Math Dude

2015-06-26 02:49:17 UTC

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I have some numerically sampled data from a system that generatesRaw Message

cyclical force - length loops.

Such a system seems to be characterized well enough here:

https://en.wikipedia.org/wiki/Work_loop

I want to seperately determine any negative and positive work being done

per cycle. If the force-length loop for a given cycle does not cross

itself, then if the loop has a counter-clock-wise direction then the

work will be positive, and vice-versa for a clock-wise direction.

If the loop crosses itself, then the cycle will contain both negative

and positive work. The animated graph on the above page shows a

sequence of loops, where the red area indicates positive work and blue

is negative.

The first method for computing power and work is stated as follows:

=================

Method 1: Instantaneous power method[edit]

Step 1) Obtain muscle velocity by numerical differentiation of muscle

length data.

Step 2) Obtain instantaneous muscle power by multiplying muscle force

data by muscle velocity data for each time sample.

Step 3) Obtain net work (a single number) by numerical integration of

muscle power data.

Step 4) Obtain net power (a single number) by dividing net work by the

time duration of the cycle.

==================

I simulated a complex loop by mathematically generating one cycle of a

figure-8 (rotated 90 degrees and shifted fully into the positive X-Y

quadrant). The cycle is then composed of the lower loop and upper loop

- one loop has a clockwise direction, the other a counter-clockwise

direction.

Numerically computing the derivative for the X axis (representing

length) to give velocity, and then multiplying by the Y-axis value will

result in a negative result for part of both the upper and lower loops.

This would indicate or imply that during the counter-clock-wise loop (a

loop that is producing work or power) that some of the instantaneous

power results will be negative. If so, then getting a negative

instantaneous power result does not tell me that I am working with that

portion of the loop that is rotating clockwise (ie - the negative loop).

Is there a straight-forward way to take a time-series f(t) and l(t)

where f is force at time t, l is length at time t, from t1 to t2, and

process the series in such a way that I can obtain either the area of

any number of loops that emerge from the path taken by the data (or just

the sum of all clockwise areas and the sum of all counter-clock-wise

areas) without manually determining where in time each loop begins/ends

and the direction of the loop, or

Is there a straight forward way to perform the instantaneous power

method and arrive at the same result - to have a sum total of all

positive work and all negative work per cycle, given a loop with a

complex path that might cross itself one or more times?