Calculate number of turns in Archimedes spiral

by evilbrent
Tags: archimedes, number, spiral, turns
evilbrent is offline
Apr29-12, 08:31 PM
P: 4

I'm an engineer designing a spring system for a garage roller door. I need to know the number of turns of the door for all the size combinations.

I've found this page which gives a good equation for finding the length if you know the number of turns, starting radius and gap between spirals:

The equation of the spiral is r=x+yθ, so x=starting radius, y=gap/2∏, and to find L we're taking the integral from a=0 to b=2∏n (where n=turns).

When you know n, this is straightfoward, and even I could work that out. But it's been a decade since I've done anything like this, so I was wondering if anyone could help me find an expression for n in this:

L=[itex]^{2∏n}_{0}[/itex] [itex]\sqrt{(a+bθ)^2+b^2}dθ[/itex]

Lord help me, my way of solving this is to find L for n=1,2,3,4,5 etc, graph it in excel and use "find trendline" to get an equation. Any help appreciated, thanks.
Phys.Org News Partner Mathematics news on
Hyperbolic homogeneous polynomials, oh my!
Researchers help Boston Marathon organizers plan for 2014 race
'Math detective' analyzes odds for suspicious lottery wins
jacobrhcp is offline
Apr30-12, 05:58 AM
P: 169

If you're only looking for the answer, you can use this.

For methodology, you can do two substitutions: first [itex]v=a+b\theta[/itex]

Then you need to make a second substitution v=b*sinh(u). Donīt forget the derivatives. For more information you can visit this topic:

hope that helps a bit!
evilbrent is offline
Apr30-12, 06:06 PM
P: 4
Ok, I'll see how I go. Thanks a lot.

Register to reply

Related Discussions
Fluids: calculate pressure loss due to turns in pipe flow Mechanical Engineering 9
How to Calculate the Number of Turns of Wire needed on a Iron Core Magnetic Solenoid General Physics 3
Coil turns number Classical Physics 17
Number of Turns on a Loop Introductory Physics Homework 3
Calculate a diver's turns in air Introductory Physics Homework 1