Could someone explain the variables of this equation to me?
April 12, 2020 9:39 AM Subscribe
Modelling an exponential curve to fit a time history of tree sway data:
a*exp(x*(t-t0))*cos(w*(t-t0)*pi/180)+b+c*(t-t0)
Above is the equation I'm using in R. It is based on this equation: . In one three-hour marathon session of data crunching with three PhDs professors, we threw it together. It isn't working, but the bigger issue is that I don't understand it. Help me understand it, please..
The changes from the original equation are because:
1) "t0 is necessary to better fit the later parts of the data on which you are focusing, allowing a decay that might start at a slightly different time than t = 0, which is built-in to the original form of this."
2) "the form of the function should be cosine rather than sine, so that it’s equal to 1 at t = t0"
This is what I've come up with so far:
am now a bit confused about what each variable is. I would expect the exponential coefficient (-x*(t-t0)) to have three variables:
d / ξ = damping ratio
x = natural frequency
t = time
t-t0 gives the period, not the frequency, so time and natural frequency don't seem to be in there at all.
I am also confused about the cosine argument. I would expect:
w = damped natural frequency
t = time [though, since this is actually the "system response", is it against for the period?]
Φ = ???
Basically, I need to find the experimentally determined damping ratio for trees that go from swaying to stillness. I have time histories for each tree. Above is the equation that we want to fit to those time histories. This is for my Master's thesis. Long story short: I'm an Urban Forestry major. When you're studying the properties of tree dynamics, it helps to have a degree in structural engineering. I do not have one. Yes, I've talked to my advisor about this. It just isn't sticking
Above is the equation I'm using in R. It is based on this equation: . In one three-hour marathon session of data crunching with three PhDs professors, we threw it together. It isn't working, but the bigger issue is that I don't understand it. Help me understand it, please..
The changes from the original equation are because:
1) "t0 is necessary to better fit the later parts of the data on which you are focusing, allowing a decay that might start at a slightly different time than t = 0, which is built-in to the original form of this."
2) "the form of the function should be cosine rather than sine, so that it’s equal to 1 at t = t0"
This is what I've come up with so far:
am now a bit confused about what each variable is. I would expect the exponential coefficient (-x*(t-t0)) to have three variables:
d / ξ = damping ratio
x = natural frequency
t = time
t-t0 gives the period, not the frequency, so time and natural frequency don't seem to be in there at all.
I am also confused about the cosine argument. I would expect:
w = damped natural frequency
t = time [though, since this is actually the "system response", is it against for the period?]
Φ = ???
Basically, I need to find the experimentally determined damping ratio for trees that go from swaying to stillness. I have time histories for each tree. Above is the equation that we want to fit to those time histories. This is for my Master's thesis. Long story short: I'm an Urban Forestry major. When you're studying the properties of tree dynamics, it helps to have a degree in structural engineering. I do not have one. Yes, I've talked to my advisor about this. It just isn't sticking
Best answer: So, the exponential portion of this equation is the decay part. The part with periodic behavior is going to be the sine or cosine portion. sometimes you will see folks represent this by using imaginary numbers (i) in the exponential, so it is possible to use an exponential for periodicity, but that doesn't seem to be what's done here.
t-t0 isn't making anything periodic. It's a straight replacement for t, just a way of defining your time axis as starting at a given point. Like, how do you put in a number for t? It's number of days, or minutes or hours since _____. Using t0 just makes "since _____" explicit.
posted by Lady Li at 10:29 AM on April 12, 2020 [4 favorites]
t-t0 isn't making anything periodic. It's a straight replacement for t, just a way of defining your time axis as starting at a given point. Like, how do you put in a number for t? It's number of days, or minutes or hours since _____. Using t0 just makes "since _____" explicit.
posted by Lady Li at 10:29 AM on April 12, 2020 [4 favorites]
Best answer: Φ is just the phase of the cycle at t0. Since you don't have that term in there and are using a cosine function, t0 is the time when it's the swaying is all the way to the positive side in your example.
t-t0 is just the time since elapsed since t0. Often times when just working out theory, people just use t, and assume t0 = 0 is when everything started.
x is just the rate of decay in terms of t^-1, so it contains both the damping ratio and the natural frequency from the equation you linked. If you don't need to know either of those things specifically, there's no real reason to split it up into those terms. So x = ζ ω_n from the linked paper.
To be honest, I'm not quite sure why there is a factor of pi/180 in the consine function. That would most likely be the ratio of radians/degree in a cycle, and so convert degrees to radians. I have never seen a w that was in terms of degrees/sec instead of radians/sec though, so it seems an odd thing to do. Then again my background is in physics and not engineering, and engineers use weird (to me) units sometimes. Is it perhaps supposed to be the other way around 180/pi because the cos function in the program uses degrees for whatever odd reason?
b and c seem to be the coefficients some arbitrary linear function. I'm not sure if there's a physical motivation for being there?
posted by Zalzidrax at 10:34 AM on April 12, 2020 [2 favorites]
t-t0 is just the time since elapsed since t0. Often times when just working out theory, people just use t, and assume t0 = 0 is when everything started.
x is just the rate of decay in terms of t^-1, so it contains both the damping ratio and the natural frequency from the equation you linked. If you don't need to know either of those things specifically, there's no real reason to split it up into those terms. So x = ζ ω_n from the linked paper.
To be honest, I'm not quite sure why there is a factor of pi/180 in the consine function. That would most likely be the ratio of radians/degree in a cycle, and so convert degrees to radians. I have never seen a w that was in terms of degrees/sec instead of radians/sec though, so it seems an odd thing to do. Then again my background is in physics and not engineering, and engineers use weird (to me) units sometimes. Is it perhaps supposed to be the other way around 180/pi because the cos function in the program uses degrees for whatever odd reason?
b and c seem to be the coefficients some arbitrary linear function. I'm not sure if there's a physical motivation for being there?
posted by Zalzidrax at 10:34 AM on April 12, 2020 [2 favorites]
b and c seem to be the coefficients some arbitrary linear function. I'm not sure if there's a physical motivation for being there?
If your data is not exactly centered at 0, or if there's a constant rate of drift in the data on top of the oscillations, then including these parameters in your target function will help the quality of the data fit.
posted by Johnny Assay at 11:11 AM on April 12, 2020
If your data is not exactly centered at 0, or if there's a constant rate of drift in the data on top of the oscillations, then including these parameters in your target function will help the quality of the data fit.
posted by Johnny Assay at 11:11 AM on April 12, 2020
Response by poster: Thank you a thousand times, Lady Li and Zalzidrax. You saved my bacon. This makes so much more sense now.
The arbitrary linear function was trying to deal with the fact that this data doesn't center on the x-axis, and so we were trying to correct for it.
posted by Alex Haist at 11:12 AM on April 12, 2020
The arbitrary linear function was trying to deal with the fact that this data doesn't center on the x-axis, and so we were trying to correct for it.
posted by Alex Haist at 11:12 AM on April 12, 2020
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posted by Alex Haist at 9:40 AM on April 12, 2020