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This dissertation presents innovative unified approaches to understand and predict the motion between potential wells. The theoretical-computational framework, based on the tube dynamics, will reveal how the dissipative and gyroscopic forces change the phase space structure that governs the escape (or transition) from potential wells. In higher degree of freedom systems, the motion between potential wells is complicated due to the existence of multiple escape routes usually through an index-1 saddle. Thus, this dissertation firstly studies the local behavior around the index-1 saddle to establish the criteria of escape taking into account the dissipative and gyroscopic forces. In the analysis, an idealized ball rolling on a surface is selected as an example to show the linearized dynamics due to its special interests that the gyroscopic force can be easily introduced by rotating the surface. Based on the linearized dynamics, we find that the boundary of the initial conditions of a given energy for the trajectories that transit from one side of a saddle to the other is a cylinder and ellipsoid in the conservative and dissipative systems, respectively. Compared to the linear systems, it is much more challenging or sometimes impossible to get analytical solutions in the nonlinear systems. Based on the analysis of linearized dynamics, the second goal of this study is developing a bisection method to compute the transition boundary in the nonlinear system using the dynamic snap-through buckling of a buckled beam as an example. Based on the Euler-Bernoulli beam theory, a two degree of freedom Hamiltonian system can be generated via a two mode-shape truncation. The transition boundary on the Poincar'e section at the well can be obtained by the bisection method. The numerical results prove the efficiency of the bisection method and show that the amount of trajectories that escape from the potential well will be smaller if the damping of the system is increasing. Finally, we present an alternative idea to compute the transition boundary of the nonlinear system from the perspective of the invariant manifold. For the conservative systems, the transition boundary of a given energy is the invariant manifold of a periodic orbit. The process of obtaining such invariant manifold compromises two parts, including the computation of the periodic orbit by solving a proper boundary-value problem (BVP) and the globalization of the manifold. For the dissipative systems, however, the transition boundary of a given energy becomes the invariant manifold of an index-1 saddle. We present a BVP approach using the small initial sphere in the stable subspace of the linearized system at one end and the energy at the other end as the boundary conditions. By using these algorithms, we obtain the nonlinear transition tube and transition ellipsoid for the conservative and dissipative systems, respectively, which are topologically the same as the linearized dynamics. / Doctor of Philosophy / 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.

The objective of this thesis is to develop a novel methodology of fractional stochastic dynamics to study stochastic stability of viscoelastic systems under stochastic loadings. Numerous structures in civil engineering are driven by dynamic forces, such as seismic and wind loads, which can be described satisfactorily only by using probabilistic models, such as white noise processes, real noise processes, or bounded noise processes. Viscoelastic materials exhibit time-dependent stress relaxation and creep; it has been shown that fractional calculus provide a unique and powerful mathematical tool to model such a hereditary property. Investigation of stochastic stability of viscoelastic systems with fractional calculus frequently leads to a parametrized family of fractional stochastic differential equations of motion. Parametric excitation may cause parametric resonance or instability, which is more dangerous than ordinary resonance as it is characterized by exponential growth of the response amplitudes even in the presence of damping. The Lyapunov exponents and moment Lyapunov exponents provide not only the information about stability or instability of stochastic systems, but also how rapidly the response grows or diminishes with time. Lyapunov exponents characterizes sample stability or instability. However, this sample stability cannot assure the moment stability. Hence, to obtain a complete picture of the dynamic stability, it is important to study both the top Lyapunov exponent and the moment Lyapunov exponent. Unfortunately, it is very difficult to obtain the accurate values of theses two exponents. One has to resort to numerical and approximate approaches. The main contributions of this thesis are: (1) A new numerical simulation method is proposed to determine moment Lyapunov exponents of fractional stochastic systems, in which three steps are involved: discretization of fractional derivatives, numerical solution of the fractional equation, and an algorithm for calculating Lyapunov exponents from small data sets. (2) Higher-order stochastic averaging method is developed and applied to investigate stochastic stability of fractional viscoelastic single-degree-of-freedom structures under white noise, real noise, or bounded noise excitation. (3) For two-degree-of-freedom coupled non-gyroscopic and gyroscopic viscoelastic systems under random excitation, the Stratonovich equations of motion are set up, and then decoupled into four-dimensional Ito stochastic differential equations, by making use of the method of stochastic averaging for the non-viscoelastic terms and the method of Larionov for viscoelastic terms. An elegant scheme for formulating the eigenvalue problems is presented by using Khasminskii and Wedig’s mathematical transformations from the decoupled Ito equations. Moment Lyapunov exponents are approximately determined by solving the eigenvalue problems through Fourier series expansion. Stability boundaries, critical excitations, and stability index are obtained. The effects of various parameters on the stochastic stability of the system are discussed. Parametric resonances are studied in detail. Approximate analytical results are confirmed by numerical simulations.