Free vibrations of an anisotropic cylinder with the Ritz method

Cylinders of elastic solids are widely used in various mechanical and electrical structures for many purposes including structural and functional elements. Vibrations of cylinders are of interests for the optimal design and enhanced performance through the consideration of structural and material parameters. By extending earlier work on free vibrations of an elastic cylinder based on the expansion of displacements in Chebyshev polynomials, we formulated the procedure with the Ritz method for an anisotropic cylinder of quartz crystal. The challenges of such an analysis include the material properties dependence on the angular position in polar coordinates, strong couplings of modes, and dense matrices for eigenvalue extraction. The vibration frequency and mode shapes are obtained from the eigenvalue problem. We intended to use the analytical procedure and software program with the RUSpec for material characterization with cylindrical samples. We found the Chebyshev polynomials are adequate to represent displacements for accurate vibration solutions.