Calf raises are a good example of applying joint mechanics to derive vertical accelerations and the resultant vertical GRF, because the hips and knees remain locked and we may easily isolate the ankle joint, calf muscles, and Achilles tendon. Walking and running involve a system of joints in each of two legs, and the motion of the COM is the combined eLect of all the joint motions. It is instructive to examine a calf raise cycle to understand how joint rotations may be broken into vertical and horizontal components, and ultimately, how footwear aLects the stress experienced in tissues. Since vertical is the dimension of gravity, the acceleration of gravity is added to the joint accelerations to produce the measurable zGRF. In the diagram, I show the rotations at the ankle and MPJ that produce the vertical rise and fall of the COM (Fig. 1).
Differentiating the vertical component of the joint angular velocity function yields the vertical acceleration as a function of time. Multiplying by mass M yields the force.
With two legs, four joints each (heel and MPJ combine to have eLect of one joint), and a period of overlap while both feet are on the ground, the vertical contributions from each joint sum to produce the measured zGRF (Fig. 3). Plotting the accelerations for each joint allows us to identify where and how accelerations change (jerk) and to understand which tendons experience the associated stresses. The sum of accelerations produces the familiar force profile for horizontal walking. The vertical change in COM elevation Δ𝑧 results from the raising and lowering of the swing leg. As depicted in the zGRF graph, the knee joint has a frequency of 4p and produces the characteristic double-hump force profile.
Since walking involves a system of rotations, the horizontal xGRF may be derived from the same angular velocities as the vertical zGRF by proper substitution of stride length for the spatial coordinate, such that the horizontal component of each joint velocity is proportional to stride length. The sum of the horizontal force components for each joint produces the characteristic GRF profile (Fig. 4). Running kinetics and kinematics may be derived similarly to walking by proper formulation of the angular velocity functions, phase of hip rotations, and determination of overlap period.
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