Effects of constants

6.1.2.1. Effects of H

It can be visualised from figures 2.6-2.9, that the time course of the mechanical events during a contraction differs for different values of normalised load, H . This is because the slope of the load acceleration-force relationship changes when H changes (Figure 2.10).

TD 0.19 0.13 0 . 0 6 0 4 2 0

Figure 2.10. Normalised load acceleration-force relationship for different levels of normalised load, S . The slopes of the different lines are E~^. Notice how as

H tends to zero, load acceleration for a given force tends to infinity and as E

6.1.2.1.1. E and contraction duration Given enough time, any force, no matter how small, can accelerate an inertial load, no matter how large its inertia is, to any desired velocity. Recall that a simulated contraction in this thesis is considered to be ‘complete’ when the CC velocity attains its final asymptotic value within six decimal places. When the load is purely inertial this final value is equal to the maximal CC shortening velocity, V cc^ • ^ order for this condition to be satisfied, L must be moving at a velocity at least equal to at the end of a contraction. Otherwise the CC will be able to ‘catch up’ with the load and keep the SEC stretched, thereby maintaining a level of force in the system to a value that is greater than zero. This force would prevent attainment of by the CC. The greater the normalised inertia, E, the longer it takes for a CC with ‘fixed’ force-velocity generating capabilities to accelerate that inertial load from zero velocity to

in order for contraction to reach ‘completion’.

The fact that it takes longer for L to be accelerated to a given velocity as its inertia increases is shown in figure 2.11. The time at which reaches the value of 0.9 is represented on the graph. This is because the value of 1 is approached asymptotically with time and it is thus easier to resolve when a certain proportion of the CC maximal shortening velocity is achieved.

4

9 0 . 5

Figure 2.11. Logarithm of normalised time for CC velocity to reach 0.9 plotted as a function of the logarithm of H . Time has a resolution of 10'^ normalised time units.

Peak (p

or

Minimal ^ { x c c )

L og(S)

Figure 2.12. Peak normalised force (black points) and minimal CC velocity (red points) plotted as a function of the logarithm of the normalised inertial load 12 (0.015 ^ Z ^ 64). The blue curve is a cumulative probability function curve containing only two natural constants (see page 150).

6.1.2.1.2. *5* and peak force The peak force achieved during a contraction rises with the load as shown in figure 2.12. In the normalised model described in this section, the peak force is equal to the maximal extension of the SEC during a contraction.

SEC extension is equal to the sum of CC and L movement (equation (2.2)). A large inertial load accelerates less compared to a smaller one in response to a given force applied to it for a given period of time. This also means that the larger inertia also moves less than the smaller one during that time. The larger the inertial load, the less L moves and the more the SEC stretches to accommodate the CC length changes. As a result the peak SEC extension and hence the peak force, will be increasing with the load. As H however tends to infinity, the contraction will initially resemble an isometric contraction and the peak SEC extension and force would be close to their maximal value, which is one. Thus the peak force-logarithm of the load curve will be approaching asymptotically the value of one as S tends to infinity. As S tends to zero, acceleration of L to Vrr will be almost instantaneous and therefore SEC extension will be approaching zero.

The shape of the peak force-load curve resembles that of a cumulative frequency distribution fimction. An attempt was made to fit this curve with a cumulative frequency distribution fimction containing two natural constants, namely e and (figure 2.12, blue trace):

I lo g ( H ) —

The position and shape of this curve can be altered by inclusion of more constants in the above equation, in a similar manner as in a normal distribution curve. No exact match between the two curves could be found when a third constant was included. It can be seen on the graph that there is an asymmetry in the difference between the force-load points and the cumulative frequency function points away from the middle axis. As a result of this asymmetry the difference between the most extreme points in the two curves is less compared to the more central ones. As the exact mathematical description of the relationship appears to be complicated by the involvement of several factors, further investigation was postponed for some later work whose scope is outside that of this thesis.

6.1.2.1.3. E and at peak force

As a result of the relationship between normalised force and velocity (equation (2.33)), the relationship of the CC shortening velocity at the time of peak force (i.e. the minimal CC velocity) and the load is simply a reflection of peak force- load relationship across the axis ç? = 0.5 or =0.5 (figure 2.12). In other words the red points on this figure are the mirror image of the black points across a horizontal axis through the value of 0.5 units in the vertical axis.

6.1.2.1.4. E and maximal rate of force rise and decline As the initial level of force in the system is zero, the CC always starts shortening at its maximal shortening velocity. As L has inertia and its initial velocity is zero the initial rate of change of SEC length and hence force is

maximal and equal to one, independently of the magnitude of the load (figure 2.13; red points).

However, the maximal rate of force decline after peak force has been achieved, declines with the load (figure 2.13; blue points). This is due to the fact that larger loads accelerate less compared to smaller loads in response to the same force, as already mentioned above. As a result, the difference between the speed at which the CC shortens and that at which L moves during the SEC recoiling phase of the contraction is smaller, the greater the inertia of the load. As this difference reflects the speed at which the SEC recoils, it also reflects the rate at which force declines in this particular model with a linear SEC force-extension relationship. It can be seen from the graph that as the load tends to zero, the maximal rate of force decline tends to one. As the inertia of the load tends to infinity, the rate tends to zero.

0.75

P e a k ^ ( ^ )

■ 2 ■ 1 0 1 2

Log(H)

Figure 2.13. Maximal rate of force rise (red) and fall (blue) plotted as functions of the logarithm of the normalised inertial load. The absolute maximal rate values are shown (otherwise the maximal rate of force decline would be plotted on a negative Y-axis scale).

0.75

(P

0.2 5

0 0.2 5 0.5 0 .7 5 1

Peak (j)

Figure 2.14. Force at maximal rate of force development (red points) and decline (blue points) expressed as proportion of the peak force achieved in that contraction {(p') plotted as a function of that peak force. The line % is also plotted (green line).

6.1.2.1.5. *5* and force at which maximal rates of force rise and decline occur As explained at the beginning of the previous section, the peak rate of force development always occurs when the force in the system is zero independently of the peak force achieved during the contraction (figure 2.14). A more interesting relationship is that between the force at which the maximal rate of force decline is observed and the peak force. When the force at which the maximal rate of force decline is expressed as a proportion of the peak force achieved in the corresponding contraction ( cp' ), and this proportion is plotted against the corresponding peak force, a linear relationship arises. The slope of this relationship is equal to one (blue points and green line in figure 2.14). This means that the force at which the maximal rate of force decline occurs is equal to the square of the peak force. In this way the normalised force at which the maximal rate of force decline is occurring can be easily estimated. For example, if the peak force in a contraction is 0.25, the force at which the maximal rate of force decline occurred in that contraction is going to be 0.25 times 0.25, which is equal to 0.0625.

6.1.2.1.6. E and final

Recall that given enough time any purely inertial load, no matter how large, can be accelerated to any desired velocity by a force and that ‘completion’ of a contraction required L to be accelerated to a speed at least equal to that of Fcc„jax (PP- 148). Also recall that the larger the inertia, the smaller the difference between the speed at which the CC shortens and the speed at which L moves towards the centre of the system (pp. 151). This can be visualised by

2.15 that shows the maximal final load velocity plotted as a function of the load. As the final velocity of the load corresponds to the load velocity at the time at which CC shortening velocity is equal to one, within six decimal places precision, L velocity may not be exactly equal to its final velocity during each contraction. The difference however between the final L velocities shown here and the actual ones is expected to be very small. When the load is very light, it can be accelerated up to approximately twice the maximal CC shortening velocity (e.g. figure 2.9). As the load tends to infinity the maximal load velocity tends to one, i.e. to the CC maximal shortening velocity.

Final ~ { x l ) d r

-0.50

Figure 2.15. Maximal load velocity plotted as a function of the logarithm of the normalised inertial load. Dotted line corresponds to a normalised load velocity o f-1 . The minus sign indicates motion of the load towards the centre of the SEC. The Y axis is inverted.

6.1.2.1.7. E and maximal and 6:^

The maximal work performed during a contraction by the CC and the kinetic energy of L increase with the load (figure 2.16). As the SEC cannot generate mechanical energy by itself and the force during a contraction rises to a peak and then falls to zero, the kinetic energy of L at the end of the contraction is equal to the total mechanical work generated by the CC (figure 2.16).

l-H O u 15 X 03

Log(E)

Figure 2.16. Logarithm of the maximal mechanical energy produced by the CC during a contraction (red) or peak kinetic energy of L (blue) during the same contraction plotted as functions of the logarithm of the normalised inertial load.

6.1.2.1.8. 5" and maximal

The part of the SEC force-extension relationship utilised in a contraction differs for different amounts of normalised inertia (graph b; figures 2.6-2.9B). The peak elastic potential energy content of the SEC is the integral of the change in force with respect to the change in SEC length from zero to the maximal elongation achieved in a particular contraction. More simply, it is the maximal area under the SEC force-extension curve that was utilised during a particular

contraction. The greater the load, the greater the peak force and SEC extension and hence the elastic potential energy of the SEC as shown in figure 2.17.

0.6 a 0.45

0.15

0 • #