Natural Frequencies and Mode Shapes of Elliptic Plates With Boundary Characteristic Orthogonal Polynomials as Assumed Shape Functions

Abstract

The natural frequencies and natural modes of vibration of uniform elliptic plates with clamped, simply supported and free boundaries are investigated using the Rayleigh-Ritz method. A modified polar coordinate system is used to investigate the problem. Energy expressions in the Cartesian coordinate system are transformed into the modified polar coordinate system. Boundary characteristic orthogonal polynomials in the radial direction, and trigonometric functions in the angular direction are used to express the deflection of the plate. These deflection shapes are classified into four basic categories, depending on their symmetrical or antisymmetrical properties about the major and minor axes of the ellipse. The first six natural modes in each of the above categories are presented in the form of contour plots.

Natural Frequencies of Orthotropic, Elliptical and Circular Plates

“…The most accurate values among the present results are fully converged values up to the figures. Close agreement may be seen to be achieved except the values for AS modes with those in reference [21]. For AS-1 mode, the present result is in close agreement with that in reference [20].…”

“…For AS-1 mode, the present result is in close agreement with that in reference [20]. The author cannot explain the discrepancy between the present results and those in reference [21]. It may be noted that the rate of convergence is relatively slower for smaller values of b/a.…”

“…The convergence study for isotropic, clamped plates are presented in Table 5, together with the comparison values by Shibaoka [19], McNitt [17], Tomar and Gupta [5], Singh and Chakraverty [20], Rajalingham et al [21], and Carrington [22]. The most accurate values among the present results are fully converged values up to the figures.…”

“…The frequency parameters for isotropic plates were presented by previous investigators. In particular, Rajalingham et al [21] presented the lowest six frequency parameters in each mode type for aspect ratio 0Á5 for all the classical free, simply supported and clamped Table 6 Frequency parameters (rh o 2 a 4 =D 1=2 for isotropic, free, elliptical plates (n ¼ 0Á3) 6Á7778 32Á817 78Á123 48Á582 101Á72 17Á389 53Á058 108Á01 25Á747 73Á638 0Á25 6Á7737 32Á779 77Á901 39Á621 86Á070 17Á377 52Á955 107Á60 20Á676 61Á181 101Á22 0Á3 6Á7654 32Á675 71Á773 33Á710 75Á839 17Á343 52Á718 95Á362 17Á301 53Á015 77Á451 0Á35 6Á7518 32Á503 53Á775 29Á520 68Á592 17Á284 52Á345 74Á353 14Á891 47Á243 0Á4 6Á7321 32Á257 41Á937 26Á393 63Á137 17Á195 51Á824 60Á261 13Á084 42Á928 0Á45 6Á7054 31Á928 33Á719 23Á963 58Á824 17Á076 50Á267 51Á179 11Á677 39Á558 0Á5 6Á6705 27Á768 31Á513 22Á015 55Á272 16Á921 42Á991 50Á276 10Á548 36Á828 0Á55 6Á6264 23Á325 30Á987 20Á411 52Á241 16Á726 37Á480 49Á183 9Á6206 34Á547 0Á6 6Á5712 19Á922 30Á337 19Á061 49Á565 16Á484 33Á239 47Á852 8Á8447 32Á590 0Á65 6Á5029 17Á264 29Á551 17Á901 43Á290 16Á185 29Á938 46Á289 8Á1848 30Á868 0Á7 6Á4185 15Á160 28Á626 16Á888 37Á800 15Á822 27Á357 44Á538 7Á6161 29Á320 43Á750 0Á75 6Á3144 13Á478 27Á580 15Á988 33Á366 15Á386 25Á345 42Á669 7Á1203 27Á898 41Á595 0Á8 6Á1861 12Á128 26Á449 15Á174 29Á743 14Á880 23Á790 40Á755 6Á6841 26Á570 39Á961 0Á85 6Á0286 11Á048 25Á277 14Á427 26Á755 14Á311 22Á599 38Á852 6Á2969 25Á311 38Á695 0Á9 5Á8381 10Á191 24Á101 13Á731 24Á274 13Á699 21Á694 36Á997 5Á9509 24Á107 0Á95 5Á6135 9Á5199 22Á949 13Á072 22Á205 13Á068 21Á005 35Á209 5Á6397 22Á949 1Á0 5Á3583 9Á0031 21Á835 the same as AS 12Á439 20Á475 33Á495 5Á3583 21Á835 boundary conditions, and Singh and Chakraverty presented the first four frequency parameters for free [24], simply supported [25] and clamped [20] plates with various aspect ratios. Nevertheless, it may be worth presenting exhaustive accurate results for all the classical boundary conditions classifying the mode types, since the values in the literature are limited to the first several (five or less) frequency parameters without distinguishing the mode types …”

“…Nevertheless, it may be worth presenting exhaustive accurate results for all the classical boundary conditions classifying the mode types, since the values in the literature are limited to the first several (five or less) frequency parameters without distinguishing the mode types [20, 24, 25, etc. ] or to the results for each mode type for a special aspect ratio [21]. In Tables 6-8, the lowest 10 frequency parameters for various aspect ratios are presented for isotropic plates with free, simply supported and clamped boundary conditions respectively.…”