# Kinematic description of a gear rack moving around a cogwheel

I am facing some issues regarding the kinematic description of a gear mechanism (or maybe more accurately gear manufacturing process). Basically I would like to obtain an animation, where the cogwheel is fixed and the gear rack moves around, but for some reason I cannot reach the result expected.

For the moment I successfully obtained this animation:

Now I got stuck with describing the movement of the gear rack (precisely hob cutting tooth) around the cogwheel. Essentially it means that the entire movement is defined with respect to the fixed coordinate system of the gear wheel, which rotates with the wheel. My best result so far is the following:

As it can be seen, the tooth goes over the gear wheel, which is obviously wrong.

My mathematical model is described in this picture:

The coordinate system $$\left[ x_c, y_c \right]$$ moves in the auxiliary coordinate system $$\left[ x_\ell, y_\ell \right]$$ along the $$x_\ell$$ axis by the vector $$\vec{l} = \left( R_{roll} + y_{\ell,O_c} \right) \tan{\psi} \cdot \vec{i}$$, where:

1. $$\vec{i}$$ is the unit vector in $$x_\ell$$ direction
2. $$\psi$$ is the parameter (and angle as well), which goes through change
3. $$R_{roll}$$ is the radius of the rolling circle

From the image it can be then derived the following:

$$x = |\vec{r_c}| \cdot \cos{\left( \beta + \psi \right)}$$

$$y =\frac{R_{roll} + y_{\ell,O_c}}{\cos{\psi}} + |\vec{r_c}| \cdot \sin{\left( \beta + \psi \right) }$$

In this model the coordinate $$y_{\ell,O_c}$$ represents the possible displacement of the hob from the gear wheel and for this reason it needs to be taken into account. Moreover the vector $$\vec{r_c} = \left[ x_{c,A}, y_{c,A} \right]$$ is the holder of the hob tooth profile and $$|\vec{r_c}|$$ is its absolute value, calculated as $$|\vec{r_c}| = \sqrt{ x_{c,A}^2 + y_{c,A}^2 }$$.

• Why dont you just rotate your original solution? Commented Sep 28, 2023 at 4:50
• @joojaa I tried using again the rotation matrix $T = [\cos(-\psi) \ \sin(-\psi); -sin(-\psi) \ cos(-\psi)]$ on the gear rack after the shift made by vector $\vec{l}$, but it again did not produce the wanted result. Basically I made the following procedure: 1) Set shift vector $\vec{l}$, 2) $\psi = \arctan{ \frac{ l_x }{ R_{roll} + y_{\ell,O_c}} }$, 3) Applying rotation matrix to gear rack profile Commented Sep 28, 2023 at 21:03
• i suggest you use a 3by 3 matrix then you can move points too. Commented Sep 30, 2023 at 8:02
• @joojaa I do not clearly understand what you mean by "moving points too". I obviously move "points" also by performing matrix multiplication with a 2x2 transformation matrix. Moreover it is not possible to make a matrix multiplication between a $3 \ x \ 3$ and $2 \ x \ \it n$. I should add a row of zeros in $z$ coordinate to make it happen, which is in my opinion useless. Commented Oct 1, 2023 at 11:25
• no but if you make your coordinates affine you can make them vectors become [x, y, 0] (for normal's and stuff) and points in space become [x, y, 1] then your matrix is [[x1, y1,0],[x2, y2,0],[tx, ty,1]] This reduces the amount fo operations by heaps. And this is the standard way in nearly everything, including openGL (in 3D you have 4x4 matrices), PDF, SVG, robotics etc. This makes it easier as you can now express rotation around a arbitrary point with T.R.T^-1 Commented Oct 1, 2023 at 13:11

Either I'm misunderstanding what you're talking about, or you're misunderstanding how gear hobbing typically works (and I tend to lean toward the latter being more accurate).

At least with most typical gear hobbs, the kinematics you seem to be trying to describe just don't happen at all. Other than turning so the cutting teeth can cut, the cutter in the hob doesn't normally move at all. It just sits in one spot, and turns on its axis.

The teeth it produces don't really fit the curve you'd get in a normal drawing. Instead, you get a few (at least three, often 5, depending somewhat on the size/pitch of the gear you're cutting) straight lines that are tangent to the curve--and that's enough that the deviation from the curve is normally pretty minimal.

The tooth you get from hobbing a gear is not this:

It's literally like a low-res preview rendering in a CAD program:

The reason you get that is specifically because nothing like the kinematics you're trying to describe will actually happen. Instead, the "rack" shaped thing is spinning to have the teeth cut, but otherwise isn't moving at all (well, other than moving toward/away from the work piece, of course).

When you do the cutting, the hobb fits in roughly like this:

So, each tooth of the hobb is cutting one straight facet. You run the cutter through to cut one set, pull it out, turn the gear one tooth, and repeat.

Depending on how you're using the gear, you frequently want to "run it in"--mesh some gears together, add some grit, and spin them for a little bit, so the grit will round off the facets into something closer to a proper involute form.

When you're cutting something like a worm gear, you do get "circular" movement, but in most cases that's more like your first picture--the hobb still just spins in place, but the angled teeth of the cutter turn the work-piece as they cut it.

• Thanks for your answer. Honestly I do not aim to describe the hobbing manufacturing process, but I appreciate your explanation. I thought the hob teeth are in threading succession, which would make this model maybe more comparable. I only wanted to apply this theoretical model in practice to help the comprehension, since this design is not even close to the classic linear actuator (rack and pinion) due to absence of clearance. Commented Sep 28, 2023 at 21:16
• "Threading succession" sounds like a worm gear, which I covered i the last paragraph. It is closer, because the gear turns continuously (rather than in steps) as it's being cut. Commented Sep 29, 2023 at 7:37