Expressing a specific motion from the point of view of a different rotating coordinate system

Question

Simplified problem presentation

I am trying to describe a specific motion from a point of view of a moving coordinate system. To make easier the comprehension I created this childish animation, where Mickey Mouse's face (expressed in the local coordinate system) is moving from left to right and the observer is looking directly at Mickey Mouse's face all the time (red coordinate system).

Geometrically the problem is described in the following image:

• $\vec{r_\ell}$ and $\beta \,$ (or $x_{A,\ell}$ and $y_{A,\ell}$) are the data holders of Mickey Mouse's face contour
• $\psi$ is the angle parameter
• $R$ is the radius of the circle
• $A$ is a specific point of the Mickey Mouse's face

The objective is to determine $\left[ x, \, y \right]$ in the rotating coordinate system. According to the image the equations should be:

1. $\quad x = | \vec{r_\ell} | \, \cos{ \left( \beta + \psi \right)}$
2. $\quad y = \frac{R}{\cos{\psi}} + | \vec{r_\ell} | \, \sin{ \left( \beta + \psi \right)}$

Real problem presentation

The real issue is involved in gear engineering, but I do not think it really matters, since it is a pure kinematic problem. Basically I want this animation with respect to the rotating coordinate system of the cogwheel.

I tried several times, but I am always getting the same wrong result. I also tried with the rotating transformation matrix: $T = \left[ \cos{\left(-\psi \right)} \ \sin{\left(-\psi \right)}; \ -\sin{\left(-\psi \right)} \ \cos{\left(-\psi \right)} \right]$ but I obtained the same problematic areas represented in the following animation and image:

The gear rack goes always over the cogwheel profile, which shall not do it. It should essentially roll as in the previous image, where both parts are moving. I really have no idea, why this is happening, so every single helpful advice is appreciated.

Answer 1

It looks like the problem here is that you've made an assumption that the $y$-axis of the rotating coordinate system always goes through the origin of the translating local coordinate system. However, this is not kinematically correct for your gear scenario:

For a rack and pinion, the translation speed of the rack is directly proportional to the angular speed of the pinion. So, there is a linear relationship between the rotation angle of the pinion and the rack's displacement. However, this is not what happens in your simplified problem presentation's diagrams. Think about what happens as the translating local coordinate system moves a very large distance, going on towards infinity. If the $y$-axis of the rotating coordinate system continues to point towards the origin of the local coordinate system, eventually the rotating $y$-axis will tend towards pointing in the global/fixed $x$-direction. It will never get there, as that would require the local coordinate system to be infinitely far away.

So the local coordinate system can freely move large distances along the global $x$-direction, but the rotating coordinate system has an angle range of no more than 180 degrees. Clearly, this cannot be describing the linear relation between translation and rotation that we get with the rack and pinion's kinematics. Another way to see this is to try calculate $|\vec{l}|$ from your geometrical problem diagram. So, we need to be more careful with how the different coordinate systems are positioned with respect to one another.

So, to fix thing, you need to relax the assumption that the rotating $y$-axis is pointing to the local coordinate system origin, and instead impose the constraint that is the linear relation between the translation and rotation of the rack and pinion:

$$|\vec{l}| = R_p \psi$$

where $R_p$ is the pitch radius of the pinion gear. (That is, the radius of the pitch circle. The pitch circle of a gear is the equivalent circle you could replace the gear with so that it rolls without slip with the pitch circle of a mating gear, or with the pitch line of a mating rack.)

It is convenient to introduce the global coordinate system $(X,Y)$ whose origin is coincident with the rotating coordinate system, as this will make it easier to describe the relationship of the local and rotating coordinate systems:

Points expressed in the local coordinate system are translated versions of those expressed in the global coordinate system:

$$ \begin{bmatrix} x_l \\ y_l \end{bmatrix} = \begin{bmatrix} X \\ Y \end{bmatrix} - \begin{bmatrix} R_p \psi\\ R_p \end{bmatrix} $$

The rotating coordinate system is of course rotated with respect to the global coordinate system, and it can be shown that the representation of points from each system are related via

$$ \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \cos{\psi} & -\sin{\psi} \\ \sin{\psi} & \cos{\psi} \end{bmatrix} \begin{bmatrix} X \\ Y \end{bmatrix} $$

These relations should be enough to express a point in the rotating coordinate system $(x, y)$ in terms of $x_l$, $y_l$, $\psi$, and $R_p$.

Also, some of the mathematics in this answer might be of interest to you: https://engineering.stackexchange.com/questions/13852/involute-gear-curve-when-root-diameter-falls-below-base-diameter/13868#13868