  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 on. 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.
