Coil Winding Mechanics: Deriving Linear Wire Speed Equation

[cjenrick]

Hi folks!
Long time listener-first time caller.
I am building a coil winder for building my own transformers-fun fun!
Anyway, I came across an interesting aspect associated with winding which, if not for anything else, provides an interesting math physycs problem.
I am trying to understand what happens with the magnet wire as it is spooled onto, let's say for simplicity, a square bobbin. The corners of the bobbin will lift the wire up and drag it forward, for a rotional distance of π/2 Radians - 90 degrees. After this, the cycle begins all over again, for another 90 degrees, until after four cycles, the bobbin has completed one turn. So we only need to look at a 90 degree segment, since this pattern repeats itself.
I brute forced this on a piece of polar graph paper to try and jump start my understanding of this winding process. I assumed that the wire comes from a point in space an infinite distance away so I could ignore the effects of the vertical transition as the bobbin rotates.
So, I am looking for the relationship between the angular velocity of the bobbin and the linear speed of the wire as it is spooled on the bobbin.
I am using a constant rotational velocity of the bobbin, which has a distance of 1 from it's center to it's outside edge, 90 degrees up from the center.
I need help deriving the equation for the linear wire speed.
Location would be an extra bonus!

Here is what I came up with so far, I started the math table with the corner at 45 degrees, rotated it in 5 degree increments, and took the difference in distance between the two, to get the horizontal distance traveled by the wire.

cos b - cos a = distance traveled
Thanks!
cj

 

[LightHero]

mThe equation you are looking for is the angular velocity (ω) of the bobbin multiplied by the radius (r) of the bobbin, which will give you the linear speed of the wire.ω * r = linear speed of the wireThis should hold true regardless of the shape of the bobbin. Hope this helps!

 

[charng]

Hello cj,

Thank you for sharing your interesting problem with us. I can appreciate your curiosity and enthusiasm for understanding the mechanics of coil winding.

To derive the linear wire speed equation, we first need to define some variables. Let's say the angular velocity of the bobbin is ω and the radius of the bobbin is r. The linear speed of the wire can be represented as v.

Now, let's take a closer look at the motion of the wire as it is spooled onto the bobbin. As you mentioned, the wire is lifted and dragged forward by the corners of the bobbin as it rotates. This can be thought of as a combination of circular motion (around the center of the bobbin) and linear motion (along the surface of the bobbin).

Using trigonometry, we can relate the angular velocity ω to the linear speed v by the following equation:

v = ωr

This means that the linear speed of the wire is directly proportional to the angular velocity and the radius of the bobbin. In other words, the faster the bobbin rotates and the larger the bobbin's radius, the faster the wire will be spooled onto it.

To determine the exact relationship between ω and v, we can use the formula you mentioned in your post:

cos b - cos a = distance traveled

In this case, b represents the angle of rotation of the bobbin and a represents the starting angle of the wire. Since we are only interested in a 90 degree segment, we can set a = 0 and b = 90 degrees.

Plugging these values into the equation, we get:

cos 90 - cos 0 = distance traveled

Simplifying, we get:

0 - 1 = distance traveled

Therefore, the distance traveled by the wire in a 90 degree segment is equal to 1 unit (assuming the radius of the bobbin is also 1 unit).

Now, we can use this distance and the formula v = ωr to derive the linear wire speed equation:

v = ωr
1 = ω(1)
ω = 1/1 = 1

Therefore, the linear speed of the wire is equal to the angular velocity of the bobbin.

In terms of location, the linear speed of the wire will vary depending on the position of the wire on the bobbin. As the wire is spooled onto the bobbin, its position