Rod Lakes

University of Wisconsin

Scaling concepts.

Consider animals or humans of different size L. Determine the scaling of the speed of walking.

Assume the leg swings as a pendulum. The muscles apply the minimal amount of effort. The torque, assumed solely due to gravity of acceleration g, is

t = -m

Here m

For a small angle q,

t = -m

Newton's second law for rotational motion is torque equals moment of inertia times angular acceleration.

t = Ia

-m

Try a solution sinusoidal in time,q(t) = q

with t as time and w as angular frequency.

-m

so w = (m f g L

But I = p mL

The stepping frequency n = w/ 2p.

n =(1/ 2p)(3fg/L

If the leg were a simple pendulum of length 1 m, f = 1 and p = 1, then for a leg 1 m long the stepping frequency is 0.5 Hz.

Considering the leg as a uniform rod, f = 1/2, p = 1/3,

for a human leg length L

As the leg goes through a full cycle, the human walks two paces of length 1.14 m as measured on a volunteer. The speed is v = 2 x 1.14 m x (0.61 / sec) = 1.4 m/sec. With 1 mph = 0.447 m/sec,

If the organism's shape does not change with size, then the leg length L

Speed v = K L w = k L^{1/2} |

Walking speed goes as the square root of organism size L, assuming that size differences are not accompanied by shape differences.