Relaxation Oscillation in Mechanics of the Horse: 2
by
James Rooney, D.V.M.
It is of interest, I think, that the mechanism described in the first essay of this little series on relaxation oscillation of the leg can be applied, as well, to the gait as a whole. As we saw during the downward displacement of the leg (about the first half of support), energy is stored in stretching tendons while that energy is suddenly released during the second half of the step.
For a stride as a whole, four steps, the body displaces downward for about the first half of the support time and springs upward during the second half. This is clearly the case for those gaits with one or more fly periods per stride. It is considerably less obvious in other cases, specifically, the walk. One can usefully think of the curve of displacement (Fig. 1 in the first essay), as a curve describing energy storage and dissipation of the leg and or the body as a whole. When a foot or feet are on the ground elastic strain energy is being generated and stored as indicated by the increasing slope of the first part of the curve.
Now I shall present a mathematical model, equation which closely simulates the curve found for the leg and/or body. I know this sort of thing is anathema to most readers, and I make no excuse. I just have to do it!
The mathematical model is:
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This equation is a combination of the Duffing and van der Pol equations and represents a self-excited system. That simply means that the horse, in this case, generates the energy for its own movement and is not moved by external forces. That does not mean that external forces do not act on the horse but that its movement over the ground is occasioned by energy generated by the animal itself. The physical meanings of the several terms of the equation are given in the Appendix at the end of this essay.
 Figure 1 shows the immediate, transient response of the equation when a is small, 0.1. The transient response is essentially a sine curve with the slope of the ascending and descending parts identical. |
 Figure 2 is the transient response when a is larger, 1.1. The transient response has less ascending slope and greater descending slope. This is the general pattern of the actual data shown in Figure 1of the first essay. The transient, immediate response of the equation corresponds to the response of the leg/body as a foot or feet impact with the surface. |
It would be nice to think that a has a small value at the walk which increases as the velocity of the gait increases. We do not have any data to suggest that is the case, however. From the data available, in fact, it seems that the relaxation pattern is present whatever the gait or velocity.
Appendix
The by3 term represents the stiffness of the system which is the deep fascia, ligaments, and tendon/muscles – the resistance of the system, the horse, to displacement /movement. As the muscles tighten the tendons more and more, the frequency of movement increases. Like a violin string which is rather loose has a low pitch, the pitch increases as the string is tightened. This term, then, can represent the increase of frequency as the horse moves faster because of the increasing tensile field of the muscle/tendons.
The term a(y2-1)y' is a “damping” term which accounts for the self-excited nature of horse movement. y'is velocity and y" is acceleration.
ay2y' is the input of energy (+) to the horse by its own musculotendinous system. This is often called negative damping and is nonlinear. Obviously the energy increases as the velocity increases and the displacement of the system increases. ay’ on the other hand is typical linear damping which dissipates energy (-) from the system and is a function of velocity (viscous damping).
The a is a measure of the degree of nonlinearity of the damping term. As a increases the system response moves steadily from near sine waves of Fig.1 to the relaxation waves of Fig.2. Thus, one can propose that a is the neural input to the musculoskeletal system which directs increases and decreases of velocity - the rate of firing of motor neurons.
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