The “associated-periodicity” extension of Fourier analysis is used to obtain an exact solution of the classical, three-dimensional elasticity problem of free vibration of the rectangular parallelepiped. This problem has been completely stated for more than a century and has been solved for only a very few special cases. The characteristic determinant yielding the eigenvalues is formulated for the completely free, rectangular parallelepiped, although the method of associated periodicity can be straightforwardly applied to arbitrary boundary conditions. Modes are classified into eight mutually exclusive and collectively exhaustive symmetry classes. Numerical results are presented for the frequency spectrum of plane-strain vibrations of completely free rectangles according to two-dimensional elasticity and are compared with classical Bernoulli-Euler beam theory and Timoshenko beam theory (including the effects of shear deformation and rotary inertia).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.