Non-Linear Normal Modes, Invariance, and Modal Dynamics Approximations of Non-Linear Systems

Abstract

Abstract Non-linear systems are here tackled in a manner directly inherited from linear ones, i.e., by denning proper normal modes of motion. These are defined in terms of invariant manifolds in the system’s phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach where the theory developed for discrete systems can be applied — are simultaneously applied to the same study case — an Euler-Bernoulli beam constrained by a non-linear spring —, and compared as regards accuracy and reliability, resulting in the abandonment of the continuous approach for lack of reliability. Numerical simulations of purely non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the nonlinear normal modes are demonstrated, and it is also found that, for a purely non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differentia] equation.