Research into Multi-mass Vibro-impact System

In the second chapter, a study is carried out of a multi-mass vibration-impact system, which is a generalization of a two-mass system and consists of \(N > 2\) identical masses connected by springs of the same stiffness. Under these assumptions, it was possible, by introducing normal coordinates, to decompose the system into N second-order equations and, in a compact form, to find in analytical form the coordinates of fixed points corresponding to single-impact n-multiple periodic movements and to study their stability depending on the stiffness values \(N\) of the connecting springs, the amplitude and frequency of the disturbing force. The problem of studying the stability of fixed points is reduced to determining the roots of some characteristic equation that satisfy the conditions \(\left| {z_{i} } \right| < 1\) . This problem is reduced to finding the maximum absolute value of the matrix \(A\) eigenvalue belonging to the interior of a circle of unit radius. The optimal values of the system parameters are found at which the maximum modulus eigenvalue of the matrix \(A\) (or the maximum modulus root of the characteristic polynomial) takes on a minimum value.