Predicting the spring constant theoretically

[hacivat]

I take a wire of metal X which has a diameter d. Let the total length of it be L and I roll it around a cylinder with diameter D to create a spring. Is it possible to predict the spring constant of this system (and relate it to the elastic constants of the metal)?

Has anybody seen/heard/know a method how to do that? Thanks in advance...

 

[pasmith]

In principle this should be possible (how else would manufacturers produce springs with a specified spring constant?). However I don't know if the geometry admits an analytical solution to the Cauchy equation, or whether a numerical method is necessary.

 

[jbriggs444]

One can get at least part way to a solution by observing that a coil spring operates as a torsion device. Compressing the coil puts torque on the wire. So one would begin by calculating the incremental twist which would result from an incremental compression.

If one ignores the curve in the wire and assumes, perhaps counter-factually, that the strain in the wire is purely rotational and is uniform (i.e. proportional to distance from the wire center) within any given cross-sectional disc then it should be simple to calculate the stress distribution across a representative cross-section for any small incremental twist based on the shear modulus for the material.

Integrating stress times an area element times a moment arm across a cross section should yield torque. Torque times twist is equal to compression distance times force. We started with the relationship between compression and twist.

But then, I've never taken a course in this stuff. So take my advice with a grain of salt.

 

[Vanadium 50]

I would split this into two problems:

The first is geometry: if the spring is stretched by an amount ΔL., how much does the metal itself stretch.

The second is physics: given the Young's Modulus for the metal, what force is required to stretch the metal a given amount.

Put the two together and you have the spring constant.

 

[jbriggs444]

[...]given the Young's Modulus for the metal, what force is required to stretch the metal a given amount.

For a coil spring, I think you want the shear modulus, not Young's modulus. Torsion will involve a shear strain.

For a leaf spring, I think you would indeed want Young's modulus, not the shear modulus. Bending will involve a lengthwise strain.

 

[jrmichler]

Yes. The calculations for coil spring design are well known to mechanical engineers. You can use search terms coil spring calculation or you can go to the definitive source - the SMI Handbook of Spring Design: https://smihq.org/store/ViewProduct.aspx?id=8525988.

The coils of a coil spring are stressed in torsion.

 

[Vanadium 50]

That's part of the reason I wanted to treat the geometry separately: coil spring, leaf spring, flat spring, Belleville spring...