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# Helical Compression Spring with Elliptical X-Sxn

## Helical Compression Spring with Elliptical X-Sxn

(OP)
Hello all,

Does anybody have a reference that gives spring rate (stiffness) of helical compression springs (valve springs) made of wire with an elliptical cross-section? I've derived a formula on my own, but it doesn't agree with a formula that I found in an ASME paper. The ASME paper doesn't give any background (it is primarily concerned with stress levels in the wire), it simply states that the formula is obtained by modifying standard formulae. When the major and minor radii are set equal, our formulae are mathematically identical, but for truly elliptical sections they are not.

I'm currently awaiting measurement info from the hard parts. That should tell me who's right, but I'd like to have another &quot;theoretical&quot; reference to compare with, especially in the case that I'm wrong. The elliptical wire case is just a stepping stone towards my true goal, which is to develop a formula for springs with a peculiar wire cross-section, so it is important that I gain an understanding of the physical principles involved. I thought I had the problem all worked out, but then I couldn't reconcile my formula with the reference material.

A derivation would be particularly appreciated.

Any help would be appreciated...

### RE: Helical Compression Spring with Elliptical X-Sxn

SAE Technical Paper 932891, Design and Testing of Ovate Wire Helical Springs is a good reference on this subject.  It was authored by 7 different individuals, representing spring manufacturers, academia, a spring wire vendor, and an automotive OEM.  It references several other technical papers and patents on ovate wire springs, including SAE Technical Paper 920775.  SAE Technical Papers can be obtained from the SAE International website: http://www.sae.org .

I have had some difficulty in verifying the patents, but you may try to find them yourself. The Japanese Patent No. is 25-13809 by Shiguma in 1950. The US Patent was in 1959 by Fuchs-- the No. listed is 814043, but that is not correct one (I checked). The No. needs to be more like 2,xxx,xxx to fit in the proper timeframe.  Full-text searching of US Patents is available at http://www.uspto.gov/patft/index.html, but it is limited to post-1975 patents.

Anyway, the equations listed for elliptical and Fuchs egg-shaped helical springs are as follows:

elliptical
taumax (shear stress) = ((8 * Dg * P) / pi * de3) * kappa

k (spring rate) = ((G * de4) / 8 * n * Dg3) * xi

where xi = (2 * r) / r2 + 1
r = aspect ratio, r = w/t
kappa = stress correction factor, kappa = a0 + a1/c + a2/c2 where a0, a1, & a2 are functions of r
Dg = (Di + Do) / 2
D = mean coil diameter
n = # of active coils
de = diameter of round wire equivalent to ovate, de = sqrt(wt)
w = width of ovate wire
t = thickness of ovate wire

Fuchs
equations are the same except kappa is different, and Dg = ((Di + Do) / 2) + {1-8/(3*pi)}(w-t)

There are a couple of figures and a table that give the values for a0, a1, & a2 as a function of r.  Let me know if you have any questions about any of this.

### RE: Helical Compression Spring with Elliptical X-Sxn

(OP)
Thanks, I'll check our library for that paper.

a couple of questions about the formulae above, just to make sure I've got them right:

is  xi = (2*r)/(r^2+1), or is it xi = ((2 * r)/r^2) + 1

is k (spring rate) = ((G * de^4) / (8 * n * Dg^3)) * xi or is it k = ((G * de^4) / 8) * n * Dg^3 * xi

thanks again...

### RE: Helical Compression Spring with Elliptical X-Sxn

(OP)
nevermind the above questions, I've got it straightened out.
Thanks.

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