Asymptotic non-linear normal modes for large-amplitude vibrations of continuous structures

Abstract

Non-linear normal modes (NNMs) are used in order to derive accurate reduced-order models for large amplitude vibrations of structural systems displaying geometrical nonlinearities. This is achieved through real normal form theory, recovering the definition of a NNM as an invariant manifold in phase space, and allowing definition of new coordinates non-linearly related to the initial, modal ones. Two examples are studied: a linear beam resting on a non-linear elastic foundation, and a non-linear clamped-clamped beam.Throughout these examples, the main features of the NNM formulation will be illustrated: prediction of the correct trend of non-linearity for the amplitude-frequency relationship, as well as amplitude-dependent mode shapes. Comparisons betwen different models -using linear and non-linear modes, different number of degrees of freedom, increasing accuracy in the asymptotic developments-are also provided, in order to quantify the gain in using NNMs instead of linear modes.

Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells

“…It is to be noted that in contrast to the linear normal modes, the NNMs are not orthogonal, but due to their invariance property they could still be suitable for accurate and efficient order reduction in nonlinear oscillatory systems ). Moreover the mode shapes corresponding to NNMs would change with amplitude and thus with time (Touze et al 2004).…”

A generalized frequency detuning method for multidegree-of-freedom oscillators with nonlinear stiffness

“…Note also that the terms h γ (u) and n uγ (u) has been introduced to account for additional terms due to the n vγ (v) term in (13). The state vector v can be eliminated from (13) using (14) and thenü can be eliminated using (15), to produce…”

“…Note that where matrices are scalar the bold notation is dropped. The third transform, the near-identity transform, uses (14) to convert this equation into a dynamic equation of the form given in (15). To evaluate h and N u , the nonlinear term N v (v) must be expressed as a power series in ε, (13).…”

“…The analytical method we use here a version of normal forms [12][13][14][15][16]. Normal forms is usually applied to first-order nonlinear oscillator equations and an assessment of their accuracy is given in [17].…”

Nonlinear Normal Modes of Nonconservative Systems

“…To derive our finite element formulation, a weighted residual approach is applied to Equations (8). The formulation is presented in Equations (10) ∇N b ).…”