The Wilberforce Pendulum

In 1896, L. R. Wilberforce, demonstrator in physics at the Cavendish Laboratory, Cambridge, constructed a pendulum that functions simultaneously as a linear oscillator coupled to a torsion pendulum [1]. If the frequencies of the translational and rotational modes of oscillation are nearly equal, energy can be transferred from one mode to the other, while each motion exhibits frequency beats depending on the coupling between the two modes [2, 3].
The graphic shows the oscillations of the pendulum for selected values of the frequency and coupling strength . On the right are plots of (in black) and (in red), the amplitudes of the linear and torsional oscillations, respectively. As shown in the plot, the out-of-phase linear and torsional motions exhibit beat oscillations as energy is exchanged back and forth between the two modes.
A YouTube video of a Wilberforce pendulum is shown in [4].


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The dynamics of an idealized Wilberforce oscillator can be represented by the Lagrangian:
where is the mass of the load on the (assumed massless) spring, is the moment of inertia of the rotating pendulum bob, is the linear force constant, is the torsional force constant and is the linear-torsional coupling strength. We obtain, thereby, two coupled equations of motion:
where , are the linear and torsional oscillation frequencies, respectively, and is the scaled coupling constant. Eliminating between the coupled equations, we obtain a fourth-order linear differential equation for :
with an identical equation for . These differential equations can be solved exactly, to give a linear combination of four complex exponential terms , where
We are interested in solutions for which , so that , assuming .
With an appropriate choice of boundary conditions, we construct the solutions
[1] L. R. Wilberforce, "XLIV. On the Vibrations of a Loaded Spiral Spring," The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 38(233), 1894 pp. 386–392. doi:10.1080/14786449408620648.
[2] R. E. Berg and T. S. Marshall, "Wilberforce Pendulum Oscillations and Normal Modes," American Journal of Physics, 59(1), 1991 pp. 32–38. doi:10.1119/1.16702.
[3] M. Hübner and J. Kröger, "Experimental Verification of the Adiabatic Transfer in Wilberforce
Pendulum Normal Modes," American Journal of Physics, 86(11), 2018 pp. 818–824. doi:10.1119/1.5051179.
[4] Wilberforce Pendulum [Video]. (Nov 15, 2018)
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