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 and Mode Shapes of Elliptic Plates With Boundary Characteristic Orthogonal Polynomials as Assumed Shape Functions
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Rajalingham, C., Bhat, R. B., and Xistris, G. D. (July 1, 1993). "Natural Frequencies and Mode Shapes of Elliptic Plates With Boundary Characteristic Orthogonal Polynomials as Assumed Shape Functions." ASME. J. Vib. Acoust. July 1993; 115(3): 353–358. https://doi.org/10.1115/1.2930356
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