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Small amplitude, transverse vibrations of circular plates elastically restrained against rotation with an eccentric circular perforation with a free edge
ARTICLE IN PRESS JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 292 (2006) 1004–1010 www.elsevier.com/locate/jsvi
Short Communication
Small amplitude, transverse vibrations of circular plates elastically restrained against rotation with an eccentric circular perforation with a free edge P.A.A. Lauraa, U. Masia´b, D.R. Avalosb, a
Department of Engineering, CONICET, Institute of Applied Mechanics, Universidad Nacional del Sur, 8000 Bahı´a Blanca, Argentina b Department of Physics, School of Engineering, Universidad Nacional de Mar del Plata, Juan B. Justo 4302, B7608FDQ Mar del Plata, Argentina Received 25 July 2005; received in revised form 25 July 2005; accepted 6 September 2005 Available online 13 December 2005
Abstract Rayleigh–Ritz variational method is applied to the determination of the first four frequency coefficients for the title problem by making use of coordinate functions which identically satisfy the boundary conditions at the outer edge. Good stability and convergence properties are found. The mathematical model seems to be realistic and accurate, within the realm of the classical theory of vibrating plates. r 2005 Elsevier Ltd. All rights reserved.
1. Introduction The title problem deals with a situation which may appear in real-life vibrating systems. A small degree of eccentricity in an internal boundary may be caused by a human error in some instances. In others, it may originate by a practical reason like passage of a cable or a conduit of small diameter. The present study makes use of the methodology employed by the senior authors (Avalos, Laura) in previous publications where the coordinate function does not satisfy the natural boundary conditions. This is admissible since Ritz variational method is used.
2. Approximate analytical solution In the case of normal modes of vibration of the vibrating system shown in Fig. 1, one takes w0 ðr0 ; y; tÞ ¼ W 0 ðr0 ; yÞeiot Corresponding author. Postal Address: Bahia Blanca 2031, B7604HGU Mar del Plata, Argentina. Fax: +54 223 481 0046.
E-mail addresses: [email protected] (P.A.A. Laura), [email protected] (D.R. Avalos). 0022-460X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2005.09.024
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ARTICLE IN PRESS P.A.A. Laura et al. / Journal of Sound and Vibration 292 (2006) 1004–1010
1005
a a1 e
Fig. 1. Vibrating mechanical system. Case of an eccentric cutout with a free edge. Table 1 Values of the first four frequency coefficients in the case of a circular plate with a circular cutout of radius a1 =a ¼ 0:1, for different values of the non-dimensional eccentricity e/a when the cutout is displaced along a radial line Eccentricity e/a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a/(fD)
O1
O2
O3
O4
0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N
4.0053 6.0053 8.7114 10.175 4.8755 6.0249 8.7407 10.206 4.9030 6.0529 8.7778 10.255 4.9182 6.0668 8.7909 10.269 4.9249 6.0706 8.7872 10.258 4.9274 6.0698 8.7744 10.235 4.9272 6.0659 8.7563 10.206 4.9268 6.0620 8.7354 10.173
13.880 14.945 18.513 21.220 13.855 14.929 18.502 21.212 13.853 14.937 18.534 21.268 13.871 14.965 18.597 21.358 13.892 14.993 18.651 21.427 13.904 15.010 18.674 21.436 13.904 15.010 18.653 21.375 13.8965 14.9958 18.5889 21.2464
25.416 26.459 30.602 34.577 25.485 26.536 30.706 34.716 25.573 26.631 30.835 34.892 25.653 26.718 30.956 35.052 25.706 26.779 31.043 35.158 25.722 26.801 31.078 35.185 25.700 26.783 31.057 35.125 25.653 26.738 30.980 34.967
29.488 30.531 34.989 39.554 29.383 30.673 35.112 39.626 29.672 30.740 35.165 39.674 29.636 30.686 35.108 39.621 29.606 30.650 35.108 39.669 29.622 30.678 35.191 39.802 29.668 30.734 35.279 39.889 29.714 30.771 35.296 39.843
for the plate transverse displacement, and then introduces the following approximation, convenient in the case of both axisymmetric and antisymmetric modes of vibration, see for example [1]. W 0 ðr0 ; yÞ ffi W 0a ðr0 ; yÞ ¼
j X j¼0
Aj0 ðajk r04 þ bjk r02 þ 1Þr02j
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þ
K X
cos ðkyÞ
j X
Ajk ðajk r04 þ bjk r02 þ 1Þr0jþk ,
ð2Þ
j¼0
k¼1
where a0 s and b0 s of each coordinate function are determined substituting each functional relation in the governing boundary conditions at the external contour, i.e., for the elastically restrained edge against rotation, qW 0 ðr0 ; yÞ ða; yÞ ¼ fM r ðr0 ; yÞ, qr
W 0 ða; yÞ ¼ 0;
(3)
Mr being the radial flexural moment, f is the flexibility coefficient of the rotational boundary spring and a is the radius of the circular plate. The Rayleigh–Ritz variational approach requires minimization of the functional J½W 0 ¼ U½W 0 T½W 0 ,
(4)
where U½W 0 is the maximum strain energy and T½W 0 is the maximum kinetic energy for the (true) displacement amplitude W 0 of the plate.
Table 2 Values of the first four frequency coefficients in the case of a circular plate with a circular cutout of radius a1 =a ¼ 0:2, for different values of the non-dimensional eccentricity e/a when the cutout is displaced along a radial line Eccentricity e/a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a/(fD)
O1
O2
O3
O4
0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N
4.7199 5.9528 8.8373 10.402 4.7272 5.9601 8.8355 10.384 4.7467 5.9784 8.8355 10.362 4.7858 6.0132 8.8483 10.354 4.8273 6.0455 8.8446 10.321 4.8569 6.0608 8.8068 10.245 4.8749 6.0608 8.7433 10.142 4.8865 6.0510 8.6676 10.040
13.606 14.661 18.152 20.741 13.673 14.760 18.407 21.192 13.756 14.899 18.738 21.699 13.833 15.020 18.975 21.981 13.882 15.099 19.072 22.007 13.895 15.125 19.031 21.829 13.869 15.094 18.858 21.488 13.830 15.027 18.580 21.034
24.881 25.904 29.936 33.753 25.003 26.030 30.112 33.998 25.311 26.365 30.639 34.795 25.664 26.899 31.481 36.001 26.141 27.315 31.958 36.361 26.224 27.396 31.905 36.096 26.125 27.218 31.594 35.582 25.774 26.941 31.151 34.878
30.878 31.973 35.440 39.656 29.899 30.910 35.290 39.770 29.396 30.149 34.365 38.616 28.983 29.989 34.295 38.710 27.591 30.213 34.861 39.695 28.910 30.552 35.397 40.216 29.638 30.754 35.529 40.134 29.629 30.781 35.374 39.782
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As has been shown elsewhere, see for example Ref. [2], in the case of a circular plate, each term in Eq. (4) can be written as 2 q2 W 0 1 qW 0 1 q2 W 0 þ 0 þ 02 r qr0 r qy2 qr02 2 0 q W 1 qW 0 1 q2 W 0 2ð1 mÞ þ 02 r qy2 qr02 r0 qr0 #) 2 0 2 1qW 1 qW 0 r0 dr0 dy, r qyqr0 r02 qy
D U½W ¼ 2 0
Z Z
ð5Þ
where D is the flexural rigidity of the plate, m its Poison’s ratio and T½W 0 ¼
ro2 h 2
Z Z
W 02 r0 dr0 dy.
(6)
The integrals in expressions (5) and (6) extend over the actual area of the double connected plate under study. Introducing the non-dimensional variables W ¼ W 0 =a;
r ¼ r0 =a.
(7)
Table 3 Values of the first four frequency coefficients in the case of a circular plate with a circular cutout of radius a1 =a ¼ 0:3 for different values of the non-dimensional eccentricity e/a when the cutout is displaced along a radial line Eccentricity e/a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a/(FD)
O1
O2
O3
O4
0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N
4.6631 6.1017 9.4525 11.312 4.6570 6.0791 9.3097 11.036 4.6503 6.0486 9.1315 10.720 4.6599 6.0181 8.9600 10.440 4.6601 6.0022 8.8257 10.231 4.6906 6.0016 8.7232 10.081 4.7296 6.0022 8.6237 9.9585
12.943 14.011 17.521 20.141 13.293 14.486 18.534 21.726 13.804 15.166 19.795 22.895 14.236 15.700 20.377 23.862 14.146 15.779 20.066 23.001 14.143 15.539 19.452 22.072 13.8727 15.2338 18.8477 21.2702
24.116 25.117 29.030 32.675 24.171 25.199 29.260 33.139 24.502 25.629 30.252 34.934 26.132 26.996 33.413 37.534 27.519 28.712 32.976 37.147 27.267 28.377 32.618 36.717 26.758 27.901 32.060 35.942
34.887 35.703 40.993 46.905 31.376 32.421 37.026 41.906 29.323 30.325 34.611 39.005 28.533 29.254 33.735 39.096 28.675 29.374 35.728 41.304 29.322 30.874 36.364 41.159 29.690 31.071 35.908 40.352
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Eqs. (5) and (6) above can be recast in a non-dimensional form. The functional for the whole system in Fig. 1 is 2J½W ¼ D
2 Z Z 2 q W 1 qW 1 q2 W þ þ 2 qr2 r qr r qy2
q2 W 1 qW 1 q2 W þ qr2 r qr r2 qy2 #) 2 2 1 q W 1 qW 2 r dr dy r qyqr r qy Z Z 2 W 2 r dr dy, O
2ð1 mÞ
ð8Þ
where aspusual, ffiffiffiffiffiffiffiffiffiffiffiffi Oi ¼ rh=Doi a2 is the non-dimensional frequency coefficient. Minimizing the governing functional with respect to the A0 jks expression (8) yields a (J K) homogeneous linear system of equations in the A0 jks. A secular determinant in the natural frequency coefficients of the system results from the non-triviality condition. The present study is concerned with the determination of the first four frequency coefficients, O1 to O4 in the case of circular plates with an eccentric circular cutout.
Table 4 Values of the first four frequency coefficients in the case of a circular plate with a circular cutout of radius a1 =a ¼ 0:4 for different values of the non-dimensional eccentricity e/a when the cutout is displaced along a radial line Eccentricity e/a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a/(FD)
O1
O2
O3
O4
0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N
4.7486 6.5515 10.689 13.048 4.7058 6.4313 10.130 12.085 4.64545 6.27936 9.57953 11.2409 4.5813 6.1292 9.1260 10.598 4.5258 6.0016 8.7818 10.142 4.4953 5.9101 8.5462 9.8560 4.5044 5.8716 8.4198 9.7107
12.253 13.497 17.699 21.026 12.989 14.530 19.849 24.148 14.116 16.008 21.908 26.074 15.121 17.057 22.259 25.816 15.310 17.006 21.372 24.431 14.821 16.328 20.144 22.835 14.194 15.605 19.068 21.455
23.218 24.234 28.203 31.953 23.507 24.625 29.237 34.248 24.284 25.778 32.373 39.173 26.056 28.123 33.807 38.105 27.324 28.554 32.890 37.065 27.367 28.400 32.615 36.768 27.257 28.266 32.406 36.461
37.293 38.255 42.481 47.231 35.506 36.224 41.105 42.594 31.137 32.196 36.827 41.686 29.212 30.433 37.166 43.816 29.620 31.781 38.230 44.215 30.365 32.316 37.799 42.757 30.354 31.979 36.830 40.276
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3. Numerical results All calculations were performed for simply supported circular plates of uniform thickness, elastically restrained against rotation at the outer edge r ¼ a while the inner border of the eccentric perforation was taken to be a free edge. In all cases, the Poisson coefficient has been taken to be m ¼ 0:3. Six tables are presented, each with a value of the ratio a1/a of the radius of the eccentric perforation with respect to the circular plate radius a. Table 1: a1 =a ¼ 0:1; Table 2: a1 =a ¼ 0:2; Table 3: a1 =a ¼ 0:3; Table 4: a1 =a ¼ 0:4; Table 5: a1 =a ¼ 0:5 and Table 6: a1 =a ¼ 0:6. In each table, in turn, four values of the nondimensional rotational spring coefficient are taken as the center of the eccentric hole is displaced along a radial line. For the double series, Eq. (2), J and K up to 16 6 terms have been used, that is to say, a secular determinant of order 96 was generated for all situations. Although satisfactory convergence is already achieved for J ¼ 8 and K ¼ 4, such high values of J and K have been used taking advantage of the speed of modern desktop computers. As usual, special care has been taken to manipulate the numerical solving of the involved determinants, and 80 bits floating point variables (IEEE-standard temporary reals) have been used to satisfy accuracy requirements. It is worth noting that computations are very stable and all frequency coefficients uniformly converge as the number of terms in the double series is increased. As a general conclusion one may say that the mathematical model seems to be quite realistic and accurate, within the realm of the classical theory of vibrating plates.
Table 5 Values of the first four frequency coefficients in the case of a circular plate with a circular cutout of radius a1 =a ¼ 0:5 for different values of the non-dimensional eccentricity e/a when the cutout is displaced along a radial line Eccentricity e/a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a/(FD)
O1
O2
O3
O4
0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N
5.0257 7.4182 12.685 15.812 4.8956 7.0447 11.329 13.683 4.7315 6.6535 10.265 12.158 4.5685 6.3172 9.4824 11.091 4.4281 6.0455 8.9203 10.352 4.3183 5.8380 8.5309 9.8633 4.2053 5.7245 8.2898 9.5581
12.028 13.802 19.979 25.154 13.161 15.669 22.971 28.003 15.067 17.773 24.168 28.495 16.370 18.691 24.044 27.967 16.491 18.399 23.055 26.544 15.872 17.459 21.556 24.591 15.139 16.395 20.024 22.634
22.566 23.751 28.556 33.520 23.388 24.957 32.201 40.367 25.576 28.369 38.129 45.338 28.769 31.065 36.162 41.042 27.891 29.158 33.606 37.932 27.113 28.253 32.575 36.749 27.019 28.072 32.340 36.4606
35.900 36.921 41.463 46.773 36.826 38.029 43.713 51.604 34.958 36.150 41.560 47.903 31.599 33.518 40.835 47.305 32.311 34.605 40.913 46.577 32.549 34.354 40.053 45.532 32.180 33.554 38.679 43.430
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Table 6 Values of the first four frequency coefficients in the case of a circular plate with a circular cutout of radius a1 =a ¼ 0:6 for different values of the non-dimensional eccentricity e/a when the cutout is displaced along a radial line Eccentricity e/a
0.1
0.2
0.3
0.4
0.5
0.6
a/(FD)
O1
O2
O3
O4
0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N
5.5689 8.8831 15.837 20.421 5.2509 7.9706 13.222 16.386 4.9072 7.2083 11.441 13.809 4.6021 6.6217 10.211 12.102 4.2920 6.1658 9.3518 10.946 4.1278 5.7953 8.7531 10.160
12.660 15.709 25.503 32.905 14.693 18.505 26.747 32.547 16.773 19.949 26.562 31.520 17.644 20.192 26.005 30.432 17.713 19.760 25.039 29.018 17.645 18.760 23.523 27.002
22.868 24.644 32.711 42.603 24.787 28.127 40.296 49.371 29.648 34.096 43.330 51.269 32.721 35.127 41.028 47.097 30.032 31.274 36.065 40.919 28.822 28.910 33.415 37.735
35.667 37.019 43.372 51.992 37.480 39.952 52.803 72.388 41.894 43.303 49.650 57.573 35.528 37.317 44.702 52.234 35.185 37.328 44.342 50.946 35.502 36.683 43.159 49.123
Acknowledgments The present study was sponsored by CONICET Research and Development Program and by Secretarı´ a General de Ciencia y Tecnologı´ a (Universidad Nacional de Mar del Plata and Universidad Nacional del Sur). References [1] D.R. Avalos, H.A. Larrondo, P.A.A. Laura, V. Sonzogni, Transverse vibrations of a circular plate with a concentric square hole with free edges, Journal of Sound and Vibration 209 (5) (1998) 889–891. [2] A.W. Leissa. Vibration of plates, NASA SP160, 1969.
Short Communication
Small amplitude, transverse vibrations of circular plates elastically restrained against rotation with an eccentric circular perforation with a free edge P.A.A. Lauraa, U. Masia´b, D.R. Avalosb, a
Department of Engineering, CONICET, Institute of Applied Mechanics, Universidad Nacional del Sur, 8000 Bahı´a Blanca, Argentina b Department of Physics, School of Engineering, Universidad Nacional de Mar del Plata, Juan B. Justo 4302, B7608FDQ Mar del Plata, Argentina Received 25 July 2005; received in revised form 25 July 2005; accepted 6 September 2005 Available online 13 December 2005
Abstract Rayleigh–Ritz variational method is applied to the determination of the first four frequency coefficients for the title problem by making use of coordinate functions which identically satisfy the boundary conditions at the outer edge. Good stability and convergence properties are found. The mathematical model seems to be realistic and accurate, within the realm of the classical theory of vibrating plates. r 2005 Elsevier Ltd. All rights reserved.
1. Introduction The title problem deals with a situation which may appear in real-life vibrating systems. A small degree of eccentricity in an internal boundary may be caused by a human error in some instances. In others, it may originate by a practical reason like passage of a cable or a conduit of small diameter. The present study makes use of the methodology employed by the senior authors (Avalos, Laura) in previous publications where the coordinate function does not satisfy the natural boundary conditions. This is admissible since Ritz variational method is used.
2. Approximate analytical solution In the case of normal modes of vibration of the vibrating system shown in Fig. 1, one takes w0 ðr0 ; y; tÞ ¼ W 0 ðr0 ; yÞeiot Corresponding author. Postal Address: Bahia Blanca 2031, B7604HGU Mar del Plata, Argentina. Fax: +54 223 481 0046.
E-mail addresses: [email protected] (P.A.A. Laura), [email protected] (D.R. Avalos). 0022-460X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2005.09.024
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ARTICLE IN PRESS P.A.A. Laura et al. / Journal of Sound and Vibration 292 (2006) 1004–1010
1005
a a1 e
Fig. 1. Vibrating mechanical system. Case of an eccentric cutout with a free edge. Table 1 Values of the first four frequency coefficients in the case of a circular plate with a circular cutout of radius a1 =a ¼ 0:1, for different values of the non-dimensional eccentricity e/a when the cutout is displaced along a radial line Eccentricity e/a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a/(fD)
O1
O2
O3
O4
0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N
4.0053 6.0053 8.7114 10.175 4.8755 6.0249 8.7407 10.206 4.9030 6.0529 8.7778 10.255 4.9182 6.0668 8.7909 10.269 4.9249 6.0706 8.7872 10.258 4.9274 6.0698 8.7744 10.235 4.9272 6.0659 8.7563 10.206 4.9268 6.0620 8.7354 10.173
13.880 14.945 18.513 21.220 13.855 14.929 18.502 21.212 13.853 14.937 18.534 21.268 13.871 14.965 18.597 21.358 13.892 14.993 18.651 21.427 13.904 15.010 18.674 21.436 13.904 15.010 18.653 21.375 13.8965 14.9958 18.5889 21.2464
25.416 26.459 30.602 34.577 25.485 26.536 30.706 34.716 25.573 26.631 30.835 34.892 25.653 26.718 30.956 35.052 25.706 26.779 31.043 35.158 25.722 26.801 31.078 35.185 25.700 26.783 31.057 35.125 25.653 26.738 30.980 34.967
29.488 30.531 34.989 39.554 29.383 30.673 35.112 39.626 29.672 30.740 35.165 39.674 29.636 30.686 35.108 39.621 29.606 30.650 35.108 39.669 29.622 30.678 35.191 39.802 29.668 30.734 35.279 39.889 29.714 30.771 35.296 39.843
for the plate transverse displacement, and then introduces the following approximation, convenient in the case of both axisymmetric and antisymmetric modes of vibration, see for example [1]. W 0 ðr0 ; yÞ ffi W 0a ðr0 ; yÞ ¼
j X j¼0
Aj0 ðajk r04 þ bjk r02 þ 1Þr02j
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þ
K X
cos ðkyÞ
j X
Ajk ðajk r04 þ bjk r02 þ 1Þr0jþk ,
ð2Þ
j¼0
k¼1
where a0 s and b0 s of each coordinate function are determined substituting each functional relation in the governing boundary conditions at the external contour, i.e., for the elastically restrained edge against rotation, qW 0 ðr0 ; yÞ ða; yÞ ¼ fM r ðr0 ; yÞ, qr
W 0 ða; yÞ ¼ 0;
(3)
Mr being the radial flexural moment, f is the flexibility coefficient of the rotational boundary spring and a is the radius of the circular plate. The Rayleigh–Ritz variational approach requires minimization of the functional J½W 0 ¼ U½W 0 T½W 0 ,
(4)
where U½W 0 is the maximum strain energy and T½W 0 is the maximum kinetic energy for the (true) displacement amplitude W 0 of the plate.
Table 2 Values of the first four frequency coefficients in the case of a circular plate with a circular cutout of radius a1 =a ¼ 0:2, for different values of the non-dimensional eccentricity e/a when the cutout is displaced along a radial line Eccentricity e/a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a/(fD)
O1
O2
O3
O4
0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N
4.7199 5.9528 8.8373 10.402 4.7272 5.9601 8.8355 10.384 4.7467 5.9784 8.8355 10.362 4.7858 6.0132 8.8483 10.354 4.8273 6.0455 8.8446 10.321 4.8569 6.0608 8.8068 10.245 4.8749 6.0608 8.7433 10.142 4.8865 6.0510 8.6676 10.040
13.606 14.661 18.152 20.741 13.673 14.760 18.407 21.192 13.756 14.899 18.738 21.699 13.833 15.020 18.975 21.981 13.882 15.099 19.072 22.007 13.895 15.125 19.031 21.829 13.869 15.094 18.858 21.488 13.830 15.027 18.580 21.034
24.881 25.904 29.936 33.753 25.003 26.030 30.112 33.998 25.311 26.365 30.639 34.795 25.664 26.899 31.481 36.001 26.141 27.315 31.958 36.361 26.224 27.396 31.905 36.096 26.125 27.218 31.594 35.582 25.774 26.941 31.151 34.878
30.878 31.973 35.440 39.656 29.899 30.910 35.290 39.770 29.396 30.149 34.365 38.616 28.983 29.989 34.295 38.710 27.591 30.213 34.861 39.695 28.910 30.552 35.397 40.216 29.638 30.754 35.529 40.134 29.629 30.781 35.374 39.782
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As has been shown elsewhere, see for example Ref. [2], in the case of a circular plate, each term in Eq. (4) can be written as 2 q2 W 0 1 qW 0 1 q2 W 0 þ 0 þ 02 r qr0 r qy2 qr02 2 0 q W 1 qW 0 1 q2 W 0 2ð1 mÞ þ 02 r qy2 qr02 r0 qr0 #) 2 0 2 1qW 1 qW 0 r0 dr0 dy, r qyqr0 r02 qy
D U½W ¼ 2 0
Z Z
ð5Þ
where D is the flexural rigidity of the plate, m its Poison’s ratio and T½W 0 ¼
ro2 h 2
Z Z
W 02 r0 dr0 dy.
(6)
The integrals in expressions (5) and (6) extend over the actual area of the double connected plate under study. Introducing the non-dimensional variables W ¼ W 0 =a;
r ¼ r0 =a.
(7)
Table 3 Values of the first four frequency coefficients in the case of a circular plate with a circular cutout of radius a1 =a ¼ 0:3 for different values of the non-dimensional eccentricity e/a when the cutout is displaced along a radial line Eccentricity e/a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a/(FD)
O1
O2
O3
O4
0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N
4.6631 6.1017 9.4525 11.312 4.6570 6.0791 9.3097 11.036 4.6503 6.0486 9.1315 10.720 4.6599 6.0181 8.9600 10.440 4.6601 6.0022 8.8257 10.231 4.6906 6.0016 8.7232 10.081 4.7296 6.0022 8.6237 9.9585
12.943 14.011 17.521 20.141 13.293 14.486 18.534 21.726 13.804 15.166 19.795 22.895 14.236 15.700 20.377 23.862 14.146 15.779 20.066 23.001 14.143 15.539 19.452 22.072 13.8727 15.2338 18.8477 21.2702
24.116 25.117 29.030 32.675 24.171 25.199 29.260 33.139 24.502 25.629 30.252 34.934 26.132 26.996 33.413 37.534 27.519 28.712 32.976 37.147 27.267 28.377 32.618 36.717 26.758 27.901 32.060 35.942
34.887 35.703 40.993 46.905 31.376 32.421 37.026 41.906 29.323 30.325 34.611 39.005 28.533 29.254 33.735 39.096 28.675 29.374 35.728 41.304 29.322 30.874 36.364 41.159 29.690 31.071 35.908 40.352
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Eqs. (5) and (6) above can be recast in a non-dimensional form. The functional for the whole system in Fig. 1 is 2J½W ¼ D
2 Z Z 2 q W 1 qW 1 q2 W þ þ 2 qr2 r qr r qy2
q2 W 1 qW 1 q2 W þ qr2 r qr r2 qy2 #) 2 2 1 q W 1 qW 2 r dr dy r qyqr r qy Z Z 2 W 2 r dr dy, O
2ð1 mÞ
ð8Þ
where aspusual, ffiffiffiffiffiffiffiffiffiffiffiffi Oi ¼ rh=Doi a2 is the non-dimensional frequency coefficient. Minimizing the governing functional with respect to the A0 jks expression (8) yields a (J K) homogeneous linear system of equations in the A0 jks. A secular determinant in the natural frequency coefficients of the system results from the non-triviality condition. The present study is concerned with the determination of the first four frequency coefficients, O1 to O4 in the case of circular plates with an eccentric circular cutout.
Table 4 Values of the first four frequency coefficients in the case of a circular plate with a circular cutout of radius a1 =a ¼ 0:4 for different values of the non-dimensional eccentricity e/a when the cutout is displaced along a radial line Eccentricity e/a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a/(FD)
O1
O2
O3
O4
0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N
4.7486 6.5515 10.689 13.048 4.7058 6.4313 10.130 12.085 4.64545 6.27936 9.57953 11.2409 4.5813 6.1292 9.1260 10.598 4.5258 6.0016 8.7818 10.142 4.4953 5.9101 8.5462 9.8560 4.5044 5.8716 8.4198 9.7107
12.253 13.497 17.699 21.026 12.989 14.530 19.849 24.148 14.116 16.008 21.908 26.074 15.121 17.057 22.259 25.816 15.310 17.006 21.372 24.431 14.821 16.328 20.144 22.835 14.194 15.605 19.068 21.455
23.218 24.234 28.203 31.953 23.507 24.625 29.237 34.248 24.284 25.778 32.373 39.173 26.056 28.123 33.807 38.105 27.324 28.554 32.890 37.065 27.367 28.400 32.615 36.768 27.257 28.266 32.406 36.461
37.293 38.255 42.481 47.231 35.506 36.224 41.105 42.594 31.137 32.196 36.827 41.686 29.212 30.433 37.166 43.816 29.620 31.781 38.230 44.215 30.365 32.316 37.799 42.757 30.354 31.979 36.830 40.276
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3. Numerical results All calculations were performed for simply supported circular plates of uniform thickness, elastically restrained against rotation at the outer edge r ¼ a while the inner border of the eccentric perforation was taken to be a free edge. In all cases, the Poisson coefficient has been taken to be m ¼ 0:3. Six tables are presented, each with a value of the ratio a1/a of the radius of the eccentric perforation with respect to the circular plate radius a. Table 1: a1 =a ¼ 0:1; Table 2: a1 =a ¼ 0:2; Table 3: a1 =a ¼ 0:3; Table 4: a1 =a ¼ 0:4; Table 5: a1 =a ¼ 0:5 and Table 6: a1 =a ¼ 0:6. In each table, in turn, four values of the nondimensional rotational spring coefficient are taken as the center of the eccentric hole is displaced along a radial line. For the double series, Eq. (2), J and K up to 16 6 terms have been used, that is to say, a secular determinant of order 96 was generated for all situations. Although satisfactory convergence is already achieved for J ¼ 8 and K ¼ 4, such high values of J and K have been used taking advantage of the speed of modern desktop computers. As usual, special care has been taken to manipulate the numerical solving of the involved determinants, and 80 bits floating point variables (IEEE-standard temporary reals) have been used to satisfy accuracy requirements. It is worth noting that computations are very stable and all frequency coefficients uniformly converge as the number of terms in the double series is increased. As a general conclusion one may say that the mathematical model seems to be quite realistic and accurate, within the realm of the classical theory of vibrating plates.
Table 5 Values of the first four frequency coefficients in the case of a circular plate with a circular cutout of radius a1 =a ¼ 0:5 for different values of the non-dimensional eccentricity e/a when the cutout is displaced along a radial line Eccentricity e/a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a/(FD)
O1
O2
O3
O4
0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N
5.0257 7.4182 12.685 15.812 4.8956 7.0447 11.329 13.683 4.7315 6.6535 10.265 12.158 4.5685 6.3172 9.4824 11.091 4.4281 6.0455 8.9203 10.352 4.3183 5.8380 8.5309 9.8633 4.2053 5.7245 8.2898 9.5581
12.028 13.802 19.979 25.154 13.161 15.669 22.971 28.003 15.067 17.773 24.168 28.495 16.370 18.691 24.044 27.967 16.491 18.399 23.055 26.544 15.872 17.459 21.556 24.591 15.139 16.395 20.024 22.634
22.566 23.751 28.556 33.520 23.388 24.957 32.201 40.367 25.576 28.369 38.129 45.338 28.769 31.065 36.162 41.042 27.891 29.158 33.606 37.932 27.113 28.253 32.575 36.749 27.019 28.072 32.340 36.4606
35.900 36.921 41.463 46.773 36.826 38.029 43.713 51.604 34.958 36.150 41.560 47.903 31.599 33.518 40.835 47.305 32.311 34.605 40.913 46.577 32.549 34.354 40.053 45.532 32.180 33.554 38.679 43.430
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Table 6 Values of the first four frequency coefficients in the case of a circular plate with a circular cutout of radius a1 =a ¼ 0:6 for different values of the non-dimensional eccentricity e/a when the cutout is displaced along a radial line Eccentricity e/a
0.1
0.2
0.3
0.4
0.5
0.6
a/(FD)
O1
O2
O3
O4
0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N 0 1 10 N
5.5689 8.8831 15.837 20.421 5.2509 7.9706 13.222 16.386 4.9072 7.2083 11.441 13.809 4.6021 6.6217 10.211 12.102 4.2920 6.1658 9.3518 10.946 4.1278 5.7953 8.7531 10.160
12.660 15.709 25.503 32.905 14.693 18.505 26.747 32.547 16.773 19.949 26.562 31.520 17.644 20.192 26.005 30.432 17.713 19.760 25.039 29.018 17.645 18.760 23.523 27.002
22.868 24.644 32.711 42.603 24.787 28.127 40.296 49.371 29.648 34.096 43.330 51.269 32.721 35.127 41.028 47.097 30.032 31.274 36.065 40.919 28.822 28.910 33.415 37.735
35.667 37.019 43.372 51.992 37.480 39.952 52.803 72.388 41.894 43.303 49.650 57.573 35.528 37.317 44.702 52.234 35.185 37.328 44.342 50.946 35.502 36.683 43.159 49.123
Acknowledgments The present study was sponsored by CONICET Research and Development Program and by Secretarı´ a General de Ciencia y Tecnologı´ a (Universidad Nacional de Mar del Plata and Universidad Nacional del Sur). References [1] D.R. Avalos, H.A. Larrondo, P.A.A. Laura, V. Sonzogni, Transverse vibrations of a circular plate with a concentric square hole with free edges, Journal of Sound and Vibration 209 (5) (1998) 889–891. [2] A.W. Leissa. Vibration of plates, NASA SP160, 1969.