Nonlinear Dynamics 8:315-346, 1995. © 1995 Kluwer Academic Publishers. Printed in the Netherlands. Non-Linear Normal Modes, Invariance, and Modal Dynamics Approximations of Non-Linear Systems

Abstract. Non-linear systems are here tackled in a manner directly inherited from linear ones, that is, by using proper normal modes of motion. These are defined in terms of invariant manifolds inthe system's phase space, on which the uncoupled system dynamics an be studied. Two different methodologies which were previously developed toderive the non-linear normal modes of continuous systems- one based on a purely continuous approach, and one based on a discretized approach towhich the theory developed for discrete systems can be applied- are simultaneously applied to the same study case- an Euler-Bernoulli beam constrained bya non-linear spring- and compared asregards accuracy and reliability. Numerical simulations of pure non-linear modal motions are performed using these approaches, and compared tosimulations of equations obtained by a classical projection onto the linear modes. The invariance properties ofthe non-linear normal modes are demonstrated, and it is also found that, for a pure 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 inthe dynamics by solving only one non-linear o dinary differential equation. Key words: Non-linear normal modes, non-linear modal dynamics, invariant manifolds. 1