Abstract
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. 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 first NNM shows an elaborate dynamics with the occurrence 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 infinite energy) toward two other Linear Normal Modes, corresponding to the system with a gap equal to zero.
Highlights
Introduction and industrial issueMany engineering systems involves components with clearance and intermittent contact
Though many researchers have examined the problem of computing the nonlinear normal modes for nonsmooth systems, few tools to analyze the complete behavior of the Nonlinear Normal Modes (NNMs) including bifurcation diagram analysis, internal resonances characterization and stability properties are available
The computation of the nonlinear normal modes of a two degrees-of-freedom with a piecewise linearity was performed using a numerical procedure, called regularization-Harmonic Balance Method (HBM)-Asymptotic Numerical Method (ANM). This procedure combine a regularization of the contact force, the harmonic balance method and the asymptotic numerical method
Summary
Many engineering systems involves components with clearance and intermittent contact. Rigid elastic stops were considered in [30] where the NNMs of a single DOF linear system with a vibro-impact attachment were obtained by employing the method of nonsmooth transformation introduced in [24] to approximate the periodic orbit in time domain. Though many researchers have examined the problem of computing the nonlinear normal modes for nonsmooth systems, few tools to analyze the complete behavior of the NNMs including bifurcation diagram analysis, internal resonances characterization and stability properties are available In this context, the objective of this paper is to demonstrate that a method combining the HBM method (to approximate the periodic responses) and the ANM (to carry out the continuation of branches of periodic orbits) to analyze the NNMs of a nonsmooth system can be efficient. We will restrict the discussion to the piecewise linear system of Eqs. (3)(4)
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