The theoretical limits to the power output of a muscle-tendon complex with inertial and gravitational loads

When a muscle delivers power to an inertial load through a spring, the peak power can exceed the maximum that the muscle alone could produce. Using normalized differential equations relating dimensionless quantities we show, by solving the equations either analytically or numerically, that one dimensionless constant (Xi), representing the inertial load, is sufficient to specify the behaviour during shortening of a muscle-tendon complex with linear force velocity and force-extension properties. In the presence of gravity, an additional constant (F), representing the gravitational acceleration, is required. Nonlinear force-velocity and force-extension relationships each introduce an additional constant, representing their curvature. In the absence of gravity the power output delivered to an inertial load is limited to approximately 1.4 times the maximum power of the muscle alone, and when gravity is present the power delivered is limited to approximately twice the power of muscle alone. These limits are found for the purely inertial load at Xi ca. 1 and with gravity acting at XiGamma = 0.5 with Xi arbitrarily small. The effects of nonlinear muscle and tendon properties tend to cancel each other out and do not produce large deviations from these optima. A lever system of constant ratio between muscle and load does not alter these limits. Cams and catches are required to exceed these limits and attain the high power outputs sometimes observed during explosive animal movement.