BOUNDS FOR THE FREE-VIBRATION FREQUENCIES OF HOMOGENEOUS ANISOTROPIC BODIES WITH CONSTRAINED BOUNDARY

A modification of the classical comparison theorem for the free vibrat ion frequencies of homogeneous linearly elastic bodies of an arbitrary anisotropy, which occupy a region of arbitrary shape with clamped bou ndary, is proved by means of Van Hove's theorem. Some other similar mo difications 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 parallelepipe d with faces parallel to the planes of symmetry, and with sliding boun dary conditions either along the faces or along their normals, are pro ved using modifications of Van Hove's theorem. On the basis of the set of proved modifications of the comparison theorem, a method for obtai ning refined bilateral bounds for all frequencies of the free vibratio n spectrum pertinent to the specified problems (for which the exact va lues of frequencies are, as a rule, unknown) is proposed. The bounds t urn out to depend in a simple manner on the least and the greatest vel ocities of propagation of elastic waves in a solid and on the characte ristic geometrical dimensions of the body. Examples are considered Ave rsion of the comparison theorem modifications and a method of obtainin g the bounds for frequencies, suitable for the linearized problem of s mall free vibrations of homogeneous uniformly strained non-linearly el astic bodies, and also for free vibrations of moderately inhomogeneous linearly elastic ones, is proposed. (C) 1997 Elsevier Science Ltd. Al l rights reserved.