Natural frequency of stressed thin-film membrane 2

[srosset]

I am looking for a formula describing the natural resonance frequency of thin circular stressed membranes with a fixed edge.
I know the formula for the situation without stress, which depends on the membrane's geometry and mechanical properties (Young Modulus, Poisson coefficient).

I have found in several different sources a formula for the resonance frequency of stressed membrane :
f(m,n)=Mu(m,n)/(2*pi*r)*sqrt(Sigma/ro)
Sigma : Stress, ro : density, m & n : mode number.

I am surprised to see that it only depends on the stress and density, but not on the thickness and the Young Modulus.

This second formula is obviously a simplification of a more complicated one, because if the stress goes to 0, then the resonance frequency is 0, but should in fact approach the "standard" expression of an unstressed membrane.

Does anyone know what are the criteria of validity of this formula? (For example Sigma/Y > given value)
Or does anyone know what is the complete formula which remain valid for small value of the stress?

Thank you in advance.

 

[btrueblood]

Mmm. No, and I don't think you'll find one.

Your 1st formula is really a "plate" formula, wherein the in-plane bending forces provide the dominant "restoring force" - there is probably a thickness term in your equation.

The 2nd formula has no thickness term (essentially the equation models an infinitely thin membrane), because the assumption is made in deriving the equation that the tension forces are much larger than any bending forces that can be generated. This assumption holds as long as the wavelengths of oscillation are much, much larger than the thickness of the material.

To model the "real" world part, with real thickness and a tension force would require numerical methods to solve* (e.g. FEA).

*Although I've been proven wrong by Mr. Timoshenko before...

 

[GregLocock]

You could probably use some sort of Rayleigh Ritz approach to get a mode shape that consists of a bending contribution and a membrane stiffness contribution, for a given geometry and pre-stress, using the mode shapes defined by those equations.

There is no criteria of validity as such, the result will shade from being bending-like to being membrane-like on a smooth curve.



Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[electricpete]

In Den Hartog on page 139 I see the case of a string (NOT a membrane) under tension. It looks very similar

First resonnatn frequency is given as

KE = PE


w = (Pi/L) * sqrt(T/mu)

Substitute
mu = mass per length = rho * Pi*R^2
and
T = sigma * A = sigma * Pi*R^2

gives

w = (Pi/L) * sqrt([sigma * Pi*R^2]/[rho * Pi*R^2])
w = (Pi/L) * sqrt(sigma/rho)

Note the dependence on radius R (analogous to plate thickness) has cancelled. If I doubled the radius (keeping stress constant), I would quadruple the force and quadruple the mass and keep the same resonant frequency.

This looks analogous to the membrane in tension.

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[electricpete]

delete "KE = PE". I didn't mean to put it there. (I tried briefly to derive this frequency equation from KE=PE but gave up).

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[electricpete]

Sorry I didn't mean to skip over the previous responses. They are on target I'm sure. I was just working talking thru the problem at my own pace.

To continue the linear analogy, the membrane with tension is comparable to the string under tension described by differential equation:

mu* d^2 y / dt^2 = T * d^2y /dx^2

The membrane without tension is comparable to the free/free beam described by the differential equation

- mu * d^2y/dt^2 = E * I * d^4 y / dx^4

On the left in both cases is the mass acceleration. On the right are forces from deformation of the deformed taught spring in the first case and forces from bending of the beam in the second case.

Both models rely on assumption of small deformation.

We might get a sense that one linear model or the other is dominant by comparing expected magnitudes of the right sides. (T * d^2y /dx^2 >> E * I * d^4 y / dx^4 or
E * I * d^4 y / dx^4 >> T * d^2y /dx^2)

In other words comparing tension forces vs bending forces as discussed in the other threads above. I'm sure you could work out a similar comparison for the membrane.

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[electricpete]

Sorry, "simply supported" beam boundary conditions would probably be more appropriate than "free/free". But doesn't change the differential equaition.

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[srosset]

Thank you very much for your replies.
I am measuring membranes which are 40 um thick and have a diameter of 2 mm. As I am exciting them at their first natural frequency, it means that the spatial wavelength of the oscillations is 4 mm. I have therefore a ratio Wavelength/Thickness of 100, and it is probably not sufficiently high to use the formula dependant only of the stress.
I'll probably have to go fetch "Vibration problems in Engineering" from Timoshenko to get a little help on formulating the problem.

Electricpete, I think your method is the way to go: Taking both contributions of bending and traction should do the trick, but in these 2-D problems, there are often diff. eq. with no analytical solutions (at least for rectangular membranes, with my circular membranes, I might have some luck.)

 

[sreid]

Blevin's "Frequency and Mode Shape" has formulas for both circular plates and membranes.

 

[Denial]

I have not seen the Blevin formulae so cannot comment on whether they are appropriate to srosset's problem. Howevever if you are happy with each formula separately, and if you are prepared to accept that the mode shapes associated with each are close to the same, then you can combine the two as follows.

Let f[sub]b[/sub] be the frequency of the circular object if bending is the totally dominant form of strain energy in the deflected shape.
Let f[sub]m[/sub] be the frequency of the circular object if membrane action is the totally dominant form of strain energy in the deflected shape.
Then the frequency of the circular object if both bending and membrane action contribute to the strain energy in the deflected shape is given by
f[sub]c[/sub][sup]2[/sup] = f[sub]b[/sub][sup]2[/sup] + f[sub]m[/sub][sup]2[/sup]

This result comes from applying a Rayleigh Ritz approach as per GregLocock's suggestion above, but relying on the people who derived the two individual formulae for the hard part.

 

[electricpete]

good point.

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[srosset]

That is the approach I used, with good correlation between predicted results using Rayleigh Ritz approximation and measured results. Thank you all for your help

 

[GregLocock]

srosset - ny chance of you publishing what you did? It would be nice to have a reference for a practical application of R-R



Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[electricpete]

I'm not adding anything new here, but just wanted to record some more basic discussion for my own benefit and possibly others like me - a "proof" of why wtotal = sqrt(w1^2 + w2^2) assuming the mode shape is unchanged.

Let's say in bending we have
PEb = m * wb^2
for the assumed mode shape.

Let's say in tension (membrane) we have
PEt = m * wt^2
for the same assumed mode shape.

If we keep the assumed mode shape, the kinetic energy doesn't change, but the PE's add.
PEtotal = m * wtotal^2
PEb + PEt = m* wtotal^2
m*wb^2 + m*wt^2 = m*wtotal^2
wtotal = sqrt(wb^2 + wt^2).

The m*w^2 on the right side is a little simplified since this is a distributed mass, but the point is the kinetic energy is the same in both cases (bending and tension) and also proportional to w^2.

I started to do it for the one-D case but I don't have enough time right now. Maybe later.

In that 1-d case I'm very sure the true mode shape would remain sin(x * Pi/L)

In the case of membrane, wouldn't the mode shape be sin([R-r] * Pi/R)?

If the exact mode shape is known, the solution is Raleigh, not Raleigh-Ritz, right?

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[electricpete]

Whoops. Membrane mode shape with coordinate r going from 0 at center to R at edge would be:
cos([R-r] * Pi/[2R])

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[electricpete]

Goofed up something else. Let me try again:
PEt = m * wt^2 [tension: string]
PEb = m * wb^2 [bending: beam]

PEb + PEt = m* wtotal^2 [total: tension and bending]
Substitute in for PEt and PEb:
m*wb^2 + m*wt^2 = m*wtotal^2
wtotal = sqrt(wb^2 + wt^2).


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[Denial]

Electricpete. You have the gist of the proof correct. The key point is your statement: "If we keep the assumed mode shape, the kinetic energy doesn't change, but the PE's add." I've never seen it proved (or even mentioned) in a text book, but have been happy to use it occasionally because it is self-evidently correct.

As for the name of the method, I suspect I was a bit sloppy with my terminology, and that we are using the Raleigh method rather than the Raleigh-Ritz method. Ready to be contradicted, though.

 

[Banqui]

Hello
I am currentely and surprisingly under the same dilema. I am trying to find out the formula of the fundamental frequency of vibration of a circular metallic film with fixed boundary conditions.

I ran into the same problem. I was able to find the formula for thin membranes, but i wanted to find out how the frequency varied with the Young modulus and with the thickness of the film. Have you made any progress with the formula? I would really appreciate it if you could guide me through the problem since I am very lost in the issue at hand.

What I had thought was how to relate the tension with the Young modulus, since one could easily relate a 2d density with a 3d density and the value of the thickness of a thin metallic film.

I hope you have the solution, or at least if someone could give me a hand.

useful link:

http://www.du.edu/~jcalvert/waves/membran.htm

there one can find the solution to the problem for a square membrane as a function of the young modulus and the density.