Shape optimization of geometrically nonlinear modal coupling coefficients: an application to MEMS gyroscopes

Micro- and nanoelectromechanical system (MEMS and NEMS) resonators can exhibit rich nonlinear dynamics as they are often operated at large amplitudes with high quality factors and possess a high mode density with a variety of nonlinear modal couplings. Their impact is strongly influenced by internal resonance conditions and by the strength of the modal coupling coefficients. On one hand, strong nonlinear couplings are of academic interest and promise novel device concepts. On the other hand, however, they have the potential to disturb the linear system behavior on which industrial devices such as gyroscopes and micro mirrors are based. In either case, being able to optimize the coupling coefficients by design is certainly beneficial. A main source of nonlinear modal couplings are geometric nonlinearities. In this work, we apply node-based shape optimization to tune the geometrically nonlinear 3-wave coupling coefficients of a MEMS gyroscope. We demonstrate that individual coupling coefficients can be tuned over several orders of magnitude by shape optimization, while satisfying typical constraints on manufacturability and operability of the devices. The optimized designs contain unintuitive geometrical features far away from any solution an experienced human MEMS or NEMS designer could have thought of. Thus, this work demonstrates the power of shape optimization for tailoring the complex nonlinear dynamic properties of MEMS and NEMS resonators.