Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes

TLDR: In this article, a nonlinear change of co-ordinates allowing one to pass from the linear modal variables to the normal ones, linked to the NNMs, defines a framework to properly truncate nonlinear vibration PDEs.

About: This article is published in Journal of Sound and Vibration.The article was published on 2004-05-21 and is currently open access. It has received 184 citations till now. The article focuses on the topics: Normal mode & Nonlinear system.

Figures

Fig. 1. Schematical representation of the phase space near the position of the system at rest, located at the origin. Orthogonal lines represent the first two linear eigenmodes, graduated by the modal co-ordinates ðXp;YpÞ: The curved heavy lines, which are tangential to the axes at the origin, are the invariant manifolds (the NNMs), graduated by the normal co-ordinates ðRp;SpÞ: Normal form theory allows one to define the dynamics into the curved grid, in the phase space spanned by the NNMs.

Table 2 Convergence of the value of Gp; which determines the hardening or softening behaviour of the mode considered, with increasing N: The first line gives the value of *Gp (single linear mode approximation). NB.: the value of G3 with two modes have been computed with modes 1 and 3. a2 ¼ 12 and a3 ¼ 1

Fig. 2. The physical two d.o.f. system for example 1.

Table 1 Internal resonance relations between the ten first eigenfrequencies

Fig. 5. First row: Comparison of analytical and backbone curves for example 1, first mode. The reference solution is given by numerically computing the amplitude/frequency relation for an initial condition onto the first invariant manifold (solid heavy line). Dashed line (- - - -): first order analytical backbone curve onto the first NNM (governed by G1; cf. Eq. (11)). Dash–dotted line (- - -): numerical backbone curve onto the first linear eigenspace. Dotted line (......): first order analytical backbone curve onto the first linear mode. Second row: positions of the two manifolds in space ðX1;X2Þ: Solid line: third order approximation; ( ): real position. (a)–(d) o1 ¼ ffiffiffiffiffiffi 0:5

Fig. 7. Mode shapes (left column) and drifts (right) for mode 2 (first row) and 3 (second row). Solid line: NNM solution. Dash–dotted line: solution of Nayfeh and Lacarbonara, using a perturbation method directly into the PDE. Dashed line: linear solution.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes" ?

Then, a single-mode motion is studied. The aim of the present work is to show that too severe trunc ature using a single linear mode c an lead to erroneous results. Two examples are studied: a disc rete system ( a mass c onnec ted to two springs ) and a c ontinuous one ( a linear Euler– Bernoulli beam resting on a non-linear elastic foundation ).