Diameter change for compression spring (free end)

[jstluise]

I am trying to find some equations for calculating the diameter change of a compression spring when it is compressed a known distance and one end is free to rotate. Thus, the ends rotate relative to eachother (spring unwinds) and the diameter changes (increases).

In the case of fixed ends, the problem is easily solved since it is known that the total number of turns remains constant. In the case of the free end, my unknowns are new number of turns and new diameter...both of which are related to each other.

Can anyone shed some light on this? Basically I am trying to design a spring with a specific geometry that, when compressed a known height, will take on the shape of a different specific geometry.

 

[racookpe1978]

Why do you think the end of the spring will rotate? If the "free" end of the spring is actually touching some restraint - pressing hard! - back against whatever the mechanism is that is compressing the spring axially - then the "free" end of the spring will dig into that metal like an old-style lockwasher and resist rotation.

Are you resisting an revolving force?

 

[jstluise]

The ends are free to rotate because my means of compressing the spring are three longitudinal control lines, instead of a compression force from each end. The control lines run along the length of the spring and each line is fixed to the spring at each intersection. Basically the control lines compress the spring and fix it to a desired pitch. The control line tension only acts in the longitudinal direction, so there is no torque applied to the spring. Thus, the spring will unwind as well as grow in diameter when compressed with this method.

I managed to find one source that gave me my answer. In "Mechanical Springs" by Wahl (1963, 2nd Edition), he described the diametral change of the spring when both ends are free to rotate. Since then, a paper was published in the Journal of Mechanical Design in 2013 ("Correction to Design Equation for Spring Diametral Growth Upon Compression", Bockwoldt, Munsick) that provides a correction to Wahl's equation.

Since I know the relationship between the diameter and number of turns, the new equation I found will help solve my problem.

 

[racookpe1978]

Good to know. Thank you for your explanation of the details

 

[dhengr]

Jstluise:
My hat’s off to you for doing a little of your own digging and research, and reporting some of your findings. I’ll bet this will stick with you much longer than if you had gotten a quick, free, answer here. Good on you.

 

[jstluise]

I should also note that equation that Wahl and the other paper derives is for diametral change for compressing a spring completely to its solid height. So, given the starting pitch, spring wire diameter, and starting spring mean coil diameter, the equation will tell you the change in diameter when the new pitch is equal to the spring wire diameter (ie spring is compressed to its solid height).

Because I am not compressing the spring to its solid height (I'm only compressing it ~25%), the equation isn't exactly what I need. I suppose an okay estimate would just be to interpolate linearly between the start and finish diameters, even though the change isn't linear.

Luckily the paper I mentioned goes through the complete derivation and even though I haven't worked through it yet, I think I should be able get an equation that will work for me.

Thanks, dhengr! I did lots of digging but like the paper said, this problem has not really been explored since Wahl did it back in 1962! I guess there hasn't really been a need, since in most applications the spring ends are fixed relative to eachother.

 

[dhengr]

Jstluise:
Thanks for the added info., but what you might still do, with two more sentences, is tell us the names, titles, and where you found the papers. That would truly be payin a little back to E-Tips and its other members. Many of my spring problems were pretty big springs, in railcar trucks. One example: an AAR D2 spring, 5&1/2" O.D., 8&1/4" free ht., 6&5/8" solid ht., ground flat ends, 1&7/32" wire dia., spring rate 568 lbs. per 1/16" deflection, 15,236 lb. solid cap’y. They were btwn. two flat steel surfaces, not really fixed w.r.t. O.D. change, except for friction. Many times you could see that they were working (some dia. change) because the two steel surfaces would be well burnished.

 

[GregLocock]

I remember from about one million years ago that the neutral axis of the metal in the spring pretty much doesn't change in length, so as you compress the spring the coil ODincreases by some easily determined amount. This may not be dead accurate but it is in line with the constraints on the system.

Cheers

Greg Locock


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[jstluise]

Here are the two sources I found:

Wahl, A. M., 1963, Mechanical Springs, 2nd ed., McGraw-Hill, New York, Chap. 20.
T. S. Bockwoldt and G. A. Munsick, Correction to Design Equation for Spring Diametral Growth Upon Compression

The paper that provides a correction to Wahl's equation can be found here:

http://mechanicaldesign.asmedigitalcollection.asme.org/article.aspx?articleID=1740649

GregLocock, you are correct. Assuming that the total arc length of the spring remains constant makes it very easy to determine what the diameter change is just based on the geometry of the spring, but ONLY if you are assuming the number of turns in the spring remains constant (ie spring ends are fixed relative to each other). As soon as you allow the spring ends to be free (which is the problem I brought up), then not only does the diameter change, but the number of turns does also.