Smruti R. Panigrahi

Degenerate “Star” Bifurcations in a Twinkling Oscillator

We have studied a nonlinear spring-mass chain loaded by a quasistatic pull. The spring forces are assumed to be cubic with intervals of negative stiffness. Depending on the parameters, the system has multiple equilibria. The normal form and the bifurcation behaviors for the single- and two-degree-of-freedom systems are studied in detail. A new type of bifurcation, which we refer to as a star bifurcation, has been observed for the symmetric two-degree-of-freedom system. This bifurcation is of codimension-four for the undamped case and codimension-three or two for the damped case, depending on the form of the damping.

“Eclipse” Bifurcation in a Twinkling Oscillator

This work regards the use of cubic springs with intervals of negative stiffness, in other words, “snap-through” elements, in order to convert low-frequency ambient vibrations into high-frequency oscillations, referred to as “twinkling.” The focus of this paper is on the bifurcation of a two-mass chain that, in the symmetric system, involves infinitely many equilibria at the bifurcation point. The structure of this “eclipse bifurcation” is uncovered, and perturbations of the bifurcation are studied. The energies associated with the equilibria are examined.

BIFURCATIONS IN TWINKLING OSCILLATORS

We present the underlying dynamics of snap-through structures that exhibit twinkling. Twinkling occurs when the nonlinear structure is loaded slowly and the masses snap-through, converting the low frequency input to high frequency oscillations. We have studied a nonlinear spring-mass chain loaded by a quasistatic pull. The spring forces are assumed to be cubic with intervals of negative stiffness. Depending on the parameters, the system has equilibria at multiple energy levels. The normal form and the bifurcation behaviors for the single and two degree of freedom systems are studied in detail. A new type of bifurcation, which we refer to as a star bifurcation, has been observed for the symmetric two degree of freedom system, which is of codimension four for the undamped case, and codimension three or two for the damped case, depending on the form of the damping.

Second-order perturbation analysis of low-amplitude traveling waves in a periodic nonlinear chain

Traveling waves in one-dimensional nonlinear periodic structures are investigated for low-amplitude oscillations using perturbation analysis. We use second-order multiple scales analysis to capture the effects of quadratic nonlinearity. Comparisons with the linear and cubical nonlinear cases are presented in the dispersion relationship, group velocity and phase velocity and their dependence on wave number and amplitude of oscillation. Quadratic nonlinearity is shown to have a significant effect on the behavior.Copyright

Harmonic Balance for Large-Amplitude Vibrations in Snap-Through Structures

A simple nonlinear Duffing oscillator has been studied for its snap-through behavior at large-amplitude vibrations. Using the harmonic balance method we have developed an algorithm to find particular amplitude and frequency relations in two-term and three-term approximations when the solution lies outside of the separatrix on the phase space, i.e. when the oscillator exhibits snap-through behavior.Copyright