0
$\begingroup$

Problem set up

I'm trying to work out the angle subtended by the stretching of a conical spring and I have a solution but I'm not 100% sure about it so would appreciate some feedback.

An Archimedes spiral spring is shown above with radii at various points. What I would like to come up with is an equation for the angle that each part of the spring gets stretched by as a function of the radius and the distance from rest $x$. I know that for a normal spiral spring of constant radius $R_{ss}$ if it is stretched by a small amount $dx$ then it will subtend an angle $d\theta$ therefore $$ dx=R_{ss}d\theta \\ \int_0^{x}dx=R_{ss}\int_0^\theta d\theta \\ \therefore \theta=\frac{x}{R_{ss}} $$

For a spiral conical spring with radius $R_{cs}$ that starts at $R_{cs0}$ and grows to $R_{cs1}$ what I did was integrate the radius: $$ dx=dR_{cs}d\theta \\ \int_0^{x}dx=\int_{R_{cs0}}^{R_{cs1}} \int_0^\theta d\theta dR_{cs} \\ \therefore \theta=\frac{x}{R_{cs1}-R_{cs0}} $$ It seems too simple though hence why I would like to check here what the thought is on the solution.

$\endgroup$
0
$\begingroup$

No, Its not right. If the cone becomes more, steep at the limit when $$ R_{cs1} \Longrightarrow R_{cs0} \quad \theta \Longrightarrow \infty $$

$\endgroup$
  • $\begingroup$ My apologies but I'm not sure I understand your answer. Since it's an Archimedean spiral the cone doesn't become more steep because the slope of the cone is constant. $\endgroup$ – enea19 Oct 16 '19 at 7:42
  • $\begingroup$ @enea19 ok fine, but lets assume we started with a more steep cone, it is going to have smaller theta, but by your calcs it will have a bigger one. $\endgroup$ – kamran Oct 16 '19 at 19:53
  • $\begingroup$ Aah, I see what you mean. But isn't that ok? I mean, a conical spiral spring that has less than a single turn is not very useful. Mathematically I understand the problem, but in practice it would never happen. $\endgroup$ – enea19 Oct 22 '19 at 16:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.