Investigation of phonon lasing like auto-parametric instability between [formula omitted]D flexural modes of electrostatically actuated microbeams

Abstract

A significant achievement of the past decade has been the realization of optomechanics phenomena in a purely mechanical resonator by utilizing intermodal coupling between vibrational modes through parametric excitation and internal resonances. For the first time here, we demonstrate the existence of phonon-lasing like auto-parametric instability in one of the simplest MEMS devices, an electrostatically actuated 1-D beam with clamped–flexible boundary conditions. Starting from the non-linear governing equation of the beam, we obtain coupled equations for the first four modes of the beam using reduced order model (ROM) and Galerkin weak form. Only two non-dimensional parameters that depend on beam geometry, beam material properties, axial spring stiffness, and external forcing parameters control the beam dynamics. We first show the existence of three-mode combination internal resonance by using ROM and finite element analysis, ω2=ω4−ω3, involving the flexural modes 2, 3, and 4 of the beam. We then demonstrate the existence of three-mode auto-parametric instability in which external excitation of driving mode-4 with frequency Ω=ω4 generates oscillations around modes 2 and 3. We report the first evidence of phonon lasing in 1−D flexural modes, implying that the inter-modal coupling here has a threshold like dependence on the excitation amplitude. The resonator also exhibits auto-parametric instability for a small detuning range ΔΩ=Ω−ω4. This work paves the way for systematic explorations of auto-parametric instability and lasing like phenomena in mechanically actuated systems.