Free vibrations of finite quartz crystal cylinders with the Rayleigh-Ritz method

We have extended the calculation procedure to cylindrical samples of anisotropic crystals for RUSpec measurements. The deformation is expressed with Chebyshev polynomials and the Rayleigh-Ritz method is used. We concentrated on anisotropic materials which can be more complicated for the formulation of vibration frequencies. The formulation is angular dependent with cylindrical coordinates. The calculation and experiment are carried out with quartz crystal samples and we found the results are accurate with frequencies and eventually obtained elastic constants which are in good agreement with known data. It is proven that the Rayleigh-Ritz method based on the deformation in Chebyshev polynomials offers simple and accurate solutions for vibration frequencies and mode shapes. With the known algorithms for matching elastic constants based on the measurement, we can obtain accurate elastic constants of anisotropic materials with a relatively simple procedure and test samples.