.956 29.760 [2] 4.21 4.09 4.20 [4] supported /1=0.25 4.872 4.860 4.865 30.889 -4.077 3.97 -/t = 0.33 . . . . . . 4.163 ,. It is important to point out that for a simply supported plate the results are more accurate than those considered as "exact" in the literature when p = 0"30. On the other hand they compare very well with values obtained by Leissa [3]. The fact should be emphasized that very few independent numerical results are available in,the case of buckling coefficients for simply supported plates. Reference [4] contains information f o r / t = 0.30. Since the Galerkin method provides an upper bound the value
390
LETTERS TO THE EDITOR TABLE 2 Frequency coefficients a/(ph/ D) Wo~a 2for the fimdamental and second axisymmetric modes of vibration ~ = 0.30)
(a/r r
N(a2/D) 0 2 5 10 15 ---4.091 --6"044 --7-394 -9"730 -14"702
9
0 4-935 30.956 6.029 32.647 7.376 35.029 9.199 38.673 10-721 42.001 0 ---.
1
2
6.063 6.777 32.173 33.188 6-984 7.616 33.666 34"546 8.167 8.715 35.790 36.491 9.820 10.274 39.077 39.524 11.227 11.614 42.110 42.344 . . . 0 --0 --. . .
5
oo
7.935 35.405 8.678 36.534 9.673 38.168 11-113 40.752 12.367 43.187 . --0
10-217 43.058 10.872 43.783 11.781 44.851 13.147 46.579 14.372 48.248 --0
obtained in the present investigation is more accurate than that available in reference [4] (almost 3Yo improvement). Even the second natural frequency can be considered very accurate from an engineering viewpoint. Institute of Applied Mechanics, Base Naval Puerto Belgrano, Argentina (Received 5 April 1975)
P. A. A. LAURA R . GELOS
REFERENCES 1. P . A . A . LAURA,J. C. PALOTO a n d R. D. SANTOS 1975 Journal of Sound and Vibration 41, 177-180. A n o t e o n the v i b r a t i o n a n d stability o f a circular plate elastically restrained against r o t a t i o n . 2. A. W. LEISSA 1969 NASA SP-160. V i b r a t i o n o f plates. 3. A. W. LEZSSA 1967 Journal of Sound and Vibration 6, 145-148. V i b r a t i o n o f a s i m p l y - s u p p o r t e d elliptical plate. 4. S. TIMOSHENKO a n d L. GERE 1961 Theory of Elastic Stability. N e w Y o r k : M c G r a w - H i l l B o o k C o m p a n y , Inc.
FURTHER RESULTS ON THE VIBRATION AND STABILITY OF A CIRCULAR PLATE ELASTICALLY RESTRAINED AGAINST ROTATION This letter deals with the analysis of axially symmetric modes of vibrations of a thin, elastic plate elastically restrained against rotation along the edge arid subjected to a hydrostatic state of in-plane stress. This investigation constitutes an extension of a recent publication [1]. 1. INTRODUCTION
It already has been shown that use of the Galerkin method yields excellent accuracy when the plate displacement function is approximated by means of a polynomial [1]. A very simple frequency equation, ideal for design purposes, is obtained in that case. The procedure is generalized in the present study for a M-term polynomial solution. Excellent accuracy is achieved for a two-term, approximate mode expansion. 2. VARIATIONALSOLUTION
In the case of normal axisymmetric modes the classical equation of motion for the displacen~ent, w(r,t), of a thin, elastic plate is D[74 W - N V 2 W - phoa2 W = 0, (1) where w(r, t) = W ( r ) e ' % D is the flexural rigidity, p is the density of the plate material and h is the thickness of the plate. The boundary conditions are: W(a) = 0, (2a)
dW(a)
w
,,dW\l
dr =-(ibn~--~rZ + 7- 711,. ,/I ~ where r is the distributed spring constant of the elastic restraint. Following reference [1 ], one can approximate W(r) by a summation of polynomials:
(2b)
M
W ( r ) " W , ( r ) = ~ A~(ajr4 + fljr 2 + 1)r2<
(3)
J.O
The coefficients aj and flj are evaluated by making use of conditions (2): 1 (a/OD) + (l + 70 + 4j as = a4 (a/d?D) + (5 + lt) + 4j' 2 (a/q~D) + (3 + ,u) + 4j fls = a2(a/c~D) + (5 +/t) + 4j" Substituting expression (3) in equation (1) results in the error function, e(r):
= J=.~r ~ ,4j ([ - , h o : - - F - ajr'+2 _ [
388
p11c~
+ ~A'( 4 + 2 j )
a~t]r "+2~]
(4a) (4b)
389
LETTERS TO THE EDITOR
The orthogonality condition 2g a
-f f e(r) Wkrdr=O
(k=O, 1. . . . M),
(6)
0 0
where Wk = (akr 4 + flkr 2 + 1)r 2k, yields
JY- Ma{jp ~ -_ ~ % a l ~% j.o
pho92
10 + 2 j + 2 k
/--~
Loj ~k + flk a j) +
-DN
(4 + 2j) 2 ~j ~k} aS§+ 2j +
2k
N
(~k + #j/~, + c,j) + ~ [~, Fj(2 + 2j) 2 + ~j 8, (4 + 2jy] - (2 + 2j)~(4 + 2j) 2 ~ ~,
Iphco2 ~
a 6+2J+2k
N
' x 6 + 2j + 2k - ["D--- (pJ + ilk) + ~ [aft % + flj fl~(2 + 2j) 2 + %(4 + 2j) 2]
- (2 + 2 j y [(4 + 2 j y ~j L + 4 f ~j e~l 4 + 2j + 2k
+ g [4f F~ + (2 + 2j) ~/~A a2J+2k+2
- (2j - 2 y 4fl ~ - 4fl(2 + 2j)~.BjB. - (2 + 2j)'- (4 + 2j)" ~j 2 j + 2k + 2 -
2j--~-~+(2j-2)~4f2j--T~-
4j~-(2j-2)~4j'~.-4.F(2+2j)~
2 =0.
(7)
Equation (7) yields a system of linear, homogeneous equations in the A/s. From the nontriviality condition one obtains a determinantal equation in the frequency coefficients: P~h D COoj a 2 .
Qoj =
Tables 1 and 2 show numerical results obtained by making use of a two-term solution (M : 1 ) . The results are compared with values available in the literature in Table 1 and it can be concluded that the accuracy of the present approach is excellent. TABLE 1
Comparison of numerical results t f
Boundary donditions
Frequency coefficient = (ph/D)cooja 2 "~ I2ot 0o2 A,
1 term 2 terms
9
Other authors
9
2 terms
,
h
a ~
Other authors
Buckling coefficient =
N,,a2 / D N 9
A
9
Other 1 term 2 terms authors
Clamped 10.33 10.2170 10.2158 [2] 43.058 39.771 [2] 16.00 14.70 14.68 [I] Simply /t=0.30 4.947 4.935 4.977[2] 30.956 29.760 [2] 4.21 4.09 4.20 [4] supported /1=0.25 4.872 4.860 4.865 30.889 -4.077 3.97 -/t = 0.33 . . . . . . 4.163 ,. It is important to point out that for a simply supported plate the results are more accurate than those considered as "exact" in the literature when p = 0"30. On the other hand they compare very well with values obtained by Leissa [3]. The fact should be emphasized that very few independent numerical results are available in,the case of buckling coefficients for simply supported plates. Reference [4] contains information f o r / t = 0.30. Since the Galerkin method provides an upper bound the value
390
LETTERS TO THE EDITOR TABLE 2 Frequency coefficients a/(ph/ D) Wo~a 2for the fimdamental and second axisymmetric modes of vibration ~ = 0.30)
(a/r r
N(a2/D) 0 2 5 10 15 ---4.091 --6"044 --7-394 -9"730 -14"702
9
0 4-935 30.956 6.029 32.647 7.376 35.029 9.199 38.673 10-721 42.001 0 ---.
1
2
6.063 6.777 32.173 33.188 6-984 7.616 33.666 34"546 8.167 8.715 35.790 36.491 9.820 10.274 39.077 39.524 11.227 11.614 42.110 42.344 . . . 0 --0 --. . .
5
oo
7.935 35.405 8.678 36.534 9.673 38.168 11-113 40.752 12.367 43.187 . --0
10-217 43.058 10.872 43.783 11.781 44.851 13.147 46.579 14.372 48.248 --0
obtained in the present investigation is more accurate than that available in reference [4] (almost 3Yo improvement). Even the second natural frequency can be considered very accurate from an engineering viewpoint. Institute of Applied Mechanics, Base Naval Puerto Belgrano, Argentina (Received 5 April 1975)
P. A. A. LAURA R . GELOS
REFERENCES 1. P . A . A . LAURA,J. C. PALOTO a n d R. D. SANTOS 1975 Journal of Sound and Vibration 41, 177-180. A n o t e o n the v i b r a t i o n a n d stability o f a circular plate elastically restrained against r o t a t i o n . 2. A. W. LEISSA 1969 NASA SP-160. V i b r a t i o n o f plates. 3. A. W. LEZSSA 1967 Journal of Sound and Vibration 6, 145-148. V i b r a t i o n o f a s i m p l y - s u p p o r t e d elliptical plate. 4. S. TIMOSHENKO a n d L. GERE 1961 Theory of Elastic Stability. N e w Y o r k : M c G r a w - H i l l B o o k C o m p a n y , Inc.