The title of my blog reflects my initial goal: promoting discussion of issues related to running efficiency. Perhaps the beginning of a new year is a good time to take stock of my current understanding of the topic. An additional reason for a review at this time is the recent protracted debate between Robert and myself waged in the comment section of my page on the Dance with the Devil (see side panel). This debate was fairly adversarial in character at times, and it prompted me to re-examine some of the issues related to the perennially thorny topic of gravitational torque. Robert’s challenges led me to do some computations, which as a by-product revealed some findings regarding linear velocity during the gait cycle. Because linear velocity is related to progress towards the finishing line, I think linear velocity is a more important aspect of running mechanics than the rotational motion arising from gravitational torque, which is largely about going around in circles (or preventing such motion). So I am grateful to Robert for re-focussing my attention on running mechanics and running style. Though first, it is important to put the issue of running style in a larger context.
Granted that races at distance ranging from 5000m to marathon are run at a paces either a little above or a little below the anaerobic threshold, the greatest determinant of efficiency is the ability to achieve a high pace at threshold so as to minimise the amount of fuel-inefficient anaerobic metabolism. If the goal is efficiency, much of one’s training efforts should be directed at this raising the threshold pace. Whether this goal is best achieved by emphasis on high volume or high intensity training (or both) remains a controversial topic, but that is not the question I will focus on in this post. Instead, I will return to the less important but nonetheless intriguing question of running style.
Almost certainly, the most important issue in considering the effects of running style on efficiency is minimization of risk of injury. Injury impairs not only performance at the time of injury but also leads to missed training and loss of aerobic fitness. Unfortunately, the evidence suggests there might be a trade-off between mechanical efficiency and safety. I think this can be illustrated most readily by examining solving the equations of motion of the running body (though if you are willing to accept my calculations, you do not need to do the maths yourself – I will illustrate the results pictorially). The complete solutions of the equations describing a multiply-jointed body made of viscoelastic tissues (i.e. tissues in which change of shape depends on how rapidly the force is applied) is of course horrendously complex. Nonetheless a great deal can be learned by focussing on the equations that describe the motion of the centre of gravity of the body (COG). If we know the time course of the external forces acting on the body – namely gravity; ground reaction force (GRF); and wind resistance – it is possible to perform an accurate computation of the motion of the COG.
On the surface of the earth, the force of gravity is constant. It is the product of mass multiplied by the acceleration due to gravity (g), which has the value 9.8 metres/second/second (or 32 feet/sec/sec in Imperial units). Ground reaction force is the reaction of the ground to the push of the body against the ground. We can measure the push of the body against the ground quite precisely using a force plate, and therefore, since action and reaction are equal and opposite, we can deduce the GRF. Estimation of wind resistance is trickier, and for the purpose of this post, I will assume that wind resistance is negligible. I have presented the equations and a description of how I solved them, in the calculations page (see the sidebar).
Ground reaction forces
For simplicity I have assumed that the vertical component of ground reaction force (vGRF) varies sinusoidally while the runner is on stance, as shown in figure 1. This is a moderately good approximation to real data for a forefoot runner, and is convenient from the computational point of view. vGRF rises rapidly from zero after footfall, reaches a peak at mid-stance and then falls away to zero as the runner approaches take-off. I do not think the main conclusions I will draw will be appreciably influenced by the exact shape of the time course of vGRF, though at the price a little more computation, I could solve the equations using real data for vGRF.
One crucial feature regarding vGRF is that the value of vGRF averaged over the entire gait cycle must equal the downward force of gravity (mg, where m is mass) since gravity acts constantly throughout the gait cycle. Otherwise, there would be a net vertical impulse that would either cause the runner to continue to float upwards after the completion of the gait cycle if average vGRF exceeded mg, or alternatively to be pulled to the ground if average vGRF was less than mg. One inevitable consequence is that when time on stance is short compared with the total period of the gait cycle, peak vGRF must be high (as is illustrated by the ochre dashed curve in figure 1) compared with the situation where time on stance is a large faction of the total duration of the gait cycle (as illustrated by the dashed blue line in figure 1).
Once vGRF is known, the horizontal GRF (hGRF) can readily be computed assuming the total GRF acts along the line from point of support to the COG (as shown in equation 5 on the calculations page). The hGRF associated with a sinusoidal time course of vGRF is depicted by the solid ochre and blue lines in figure 1. In early stance, vGRF is negative, indicating that it exerts a braking effect on the runner. Early in the stance phase magnitude of hGRF increases as vGRF increases, but because hGRF is only appreciable when the line joining point of support to COG is oblique, the magnitude of hGRF begins to decrease despite the continued rise in vGRF as the runner approaches mid-stance. By mid-stance, the COG is directly above the point of support, total GRF is vertical and hGRF is zero. After mid-stance, the line from COG to point of support is directed obliquely backwards, so hGRF is now directed forwards and has an accelerating effect on the runner that reveres the braking effect in early stance phase. One feature of interest is that when the runner spends a long time on stance, the peak magnitude of hGRF is almost the same as when the runner spends only a short time on stance, despite the much greater peak vGRF when stance is short. The reason is that when time on stance is short, the line from COG to point of support is never far from vertical so hGRF does not rise as high is it would if this line was more obliquely inclined. The fact that the magnitude of peak hGRF is similar for both short and long times on stance means that the braking effect is actually much greater when time on stance is longer, because the braking force acts for a longer time.
Vertical and horizontal components of velocity
Figure 2 depicts the time course of the velocity of the body in both vertical and horizontal direction throughout the gait cycle, based on the solution of equations 1 and 2 shown on the calculation page.
If we focus first of all on the vertical velocity in the case where time on stance is a large proportion of the total gait cycle (the dashed blue line), we see that starting from the high point at mid-flight, downwards velocity increases at a steady rate under the influence of the uniform accelerating effect of gravity. After foot-fall, as vGRF rises, the rate of acceleration slows and once vGRF exceeds mg, the downwards acceleration ceases, though the body still continues to move downwards at a decreasing rate until mid-stance, by which time vertical velocity is zero. After mid-stance, the body accelerates upwards under the influence of vGRF. Once vGRF has fallen below mg, the acceleration diminishes, though the velocity remains upwards. After the body becomes airborne, vGRF is zero and the upwards velocity continues to decrease at a constant rate as gravity retards the ascent. By the middle of the airborne period (the end of the cycle in figure 2) the vertical velocity is zero.
In the case in which time on stance is short (the ochre dashed line in figure 2), the constant increase in downwards velocity during the airborne phase continues for a longer period than when time on stance is long (blue dashed line). Consequently, when stance time is short, the downwards velocity is much greater at foot-fall. As vGRF rises in the first half of the stance phase, the downwards velocity decreases reaching zero at mid-stance. After mid-stance, the high vGRF causes a greater upwards acceleration than in the case where time on stance is longer, so that upward velocity at take off is higher. The body rises to a greater height before its ascent is arrested by gravity in the middle of the airborne phase. Using equation 3 to compute distances travelled, in the case where horizontal velocity at mid-stance is 4 m/sec, it can be shown that in the case when peak vGRF is 2mg, the total vertical distance travelled between mid-stance and airborne peak is 5.8 cm whereas it is 9.8 cm when peak vGRF is 4mg.
In contrast, in the case of horizontal velocity (solid lines in figure 2) the amount of slowing between footfall and mid-stance is appreciably greater when time on stance is longer, because, as we have seen, the braking force (hGRF) is of similar magnitude but acts for a longer period of time.
Implications for efficiency
What do these calculations tell us about mechanical efficiency? It is important to note that a substantial proportion of the kinetic energy of the falling body is absorbed and stored as elastic energy during the first half of stance, and is recovered by elastic recoil after mid-stance. The proportion that is recovered is likely to be higher when time on stance is short because tendons and muscle as viscoelastic, meaning that up to a certain point, they are more elastic when the force is applied over a shorter period of time. In similar manner, some of the kinetic energy lost due to the braking effect of hGRF in early stance can be stored as elastic energy and recovered after mid-stance. Again, the proportion recovered is likely to be higher when time on stance is shorter. However, irrespective of whether time on stance is short or long, only a proportion of the kinetic energy lost during the first half of stance can be recovered. Thus, in general efficiency will be less when the total amount of work that must be done to reverse the braking effect and to elevate the body back to its peak height is large. We have already seen that the braking effect is greater when time on stance is long, whereas the amount of upwards acceleration required to elevate the body to its peak height is greater when time on stance is shorter. Which of these effects demands more energy?
The amount of work done when a force is applied can be computed using equation 6. The results are shown in table 1 for a running speed of 4 m/sec.
At both short and long times on stance, the energy required to overcome braking is greater than the energy required to elevate the body from its low point at mid-stance to its high point in the airborne phase. Thus, the sum of the amounts of energy required to overcome braking and to elevate the body is substantially greater when time on stance is longer. Since the proportion of energy recovered by elastic recoil is likely to be less under these circumstances, it is clear that mechanical efficiency is less when time on stance is long. It should be noted that these calculations refer to the work done to counteract the effects of external forces acting on the body. Some additional work is also done repositioning the limbs, and at very high running speeds this can become appreciable, but is beyond the scope of this discussion.
The calculations confirm that mechanical efficiency is increased by shorter time on stance. Although many coaches believe this, it is not universally accepted, so it is re-assuring to see that the equations provide a clear confirmation. In practice a shorter time on stance can be achieved though stiffening the hip, knee and ankle joints by applying greater tension in the muscles that flex and extend these joints, especially the hamstrings and quads. The BK method of running developed by Frans Bosch and Ronald Klomp focuses on decreasing time on stance via plyometric drills that develop the strength necessary to maintain adequate leg stiffness. However, the equations also provide a clear warning regarding the increased ground reaction force. As shown in figure 1, when time on stance is short, the peak vertical forces acting on the body are much larger, and the potential risk of injury is potentially greater.
It is noteworthy that in the late stages of a marathon, many runners automatically increase their time on stance. This is probably due in part to the fact that as muscle strength diminishes it is harder to maintain the required tension in the hamstrings and quads, but also might be an unconscious defensive reaction to protect the body from injury at a stage where tired muscles are less able to withstand stress.
So far we have not addressed the issue of cadence. For a given proportion of the gait cycle spent on stance, the magnitude of peak vGRF required to achieve a specified proportion of the gait cycle airborne is lower at high cadence because both airborne time and stance time decrease at higher cadence. In my next post I will discuss cadence. The interim conclusion is that there is a trade off between mechanical efficiency and risk of injury. Efficiency can be increased by spending a shorter period of the gait cycle on stance, but the risk of injury is greater. Therefore, a style which entails greater leg stiffness should be adopted cautiously, and requires careful conditioning of the muscles, tendons and joints to allow them to withstand the greater forces.
Note added 27th Feb 2112:
In the discussion below, Simon drew attention to the fact that by emphasising the trade off between increased efficiency and increased risk of injury that appears to arise in the circumstances described in this post, I do not acknowledge that for a recreational runner who habitually runs at a cadence in the vicinity of 160, it is possible to increase efficiency with no appreciable increase in injury risk simply by increasing cadence from 160 to 180 steps per minute. I agree that such an increase in cadence will lead to a decrease in braking cost without the need to increase vGRF. The potential benefits of increasing cadence from 180 to 200 are less clear-cut, and are discussed in my posts of Feb 6th and Feb 27th.