Geometrical Investigation on Escape Dynamics in the Presence of Dissipative and Gyroscopic Forces.
Degree: PhD, Engineering Mechanics, 2020, Virginia Tech
Transition or escape events are very common in daily life, such as the snap-through of plant leaves and the flipping over of umbrellas on a windy day, the capsize of ships and boats on a rough sea. Some other engineering problems related to escape, such as the collapse of arch bridges subjected to seismic load and moving trucks, and the escape and recapture of the spacecraft, are also widely known. At first glance, these problems seem to be irrelated. However, from the perspective of mechanics, they have the same physical principle which essentially can be considered as the escape from the potential wells. A more specific exemplary representative is a rolling ball on a multi-well surface where the potential energy is from gravity. The purpose of this dissertation is to develop a theoretical-computational framework to understand how a transition event can occur if a certain energy is applied to the system.
For a multi-well system, the potential wells are usually connected by saddle points so that the motion between the wells generally occurs around the saddle. Thus, knowing the local behavior around the saddle plays a vital role in understanding the global motion of the nonlinear system. The first topic aims to study the linearized dynamics around the saddle. In this study, an idealized ball rolling on both stationary and rotating surfaces will be used to reveal the dynamics. The effect of the gyroscopic force induced by the rotation of the surface and the energy dissipation will be considered.
In the second work, the escape dynamics will be extended to the nonlinear system applied to the snap-through of a buckled beam. Due to the nonlinear behavior existing in the system, it is hard to get the analytical solutions so that numerical algorithms are needed. In this study, a bisection method is developed to search the transition boundary. By using such method, the transition boundary on a specific Poincar'e section is obtained for both the conservative and dissipative systems.
Finally, we revisit the escape dynamics in the snap-through buckling from the perspective of the invariant manifold. The treatment for the conservative and dissipative systems is different. In the conservative system, we compute the invariant manifold of a periodic orbit, while in the dissipative system we compute the invariant manifold of a saddle point. The computational process for the conservative system consists of the computation of the periodic orbit and the globalization of the corresponding manifold. In the dissipative system, the invariant manifold can be found by solving a proper boundary-value problem. Based on these algorithms, the nonlinear transition tube and transition ellipsoid in the phase space can be obtained for the conservative and dissipative systems, respectively, which are qualitatively the same as the linearized dynamics.
Advisors/Committee Members: Ross, Shane David (committeechair), Virgin, Lawrie (committee member), Plaut, Raymond H. (committee member), Abaid, Nicole Teresa (committee member), Tarazaga, Pablo Alberto (committee member).
Subjects/Keywords: Escape dynamics; Invariant manifold; Transition tube; Transition ellipsoid; Hamiltonian system; Dissipative system; Gyroscopic system
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APA (6th Edition):
Zhong, J. (2020). Geometrical Investigation on Escape Dynamics in the Presence of Dissipative and Gyroscopic Forces. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/97363
Chicago Manual of Style (16th Edition):
Zhong, Jun. “Geometrical Investigation on Escape Dynamics in the Presence of Dissipative and Gyroscopic Forces.” 2020. Doctoral Dissertation, Virginia Tech. Accessed April 09, 2020.
MLA Handbook (7th Edition):
Zhong, Jun. “Geometrical Investigation on Escape Dynamics in the Presence of Dissipative and Gyroscopic Forces.” 2020. Web. 09 Apr 2020.
Zhong J. Geometrical Investigation on Escape Dynamics in the Presence of Dissipative and Gyroscopic Forces. [Internet] [Doctoral dissertation]. Virginia Tech; 2020. [cited 2020 Apr 09].
Available from: http://hdl.handle.net/10919/97363.
Council of Science Editors:
Zhong J. Geometrical Investigation on Escape Dynamics in the Presence of Dissipative and Gyroscopic Forces. [Doctoral Dissertation]. Virginia Tech; 2020. Available from: http://hdl.handle.net/10919/97363