Abstract
Approximations of the resonant non-linear normal modes of a general class of weakly non-linear one-dimensional continuous systems with quadratic and cubic geometric non-linearities are constructed for the cases of two-to-one, one-to-one, and three-to-one internal resonances. Two analytical approaches are employed: the full-basis Galerkin discretization approach and the direct treatment, both based on use of the method of multiple scales as reduction technique. The procedures yield the uniform expansions of the displacement field and the normal forms governing the slow modulations of the amplitudes and phases of the modes. The non-linear interaction coefficients appearing in the normal forms are obtained in the form of infinite series with the discretization approach or as modal projections of second-order spatial functions with the direct approach. A systematic discussion on the existence and stability of coupled/uncoupled non-linear normal modes is presented. Closed-form conditions for non-linear orthogonality of the modes, in a global and local sense, are discussed. A mechanical interpretation of these conditions in terms of virtual works is also provided.
Highlights
Non-linear modal couplings due to internal resonances are possible in distributed-parameter systems depending on some geometrical and=or mechanical control parameters
There is a theoretical interest in exploring the bifurcation behavior of the non-linear normal modes of continuous systems per se because this leads to a deeper understanding of the forced resonant dynamics when these modal interactions are activated
On the other hand, when studying modal interactions in the non-planar dynamics of cables described by two displacement components, the full-basis Galerkin discretization leads to two sets of coupled non-linear ordinary-di erential equations, as documented in [11]
Summary
Non-linear modal couplings due to internal resonances are possible in distributed-parameter systems depending on some geometrical and=or mechanical control parameters. Investigating non-linear normal modes in clamped–clamped buckled beams [10], it was found that some modes could not interact at all, in spite of proper integer ratios between the associated frequencies, due to vanishing of the non-linear interaction coe cients in the normal forms Inspired by these results, we attempt to develop a general and systematic approach to determine a priori conditions for activation=non-activation of the modes under speciÿc internal resonance conditions. The objective of Part I is twofold: (i) to study the existence and stability of coupled=uncoupled non-linear normal modes over variation of the internal resonance detuning in a general and systematic fashion using a set of partial-di erential equations of motion and boundary conditions with general linear, quadratic, and cubic geometric operators; (ii) to determine closed-form conditions for the non-linear global and local orthogonality of the modes thereby extending the linear orthogonality concept applicable to self-adjoint systems entering the ÿnite-amplitude vibration regime. The obtained closed-form non-linear orthogonality conditions can save an enormous computational e ort when it is needed to ascertain a priori whether orthogonality=non-orthogonality occurs in self-adjoint systems, the latter leading to modal interactions
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