On the equivalent systems for concurrent springs and dampers – Part 2: Small out-of-plane oscillations

Abstract

Concurrent linear springs belonging to systems that perform small out-of-plane oscillations around a stable equilibrium position are considered with a view to obtaining equivalent systems of three mutually orthogonal linear springs. Theorems defining their stiffness coefficients as well as their position, i.e. the position of the principal stiffness axes for which the potential energy does not contain mixed terms, are stated and proven. So far unknown invariants related to the sum of original and new stiffness coefficients are provided. In addition, the equivalent system of three mutually orthogonal dampers is obtained for any system of out-of-plane concurrent linear viscous. The theorem defining their damping coefficients and their directions, collinear with the principal damping axes for which the dissipative function does not contain mixed terms, is provided. The corresponding invariant for damping coefficients is presented, too. An ellipsoid of displacement and an ellipsoid of stiffness are discussed. Three illustrated examples are given.