Dynamics and Nonlinear Interactions of Macro and Micro Structures: Inclined Marine Risers and MEMS Resonators

Abstract

This work presents a combination of analytical and numerical approaches to gain an insight of the dynamics of marine risers and micro machined resonators. Despite their scale difference, we show that both systems share similarities in terms of initial static deformation, quadratic and cubic nonlinearities, and internal resonances. First, we utilize the state space method to study the eigenvalue problem of vertical riser. An orthonormalization step is introduced to recover the numerical scheme during numerical integration and we investigate the effect of applied tension, apparent weight, and higher-order modes on the accuracy of the scheme. We show that the method is advantageous to find eigenvalues and mode shapes of vertical risers in comparison to other methods. The work is extended to study the eigenvalue problem of inclined risers considering the influence of static deflection, self-weight and mid-plane stretching. The linear dynamics is solved using Galerkin method. The results demonstrate that under the influence of tension and configuration angle, the frequencies exhibit commensurate ratio with respect to the first natural frequency leading to the possible activation of internal resonances. Next, we study the nonlinear interactions of inclined risers considering two-to-one and three-to-one internal resonances under single and multifrequency excitations. The multiple times scale method is applied to study the nonlinear interaction and results are compared to those from a Galerkin solution showing good agreement. Time histories and perturbation’s response curves, in addition to the dynamical solution obtained by Galerkin and stability analysis using Floquet theory are utilized to examine the system. These results feature nonlinear energy exchange, saddle node jumps, and Hopf bifurcations leading to complex dynamic motion that can endanger the riser structure. Finally, the analysis using pertubation is extended to investigate the two-to-one internal resonance in micromachined arch beams between its first two symmetric modes. The response is analyzed using the perturbation method considering the nonlinear interaction and two simultaneous excitations at higher AC voltages. Good agreement is found among the results of pertubations, Galerkin and experimental data from fabricated Silicon arch beam. Different types of bifurcations are observed which can lead to quasi-periodic and potentially chaotic motions.