Relaxation Oscillation in Mechanics of the Horse

by
James Rooney, D.V.M.

Continuing investigation of the gaits of horses, together with the availability of data not readily, if at all, available in past years, leads me to the following considerations^[1].

Earlier in vitro studies of horse joints showed clearly that energy was stored in collateral ligaments as the ligaments were stretched under load and that the energy was released in an abrupt manner when the loading ceased. This type of behavior is characteristic of so-called relaxation oscillation. At least one interpretation of the behavior of the digital cushion during support indicated that it, too, undergoes relaxation oscillation.

Figure 1: The saw tooth pattern is characteristic of relaxation phenomena.

Relaxation, in this context, has a specific meaning or definition and is not the usual everyday usage. An example might be helpful,. When a bowstring is drawn, energy is stored at a moderate rate, as elastic strain energy, in the bending of the material of the bow itself. When the bowstring is released, the strain energy is almost instantly converted to kinetic energy propelling the arrow. This is shown in Figure 1.

Examination of the curves of angular displacement of the fore and hind fetlock joints and coffin joints of horses trotting on a treadmill^[2] suggested that relaxation was occurring. I copied this data and took off the coordinates for the two joints while in support. From inspection it is quite obvious that the slope of the displacement curves for the first half of support is less than the slope of the curves for the second half of support, Figure 2.

Figure 2: Angular displacement of hind fetlock. The displacements of the fore fetlock and both coffin joints were similar.

This means, of course, that the displacement during the loading phase – first half of support – is slower than the unloading during the second half of support just like the bending and release of the bow. This is obvious as well when the slopes of the two parts of support are plotted as in Figure 3.

Figure 3: Slopes of the first and second half of support for both fetlock and coffin joints.

The value of a relaxation oscillation is that it is: “…ideally suited to control systems in which an input stimulus should produce a response of fixed amplitude but adaptable frequency or repetition rate.” Thompson and Stewart^[3]. The response, in this case, is to maintain a near-constant amplitude (in the vertical sense, minimal work against gravity) allowing the change of frequency necessary for the range of velocity and gait. The relaxation does not apply to the stride length.

• ^[1] I have availed myself of the published data from the Faculty of Veterinary Medicine of the University of Utrecht, one of the most active and productive centers for the study of equine gait.
• ^[2] Beck and Clayton. Equine Locomotion .Saunders. 2001.
• ^[3] Thompson and Stewart. Nonlinear Dynamics and Chaos. Wiley.2002.

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