Studies in resonant responses of nonlinear structures with cyclic symmetry

Sobitha Wasantha Samaranayake, Purdue University

Abstract

Nonlinear vibratory responses of a simplified model of cyclic structures to harmonic excitations are studied using the asymtotic method of averaging. The dynamic response of these structures for both the case of strong and weak coupling cases is examined. In their linear approximation, strongly-coupled cyclic structures possess pairwise double-degenerate natural frequencies with orthogonal normal modes, resulting in 1:1 internal resonances. The method of averaging is used to study the nonlinear interaction between the pairs of modes with identical natural frequencies. It is shown that with no loss in generality, the external harmonic excitation can be assumed to be spatially distributed like one of the two orthogonal modes in resonance. Depending on the parameter values, coupled-mode responses arise due to pitchfork bifurcations from the directly excited single-mode responses. The occurrence of Hopf bifurcations in the coupled-mode responses leading to complicated amplitude-modulated motions of the structure depend on the amount of damping, the excitation frequency, and the internal mistuning. The averaged equations for the subharmonic response of order-three are also found to exhibit complicated dynamics, including periodic and chaotic motions. Using an appropriate form, global behavior of the averaged system is examined in the case of primary resonance. When the system satisfies a certain "nonresonance" condition, a higher-order generalization of the Melnikov technique is utilized and a criterion to determine the dependence on system parameters for the existence of chaotic behavior is derived. When the linear coupling stiffness is $O(\varepsilon),$ the cyclic system consists of n weakly-coupled identical nonlinear oscillators. A careful bifurcation analysis of the amplitude equations is performed in the fundamental resonance case for an illustrative example consisting of a three particle system. In case of a uniformly distributed excitation, a localized response is identified, in which one of the particles exhibits large amplitude motions compared with that of the other particles. In the case when only one particle is excited, it is found that increase in coupling strength and decrease in damping results in isolated asymmetric solution branches, which bifurcate from the symmetric solutions via pitchfork symmetry-breaking bifurcations.

Degree

Ph.D.

Advisors

Nwokah, Purdue University.

Subject Area

Mechanical engineering

COinS