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Research Papers

Degenerate “Star” Bifurcations in a Twinkling Oscillator

[+] Author and Article Information
Smruti R. Panigrahi

Department of Mechanical Engineering
Michigan State University
East Lansing, MI 48823
e-mail: smruti@msu.edu

Brian F. Feeny

Department of Mechanical Engineering
Michigan State University
East Lansing, MI 48823
e-mail: feeny@egr.msu.edu

Alejandro R. Diaz

Department of Mechanical Engineering
Michigan State University
East Lansing, MI 48823
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 February 22, 2013; final manuscript received August 4, 2013; published online December 9, 2013. Assoc. Editor: Brian P. Mann.

J. Vib. Acoust 136(2), 021004 (Dec 09, 2013) (10 pages) Paper No: VIB-13-1058; doi: 10.1115/1.4025442 History: Received February 22, 2013; Revised August 04, 2013

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.

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Figures

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Fig. 1

n-degree-of-freedom spring-mass chain connected by n masses, (n + 1) nonlinear springs, and n dash-pots. As shown in this figure the left spring is fixed to a base and the right-most spring is pulled quasistatically to a distance y0.

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Fig. 2

(a) A possible arrangement of masses with magnets and linear springs to produce a chain of bistable elements. The masses consist of magnets that are arranged in a way that they repel each other. Each mass consists of three magnets (filled circles), two on the right (sleeve) and one on the left (tongue) except for the first and the last mass. Here y is the quasistatic pull distance of the end spring. (b) The repelling forces fm between the magnets on the tongue and the sleeves are shown. S and N represent the south and the north pole of the magnets, respectively, h is the separation distance between the two sleeves, and 2fmx is the resultant magnetic force in the direction of motion of the tongue and sleeves. (c) Total nonlinear magnetic spring force (solid curve) that can be achieved from combining the forces due to the magnets (dashed curve) and the linear spring (dotted line).

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Fig. 3

The characteristic spring force fi(si) of the nonlinear spring as a function of the spring deformation si, and an example of the quasistatic pull as a function of time

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Fig. 4

This plot shows the global picture of the bifurcation behavior of a SDOF system with respect to the pull parameter y0. (a) Bifurcations of equilibria and potential energy in a SDOF symmetric system. (b) Bifurcations in symmetry breaking SDOF system. The energy plot in figure (a) has two overlapping energy levels in the negative stiffness region that are revealed with an applied perturbation in figure (b), where a2 = a1 + δ and b2 = b1 + δ, for a1 = 0.5, b1 = 3, and δ = 0.2.

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Fig. 5

The bifurcation diagram for the equilibrium solutions of the lightly damped symmetric 2DOF 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). The vertical dotted lines show infinitely many solutions at y0 = a+b, where at the bifurcation points B7B10, two of the four eigenvalues are complex conjugates with zero real parts and the other two are zeros for undamped system, whereas with light damping there is one zero, one purely real negative, and the other two are complex conjugate eigenvalues with negative real parts. The bifurcation points B3B6, are saddle-nodes with two zero and two complex conjugate eigenvalues with zero real parts for undamped system, and with light damping there are two complex conjugate eigenvalues with negative real parts, one zero, and one purely real negative eigenvalues.

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Fig. 12

In top figures {h∧1+ε=0,h∧2-ε=0}, the star breaks into pitchfork bifurcations on both the u∧1 and u∧2 plane, whereas in the bottom figures it breaks into saddle-nodes, where {h∧1-ε=0,h∧2-ε=0}. Here the symmetry breaking on the (p∧,u∧1) plane is qualitatively similar to the perturbation case {h∧1-ε=0,h∧2+ε=0} shown in Fig. 11. However, this presents a different symmetry breaking configuration on (p∧,u∧2) plane.

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Fig. 11

(top) The perturbation is such that {h∧1-ε = 0,h∧2 = 0}, where the branches with positive slope on the (p∧,u∧1) plane are both two distinct, overlapping projected branches that are separated in the projection on the (p∧,u∧2) plane. (bottom) {h∧1-ε = 0,h∧2+ε = 0}, where all the branches in the (p∧,u∧1) and (p∧,u∧2) planes are revealed unfolding the star bifurcation into saddle nodes.

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Fig. 10

Symmetric case of the star bifurcation is shown on the top, and the bottom figures show the symmetry breaking of the star bifurcation, where {h∧1+ε=0,h∧2=0}. With reference to the symmetric case, overlapping projected branches are revealed.

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Fig. 9

The state-variable eigenvalues α0 of the symmetric damped (with two dampers) 2DOF system near the star bifurcation in the first-order form at equilibrium R0. Single and double arrows represent slow and fast approach, respectively, to show relative dynamics of the eigenvalues. Solid squares and circles represent the initial positions (i.e., for p∧ < 0) and the hollow squares and circles represent the final positions (i.e., for p∧ > 0) of the eigenvalues as p∧ goes from negative to positive in the direction of the arrows. Both the square and the circle on the positive x-axis pass through the origin at p∧ = 0, hence making p∧ = 0 a codimension-two bifurcation point.

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Fig. 8

The vertical axis represents the eigenvalues λ at the equilibria in the normal form at the star bifurcation local to p∧=0. The points R1, R2, and R3 remain unstable whereas the stability of R0 changes from stable to unstable as p∧ goes from negative to positive.

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Fig. 7

The equilibrium solutions projected onto the u1u2 plane and denoted as Ri = (u1, u2)i. The solid and dashed ellipses and straight lines satisfy g1(u1, u2, p) = 0 and g2(u1, u2, p) = 0, respectively. As p approaches zero from both directions the equilibrium solutions R1, R2, and R3 converge into R0. For p < 0 the points R1, R2, and R3 are unstable and become stable when p > 0. R0 changes from stable to unstable as p goes from negative to positive. The stabilities of the points R4, R5, and R6 remain stable on both sides local to p = 0.

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Fig. 6

The bifurcation diagram for the equilibrium solutions (u1, u2) in terms of the pull parameter p are qualitatively similar to that of Fig. 5, hence the stabilities of the solution curves and the degree of degeneracy of the bifurcation points are inferred

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Fig. 15

Total final potential energies of the 2DOF system. The left figure represents the symmetric case corresponding to Fig. 6 and the right figure corresponds to a symmetry breaking, which reveals the overlapping energy levels.

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Fig. 13

Columns (a) and (b) represent eigenvalues at equilibria R0 and R1, respectively, i.e., the real and imaginary parts of the eigenvalues at equilibria R0 (α0) and R1 (α1) of the symmetric damped 2DOF system near the star bifurcation in the first-order form as p∧ goes from negative to positive. Each equilibrium has four different eigenvalues that are designated as αji on the plot for j = 0, 1 and i = 1, 2, 3, 4. The dashed and dotted curves represent two and four overlapping eigenvalues, respectively.

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Fig. 14

Columns (a) and (b) represent eigenvalues at equilibria R2 and R3, respectively, i.e., the real and imaginary parts of the eigenvalues at equilibria R2 (α2) and R3 (α3) of the symmetric damped 2DOF system near the star bifurcation in the first-order form as p∧ goes from negative to positive. Each equilibrium has four different eigenvalues that are designated as αji on the plot for j = 2, 3 and i = 1, 2, 3, 4. The dashed and dotted curves represent two and four overlapping eigenvalues, respectively.

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