Dynamics and Stability of Parametrically Excited Gyroscopic Systems
Vedula, Narayana L.
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https://hdl.handle.net/2142/85096
Description
Title
Dynamics and Stability of Parametrically Excited Gyroscopic Systems
Author(s)
Vedula, Narayana L.
Issue Date
2005
Doctoral Committee Chair(s)
Namachchivaya, N. Sri
Department of Study
Aerospace Engineering
Discipline
Aerospace Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Aerospace
Language
eng
Abstract
We study the reduction, dynamics and stability of two-degree-of-freedom mechanical systems. We are particularly interested in understanding energy transfer between modes in such systems. The first part of this research is concerned with the stochastic stability of a two-degree-of-freedom linear system: (a) with one asymptotically stable and one critical mode, (b) with both modes critical and one of the modes corresponding to a nilpotent structure. We obtain asymptotic expansions for the moment and maximal Lyapunov exponents which characterize the exponential growth rate of the amplitude. The results from (a) indicate that the presence of noise may have a stabilizing effect and are applied to explain experimental observations on fluid flow over tube bundles. The results from (b) are applied to show that the effects of noise on a pipe conveying fluid close to divergence are always destabilising in nature. The second part of this research involves the reduction of two-degree-of-freedom randomly perturbed nonlinear gyroscopic systems close to a double zero resonance. It is shown that the long term behaviour of the original four-dimensional system can be approximated by a one dimensional Markov process which take values on a line or a graph. These results are applied to study the dynamics and stability of a rotating shaft subjected to fluctuating axial load. In the final part of this research, we study the dynamics and stability of nonlinear delay gyroscopic systems with periodically varying delay. The center manifold and normal form methods are used to obtain an approximate and simpler two dimensional system. Analysis of this simpler system shows that periodic variations in the delay may lead to larger stability boundaries. These results are applied to demonstrate that greater depths of cut may be achieved in a boring process when the speed of the spindle is modulated sinusoidally instead of being kept constant. A detailed knowledge of the machine-tool structure dynamics, experiments and numerical simulations are required to determine the frequency and amplitude of the sinuosoidal spindle speed variation. The method of random spindle speed variation is more robust with respect to the cutting conditions. We compute the Lyapunov exponent for the machine-tool system and show that the stability boundary when the spindle speed is modulated randomly is larger in comparison with the stability boundary for the constant spindle speed case.
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