About the Free-vibration Mode Shapes of Elastoplastic Dissipative Systems

The author presents the conditions of the generalized orthogonality of the free-vibration mode shapes of an elastic dissipative system, for which traditional classical orthogonality conditions are a private case. As opposed to these conditions, the above ratios contain the mass matrix, the damping matrix, and the diagonal form of the spectral characteristics (damping coefficients and mode-shape frequencies). Within the theory of time analysis, free-vibration mode shapes of an elastoplastic system are built on the basis of using a schematized diagram of strain with hardening. The author proposes a design scheme that reduces the process of nonlinear vibrations to a sequence of processes flowing according to a linear scenario within the time intervals called quasilinear. In these intervals, the parameters of the dynamic model (elements of the stiffness matrix and the damping matrix) remain unchanged, all the changes occur only when passing through the critical points. As a result, the author formulated the condition for the nondegenerate state of an elastoplastic dissipative system. According to the condition, local plastic zones characterized by the size, the number and location of the zones on the design scheme of the structure correspond to each quasilinear interval. Since within the intervals, the parameters of the plastic zones are unchanged, the conditions of the generalized orthogonality of the mode shapes of the elastoplastic system are satisfied by analogy with the vibration mode shapes of an elastic dissipative system. The free-vibration motion of a hinged beam with three degrees of freedom are analyzed taking into account local plastic zones with different lengths and the location of zones in different nodes. It is shown that the configuration of the forms of elastoplastic oscillations differs qualitatively from the configuration of the corresponding forms of elastic vibrations.