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TRANSVERSE VIBRATIONS OF A CIRCULAR PLATE OF RECTANGULAR ORTHOTROPY WITH A CONCENTRIC CIRCULAR SUPPORT AND CARRYING A CONCENTRATED MASS
Journal of Sound and Vibration (1999) 221(3), 537–544 Article No. jsvi.1998.2001, available online at http://www.idealibrary.com on
TRANSVERSE VIBRATIONS OF A CIRCULAR PLATE OF RECTANGULAR ORTHOTROPY WITH A CONCENTRIC CIRCULAR SUPPORT AND CARRYING A CONCENTRATED MASS E. R P. A. A. L Department of Engineering, Universidad Nacional del Sur and Institute of Applied Mechanics (CONICET), 8000 Bahı´ a Blanca, Argentina (Received 7 October 1998)
1. A recent publication [1] dealt with vibrating clamped and simply supported circular plates of rectangular orthotropy carrying a central, concentrated mass. The present letter is an extension of the previously quoted paper. A concentric, circular support is provided in order to increase the stiffness of the system; see Figure 1. As the radius of the concentric support, Ro , approaches zero, the structural system degenerates into the central point support situation for which an exact solution is available in the case of isotropic plates. The fundamental eigenvalues for clamped and simply supported isotropic plates obtained in the present study are in good agreement with those determined exactly in terms of Bessel’s functions [2]. On the other hand, an exact solution seems out of the question in the case of circular plates of rectangular orthotropy. The optimized Rayleigh–Ritz method is used whereby three exponential optimization parameters are employed, extending the procedure developed in reference [3] where two optimization exponents were used. 2. Using Lekhnitskii’s well known notation [4], the maximum strain energy is expressed in the form Umax =
1 2
gg$ 0 1 D1
+ 4Dk
1 2W 1x2
2
+ 2D1 n2
0 1% 1 2W 1x 1y
0 1
1 2W 1 2W 1 2W + D 2 1x2 1y2 1y2
2
2
dx dy,
(1)
while the maximum kinetic energy is given by Tmax =
$ gg
1 rv 2h 2
0022–460X/99/130537 + 08 $30.00/0
%
W2 dx dy + Mv 2W=2x = y = 0 .
(2)
7 1999 Academic Press
538
a
D1 D2
M
R0 M Clamped edge
M
Simply supported edge
Figure 1. Structural system executing transverse vibrations, considered in the present study.
The energy functional is then defined as J[W] = Umax − Tmax .
(3)
The displacement amplitude is approximated by means of W 3 Wa = C1 (ap rp + aq rq + as r s + 1) + C2 (bp r p + 1 + bq r q + 1 + bs r s + 1 + 1), (4) where p, q and s are Rayleigh’s optimization parameters and a’s and b’s are determined substituting each co-ordinate function into W(a) =
dW (a) = W(Ro ) = 0. dr
(5a)
in the case of a clamped edge, Figure 1, and into W(a) =
0
d2W n2 dW + dr2 r dr
1b
= W(Ro ) = 0
(5b)
r=a
when the plate is simply supported. Clearly, the natural boundary conditions are not satisfied in the case of rectangular orthotropy but when n2 = n one has the isotropic situation and in this circumstance the polynomial co-ordinate functions expressed in equation (4) yield excellent accuracy [5]. Substituting equation (4) into equations (1), (2) and (3) and applying the classical Rayleigh–Ritz method 1J[W] 1Umax 1Tmax = − = 0, 1Ci 1Ci 1Ci
(6)
539
T 1 Values of the frequency coefficient V1 = zrh/Dv1 a2 for an isotropic circular plate with an intermediate circular support and a central, concentrated mass M: concentrated mass; Mp : plate mass; n = 0·30 Ro /a ZXXXXXXXXXXXCXXXXXXXXXXXV :0 0·2 0·4 0·6 0·8 :1
M/Mp Clamped
Simply supported
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
22·963 22·738 4·6–2·7 1·7 22·963 22·738 4·6–2·7 1·7 22·963 22·738 4·6–2·7 1·7 22·963 22·738 4·6–2·7 1·7 22·963 22·738 4·6–2·7 1·7
30·297 29·612 3·6–3·4 3·1 26·355 25·878 3·5–2·2 1·9 22·792 20·731 2·5–2·2 1·9 20·120 17·450 2·4–2·1 2·0 18·136 15·315 2·4–2·1 2·0
39·838 39·348 4·3–4·0 2·2 15·966 15·816 3·3–3·2 1·8 11·677 11·533 3·4–3·1 1·8 9·646 9·515 3·4–3·1 1·8 8·403 8·284 3·4–3·0 1·8
23·956 23·123 5·5–5·1 2·2 10·869 10·763 5·3–4·9 1·7 8·082 8·009 6·4–4·0 1·7 6·717 6·658 6·7–3·8 1·7 5·870 5·819 6·9–3·7 1·7
14·618 14·455 13·3 7·9–2·0 8·209 8·083 19·0 6·3–1·7 6·273 6·172 18·1 6·3–1·6 5·268 5·181 19·5 6·0–1·6 4·629 4·551 20·2 5·8–1·6
11·504 10·218 21·400 3·9–2·1 7·044 6·424 20·900 2·7–1·8 5·474 5·022 21·200 2·5–1·8 4·628 4·257 21·200 2·4–1·8 4·082 3·759 21·400 2·4–1·8
14·973 14·816 5·9–2·4 1·8 14·973 14·816 5·9–2·4 1·8 14·973 14·816 5·9–2·4 1·8 14·973 14·816 5·9–2·4 1·8 14·973 14·816 5·9–2·4 1·8
19·428 19·134 3·6–3·3 2·9 18·411 18·386 4·4–3·9 1·8 17·327 17·103 4·4–2·3 1·8 16·267 15·518 2·4–2·1 2·0 15·287 14·086 2·4–2·1 2·0
29·054 28·769 6·2–5·4 2·1 15·210 15·030 3·2–3·1 1·8 11·260 11·084 3·3–3·0 1·8 9·337 9·176 3·3–3·0 1·8 8·149 8·001 3·3–2·9 1·8
22·508 21·906 5·3–4·9 2·2 10·504 10·430 5·1–4·8 1·7 7·833 7·780 6·1–3·9 1·7 6·517 6·473 6·4–3·7 1·7 5·699 5·659 6·6–3·6 1·7
14·138 14·047 12·3 7·6–2·2 8·007 7·926 18·9 6·0–1·7 6·130 6·063 17·5 6·0–1·6 5·152 5·093 19·4 5·7–1·6 4·529 4·475 20·4 5·5–1·6
11·504 10·218 21·400 3·9–2·1 7·044 6·424 20·900 2·7–1·8 5·474 5·022 21·200 2·5–1·8 4·628 4·257 21·200 2·4–1·8 4·082 3·759 21·400 2·4–1·8
(a) Eigenvalue determined taking p = 4, q = 3 and s = 2; (b) eigenvalue determined by optimization with respect to p, q and s; (c) optimum values of p, q and s.
540
T 2 Values of the frequency coefficient V1 = zrh/Dv1 a 2 for an orthotropic circular plate with an intermediate circular support and a central, concentrated mass M: concentrated mass; Mp : plate mass; D2 /D1 = 1/2; Dk /D1 = 1/2; n2 = 0·30 Ro /a ZXXXXXXXXXXXCXXXXXXXXXXXV :0 0·2 0·4 0·6 0·8 :1
M/Mp Clamped
Simply supported
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
21·633 21·421 4·6–2·7 1·7 21·633 21·421 4·6–2·7 1·7 21·633 21·421 4·6–2·7 1·7 21·633 21·421 4·6–2·7 1·7 21·633 21·421 4·6–2·7 1·7
28·542 27·898 3·8–3·5 2·9 24·828 24·379 3·5–2·2 1·9 21·472 19·530 2·5–2·2 1·9 18·955 16·440 2·3–2·1 2·0 17·086 14·428 2·3–2·1 2·0
37·530 37·070 4·4–3·9 2·2 15·041 14·900 3·3–3·2 1·8 11·001 10·865 3·4–3·1 1·8 9·087 8·964 3·4–3·1 1·8 7·916 7·804 3·1–3·0 1·8
22·569 21·784 5·5–5·1 2·2 10·239 10·139 5·3–4·9 1·7 7·614 7·545 6·4–4·0 1·7 6·328 6·272 6·7–3·8 1·7 5·530 5·481 6·9–3·9 1·7
13·771 13·617 14·0 7·6–2·2 7·734 7·615 19·0 6·3–1·7 5·910 5·814 18·1 6·3–1·6 4·963 4·881 19·5 6·0–1·6 4·361 4·288 20·2 5·8–1·6
10·837 9·626 21·400 3·9–2·1 6·636 6·052 20·900 2·7–1·8 5·257 4·731 21·200 2·5–1·8 4·360 4·010 21·200 2·4–1·8 3·845 3·541 21·400 2·4–1·8
13·960 13·806 4·9–2·4 1·8 13·960 13·806 4·9–2·4 1·8 13·960 13·806 4·9–2·4 1·8 13·960 13·806 4·9–2·4 1·8 13·960 13·806 4·9–2·4 1·8
18·115 17·858 3·9–3·3 1·7 17·190 17·171 4·4–3·9 1·8 16·203 16·002 4·3–2·3 1·8 15·231 14·552 2·4–2·1 2·0 14·330 13·229 2·3–2·1 2·0
27·176 26·935 6·4–5·3 2·1 14·313 14·151 3·2–3·1 1·8 10·600 10·433 3·3–3·0 1·8 8·790 8·637 3·3–2·3 1·8 7·672 7·532 3·3–2·9 1·8
21·169 20·621 5·4–4·8 2·2 9·889 9·821 3·4–3·0 1·8 7·375 7·326 6·0–3·9 1·7 6·136 6·095 6·3–3·7 1·7 5·366 5·329 6·5–3·6 1·7
13·314 13·231 12·5 7·5–2·0 7·542 7·466 18·7 6·1–1·7 5·774 5·711 17·5 6·0–1·6 4·853 4·797 17·4 5·5–1·6 4·266 4·216 20·4 5·5–1·6
10·837 9·625 21·400 3·9–2·1 6·636 6·052 20·900 2·7–1·8 5·157 4·731 21·200 2·5–1·8 4·360 4·010 21·200 2·4–1·8 3·845 3·541 21·400 2·4–1·8
(a) Eigenvalue determined taking p = 4, q = 3 and s = 2; (b) eigenvalue determined by optimization with respect to p, q and s; (c) optimum values of p, q and s.
541
T 3 Values of the frequency coefficient V1 = zrh/Dv1 a 2 for an orthotropic circular plate with an intermediate circular support and a central, concentrated mass M: concentrated mass; Mp : plate mass; D2 /D1 = 1; Dk /D1 = 1/2; n2 = 0·30 Ro /a ZXXXXXXXXXXXCXXXXXXXXXXXV :0 0·2 0·4 0·6 0·8 :1
M/Mp Clamped
Simply supported
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
23·809 23·575 4·6–2·6 1·7 23·809 23·575 4·6–2·6 1·7 23·809 23·575 4·6–2·6 1·7 23·809 23·575 4·6–2·6 1·7 23·809 23·575 4·6–2·6 1·7
31·413 30·703 3·8–3·5 2·9 27·325 26·831 3·5–2·2 1·9 23·631 21·494 2·5–2·2 1·9 20·861 18·093 2·4–2·1 2·0 18·804 15·879 2·3–2·1 2·0
41·305 40·798 4·3–3·9 2·2 16·554 16·399 3·3–3·2 1·8 12·107 11·957 3·4–3·1 1·8 10·001 9·866 3·4–3·1 1·8 8·712 8·589 3·4–3·0 1·8
24·838 23·975 5·5–5·1 2·2 11·269 11·159 5·3–4·9 1·7 8·380 8·304 6·4–4·0 1·7 6·964 6·903 6·7–3·8 1·7 6·066 6·033 6·9–3·7 1·7
15·156 14·986 14·0 7·6–2·0 8·511 8·380 19·1 6·2–1·7 6·504 6·399 18·1 6·3–1·6 5·463 5·372 19·5 6·0–1·6 4·800 4·719 20·2 5·8–1·6
11·927 10·594 21·400 3·9–2·1 7·303 6·660 20·900 2·7–1·8 5·676 5·207 21·200 2·5–1·8 4·799 4·413 21·200 2·4–1·8 4·232 3·897 21·400 2·4–1·8
15·400 15·232 4·9–2·8 1·7 15·400 15·232 4·9–2·8 1·7 15·400 15·232 4·9–2·8 1·7 15·400 15·232 4·9–2·8 1·7 15·400 15·232 4·9–2·8 1·7
19·984 19·695 3·6–3·3 2·9 18·957 18·935 4·4–3·9 1·8 17·863 17·639 4·4–2·3 1·8 16·786 16·033 2·4–2·1 2·0 15·789 14·570 2·3–2·1 2·0
29·957 29·635 6·1–5·5 2·1 15·757 15·578 3·2–3·1 1·8 11·668 11·485 3·3–3·0 1·8 9·676 9·507 3·3–2·9 1·8 8·445 8·304 4·1–2·8 1·7
23·307 22·699 5·3–4·9 2·3 10·886 10·810 5·1–4·7 1·7 8·118 8·064 6·0–3·9 1·7 6·755 6·709 6·3–3·7 1·7 5·906 5·866 6·5–3·6 1·7
14·655 14·562 12·5 7·5–2·0 8·301 8·217 18·60 6·1–1·7 6·355 6·286 17·5 6·0–1·6 5·341 5·280 19·4 5·7–1·6 4·695 4·640 20·4 5·5–1·6
11·927 10·594 21·400 3·9–2·1 7·303 6·660 20·900 2·7–1·8 5·676 5·207 21·200 2·5–1·8 4·799 4·413 21·200 2·4–1·8 4·232 3·897 21·400 2·4–1·8
(a) Eigenvalue determined taking p = 4, q = 3 and s = 2; (b) eigenvalue determined by optimization with respect to p, q and s; (c) optimum values of p, q and s.
542
T 4 Values of the frequency coefficient V1 = zrh/Dv1 a 2 for an orthotropic circular plate with an intermediate circular support and a central, concentrated mass M: concentrated mass; Mp : plate mass; D2 /D1 = 1/2; Dk /D1 = 1/2; n2 = 1/3 Ro /a ZXXXXXXXXXXXCXXXXXXXXXXXV :0 0·2 0·4 0·6 0·8 :1
M/Mp Clamped
Simply supported
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
20·699 20·496 4·6–2·7 1·7 20·699 20·496 4·6–2·7 1·7 20·699 20·496 4·6–2·7 1·7 20·699 20·496 4·6–2·7 1·7 20·699 20·496 4·6–2·7 1·7
27·309 26·694 4·0–3·4 2·8 23·756 23·326 3·5–2·2 1·9 20·545 18·687 2·5–2·2 1·9 18·136 15·730 2·4–2·1 2·0 16·348 13·805 2·3–2·1 2·0
35·909 35·469 4·3–3·9 2·2 14·392 14·257 3·5–3·1 1·8 10·526 10·397 3·4–3·1 1·8 8·694 8·577 3·4–3·1 1·8 7·574 7·467 3·4–3·0 1·8
21·594 20·843 5·6–5·0 2·2 9·797 9·701 5·3–4·9 1·7 7·285 7·219 6·4–4·0 1·7 6·055 6·001 6·7–3·8 1·7 5·291 5·245 6·9–3·7 1·7
13·176 13·029 14·0 7·6–2·0 7·400 7·286 19·0 6·3–1·7 5·655 5·563 18·1 6·3–1·6 4·749 4·670 19·5 6·0–1·6 4·173 4·103 20·2 5·8–1·6
10·369 9·210 21·400 3·9–2·1 6·349 5·790 20·900 2·7–1·8 4·934 4·527 21·200 2·5–1·8 4·172 3·837 21·200 2·4–1·8 3·679 3·388 21·400 2·4–1·8
13·537 13·395 4·9–2·4 1·8 13·537 13·395 4·9–2·4 1·8 13·537 13·395 4·9–2·4 1·8 13·537 13·395 4·9–2·4 1·8 13·537 13·395 4·9–2·4 1·8
17·559 17·294 4·0–3·3 2·7 16·635 16·613 4·4–3·9 1·8 15·652 15·446 4·3–2·3 1·8 14·690 14·006 2·4–2·1 2·0 13·802 12·709 2·3–2·1 2·0
26·235 25·978 6·1–5·5 2·1 13·715 13·562 3·4–3·0 1·8 10·152 9·994 3·3–3·0 1·8 8·418 8·273 3·5–2·8 1·7 7·347 7·214 3·5–2·8 1·7
20·294 19·751 5·4–4·8 2·2 9·470 9·403 5·1–4·8 1·7 7·062 7·013 6·1–3·9 1·7 5·875 5·835 6·4–3·7 1·7 5·137 5·102 6·6–3·6 1·7
12·744 12·663 12·3 7·6–2·0 7·218 7·145 18·7 6·1–1·7 5·526 5·465 17·5 6·0–1·6 4·644 4·591 19·4 5·7–1·6 4·083 4·034 20·4 5·5–1·6
10·369 9·210 21·400 2·1 6·349 5·790 20·900 3·9–2·1 4·934 4·527 21·200 2·5–1·8 4·172 3·837 21·200 2·4–1·8 3·679 3·388 21·400 2·4–1·8
(a) Eigenvalue determined taking p = 4, q = 3 and s = 2; (b) eigenvalue determined by optimization with respect to p, q and s; (c) optimum values of p, q and s.
543
one obtains a linear system of homogeneous equation in C1 and C2 . The non-triviality conditions yield a determinantal equation whose lowest root constitutes the fundamental frequency coefficient V1 = zrh/D1 v1 a2. Obviously, the fundamental mode shape must be defined as a quasi-axisymmetric shape in view of the orthotropy of the material (the fundamental mode will be completely independent of the azimuthal variable only in the case of an isotropic plate). On the other hand, the existence of the circular inner support contributes to a higher degree of radial symmetry than in the case of reference [1]. Since, in view of equation (4), V1 = V1 (p, q, s),
(7)
by minimizing equation (7) with respect to p, q and s one is able to optimize the frequency coefficient. 3. Frequency coefficients have been determined for isotropic plates (n = 0·30), Table 1, and for orthotropic plates: (1) D2 /D1 = 1/2, Dk /D1 = 1/2, (n2 = 0·30), Table 2; (2) D2 /D1 = 1, Dk /D1 = 1/2, (n2 = 0·30), Table 3; (3) D2 /D1 = 1/2, Dk /D1 = 1/3, (n2 = 1/3), Table 4. The values of the exponential parameters which minimize the fundamental eigenvalue are also indicated in the tables. The procedure is essentially a non-linear optimization process but it has been performed by a simple numerical searching process. As Ro :a, one immediately notices the fact that the optimized eigenvalue corresponds to very large values of the parameter p and when this situation occurs, the plate behaves as clamped. In the case of a clamped, isotropic plate with a central point support one has V1 = 22·7 and when the plate is simply supported V1 = 14·8, where Poisson’s ratio has been taken equal to 0·30 [2]. The present study yields 22·738 and 14·816, respectively, for Ro /a:0. In each table the eigenvalues has been listed as a function of Ro /a and M/Mp , where Mp is the plate mass. Also shown in each case, and in the first line, is the eigenvalue obtained using the classical Rayleigh–Ritz method with fixed values of the exponents appearing in the co-ordinate functions. One can notice the fact that in some instances the eigenvalues are lowered considerably by means of the optimization process. The present study has been sponsored by CONICET Research and Development Program, at the Institute of Applied Mechanics, Secretarı´ a General de Ciencia y Tecnologı´ a of Universidad Nacional del Sur, and Comisio´n de Investigaciones Cientı´ ficas, Buenos Aires Province. 1. P. A. A. L, E. R, V. S and S. I 1999 Journal of Sound and Vibration (in press). Numerical experiments on vibrating circular plates of rectangular orthotropy and carrying a central, concentrated mass.
544
2. A. W. L 1969 NASA SPI60. Vibration of plates. 3. R. O. G, P. A. A. L and Y. N 1986 Journal of Sound and Vibration 106, 181–186. A note on vibrating polar orthotropic circular plates carrying concentrated masses. 4. S. G. L 1968 Anisotropic Plates. New York: Gordon and Breach Science Publishers. 5. P. A. A. L, J. C. P and R. D. S 1975 Journal of Sound and Vibration 41, 177–180. A note on the vibration and stability of a circular plate elastically restrained against rotation.
TRANSVERSE VIBRATIONS OF A CIRCULAR PLATE OF RECTANGULAR ORTHOTROPY WITH A CONCENTRIC CIRCULAR SUPPORT AND CARRYING A CONCENTRATED MASS E. R P. A. A. L Department of Engineering, Universidad Nacional del Sur and Institute of Applied Mechanics (CONICET), 8000 Bahı´ a Blanca, Argentina (Received 7 October 1998)
1. A recent publication [1] dealt with vibrating clamped and simply supported circular plates of rectangular orthotropy carrying a central, concentrated mass. The present letter is an extension of the previously quoted paper. A concentric, circular support is provided in order to increase the stiffness of the system; see Figure 1. As the radius of the concentric support, Ro , approaches zero, the structural system degenerates into the central point support situation for which an exact solution is available in the case of isotropic plates. The fundamental eigenvalues for clamped and simply supported isotropic plates obtained in the present study are in good agreement with those determined exactly in terms of Bessel’s functions [2]. On the other hand, an exact solution seems out of the question in the case of circular plates of rectangular orthotropy. The optimized Rayleigh–Ritz method is used whereby three exponential optimization parameters are employed, extending the procedure developed in reference [3] where two optimization exponents were used. 2. Using Lekhnitskii’s well known notation [4], the maximum strain energy is expressed in the form Umax =
1 2
gg$ 0 1 D1
+ 4Dk
1 2W 1x2
2
+ 2D1 n2
0 1% 1 2W 1x 1y
0 1
1 2W 1 2W 1 2W + D 2 1x2 1y2 1y2
2
2
dx dy,
(1)
while the maximum kinetic energy is given by Tmax =
$ gg
1 rv 2h 2
0022–460X/99/130537 + 08 $30.00/0
%
W2 dx dy + Mv 2W=2x = y = 0 .
(2)
7 1999 Academic Press
538
a
D1 D2
M
R0 M Clamped edge
M
Simply supported edge
Figure 1. Structural system executing transverse vibrations, considered in the present study.
The energy functional is then defined as J[W] = Umax − Tmax .
(3)
The displacement amplitude is approximated by means of W 3 Wa = C1 (ap rp + aq rq + as r s + 1) + C2 (bp r p + 1 + bq r q + 1 + bs r s + 1 + 1), (4) where p, q and s are Rayleigh’s optimization parameters and a’s and b’s are determined substituting each co-ordinate function into W(a) =
dW (a) = W(Ro ) = 0. dr
(5a)
in the case of a clamped edge, Figure 1, and into W(a) =
0
d2W n2 dW + dr2 r dr
1b
= W(Ro ) = 0
(5b)
r=a
when the plate is simply supported. Clearly, the natural boundary conditions are not satisfied in the case of rectangular orthotropy but when n2 = n one has the isotropic situation and in this circumstance the polynomial co-ordinate functions expressed in equation (4) yield excellent accuracy [5]. Substituting equation (4) into equations (1), (2) and (3) and applying the classical Rayleigh–Ritz method 1J[W] 1Umax 1Tmax = − = 0, 1Ci 1Ci 1Ci
(6)
539
T 1 Values of the frequency coefficient V1 = zrh/Dv1 a2 for an isotropic circular plate with an intermediate circular support and a central, concentrated mass M: concentrated mass; Mp : plate mass; n = 0·30 Ro /a ZXXXXXXXXXXXCXXXXXXXXXXXV :0 0·2 0·4 0·6 0·8 :1
M/Mp Clamped
Simply supported
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
22·963 22·738 4·6–2·7 1·7 22·963 22·738 4·6–2·7 1·7 22·963 22·738 4·6–2·7 1·7 22·963 22·738 4·6–2·7 1·7 22·963 22·738 4·6–2·7 1·7
30·297 29·612 3·6–3·4 3·1 26·355 25·878 3·5–2·2 1·9 22·792 20·731 2·5–2·2 1·9 20·120 17·450 2·4–2·1 2·0 18·136 15·315 2·4–2·1 2·0
39·838 39·348 4·3–4·0 2·2 15·966 15·816 3·3–3·2 1·8 11·677 11·533 3·4–3·1 1·8 9·646 9·515 3·4–3·1 1·8 8·403 8·284 3·4–3·0 1·8
23·956 23·123 5·5–5·1 2·2 10·869 10·763 5·3–4·9 1·7 8·082 8·009 6·4–4·0 1·7 6·717 6·658 6·7–3·8 1·7 5·870 5·819 6·9–3·7 1·7
14·618 14·455 13·3 7·9–2·0 8·209 8·083 19·0 6·3–1·7 6·273 6·172 18·1 6·3–1·6 5·268 5·181 19·5 6·0–1·6 4·629 4·551 20·2 5·8–1·6
11·504 10·218 21·400 3·9–2·1 7·044 6·424 20·900 2·7–1·8 5·474 5·022 21·200 2·5–1·8 4·628 4·257 21·200 2·4–1·8 4·082 3·759 21·400 2·4–1·8
14·973 14·816 5·9–2·4 1·8 14·973 14·816 5·9–2·4 1·8 14·973 14·816 5·9–2·4 1·8 14·973 14·816 5·9–2·4 1·8 14·973 14·816 5·9–2·4 1·8
19·428 19·134 3·6–3·3 2·9 18·411 18·386 4·4–3·9 1·8 17·327 17·103 4·4–2·3 1·8 16·267 15·518 2·4–2·1 2·0 15·287 14·086 2·4–2·1 2·0
29·054 28·769 6·2–5·4 2·1 15·210 15·030 3·2–3·1 1·8 11·260 11·084 3·3–3·0 1·8 9·337 9·176 3·3–3·0 1·8 8·149 8·001 3·3–2·9 1·8
22·508 21·906 5·3–4·9 2·2 10·504 10·430 5·1–4·8 1·7 7·833 7·780 6·1–3·9 1·7 6·517 6·473 6·4–3·7 1·7 5·699 5·659 6·6–3·6 1·7
14·138 14·047 12·3 7·6–2·2 8·007 7·926 18·9 6·0–1·7 6·130 6·063 17·5 6·0–1·6 5·152 5·093 19·4 5·7–1·6 4·529 4·475 20·4 5·5–1·6
11·504 10·218 21·400 3·9–2·1 7·044 6·424 20·900 2·7–1·8 5·474 5·022 21·200 2·5–1·8 4·628 4·257 21·200 2·4–1·8 4·082 3·759 21·400 2·4–1·8
(a) Eigenvalue determined taking p = 4, q = 3 and s = 2; (b) eigenvalue determined by optimization with respect to p, q and s; (c) optimum values of p, q and s.
540
T 2 Values of the frequency coefficient V1 = zrh/Dv1 a 2 for an orthotropic circular plate with an intermediate circular support and a central, concentrated mass M: concentrated mass; Mp : plate mass; D2 /D1 = 1/2; Dk /D1 = 1/2; n2 = 0·30 Ro /a ZXXXXXXXXXXXCXXXXXXXXXXXV :0 0·2 0·4 0·6 0·8 :1
M/Mp Clamped
Simply supported
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
21·633 21·421 4·6–2·7 1·7 21·633 21·421 4·6–2·7 1·7 21·633 21·421 4·6–2·7 1·7 21·633 21·421 4·6–2·7 1·7 21·633 21·421 4·6–2·7 1·7
28·542 27·898 3·8–3·5 2·9 24·828 24·379 3·5–2·2 1·9 21·472 19·530 2·5–2·2 1·9 18·955 16·440 2·3–2·1 2·0 17·086 14·428 2·3–2·1 2·0
37·530 37·070 4·4–3·9 2·2 15·041 14·900 3·3–3·2 1·8 11·001 10·865 3·4–3·1 1·8 9·087 8·964 3·4–3·1 1·8 7·916 7·804 3·1–3·0 1·8
22·569 21·784 5·5–5·1 2·2 10·239 10·139 5·3–4·9 1·7 7·614 7·545 6·4–4·0 1·7 6·328 6·272 6·7–3·8 1·7 5·530 5·481 6·9–3·9 1·7
13·771 13·617 14·0 7·6–2·2 7·734 7·615 19·0 6·3–1·7 5·910 5·814 18·1 6·3–1·6 4·963 4·881 19·5 6·0–1·6 4·361 4·288 20·2 5·8–1·6
10·837 9·626 21·400 3·9–2·1 6·636 6·052 20·900 2·7–1·8 5·257 4·731 21·200 2·5–1·8 4·360 4·010 21·200 2·4–1·8 3·845 3·541 21·400 2·4–1·8
13·960 13·806 4·9–2·4 1·8 13·960 13·806 4·9–2·4 1·8 13·960 13·806 4·9–2·4 1·8 13·960 13·806 4·9–2·4 1·8 13·960 13·806 4·9–2·4 1·8
18·115 17·858 3·9–3·3 1·7 17·190 17·171 4·4–3·9 1·8 16·203 16·002 4·3–2·3 1·8 15·231 14·552 2·4–2·1 2·0 14·330 13·229 2·3–2·1 2·0
27·176 26·935 6·4–5·3 2·1 14·313 14·151 3·2–3·1 1·8 10·600 10·433 3·3–3·0 1·8 8·790 8·637 3·3–2·3 1·8 7·672 7·532 3·3–2·9 1·8
21·169 20·621 5·4–4·8 2·2 9·889 9·821 3·4–3·0 1·8 7·375 7·326 6·0–3·9 1·7 6·136 6·095 6·3–3·7 1·7 5·366 5·329 6·5–3·6 1·7
13·314 13·231 12·5 7·5–2·0 7·542 7·466 18·7 6·1–1·7 5·774 5·711 17·5 6·0–1·6 4·853 4·797 17·4 5·5–1·6 4·266 4·216 20·4 5·5–1·6
10·837 9·625 21·400 3·9–2·1 6·636 6·052 20·900 2·7–1·8 5·157 4·731 21·200 2·5–1·8 4·360 4·010 21·200 2·4–1·8 3·845 3·541 21·400 2·4–1·8
(a) Eigenvalue determined taking p = 4, q = 3 and s = 2; (b) eigenvalue determined by optimization with respect to p, q and s; (c) optimum values of p, q and s.
541
T 3 Values of the frequency coefficient V1 = zrh/Dv1 a 2 for an orthotropic circular plate with an intermediate circular support and a central, concentrated mass M: concentrated mass; Mp : plate mass; D2 /D1 = 1; Dk /D1 = 1/2; n2 = 0·30 Ro /a ZXXXXXXXXXXXCXXXXXXXXXXXV :0 0·2 0·4 0·6 0·8 :1
M/Mp Clamped
Simply supported
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
23·809 23·575 4·6–2·6 1·7 23·809 23·575 4·6–2·6 1·7 23·809 23·575 4·6–2·6 1·7 23·809 23·575 4·6–2·6 1·7 23·809 23·575 4·6–2·6 1·7
31·413 30·703 3·8–3·5 2·9 27·325 26·831 3·5–2·2 1·9 23·631 21·494 2·5–2·2 1·9 20·861 18·093 2·4–2·1 2·0 18·804 15·879 2·3–2·1 2·0
41·305 40·798 4·3–3·9 2·2 16·554 16·399 3·3–3·2 1·8 12·107 11·957 3·4–3·1 1·8 10·001 9·866 3·4–3·1 1·8 8·712 8·589 3·4–3·0 1·8
24·838 23·975 5·5–5·1 2·2 11·269 11·159 5·3–4·9 1·7 8·380 8·304 6·4–4·0 1·7 6·964 6·903 6·7–3·8 1·7 6·066 6·033 6·9–3·7 1·7
15·156 14·986 14·0 7·6–2·0 8·511 8·380 19·1 6·2–1·7 6·504 6·399 18·1 6·3–1·6 5·463 5·372 19·5 6·0–1·6 4·800 4·719 20·2 5·8–1·6
11·927 10·594 21·400 3·9–2·1 7·303 6·660 20·900 2·7–1·8 5·676 5·207 21·200 2·5–1·8 4·799 4·413 21·200 2·4–1·8 4·232 3·897 21·400 2·4–1·8
15·400 15·232 4·9–2·8 1·7 15·400 15·232 4·9–2·8 1·7 15·400 15·232 4·9–2·8 1·7 15·400 15·232 4·9–2·8 1·7 15·400 15·232 4·9–2·8 1·7
19·984 19·695 3·6–3·3 2·9 18·957 18·935 4·4–3·9 1·8 17·863 17·639 4·4–2·3 1·8 16·786 16·033 2·4–2·1 2·0 15·789 14·570 2·3–2·1 2·0
29·957 29·635 6·1–5·5 2·1 15·757 15·578 3·2–3·1 1·8 11·668 11·485 3·3–3·0 1·8 9·676 9·507 3·3–2·9 1·8 8·445 8·304 4·1–2·8 1·7
23·307 22·699 5·3–4·9 2·3 10·886 10·810 5·1–4·7 1·7 8·118 8·064 6·0–3·9 1·7 6·755 6·709 6·3–3·7 1·7 5·906 5·866 6·5–3·6 1·7
14·655 14·562 12·5 7·5–2·0 8·301 8·217 18·60 6·1–1·7 6·355 6·286 17·5 6·0–1·6 5·341 5·280 19·4 5·7–1·6 4·695 4·640 20·4 5·5–1·6
11·927 10·594 21·400 3·9–2·1 7·303 6·660 20·900 2·7–1·8 5·676 5·207 21·200 2·5–1·8 4·799 4·413 21·200 2·4–1·8 4·232 3·897 21·400 2·4–1·8
(a) Eigenvalue determined taking p = 4, q = 3 and s = 2; (b) eigenvalue determined by optimization with respect to p, q and s; (c) optimum values of p, q and s.
542
T 4 Values of the frequency coefficient V1 = zrh/Dv1 a 2 for an orthotropic circular plate with an intermediate circular support and a central, concentrated mass M: concentrated mass; Mp : plate mass; D2 /D1 = 1/2; Dk /D1 = 1/2; n2 = 1/3 Ro /a ZXXXXXXXXXXXCXXXXXXXXXXXV :0 0·2 0·4 0·6 0·8 :1
M/Mp Clamped
Simply supported
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
0
(a) (b) (c)
0·25
(a) (b) (c)
0·50
(a) (b) (c)
0·75
(a) (b) (c)
1
(a) (b) (c)
20·699 20·496 4·6–2·7 1·7 20·699 20·496 4·6–2·7 1·7 20·699 20·496 4·6–2·7 1·7 20·699 20·496 4·6–2·7 1·7 20·699 20·496 4·6–2·7 1·7
27·309 26·694 4·0–3·4 2·8 23·756 23·326 3·5–2·2 1·9 20·545 18·687 2·5–2·2 1·9 18·136 15·730 2·4–2·1 2·0 16·348 13·805 2·3–2·1 2·0
35·909 35·469 4·3–3·9 2·2 14·392 14·257 3·5–3·1 1·8 10·526 10·397 3·4–3·1 1·8 8·694 8·577 3·4–3·1 1·8 7·574 7·467 3·4–3·0 1·8
21·594 20·843 5·6–5·0 2·2 9·797 9·701 5·3–4·9 1·7 7·285 7·219 6·4–4·0 1·7 6·055 6·001 6·7–3·8 1·7 5·291 5·245 6·9–3·7 1·7
13·176 13·029 14·0 7·6–2·0 7·400 7·286 19·0 6·3–1·7 5·655 5·563 18·1 6·3–1·6 4·749 4·670 19·5 6·0–1·6 4·173 4·103 20·2 5·8–1·6
10·369 9·210 21·400 3·9–2·1 6·349 5·790 20·900 2·7–1·8 4·934 4·527 21·200 2·5–1·8 4·172 3·837 21·200 2·4–1·8 3·679 3·388 21·400 2·4–1·8
13·537 13·395 4·9–2·4 1·8 13·537 13·395 4·9–2·4 1·8 13·537 13·395 4·9–2·4 1·8 13·537 13·395 4·9–2·4 1·8 13·537 13·395 4·9–2·4 1·8
17·559 17·294 4·0–3·3 2·7 16·635 16·613 4·4–3·9 1·8 15·652 15·446 4·3–2·3 1·8 14·690 14·006 2·4–2·1 2·0 13·802 12·709 2·3–2·1 2·0
26·235 25·978 6·1–5·5 2·1 13·715 13·562 3·4–3·0 1·8 10·152 9·994 3·3–3·0 1·8 8·418 8·273 3·5–2·8 1·7 7·347 7·214 3·5–2·8 1·7
20·294 19·751 5·4–4·8 2·2 9·470 9·403 5·1–4·8 1·7 7·062 7·013 6·1–3·9 1·7 5·875 5·835 6·4–3·7 1·7 5·137 5·102 6·6–3·6 1·7
12·744 12·663 12·3 7·6–2·0 7·218 7·145 18·7 6·1–1·7 5·526 5·465 17·5 6·0–1·6 4·644 4·591 19·4 5·7–1·6 4·083 4·034 20·4 5·5–1·6
10·369 9·210 21·400 2·1 6·349 5·790 20·900 3·9–2·1 4·934 4·527 21·200 2·5–1·8 4·172 3·837 21·200 2·4–1·8 3·679 3·388 21·400 2·4–1·8
(a) Eigenvalue determined taking p = 4, q = 3 and s = 2; (b) eigenvalue determined by optimization with respect to p, q and s; (c) optimum values of p, q and s.
543
one obtains a linear system of homogeneous equation in C1 and C2 . The non-triviality conditions yield a determinantal equation whose lowest root constitutes the fundamental frequency coefficient V1 = zrh/D1 v1 a2. Obviously, the fundamental mode shape must be defined as a quasi-axisymmetric shape in view of the orthotropy of the material (the fundamental mode will be completely independent of the azimuthal variable only in the case of an isotropic plate). On the other hand, the existence of the circular inner support contributes to a higher degree of radial symmetry than in the case of reference [1]. Since, in view of equation (4), V1 = V1 (p, q, s),
(7)
by minimizing equation (7) with respect to p, q and s one is able to optimize the frequency coefficient. 3. Frequency coefficients have been determined for isotropic plates (n = 0·30), Table 1, and for orthotropic plates: (1) D2 /D1 = 1/2, Dk /D1 = 1/2, (n2 = 0·30), Table 2; (2) D2 /D1 = 1, Dk /D1 = 1/2, (n2 = 0·30), Table 3; (3) D2 /D1 = 1/2, Dk /D1 = 1/3, (n2 = 1/3), Table 4. The values of the exponential parameters which minimize the fundamental eigenvalue are also indicated in the tables. The procedure is essentially a non-linear optimization process but it has been performed by a simple numerical searching process. As Ro :a, one immediately notices the fact that the optimized eigenvalue corresponds to very large values of the parameter p and when this situation occurs, the plate behaves as clamped. In the case of a clamped, isotropic plate with a central point support one has V1 = 22·7 and when the plate is simply supported V1 = 14·8, where Poisson’s ratio has been taken equal to 0·30 [2]. The present study yields 22·738 and 14·816, respectively, for Ro /a:0. In each table the eigenvalues has been listed as a function of Ro /a and M/Mp , where Mp is the plate mass. Also shown in each case, and in the first line, is the eigenvalue obtained using the classical Rayleigh–Ritz method with fixed values of the exponents appearing in the co-ordinate functions. One can notice the fact that in some instances the eigenvalues are lowered considerably by means of the optimization process. The present study has been sponsored by CONICET Research and Development Program, at the Institute of Applied Mechanics, Secretarı´ a General de Ciencia y Tecnologı´ a of Universidad Nacional del Sur, and Comisio´n de Investigaciones Cientı´ ficas, Buenos Aires Province. 1. P. A. A. L, E. R, V. S and S. I 1999 Journal of Sound and Vibration (in press). Numerical experiments on vibrating circular plates of rectangular orthotropy and carrying a central, concentrated mass.
544
2. A. W. L 1969 NASA SPI60. Vibration of plates. 3. R. O. G, P. A. A. L and Y. N 1986 Journal of Sound and Vibration 106, 181–186. A note on vibrating polar orthotropic circular plates carrying concentrated masses. 4. S. G. L 1968 Anisotropic Plates. New York: Gordon and Breach Science Publishers. 5. P. A. A. L, J. C. P and R. D. S 1975 Journal of Sound and Vibration 41, 177–180. A note on the vibration and stability of a circular plate elastically restrained against rotation.