Theoretical analysis and experiment of multi-modal coupled vibration of piezo-driven Π-shaped resonator

Nonlinear modal interactions have recently become the focus of intense research in mechanical resonators for their use to improve oscillator performance and probe the frontiers of fundamental physics. This paper reports for the first time a piezoelectric driven Pi-shaped beam with 1:1:2 internal resonance and systematically studies the multi-mode coupling dynamic behavior. Typically, the influence mechanism of driving voltage on the energy transfer path of the multi-modal vibration is experimentally observed. With the increase of the driving voltage, number of frequency intervals of modal coupled vibration increases from 1 to 3, and then decreases to 2, which is unique to multi-mode coupled resonators. To explain the various complex nonlinear phenomena observed experimentally, a theoretical framework for multi-modal coupled vibration analysis of Pi-shaped resonator is proposed for the first time. Perturbation and bifurcation analysis are introduced to deduce the discriminant formula of multi-modal coupled vibration theoretically. Both experimental measurements and theoretical predictions show that the multi-modal coupling greatly complicates the dynamic behavior of the resonant system and induces multiple amplitude jumps. In particular, based on the multi-modal coupled vibration behavior of the Pi-shaped resonator, a conceptual model of trigger resonance sensor with multiple mass warnings is creatively proposed. The experimental results show that the amplitude of the resonant system has six distinct changes with the increase of the added mass, accompanied by bifurcation phenomenon. The research results of this paper provide a systematic framework for the development of multi-mode coupled resonant devices.