On the normal modes of free vibration of inhomogeneous and anisotropic elastic objects

The Hamilton’s principle approach to the calculation of vibrational modes of elastic objects with free boundaries is exploited to compute the resonance frequencies of a variety of anisotropic elastic objects, including spheres, hemispheres, spheroids, ellipsoids, cylinders, eggs, shells, bells, sandwiches, parallelepipeds, cones, pyramids, prisms, tetrahedra, octahedra, and potatoes. The paramount feature of this calculation, which distinguishes it from previous ones, is the choice of products of powers of the Cartesian coordinates as a basis for expansion of the displacement in a truncated complete set, enabling one to analytically evaluate the required matrix elements for these systems. Because these basis functions are products of powers of x, y, and z, this scheme is called the xyz algorithm. The xyz algorithm allows a general anisotropic elastic tensor with any position dependence and any shape with arbitrary density variation. A number of plots of resonance spectra of families of elastic objects are displayed as functions of relevant parameters, and, to illustrate the versatility of the method, the measured resonant frequencies of a precision machined but irregularly shaped sample of aluminum (called a potato) are compared with its computed normal modes. Applications to materials science and to seismology are mentioned.