The Mechanics of Walking: Levers And Pendulum Swings

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The Mechanics of Walking: Levers And Pendulum Swings

May 30 2011 11:51:43 am EST

Topics: Evolution, Morphology,

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Walking and running has evolved to generate as much thrust as possible while using the least amount of energy. The process of human walking recovers approximately sixty per cent of the energy used due to pendulum dynamics and ground reaction force. However, the legs are not simple straight poles that swing uniformly like the pendulum in a clock; a leg swings and thrusts at the hip, knee, ankle, and toes, utilizing a dizzying array of muscles, ligaments, tendons, bones, and other tissues.

Looking at a human gait cycle, it consist of 60% stance, 40% swing. In other words, the leg spends more time thrusting than it does swinging forward to the next step. Yet one foot has to always be in contact with the ground. In order to maximize thrust while minimizing swing time, the leg is extended while it is in contact with the ground, while the various levers are contracted while the leg is swinging in order to move more quickly to the next step.

In other words, if the leg were a simple pendulum that swings at the hip, the length of the pendulum is increased during contact with the ground, making a wider, slower arc, while the length of the swinging leg is shortened, making a shorter, faster arc.

Increasing or decreasing the size of an animal has an effect on the length of these various lever and pendulum swings, and has an overall impact on the maximum speed attained. The smaller the animal, the slower it tends to go, in ratio to the square root of its linear dimensions.

D’Arcy Wentworth Thompson covers the issues of scaling nicely in the following passage of On Growth and Form (39-41):

An apparently simple problem, much less simple than it looks, lies in the act of walking, where there will evidently be great economy of work if the leg swing with the help of gravity, that is to say, at a pendulum-rate. The conical shape and jointing of the limb, the time spent with the foot upon the ground, these and other mechanical differences complicate the case, and make the rate hard to define or calculate. Nevertheless, we may convince ourselves by counting our steps, that the leg does actually tend to swing, as a pendulum does, at a certain definite rate. So on the same principle, but to the slower beat of a longer pendulum, the scythe swings smoothly in the mower’s hands.

To walk quicker, we “step out”; we cause the leg-pendulum to describe a greater arc, but it does not swing or vibrate faster until we shorten the pendulum and begin to run. Now let two similar individuals, A and B, walk in a similar fashion, that is to say with a similar angle of swing (Fig. 1). The arc through which the leg swings, or the amplitude of each step, will then vary as the length of leg (say as a/b), and so as the height or other linear dimension (l) of the man. But the time of swing varies inversely as the square root of the pendulum-length, or √a/√b. Therefore the velocity, which is measured by amplitude/time, or a/b x √b/√a, will also vary as the square root of the linear dimensions; which is Froude’s law over again.

The smaller man, or smaller animal, goes slower than the larger, but only in the ratio of the square roots of their linear dimensions ; whereas, if the limbs moved alike, irrespective of the size of the animal—if the limbs of the mouse swung no faster than those of the horse—then the mouse would be as slow in its gait or slower than the tortoise. M. Delisle saw a fly walk three inches in half-a-second ; this was good steady walking. When we walk five miles an hour we go about 88 inches in a second, or 88/6 = 14*7 times the pace of M. Delisle’s fly. We should walk at just about the fly’s pace if our stature were 1/(14*7)², or 1/216 of our present height—say 72/216 inches, or one-third of an inch high. Let us note in passing that the number of legs does not matter, any more than the number of wheels to a coach; the centipede runs none the faster for all his hundred legs.

But the leg comprises a complicated system of levers, by whose various exercise we obtain very different results. For instance, by being careful to rise upon our instep we increase the length or amplitude of our stride, and improve our speed very materially; and it is curious to see how Nature lengthens this metatarsal joint, or instep-lever, in horse and hare and greyhound, in ostrich and in kangaroo, and in every speedy animal. Furthermore, in running we bend and so shorten the leg, in order to accommodate it to a quicker rate of pendulum-swing . In short the jointed structure of the leg permits us to use it as the shortest possible lever while it is swinging, and as the longest possible lever when it is exerting its propulsive force.

Dates:

2011

2010