Analytical Modeling and Experimental Verification of Nonlinear Mode Coupling in a Decoupled Tuning Fork Microresonator

This paper provides the analytical and experimental studies into the nonlinear mode coupling in a tuning fork microresonator, with electrostatic actuation at forced and internal resonance frequencies. The analytical investigation leads to modal equations that take into account kinematic and electrostatic nonlinearities. The equations of motion are derived using Lagrange's energy method. The theoretical resonance curves are determined by using the two-variable expansion perturbation technique near simultaneous subharmonic and internal resonances. The two desired mode shapes of the device are tailored to establish an approximate two-to-one frequency ratio between their natural frequencies. A sample device design is fabricated in a commercial foundry process. The influence of AC driving levels and a detuning parameter, due to inevitable fabrication imperfections, on the system dynamics are investigated theoretically and experimentally. It is shown that the deviation from ideal internal resonance condition separates the region of the nonlinear vibration into two distinct excitation frequencies. The model and simulated results obtained by the perturbation analysis are validated by comparing them with the experimental results.