This work presents a combination of analytical and numerical approaches to gain in depth insight of the dynamical behaviors of marine risers and micro machined vibrating structures (resonators). Despite their scale difference, we show in this dissertation that both systems share considerable similarities in terms of initial static deformation, quadratic and cubic nonlinearities, and nonlinear internal resonances.
In the first part, we utilize the state space method to study the eigenvalue problem of vertical riser. The modified Gram-Schmidt orthonormalization process is applied as an intermediate step recovering the numerical scheme during the numerical integration process using the Runge-Kutta method. Also, we investigate the effect of high applied tensions, the high apparent weight, and the higher-order modes on the accuracy of the numerical scheme. We show that the method is advantageous to find the eigenvalues and mode shapes of the riser in comparison to finite-element, Galerkin, and power-series 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 of the inclined riser is solved using Galerkin method utilizing beam mode shapes. The results demonstrate that under the influence of tension and configuration angle, the natural frequencies can exhibit commensurate ratio with respect to the first natural frequency leading to the possible activation of internal resonances.
In the next part, we study the nonlinear interactions of inclined risers considering two-to-one and three-to-one internal resonances under single and multifrequency excitations. The risers are modeled as Euler-Bernoulli beams accounting for variable axial load due to self-weight, static deflection and nonlinear mid-plane stretching. Then, 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 response curves of the perturbation solution, 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 mechanisms, saddle node jumps, and Hopf bifurcations leading to more complex dynamic motion that can endanger the riser structure.
Finally, the analysis using multiple times scale is extended to investigate the two-to-one internal resonance in micromachined arch beams between its first two symmetric modes. The nonlinear response of the arch beam is analyzed using the perturbation method considering the nonlinear interaction and the two simultaneous excitations at higher AC voltages. Good agreement is found among the results of pertubations, Galerkin and experimental data based on deliberately fabricated Silicon arch beam. Different types of bifurcations are observed, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.
Feras is a Ph.D. candidate in Mechanical Engineering (ME) at King Abdullah University of Science and Technology (KAUST). He received his BSc degree in Mechanical Engineering from KFUPM in 2007 after which he joined Saudi Aramco to work as a field operations engineer in various engineering aspects related to offshore engineering. In 2013, he Joined KAUST where he pursued and completed his MS degree in 2015 in Mechanical Engineering program. In the early work of his Ph.D. thesis, he worked closely with Professor Ali H. Nayfeh developing his skills in mathematical modeling, numerical simulations and perturbation methods. His research interests includes modeling and simulations of dynamic systems, perturbation methods, and analysis of nonlinear systems.