Time Analysis of Dissipative Systems Elastic Response. Part 1

AbstractThe article presents the results for the theory of time analysis of discrete dissipative systems, the vibrations of which are described with the help of ordinary differential equations with constant coefficients. The theory is based on the algebraic approach centered around the research of matrix quadratic equation which is characteristic for the homogeneous system of ordinary differential equations. The basic concepts of the theory applied to the analysis of elastic systems: mathematical apparatus, solution structure of matrix quadratic equation and properties of matrix ratios, are presented. The conditions, under which the structure of solutions of the quadratic equation corresponds coupled and uncoupled differential equations of motion, are considered. The scheme of the solution of solving dissipative system elastic response equations, that allows determining the displacement and speeds of the nodes of the construction as the result of the effects of arbitrary dynamic load is given. These equations are written in a non-trivial matrix form of Duhamel's integral