M2 Elastic Locomotion

Topic

During running and trotting, the body bounces on the supporting limbs as if it were a mass held by a spring: the potential and kinetic energy of the center of mass are at a minimum midway through the stance phase—when the spring is most compressed—and at a maximum at the beginning and end of the stance, the exact opposite of what occurs during walking.

This spring-like behavior relies heavily on the limb's elastic tendons—such as the Achilles or patellar tendons—which store energy during the first half of ground contact and release it during the second half, thereby reducing the amount of active work muscles must perform with each stride. In mammals specialized for this type of locomotion, the efficiency with which these tendons return stored energy ranges from 85 to 95 percent.

The stiffness of the supporting limb—that is, how much this effective spring compresses under a given load—is the key variable governing this entire elastic behavior. When this stiffness is combined with limb length, animal mass, and gravity into a single dimensionless number, that number remains virtually identical across animals of vastly different sizes, provided they are compared at the same dimensionless speed relative to their own size.

As this dimensionless speed increases sufficiently, the trot gives way to the gallop; the gallop introduces aerial phases—where no limb touches the ground—and a different pattern of footfalls among the four legs. In most quadrupedal mammals, the transition from trot to gallop occurs when this dimensionless number falls within a range of approximately two to three. The savings in active muscular work achieved by elastic tendons—as described in the previous mechanism—are ultimately reflected in the cost of running transport: the energy required to move each unit of body mass over a given distance is not the same for all body sizes; instead, it decreases as the animal's mass increases, following a negative power function of body mass with an exponent close to one-third. Consequently, for every kilogram of body mass, large animals are more efficient runners than small ones.

Certain mammals, such as the cheetah and the kangaroo, take the use of these elastic tendons to an extreme: their tendons store and return energy with minimal loss during each stride, functioning almost like purely passive springs that handle a significant portion of the mechanical work of running, thereby further reducing the active muscular effort the animal must exert with each step.

The spring-mass system formed by the limb and the body—mentioned earlier—also has a specific bouncing frequency at which running is most efficient. This optimal frequency coincides with the system's natural resonant frequency, and running close to this resonance minimizes the active energy expenditure required by the animal during the run.

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