Investigating nonlinear dynamic properties of an inertial sensor with rotational velocity-dependent rigidity

This study investigates the nonlinear dynamics of a system with frequency-dependent stiffness using a MEMS-based capacitive inertial sensor as a case study. The sensor is positioned directly on a rotating component of a machine and consists of a microbeam clamped at both ends by fixed supports with a fixed central proof mass. The nonlinear behavior is determined by electrostatic forces, axial and bending motion coupling, and frequency-dependent stiffness. The numerical Galerkin approach is employed for discretization of the coupled differential equations in spatial coordinates. To obtain the sensor response as a function of frequency, a continuation arc-length method based on weak formulation energy balance method is used. This approach uses a physical gradient descent learning based method to obtain unknown coefficients of the considered response. The presented method computes the periodic steady-state solution of the design by considering different frequency contents within the response, including different harmonics of the response expansion. However, the primary and secondary resonances at various harmonic frequencies within the response can be predicted. In this article, the dynamic behavior of the design is investigated by testing different levels of shaft acceleration and different bias voltages. The purpose of the evaluation is to determine the frequency-dependent stiffness of the accelerometer for different levels of shaft speed, taking into account various sensor geometric parameters. The analysis demonstrates the behavior of the sensors under different conditions, particularly highlighting the nonlinearity of the sensor, which causes primary and secondary resonances in the first, second, and third harmonic responses of the accelerometer, especially at higher bias voltages. Fast Fourier analyses indicated that the transversal vibration amplitude of the structure in the second and third harmonies are so considerable that cannot be neglected. The mathematical method used in this study can be applied by researchers and engineers in the design of such sensors to effectively create these devices.