# Calculate number of turns in Archimedes spiral

• evilbrent
In summary, the conversation is about an engineer designing a spring system for a garage roller door. They need to know the number of turns for various door sizes and have found an equation for calculating the length based on the number of turns, starting radius, and gap between spirals. They are looking for help with finding an expression for the number of turns in the equation. A suggestion is made to use two substitutions for the methodology.
evilbrent
Hi,

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=$^{2∏n}_{0}$ $\sqrt{(a+bθ)^2+b^2}dθ$

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.

Last edited:
Hey!

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

For methodology, you can do two substitutions: first $v=a+b\theta$

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

hope that helps a bit!

Ok, I'll see how I go. Thanks a lot.

## What is an Archimedes spiral?

An Archimedes spiral is a type of spiral that was discovered by the ancient Greek mathematician Archimedes. It is formed by drawing a series of tangent lines from a point on a line as it rotates around a fixed point at a constant rate.

## How do you calculate the number of turns in an Archimedes spiral?

The number of turns in an Archimedes spiral can be calculated using the formula n = r / a, where n is the number of turns, r is the distance from the fixed point to the starting point of the spiral, and a is the distance between each turn of the spiral.

## What is the significance of the number of turns in an Archimedes spiral?

The number of turns in an Archimedes spiral is important because it determines the shape and size of the spiral. It also has practical applications in fields such as architecture, engineering, and art.

## Can the number of turns in an Archimedes spiral be a decimal or a fraction?

Yes, the number of turns in an Archimedes spiral can be a decimal or a fraction. This is because the formula for calculating the number of turns allows for any real number value for n.

## How is an Archimedes spiral different from other types of spirals?

An Archimedes spiral is different from other types of spirals because it has a constant distance between each turn, whereas other spirals may have varying distances between turns. Additionally, the Archimedes spiral has a linear relationship between the distance from the fixed point and the angle of rotation, whereas other spirals may have a more complex relationship.

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