Bifurcation and symmetry in cyclic structures
Abstract
A planar ring of n identical single degree-of-freedom oscillators with cubic nonlinearity, damping, and external excitation, is considered. Each oscillator is coupled to its nearest-neighbors by linear extensional springs. The damping, amplitude of external excitation, cubic nonlinearity stiffness, and nearest-neighbor coupling stiffness are assumed weak in comparison to the linear stiffness and inertia of each oscillator. The external forcing is a mono-frequency excitation near primary resonance with each oscillator. Due to the planar geometry of the ring, the external excitation is applied in tangential and radial directions. An averaging procedure is used to compute the normal form for each excitation case, creating rings of weakly coupled Duffing and Mathieu-Duffing oscillators due to the tangential, respectively radial, excitation cases. It has been shown in previous works that, for such weakly coupled networks, each oscillator is in 1:1 resonance with the remaining (n-1) oscillators. Some results from Equivariant Bifurcation Theory are applied to determine all possible classes of fixed-point solutions of the normal form equations for arbitrary numbers of oscillators. These solutions are characterized by several classes of standing waves, traveling waves, and motions in-phase with the external forcing, regardless of the type of excitation. A linear stability analysis is performed for each solution class containing the highest degree of symmetry. To continue the study to motions with less symmetry, numerical simulations using the bifurcation analysis and branch continuation software AUTO 97 is used to study the case of three Mathieu-Duffing oscillators. Preliminary results for the case of three Duffing, and four and five Mathieu-Duffing oscillators comprising the ring is also presented. The behavior is studied as a function of detuning frequency and coupling stiffness for fixed damping. Unfortunately, due to the symmetry contained in the normal form equations and solutions, AUTO is not able to discover all the possible fixed-point solutions, or branch points and associated solutions. Nevertheless, the simulations predict quite complicated dynamics, including modulated wave motions that arise from Hopf bifurcations, and an abundance of mode localized dynamics.
Degree
Ph.D.
Advisors
Bajaj, Purdue University.
Subject Area
Mechanical engineering|Mechanics
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