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Research Papers

Twinkling Phenomena in Snap-Through Oscillators

[+] Author and Article Information
Brian F. Feeny1

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226feeny@egr.msu.edu

Alejandro R. Diaz

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226

1

Corresponding author.

J. Vib. Acoust 132(6), 061013 (Oct 12, 2010) (7 pages) doi:10.1115/1.4000764 History: Received April 18, 2007; Revised July 15, 2009; Published October 12, 2010; Online October 12, 2010

Oscillatory behavior in a chain of masses connected by springs with continuous but nonmonotonic spring forces can be induced under quasistatic loading. An insight into the birth of this behavior is obtained from a single mass system. A bifurcation study shows the potential for equilibrium jumps between multiple equilibria. As such, the transients occurring under quasistatic loading do not converge to the static loading case. Transients during dynamic loading show sensitivity to the loading parameters.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 2

Equilibrium condition that yields multiple equilibrium solutions

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Figure 3

Values of y enabling equilibria of x

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Figure 4

Equilibrium positions in x as functions of imposed extensions y. Solid lines are stable. (a) and (b) are two characteristics, depending on the form of f(x). Part (a) shows examples of branches x11 (solid line), x12 (dotted), and x13 (solid), corresponding to those shown in Fig. 3. A branch such as x23 contains both stable and unstable intervals (dotted and dashed).

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Figure 9

Force-displacement relation for springs in the four-mass chain

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Figure 10

Positions of the four-mass train when the right mass moves as L(t)

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Figure 11

Total energy as a function of total elongation for a full cycle of variation in L(t)

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Figure 12

Hysteresis caused by damping for a full cycle of variation in L(t)

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Figure 13

The equilibrium position settled to by each mass when L=5 displacement units is reached at various speeds v

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Figure 14

The equilibrium deflection of each spring when L=5 displacement units is reached at various speeds v. For any value of v, the deflections add up to five displacement units.

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Figure 1

Nonmonotonic but continuous force characteristic

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Figure 5

(a) Graph of f(x). (b) Equilibria associated with springs characterized by f(x), as a function of y.

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Figure 6

Graphs of f(x)−f(y−x) show zeros which correspond to equilibria, and how they change as y varies. (a) At y=1, there is one zero; (b) at y=3, there are three zeros; (c) at y=4, there are also three zeros; (d) at y=4.2, we see five zeros; (e) y=4.25 has five zeros; and (f) at y=4.35, we see one zero after a double saddle-node bifurcation.

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Figure 7

Phase portraits for various values of fixed y illustrate the multiple equilibria, their stabilities, and imply separatrices for the stable equilibria. (a) y=1, (b) y=3, (c) y=4, (d) y=4.2, (e) y=4.25, and (f) y=4.35. Between (a) and (b), there is a pitchfork bifurcation near y=2. Between (c) and (d) there is a pitchfork bifurcation near y=4. Between (e) and (f) there is a pair of saddle-node bifurcations.

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Figure 8

Configuration of four-mass chain. The motion of the fourth “mass” is imposed.

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