Free Vibrations Of Anisotropic Ellipsoids

An elastic ellipsoid is one of popular configurations of materials as samples and components in various engineering applications.An ellipsoid can be used for the analytical representation of both spheres and ellipsoids of elastic materials,creating a more generalized model for the needed analysis of deformations and stresses.In this study,we are interested in free vibrations of elastic ellipsoids of generalized anisotropic materials.For simple formulation and efficient calculation,the Cartesian coordinate is utilized as the framework of this study.It is hoped that the formulation,ostensibly with lengthy and strange expressions and coupled terms,can be solved with fewer terms of displacement functions in ellipsoidal representation for simple evaluations of stiffness and mass matrices in 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 Cartesian coordinate.The equations are formed in a systematic procedure as we have been shown in earlier studies with Cartesian and cylindrical coordinates for isotropic solids of different configurations,like cubes and cylinders.The formulation,procedure,elements of matrices,and numerical results are compared with similar studies for ellipsoids in other coordinates such as the spherical systems.The case of samples of isotropic ellipsoids is calculated with geometric parameters for the validation of the procedure and formulation.The results have been compared with known studies from other coordinate systems and methods to validate the current formulation and procedure.The calculations from isotropic ellipsoid samples are consistent with known results,and the extension of the formulation and procedure for anisotropic ellipsoids of novel crystals will be straightforward.Then we also analyzed the relation between the frequencies of free vibration with different size of quartz crystal ellipsoids and the associated vibration modes.