'Wheel-like' Mathematics (Modulating Trig Functions?)

[Chrono G. Xay]

As part of a personal musicology project I found myself with the mathematical model of a geometry which utilizes the equation

a*(a/b)^sin(pi*x)

The only problem with this is that I need to take the integral from -1/2 <= x <= 1/2, and according to Wolfram Alpha no such integral exists. I can take the derivative, of course, which is

cos(x)*ln(a)*a^sin(x)

and I can graph the thing and approximate the area under the curve, but it's incredibly annoying. Is it for situations such as these why Riemann Sums are even taught? (Not bashing on Riemann, it's just incredibly irritating.)

I can approximate the equation itself with

(1/2)[(a-a^-1)*sin(x)+(a+a^-1)

but it doesn't do it justice. Math people, please rectify this. XD

It's like it has a modulating frequency or something... like a 'chorus' effect for guitar.

Imagine drawing 'a/b' on a circle, including the dividing line between the numerator and denominator, and then slowly rotating it.

The equation came about when I was trying to calculate the length of chord segments when pivoting the initial chord around a point a certain distance from the center of a circle. What it results in is a fraction that changes like a rotating wheel... as though the concept of the equation has a 'visio-mechanical' aspect to it...

 

[jedishrfu]

While I couldn't integrate it, I found this site that plots it if you replace the a*(a/b) with some number like 3 for 3^sin(pi*x)

https://graphsketch.com

 

[Chrono G. Xay]

I think I found an article that will help: http://www.gigatronics.com/uploads/document/AN-GT102A-Getting_the_Most_Out_of_FM_in_the_Giga-tronics_2500.pdf

 

[Mark44]

As part of a personal musicology project I found myself with the mathematical model of a geometry which utilizes the equation

a*(a/b)^sin(pi*x)

This is not an equation. An equation always has = between two expressions.

The only problem with this is that I need to take the integral from -1/2 <= x <= 1/2, and according to Wolfram Alpha no such integral exists. I can take the derivative, of course, which is

cos(x)*ln(a)*a^sin(x)

And this isn't the derivative. I get the derivative as $a\pi \ln(\frac a b)\cos(\pi x) (\frac a b)^{\sin(\pi x)}$. The easiest you to do this is to write $y = a (\frac a b)^{\sin(\pi x)}$ and then take the ln of both sides of this equation. The derivative, with respect to x, of ln(y) is $\frac {y'} y$. The right side is easy to differentiate. After differentiation, multiply both sides by y to get y', the derivative.

and I can graph the thing and approximate the area under the curve, but it's incredibly annoying. Is it for situations such as these why Riemann Sums are even taught? (Not bashing on Riemann, it's just incredibly irritating.)

I can approximate the equation itself with

(1/2)[(a-a^-1)*sin(x)+(a+a^-1)

but it doesn't do it justice. Math people, please rectify this. XD

It's like it has a modulating frequency or something... like a 'chorus' effect for guitar.

Imagine drawing 'a/b' on a circle, including the dividing line between the numerator and denominator, and then slowly rotating it.

The equation came about when I was trying to calculate the length of chord segments when pivoting the initial chord around a point a certain distance from the center of a circle. What it results in is a fraction that changes like a rotating wheel... as though the concept of the equation has a 'visio-mechanical' aspect to it...

 

[Chrono G. Xay]

Whoops, that was the derivative of a^sin(x). I had posted this same basic question elsewhere, where the base had been further simplified to just 'a'.

 

[Chrono G. Xay]

The original purpose of this pursuit was to calculate the area of a circle using a non-radial(?) point.

 

[Mark44]

The original purpose of this pursuit was to calculate the area of a circle using a non-radial(?) point.

What is a non-radial point? Any point on a circle is at one end of a line segment from the center to that point; i.e., is on a radius of the circle.

 

[Chrono G. Xay]

A point that is not at the center, yet inside the circle; i.e., a method that does not use πr².

 

[Chrono G. Xay]

Anyway, I think it could be re-written as a wave with an amplitude modulating between a minima (below 'unity'(?) amplitude) of '(1/n)-1' and a maxima (above 'unity'(?) amplitude) of 'n' at intervals of π...

 

[Mark44]

What is a non-radial point? Any point on a circle is at one end of a line segment from the center to that point; i.e., is on a radius of the circle.

A point that is not at the center, yet inside the circle; i.e., a method that does not use πr².

Your description doesn't contain enough details for me to understand what you're trying to do. If you're given a circle of some unknown radius, and a point inside the circle, I don't see how you can find the area inside the circle. BTW, every point inside a circle lies along some radius.

 

[Chrono G. Xay]

How could one go about calculating the area of a circle without using πr², but knowing the radius and a point a distance from the center of the circle that lies within the circle?

Get your calculator out, run your graphing program, switch to polar graphing, plug in

"r=sqrt(6^2-2^2)*(sqrt(6^2-2^2)/(6-2))^sin(θ)"

then plug in

"r=12sin(θ)"

and compare the two.

Then switch back to Cartesian graphing and plug the first equation into y=, where θ=x. What I am looking for is:

1. An educated opinion on the behavior of the wave and

2. How to get the same result with a equation that doesn't have a trigonometric function in the exponent, as well as

3. A theory on how the first polar graph might need to be modified in order to transform into the equivalent of 12sin(θ) as 2 approaches 6.

 

[Mark44]

How could one go about calculating the area of a circle without using πr², but knowing the radius and a point a distance from the center of the circle that lies within the circle?

I realize this is obvious, but if you know the radius, that's all you need to calculate the area of the circle. You don't need a point inside the circle.

Get your calculator out, run your graphing program, switch to polar graphing, plug in

"r=sqrt(6^2-2^2)*(sqrt(6^2-2^2)/(6-2))^sin(θ)"

Where does this formula come from, and what is the significance of 2 and 6 in it? You mentioned that you know a point inside the circle. What are its coordinates?

then plug in

"r=12sin(θ)"

Where does the 12 come from?

and compare the two.

Well, they are either equal or unequal. More interesting to me is why you would want to do this in the first place, given that we know how to calculate the area of a circle, using only its radius (which you state is already known). You can also use integration to calculate the area under, say, the upper half of a circle, provided that you know the equation that represents the circle. Once you know the area of half of the circle, it's a simple matter to get all of the area, just by doubling the value you got for half the area.

Then switch back to Cartesian graphing and plug the first equation into y=, where θ=x. What I am looking for is:

1. An educated opinion on the behavior of the wave and

2. How to get the same result with a equation that doesn't have a trigonometric function in the exponent, as well as

3. A theory on how the first polar graph might need to be modified in order to transform into the equivalent of 12sin(θ) as 2 approaches 6.

What does "2 approaches 6" mean to you? This phrase doesn't mean anything to me, since is constant, and doesn't "approach" anything. It pretty much just sits there on the number line, midway between 1 and 3. You can talk about a variable approaching some number, though.

 

[Chrono G. Xay]

For my purposes I need to be able to calculate the area of a circle without using πr^2 (Just play along.)

The equation for the length of a chord running through a circle can be expressed as:

c=2*sqrt(r^2-d^2)

Next we can expand it into the sum of the lengths of two individual chord segments:

c=sqrt(r^2-d^2)+sqrt(r^2-d^2)

From there, being two individual chord segments, if we were to pivot a line about a point 'd' from the circle's center, we would notice the individual line segments on either side of the pivot point change in length at a particular rate:

(sqrt(r^2-d^2)/(r-d))^sin(x)

It took a lot of experimentation and meditation on the patterns I was seeing to deduce how I would need the line segments' lengths to change to come up with that equation; to the best of my existing knowledge it's purely of my own creation, and it works.

In my earlier example, r=6 and d=2

L=sqrt(6^2-2^2)*[sqrt(6^2-2^2)/(6-2)]^sin(x)

L=sqrt(36-4)*[sqrt(36-4)/4]^sin(x)

L=4*sqrt(2)*[(4*sqrt(2))/4]^sin(x)

L=4*sqrt(2)*sqrt(2)^sin(x)

When x=π/2...

L1=4*sqrt(2)*sqrt(2)^sin(π/2)

L1=4*sqrt(2)*sqrt(2)^1

L1=4*sqrt(2)*sqrt(2)

L1=4*2

L1=8

And when x=-π/2...

L2=4*sqrt(2)*sqrt(2)^sin(-π/2)

L2=4*sqrt(2)*sqrt(2)^-1

L2=4

L1+L2 = 12 = 2r

A circle of radius 6 has a diameter of 12

As d -> r (as "2 approaches 6"), c=0

As d -> 0, c=2r

 

[Chrono G. Xay]

When x=π/2,

L1=sqrt(r^2-d^2)*(sqrt(r^2-d^2)/(r-d))^sin(π/2)

L1=sqrt(r^2-d^2)*(sqrt(r^2-d^2)/(r-d))^1

L1=sqrt(r^2-d^2)*sqrt(r^2-d^2)/(r-d)

L1=(r^2-d^2)/(r-d)

L1=((r-d)(r+d))/(r-d)

L1=r+d

When x=0

L=sqrt(r^2-d^2)*(sqrt(r^2-d^2)/(r-d))^sin(0)

L=sqrt(r^2-d^2)*(sqrt(r^2-d^2)/(r-d))^0

L=sqrt(r^2-d^2)*(1)

L=sqrt(r^2-d^2)

When x=-π/2

L2=sqrt(r^2-d^2)*(sqrt(r^2-d^2)/(r-d))^sin(-π/2)

L2=sqrt(r^2-d^2)*(sqrt(r^2-d^2)/(r-d))^-1

L2=sqrt(r^2-d^2)*((r-d)/sqrt(r^2-d^2))

L2=r-d

L1 + L2 = 2r

(r+d)+(r-d) = 2r
r+d+r-d = 2r
r+r+d-d = 2r
2r+0 = 2r
2r = 2r

Thus, sqrt(r^2-d^2)*(((sqrt(r^2-d^2)/(r-d))^sin(π/2)+(sqrt(r^2-d^2)/(r-d))^sin(-π/2))=2r

 

[Mark44]

I'm glad you finally stated what you're doing, which makes things a lot clearer.

The equation for the length of a chord running through a circle can be expressed as:

c=2*sqrt(r^2-d^2)

I'm not able to look at your work now, but will do so later today or tomorrow.

 

[Chrono G. Xay]

. . .

 

[Chrono G. Xay]

Anyway, for my purposes I need to take the integral of (in terms of its form):

a*(a/b)^sin(x)

However, it seems you simply cannot take the integral of a function with such an exponent, so I've been working out an alternative *integratabtle* way to write it using 'n' instead of 'a/b':

I'm not entirely sure what 'q' is, but stopped here for now because it, too, seems to vary periodically (and I was getting frustrated). Another concern of mine has to do with how the earlier equation

y(x)=sqrt(r^2-d^2)*(sqrt(r^2-d^2)/(r-d))^sin(x)

when written as a polar equation (i.e. where y -> r and x -> θ) breaks down more and more as 'd' approaches 'r', which I interpret to mean that something needs to be 'tacked on' so it will or rather could theoretically transition into an equivalent of

r(θ)=2*r*sinθ

 

[phion]

I would just break it down into quadrants and find the integral for an ellipse with half axes a and b?

 

[Mark44]

For my purposes I need to be able to calculate the area of a circle without using πr^2 (Just play along.)

The equation for the length of a chord running through a circle can be expressed as:

c=2*sqrt(r^2-d^2)

You show a lot of work on this page, but if you want others to be able to follow it, you need to explain what the symbols represent. I can infer that c is the length of the chord, and r is the radius of the circle, but what is d? This is the first equation in this post, and I'm already lost.

Next we can expand it into the sum of the lengths of two individual chord segments:

c=sqrt(r^2-d^2)+sqrt(r^2-d^2)

From there, being two individual chord segments, if we were to pivot a line about a point 'd' from the circle's center, we would notice the individual line segments on either side of the pivot point change in length at a particular rate:

Still no help. A point in the plane has two coordinates. Is d the label for the point or does it represent a number? If it's a number, is it the x-coordinate or y-coordinate or what?

(sqrt(r^2-d^2)/(r-d))^sin(x)

It took a lot of experimentation and meditation on the patterns I was seeing to deduce how I would need the line segments' lengths to change to come up with that equation; to the best of my existing knowledge it's purely of my own creation, and it works.

In my earlier example, r=6 and d=2

L=sqrt(6^2-2^2)*[sqrt(6^2-2^2)/(6-2)]^sin(x)

L=sqrt(36-4)*[sqrt(36-4)/4]^sin(x)

L=4*sqrt(2)*[(4*sqrt(2))/4]^sin(x)

L=4*sqrt(2)*sqrt(2)^sin(x)

When x=π/2...

L1=4*sqrt(2)*sqrt(2)^sin(π/2)

L1=4*sqrt(2)*sqrt(2)^1

L1=4*sqrt(2)*sqrt(2)

L1=4*2

L1=8

And when x=-π/2...

L2=4*sqrt(2)*sqrt(2)^sin(-π/2)

L2=4*sqrt(2)*sqrt(2)^-1

L2=4

L1+L2 = 12 = 2r

A circle of radius 6 has a diameter of 12

As d -> r (as "2 approaches 6"), c=0

As d -> 0, c=2r

 

[Chrono G. Xay]

https://www.easycalculation.com/area/learn-chord-length-circle.php

 

[Chrono G. Xay]

The polar equation

r=12sinθ

Implicitly follows the form shown in the link, below, which is another way to calculate chord length:

http://mathcentral.uregina.ca/QQ/database/QQ.09.05/bes1.html

 

[Chrono G. Xay]

One observation I made was that if you write the polar equation

r=sqrt(r^2-d^2)*[sqrt(r^2-d^2)/(r-d)]^sin(Θ)

add the expression "2d*sin(Θ)" to it for

r=sqrt(r^2-d^2)*[sqrt(r^2-d^2)/(r-d)]^sin(Θ)+2d*sin(Θ)

and vary 'd' between 'r' and '-r', you have what seems to be the beginnings of an equation that will translate a polar equation along the plane, theoretically without distortion (though *some* distortion is obviously present). I feel rather confident that by reworking it to include 'i' at the proper time variable 'd' could then be taken *beyond* 'r' or '-r'.

(You could, of course, change the direction of translation by simply adding or subtracting from Θ.)

 

[Chrono G. Xay]

Mark44, the lack of a reply would seem to imply a continued dissatisfaction with my answer to your earlier question. If that is in fact the case, perhaps it is the question itself that needs further explanation. To the best of my understanding I believe I have answered it.

 

[Mark44]

Chrono, I will take a look, but I've been pretty busy this weekend, with guests at the house. I'll try to take another look later tonight or tomorrow.

 

[Chrono G. Xay]

Oh, ok. Take care.

 

[Mark44]

Quote from post #13 in this thread:

The equation for the length of a chord running through a circle can be expressed as:

c=2*sqrt(r^2-d^2)

Next we can expand it into the sum of the lengths of two individual chord segments:

c=sqrt(r^2-d^2)+sqrt(r^2-d^2)

The step above is so trivial that it doesn't even need an explanation. You could just have said this:
c=2*sqrt(r^2-d^2) = sqrt(r^2-d^2) + sqrt(r^2-d^2)

From there, being two individual chord segments, if we were to pivot a line about a point 'd' from the circle's center, we would notice the individual line segments on either side of the pivot point change in length at a particular rate:

(sqrt(r^2-d^2)/(r-d))^sin(x)

This I don't follow. From the figure at this site, https://www.easycalculation.com/area/learn-chord-length-circle.php (a link you posted earlier), d is the length of the line segment from the center of the circle to the center of chord. Is d also a point? It's very bad practice to use one letter to mean two different things.

So where is point 'd'? What does "pivot a line about a point 'd' from the circle's center" mean? If you're going to describe an image you're working from in words, you need more clarity in your descriptions than I am seeing.

Also, you went from a very detailed (and unneeded) explanation of how 2*sqrt(r^2-d^2) can be turned into sqrt(r^2-d^2) + sqrt(r^2-d^2), and now you show an expression raised to the power sin(x), with no explanation that I can see. Where in the world did that exponent of sin(x) come from?

 

[Chrono G. Xay]

There is no need to criticize so harshly my honest attempts to help you better understand where I'm coming from. Even if I forget something or don't use a word quite right, I would like to think that there is still enough for whomever is reading to get the gist of where I'm trying to go; reasonable deductions can be made.

The phrase "a point 'd' from the center..." implied a unit measure of distance. There was no contradiction.

I needed a way for the expression 'r-d' to change to 'sqrt(r^2-d^2)' and then 'r+d' (or vice versa) in a periodic manner. I needed to model the change in values I required. I noticed that when r=>6 and d=>2 the base 'sqrt(2)' worked perfectly, while when d=>3 the base 'sqrt(3)' worked, so at first I thought that so long as the base was 'sqrt(d)' I would be fine. That was wrong. It was only through further experimentation and thought on how to model the change in values that I realized when the base of 'sin(x)' has a numerator of 'sqrt(r^2-d^2)' and a denominator of 'r-d', and when for d=>2 or d=>3 that when the square root expression was simplified resulted in the exact same values (as well as form) that had worked earlier. Before I had this realization I was about ready to use the following quadratic equation, determined by the 'System of Equations' method (for the third point required to be able to use this method I evaluated 'sqrt(r^2-d^2)' at d=>4 for 2*sqrt(5) ):

Most recently I realized that the chord segments could have also been modeled as

l1=(r-d)*[sqrt(r^2-d^2)/(r-d)]^[sin(x)+1]

l2=(r+d)*[sqrt(r^2-d^2)/(r+d)]^[sin(x)+1]

 

[Mark44]

There is no need to criticize so harshly my honest attempts to help you better understand where I'm coming from.

I don't mean to come across quite so harshly, but this thread is frustrating to me. It might be that you have a good idea here, but you are not communicating it well. You started off with a question about how to integrate $a*(a/b)^{sin(\pi*x)}$, with no explanation of where this came from. You then said that the derivative "of course" is $cos(x)*ln(a)*a^{sin(x)}$, which I checked and found not to be the case.

In post #6 you said that your goal was to calculate the area of a circle using a "non-radial" point (your terminology). That threw me off, since every point inside a circle is on some radius of the circle.

You then went on to state that you wanted to calculate the area of the circle, but not by using the standard formula $A = \pi r^2$.

In posts #13 and #14 you included a lot of equations, without any explanation of what your variables represented. After I asked for clarification, you posted a link (post #20) to a site that shows how to calculate the length of a chord of a circle.

When I was able to take a look at what you had written in posts #13 and #14, I was stumped fairly early on, where you come up with this expression: $\sqrt{r^2-d^2}/(r-d))^{sin(x)}$ completely out of the blue, after considerable explanation (where none was needed, really, that $2\sqrt{r^2 - d^2} = \sqrt{r^2 - d^2} + \sqrt{r^2 - d^2}$. That part is obvious. The part about $\sqrt{r^2-d^2}/(r-d))^{sin(x)}$ is far from obvious.

Even if I forget something or don't use a word quite right, I would like to think that there is still enough for whomever is reading to get the gist of where I'm trying to go; reasonable deductions can be made.

No, there is not. The reason I've been asking questions throughout this thread is that I genuinely don't understand what you're trying to do, and I'm trying hard to get you to fill in the gaps. There are lots of people who know more mathematics than I do, but I taught mathematics in college for 18 years, so I have some cred in the area.

The phrase "a point 'd' from the center..." implied a unit measure of distance. There was no contradiction.

I didn't say there was a contradiction. What you said was ambiguous.

if we were to pivot a line about a point 'd' from the circle's center

There's no indication that you're talking about a point d units away from the center. I took 'd' to mean that you were giving the label 'd' to some point. My confusion was in the drawing in the link you posted, where d was the distance from the center of the circle to the midpoint of the chord.

I needed a way for the expression 'r-d' to change to 'sqrt(r^2-d^2)' and then 'r+d' (or vice versa) in a periodic manner. I needed to model the change in values I required. I noticed that when r=>6 and d=>2 the base 'sqrt(2)' worked perfectly, while when d=>3 the base 'sqrt(3)' worked, so at first I thought that so long as the base was 'sqrt(d)' I would be fine. That was wrong. It was only through further experimentation and thought on how to model the change in values that I realized when the base of 'sin(x)' has a numerator of 'sqrt(r^2-d^2)' and a denominator of 'r-d', and when for d=>2 or d=>3 that when the square root expression was simplified resulted in the exact same values (as well as form) that had worked earlier. Before I had this realization I was about ready to use the following quadratic equation, determined by a typical approach where I had to plug in the values of 'y' I needed (among other things) when I needed them and slowly solve for 'a', 'b', and 'c':

View attachment 84218

Most recently I realized that the chord segments could have also been modeled as

l1=(r-d)*[sqrt(r^2-d^2)/(r-d)]^[sin(x)+1]

l2=(r+d)*[sqrt(r^2-d^2)/(r+d)]^[sin(x)+1]

 

[Chrono G. Xay]

My 'project' is, to use music terms: predict the 'feel' of a drumhead.

That is to build up a realistic mathematical model of a pre-loaded 'cleared' (i.e. equal tension all around) circular membrane. There is such an equation, derived by rewriting the equation for tension on a string, a long, thin circular cylinder, to accept instead the geometry of a thin, large diameter cylinder),

T = ( π ρ h f^2 D^3 ) / G

T - Tension (lbs/kg)
ρ - Material density
h - Membrane thickness (height)
f - frequency
D - membrane Diameter
G - Force of gravity

model the deformation of the drumhead in response to a point force a distance 'd' from the center, so long as d<=r, and use physics and materials science to calculate the restoring force.(I compared the values the above equation computed with the breaking strength of Mylar, and they were at the very least believable.)

In your last post you used a large block quote at the very bottom, and I'm not sure what you were trying to do.

 

[phion]

My 'project' is, to use music terms: predict the 'feel' of a drumhead.

That is to build up a realistic mathematical model of a pre-loaded 'cleared' (i.e. equal tension all around) circular membrane. There is such an equation, derived by rewriting the equation for tension on a string, a long, thin circular cylinder, to accept instead the geometry of a thin, large diameter cylinder),

T = ( π ρ h f^2 D^3 ) / G

T - Tension (lbs/kg)
ρ - Material density
h - Membrane thickness (height)
f - frequency
D - membrane Diameter
G - Force of gravity

model the deformation of the drumhead in response to a point force a distance 'd' from the center, so long as d<=r, and use physics and materials science to calculate the restoring force.(I compared the values the above equation computed with the breaking strength of Mylar, and they were at the very least believable.)

In your last post you used a large block quote at the very bottom, and I'm not sure what you were trying to do.

Cool!

 

[Chrono G. Xay]

@phion

I have personally spoken with the drum specialists at Evans--one of the big names in the drumhead industry--in the past and pitched such an idea to them- even just predicting the tension on a drum--since one of their partners, a string company called D'Addario, already has on their website an extensive table displaying values of string tension at several different tunings for dozens of the strings they sell--and they weren't interested because they were afraid of basically 'overwhelming their customers with data'. (Their exact phrase was "paralysis by analysis".) What? That's seems like a rather hypocritical thing to say, considering their partner's business practices. I've spoken with professional performers, public school and higher education percussion teachers, and they would be excited to have such a tool at their disposal.

Imagine being able to mentally prepare yourself for how your drums are going to feel before you've even submitted an order to your local music business for those new heads you want to try. Or, if you already have the feel you want and don't want to change it, what if you would like to know, scientifically, what your preferred 'feel' is? or another artist's preferred 'feel'?

 

[phion]

@phion

I have personally spoken with the drum specialists at Evans--one of the big names in the drumhead industry--in the past and pitched such an idea to them- even just predicting the tension on a drum--since one of their partners, a string company called D'Addario, already has on their website an extensive table displaying values of string tension at several different tunings for dozens of the strings they sell--and they weren't interested because they were afraid of basically 'overwhelming their customers with data'. (Their exact phrase was "paralysis by analysis".) What? That's seems like a rather hypocritical thing to say, considering their partner's business practices. I've spoken with professional performers, public school and higher education percussion teachers, and they would be excited to have such a tool at their disposal.

Imagine being able to mentally prepare yourself for how your drums are going to feel before you've even submitted an order to your local music business for those new heads you want to try. Or, if you already have the feel you want and don't want to change it, what if you would like to know, scientifically, what your preferred 'feel' is? or another artist's preferred 'feel'?

I know a couple drummers personally. I'll probably talk about this thread to them.

 

[phion]

It just occurred to me that this is a pretty common problem within functional analysis. It concerns modes of vibration.

One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis.

 

[Chrono G. Xay]

That's why, as much as I'd love to calculate the damping coefficient of a membrane-shaped spring that doubles as its own mass in the mass-spring-damper system (and so be able to ascribe an actual value to 'sustain' when it comes to just the drumhead and not resonances of the air cavity or the drum shell itself), I can't--or at least *I* can't--because, for one, I don't have the knowledge to even START looking at eigenvalues/eigenvectors/etc., myself, if that's even the right road to go down!

For now I'm just content with trying to obtain a theoretical value for 'feel' (restoring force), which for what I'm trying to do is more of a static condition (for example: At 'x' displacement from rest, what is the restoring force 'F'?)

However, you can obtain an empirical value for the damping coefficient just by striking a drum at a point, noting the amount of kinetic energy imparted to the drumhead at that point (which might be best done by a machine sporting an apparatus that wields a drumstick so as to better control the striking point, velocity, impulse, et al), and as such how far the membrane is initially displaced when struck (in my opinion, the drumhead should be struck in the very center, so as to hopefully better excite the 0,1 mode as opposed to others). From there you'd analyze how the amplitude of motion the membrane makes changes as the sound decays (which might be best done with some sort of laser distance probe). When comparing data in terms of the logarithmic measure of sound pressure levels, is the damping linear, is it quadratic, or is it something else?

 

[Chrono G. Xay]

Well, *this* thread seems to have died...