Gradient-based optimization in nonlinear structural dynamics

The intrinsic nonlinearity of mechanical structures can give rise to rich nonlinear dynamics. Recently, nonlinear dynamics of micro-mechanical structures have contributed to developing new Micro-Electro-Mechanical Systems (MEMS), for example, atomic force microscope, passive frequency divider, frequency stabilization, and disk resonator gyroscope.

For advanced design of these structures, it is of considerable value to extend current optimization in linear structural dynamics into nonlinear structural dynamics. In this thesis, we present a framework for modelling, analysis, characterization, and optimization of nonlinear structural dynamics. In the modelling, nonlinear finite elements are used.

In the analysis, nonlinear frequency response and nonlinear normal modes are calculated based on a harmonic balance method with higher-order harmonics. In the characterization, nonlinear modal coupling coefficients are calculated directly from a nonlinear finite element model. Based on the analysis and the characterization, a new class of optimization problems is studied.

In the optimization, design sensitivity analysis is performed by using the adjoint method which is suitable for large-scale structural optimization. The optimization procedure is exemplified by the design of plane frame structures. The work has demonstrated the following results: the amplitude and the frequency of nonlinear resonance peak can be effectively optimized; the super-harmonic resonances can be either suppressed or enhanced; the hardening/softening behavior can be qualitatively changed; and an order-of-magnitude improvement of some essential modal coupling coefficients can be achieved.

The study has shown promising applications in nonlinear micro-mechanical resonators. It also paves the way for topology optimization of complex nonlinear dynamics