# Relation between torsional and linear spring constants for a cantilever beam?

Hello,

This should be a straight one for most of you. Given a cantilevered beam that has a force F applied across it (or at one end), causing a displacement d and deflection $$\theta$$ , what is the relationship between the torsional spring constant k$$_{theta}$$ and the linear spring constant k?

What I do know is that the linear spring constant can be expressed in terms of the moment of inertia and modulus of elasticity as follows:

k = $$\frac{3EI}{l^{3}}$$​

I would like to know how to derive the relation between the two spring constants. In my problem, the parameters that I have are E, I, length, and $$\theta$$ .

If you could point to some website that derives this, that would be good too.

Mapes
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What does the cross-section look like? The relationship depends on this parameter.

The beam is cylindrical.

I'm trying to understand the effects that various force distributions will have on the beam: from
1) a constant force acting on the free end,
2) a force that varies both across the beam length and across space.

And other force distributions.

But simple cases first.

Mapes
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Do you know the equation that relates torsion and twist for a rod? It should be pretty to combine this with the cantilever deflection equation to relate the two spring constants. http://www.engineersedge.com/beam_calc_menu.shtml" [Broken] might be useful.

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Unfortunately, I don't have a background in this subject. Not since freshman year in college.

The site that you linked to was good. But I'm not aware of the relationship that you mentioned.

There should be some scientific paper on this subject, if not a basic derivation, that I could use to pick up ideas from?

Mapes
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This level of mechanics is well established enough to appear in introductory textbooks and reference books. There's some information on the Wikipedia page http://en.wikipedia.org/wiki/Torsion_%28mechanics%29" [Broken], and any mechanics of materials book (e.g., Beer and Johnston) will contain the derivations. What exactly are you trying to do?

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Among other things, I'm currently plotting the variation of strain energy 0.5K$$\theta^{2}$$ with the angle. But I have a limited number of parameters describing the beam, which I believe should be sufficient to get me the torsional spring constant.

Since I have the linear spring constant, I'm hoping that the torsional spring constant would pop out.

From the following page, http://en.wikipedia.org/wiki/Torsion_spring#Applications", I do have an expression, which is simply the torque divided by the deflection angle. I could substitute things along the way, to get this in terms of E, moment of inertia, linear spring constant, deflection angle.

If anybody has any other ideas, do post it here.

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Mapes
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From the following page, http://en.wikipedia.org/wiki/Torsion_spring#Applications", I do have an expression, which is simply the torque divided by the deflection angle. I could substitute things along the way, to get this in terms of E, moment of inertia, linear spring constant, deflection angle.

Sounds good to me!

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Seeing as this thread has had a large number of views, here's the expression that I finally got:

$$\kappa_{\theta}$$ = $$\frac{3EI}{L}$$$$\frac{tan\theta}{\theta}$$

The units seem to match.

This is for a cantilevered beam with a force applied at the free end.

When the angle $$\theta$$ is really small, the $$\frac{tan\theta}{\theta}$$ cancel out, leaving just $$\frac{3EI}{L}$$ in the expression.

Mapes
I agreed too fast before, and without looking carefully at your link. The page you linked to describes torsional springs (coils), not a straight cantilevered rod with a torsional load on the end. If you're interested in a straight rod, I believe the http://en.wikipedia.org/wiki/Torsion_%28mechanics%29" [Broken] you're looking for is $GJ/L$, where G is the shear modulus, J is the torsion constant / polar moment of inertia, and L is the length.