Input-output systems with time periodic parameters are commonly found in nature (e.g., oceanic movements) and engineered systems (e.g., vibrations due to gyroscopic forces in vehicles). In a broader sense, periodic behaviors can arise when there is a dynamic equi- librium between inertia and various balancing forces. A classic example is a structure in a steady wind or current that undergoes large oscillations due to vortex shedding or flutter. Such phenomena can have either positive or negative outcomes, like the efficient operation of wind turbines or the collapse of the Tacoma Narrows Bridge. While the systems mentioned here are typically all modeled as systems of nonlinear partial differential equations, the pe- riodic behaviors of interest typically form part of a stable "center manifold," the analysis of which prompts linearization around periodic solutions. The linearization produces linear, time periodic partial differential equations. Discretization in the spatial dimension typically produces large scale linear time-periodic systems of ordinary differential equations. The need to simulate responses to a variety of inputs motivates the development of effective model re- duction tools. We seek to address this need by investigating partial Floquet transformations, which serve to simultaneously remove the time dependence of the system and produce effec- tive reduced order models. In this thesis we describe the time-periodic analogs of important concepts for time invariant model reduction such as the transfer function and the H2 norm. Building on these concepts we present an algorithm which converges to the dominant poles of an infinite dimensional operator. These poles may then be used to produce the partial Floquet transform. / Master of Science / Systems that exhibit time periodic behavior are commonly found both in nature and in human-made structures. Often, these system behaviors are a result of periodic forces, such as the Earth's rotation, which leads to tidal forces and daily temperature changes affecting atmospheric and oceanic movements. Similarly, gyroscopic forces in vehicles can cause no- ticeable vibrations and noise.
In a broader sense, periodic behaviors can arise when there's a dynamic equilibrium between inertia and various balancing forces. A classic example is a structure in a steady wind or current that undergoes large oscillations due to vortex shedding or flutter. Such phenomena can have either positive or negative outcomes, like the efficient operation of wind turbines or the collapse of the Tacoma Narrows Bridge.
Linear Time-Periodic (LTP) systems are crucial in understanding, simulating, and control- ling such phenomena, even in situations where the fundamental dynamics are non-linear. This importance stems from the fact that the periodic behaviors of interest typically form part of a stable "center manifold," especially under minor disturbances. In natural systems, the absence of this stability would mean these oscillatory patterns would not be commonly observed, and in engineered systems, they would not be desirable. Additionally, the process of deriving periodic solutions from nonlinear systems often involves solving large scale linear periodic systems, raising the question of how to effectively reduce the complexity of these models, a question we address in this thesis.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/117285 |
| Date | 02 January 2024 |
| Creators | Bender, Sam |
| Contributors | Mathematics, Beattie, Christopher A., Gugercin, Serkan, Embree, Mark P. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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