Exact Modal Characterization of the Non Conservative Non Linear Radial Mass Spring System

Since the spread of robotic systems embedding in their mechanics purposefully designed elastic elements, the interest in characterizing and exploiting non-linear oscillatory behaviors has progressively grown. However, few works so far looked at the problem from the point of view of modal analysis. This is particularly surprising if considered the central role that modal theory had in the development of classic results in analysis and control of linear mechanical systems. With the aim of making a step toward translating and extending this powerful tool to the robotic field, we present the complete modal characterization of a simple yet representative non-linear elastic robot: the 2D planar mass-spring-damper system. Generic non-linear elastic forces and dissipative effects are considered. We provide here exact descriptions of the two non-linear normal modes of the system. We then extend the analysis to generic combinations of the modes in conservative case and for small damping. Simulations are provided to illustrate the theoretical results. This is one of the very firsts applications of normal mode theory to dynamically coupled non-linear systems, and the first exact result in the field.