Dancing with the devil: the laws of the dance, part 1

As discussed in the introduction to this series (posted on 18th March 2008) the essence of running is locomotion in which the length of stride is increased by becoming airborne for a part of each stride. In this article e will consider the constraints that the laws of Newtonian mechanics place on how we run. These laws apply whatever running style we adopt. The laws do not tell us which muscles we should use to achieve our goal, but they do provide guidance to help answer questions about optimum cadence and stride length and the relative proportion of each stride that should be spent airborne for optimum efficiency and safety. Before starting, we should define a few of the terms we will use.

The gait cycle
The full gait cycle covers the period from the time at which one foot contacts the ground (foot-fall) to the next time point at which that same foot contacts the ground. For ease of description, we will assume that this foot is the right foot. During the cycle, there are several phases. At first, the right foot remains stationary on stance while the torso passes forwards over it. Once the torso has passed over the point of support (usually located under the forefoot), the hip extends backwards until the point of lift-off is reached, initiating the swing phase for the right leg. The first airborne phase continues until foot fall of the left leg. While the left leg is on stance, the right leg continues to swing forwards. Shortly after the left leg lifts from the ground, the right leg reaches it forward most point of travel relative to the torso and then drops to the ground. At footfall of the right foot, the full cycle is completed. It contains one period of stance for each foot and two airborne phases. Note that the swing phase for one leg includes two airborne phases and also the period while the other leg is on stance.

Stride length is the distance on the ground from where the right foot contacts the ground to the point where the left foot contacts the ground. Cadence is the number of strides per minute. (Note that some people define cadence as number of gait cycles per minute, giving numerical values half as large as the values we will quote.) Speed is obtained by multiplying stride length by cadence. For most runners, cadence is approximately constant through the much of their range of speeds, and is typically 180 strides per minute. At this cadence, a stride length of 1 metre corresponds to a speed of 180 metres per minute (which is a little slower than 1Km in 5 minutes or 1 mile in 8 minutes.) Speed can be increased at constant cadence by increasing stride length. At a cadence180, a stride length of 2.2 metres corresponds to 4 minute mile pace is .

Efficiency and safety
We run more efficiently when we consume less energy per kilometre at a given speed. Efficiency can be quantified as the energy required to run a fixed distance at a particular speed. The higher the efficiency the lower the energy required.

Safety refers to running with low risk of injury. In general, risk of injury increases with increasing magnitude of forces applied to body tissues, though factors such as the direction of application of force, and the rate at which forces are applied play a large part. Also, in light of the fact that running often involves thousands of repeated impacts, that it is important to note that repeated application of relatively small forces than are well below the level required to break a bone or tear a muscle can cause stress fractures of bones or repetitive strain injures to muscles and other connective tissues. Nonetheless, if our goal is running safety, in general we are aiming to minimise the size of forces exerted on body tissues and the abruptness with which they are applied.

The tasks of running

Running at constant speed on a level surface demands the execution of three main tasks:

1) Maintaining constant forwards velocity

2) Propelling the body upwards to initiate each airborne phase

3) Moving the legs forward to provide support on landing.

Maintaining constant forwards velocity
Newton’s first law of motion a states that a body will continue in a state of uniform motion at constant velocity unless acted upon by a force. That is, the body maintains constant forward directed momentum unless acted upon by external forces. To influence forward momentum, these forces must have a component acting the either the forwards or backwards direction.

The external forces that act on a running body are:

Gravity
Although gravity acts on each part of the body, for the purpose of estimating the overall effect of gravity on the body, the force of gravity can be treated as acting through the general centre of mass (gcm) of the body, which is also called the centre of Gravity (COG). Although the anatomical location of the cgm moves slightly within the body as the legs move relative to the torso, the COG is always in the vicinity of the midpoint of the line that joins the iliac crests (the prominent curved bony ridge above the hip and just below the waist level on each side of the body); it is the top edge of the side of the pelvis.) Because gravity act downwards, it cannot directly produce acceleration or deceleration of the body in a forwards or backwards direction. Thus, gravity does not directly produce any change in forward momentum.

Vertical ground reaction force
The vertical ground reaction force (vGRF) arises as a reaction by the ground to the downwards forces exerted by the body via the legs and feet on the ground. The downwards forces exerted by the body arise from the body’s weight; from active contraction of muscles pushing down; and from elastic recoil of stretched muscles and connective tissues. The upwards vGRF is due to elastic reaction by the ground as it resists compression by the body. Upwards ground reaction force is equal and opposite to the downwards force exerted by the body according to Newton’s third law. By virtue of acting vertically, vGRF cannot directly alter the forwards momentum of the body.

Horizontal ground reaction force along the y axis
When the legs are directed obliquely forwards and down as is the case in the early part of the stance phase, the ground reaction has a backward directed horizontal component (hbGRF) that exerts a braking effect on the body. The horizontal component of the force that the body exerts on the ground arises from muscle contraction pushing obliquely and/or from elastic recoil forces acting obliquely. The ground reaction force is generated by the force of friction, which we assume for the present discussion is adequate to stop the foot slipping. Similarly, when the legs are directed obliquely down and back as in the latter part of stance, the ground reaction generates a forward directed hfGRF, that tend to accelerate the body forwards. By convention, we regard the from front to back of the body as the direction of the y axis. Relative to this axis, hbGRF has positive values while hfGRF has negative values

Figure 1 is a diagrammatic illustration of typical force plate data demonstrating the ground reaction forces along the y axis for a mid-foot runner.

HorizontalGRF

The figure shows that in the early part of stance hbGRF rises initially and, after a brief drop at around 20 milliseconds, continues to rise to a peak at around 50 milliseconds and then falls to zero at 90 milliseconds after footfall. The drop at around 20 milliseconds is due to the fact that usually for a mid-foot runner, the point of support at foot fall is on the lateral edge of the foot a little in behind the ball of the foot. During the first 10 milliseconds the foot rolls inwards and the point of support moves towards the centre of the foot before the heel descends to the ground and the point of support moves backwards. As the point of support moves backwards, there is a brief drop in the forward component of force exerted by the foot and GRF exhibits the notch observed around 20 milliseconds, Then the point of support shifts forwards to the ball of the foot as the COG passes over it (at around 90 milliseconds). At that point the GRF is purely vertical and hbGRF is zero. In the remaining period before lift-off, the leg is angled down and backwards as the runner’s hip extends, and a hfGRF rises to a peak before finally falling to zero at lift-off. A more complete description of ground reaction forces is provided in the paper by Cavanagh and LaFortune (Journal of Biomechanics, 1980)

Horizonatal GRF along the x axis
Because the foot must be angled inwards if the point of support is to be under the centre of mass, the foot exerts a sideways (x axis) force on the ground that elicits an opposing sideways ground reaction force, hxGRF. However, assuming symmetry, the sideways forces exerted by one foot exactly balance those exerted by the other foot so there is no net sideways impulse averaged over the full gait cycle. In any case, because it acts sideways, hx GRF cannot affect forwards momentum.

Wind resistance
Except when running with a strong following wind, wind resistance mainly acts in a backwards direction on the body and tend to produce deceleration. Movement of the body in the vertical direction and movement of the limbs will also generate air drag, but these forces are usually very small, and in any case, tend to be reversed and therefore to cancel out over the duration of the gait cycle.

Other forces
The other forces that act during running, naming the forces generated by muscle contraction, and the elastic recoil forces generated when muscles, tendon and ligaments are passively stretched, do not act on the body. Rather than act either within the body or they act on the ground thereby generating the GRF. Because they do not act on the body, they do not directly cause acceleration or deceleration of the body.

Impulse
If a force F acts for a time t, then it can readily be shown from Newtons second law of motion (F=ma) that the force produces a change in momentum given by Ft. This product of force and time is known as the impulse delivered by the force.

Balance of forces
When running at a constant velocity, Newton’s first law requires that the impulse due to forward directed forces acting over the duration of each gait cycle must exactly balance the impulse due to backwards directed forces. (It should be noted that within a single cycle, the body is often off balance. Being off balance is probably one of the major stimuli to automatic movement of the legs to stop a face down crash.)

Backward directed forces are wind resistance and the backward component of ground reaction, hbGRF, which acts while the leg is directed obliquely forwards and down between footfall and the point at which the COG passes over the point of support. The only forward directed force is the forward directed component, hfGRF, which acts when the leg is directed obliquely down and back after the COG has passed over the point of support until lift-off.

These considerations reveal two important principles.

a) In the absence of wind resistance, the impulse due to backwards directed GRF must equal that due to forwards directed GRF. The period that the foot is on the ground before GOG passes over the point of support must be approximately equal to the period after the COG passes over the point of support.

If one lands with point of support directly under the COG, impulse due to hfGRF will not be balanced by a braking impulse , and the body will accelerate out of control. Thus, except in the presence of a substantial head wind, the advice to aim to land under the COG, commonly given by advocates of efficient running, is misleading. In fact, video recording of runners demonstrate that the foot does land in front of the COG even in individuals who aim to land under the COG.

b) In the presence of wind resistance, the drag due to the wind must be compensated for hfGRF which is a reaction to a downwards and backwards push by the leg on the ground. The push against the ground must be provide either by elastic recoil of muscles and connective tissues muscle releasing energy stored as elastic potential energy following the impact of footfall, or by active muscle contraction.

Gravitational torque
When the long axis of the body is leaning forwards (i.e. when the COG is in front of the point of support) gravity acts obliquely relative to the axis of the body. Therefore, there is a component of gravity at right angle to the body that can be considered to be acting at the site of the COG. If the body is on stance the foot is fixed, the component of gravity at right angles to the body is will exert a torque that tends to cause the body to rotate in a face forwards and downwards direction. This situation exists during the latter half of the stance phase.

If a face-down crash is to be avoided, this rotation must be reversed at some other point in the gait cycle. Torque can only be applied by an external force acting on the body. The only tow such forces that might reverse the rotation are wind resistance and the oppositely directed gravitational torque that will be generated when the COG is behind the point of support.

We have already seen that in order to avoid uncontrolled acceleration in the absence of a strong head wind, it is essential to land in front of the COG. The need to cancel rotation provides an additional reason why we must land in front of the COG.

Running on Ice
Horizontal ground reaction force is due to friction which arises in reaction to the horizontal component of forces acting obliquely down the leg . Friction is minimal on ice. Therefore, in order to run on ice, it is essential to spend a very short time on stance, so that the long axis of the body never becomes more than very slightly oblique while on stance. This demands a very short time on stance. As well shall see in the next section, very short time on stance is associated with large vertical ground reaction forces. Nonetheless, running on ice is possible, though it is a stringent test of the ability to lift the foot from stance quickly.

This article will be continued in a subsequent post.

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One Response to “Dancing with the devil: the laws of the dance, part 1”

  1. richh Says:

    Canute,
    Sounds quite reasonable so far, I eagerly await the next section!
    –richh

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