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.

翻译：微纳机电系统（MEMS和NEMS）谐振器通常在高品质因子下以大振幅运行，且具有高模态密度和多种非线性模态耦合，因此能够展现丰富的非线性动力学特性。内部共振条件和模态耦合系数强度对其影响显著。一方面，强非线性耦合具有学术研究价值，并有望催生新型器件概念。另一方面，它们可能干扰基于线性系统行为的工业器件（如陀螺仪和微镜）的运行。无论哪种情况，通过设计优化耦合系数无疑具有重要价值。几何非线性是产生非线性模态耦合的主要来源。本研究采用基于节点的形状优化方法，对MEMS陀螺仪的几何非线性三波耦合系数进行调谐。结果表明，通过形状优化可在满足器件可制造性和可操作性典型约束的前提下，将单个耦合系数调谐数个数量级。优化后的设计包含违反直觉的几何特征，远超经验丰富的MEMS或NEMS设计师所能构思的方案。因此，本研究展示了形状优化在定制MEMS和NEMS谐振器复杂非线性动态特性方面的强大能力。