Traction-free vibrations of finite trigonal elastic cylinders

The unrestrained, traction-free vibrations of finite elastic cylinders with trigonal material symmetry are studied using two approaches, based on the Ritz method, which formulate the weak form of the equations of motion in cylindrical and rectangular coordinates. Elements of group theory are used to divide approximation functions into orthogonal subsets, thus reducing the size of the computational problem and classifying the general symmetries of the vibrational modes. Results for the special case of an isotropic cylinder are presented and compared with values published by other researchers. For the isotropic case, the relative accuracy of the formulations in cylindrical and rectangular coordinates can be evaluated, because exact analytical solutions are known for the torsional modes. The calculation in cylindrical coordinates is found to be more accurate for a given number of terms in the series approximation functions. For a representative trigonal material, langatate, calculations of the resonant frequencies and the sensitivity of the frequencies on each of the elastic constants are presented. The dependence on geometry (ratio of length to diameter) is briefly explored. The special case of a transversely isotropic cylinder (with the elastic stiffness C14 equal to zero) is also considered.