Nonlinear normal modes of forced vibrations in piecewise linear systems under superharmonical resonances

Abstract

The paper describes a new technique for analysis of forced oscillations in strongly nonlinear piecewise linear systems considering superharmonic resonances. Nonlinear oscillations of piecewise linear systems have complex behavior including bifurcations, chaotic oscillations, sub- and superharmonic responses. Extreme importance of piecewise linear systems analysis due to their abundance in machinery and, particularly, engines makes the problem of nonlinear oscillatory dynamics in such systems highly topical. Nonlinear normal modes as an approach for analysis of nonlinear oscillations were developed by Rosenberg. Shaw and Pierre amended this approach using an invariant manifolds ideology. This paper utilizes and modifies Shaw-Pierre nonlinear normal modes approach to analyze superharmonical oscillations occurring in piecewise linear mechanical systems under harmonic excitation. The Rauscher technique is used to bring a non-autonomous dynamical system to an equivalent pseudo-autonomous one. To commit analysis of superharmonic resonances in the system, a modification to the Rauscher method is proposed. Eventually, an analysis of a mechanical system modeling a circuit of a power transmission of an internal combustion engine is performed. Amplitude-frequency diagram is obtained for the second superharmonical resonance. It is discovered that in the configuration space the second superharmonical nonlinear normal mode contains delamination that prevents it to be found using Rosenberg nonlinear normal modes technique.