The mechanics of running: How does stiffness couple with speed?

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The sources of external work in level walking and running.

H Thys, G Zamboni, G. A. Cavagna (1976)

The work done at each step during level walking and running to lift the centre of mass of the body, Wv, and to increase its forward speed, Wf, and the total mechanical energy involved (potential + kinetic) Wext, have been measured at various 'constant' speeds (2-32 km/hr) with the technique described by Cavagna (1975). 2. At intermediate speeds of walking (about 4 km/hr) Wv = Wf and Wext/km is at a minimum, as is the energy cost. At lower speeds Wv greater than Wf whereas at higher speeds Wf greather than Wv: in both cases Wext/km increases. 3. The recovery of mechanical energy, through the pendular motion characteristic of walking, was measured as (/Wv/ + /Wf/ - Wext)/(/Wv/ + /Wf/): it attains a maximum (about 65%) at intermediate speeds. 4. A simple model, assuming that in walking the body rotates as an inverted pendulum over the foot in contact with the ground, fits the experimental data better at intermediate speeds but is no longer tenable above 7 km/hr. 5. In running the recovery defined above is minimal (0-4% independent of speed), i.e. Wext congruent to /Wv/ + /Wf/: potential and kinetic energy of the body do not interchange but are simultaneously taken up and released by the muscles with a rate increasing markedly with the speed (from about 1 to 4 h.p.). 6. Wext increases linearly with the running speed Vf from a positive y intercept owing to the fact that Wv is practically constant independent of Vf. On the contrary, Wf = aVf2/(1 + bVf), where b is the ratio between the time spent in the air and the forward distance covered while on the ground during each step.

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The determinants of the step frequency in running, trotting and hopping in man and other vertebrates.

G. A. Cavagna, P Franzetti, N Heglund … (1988)

1. During each step of running, trotting or hopping part of the gravitational and kinetic energy of the body is absorbed and successively restored by the muscles as in an elastic rebound. In this study we analysed the vertical motion of the centre of gravity of the body during this rebound and defined the relationship between the apparent natural frequency of the bouncing system and the step frequency at the different speeds. 2. The step period and the vertical oscillation of the centre of gravity during the step were divided into two parts: a part taking place when the vertical force exerted on the ground is greater than body weight (lower part of the oscillation) and a part taking place when this force is smaller than body weight (upper part of the oscillation). This analysis was made on running humans and birds; trotting dogs, monkeys and rams; and hopping kangaroos and springhares. 3. During trotting and low-speed running the rebound is symmetric, i.e. the duration and the amplitude of the lower part of the vertical oscillation of the centre of gravity are about equal to those of the upper part. In this case, the step frequency equals the frequency of the bouncing system. 4. At high speeds of running and in hopping the rebound is asymmetric, i.e. the duration and the amplitude of the upper part of the oscillation are greater than those of the lower part, and the step frequency is lower than the frequency of the system. 5. The asymmetry is due to a relative increase in the vertical push. At a given speed, the asymmetric bounce requires a greater power to maintain the motion of the centre of gravity of the body, Wext, than the symmetric bounce. A reduction of the push would decrease Wext but the resulting greater step frequency would increase the power required to accelerate the limbs relative to the centre of gravity, Wint. It is concluded that the asymmetric rebound is adopted in order to minimize the total power, Wext + Wint.