friedlander's equation calculations

[AnswersPlease]

I am having trouble understanding how D-Plot calculates it's variables for the Friedlander's eqn. When I use my own experimental peak pressure and time of arrival (not the ones automatically generated, defaults), my curve is vastly different than the one D-Plot generates. Any help on what's going on or possible resources available?

Thanks

[DPlotAdmin]

The most important aspect of getting a good fit with that equation form is to uncheck the Fit all points box and use the arrival time for the From X value. If you don't do that, data values before the arrival time will be included when attempting to find a fit and the answer will be nonsense, usually with a very low correlation coefficient.

If this is no help then it might be helpful for you to send me a data file that's giving you problems.

As an aside, this comes up often enough that I probably need to modify the plugin to ignore values before a given iteration's assumed arrival time. I'll give that a try before the next release.

[DPlotAdmin]

PS: Something else I should have mentioned: If your pressure record has a substantial negative phase then you should also limit the To X value to the arrival time + positive phase duration.

[AnswersPlease]

Thanks so much for your help. Is there any way you could help me understand how D-Plot calculates the four parameters. I still don't understand that as much as I would like.

[DPlotAdmin]

If you mean the initial values, that's fairly simple. The maximum pressure is exactly that - the maximum of all Y values. The initial time of arrival is the time at which Y(i+1) - Y(i) is greater than 5% of the maximum pressure. This falls apart for noisy records but generally works well. The initial value for the positive phase duration is the time after the arrival at which the pressure goes negative (if it ever does) or the maximum X. The initial value of the decay coefficient is the only one that requires much processing: first the plugin integrates the input to get impulse, then finds the decay coefficient that, given the initial values for peak pressure, time of arrival, and duration will give that impulse. Keep in mind that these are guesses only and will be improved on by the solution procedure, assuming that your data is really a pressure-time history and isn't exceptionally noisy.

If instead you mean the solution procedure, that's quite a bit more complicated and is way more than you want to read here. The short version is the same technique (the name of which escapes me at the moment) that can be found in Numerical Recipes and other math references is used for all equation forms. It makes use of the derivative of the equation with respect to each coefficient at every input X. The goal of the solution procedure is to minimize the sum of the squares of the errors at each data point. It's an iterative procedure that will stop once it can no longer improve on the solution or when the maximum number of iterations is exceeded. If the initial guesses are reasonably good, it will generally find the "correct" answer in a small number of iterations. On the other hand if the initial guesses are bad it will likely blow up with an overflow or underflow error, particularly with exponential forms like Friedlander's equation.

[AnswersPlease]

I was inclined to think that it was a noise issue, and I think you've confirmed my thoughts. I appreciate your help on this matter. Thank you very much.