Title page for ETD etd-02142007-140350


Type of Document Dissertation
Author Thota, Phanikrishna
Author's Email Address thota@vt.edu
URN etd-02142007-140350
Title Analytical and Computational Tools for the Study of Grazing Bifurcations of Periodic Orbits and Invariant Tori
Degree PhD
Department Engineering Science and Mechanics
Advisory Committee
Advisor Name Title
Batra, Romesh C. Committee Co-Chair
Dankowicz, Harry J. Committee Co-Chair
Hendricks, Scott L. Committee Member
Inman, Daniel J. Committee Member
Nayfeh, Ali H. Committee Member
Paul, Mark R. Committee Member
Keywords
  • Grazing Bifurcations
  • Co-dimension-one
  • Hybrid Dynamical Systems
Date of Defense 2007-02-02
Availability unrestricted
Abstract
The objective of this dissertation is to develop theoretical and

computational tools for the study of qualitative changes in the dynamics of

systems with discontinuities, also known as nonsmooth or hybrid dynamical

systems, under parameter variations. Accordingly, this dissertation is

divided into two parts.

The analytical section of this dissertation discusses mathematical tools for

the analysis of hybrid dynamical systems and their application to a series

of model examples. Specifically, qualitative changes in the system dynamics

from a nonimpacting to an impacting motion, referred to as grazing

bifurcations, are studied in oscillators where the discontinuities are

caused by impacts. Here, the study emphasizes the formulation of conditions

for the persistence of a steady state motion in the immediate vicinity of

periodic and quasiperiodic grazing trajectories in an impacting mechanical

system. A local analysis based on the discontinuity-mapping approach is

employed to derive a normal-form description of the dynamics near a grazing

trajectory. Also, the results obtained using the discontinuity-mapping

approach and direct numerical integration are found to be in good agreement.

It is found that the instabilities caused by the presence of the square-root

singularity in the normal-form description affect the grazing bifurcation

scenario differently depending on the relative dimensionality of the state

space and the steady state motion at the grazing contact.

The computational section presents the structure and applications of a

software program, TC-HAT, developed to study the bifurcation analysis of

hybrid dynamical systems. Here, we present a general boundary value problem

(BVP) approach to locate periodic trajectories corresponding to a hybrid

dynamical system under parameter variations. A methodology to compute the

eigenvalues of periodic trajectories when using the BVP formulation is

illustrated using a model example. Finally, bifurcation analysis of four

model hybrid dynamical systems is performed using TC-HAT.

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