Equivalent spring equations for non-helical coil shapes?

Question

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

Answer 1

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.

Answer 2

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:

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.

Answer 3

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 $$\delta = \frac{{\rm d}U}{{\rm d}P}$$.