Date of Award
2011
Document Type
Thesis
Publication Status
Version of Record
Submission Date
October 2026
Abstract
Sustained resonance in a linear oscillator is achievable with a drive whose constant frequency matches the resonant frequency of the oscillator. In oscillators with nonlinear restoring forces, i.e., Dung-type oscillators, resonant frequency changes with amplitude, so a constant frequency drive generates a beat oscillation instead of sustained resonance. Dung-type oscillators can be driven into sustained resonance, called autoresonance (AR), when drive frequency is swept in time to match the changing resonant frequency of the oscillator. It is found that near-optimal drive linear sweep rates for autoresonance can be estimated from the beat oscillation resulting from constant frequency excitation. Specically, a least squares estimate of the slope of the Teager-Kaiser instantaneous frequency versus time plot for the rising half-cycle of the beat response to a stationary drive provides a near-optimal estimate of the linear drive sweep rate that sustains resonance in the pendulum, Dung and Dung-Van der Pol oscillators. These predictions are confirmed with model-based numerical simulations. A closed-form approximation to the AM-FM nonlinear resonance beat response of a Dung oscillator driven at its low-amplitude oscillator frequency is obtained from a solution to an associated Mathieu equation. AR time responses are found to evolve along a Mathieu equation primary resonance stability boundary. AR breakdown occurs at sweep rates just past optimal and map to a single stable point just off the Mathieu equation primary resonance stability boundary. Optimal AR sweep rates produce oscillating phase dierences with extrema near 90 degrees, allowing extended time in resonance. AR breakdown occurs when phase difference equals 180 degrees. Nonlinear resonance of the van der Pol type may play a role in the extraordinary sensitivity of the human ear.
Extent
xv, 89 p. : ill. (some col.)
Recommended Citation
Witkov, Carey., "Nonlinear resonance" (2011). Electronic Theses and Dissertations 2010 - 2019. 347.
https://digitalcommons.fau.edu/etd_general_6/347
Identifier
FADT3174314
Comments
by Carey Witkov.|Thesis (Ph.D.)--Florida Atlantic University, 2011.|Includes bibliography.|Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.|by Carey Witkov.