Free Vibration of a Class of Homogeneous Isotropic Solids

A Ritz approach, with simple polynomials as trial functions, is used to obtain the natural frequencies of vibration of a class of solids. Each solid is modeled by means of a segment which is described in terms of Cartesian coordinates and is bounded by the yz, zx, and xy orthogonal coordinate planes as well as by a fourth curved surface, which is defined by a polynomial expression in the coordinates x, y, and z. By exploiting symmetry, a number of three-dimensional solids previously considered in the open literature are treated, including a sphere, a cylinder and a parallelepiped. The versatility of the approach is then demonstrated by considering several solids of greater geometric complexity, including an ellipsoid, an elliptical cylinder, and a cone.