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.