Sub-harmonic resonance in a nearly pre-loaded mechanical oscillator

Abstract

In this paper, the sub-harmonic resonance of a single degree of freedom system with a nearly preloaded non-linearity (gain-changing clearance) is studied. First, a new perturbation approach is incorporated in the traditional multi-term harmonic balance method to calculate the sub-harmonic resonance. Our approach significantly reduces the work as it computes the sub-harmonic responses with just one run. Initial guesses in the vicinity of sub-harmonic regime are relaxed compared to prior approaches. Second, a parametric study is conducted to examine the occurrence and characteristics of sub-harmonic resonance. The possibility of the sub-harmonic occurrence increases with an increase of dynamic excitation or the stiffness ratio. For instance, our analysis shows that the sub-harmonic resonance typically occurs when the mean load is close to the stiffness transition point. In the extreme case, a very small excitation would generate a sub-harmonic resonance. With a higher mean load, the resonant peak appears at lower frequencies as a result of the reduced equivalent stiffness. Finally, our analytical formulation is successfully validated using numerical integration results.