The figure below from McMahon compares step length (

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) on a rigid surface (left panel) to step length (

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) on a compliant surface (right panel). The illustration shows the limb in multiple positions during stance phase: (i) at initial ground contact with the knee fully extended; (ii) in mid-stance with the knee flexed; and (iii) at toe-off with the knee again fully extended.

In mid-stance on the rigid surface (left panel), the height of the hip above the ground in mid-stance is

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, where

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is the leg length and

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is modeled as a constant, independent of running speed, set by the central nervous system to control the body’s trajectory.

The additional deflection of a compliant surface is characterized by

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, which represents the peak deflection of the surface in mid-stance. However,

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can be approximated as the static deflection that would be observed if the runner were simply standing on the surface.

Figure 3.1 Schematic of a step on a rigid surface (left) and compliant surface (right). Solid line shows the stance leg, broken line shows the swing leg moving forward. Because the foot descends a distance into the compliant track, the step length on the compliant track is necessarily greater. Figure adapted from Figure 5 (McMahon, T. A., & Greene, P. R. (1979). The influence of track compliance on running. Journal of Biomechanics 12(12), 893–904.)

(a) Using the figures and description of

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above, derive the expression below for the step length of the runner on a compliant surface,

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, in terms of the runner’s mass,

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, runner’s leg length,

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, the rigid surface step length,

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, surface stiffness,

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, and local gravitational acceleration,

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.

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(b) Choose some realistic values for

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,

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,

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,

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, and

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. What is the predicted value for

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? How does this equation predict how

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will vary with decreasing

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? Discuss how this prediction compares with your intuition.

Solution (only visible by instructors; please contact us to request access)

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