Nonlinear Interactions in Structures

Abstract of Ph.D. Dissertation (December 1997)

Analytical, numerical, and experimental investigations into nonlinear interactions in structures are conducted by using higher-order spectra. In the analytical component of this work, single nonlinear oscillators and coupled nonlinear oscillators are considered. All of the coupled nonlinear oscillators considered have quadratic nonlinearities and a two-to-one internal resonance. For different harmonic excitations of weakly nonlinear systems, explicit expressions are derived for the considered higher-order spectra. The expressions help in understanding the dependence of the higher-order spectral quantities on the nonlinearities in the system. A finite-dimensional model is developed to study the nonlinear motions of an L-shaped structure. In the experimental component of this work, a flexible L-shaped structure with a two-to-one frequency relationship between its first two natural frequencies and a flexible cantilever beam with widely spaced natural frequencies are considered. Nonlinear oscillations in these structures are studied in the presence of harmonic excitations and bispectral and trispectral quantities are computed to understand the role of phase coupling in the observed interactions. The different spectral quantities are also studied in transitions from periodic motions to periodically modulated motions to chaotically modulated motions. In another analytical component of this work, a novel procedure based on a synthesis of Volterra kernel representation and relationships among higher-order transfer functions and higher-order spectral quantities is developed for parametrically identifying characteristics of systems, which exhibit nonlinear interactions. Throughout the dissertation, numerical simulations are used to support the analytical predictions and experimental observations. The efforts of this dissertation help in understanding the role of higher-order spectra in characterizing nonlinear interactions in structural systems and identifying systems that exhibit nonlinear interactions.