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This article discusses free transverse vibrations of a homogeneous rod. The left end of the rod is clamped, and a cylindrical weight is concentrated at the right end. The eigenfrequencies of the rod vibration are known. The purpose of this work is to determine the parameters of the end cylindrical weight of the rod (mass, moment of inertia, length and radius) by the natural frequencies of the rod vibrations. We use a partial differential equation derivative of the fourth order to solve this problem. This equation and boundary conditions are reduced to a spectral problem. To find the mass and moment of inertia of the weight, the «Method of an additional unknown» was applied. In the characteristic determinant of the spectral problem, there are terms that contain products of unknown coefficients. The essence of the «Method of an additional unknown» is that some of these products are proposed to be considered new additional unknowns, through which the rest can be expressed. It is shown that the mass and moment of inertia of the weight can be found using the three natural frequencies of the rod vibrations. Formulas for finding the length and radius of a cylindrical weight are obtained, and corresponding examples of finding unknown parameters are considered.

Determination of the dimensions of the cylindrical weight at the end of the rod