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natural frequency of a radially compressed ring 3

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cspkumar

Structural
Jul 30, 2002
34
Hi all,

I am looking at the eigen-value analysis of a ring compressed radially using nodal forces. The natural frequencies increased when I used load stiffening option in Algor when compared to a ring with no loads on it. The constraints were the same for both. Is this correct? I am not sure if this is right.

I think for a beam in tension, the natural frequencies would increase and decrease when in compression (when the load is less than the buckling load), but here, in the case of the ring, the frequencies are increasing in compression. Could someone explain this to me?

Thanks,
Kumar
 
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Hi Kumar,
It sounds like a glitch to me. Why don't you try putting the ring in radial tension and see what result you get. That will give an indication of what's going on.
Good luck.
 
A beam in tension has a uniaxial stress but a ring has both hoop and radial stresses. With an external load on the ring the hoop stresses are compressive but the radial stresses are tensile. It may be this tensile component that increases the natural frequency.

corus
 
If you have a ring that is loaded with equally spaced loads that are equal in magnitude the stress pattern will vary. The stresses are calculated as P/A ± M/S. P = axial compression. M = bending moment (+M is outer fiber compression). P and M will vary at each location. Roark's book has this analysis method in it.

Steve Braune
Tank Industry Consultants
 
In a linear homogenous system the resonant frequency is controlled entirely by the geometry, constraints, density and the Young's modulus.

The static loads superimposed on the structure should have no effect.

As an example, consider the surge mode of a coil spring, with varying amounts of preload. So long as the geometry is unchanged the preload is not part of the equation.

The reason ( I think) is that the resonant frequency is an energy balance, and as much work is done against the preload in one half of the cycle as is regained from it in the other, so it has no net effect.

Cheers

Greg Locock
 
cspkumar is certainly correct when he says that the natural frequency of a beam increases under tension, and Blevins is one place you can find the exact equations. The mathematics of this is non-linear, so this does not conflict with GregLococks assertion. Interestingly, if you take a closed hollow tube and pressurize it internally, the natural frequency stays exactly the same, even though the axial force applied should have caused it to rise. It turns out that radial effects exactly cancel out the axial force effect. In the case of a circular ring with externally applied radial pressure, my initial inclination would be to say that the frequency of the uniform radial mode should remain unchanged, in agreement with GregLocock, since the mathematics appears linear, in which case the FEA result would be an artifact. But I could be dead wrong - I'd be interested to hear more - effect of number of nodes, how big is the effect etc.
 

Believe elastic deflections are being dealt with; the equations are non-linear only where the deflections are large.

cspk: You need to specify which modes are involved: extensional, transverse, torsional?

The amount of force required to produce a unit of deflection is greater under radial compression, so your critical frequencies should increase. The analogy with the axial loading of a beam is flawed. It would be more correct to compare your case with a beam with lateral supports.
 
Hacksaw: "The amount of force required to produce a unit of deflection is greater under radial compression"
Explain why you think this should be true for the ring, and if it is true, why it does not imply non-linearity.
 
Hacksaw: Perhaps I should further clarify something I said in the previous post. When I stated that the mathematics of a beam under tension is non-linear, I meant exactly that - I did not mean to imply that the transverse deflection is not approximately a linear function of the transverse force for small deflections. But since one cannot apply simple superposition to the case, it must in some sense be a non-linear problem. (The sequence in which the axial and transverse loads is applied is very important). In the case of the ring, however, I can see no obvious reason why superposition should not apply, assuming uniform external pressure, although this is possibly not true when approximating with discrete nodes, but I most certainly agree with you that the analogy is flawed.
 
I don't go along with much of the above I have to say, but there again I'm not too old to learn !. If kumar is using some form of 'load stiffening' option it's basically a departure from a straightforward linear analysis and it will be only the axial load that makes any difference to the natural frequency. And I would be suspicious of results that give increased frequency (ie stiffening ) under axial compression of any structure !!
 
JWB46: What don't you go along with exactly ? Anything I said ? I can envision a number of situations where the natural frequency of a structure could increase under axial compression, although that would imply some type of non-linear behavior, which should not be the case here, at least as the problem has been defined.
 


believe the radial load stiffens motion in the the transverse(radial) direction. but before we get to far along we need a more precise statement from cspk as to what modes he has modeled.

agreed axial loads(compressive) do not increase the criticals. however, this is not the case with transverse stiffening.

linearity in the sense of hooks law.

 
My comments were intended to apply only to a uniform radial mode, as I stated. (ie the component remains circular at all times). If the component obeys the Lame thick ring equations, there is no way that I can see a classical solution producing a change in frequency under a superimposed uniform external or internal pressure for such a mode. But I wouldn't like to say what happens for more complex modes. It is also the case that finite element methods do not always produce results which converge to the correct classical solution, even if the number of nodes is increased ad infinitum. An example of this would be the famous "Babuska Paradox" (not Babushka!). It is just possible something similar could be happening here.
 
Incidentally my list above should include Poisson's ratio.

"the so called Babuska's paradox may be mentioned. Whereas the solution of a bending problem for the thin elastic circular plate is unique, it differs from the case of a plate with polygonal boundary even if the number of sides of the polygon tends to infinity. The solution becomes non-unique in this limit."

I assume this is for distributed loading - do you know how big the difference is, typically? What does it depend on?


Here's Roark's formula for an infinite sided polygon, fixed edges, nu=.3

max y= -.171*q*a^4/E/t^3

where a is the inscribed radius

and here's the formula for a disk (table 24 case 10b)

y=-.01563*qa^4/D

and D=Et^3/12/(1-nu^2)

which is an error of 0.2%

Good. I always knew there was something wrong with FEA!






Cheers

Greg Locock
 
Well, that is not much good, the error is smaller than the number of significant figures in the constants. If anyone has access to Leisser Lo and Niedenfuhr's seminal paper "Uniformly Loaded Plates of Regular Polygonal Shape" then we might chase this one down a bit further.



Cheers

Greg Locock
 
I think from memory that the paradox has to do with the fact that the deflection behavior of a circular plate using plate elements converges to an encastre boundary instead of a simply supported one as the number of nodes increases, or something vaguely like that - but I've probably got it wrong - please feel free to correct me anyone. What we need is "The Finite Element Method and Its Reliability" by Ivo Babuska. It's probably got nothing to do with this problem of course, but it was just something that came to mind. It's worth noting that the finite element method was originally pioneered by aerospace engineers in the 1950's, not mathematicians, long before it was rigorously investigated. When mathematicians became aware of it, the first reaction seems to have been that it was not necessarily theoretically valid in all cases, and it was a long time before they convinced themselves that it was, on the basis of some rather abstract theorems. And in some cases, it seems that their original doubts were justified.
 
Agreed than modern FEA began in the 50's, but it had traceable origins in the 30's and 40's in the modeling of air frames and wings. Myklestad one such pioneer.
 
The problem kumar posed was a radially compressed RING (presumbly linearly elastic). If the result from the analysis indicates an increased natural frequency what result would you expect if the radial loads were reversed to produce axial tension in the ring ? and what sort of result would you expect for the elastic buckling load for the same cases ?

As I said I'm not too old to learn ! (I'm serious about this) What sort of structures stiffen under compression ?


By the way Kumar have you lost interest ??
 
Yes, I don't deny we have gone a little off-topic here. "What sort of structures stiffens under compression". Well, to reduce it to the simplest case I can think of, although I don't suppose you'd call this a structure, consider first say a 2" diameter 6" long steel rod. Under axial compression, the transverse natural frequency decreases, according to classical theory. Now suppose you did this with a material like hard rubber, which can deflect a lot. You would get stress stiffening partly because of the material non linearity, and partly because the very large deflection would increase the second moment of area and shorten the rod. Both these effects would completely overwhelm the weakening effect and the transverse natural frequency would increase. It is also possible to envisage mechanical analogs of this in which large deflections of structural components lead to increased stiffness. But all these effects require some kind of non linearity, either from large deflections , material non lenearity, or the effect whereby a force in one direction changes the geometry such that forces in another direction suddenly have a marked effect. I cannot see how kumar's case should lead to stiffening if none of these effects is present, and for a purely radially compressed ring of linear material, I don't believe any of them are. But I too am ready to learn.
 
Thanks Englishmuffin! That's a very interesting answer to my last question. I would certainly agree your example is a structure and I daresay you could devise some kind of structural/mechanical system with the same sort of behaviour at large deflections. The classic case is post- buckling of a flat compressed plate, I suppose, but this is post-buckling response, not Kumar's simple eigenvalue problem.

I'll watch this space!
 
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