RECIPROCITY LAWS FOR OSCILLATIONS OF DISSIPATIVE SYSTEMS

Abstract

) is presented. The method is based on the use of the algebraic theorem of P.L. Pasternak and on the new properties of the Duhamel integral, which are obtained for a dissipative system with internal friction of the material, which is taken into account on the basis of the non-proportional damping model. For displacements, velocities and accelerations, the dynamic reaction equations are written in the form of systems of linear equations and their symmetrical structure is shown. The functional dependence of the force parameters of the calculation model and the corresponding kinematic parameters of the reaction is determined by an arbitrary scalar function of time. An extended interpretation of the reciprocity theorems is given and sufficient conditions for their fulfillment are formulated, which consist in the requirement that the matrix differential operator of the equation of motion be symmetrical. New laws of reciprocity in dissipative systems are formulated and proved. The reciprocity of the product between the velocities / accelerations of masses and nodal forces is established. In contrast to the well-known theorem on the reciprocity of possible work, these laws are theorems on the 1st / 2nd derivative of possible work with respect to time and therefore go beyond the Betti principle. For particular cases of these theorems, the reciprocity of velocities and reciprocity of accelerations is shown. Expressions of general and particular theorems have a fairly simple mathematical form that does not require recourse to integral transformations, and are presented in an analytical form.