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Normal Modes for Non-Linear Vibratory Systems

Abstract : A methodology is presented which extends to non-linear systems the concept of normal modes of motion which is well developed for linear systems. The method is constructive for weakly non-linear systems and provides the physical nature of the normal modes along with the non-linear differential equations which govern their dynamics. It also provides the non-linear co-ordinate transformation which relates the original system co-ordinates to the modal co-ordinates. Using this transformation, we demonstrate how an approximate non-linear version of superposition can be employed to reconstruct the overall motion from the individual non-linear modal dynamics. The results presented herein for non-linear systems reduce to modal analysis for the linearized system when non-linearities are neglected, even though the approach is entirely different from the traditional one. The tools employed are from the theory of invariant manifolds for dynamical systems and were inspired by the center manifold reduction technique. In this paper the basic ideas are outlined, a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions.
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Steven Shaw, Christophe Pierre. Normal Modes for Non-Linear Vibratory Systems. Journal of Sound and Vibration, Elsevier, 1993, 164 (1), pp.85-124. ⟨10.1006/jsvi.1993.1198⟩. ⟨hal-01544500⟩

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