Trying to find an equation for spiral springs

I have been looking at using spiral springs to constrain movement of a rod so it can only move linearly, without the friction of linear bearings. I’m curious if anyone has run across a program to generate the spiral cuts required. If nobody has I’ll probably end up generating my own, but I feel like this is something that should exist, where you can provide material thickness and Young’s modulus and the travel you need, and it’ll generate the code to make the spring. Any thoughts?

Not sure if this is what you are looking for but if you have access to a 3D printer, this would be the quickest way to do it using openscad.

Here is another solution in openscad

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@HalfNormal these are different springs. The goal is to create a constraint, so it’s cutting a spiral out of flat stock, such that the material is held in two axes and can move in one. But instead of being the normal spiral torsion spring, being used to constrain motion to one direction.

@John_Bump since this is to replace a bearing, I assume you mean to rigidly attach the rod to the spring(s). Does the slight rotation of the rod when the spring is expanded in Z (where X,Y is the flat plane of your stock) not cause a problem for your application? See Ortho-Planar Linear-Motion Springs for whole classes of non-spiral flat springs which have no rotational moment.


Unlike geophone (spider) springs, these ortho-planar springs do not experience significant rotation of the platform.

That patent is dated 18 years ago and is assigned to the US Government so I presume it’s open without looking further. Look at figures 13-15 to see examples of how those springs could stack. For your case, if rotation would cause a problem, there’s a pretty obvious solution of stacking a set of zig-zag shapes in a Tri-n-1R for large n (10? 20?) using the nomenclature on page 4 of that patent. The goal of their patent was a little different from what you are looking at; they wanted the axis to be stable with a single spring; I’m inferring that you plan to use sets. It would look like three living hinges arranged radially.

That doesn’t actually answer your question about existing code. :roll_eyes:

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Not what you want but useful for spring design…

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That patent led me to geophone springs, which are generally very similar to what I’m trying to make, so that gives me a lot more material to search through. Thanks, @mcdanlj! This is for a crosshead for a high frequency stirling engine, so slight rotation is not a problem at all. (In fact, some use air bearings that rely on rotation to provide the air bearing seal, but I don’t have the facilities to fabricate pistons and cylinders with the tolerances required for that kind of shenanigans.)


I definitely saw examples of planar springs for stirling engines, so you wouldn’t be alone. Looking forward to project pictures if you are willing to share. It makes sense to me that a geophone spring is similar in purpose.

As in all the geophone springs I saw, I think you would want a symmetric (at least double-spiral) spring to balance the side forces. An asymmetric (e.g. single spiral) will pull a little to one side. (Thought experiment: pull a single spiral out until it is completely straight, and it will rotate for each spiral turn and center attach point in alignment with the outer attach point. A double-spiral will have equal and opposing twist forces at the center, and stretched all the way out will end up with matching helical twists in the arms and have the center attach point still centered.)

Here are some shapes of planar springs but I don’t see words to accompany them. Most of those are symmetric.

For others curious about “geophone springs” see this patent which has what I believe are typical applications.

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I’ve made coiled springs with a jigsaw, being just a single spiral, and they were kinda terrible. I’d definitely do a symmetric one. (Well, in fact, I’m printing one right now on the 3d printer, and if I like it I’ll make a larger one out of acrylic on the laser cutter tomorrow.)
I’ve spent a lot of the evening reading other patents about them, and am trying to get FreeCAD to perform a finite element analysis on one, but getting the constraints set up is challenging.