Hanging Chain Resonance Lab

Why a chain is not a string

On a string the tension is the same everywhere. On a hanging chain of length L the tension at distance s above the free end is T = mu g s, so the wave speed sqrt(g s) falls to zero at the bottom. The wave equation for small sideways displacement y(s,t) becomes

y_tt = g (s y_s)_s.

Bessel modes

Separating y = Y(s) cos(omega t) gives Bessel's equation, whose solution regular at the free end is Y(s) = J0(2 omega sqrt(s / g)). With the top fixed, J0(2 omega sqrt(L/g)) = 0, so the normal-mode frequencies are

omega_n = (j_(0,n) / 2) sqrt(g / L), with j_(0,n) = 2.405, 5.520, 8.654, 11.792, ... the zeros of J0. They are not whole multiples of the fundamental, unlike a string. Daniel Bernoulli found the first of them in 1732.

The simulated chain

The model has N = 40 equal beads joined by massless links, with the tension in link j equal to the weight of everything below it. Each bead obeys y_j'' = (g/l) [ (N-j) (y_(j+1) - y_j) + (N-j+1) (y_(j-1) - y_j) ] - 2 gamma y_j', where the top neighbour is the pivot displacement. The same equation applies independently to the second horizontal direction z. The exact eigen-decomposition of the 40 x 40 matrix gives the chain's own mode frequencies, which agree with the Bessel values to about 0.6 %.

Driving and the response

The pivot moves as p(t) = A sin(2 pi f t) (planar) or in a circle (whirl, y and z a quarter period apart). The steady-state response is computed independently by solving the complex tridiagonal system (K - omega^2 + 2 i gamma omega) y = force for every frequency, which gives the green curve; the dot is the measured tip amplitude from the time simulation. At resonance the tip gain is limited only by damping, roughly proportional to Q.

Whirling

If the pivot moves in a circle at a mode frequency, the y and z motions combine into a rotating arch: the beads trace circles seen from above, with the amplitude of each circle following the mode shape. Modes above the first have nodes where the circle shrinks to a point.

Simplifications

The equations are linear in the displacement, so they hold for slopes below about 0.3; the readout flags steeper motion. Link rigidity, air drag and rotational inertia of the beads are not modelled beyond a single viscous damping constant. Displacements are drawn magnified (default 4x) so centimetres are visible.

1. Press Fundamental, then switch to the Whirl-from-above camera: the whole chain rotates as one arch and the tip trail is a circle. Compare with the Circular drive button.
2. Second mode: the chain has one node, and the amplitude profile graph follows the dashed J0 curve. Notice the nodes are not equally spaced.
3. Frequency sweep: the tip gain curve peaks at the grey Bessel-zero lines. Raise Q to sharpen them.
4. Pluck mode 4 and stop the drive: the ringing is a single clean frequency, and the energy bar for mode 4 decays alone.
5. Change gravity or length: all frequencies scale as sqrt(g/L). The Moon preset slows everything by a factor 2.5.