Dynamics of a non-linearly damped microresonator under parametric excitation and its application in developing sensitive inertial sensors with ultra-wide dynamic ranges

We model and investigate the response of a nonlinear cantilever beam under principal parametric excitation. The design is initially assessed, optimized, and tuned using three-dimensional finite element analysis (FEA) to ensure the presence of fundamental parametric resonance and the absence of other internal and higher-order parametric resonances. The derived governing differential equation represents a modified generalized parametrically excited dynamic system under principal parametric excitation. The nonlinear dynamic system is developed and presented in the context of resonators with extensive applications in developing sensors, filters, and switches. The quadratic and cubic nonlinearities include second- and third-order deflection, velocity, acceleration terms describing stiffness, damping, and inertial nonlinearities. To explore and investigate the corresponding generalized nonlinear Mathieu equation, the method of multiple-scales along with the reconstitution method are used and modulation equations are obtained and solved to obtain closed-form amplitude and phase equations. The quadratic damping is modeled and approximated using a Fourier series and analytical models are generated in both Cartesian and Polar frames. To further explore the dynamic system and its applications, a resonator is designed to measure external acceleration and investigated for two cases. It is discussed and shown how the external acceleration modifies the dynamic system, the corresponding reduced-order model, and the modulation equations. The external acceleration affects the amplitude, phase, and frequency of oscillation providing means to estimate the input. These results indicate that the proposed resonator design (dynamic system) is able to significantly improve the dynamic range of shock/acceleration sensors.