Dynamic analysis of preload nonlinearity in a mechanical oscillator

New work on the dynamics of preload nonlinearity in a single degree of freedom mechanical system is described in this article. Significant computational issues that are encountered in the application of direct harmonic balance method are avoided by flipping over the force–displacement nonlinear relationship. An indirect multi-term harmonic balance method is then proposed. Unlike the traditional direct harmonic balance method, our effort is targeted toward the determination of periodic solutions of nonlinear force instead of displacement. The indirect method also allows us to evaluate the stability of periodic solutions by employing the Hillʹs scheme. The primary harmonic responses as exhibited by the preload nonlinearity are validated by the describing function method. Results show that, in general, the nonlinear responses depend on the value of mean load and they differ considerably from those based on linear system analysis. Primary resonance typically shows the hardening spring effect. Unstable solutions are observed in the vicinity of primary resonance as the oscillator makes a transition from a linear to a nonlinear system. Super-harmonic resonances are found under the light mean load conditions. A new instability, in the form of quasi-periodic or chaotic responses at or near the anti-resonances, is also found in our work. Finally, we successfully compare our analysis with one specific experiment that is reported in the literature.