The parametrical free vibrations of elastic systems. the analytical exact solutions

Abstract

The movements of many material systems can be described by differential equations that have coefficients that depend on time. It is difficult to determine the solutions of these equations. Although many concrete problems lead to non-linear differential equations where we can also find the term of variable damping, the classic equations that have been studied more were Hill or Mathieu, with periodic coefficient, that do not contain derivations of the first order. At this kind of equations, we reduce them to second order using substitutions as we will demonstrate. In this work, we show that we can obtain analytical exact solutions for non-linear homogeneous differential equations with a variable coefficient which describes the parametric free vibrations of mechanical systems.