Chaotic and Stochastic Dynamics of Nonlinear Structural Systems
Tien, Win-Min
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https://hdl.handle.net/2142/72370
Description
Title
Chaotic and Stochastic Dynamics of Nonlinear Structural Systems
Author(s)
Tien, Win-Min
Issue Date
1993
Doctoral Committee Chair(s)
Namachchivaya, N. Sri
Department of Study
Aeronautical and Astronautical Engineering
Discipline
Aeronautical and Astronautical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Applied Mechanics
Abstract
This research investigates the dynamic response, stability and bifurcation behavior of nonlinear dynamical systems subjected to deterministic and/or stochastic excitations which includes the following systems:
1. The weakly nonlinear resonance response of a two-degree-of-freedom structural systems subjected to simple harmonic excitation is examined in detail for the cases of 1:2 and 1:1 internal resonance. It is found that by varying the detuning parameters from the exact external and internal resonance, the coupled mode response can undergo a Hopf bifurcation to limit cycle motions. It is also shown that the limit cycles quickly undergo period-doubling bifurcations giving rise to chaos. The method of Melnikov and a new global perturbation technique are used to analytically predict results for the critical parameter at which the dynamical system possesses a Smale horseshoe type of chaos for 1:2 and 1:1 internal resonance, respectively.
2. The dynamic stability of general linear non-conservative systems under stochastic parametric excitation is examined. Conditions for mean square stability of the dynamic response are obtained. Results are shown to depend only on those values of the excitation spectral density near twice the natural frequencies, the difference and combination frequencies of the system. The results are applied to the problem of a cantilever column subjected to stochastic follower force.
3. For nonlinear systems with strong cubic nonlinearities, a new scheme of stochastic averaging using elliptic functions is presented that approximates nonlinear dynamical systems with strong cubic nonlinearities in the presence of noise by a set of Ito differential equations. This is an extension of some recent results presented in deterministic dynamical systems. The resulting equations are Markov approximations of amplitude and phase involving integrals of elliptic functions. This method retains those nonlinear terms in the response equations as opposed to the regular averaging technique.
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