Determination of Dissipative Characteristics for Solving Problems of Dynamic Loading of Elastic Structures Using a Modal Approach

A method is proposed for solving non-stationary dynamic problems with complex elastic moduli or stiffness matrices, which makes it possible to take into account the local dissipative properties of structural elements. It is shown that in synthesized structures, even with significantly different dissipative properties, the dynamic behavior and loading parameters at resonant frequencies are determined by single integral logarithmic decrements obtained by mass-energy averaging of the dissipative properties of structures. Two concepts for estimating the integral quality factors and logarithmic vibration decrements of structures with different dissipative properties are proposed, one of which is based on the physical features of resonant phenomena in mechanical systems, and the second is based on the analysis of the behavior of damped fundamental solutions of the corresponding homogeneous equations. The coincidence of the integral logarithmic decrements obtained on the basis of different concepts makes it possible to solve non-stationary problems for structures with different dissipative properties by expanding the kinematic parameters in terms of natural vibration modes (modal method). The results of calculations of the dynamic behavior of structures with different dissipative properties are demonstrated on model problems.