Possible Explanation for Behaviour Non-Linear Bead Knott Experiment I have a very simple setup to study non-linear phenonenon with plastic beads, just like described here:
E. Ben-Naim, Z.A. Daya, P. Vorobieff, and R.E. Ecke, “Knots and Random Walks in Vibrated
Granular Chains”, Phys. Rev. Lett. 86 (2001) 1414;
Essentially, a long string of connected beads is placed on a vibrating plate with a knot. Over time, the knot untangles. The time it takes for the knot to untangle is related to the frequency, amplitude of the plate and length of the beads.
When I keep the amplitude constant, and bead length constant, but alter the frequency, when I read $20Hz$ the bead is no longer able to untangle (just sits there vibrating as appose to slowly twitching around and untangling). I know the system of the plate has a resonance at $20Hz$, but this is the opposite effect I would expect. I am checking the external accelerometer so that the amplitude remains constant with all frequencies tested.
Could someone propose a possible reason why this occurs?
 A: There are a couple of regimes where the knot will be static.
The first is the synchronous phase-stable mode where the whole chain bounces up and down barely changing shape. This happens when:

*

*The kinetic energy gained by the chain when colliding with the plate equals the energy loss due to inelasticity.

*The period of bouncing matches the period of the plate (or some even multiple).

*The bounces occur when the plate is rising but slowing down. If the chain has bit too little speed, its bounce period will be a bit less than the period of the plate. It will arrive a bit early for the next bounce, be hit a bit harder which lengthens its period a bit moving it closer to the period of the plate. Conversely, if the chain has a bit too much speed, it will arrive a bit late and receive less energy, slowing it down and its bounce period will shorten a bit becoming closer to the plate period.

For a more detailed discussion of the above situation for a single bouncing ball, see "Mechanical analog of the synchrotron, illustrating phase stability and two- dimensional focusing".
Since your problem is at the relatively high frequency of 20 Hz, however, I expect the issue is different.  You "keep the amplitude constant", and I assume this means you are keeping the peak acceleration of the plate constant, not that the vertical movement amplitude is constant.  (The latter is eventually impossible since you run out of power as the frequency increases.)  For constant peak acceleration, the height the chain will be thrown up by the plate gets smaller and smaller as the frequency gets higher. At 20 Hz the beads might only be jumping a millimetre or so for typical (of order 1g) peak accelerations.  If your beads are a few millimetres in diameter, at high enough frequencies they won't jump high enough for the knot crossing points to move.  The chain just jitters and wiggles, which sounds like what you describe.
