Bounds for the free vibration frequencies of homogeneous anisotropic bodies with constrained boundary

Abstract

A modification of the classical comparison theorem for the free vibration frequencies of homogeneous linearly elastic bodies of an arbitrary anisotropy, which occupy a region of arbitrary shape with clamped boundary, is proved by means of Van Hove's theorem. Some other similar modifications of the comparison theorem for homogeneous linearly elastic bodies of special types of anisotropy (characterized by the presence of specular symmetry), having the shape of a rectangular parallelepiped with faces parallel to the planes of symmetry, and with sliding boundary conditions either along the faces or along their normals, are proved using modifications of Van Hove's theorem. On the basis of the set of proved modifications of the comparison theorem, a method for obtaining refined bilateral bounds for all frequencies of the free vibration spectrum pertinent to the specified problems (for which the exact values of frequencies are, as a rule, unknown) is proposed. The bounds turn out to depend in a simple manner on the least and the greatest velocities of propagation of elastic waves in a solid and on the characteristic geometrical dimensions of the body. Examples are considered. A version of the comparison theorem modifications and a method of obtaining the bounds for frequencies, suitable for the linearized problem of small free vibrations of homogeneous uniformly strained non-linearly elastic bodies, and also for free vibrations of moderately inhomogeneous linearly elastic ones, is proposed.