Perhaps some are not aware of this effect, but when an individual runs they prefer a step frequency that minimizes their oxygen consumption by a few percent at that speed. Some refer to this as the Preferred Step Frequency (PSF). If the runner tries to run with a faster or slower step frequency, their O2 consumption increases (1a, 1b)

Through the help of a paper by G. Cavagna (2) I developed a physics-based analog that may help to qualitatively explain this effect and two others. Unfortunately, I have a problem verifying the required rotational power. I hope someone might help me with their own measurements or direct me to some published research in this area.

The approach relies on combining two Mass Specific Power or MSP (w/kg) functions as shown in the attached figure, Min_MSP. Here, the MSP_SMM is based on the Spring Mass Model (SMM) with a leg-stiffness variation reflected in a paper by Farley and Gonzalez (3). The MSP_Rot is due to rotational dynamics.

The MSP_SMM curves are similar to the hyperbolic-like curves in the attached Figure3 from G. Cavagna. Since this is a lossless system, the computed MSP_SMM (8 w/kg) is greater than the experimental power (4 w/kg) in Figure3 and less than the metabolic power estimate based on O2 consumption (10 w/kg). (See ACSM relations and conversion below.)

The lower power curve or MSP_Rot is a linear function of step frequency. It is mathematical fudge. It is based on a reverse calculation of the rotational power required to generate a minimum in MSP_Total at the PSF.

Unfortunately, I am piteously stuck. I can’t guess at the inertia or spin tensors, so I cannot legitimately compute the MSP_Rot based on anthropometric parameters.

Can anyone help with any comments, by sharing their lab measurements of a runner’s rotational power or by referring me to additional published work on this topic?

Ted

(1a) Iain Hunter and Gerald A. Smith, Preferred and optimal stride frequency, stiffness and economy: changes with fatigue during a 1-h high-intensity run Eur J Appl Physiol (2007) 100:653–661, DOI 10.1007/s00421-007-0456-1

(1b) Cavanaugh PR, Williams KR, The effect of stride length variation on oxygen uptake during distance running. Medicine and Science in Sports and Exercise [01 Jan 1982, 14(1):30-35]

(2) Giovanni Cavagna, The two power limits conditioning step frequency in human running, https://doi.org/10.1113/jphysiol.1991.sp018586

(3) Farley C T, Gonzalez O. Leg stiffness and stride frequency in human running. J Biomech. 1996;29:181–186

(ACSM relations and conversion)

VO2 = (0.2 * speed (m/min) + 0.9 speed (m/min) * grade + 3.5) (ml of O2/(kg-min))

Speed is in m/min. If the grade is set to zero,

VO2 = (0.2 * speed*(m/min) + 3.5) (ml of O2/(kg-min))

The relation requires speed in m/min. If we want to insert speed in m/s, we must multiply the speed in m/s by 60 to obtain speed in m/min.

VO2 = ((0.2 * 60) * speed* (m/s) + 3.5)* (ml of O2/(kg-min))

VO2 = (12 * speed* (m/s) + 3.5)* (ml of O2/(kg-min))

So, at 2.2 m/s in the attachment, the ACSM relation for VO2 would give 30 ml of O2/(kg-min)

Now, we can convert the entire relation’s units to J/(s-kg) or w/kg

VO2 = (12 * speed* (m/s) + 3.5)* (ml of O2/(kg-min))*(4.82 Kcal/1000 ml of O2)*(4186 J/1 Kcal) *(1 min/60 s)

VO2 = (12 * speed* (m/s) + 3.5) * 0.336 (J/(kg/s))

VO2 = (12 * speed* (m/s) + 3.5) * 0.336 (w/kg)

VO2 = (4 * speed (m/s) + 1.2) w/kg

At 2.2 m/s in the attached Figure3, the metabolic MSP would be:

V02 = 10 w/kg, or numerically about 1/3 of the VO2 value in ml of O2/(kg-min)