On the equivalent systems for concurrent springs and dampers – Part 1: Small in-plane oscillations

Abstract

In this work, concurrent linear springs placed in the system that performs small in-plane oscillations around the stable equilibrium position are considered. New theorems defining how they can be replaced by two mutually orthogonal springs are provided. The same concept is applied to find two mutually orthogonal linear viscous dampers that can replace a system of concurrent linear viscous dampers. The directions of such springs and dampers correspond to the principal stiffness and damping axes, respectively. So far unknown invariants related to the sum of stiffness coefficients and damping coefficient of the original and equivalent systems are presented. A few examples are given to illustrate the use and benefits of this approach. In addition, it is shown how the concept of two mutually orthogonal springs can be beneficially used for analysing problems concerned with oscillations of a particle on elastic frames.