J. Peter Campbell
The Wilberforce pendulum combines longitudinal motion with rotational motion about a vertical axis. We have assumed that the motion of the system is characterized by a linear restoring force in both the z direction, and the q direction, with a slightly complicating coupling factor, c. As the moment of inertia approaches the resonant value, the beat frequency shortens. At the resonant I, where the resonant frequencies are identical in the two degrees of motion, there is a perfect transfer of energy between the longitudinal and rotational modes. Due to damping and limited precision, our coupling coefficient never becomes completely insignificant.
Because there are two degrees of freedom, there can be two normal modes of operation, in which energy is not transferred between the z and q components. Our results have indicated that in this case, only at the point of resonance are there two normal modes. For all other conditions, there is only one.
Our mathematical models matched up very well with the experimental results. The two "C vs. I" graphs demonstrate that the system, both theoretically and experimentally, behaves as predicted. Specifically, around the point at which the two resonant frequencies are identical, the coupling is minimized. The fact that there is less than 1% error between the theoretically predicted wz and wq and the observed frequencies is remarkable, considering the several possible sources of error. First, the assumption that wz can be approximated in terms of k and m, and wq in terms of delta and I is based on the assumption of simple harmonic oscillation. Should q or z exceed the point at which there is no longer essentially a linear restoring force, this approximation no longer necessarily holds. Further, the measurement of delta was indirect, and the estimation of I was very simplistic. Nevertheless, the mathematical models (both analytical and numerical) represent the experimentally observed data quite satisfactorily.
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R.E. Berg and T. S. Marshall, “Wilberforce pendulum oscillations and normal modes,” Am. J. Phys. 59, 32-38 (1991).
M. L. Boas, Mathematical Methods in the Physical Sciences, (John Wiley & Sons, New York, 1983).