Kinematic description of a gear rack moving around a cogwheel

Question

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 $|\vec{r_c}| = \sqrt{ x_{c,A}^2 + y_{c,A}^2 }$.

Thank you in advance for any helpful advice

Answer 1

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.