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natural frequency of a radially compressed ring
3

natural frequency of a radially compressed ring

natural frequency of a radially compressed ring

(OP)
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

RE: natural frequency of a radially compressed ring

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.

RE: natural frequency of a radially compressed ring

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

RE: natural frequency of a radially compressed ring

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
www.tankindustry.com

RE: natural frequency of a radially compressed ring

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

RE: natural frequency of a radially compressed ring

2
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.

RE: natural frequency of a radially compressed ring


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.   

RE: natural frequency of a radially compressed ring

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.

RE: natural frequency of a radially compressed ring

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.

RE: natural frequency of a radially compressed ring

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 !!

RE: natural frequency of a radially compressed ring

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.

RE: natural frequency of a radially compressed ring



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.

RE: natural frequency of a radially compressed ring

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.

RE: natural frequency of a radially compressed ring

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

RE: natural frequency of a radially compressed ring

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

RE: natural frequency of a radially compressed ring

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.

RE: natural frequency of a radially compressed ring

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.

RE: natural frequency of a radially compressed ring

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 ??

RE: natural frequency of a radially compressed ring

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.

RE: natural frequency of a radially compressed ring

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!

RE: natural frequency of a radially compressed ring

(OP)
Hi,

Thanks all. I was out of my office and didn't get a chance to access this forum. I am very excited to see so many replies.

I ran the eigen value analysis with different loads and these are the results.

Nodal force: -1.75 lbs (radial compression)
      1st Frequency: 3.76 Hz (In-plane bending mode)
Nodal force: -3.5342 lbs (radial compression)
      1st Frequency: 5.04 Hz (In-plane bending mode)
Nodal force: -4.36 lbs (radial compression)
      1st Frequency: 3.22 Hz (In-plane bending mode)
Nodal force: 4.36 lbs (radial tension)
      1st Frequency: 16.1 Hz (In-plane bending mode)

Btw, the carbon-epoxy composite ring is 3 meters in diameter. Cross-section of the ring: O.D: 0.726", thickness: 0.03"

I tried the elastic buckling analysis too. I designed the ring for a total radial compressive load of 558 lbs using the elastic stability equation given by Timoshenko in "Theory of Elastic stability" (page 291). The finite element model had a load of 279 lbs. I was expecting a buckling load multiplier of about 2, but the analysis gave a multiplier of 1.1. I think I am missing something. Thanks for the help.

Regards,
Kumar

RE: natural frequency of a radially compressed ring

I think you've got a most unusual system, looking at the trend in the first three results. How many nodes are you applying these forces to?

What sort of FEA analysis are you doing?

What is the predicted linear first mode, by analysis?

Cheers

Greg Locock

RE: natural frequency of a radially compressed ring

With such a small thickness, I wonder if the elastically calculated deflections are so high that the system results may not be valid.  I'm assuming you are analyzing it on basis of small deflection theory.  If deflections are of same magnitude as ring thickness perhaps it is a large deflection theory problem.

Steve Braune
Tank Industry Consultants
www.tankindustry.com

RE: natural frequency of a radially compressed ring

Well, at least the increased frequency under tension might make sense.
Questions:
What does the mode shape look like ? I assume it's some sort of elliptical deformation, not the pure radial deformation that most of my comments referred to.
What's the frequency with no load ?

RE: natural frequency of a radially compressed ring

Hi Kumar,
Yes you certainly started an interesting thread ! It looks even more like a glitch to me. The proportions of the structure are a bit unusual, but that shouldn't make any difference in the sort of eigenvalue analyses you're doing. I would suspect some kind of numerical problem in Algor. Why not contact the supplier to see what they say, unless you have access to another analysis package to do a comparison with ?  Or you could try running the analysis again using different units, that might shed some light on  a numerical problem, if there is one.

There is another possibility, and I don't have time to do any arithmetic myself on this (it's a long shot too), if the total radial compression you are using in any of the nat freq analyses is anywhere near the buckling load, you could get strange results.

Best of luck

RE: natural frequency of a radially compressed ring


Now that I've read the question (again), my comments do not apply to a ring loaded at the nodes. You might look into  "shear locking" if your stiffness is increasing with node count.



RE: natural frequency of a radially compressed ring

(OP)
Hi,
The mode shape is elliptical deformation. One thing which I forgot to mention is the constraints in the ring. The ring is in the XY plane. The model (beam elements) was constrained with Txz at the top and bottom nodes and Tyz at the two side nodes. The load's been distributed in 64 nodes on the circumference.

Frequency with no load is about 0.008 Hz (a 3-m ring vibrating at 0.48 cycles every minute, I don't understand). Unfortunately, I don't have access to any other software. I will try to post my findings as I work on this.

Thanks all..
Kumar

RE: natural frequency of a radially compressed ring

One silly question regarding the (apparently) rather low frequency - I assume your units are consistent ? If you are working in metric they probably are - but in lbm/lbf units you could have a gc multiplier missing.

RE: natural frequency of a radially compressed ring

(OP)
I am working in English units (lbf, inches..). I am positive that the units are consistent. What's a gc multiplier? How would that affect the frequency?

RE: natural frequency of a radially compressed ring

Well, if they are consistent, that's OK. By consistent, I mean of course that F=M*a is numerically true when using those units. In lbf/lbm units, this is not the case, and F*gc=M*a, so natural frequency is sqrt(K*gc/m). In the lbf/lbm/in system, gc would be 386.4. But I expect you know all this , in which case my apologies - although it's always a good idea to ask silly questions just in case.

RE: natural frequency of a radially compressed ring

I suppose I should also have mentioned that consistency is often achieved by using the so called "weight density" instead of the "mass density". I don't know much about Algor, but FEMAP has a place you can enter gc directly for dynamics problems.

RE: natural frequency of a radially compressed ring

(OP)
Sorry, my brain could not figure out that gc stands for gravitational constant. I am using the mass density here (lbs/in^3)/(gcc). I tried the analysis running with zero gravity and the result is still the same. Anyways, I think all the analysis cares is whether the acceleration units are in/sec^2.  Please correct me if I am wrong.

RE: natural frequency of a radially compressed ring

Well, if you are using lbf/lbm/in inits, and you used lbm/in^3/386.4 for the mass density, that should be OK as far as I can see. I don't quite follow you when you say you ran with "zero gravity" (which of course has nothing to do with the units). But it doesn't surprise me that the natural frequency is unaffected by the presence of absence of a gravity field, if that's truly what you meant.
Have you tried increasing the number of nodes yet ? My guess would be that at least some of the anomalies would disappear. But none of it makes sense to me so far.

RE: natural frequency of a radially compressed ring

By the way, Blevin's gives a closed form solution for the in-plane flexure modes for the ring.

f(Hz) = i*(i^2-1)/(2*pi*R^2*sqrt(i^2+1))*sqrt(E*Iz*gc/m)

where i = 1,2,3 etc (in this case i=2, since i=1 is pure translation)
R = radius of ring
E = youngs modulus
Iz = second moment of area of ring section (using your xyz)
m = mass density per unit length
I put the gc in for clarity.

How does that stack up against your .008 Hz ?

RE: natural frequency of a radially compressed ring

Sorry, I meant m = mass per unit length.

RE: natural frequency of a radially compressed ring

Table 9.1 for those who are interested.

I get 12.75 Hz for a steel ring - carbon won't be much better.

I think the non linear solver is probably getting confused by the relatively large static forces compared with the stiffness of the structure, small number of nodes, and rather odd constraints.

Anyway, it sounds as though either you have very fundamental modelling problems, or (Heaven forbid) I've made a mistake.

Cheers

Greg Locock

RE: natural frequency of a radially compressed ring

Well, I get about 10.7 Hz for steel - probably because I used a slightly different density. If he does a solution without the radial load, that shouldn't require a non-linear solver. Looks like we both assumed that this is a hollow tubular ring. I hope this isn't one of those Morton Thiokol solid rocket booster "O" rings or something!

RE: natural frequency of a radially compressed ring

Actually, I think I made a mistake - I get 6.2Hz for steel. But anyway, it sounds in the ballpark and could be somewhat consistent with the radially loaded results. Awaiting further info.

RE: natural frequency of a radially compressed ring

I have replicated this problem as best I can using Strand7.

I get 5.994 Hz for an unloaded STEEL ring, using a diameter of 3 metres, cross section is circular tube, OD 0.726" (18.44 mm), wall thickness is 0.03" (0.762 mm). (I used steel because it has relatively "standard" material properties. E = 200 GPa, Density = 7,870 kg/m3, Poisson's Ratio = 0.25.) The first mode shape is elliptical in-plane vibration, as expected.

My value for the frequency of the same ring using EnglishMuffin's formula is 5.986 Hz, which agrees to 0.13% with my FEA result.

In my FEA analysis, the frequency drops SLIGHTLY when the ring goes into compression, and rises SLIGHTLY when the ring goes into tension, all as expected. I kept my ring compression load significantly less than the buckling capacity of the ring. You would expect to see a dramatic drop in frequency if the ring compression approaches the ring buckling load. Similarly, a significant increase in frequency could arise if the ring tension becomes “significant”. I kept my axial loads (tension and compression) to less than 15% of the buckling load. My frequencies only changed by about 1.5%. I didn’t see any frequency changes of the order of magnitude of those reported by cspkumar.

I suspect the very low frequency of 0.008 Hz is actually a free-body mode. Check your constraint conditions to make sure your model is actually constrained properly. When I analyse my ring using 2D beam DOFs, and no other restraints at all, I get 3 zero natural frequencies (X translation, Y translation, and Z rotation), before my first “real” frequency of 5.994 Hz. If I constrain my ring with minimum constraints to permit normal linear static analysis (similar to cspkumar’s description), my first reported mode shape is an elliptical mode at 5.994 Hz.

Apart from this, I can't account for the apparent behaviour of frequency increasing under slight compression, and then decreasing again. It sounds like a modelling problem to me. I would check all units carefully, as well as model constraints.

In particular, is there a possible problem with confusing mass, weight and force units? In the metric world, we are lucky that the only mass unit we need to know is the kilogram, and the only force/weight unit we need is the Newton. In the foot-pound-second / inch-pound-second world, you need to be VERY careful to not confuse the pound-mass, and the pound-force – they are NOT equivalent. I believe that the “pound mass” is defined as that mass which has a weight of one “pound force” when accelerated at one inch/s/s. This is approximately equivalent to 386.4 “pounds” (as in “a pound of sugar”), or 175.24 kilograms. An error in density or force of this order of magnitude could result in alls orts of unexpected results!

(My understanding of the “pound mass” could be wrong, or perhaps there is more than one “common” definition of the “pound mass”. My understanding comes from “Building Better Products with Finite Element Analysis” by Adams & Askenazi. As I said, in the metric world, we rarely have to deal with this confusion.)

RE: natural frequency of a radially compressed ring

JulianHardy: Well, your results appear to make sense, and your frequency for steel roughly agrees with mine. Greg must have screwed up (as I did)! It occurred to me also that the .008 Hz could be a free body mode - in other words it should have been zero. But it all depends on the inner vagaries of Algor, which you probably did not use. I think he's OK on the units, although I could be wrong.
As far as the definition of pound mass goes, the situation is confused because there are two different systems employing english units, one based on force and the other on mass. In the UK, the lbm used to be based on a physical standard made of platinum. Today, for the f lbm sec system it is simply defined in terms of the Kilogram. In the case of the f lbf sec system, your definition is not even approximately correct. You should have said that a pound mass is defined as that mass to which a pound force imparts an acceleration of 386.088 in/sec^2. However, we seem to be in basic agreement. I suspect that with more nodes, kumars results will look better.

RE: natural frequency of a radially compressed ring

(OP)
Hi all,
It was a software glitch. Somehow it got corrupt and I had to delete and load it again. Now, the first real mode for an unloaded ring is out-of-plane flexure mode with a frequency of 11.75 Hz. The first in-plane flexure mode frequency is 12.1 Hz, which agrees with Englishmuffin's equation(3% error, I am sure like what englishmuffin said, if I use more nodes, I would converge to the theory). Even the buckling analysis worked. The code came back with 2.013, which is what I expected.
Thanks for all the replies. It was very very informative. Thanks again,
Kumar

RE: natural frequency of a radially compressed ring

Interesting that GregLocock got close to the right frequency using the wrong material with an apparent additional error in his calculation. Some people just can't go wrong!

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