Online Supplements for “Ratcheting and Transitions: Short Granular Chain in a Gradient of Vibration ”[pdf]

S0) Movie clips showing different states of the chain, with high-speed videos and long-time recordings.
Definition of the overall vibration strength: VS=Γ(x=25cm). Total length of the track(s) = 50cm. The right-hand side of these images is defined as the direction of +x, associated with the increase of Γ. The distance between black marks (on the white strip) is 1cm each.

S1) Robustness of the ratcheting --- the tilt-resistance:

(a) We find that the ground-state creeping (the regular ratcheting) can survive a considerable tilting against it --- see schematics and the full descriptions for measuring the critical slope |tanθC| (or |tanθC-| ). The results reveal two important messages:The measured values significantly exceed the small angular amplitudes δθ of the substrate. That is, when the angular offset is close to the critical value, the chain can creep steadily upwards whereas the slope of the substrate never changes its sign throughout the entire vibration cycle. This naturally removes the suspicion that the periodic alternation of the normal vector of substrate might be the source of this steady ratcheting. In addition, the values of | tan θC| show a strong correlation to the locations of the activity zone and, in turn, to the range of the intensity of gradient grad Γ/Γ = 1/x around the chain. (2) The critical slope is insensitive to the change of frequency and to the reduction of N from 8 to 4.

(b) These measurements are also extended to characterize the tilt-resistance of reversed ratcheting as well. For this purpose, two critical angles |tanθC-| vs |tanθC+| for the regular (-x) and reversed (+x) ratcheting are defined, respectively.

S2) Theoretical considerations on the observed "regular ratcheting"

Analyses on the "parabolic flights" of a non-cohesive point mass starting at rest from a vibrating substrate, using elementary calculations: we solve the take-off (φO or φOff) and landing (φL or φLanding), respectively, and the normalized landing speed (for the numerical demonstration below), as functions of Gamma -- all results are frequency-independent.

Numerical demonstrations suggesting the importance of coherence behind the ground-state ratcheting: we have found that simple models ignoring the coherence along the chain (by using merely ramdomized momentum-impacts) would severely under-estimate the persistence and speed of the creeping seen in our experiments -- even with the resitution coefficient set as high as 0.5.

S3) Further information on the regular ratcheting

Assessments on the potential effects of unintended XZ-coupling.

Peculiar exceptions: cases of "resonances"

Contraction of the chain during the creeping.

S4) Further considerations on the head-swinging modes (Mode 1 and Mode 2)

Role of bending : our assessments with a much reduced "theta_max" show that the onset values of Gamma for the occurence of these modes can be changed by deliberately suppressing the bending of the chain.

One may anticipate the breakdown of these modes at excessive excitations, based on the fact that φ2 almost coincides φO (as shown by Fig.3b in the Manuscript).