You also seem to be questioning the validity of an equation describing displacement/time,
No, just one that uses sums of displacements with individual magnitudes ranging into the hundreds of miles, to derive actual displacements of up to about 300 feet.
which is a bit odd bearing in mind that NIST calculated one...
z(t) = 379.62{1-exp[-(0.18562t)3.5126]}
Which Norse Gods is that one a battle between ?![]()
Glad you asked.
The argument of the exp function will range from zero, at t = 0, to a very large negative number at very large values of t.
exp(0) = 1, and
exp(a very large negative number) -> 0
and the function remains inside that interval for all values in between.
So, {1 - the exponential part} is 0 at t = 0, and 1 at a large t.
This makes it obvious right away what that first parameter 379.62 represents: the maximum displacement the model can generate, while the rest represents the fraction of that maximum displacement that has occurred as a function of time t. In other words, the displacement scale. We can regard the displacement units as being attached to that coefficient.
Now consider the .18562. We can see right away that it's multiplied by t. So if the number were doubled, it would have the same effect as doubling all values of t in the data. So that parameter establishes our time scale. Changing it would be like stretching or squeezing the x (time) axis of the resulting graph.
But what does the value actually mean? One to any power is one, so when (.18652 * t) = 1, the exponential is exp(-1), regardless of the value of the third parameter. Exp(-1) is 1/e or 0.3679. So after 1/.18652 = 5.36 seconds, the displacement will have reached about (1 - 0.3679) = .6321 of the maximum. This establishes the time it takes the model to displace to 63% of the maximum (a standard benchmark, closely related to the "time constant" of decay functions): in this case about 5 seconds.
Since those two parameters establish the x and y (time and displacement) scales of the model, only the third remains to refine the shape of the curve. The higher that value, the slower the displacement increases initially and the faster it increases near and after that 63% benchmark. That value is dimensionless (time and displacement already being accounted for in the other two parameters) but is proportional to the slope of the curve where t = the "time constant." If it were zero, the whole displacement curve would be a horizontal line at 63% of maximum displacement (never dropping off at all); if it were a large positive value, the displacement would stay very close to 0 up until t reached the "time constant" and then increase suddenly to the maximum.
So to sum up, the model's three parameters represent displacement scale, time scale, and the general distribution of movement over the time period.
Of course, their linear model is better, for the time period for which it was derived, because it represents not only the characteristics of the displacement curve but the main physical mechanism causing it.
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Okay, your turn. What does the subtraction of 162.215785114699 * t3 from the total displacement mean in your model?
Respectfully,
Myriad


