Abstract

Nonlinear systems are known to exhibit widely differing steady-state behaviors based on small modifications to the control parameters within the equations. These small modifications may be the difference between a chaotic output and a periodic output. Many investigators choose to study the varying behaviors through varying forcing conditions, specifically the forcing amplitude or frequency. However, from a linear vibration theory standpoint, systems are often "tuned" to minimize system response to a known forcing input by varying the strength of the damping and stiffness elements within the system. It may also be the case that the parameters governing the strengths of these elements are constant, but uncertain within a specific range. In these cases, it is more advantageous to understand how the response will vary based on these design parameters or uncertain constants.

The parameters defining the potential field for a nonlinearly coupled Duffing oscillator system were used as the control parameters in this study. The steady-state system response was investigated through the techniques of Poincaré maps, bifurcation diagrams, Lyapunov exponents and spectra, power spectra estimates, and phase portrait projections. The system of equations was integrated through a semi-discrete algorithm based on continuous transformation group theory, which improved the accuracy of the integrated trajectories and the accuracy of the Lyapunov exponents. Additionally the Poincaré maps, power spectra estimates, and phase portrait projections were animated to simplify the analysis of the varying parameters. The use of these animations saved countless hours of analysis time, and revealed details of the parameter-based variations that would not have been observed otherwise. This technique has not been used in any of the known literature.

Two separate forcing conditions were considered; synchronous sinusoidal forcing on each oscillator and nonsynchronous forcing. Each system exhibits a wide variety of nonlinear phenomena including period-doubling sequences, quasiperiodicity, and chaos. The existence of the merging of chaotic attractors and hysteresis was also confirmed. This study also suggests the hyperchaotic attractors and chaotic tori may be present under certain parameter combinations.

Library of Congress Subject Headings

Nonlinear oscillators--Mathematical models; Nonlinear systems--Mathematical models; Duffing equations; Lyapunov exponents

Publication Date

10-2005

Document Type

Thesis

Student Type

Graduate

Degree Name

Mechanical Engineering (MS)

Department, Program, or Center

Mechanical Engineering (KGCOE)

Advisor

Josef S. Török

Advisor/Committee Member

Agamemnon L. Crassidis

Advisor/Committee Member

Daniel B. Phillips

Comments

Physical copy available from RIT's Wallace Library at TK7872.O7 O42 2005

Campus

RIT – Main Campus

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