1 research outputs found
One-Dimensional Solution Families of Nonlinear Systems Characterized by Scalar Functions on Riemannian Manifolds
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