natural frequency of a radially compressed ring 3

[cspkumar]

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

 

[cspkumar]

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

 

[GregLocock]

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

 

[JoeTank]

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

http://www.tankindustry.com

 

[EnglishMuffin]

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 ?

 

[JWB46]

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 ;-)

 

[hacksaw]

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.

 

[cspkumar]

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

 

[EnglishMuffin]

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.

 

[cspkumar]

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?

 

[EnglishMuffin]

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.

 

[EnglishMuffin]

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.

 

[cspkumar]

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.

 

[EnglishMuffin]

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.

 

[EnglishMuffin]

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 ?

 

[EnglishMuffin]

Sorry, I meant m = mass per unit length.

 

[GregLocock]

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

 

[EnglishMuffin]

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!

 

[EnglishMuffin]

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.

 

[jhardy1]

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.)

 

[EnglishMuffin]

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.