Can't Find 2D Elastic Modulus Equation

[Chrono G. Xay]

I've tried searching online for one in what I can only guess would be called a reduced algebraic form, and I cannot find it.

To make matters worse, for me, at least, I do not have the mathematics knowledge necessary to understand advanced functions, series, and transformations of mathematics such as Gamma, Fourier, Bessel, or Laplace, and have only some knowledge of derivatives, integrals, and introductory knowledge of differential equations from a high school Calculus AB class I took seven years ago.

The reason I am searching this is because there is a 2D equation for surface tension on a membrane: http://hyperphysics.phy-astr.gsu.edu/hbase/music/cirmem.html#c2

and I'm not sure if I can use what I'm guessing is the 1D(?) equation for elastic modulus:

E = ( F / A_0 ) / ( ΔL / L )

for the purpose of building a mathematical model of sorts describing a clamped edge, pre-tensioned, radially-symmetric membrane experiencing a transverse displacement 'z' at a point a set distance 'd' from its center in response to a point force (and 0 =< d < membrane radius). I know that in the case of such a deformation the membrane's maximum deformed shape could be said to resemble sqrt(x) revolved about the origin (when d=0).

The desired end result would be a function of 'z' with respect to 'd' on the interval [0,r]. It would be implied that the graph of the function would be even.

 

[Nidum]

Can probably shed some light on your problem but need to understand more clearly what you are trying to do .

Are you just trying to find the deflection or are you trying to find the vibrational response ?

 

[Chestermiller]

See the final equations in this reference: http://emweb.unl.edu/NEGAHBAN/Em325/17-Hooke's-law-for-plane-stress/Hooke's law for plane stress.htm

In your application, the x and y strains are equal, and the shear strain (gamma) is zero. Also, in your application, the equations would be expressed in cylindrical coordinates, in terms of the radial/circumferential strain.

Chet

 

[Chrono G. Xay]

@Nidum - I was trying to either find or work out an equation for the deflection of the membrane as a function of the distance from its center. At the same time, though, the generalized(?) equation at hyperphysics.com is for 'surface tension', and I am hoping that there is one for radial/uniaxial(?) tension.

@chet - Funny that you mention those equations, because just last night I was looking at them, albeit a different article, and hoping I could solve for E and then set the whole thing as being equal to the formula in my last post, and... from there I'm lost.

Here's the thing: people who can play string instruments can know the tension on their strings using a simple general equation. (Shown here: http://www.daddario.com/balanced_tension.page?sid=f7bd5a9f-991e-4539-a24a-2719f97d53a9)

Here's the thing- I have spoken with the percussion specialists at Evans, which is a major company in drumhead manufacture, and they were afraid of the possible "paralysis by analysis" (their words) of enabling the layman to predict the approximate tension on their drums, and thereby select drumheads and tunings based on the 'feel' they want (which is what I am hoping to quantify here- the deflection of the drum, given the radial/uniaxial tension, dimensions, materials science values, and, most importantly, the distance from the membrane center that the deflection originated).

What I find incredibly ironic about this is that Evans is a conglomerate of D'Addario, a major company in the manufacture of strings for musical instruments, and on D'Addario's website they have, for everyone to see, a detailed spreadsheet including what is probably just about every string gauge they sell, and the tension values (at a scale length of 25.5" for guitar, 34" for bass) of an octave's worth of pitches in each string's tuning range. Therefore, what argument do they have that percussionists will experience such a thing? For those, such as Steve Gadd, who have established preferences and methods by which to tune drums, this affects them and their craft in no way, shape or form. This is merely another tool- a tool which can be used to quantify exactly what different players or musical genre prefer.

 

[nasu]

The Hyper-physics formula is for the membrane model. There is no elastic constant (like Young's modulus or shear modulus in this model) so I don't understand why are you mentioning this.
On the other hand, Young's modulus is a material parameter. There is no 1D or 2D in it. If you know it, you can apply it to any model that it may appear as a parameter.

It may help to realize that there are two main models describing vibration of elastic thin sheets: "membrane" and "plate".
In the membrane model, the frequency and such depend only on the tension and density of the medium. Like for a string. You need to apply some tension to the membrane to work.
In the plate model you do have these elastic constants, the wave velocity and frequencies of the modes depend on the bending elasticity of the plate.

The membrane model is usually applied to drums.
The plate apply to things like cymbals or gongs or such.

 

[Andy Resnick]

@Nidum - I was trying to either find or work out an equation for the deflection of the membrane as a function of the distance from its center. At the same time, though, the generalized(?) equation at hyperphysics.com is for 'surface tension', and I am hoping that there is one for radial/uniaxial(?) tension.

<snip>

I think by now you have realized that there is no simple solution- there are too many variables. *If* the head is tensioned uniformly, *and* the head- shell coupling is uniform, *and* the bottom head is likewise tensioned uniformly, etc; the problem is still very complex and (AFAIK) does not have analytic solutions. Drums are atonal instruments.

Now add in: the tension may not be uniform- a standard trick is to de-tune one lug. And there are discrete lugs, not a continuous load exerted onto the shell. Or there are damping pads. Or the head is not struck on center. Don't forget the air vent! And the shell material is non-uniform: just painting the inside of the shell will alter the tone. Brass/wood/steel snares sound different due to the shell material. Now there's the bottom head- maybe it's a different material, tensioned differently as well.

And besides, nobody is going to tune a drum with a precision torque wrench. Drums are tuned to each other.

You should be starting with a timpani- those are single-head instruments, tuned to specific notes, and struck with a mallet:

http://www.aes.org/e-lib/browse.cfm?elib=16967

 

[Chrono G. Xay]

@Andy - I do not believe you understand my question. I am not asking about the resulting waves at all. The point that there is even a general equation for radial tension on a drum (when the drumhead has been 'cleared') is the goal. Unless the drums you're using are some of those made by Sleishman, where the drum's shell is, in effect, suspended between the two drumheads, the effect of the tension on the bottom head has a negligible effect on the tension of the batter head. Also, it is the second question of mine which asks about a graph of deflection as the point of contact is moved farther from the membrane's center.

Think of my question this way: What is the radial tension on a roto-tom? Following this, what is a graph of the roto-tom's drumhead's deflection (that is, its 'feel') as a drumstick with the same final angular velocity strikes a point on the drumhead farther and farther from its center?

 

[Chestermiller]

OK, let's see where this takes us. If the strain in the drum head membrane is the same in all directions (aka, transversely isotropic), then the stress tensor is transversely isotropic also, and given in magnitude by:

$$σ=\frac{E}{(1-ν)}ε$$
where E is the Young's modulus, ν is the Poisson ratio, and ε is the isotropic strain. If expressed in terms of the radial stretching of the drum head, the tensile strain the the membrane is given by

$$ε=\frac{(r-r_0)}{r_0}$$ where r is the stretched radius of the drum head membrane and r₀ is the original unstretched radius. The stress is not as important as the so-called Stress Resultant N, which is defined as

$$N=σh=\frac{E}{(1-ν)}εh$$,
where h is the thickness of the membrane. The stress resultant N takes the place of the tension in 2D membrane analysis. If you evaluate the internal force per unit length along a fictitious cut in the drum head, that is the same as the Stress Resultant (i.e., the tension per unit length).

Does that in any way seem of use to you?

Chet

 

[Andy Resnick]

@Andy - I do not believe you understand my question. I am not asking about the resulting waves at all. The point that there is even a general equation for radial tension on a drum (when the drumhead has been 'cleared') is the goal. Unless the drums you're using are some of those made by Sleishman, where the drum's shell is, in effect, suspended between the two drumheads, the effect of the tension on the bottom head has a negligible effect on the tension of the batter head. Also, it is the second question of mine which asks about a graph of deflection as the point of contact is moved farther from the membrane's center.

Think of my question this way: What is the radial tension on a roto-tom? Following this, what is a graph of the roto-tom's drumhead's deflection (that is, its 'feel') as a drumstick with the same final angular velocity strikes a point on the drumhead farther and farther from its center?

Ah- so you are only interested in a (uniformly) clamped circular elastic sheet. Remember, tho- the elastic sheet is analogous to a 'neutral axis' in 1-D problems: oil-filled drumheads are potentially more difficult to model, as are Remo's 'controlled sound' heads.

Impacts to thin elastic sheets have a long history of study:

http://naca.central.cranfield.ac.uk/reports/1942/naca-report-744.pdf
http://rspa.royalsocietypublishing.org/content/465/2103/823
http://www.princeton.edu/~stonelab/...d/PRLandPREandPRB/CourbinAjdariStonePRL06.pdf
http://www.sciencedirect.com/science/article/pii/S0022509606000068

Related problems include trampolines and impact damage.

http://www.sciencedirect.com/science/article/pii/S0734743X15001268

 

[Chrono G. Xay]

@chet - I think so... I do have a concern with Δr / r_0 , though, in that the layman won't be able to tell what the change in radius is... Before I started this thread I had worked at least this much up (below). Let me know what you think.

E = σ / ε = ( F / A_i ) / ( ΔL / L_i )

A_i - cross-section of membrane = D h

D - membrane diameter
h - membrane height (thickness)

ΔL = L_f - L_i

ΔL / L_i = ( L_f / L_i ) - 1

L_f = L_{1f} + L_{2f}

L_i = L_{1i} + L_{2i}

L_{1i} = r + d

L_{2i} = r - d

=> L_i = 2 r = D

Where 'r' is the membrane's radius and 'd' the distance from the striking point to the membrane's center.

Considering the deformation of the membrane looks similar to the graph

f( x ) = sqrt( x )

revolved about the y-axis (at least when d = 0), I chose to model the deformed membrane as such, where function to be integrated for deformed length, then for deformed area is appropriately scaled by the z-axis deflection, z, and the length L_n .

The vertical scaling can obviously be solved by multiplying f( x ) by z, but I needed a formula which would describe the distance from a point on a circle to any point along its circumference. I found this one in polar coordinates:

r^2 - 2 ( h cosθ + k sinθ ) r = R^2 - h^2 - k^2

Where 'h' and 'k' are the horizontal and vertical coordinates of the translated circle, and 'R' is the circle's radius.

I then set it as a quadratic equation and solved for 'r' :

r = { - [ 2 ( h cosθ + k sinθ ) ] +/- sqrt[ [ -2 ( h cosθ + k sinθ ) ]^2 - 4 ( h^2 + k^2 - R^2 ) ] } / 2

By setting
k = 0,
r = y,
θ = r,
h = d,

and simplifying, we get the rectangular function:

g ( x ) = d cos( x ) + sqrt{ 6^2 - [ d sin( x ) ]^2 }

The composition 'f o g' gives us the formulas for L_{1f} and L_{2f} :

L_{1f}( x ) = z sqrt( x / ( d cosθ + sqrt{ 6^2 - [ d sinθ ]^2 } )

L_{2f}( x ) = z sqrt( x / ( d cos( θ + π ) + sqrt{ 6^2 - [ d sin( θ + π ) ]^2 } )

However, considering g( x ) is a periodic function that doesn't vary in frequency, it becomes possible when integrating for area (which I believe would come after integrating for length) to write something like:

L_f = 2 [integral; 0, π] L_{1f} dθ

What it boils down to is :

2 [integral; 0, π] L_i dθ = A_0 (not to be confused with A_i ), which is equal π r^2,

and the expression for the integral of L_f is supposed to, in effect, integrate to A_f, meaning the substituted expression is more like:

( A_f / A_0 ) - 1

I believe that should give us all of our variables... from there all we would need to do is solve for 'z'... that is also assuming that inserting 'surface tension' for T will give an answer in meters (of which I am skeptical)

 

[Chrono G. Xay]

Question: It is 'weird' when, as a part of a problem, one or even both of the bounds of an integral are expressions?

 

[Chrono G. Xay]

@Andy - A couple of your articles would seem to not be relevant to this thread. We are dealing with membranes, not plates (re-read nasu's comment, #5); a clamped edge, not a free edge (i.e. drums, not cymbals); a point force, not a change in pressure on a side (ex. not the ear drum).

 

[Chrono G. Xay]

Correction to my last post:

"However, considering g( x ) is periodic, it becomes possible when integrating for area (which I believe would come after integrating for length) to write:

L_f( x ) = 2 [integral; 0, π] L_{1f} dθ"

That expression is very wrong, and I apologize.

There is a way to take the integral of a 1-D graph such that you obtain the length of the line from f(a) to f(b).

The expression defining the length of the desired portion of the graph is supposed to be more like:

L_nF = [integral; 0, g( θ )] sqrt[ 1 + f'( L_{nf} )^2 ] dx

(Note that the upper bound of the integral is a function, hence the question in #11)

It would be from there that we take the polar(?) integral of the above expression, which, because the 2π periodicity of g( θ ), can be more simply written as:

[integral; 0, π] L_F dθ = 2 [integral; 0, π] [integral; 0, g( θ )] L_{1f} dx dθ = A_f

 

[Chrono G. Xay]

Also, the fact of the matter is:

g( x ) = L_{10}, and
g( x + π ) = L_{20},

and when x_0 = 0...

L_{10} = d + r, and
L_{20} = -d + r,

which comes right back to:

L_0 = L_{10} + L_{20}
= ( d + r ) + ( -d + r ) = 2 r = D

 

[Chestermiller]

Hi Chrono G. Xay,

I wasn't able to figure out what you were trying to do in your analysis, but here's what I think it might have been:

You have a drum head membrane that is initially not under tension. You apply a point static vertical drum stick load at a location not at the center of the drum head and you wish to find (1) the deflection of the drum head, (2) the stress distribution within the drum head membrane, (3) the force exerted by the drum stick on the drum head, and (4) the corresponding force exerted by the drum head on the drum stick. Is this correct?

Chet

 

[Chrono G. Xay]

The drumhead would be under an initial tension, implying that it had been tuned to where the surface tension--or rather 'radial tension', which is what I *think* I am looking for--all over is equal (or at least arguably so). (Meaning no detuned lugs for disrupting higher modes of vibration, Andy.) The location of the point static force *could* be at the center of the membrane, but it could instead be applied any distance 'd' between the center and the edge of the 'speaking area' of the membrane, where 0 =< d < r , for the purpose of generating a rectangular (symmetric?) graph of the deflection on the interval [0, r) when solved for the maximum transverse deflection, 'z'.

(1) Yes.
(2) By modeling the deformed drumhead's area, yes.
(3) and (4) Yes, I think. Just to make I'm understanding, would these two not be equal?

 

[Chrono G. Xay]

Here are some pictures of the 2-D graphs I mentioned:

 

[Chestermiller]

The drumhead would be under an initial tension, implying that it had been tuned to where the surface tension--or rather 'radial tension', which is what I *think* I am looking for--all over is equal (or at least arguably so). (Meaning no detuned lugs for disrupting higher modes of vibration, Andy.) The location of the point static force *could* be at the center of the membrane, but it could instead be applied any distance 'd' between the center and the edge of the 'speaking area' of the membrane, where 0 =< d < r , for the purpose of generating a rectangular (symmetric?) graph of the deflection on the interval [0, r) when solved for the maximum transverse deflection, 'z'.

(1) Yes.
(2) By modeling the deformed drumhead's area, yes.
(3) and (4) Yes, I think. Just to make I'm understanding, would these two not be equal?

OK. Please be patient, and I'll be back with you in a little while.

The analysis you have done so far is an admirable attempt, but, unfortunately, is both incorrect and simplistic. Unfortunately also, it is not possible to present here the volume of material you will have to learn before you are able to do what you want correctly. At the very least, you need to study Strength of Materials or, preferably, Theory of Elasticity. There are many books on Strength of Materials available.

What I will do is work out the solution to the static loading problem you have been analyzing to illustrate how different the results are from what you have done above, and also to give you a starting point for further analysis.

Chet

 

[Chestermiller]

The figure above shows an exaggerated view (not to scale) of the shape of the drum head membrane when a drum stick is pressed vertically downward at the centerline of the drum head. Rather than making point contact with the membrane, the drum stick exhibits a circle of contact at its periphery, as the bottom of the membrane wraps around the bottom of the drum stick. The force of the drum stick pressing down on the drum head membrane is indicated by the force F in the figure, and this force is distributed summetrically around the circle of contact. The depth of penetration of the drum stick is indicated by the symbol δ.

In the analysis I did, I used the following symbols:

$r_i$ = radius of outer circle of contact with drum stick

$r_o$ = radius of rim

$σ_p$ = initial isotropic pre-load stress in membrane, prior to lowering the drum stick

$h$ = thickness of membrane

Based on the axisymmetric membrane analysis I did, the force exerted by the drum stick on the membrane (which is equal and opposite to the force exerted by the membrane on the drum stick) is given by:
$$F=\frac{2πσ_ph}{\ln(r_o/r_i)}δ$$

Note that, analogous to a pre-tensioned guitar string subjected to a load at its center, the displacement here is proportional to the applied load. However, the geometric factors are quite different.

Chet

 

[Chrono G. Xay]

Ok. You know I really, REALLY appreciate your help, Chet. I will definitely be patient.

For the record, I went in knowing I wouldn't be able to do the true model justice; that the drumstick actually depresses a circular/semi-spherical area, and I would have to refine my model a few more times as I understood better how to implement the appropriate contact area.

 

[Chestermiller]

Ok. You know I really, REALLY appreciate your help, Chet. I will definitely be patient.

For the record, I went in knowing I wouldn't be able to do the true model justice; that the drumstick actually depresses a circular/semi-spherical area, and I would have to refine my model a few more times as I understood better how to implement the appropriate contact area.

Chrono,

In my judgement, you're a born modeller. You have the key characteristic that I always look for: being optimistic and always believing that you will be able to do it. I have never seen a pessimist that can do modelling. I feel that you are going to be just amazing once you get broader knowledge and more experience.

Chet

 

[Chrono G. Xay]

Your compliments are well-received, Chet, and I will take them to heart. I am honored.

 

[Chrono G. Xay]

I have a question about the pre-load stress, but I don't want to ask until you're done.

$σ_p$ = initial isotropic pre-load stress in membrane, prior to lowering the drum stick

 

[Chestermiller]

I have a question about the pre-load stress, but I don't want to ask until you're done.

Go ahead and ask. I wasn't going to present anything else, unless you want the shape of the drum head in the deformed configuration.

Chet

 

[Chrono G. Xay]

Ok. (Some times I ask a question to at least have asked, no matter how silly it may feel at the time.)

Anyway, the variable $σ_p$ for pre-load stress – is it supposed to be the surface tension, radial tension, or neither?

I have designs for a physical apparatus so I check an *aspect* of membrane tension, but it's really just 100lb capacity spring-scales attached to #10-24 thread turnbuckles via short eye-bolts, which are then attached to the tuning bolts of the drumhead (which apparently use #10-24 threads, hence the earlier number) so I can tune it normally and then check my force values. All of this would be mounted to a 14" diameter steel barrel of, which is the typical snare drum diameter.

The equation you just worked out is incredible to me, in that it actually takes into account a general contact area, and I can not thank you enough. It is, indeed, a great starting point on which to base further analysis.

This just occurred to me... There are 'general purpose' drumsticks (not just for students, per se) where the tip is purposefully made to be spherical (and one can reasonably assume that brands do not make this sphere with the same radius)... I can understand that this would require a more rigorous analysis, involving how the contact point on the membrane begins to deform around the tip as one strikes with increasing force, meaning the contact area, itself, begins to change, up to even a half-sphere of the tip (which is likely to occur more by stabbing the membrane with great force). The reason I ask this is because it would also address an actual drumstick tip shape without having to necessarily factor in the contact angle of the drumstick (that is, we assume the contact angle is 0-degrees).

 

[Chestermiller]

Ok. (Some times I ask a question to at least have asked, no matter how silly it may feel at the time.)

Anyway, the variable $σ_p$ for pre-load stress – is it supposed to be the surface tension, radial tension, or neither?

It is the force per unit area acting on a tiny vertical planar element of area situated within the membrane. If you stretched a square sheet of the drum head membrane material equally in both directions, it would be the force that you would apply at each edge divided by the cross sectional area of the edge. Unfortunately, since, in the actual system, the membrane is circular, it would be difficult (although not impossible) to measure.

I have designs for a physical apparatus so I check an *aspect* of membrane tension, but it's really just 100lb capacity spring-scales attached to #10-24 thread turnbuckles via short eye-bolts, which are then attached to the tuning bolts of the drumhead (which apparently use #10-24 threads, hence the earlier number) so I can tune it normally and then check my force values. All of this would be mounted to a 14" diameter steel barrel of, which is the typical snare drum diameter.

The equation you just worked out is incredible to me, in that it actually takes into account a general contact area, and I can not thank you enough. It is, indeed, a great starting point on which to base further analysis.

This just occurred to me... There are 'general purpose' drumsticks (not just for students, per se) where the tip is purposefully made to be spherical (and one can reasonably assume that brands do not make this sphere with the same radius)... I can understand that this would require a more rigorous analysis, involving how the contact point on the membrane begins to deform around the tip as one strikes with increasing force, meaning the contact area, itself, begins to change, up to even a half-sphere of the tip (which is likely to occur more by stabbing the membrane with great force). The reason I ask this is because it would also address an actual drumstick tip shape without having to necessarily factor in the contact angle of the drumstick (that is, we assume the contact angle is 0-degrees).

I'm not sure what your question is here. If it is, "can the analysis be done," the answer is yes, although it would require more work and thinking.

 

[Chrono G. Xay]

Do you know if the apparatus described above would be able to measure the force per unit area, or would I even be able to compute the force per unit area from the values obtained by such a device?

As the second part of my post, my question is if you would be *willing* to do such an analysis. I understand that these things take time–personal time–and you probably have other things of greater importance or things that you might rather be doing than that, though I do not mean to patronize you in mentioning it. If it would mean waiting a few days or even weeks (arguably at the worst), just know that I am more than willing.

 

[Chrono G. Xay]

It is the force per unit area acting on a tiny vertical planar element of area situated within the membrane. If you stretched a square sheet of the drum head membrane material equally in both directions, it would be the force that you would apply at each edge divided by the cross sectional area of the edge. Unfortunately, since, in the actual system, the membrane is circular, it would be difficult (although not impossible) to measure.

Considering this is a circular membrane, meaning it has but one edge, could we not then use something like the surface tension divided by the circumference? Surface tension, at least, is in N/m...

 

[Chrono G. Xay]

I just realized that the diagram you drew earlier doesn't include the distance--if any--between the center of the membrane and the center of the contact area...

 

[Chestermiller]

Do you know if the apparatus described above would be able to measure the force per unit area, or would I even be able to compute the force per unit area from the values obtained by such a device?

I am not able to understand the geometry of your setup.

As the second part of my post, my question is if you would be *willing* to do such an analysis. I understand that these things take time–personal time–and you probably have other things of greater importance or things that you might rather be doing than that, though I do not mean to patronize you in mentioning it. If it would mean waiting a few days or even weeks (arguably at the worst), just know that I am more than willing.

I really wouldn't want to do this. At Physics Forums, we are supposed to help you solve your own problem by asking you leading questions and giving you hints. I've already gone way beyond this criterion by providing you with the answer to the case we discussed. My motivation was only to show you how the actual result differed from the result of your analysis.

To comply with the PF guidelines, I would have to wait until you develop significantly more background, so that my hints and leading questions would be meaningful to you. I hope you understand.

The contact problem with a spherical probe would be difficult because of conformability issues. You can't wrap a flat sheet of paper on the surface of a sphere without developing wrinkles in the sheet. So the membrane would have to deform (i.e., strain in-plane) within the contact patch in order to conform to the sphere. This deformation would have to be analyzed. There is also the question of frictional slippage between the membrane and the sphere as the membrane lays on to the contact patch.

I'm confident that I can analyze all this, but then it would feel like I'm back at work again, after retiring.

Chet