Nonlinear normal modes of a two degrees-of-freedom piecewise linear system

El Hadi Moussi (1, 2) , Sergio Bellizzi (3) , Bruno Cochelin (4) , Ionel Nistor (2)

1 LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]
2 LAMSID - UMR 8193 - Laboratoire de Mécanique des Structures Industrielles Durables
3 Sons
4 M&S - Matériaux et Structures

A study of the Nonlinear Normal Modes (NNMs) of a two degrees of freedom mechanical system with a bilateral elastic stop for one of them is considered. The issue related to the non-smoothness of the impact force is handled through a regularization technique. In order to obtain the NNM, the Harmonic Balance Method (HBM) with a large number of harmonics, combined with the Asymptotic Numerical Method (ANM), is used to solve the regularized problem. The results are validated from periodic orbits obtained analytically in the time domain by direct integration of the non-regular problem. The two NNMs starting respectively from the two Linear Normal Modes (LNMs) of the associated underlying linear system are discussed. The frequency-energy plot is used to present a global vision of the behaviour of the two modes. Local vision of the dynamic of the mode are also provided using modal line plots. The first NNM shows an elaborate dynamics with the occurence of multiple impacts per period, internal resonance and instabilities. On the other hand, the second NNM presents a more simple, almost linear, dynamics. The two NNMs converge asymptotically (for an in finite energy) toward two other LNMs, corresponding to the system with a gap equal to zero.

Mots clés: Asymptotic Numerical Method; Nonlinear Normal Mode; Piecewise linear system; Periodic orbit; Stability; Harmonic Balance Method; Asymptotic Numerical Method.

Domaines: Vibrations [physics.class-ph]; Vibrations [physics.class-ph]

El Hadi Moussi, Sergio Bellizzi, Bruno Cochelin, Ionel Nistor. Nonlinear normal modes of a two degrees-of-freedom piecewise linear system. 2013. ⟨hal-00783088v1⟩