Online Supplements for “Short granular chain under vibration: Spontaneous switching of states ”[pdf]

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High-speed demo of three main dynamics (Vibration=25Hz) @ @ @ @ @ @ @ @ @ @
(a) uniform bouncing (Γ=1.45, State 0)(b) dominance by two ends (Γ=1.80)(c) chaos (Γ=2.20)

High-speed demo of three states (Vibration=25Hz, Γ=1.65) @ @ @ @ @ @ @ @ @ @ @ @
State 0
State 1
State 2

Spontaneous switching at Γ=1.65 (120s in 10s) @ @ @

More movies cited by Y.-C. Sun, Master Thesis, National Taiwan Normal University, June 2015, Taipei, Taiwan.

Figures from previous manuscript [pdf]
Distance of individual particles from the substrate, as functions of time t, for different values of F. 25Hz, N = 8, in periods of State 0. (a) Sample trajectories, with the end particles displayed by two different colors and others in gray. The phase of the substrate and the hypothetical parabolic flight are also displayed above each sub-panels. (b) Trajectory of one particle accumulated from 18 vibration cycles, projected onto one single period. The thick line shows the average, with the hypothetical parabolic flight displayed by the thin line. (c) The variance among the trajectories shown in (b), _Z12, normalized by the mean separation with the substrate over the entire cycle, as a function of Γ
Image-based definition of states V (a) Sample images and tracking results, with small crosses indicating the center of the first and last segments, the large cross showing the center of the middle particles. The long dashed line indicates the threshold height hc = d/4 above the center of the middle particles, for defining the states. (b) The time fraction of each state for an accumulation of 2500 vibrations, using different choice of hc. The results are insensitive to the change of hc within the range marked by stripe. (c) Sample of an blurred frame and its result.
Durations of events defined by the switch of states. N = 8, 25Hz, hc = d/4, based on about 40 minutes of recording in setup β. (a) Histogram of the durations for State 0 and State 2, 0 and 2, with the bin width indicated by the vertical grid lines, at three different vibration intensity. (b) Mean values of τ0, τ2, and the average time for either end of the chain staying active τ(1).
Trajectories of three of the particles in a chain with N = 8 at State 0, at the vibration intensity around 1.45, showing clear deviations from the hypothetical parabolic flight and increasingly complicated behaviors, as results of the smaller amplitudes.