Biomechanics BME 315
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 = -mLf gLL sin q
Here mL is the mass of the leg; LL is its length; f is the fraction of length from the pivot to the center of mass.
For a small angle q,
t = -mLf g LL q
Newton's second law for rotational motion is torque equals moment of inertia times angular acceleration.
t = Ia
-mLf g LL q= I d2 q / d t2
Try a solution sinusoidal in time,q(t) = q0 sin (wt)
with t as time and w as angular frequency.
-mLf g LL q0= - q0I w2
so w = (m f g LL/I)1/2
But I = p mL2 with p as a geometrical factor, 1 for a simple pendulum, and 1/3 for a thin uniform rod rotating about one end.
The stepping frequency n = w/ 2p.
n =(1/ 2p)(3fg/LL)1/2
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 LL = 1 m, the frequency is 0.61 Hz. So the stepping frequency does not vary strongly with leg mass distribution as a result of the square root.
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, v = 3.1 mph. This is a remarkably reasonable estimate of a comfortable walking speed for an adult human.
If the organism's shape does not change with size, then the leg length LL is proportional to the overall organism size L. In the following, k and K are constants of proportionality.
 Speed v = K L w = k L1/2

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