One-Dimensional Solution Families of Nonlinear Systems Characterized by Scalar Functions on Riemannian Manifolds

TL;DR

This paper explores the existence and properties of nonlinear normal modes in complex Riemannian systems, extending classical linear concepts to nonlinear, non-Euclidean contexts with applications to biomechanics and robotics.

Contribution

It introduces a theorem with necessary and sufficient conditions for strict nonlinear normal modes in Riemannian systems and provides a constructive example involving an elastic double pendulum.

Findings

Necessary and sufficient conditions for nonlinear normal modes

Existence of low-dimensional invariant manifolds in nonlinear systems

Design of nonlinear oscillation modes in elastic double pendulum

Abstract

For the study of highly nonlinear, conservative dynamic systems, finding special periodic solutions which can be seen as generalization of the well-known normal modes of linear systems is very attractive. However, the study of low-dimensional invariant manifolds in the form of nonlinear normal modes is rather a niche topic, treated mainly in the context of structural mechanics for systems with Euclidean metrics, i.e., for point masses connected by nonlinear springs. Newest results emphasize, however, that a very rich structure of periodic and low-dimensional solutions exist also within nonlinear systems such as elastic multi-body systems encountered in the biomechanics of humans and animals or of humanoid and quadruped robots, which are characterized by a non-constant metric tensor. This paper discusses different generalizations of linear oscillation modes to nonlinear systems and proposes a definition of strict nonlinear normal modes, which matches most of the relevant properties of the linear modes. The main contributions are a theorem providing necessary and sufficient conditions for the existence of strict oscillation modes on systems endowed with a Riemannian metric and a potential field as well as a constructive example of designing such modes in the case of an elastic double pendulum.