The force–velocity relationship for a single muscle fibre is commonly described using a hyperbolic equation. In contrast, in-vivo experiments on movements requiring coordination of multiple joints have found a linear relationship for leg press and cycling.
To investigate this relationship, cycling was modelled using a forward dynamic model incorporating four muscles: rectus femoris, a single vastus muscle, a single hamstring muscle and gluteus maximus. The model was used to simulate muscle tendon forces, the torque exerted about the hip and knee by those muscles, and the corresponding torque applied to the cycle cranks at pedalling cadences between 10 and 120 rpm.
Modelled results found a small upward concavity in the crank torque–angular velocity relationship. The knee torque–velocity curve was concave upwards as would be expected, however the hip torque-velocity curve was linear. This is because hip torque was dominated by the hamstring muscles, a two-joint muscle. As cycling cadence increased, the rate of hamstring shortening across the hip increased, however the rate of hamstring lengthening across the knee also increased. This resulted in a much smaller rate of increase in hamstring fibre shortening velocity than for other muscles, with a correspondingly smaller rate of force decline.
The results of this study are dependent on the model parameters used. It is possible to imagine differing results where the concave upward knee torque–velocity curve could be exactly matched by a concave downward hip torque–velocity curve; resulting in a linear crank torque–velocity curve as previously measured in-vivo.