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Symmetrization of Ritz Approximation Functions for Vibrational Analysis of Trigonal Cylinders



Ward L. Johnson, Paul R. Heyliger


In the Ritz method of calculating vibrational normal modes, a set of finite series approximation functions provides a matrix eigenvalue equation for the coefficients in the series and the resonant frequency. The matrix problem usually can be block-diagonalized by grouping the functions into subsets according to the their properties under the symmetry operations that are common to the specimen geometry and crystal class. This task is addressed, in this study, for the case of cylindrical specimens of crystals belonging to one of the higher trigonal crystal classes. The existence of two-fold degenerate resonant modes significantly complicates the analysis. Group theoretical projection operators are employed to extract, from series approximation functions in cylindrical coordinates, the terms that transform according to each irreducible representation of the point group. This provides a complete symmetry-based block-diagonalization and categorization of the modal symetries. Off-diagonal projection operators are used to provide relations between the displacement patterns of the degenerate modes. The method of analysis is presented in detail to assist in its application to other geometries, crystal structures, or forms of ritz approximation functions.
Journal of the Acoustical Society of America
No. 4


acousic resonance, group theory, resonant ultrasound spectroscopy, Ritz method, trigonal crystals


Johnson, W. and Heyliger, P. (2003), Symmetrization of Ritz Approximation Functions for Vibrational Analysis of Trigonal Cylinders, Journal of the Acoustical Society of America, [online], (Accessed May 23, 2024)


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Created April 1, 2003, Updated June 2, 2021