Solving process

With the three initial conditions described in the previous section, the equations on the loop diagram (figure 2.5) below can be solved numerically over pre-set simulation times, in order to determine the time-course of the behaviour for a particular system. A diagram of the solving process is only shown for the simplest one of the MTI models presented in this thesis (linear motion load with linear properties) (figure 2.5). Although the solution process is explained for this simplest model only, more complex models including CC activation and hyperbolic CC force-velocity and SEC force- extension relationships and gravity have been solved in exactly the same way. The explanation given here provides a fi*amework onto which the solving process of the more complex models in this work can be understood.

Firstly, the duration of the simulated contraction is chosen such that the value of

— ( Xc c ) its final asymptotic value within at least six decimal places. A

d z

simulated contraction, which is long enough for this to occur, is called in this thesis a ‘complete contraction’. It is recognised that anatomical constraints are important in setting limits to the extent of MTC shortening inside the body. Such constraints were

the mechanical output of shortening MTI systems. The potential importance of anatomical constraints in the conclusions drawn in this thesis is discussed in later sections. The number of observations {N) to be generated within the selected duration is then chosen so that the peak force is within at least three decimal places of its peak value for that contraction. The time interval, Ar, between subsequent observations is equal to the selected duration divided by 7V-1.

= 0 otherw ise

d z

d z _{d z}

• d z

n -

Figure 2.5. Flow diagram of the numerical solution process for the MTI system with linear motion and properties. « is a range variable indicating the order of the terms of the sequence generated by the solving process. «=0,I..,iV-l, where Id is the total number of terms to be generated for a given duration of simulation. The values of L acceleration and velocity, SEC extension and time are retained (as indicated by the black arrows).

Calculation starts using the values set in the initial conditions at time Tq= 0 . According

to the third condition SEC extension is zero. The top equation of the loop calculates the magnitude of the dimensionless force from the dimensionless SEC extension (equation (2.32)). As already shown in the initial conditions the initial value of force in the system will be zero. This value of force is then fed into the second equation from the top (equation (2.34)), which calculates the normalised acceleration of L as a function of the dimensionless force and inertia. Notice how the first value obtained from this equation is zero, because the initial force in the system is zero at this time. The third equation from the top represents the dimensionless velocity of L and it is simply the integral of equation (2.34) with respect to time. The first value at that level is also zero because the initial values of force and load velocity are zero. The first results from these three steps in the loop apply for time Tq= 0 . Once the values of load

acceleration and velocity have been obtained for the onset of the contraction, the extension of the SEC at the next time instant can be calculated. The fourth equation initiates calculation of the next term of the calculated variables. Thus, after the results at Tq= 0 have been obtained, calculation at ri= A r starts, as shown by the fifth equation

which calculates the new time by adding on the previous time the value of one time interval. The bottom equation in this diagram is a rearranged form of equation (2.31) calculating the new dimensionless SEC length change. In this equation, the dimensionless CC and L movements are obtained by integration of the corresponding velocities with respect to time over the duration of the simulated contraction. Notice that the first result value of this calculation is not going to be zero, but T length units instead. The new value of the dimensionless SEC length change is then fed back to beginning of the loop to calculate the new dimensionless force in the system. Notice

how the second iteration gives values different to zero for all equations, as a result of starting from a level of force greater than zero. In this iterative manner a pre-set number of points can be generated over the desired simulated contraction duration thereby obtaining the time course of the SEC extension, force, load acceleration and velocity. This process is carried out using the mathematical software mathcad version 7. This process of obtaining the time course of SEC extension, L acceleration and velocity is common for all models, although the number sequences of these variables are generated from equations which are specific to each model.

It was mentioned in a previous paragraph that the duration of each simulated contraction is based on the final value of the CC shortening velocity. As shown above the CC shortening velocity is not calculated in the loop. However, the values of all the dimensionless variables of interest during the time course of the simulated contraction, including the CC shortening velocity, can be calculated from the dimensionless SEC extension and L velocity results. This is achieved as follows:

CC shortening. By inputting the values o f Xi and XsEC equations (2.31) and solving for Xcc •

Force in the CC and SEC. From equations (2.32) by inputting the values of XsEC • CC shortening velocity. By inputting the CC (or SEC) force values in equations (2.33).

SEC velocity: As the sum of CC and L velocity. (In the rotational models L velocity must be expressed relative to the point of SEC attachment on the lever).

Mechanical work generated by the CC {Scc)- From the force and CC shortening according to equation:

X cc

% C - I^CC ■ ^ Z c c (2.56)

SEC elastic potential energy { s From the force and SEC length change according to equation (2.57):

X S E C

^SEC - l^SEC ' ^XSEC (2-57) 0

Kinetic energy of L{ Si ). From the velocity of L according to equation (2.58):

S r = —

d r i x i )_/

\2 (2.58)

Power generated by the CC ( Pqq ). As the product of force and CC shortening velocity:

PCC = ^CC ' ~ ^ { X c c ) (2 59)

d r

SEC Power ( Pspc )- Product of force and SEC extension velocity:

P S E C - 9SEC * ~T~ { Z s E C ) (2-60)

d t

Power delivered to L{ p ^ ). Product of force and L velocity:

Pl ~ ^l ' ~T^^L ) (2 61)

d r

A similar approach can be used to obtain solutions for the more complex models by using the appropriate equations. All calculated variables are dimensionless. The time course of the mechanical behaviour of the components of MTI systems is presented

required to illustrate this behaviour over a wide range of different systems. It is, however, easy to revert back to the dimensioned system simply by multiplying each variable by the appropriate normalising factor.