Free Vibration of the Rectangular Parallelepiped

Abstract

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).

On Coupled Longitudinal and Lateral Vibrations of Elastic Rods

“…This expression is obtained by simplifying the expression for the kinetic energy of the directed curve recorded by Naghdi [20] and using equations (12). It may be used in a standard manner to construct orthogonality relations for the eigenmodes.…”

“…Most of their results pertain to large wavelength vibrations. For parallelepipeds of "nite length, and using the three-dimensional theory of linear elasticity, exact solutions were obtained for certain cases by LameH [7] and Mindlin [8,9] and approximate solutions have been obtained by Ekstein and Schi!man [10], Hutchinson and Zillmer [11], Leissa et al [12,13], and Lim [14]. Recently, Rubin [15] employed his theory of a Cosserat point to examine this problem.…”

Uniqueness and properties of an interesting class of natural oscillations of a solid rectangular parallelepiped

“…For stress-free surfaces, an infinite matrix system arises [1,2]. Physically [3], at the surfaces a coupling of transverse and longitudinal waves occurs, so individual component waves cannot in general be treated independently.…”

Three-Dimensional Vibration Analysis of Rectangular Thick Plates on Pasternak Foundation with Arbitrary Boundary Conditions

“…Hutchinson and Zillmer [11] used the series solution method to analyze the free vibration of a completely free parallelepiped. Fromme 2 Shock and Vibration and Leissa [12] extended the Fourier method to investigate the free vibration of the rectangular parallelepiped with simple classical boundary conditions based on the threedimensional elasticity theory. On the basis of the threedimensional elasticity theory and differential quadrature method (DQM), Malik and Bert [13], Liew and Teo [14], and Liew et al [15] investigated the free vibration characteristics of rectangular plates with some selected classical boundary conditions.…”