Free vibrations of an anisotropic elastic ellipsoid

An elastic ellipsoid is one of popular configurations of materials as samples and components in various engineering applications. In this study, we are interested in free vibrations of anisotropic elastic ellipsoids. For simple formulation and efficient calculation, the Cartesian coordinates are utilized as the framework of this study. It is hoped that the formulation, ostensibly with lengthy expressions and coupled terms, can be solved with fewer terms of displacement functions in the ellipsoidal representation for simple evaluations of stiffness and mass matrices with the Rayleigh-Ritz solution procedure. By applying the variational principle of elasticity, equations of motion are given in terms of vibration amplitudes with matrices containing the displacement components in Chebyshev polynomials of coordinates. The free vibration frequency and mode of an ellipsoid of anisotropic material are studied. The accuracy of the analysis is improved through the increase of orders of Chebyshev polynomials.