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.)