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The purpose of this webpage is to present and explain McNeil Alexander's analysis of animal running. This analysis arose in connection with estimating the running speed of dinosaurs. One of the few, perhaps the only, bits of data on the dynamics of dinosaurs is the tracks of dinosaurs in mud which got turned into stone. The dinosaur tracks provide the stride length of various dinosaurs and their skeletons would provide their leg lengths. Where skeletons are not available the foot size can be used to estimate leg length. The ratio of stride length to leg length is called the relative stride. The relative stride is a dimensionless number that is likely to be a crucial variable in the dynamics of an animals running.
McNeil Alexander took an engineering approach to analyzing the dynamics of animal running. Relative stride is one dimensionless variable in the problem. He looked for another dimensionless variable and found it in the Froude number. Froude in analyzing the problems of ship motion and design constructed the ratio of speed squared to the product of a characteristic length times the acceleration due to gravity; i.e.,
where g = 32 feet/sec/sec if length is in feet and speed is in feet/sec.
What McNeil Alexander found is that for a wide variety of animals there is a relationship between the Froude number for their running and their relative stride. Specifically he found the statistical relationship
This relationship was found by fitting a regression line to the logarithms of the Froude numbers and relative strides for his data sample.
With this relationship McNeil Alexander was able to compute the Froude number for the dinosaurs for which he had data on relative stride. From the Froude numbers he could compute the speeds of the different dinosaur species he had data for. The estimated speeds turned out to not be very large, about comparable to the walking speeds of humans.
In his book McNeil Alexander characterizes the conjecture that there would be a relationship between the relative strides of different animals and the Froude numbers for their running as being a guess. But in his article for Scientific American he notes that the Froude number is closely related to the ratio of kinetic energy and potential energy for the running creatures. This might provide an explanation of why there should be a relationship between Froude numbers and relative strides of different animals.
The relative stride length determines the maximum angle between the legs and, consequently, the variation in height of the center of gravity during the course of a stride. If the stride length is S and the leg length is L then the maximum leg angle α is 2sin-1((S/2)/L). The height of the top of the leg at mid-stride is L and at maximum stride is Lcos(α/2). Thus the difference in the top of the leg height and consequently also in the height of the center of gravity of the creature during a stride is L(1-cos(α/2).
Since sin(α/2)=(S/2)/L=(S/L)/2,
The function 1-(1-x)1/2 has a Maclaurin series of the form
The loss of energy during a complete stride is twice the work done in reducing the angle between the legs from α to zero. If M is the mass of the running creature and thus Mg is its weight (where g is the acceleration due to gravity) then this work is equal to MgL[1-(1-(S/L)2/4)1/2] which to the first approximation is MgL(S/L)2/8.
The maximum running speed of a creature may be related to the forces generated during running in relation to the strength of the creature but the comfortable running speed for a creature is probably related to other characteristics.
Any structure or object has a natural frequency of oscillation. For example a pendulum formed by a weight at the end of string of length L has a natural frequency ω equal to (g/L)1/2. The legs of a creature may be consider pendula of roughly cylindrical form. The frequency of an object pivoted about some point is given by the equation
where m is the mass of the object, I is its moment of inertia and h is the distance between the pivot point and the center of gravity of the object. For a thin cylinder of length L and mass m the moment of inertia is equal to mL2/3. When the cylinder swings about one end the distance h is L/2. Thus the natural frequency of the cylinder is
If the frequency of strides ω of a running creature is equal to the natural frequency of its legs as pendula then
Multiplying the numerator and denominator of the fraction on the left by the square of the stride length S gives
Dividing the numerator and denominator by L and a bit of rearrangement gives the equation
But ωS is the same as the running speed of the creature v. Thus
The fraction on the left is the Froude number F and this equation give the Froude number of a running creature as a function of its relative stride (S/L); i.e.,
This is not the same relationship between the Froude number and relative stride found by Alexander but it is such a relationship.
Note that the previous equation can be put into the form
where M is the mass of the creature. The numerator of the fraction on the left is the kinetic energy of the creature and the denominator is the potential energy change during a stride and the first approximation of the work lost due to the up-and-down motion of the creature during a stride.
If MgL(S/L)2/8 is replaced by the exact value of the work loss the resulting formula is
The exponent 0.3 in the equation found by Alexander corresponds to the derivative of ln(F) with respect to ln(S/L). In the above relation this derivative would depend upon the level of (S/L), but ranges between 1.0 and 2.0 and thus cannot match Alexander's measured figure of 0.3.
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