How to determine curvature of face and flank of a spur gear? What is the involute in gear design?

I have difficulty to understand the profile of a spur gear, the part I put in a red box line inside a box, line number 2 in the picture, or the part number 6 and 7. How it is determined? I read or watched many about spur gear. When talking about spur gear, many times there are involute also mentioned. But so far I don't understand what is its relation to gear profile?

How is a spur gear profile is determine?

You could device other gear shapes, in practice only involutes (of a circle) and cycloidal gears are used. However, there are a lot of shape modifications to gears you can make making the subject complex indeed.

The reason why you find the shape hard to understand is because its not one of the commonly used basic shapes (line, circular arc) nor one of the easy graphs of one dimension (paraboloa...) that usually makes up of most designs.

Instead the shape is the involute of a circle. Easiest way for you to make it is to tie a pen on a string around a round object and keeping the string tight as you draw the curve. This is the involute for that particular circle. This makes the curve quite simple in cartesian vector form. Its essentially a rotating vector + a perpendicular vector to that that is as long as the distance traveled. *

Image 1: Formation of the involute shape.

This in mathematical parametric terms forms a function for x and y as for example as follows, you get the following parametric function:

x = r * sin(t) - r * t * cos(t)
y = r * cos(t) + r * t * sin(t)


Where t is the angle in radians and r is the radius of the base circle. Yes that means the involute is different for different circles.

But why this shape? Well, you want generally the surfaces to be rolling instead of sliding since rolling has less friction. So you want the contact to be point like in all possible cases. Now the easiest even contact is a line. if you draw a line for the contact you get either an cycloid or a involute depending on wether the contact line is perpendicular or not to the rotating surface.

The main benefit of the involute is that if the line starts a bit on a different position but moving in same direction its still the same involute. Which means your gear has some tolerance for the gear center distances.

* although you can express the curve as a polar formulation with a one dimensional function. I find that its really hard to understand while the cartesian formulation is easy enough to describe and syhesize from the description especially if your comfortable working with vectors.

• Thank you friend. I just understood their relation after I found one simple explanation with step by step action to draw even it is in Hindi :). But probably should be using the formula the right way to determine the teeth profile. Jan 19, 2023 at 14:37
• Btw, if we see in that picture, its addendum (a) = diameter pitch (Dp)+1m while its dedendum (d) = Dp-1.2m. But I found in my explanations that addendum = Dp-2m while dedendum = Dp-2m. In another more I found, addendum=Dp-2m while dedendum=Dp-(1.5*m)*2, or Dp-3m. Do we have freedom to determine both those addendum and dedendum? Jan 19, 2023 at 14:48
• @AirCraftLover if you must. but check where the trochioid that the oppsite corner makes hits your curve see: engineering.stackexchange.com/questions/13852/… Jan 19, 2023 at 15:20
• Thanks. It makes me have better understanding. Jan 20, 2023 at 15:24