Technical Brief

“Eclipse” Bifurcation in a Twinkling Oscillator

[+] Author and Article Information
Smruti R. Panigrahi

Dynamics and Vibrations Research Lab,
Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: smruti@msu.edu

Brian F. Feeny

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: feeny@egr.msu.edu

Alejandro R. Diaz

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: diaz@egr.msu.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 10, 2013; final manuscript received February 7, 2014; published online April 1, 2014. Assoc. Editor: Philip Bayly.

J. Vib. Acoust 136(3), 034504 (Apr 01, 2014) (5 pages) Paper No: VIB-13-1153; doi: 10.1115/1.4026850 History: Received May 10, 2013; Revised February 07, 2014

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.

Copyright © 2014 by ASME
Topics: Bifurcation , Springs
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Grahic Jump Location
Fig. 2

The bifurcation diagram for the equilibrium solutions of the lightly-damped symmetric 2-DOF system with respect to the pull parameter y0, where B1,B2,…,B10 are the bifurcation points. The dashed lines represent unstable solutions, and the solid lines represent the stable equilibrium solutions (neutrally stable for the undamped system). B1 and B2are the star bifurcation points (Ref. [7]) and B3,…,B6, are saddle-nodes. The vertical dotted lines in (a) and (b) show infinitely many solutions at p = 3 (or y0 = a+b). Also, at p = 3 we observe the bifurcation points B7…,B10 and refer to it as an “eclipse bifurcation”. At these four bifurcation points in the undamped case, two of the four eigenvalues, α, are zero and the other two are complex conjugates with zero real parts. With light damping, the four eigenvalues at the bifurcation points include a zero, a negative real, and two complex-conjugate eigenvalues with negative real parts. Figure 2(c) represents the total final potential energy. For a given value of p, all stable solutions have the same potential energy in the symmetric system.

Grahic Jump Location
Fig. 1

2-DOF spring-mass chain connected by two masses, three nonlinear springs, and two dash-pots. The left spring is fixed to a base and the right most spring is pulled quasistatically to a distance y0.

Grahic Jump Location
Fig. 3

The equilibrium solutions projected onto the (u1,u2) plane and noted as Rj = (u1,u2)j, for j = 0,1,2,…,6. The solid and dashed ellipses and straight lines satisfy h1(u1,u2,q) = 0 and h2(u1,u2,q) = 0, respectively. As q approaches zero from both directions, the two ellipses coincide, resulting in an infinite number of solutions on the ellipse. All of the solutions on the ellipse are marginally stable in the undamped case.

Grahic Jump Location
Fig. 4

The vertical axis represents the eigenvalues at the equilibria in the normal form at the eclipse bifurcation local to q = 0. The points R1, R4, and R6 are stable when q < 0 and become unstable when q > 0, whereas R2, R3, and R5 change from unstable to stable as q goes from negative to positive. The point R0 remain unstable on both sides local to q = 0.

Grahic Jump Location
Fig. 5

Symmetric case and three different symmetry-breaking bifurcations are presented. The solid and dashed curves represent the stable and unstable branches, respectively, and the dots represent the bifurcation points. Top and bottom rows show the projections on u1-q, and u2-q planes, respectively. The first two symmetry-breaking cases unfold the eclipse bifurcation into transcritical bifurcations and the last figure shows the breaking of the eclipse into stiff pitchfork bifurcations. (a) is the symmetric case. The broken symmetry in (b) results in four transcritical bifurcations, while that in (c) results in two transcritical bifurcations and that of (d) results in two stiff pitchforks.

Grahic Jump Location
Fig. 6

For symmetric system at q = 0, infinitely many equilibria exist on the overlapping ellipses that are contained within the circle of radius 22/3. This circle is used as a criterion to determine that these equilibria are marginally stable.

Grahic Jump Location
Fig. 7

Columns (a) and (b) represent the bifurcation diagram for the equilibrium solutions on the entire (p,u1) and (p,u2) planes, in various symmetry breaking configurations. The dots represent the bifurcations that involve a stable branch. The perturbation is applied such that the end spring force f3(y0 − x2) of Eq. (2) is perturbed by making a3 = a+ɛ, b3 = b+ɛ on the top row, a3 = a+ɛ, b3 = b-ɛ on the middle row, and a3 = a-ɛ, b3 = b+ɛ on the bottom row, for ɛ = 0.15. The star bifurcations at p = 0 and p = 6 and the eclipse bifurcation at p = 3 are broken into combinations of pitchfork and saddle-node bifurcations. Column (c) shows the corresponding energy levels in the perturbed systems.



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