The laws of the dance: part 2

Preamble

This posting is a continuation of the article examining the constraints that the laws of Newtonian physics place on how we run. In the introduction to this series of articles, posted on 18th March 2008, it was pointed out that being airborne for part of each stride is the defining feature of running. A posting on 21st March examined the nature of the forces acting on the body of a runner, and in particular, addressed the question of what is required to maintain forwards momentum. This posting addresses the implications of Newtonian physics for how a runner gets airborne.

The full series of postings is assembled in order of posting in the page ‘Running: a dance with the devil’ accessible via the side bar of this blog.

 

 

Getting airborne

While forward momentum can be maintained fairly easily provided the braking forces in the first half of stance are kept within reasonable limits, getting airborne is both consumptive of energy and risky.

Acceleration upwards requires a vertically directed force. The only external force that acts upwards on the body is vGRF (apart from a trivial contribution from air drag as the body falls). As discussed above, vGRF is the reaction of the ground as it resists compression by the vertical downwards force exerted by the body via the legs. These vertically downwards forces are body weight, and downwards push by contracting muscles or elastic recoil of stretch tissues.

 

How large is vGRF? When running on a level surface, the body is at the same height at the end of each gait cycle as it was at the beginning. Therefore, the net impulse due to vertical forces acting on the body must be zero. Apart form a trivial contribution from air drag as the body rises, the only external force acting downwards on the body is gravity (i.e body weight). If Ta denotes airborne time and Ts denotes time on stance, while W denotes body weight, the downwards directed impulse arising from gravity during each stride is W(Ta+Ts). vGRF acts only during stance. vGRF is not constant during the time on stance. Nonetheless, the upwards directed force is the product of average vGRF during stance by Ts. Thus, the requirement that net vertical impulse over the entire gait cycle is zero requires that:

average(vGRF) . Ts = W.(Ts+Ta).,

Therefore, average (vGRF) = W. (Ts+Ta)/Ts

 

If time on stance is equal to time airborne, average vGRF is twice body weight.

If time on stance is 66 milliseconds, as is recommended in the Pose Method, while cadence is 180 strides per minute (corresponding to stride duration (Ta+Ts) = 333 milliseconds, average vGRF is five times body weight. Peak vGRF will be even greater

 

Conclusion: If time on stance is only a small fraction of total stride duration, average vGRF is many times greater than body weight.

 

The central challenge of the dance with the devil
To elicit such a vGRF, a powerful downwards push by the leg on the ground is required. At least some of this downward push can be supplied by elastic recoil of tissues that were stretched at impact. However, it is likely that capture of the energy of impact and its subsequent recovery can only be performed efficiently if time on stance is quite short .

The improvement in efficiency with decreased time on stance (at all except very slow speeds) presents us with the central challenge of the dance with the devil. When time on stance is short, vGRF is high. Exerting a push that is several times body weight is likely to demand strong muscles and is very consumptive of energy unless a substantial proportion of the energy released by the impact at foot fall can be stored as elastic potential and recovered in late stance to hep generate the vGRF required to propel the body upwards. However, efficient capture of the impact energy and its subsequent recovery is likely to place substantial stress on muscles, tendons and ligaments, unless it is done very skilfully.

If forces many times body weight are applied very abruptly, the risk of injury is likely to be high. What is required is a finely controlled foot fall that results a large rise in vGRF over a relatively short time while avoiding a very abrupt rise.

Strictly speaking, this principle is not a direct consequence only of Newtonian mechanics, but also depends on the science of materials. The way in which structures fail when stress is applied is determined in part by the intrinsic properties of the material. For example, kangaroo tail tendon is more capable of absorbing stress without failure than any other material, but all mammalian tendon is fairly tough due to the properties of the collagen protein. The failure of a structure under load is also determined by extrinsic design of the structure. The way in which load can be transferred from the lateral edge to the medial arch of the foot in the first 20-30 milliseconds after foot fall provides increased ability to absorb stress. In the subsequent section on biodynamics we will address the question of what specific orientation of joints and tensioning of muscles might facilitate a relatively slow rate of rise of stress on foot fall. Another major factor to consider is cadence.

 

Optimum cadence

Because a freely falling body accelerates steadily under the constant influence of gravity, the downwards speed of the body at the end of the airborne period is much greater for a longer airborne duration. Thus, the total distance of fall during a series of many short duration airborne periods is less than that for a series of fewer longer duration airborne period of the same total duration, (This is demonstrated mathematically in the calculations page on the side bar of this blog). Thus a high cadence requires less energy expenditure in raising the body and less severe impact forces.

 

Interim conclusion

It is most efficient to run with a high cadence and short period on stance relative to airborne time, but the risk of injury is likely to be high unless joints are positioned and muscles tensioned at foot fall in a way that avoids very abrupt rise in vGRF

 

 

Moving the legs forwards to provide support on landing

The foot and leg must be accelerated forwards relative to the torso (and relative to the ground) in the early part of the swing phase, but then decelerated in the second half of swing so that at footfall the foot is travelling backwards relative to the torso. Once the foot is on stance, speed of the foot relative to the ground must be zero. Thus, during each gait cycle, the foot is accelerated from rest to a speed somewhat greater than the speed of the torso and then decelerated to rest once again.

The acceleration and deceleration can in principle be performed either by external forces acting on the foot (horizontal GRF) or by internal forces. Unless the body is on stance for an infinitesimally small time (which would necessarily be associated with huge values of vGRF) the legs must be angled obliquely forward and down during the first part of stance, resulting in a braking force acting on the foot and similarly, obliquely backwards and down during the second half of resulting in forward acceleration of the foot. Therefore at least some of the impulse required to accelerate and decelerate the foot and leg will be provided by external forces.

As discussed in the previous section, a short time in stance is preferable with a view to efficient capture of the energy of impact as elastic energy and the subsequent re-use of that energy to help propel the body upwards for the next airborne phase. When time on stance is short compared to airborne time, the amount of acceleration of the foot and leg necessary to allow the foot to overtake the torso in mid-swing is lower than when time on stance is a large part of total stride duration, For example, if time on stance is half of stride duration the average velocity of the foot must be twice that of the torso during the time from lift off to mid-swing when the swinging foot passes under the COG, whereas if the time on stance is one fifth of total stride duration, the average velocity of the swinging foot during the first part of swing need only be about 20% greater than the velocity of the torso.

 

These consideration demonstrate that despite the fact that some of the acceleration and deceleration of the foot and leg must be done by hGRF (which is a reaction to the horizontal component of push upon the ground during stance) it is preferable that the majority of the acceleration and deceleration should be done by contraction of muscles so as to pull the foot towards the torso. In the first half of swing, such a pull will accelerate the foot and leg; in the second half of swing, such a pull will decelerate the leg. It is also important to note that just as the magnitude of the forward impulse by external forces (hfGRF) must be equal to the backward directed impulse by external forces (nbGRF), similarly , the acceleration due to pull in the first half of swing must be matched by an equal deceleration by pull of the foot back towards the torso in the second half of swing.

 

Overall conclusions derived from the implications of Newtonian physics:

1) High cadence is beneficial

2) Time on stance should be small compared with airborne time (though at very slow speeds total energy cost actually increases as time on stance decreases while high cadence is maintained).

3) If time on stance is substantially shorter than airborne time, vGRF will be at least several times body weight

4) The impulse require to lift the body against gravity must be provided by a downwards push of the leg against the ground.

5) Skilful orientation of joints and tensioning of muscles at foot fall is required to minimize abrupt rise in vGRF

6) Landing in front of the COG is inevitable to balance the effects of the forward directed hGRF acting in the second half of stance.

 

7) The majority of the impulse required to accelerate the legs forward past the torso and then decelerate them to provide support at foot fall, is best provided by internal muscle action that pulls the foot forward towards the torso in early swing and backwards towards the torso in late swing.

 

 

These principles are essential for efficient running irrespective of the specific running style adopted. In the next section, we will address the question of which specific muscle actions are mostly likely to achieve these principles efficiently and safely

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9 Responses to “The laws of the dance: part 2”

  1. Simon Says:

    That’s all looking good – well done.
    The part about accelerating and decelerating the swing leg and its relationship with time on support has been very useful for me as it fills in a ‘blank’ that I understood intuitively but did not appreciate the actual relationship.

    One thing that might be considered is the swing leg acting as a pendulum and thus relating gravity in to the equation (at least to see if the effect is more than negligible) as well as the length of the pendulum effecting the angular velocity of the leg.
    The pendulum motion could be viewed to start when the foot leaves the ground and the leg will have a rearwards angular momentum pivoting from the hip. Assuming the mass of the body is much greater than the mass of the leg, we can ignore effects on the body and approximate it as an anchor point for the leg. The angular momentum will be acting against a gravitational torque and so will naturally lose momentum. Depending on the speed of running, the momentum may be reversed before the leg reaches a large rearward angle.
    ‘Pulling’ the foot has the interesting effect of shortening the pendulum (flexing the knee joint) which could, in theory, provide a much faster forward swing as its pendulum length has been shortened relative to the rearward swing.
    The leg is again extended during the front phase of the swing so is slowed by pendulum action.
    Unfortunately, the maths and inter-relationships for this would be well out of my league so I can only guess at the effects.

    Perhaps a more important factor is tissue impedance to the swinging of the leg at the hip. I believe this will happen naturally both forwards and rearwards especially at higher paces.
    As the leg swings rearward of the hip, the hip flexors will be elastically stretched which will slow and reverse the rearward swing. An efficient stretch shortening cycle could then be used to swing the leg forward. The forward swing would also eventually elastically stretch the glutes and hamstrings. This movement could also make efficient use of the elastically stored energy.

    In summary, I think the leg’s extension and flexion at the knee needs to be well orchestrated to benefit from the pendulum effects for controlling speed (angular velocity). I also think that muscle elasticity probably plays a large role in efficient leg swinging.

  2. canute1 Says:

    Simon, Thanks for your comment,

    Yes the swing leg might swing like a pendulum. at least while the opposite leg is on stance. When the body is in free fall during airborne phase the torso is falling, so gravity will not accelerate the leg relative to the torso as would happen with a pendulum suspended from a fixed pivot. However, while the opposite leg is on stance, the torso is supported, and the swing leg could simply swing like a pendulum. As you point out, the shorter the length of the pendulum the more rapid the swing. However, an approximate calculation (taking account of the fact that the leg is a complex pendulum in which the weight is distributed along its length with more mass per unit length near to pivot point at the hip) indicates that the natural periodicity of the human leg is probably a little to slow to match a cadence of 180 strides per min. So I think it is necessary to assist gravity by an active contraction of hip flexors until mid swing and then an active contraction of hip extensors (mainly hamstrings) in late swing. Thus I think the swinging leg is a ‘power assisted’ pendulum
    Canute

  3. richh Says:

    Canute,
    I hope you will consider the issue of why we get more tired running quickly than running slowly, even if cadence is the same. Is the energy expended per unit distance similar, but fast running taxes our ability to generate that energy at a rapid rate? Or do we expend more energy per unit distance when we run faster, and, if so, why?
    –richh

  4. Simon Says:

    Canute,
    Thanks for that – you made the point that support is needed for the pendulum effect which cannot happen whilst airborne which I had overlooked.
    Looking at a sequence of stills of a runner shows that there is no point of support when the trailing foot is removed from the ground and support is usually not present again until the leg is swinging with an angle of 90* to the horizontal. This means the entire rearward part of the swing is supportless and that just leaves the front half of the swing as a supported swing.
    We can see the muscle activity here:
    http://books.google.co.uk/books?id=8MxEuq7OEqQC&printsec=frontcover#PRA1-PA315,M1
    Looking at the top graph (hip) in between toe off and foot strike we see that the muscular activity does indeed shift from hip flexion in the first half of swing (pulling the leg forwards) to hip extension in the second half of swing (pulling the leg backwards).
    Another striking thing about the graph is that hip flexion begins well before support is removed at ‘toe off’. This may come in useful for your next chapters where you look at muscular activity (note though that the graphs are for sprinting and may be accelerating rather than constant speed).

    All that is to say I agree with you – it’s a power assisted pendulum rather than one that acts only from the forces present in support.

  5. canute1 Says:

    Simon, Thanks. Sprinters will exert greater force that distance runners.
    I think that most of the patterns of muscle activity will be similar, but less extreme in distance runners.

    As you point out, the diagrammatic representation of muscle activity during the gait cycle in Vescovi’s chapter in ‘Conditioning for Strength and Human Performance’ demonstrates that there is hip flexion in early swing and hip extension in late swing, in accord with the concept of the power assisted pendulum.

    Also you point out that hip flexion begins well before toe off. This is an eccentric contraction of the hip flexors. In late stance, momentum carries the torso forwards over the anchored foot, causing a passive extension of the hip, so there is the apparent paradox of active contraction of the hip flexors while the hip is still extending. This eccentric contraction of the hip flexors primes them to perform a very rapid concentric contraction as soon as the foot lifts off, thereby accelerating the leg forwards to overtake the torso. I would expect this action to be much more exaggerated in sprinters than in distance runners
    Canute

  6. richh Says:

    Canute,
    As a test of your way of looking at efficient running, it would be interesting to explain why we get more tired running quickly than running slowly, at the same cadence.
    –richh

  7. canute1 Says:

    Richh
    At higher speed but same cadence, stride length must be longer. The increase in stride length might be achieved by one (or both) of two adjustments: 1) spend a higher proportion of time airborne, This will require more work against gravity to compensate for the increased time of free fall; or 2) accelerate the leg forwards more forcefully at lift-off and then decelerate it in late swing. Both of these adjustments will consume additional energy. So we will use more energy and feel more tired.

    Canute

  8. richh Says:

    Canute,
    I suspect the first factor is more significant, since the entire mass of the body is involved rather than just one leg. Could increased time on support give longer stride length, and hence more speed, without increasing air time? Is the height of vertical displacement – “bouncing” – a major factor to be minimized in order to reduce energy use?
    –richh

  9. canute1 Says:

    Richh
    I agree with you that adjustment of airborne time is likely to be more useful than adjusting forward acceleration of the legs.

    I am not sure that we can increase speed by increasing time on support, except at very slow speeds when a short time on support is inefficient. If a constant cadence and constant short time on stance are both maintained at low speed, then airborne time per stride will remain the same as at higher speed and hence work against gravity will remain the same despite a shorter stride. This is likely to be inefficient. If time on stance is increased, airborne time will be less, and the and less work will be required to compensate for free fall, so efficiency would improve. Therefore, a higher speed could be maintained at a given energy consumption. However, this situation only applies at very low speeds.

    At higher speeds, the energy consumed in generating horizontal GRF becomes appreciable and this energy cost increases with increasing time on stance. So total energy consumption is minimized by employing a shorter time on stance. Although we can decrease the amount of energy wasted in generating horizontal GRF by decreasing time on stance, the increased airborne time does demand more work against gravity, so maintaining a high speed necessarily entails high energy costs. In our dance with the devil there is no free lunch

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