In my post yesterday, I raised the question of the optimum ratio of stride length to cadence. Speed is the product of stride length and cadence, but if we wish to increase speed efficiently, it is not simply a matter of increasing one or the other. There is little doubt that efficient running requires a fairly rapid cadence. Video recordings of elite athletes demonstrate that most run with a cadence of at least 180 steps /min (90 per foot), and many recordings of elite 10K runners demonstrate a cadence of 200 steps per minute or slightly more.
However, there are several reasons why there is likely to be a point beyond which increase in cadence become inefficient. First, there are factors related to the internal (molecular) mechanism of muscle contraction. A muscle contraction involves formation and breakage of chemical cross- links as the chain-like actin and myosin molecules slide past each other. The evidence suggests that maximum efficiency is achieved when the rate of contraction is about one third of the maximum velocity of shortening of the muscle (Koushmerick and Davies, 1969). Secondly, there are external mechanical factors that would make very high cadence inefficent. Very rapid starting and stopping of the actions of repositioning the limbs is inefficient (Cavagna and Franzetti, 1986).
Therefore, as speed increases there will come a point beyond which it is no longer efficient to increase cadence. Observational evidence suggests that this optimum cadence is around 180-200 steps per minute (90-100 per minute for each foot). Beyond this stage increase in stride length is essential for any further increase in speed. It is almost certain that my current step rate of around 245 steps per minute when sprinting is inefficient. I need to increase my stride length.
However, the issue of the most efficient way to increase stride length is a challenging question. Reaching out with the swinging leg so that the foot lands too far in front of the centre of gravity (COG) will result in a wasteful and jarring braking action that decreases efficiency and is likely to increase the risk of injury to knees, hips and spine.
This might suggest that we should aim to have the point of support behind the COG throughout stance. However, this would also present problems. While point of support is behind the COG there will be a gravitational torque producing a head-forwards and downwards rotation. If this is not reversed at some point in the gait cycle, we will end up flat on our face.
It is therefore tempting to think that we should spend as little time on stance as possible. However once we are airborne, we are subject to the downwards pull of gravity. The impulse that gets us airborne will cause us to follow an arch-like trajectory. Gravity will nullify our ascent by mid-flight and for half of each airborne period we will accelerate earthwards at 9.8 metres/sec/sec. If cadence has an upper limit, the duration of each step must have a lower limit. If we spend half of this time in free fall, we must necessarily lose height and regain it in the next airborne phase. So being airborne extracts a large energy cost.
In addition, because the average upwards force acting throughout the gait cycle must be equal to body weight, a very short time on stance demands a very large upwards force during time on stance to provide the impulse necessary to get us airborne.
Therefore despite the immediate attraction of the proposals that we should land with the foot under the centre of gravity and that we should spend as little time on stance as possible, these proposals present serious problems. The challenge of working out how to run most efficiently has no easy answer.
There are many things about running mechanics which cannot be described precisely in terms of either physics or physiology, but there are a few important things that can be established with confidence from the laws of physics. Before attempting to reach a decision about the best way of increasing my stride length, it is worthwhile to step back and examine a few of the conclusions that can be drawn with confidence. Unfortunately, this discussion does require that we grapple with some of the principles of Newtonian Mechanics. As I remarked in the previous paragraph, there is no easy answer, but I think the intellectual effort is worthwhile if it leads to a constructive plan for the application of physical effort in training.
Gravity and getting the cost of getting airborne
First, it should be stated that despite the impression created by some recent theories, such as Pose and Chi, that emphasize the benefits of gravitational torque, gravity can provide no net energy when running on a level surface. The law of conservation of energy dictates that gravity can only provide energy if the centre of mass suffers a net fall. When running on a level surface, a fall in one part of the gait cycle must be compensated for by a rise at another phase of the cycle. If gravitational torque plays any useful role at all, it might be that the unbalancing associated with gravitational torque triggers a reflex muscle action that gets the foot off stance and initiates the swing phase.
Although gravity does not provide net energy, it plays a major role in running because the essence of running is getting airborne. At low speeds, walking is a very efficient form of locomotion. A walker maintains unbroken contact with the ground and relatively little energy is spent lifting the body. Most of the energy costs of walking are consumed by repositioning the limbs at each step, and in compensating for the braking effect that arises when the leading leg presses obliquely against the ground at footfall. As speed increases, the leg and foot must undergo increasingly rapid acceleration to catch up with the torso, and then suffer a corresponding deceleration in late swing phase. As a result, the energy costs of repositioning the limbs increase rapidly with increasing speed, and there comes a point where walking fast becomes inefficient. At this point, it is better to pay the price of getting airborne. Perhaps surprisingly, the costs (per Km) of getting airborne actually decrease with increasing speed when running. This is because when moving at a higher speed, momentum carries us further during the arch-like trajectory of a given height. If we maintain a trajectory of the same height on each step, we need fewer steps per Km and hence the energy cost of getting airborne (per Km) actually decreases as speed increases. However, in considering the overall energy requirements of running we must also take account of the costs of braking and the costs of re-positioning the limbs.
Gravitational torque and braking
While it is tempting to think that we might eliminate the costs of braking by landing with the foot immediately under the centre of gravity (COG), as mentioned above, under most circumstances, this hope is illusory because we must reverse the head-forwards-and-downwards angular momentum generated during the second part of stance when the COG is in front of the point of contact between foot and ground. If we ignore wind shear, the reversal this head-forwards this must occur when the foot is on the ground because the angular momentum of a body can only be changed by external torque acting on the body. (This is illustrated, apparently paradoxically, by the fact that an airborne high-board diver spins faster as he wraps his body into compact ball. A compact ball has a smaller moment of momentum – the equivalent of mass for a rotating body – and hence must spin faster if it cannot transfer momentum to an external object.). Gravity cannot exert torque on a freely falling body. The wikipedia article on angular momentum states: ‘Angular momentum is conserved in a system where there is no net external torque.’
The observational evidence that runners land at least a short distance in front of their centre of mass suggests that most of the compensation for the head-forwards angular momentum generated by gravitational torque in the second half of stance is provided by an opposite torque acting the first half of stance. An alternative possibility, suggested by Simbil in his comment on my blog posting on 24th February is that a backwards push in late stance reverses the head-forwards angular momentum generated in the second part of stance. I believe Simbil’s proposal would lead to a violation of the law of conservation of linear momentum, but for the time being, this debate between Simbil and myself has not been completely resolved. Nonetheless, there is little doubt from observational evidence, that all runners do land with the initial point of support in front of the COG, thereby providing for at least part of the reversal of the head-forwards rotation.
Landing in front of the COG inevitably results in a braking force because the leg is angled forwards and down at the point of impact, and therefore pushes obliquely against the ground. The ground reaction force pushes back and up. The backwards component of this push acts as a brake. While the energy cost of braking can be reduced by spending only a short time of stance, and thereby producing less braking and angular momentum due to gravitational torque, the price is the need for a greater vertical push on the ground. Because the upwards impulse exerted by vertical ground reaction force, averaged over the entire gait cycle must be equal to body weight, a shorter proportion of the time on stance demands a greater the vertical ground reaction force during the time on stance. If one third of the gait cycle is spent on stance, the average vertical ground reaction force during stance must be three times body weight; the peak might be substantially higher.
An additional crucial issue is the fact that the muscles and tendons can absorb some of the energy of impact at footfall, and this elastic energy can be recovered at lift off, thereby providing part of the energy necessary to get airborne. Typically, about 35% of the stored energy can be recovered. However the capture and release of elastic energy takes time as the tissues must stretch and then contract again. It is difficult to calculate the minimum time required for this stretching and contraction because muscle and tendons are actually viscoelastic- that is, their elastic properties change as the load changes. They are quite stiff when loading is rapid but less stiff when loading is slower. Observational evidence suggests that it is necessary to spend at least 70-80 milliseconds on stance if elastic energy is to be recovered and re-used efficiently.
The conclusion so far is that time on stance should be as short as possible down to a limit of around 70-80 milliseconds. However, a stance time as short of this will necessitate the strength necessary to sustain high vertical forces. If cadence is 200 steps per minute, step time is 60/200 sec (= 0.3 sec = 300 milliseconds). If time on stance is 70 milliseconds at this cadence, the average vertical ground reaction force during stance is 300/70 (= 4.3) times body weight. Because Newton’s third law demands that action and reaction are equal and opposite, the foot must push against the ground with a force of at least 4.3 times body weight
Repositioning the limbs
The energy cost of repositioning the limbs will increase as running speed increases, although a full explanation requires some mathematics. Peluko presented the mathematics in a comment on my blog a few weeks ago. The essence of the argument is that at higher running speeds (and approximately fixed airborne time) the foot must accelerate faster to catch up with the torso because airborne body follows a longer trajectory. Rapid acceleration requires a large force. As this larger force acts for a similar length of time, the transfer of momentum to the leg and foot will be greater and will consume a larger amount of energy. I believe that in his calculations Peluko over-estimated the magnitude of the increase because he did not allow for the fact that the action of pulling the foot towards the torso actually slows the torso a little. Thus some of the energy required for accelerating the foot can be extracted from the kinetic energy of the moving torso. This kinetic energy is then returned to the torso in late swing when contraction of the hip extensors pulls the swinging leg back towards the torso. Nonetheless, although the cost of repositioning the legs is not as large as Peluko estimated, I agree that there will be a substantial relative increase in energy cost per Km with increasing speed.
Direct observation indicates that the total energy cost of running per Km (which can be computed fairly accurately by measuring fuel consumption) remains almost constant as speed increases. This suggests that the saving on energy (per Km) required for getting airborne as speed increases is consumed in the increased energy cost of repositioning the limbs.
There are at least three major factors that create a demand for muscle strength and power if we wish to run fast.
1) the ability of the extensor and flexors of the hip, knee and ankle to absorb a substantial portion of impact energy as elastic energy and to release this by a controlled approximately isometric contraction in late stance to help propel the body upwards.
2) A powerful concentric contraction of the hamstrings (hip extensor and knee flexor) to initiate the upward swing of the foot towards the buttocks followed by powerful concentric contraction of the hip flexors to accelerate the trailing leg and foot forwards to overtake the torso in mid-swing.
3) A powerful eccentric contraction of the hip extensors to arrest the forward swing of the leg so that at footfall the foot is only a short distance in front of the COG.
Of course we require not only strength and power, but also exquisite neuromuscular coordination to achieve these actions in a timely manner. I believe that when we are running, we should not focus our attention on these individual actions, but instead devote our conscious attention to maintaining a sufficiently rapid cadence while relaxing all non-essential muscles. Provided we have developed the required strength, power and coordination, stride length will be adjusted automatically.
It is clear that despite being able to run fairly efficiently at slow speeds at present, I do not have the strength, power and coordination required to maintain an adequate stride length at high speed. I had originally intended to outline my plans for developing the required strength, power and neuromuscular coordination in this posting, but the story is already more than long enough to be digested at a single sitting, so I will defer the remainder of the story until early next week.