The construction of non-linear normal modes for systems with internal resonance

Dongying Jiang (1) , Christophe Pierre (1) , Steven Shaw (2)

1 Department of Mechanical Engineering
2 Michigan State University [East Lansing]

A numerical method, based on the invariant manifold approach, is presented for constructing non-linear normal modes for systems with internal resonances. In order to parameterize the non-linear normal modes of interest, multiple pairs of system state variables involved in the internal resonance are kept as 'seeds' for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are then constrained to these 'seed', or master, variables, resulting in a system of non-linear partial differential equations that govern the constraint relationships, and these are solved numerically. The computationally-intensive solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two non-linear normal modes is constructed, resulting in a reduced order model that accurately captures the system dynamics. The methodology is then applied to a larger order system, specifically, an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the non-linear two-mode reduced order model is verified by comparing time-domain simulations of the two DOF model and the full system equations of motion.

Mots clés: invariant manifold; nonlinear normal mode; internal resonance

Domaines: Vibrations [physics.class-ph]

Dongying Jiang, Christophe Pierre, Steven Shaw. The construction of non-linear normal modes for systems with internal resonance. International Journal of Non-Linear Mechanics, 2005, 40 (5), ⟨10.1016/j.ijnonlinmec.2004.08.010⟩. ⟨hal-01350807⟩