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The compression spring equations are generally given for helical coil. What are the equivalent equations for alternative coil shapes, like oval?

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Just to get the Nomenclature straight, the spring is still a helical spring, just the cross section can change between circular, ovate, square or trapezoidal. If the profile of the outside diameter changes along the spring, then it is called a beehive spring. –  ja72 Nov 13 '12 at 0:25
    
I'm not referring to cross section, but coil shape. –  ThePeopleWantToKnow Nov 26 '12 at 6:36
    
So what you are looking really is deflection of curved beam loaded perpendicular to the coil? –  ja72 Nov 26 '12 at 14:28
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3 Answers

The mechanics of all spring problems is derived from Hooke's law which models the properties of springs for small changes in length. Even for helical springs this law is a (very good) approximation. You can consider different spring geometries, different materials, larger changes in length... and this approximation will obviously worsen. Of course one can try to create more complicated models to take this matters into account, but it's generally not worth it and, almost certainly, you are never going to see different equations for spring dynamics.

For further reading on the limitations of Hooke's law click here.

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What would you recommend as a good translation factor from a circular coil to an oval? –  ThePeopleWantToKnow Nov 26 '12 at 6:38
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I'm guessing that the "equations that are generally given" you refer to the ones calculating deflection of a helical spring, based on the torsional deformation of the coil, such as what is found here. The following picture is taken from there:

enter image description here

As you can see, any cross section of the coil is subject to shear, as well as to torsion. If your coil is not perfectly circular, it will have the same shear, but different torsion, depending on the point around the coil you are considering, $T=Fr$, with $T$ the torsion torque, $F$ the axial load, and $r$ the effective radius of the coil.

In the reference above, they compute the deformation from the total strain energy, which is made up of a shear term and a torsional term. In their case it is relatively easy, since the torsion is constant throughout the coil. You will have to integrate around the coil based on the shape of your coil. The exercises here should help in figuring out how to do that.

If you get stuck, shout in the comments, and I can guide you through the calculaitons for a particular case.

Or you can go the quick, dirty way, and copy the formulas found in this discussion.

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I understand the issues with cross section--NASA has a good paper on the benefits of an egg shape. My problem is with coil shape. Think rectangular / magazine springs--which are typically oval in form today. –  ThePeopleWantToKnow Nov 26 '12 at 6:40
    
So the link I gave deals with round helical springs of non-circular cross-section, did not ready it carefully enough. But my explanation should allow you to calculate the deflection for a non circular helicoid. It's the same method proposed by in ja72's answer, by the way. –  Jaime Nov 27 '12 at 0:32
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Start by reading the deflection of curved beams, especially the part on how to solve this with strain energy method.

http://www.codecogs.com/reference/engineering/materials/curved_beams.php

Next apply this method to combined torsion, shear and bending for your geometry. The strain energy is $$ U = \int \frac{T^2}{2 G J}\,{\rm d}l + \int \frac{F^2}{2 G A}\,{\rm d}l + \int \frac{M^2}{2 E I} {\rm d}l$$ where $T$ is torque, $F$ shear force, $G$ shear modulus, $J$ polar second moment of area, $A$ area, $M$ is bending moment, $E$ modulus of elasticity, $I$ second moment of area and ${\rm d}l$ a small section of the spring.

Typically for helical spring the moment component is negligible.

Once you have the total strain energy equation as a function of the applied loading $P$ the the deflection along the load axis is $$\delta = \frac{{\rm d}U}{{\rm d}P}$$.

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