The dynamics of mechanical systems such as turbomachinery and vibration energy harvesting systems (VEH) consisting of one or multiple cantilever structures is often modeled by arrays of periodically driven coupled nonlinear oscillators. It is known that such systems may have multiple stable vibration steady states. Some of these steady states are localized vibrations that are characterized by high amplitude vibrations of a subset of the system, with the rest of the system being in a state of either low amplitude vibrations or no vibrations. On one hand, these localized vibrations can be detrimental to mechanical integrity of turbomachinery, while on the other hand, the vibrations can be potentially desirable for increasing energy yield in VEHs. Transitions into or out of localized vibrations may occur under the influence of random factors.

A combination of experimental and numerical studies has been performed in this dissertation to study the associated transition times and probability of transitions in these mechanical systems. The developments reported here include the following: (i) a numerical methodology based on the Path Integral Method to quantify the probability of transitions due to noise, (ii) a numerical methodology based on the Action Plot Method to quantify the quasipotential and most probable transition paths in nonlinear systems with periodic external excitations, and (iii) experimental evidence and stochastic simulations of the influence of noise on response localizations of rotating macro-scale cantilever structures. The methodology and results discussed in this dissertation provide insights relevant to the stochastic nonlinear dynamics community, and more broadly, designers of mechanical systems to avoid potentially undesirable stochastic nonlinear behavior.