Gears:Simultaneous Meshing with Rack
Release status: Working
Description  A method for calculating gear center distances

License  unknown

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Introduction
Recently, Peter posted this great prototype of a linear motion mechanism using lasercut gears.
He stated: "there's probably an ideal solution when solving simultaneously for (a) a rack, with (b) two gears of radius R1, being driven by (c) another larger gear of radius R2; all for a given tooth size."
Matt Moses solved this problem while working on a linear motion mechanism of his own, which uses a similar arrangement of gears. This page has a brief description of how he solved the problem, as well as MATLAB code you can download to generate your own gearing arrangements if you are using this particular mechanism. If you don't have access to MATLAB, the code should probably work with GNU Octave with minor modification.
FDavies posted an Octaveready version of the script here. You will need the zipped code from this page and FDavies's file to run under Octave.
Problem Description
There are two constraints on the system, which can be found by starting with two gears in an initial configuration and then making a sequence of three "virtual rolling motions". This is shown in the figure below.
We are given the pitch radii of each gear, <math>r_1</math> and <math>r_2</math>, and the diametral pitch <math>DP</math> (which must be the same for both gears). Then we search for values of <math>\theta_1</math> that satisfy both constraints.
We start with the gears arranged as shown at the top of the figure. A radial line marks each gear so we can track their angular position. For the first rolling motion, both gears are rotated together through an angle of <math>\theta_1</math>.
We remember this position (indicated by the dotted outline), and then for the second motion both gears continue to rotate for a total angle of <math>\theta_2</math>. This gives us the first constraint, which provides <math>\theta_2</math> as a function of <math>r_1</math>, <math>r_2</math> and <math>\theta_1</math>
In the "x" direction, the top gear is centered between the bottom gears.
<math>r_1 \theta_2  r_1 \theta_1 = 2 \left(r_1 + r_2\right) \sin(\theta_1)</math>.
Now for the third "virtual rolling motion" we leave the large gear fixed and roll the small gear back towards the top, rolling along the large gear. The solid line on the small gear indicates the angular position of the gear when it comes to rest at the top. <math>\alpha</math> is the angle between the solid line and the vertical.
In order for all gears to mesh, the angle between the solid and dotted lines on the small gear must be an integer multiple of the angle between teeth. This gives the second constraint and shows us also that the solution space is a set of discrete values, rather than a continuum of possible positions. The second constraint is
<math>\theta_1  \alpha = k \frac{2 \pi}{n_2}</math>
where <math>k = 0, 1, 2, 3, ...</math>, and <math>n_2</math> is the number of teeth on gear 2.
The final position of the mark on the small gear in Fig. 4 (α) = the starting position of the mark in Fig. 3 (θ_{2}), minus the amount it turned from the distance traveled around the lower gear, minus the amount it turned from the angle traveled around the lower gear.
<math>\alpha = \theta_2  \frac{r_1}{r_2} \left( \theta_1 + \theta_2 \right)  (\theta_1 + \theta_2) = \theta_1  \frac{r_1}{r_2} \left( \theta_1 + \theta_2 \right) </math>,
The number of teeth is related to the radius <math>r_2</math> and diametral pitch <math>DP</math> by
 <math>n_2 = 2 \left(DP\right) r_2</math>.
 <math>n_1 = 2 \left(DP\right) r_1</math>.
Code Overview
The code simply checks for values of <math>k</math> and <math>\theta_1</math> that satisfy the constraints and then draws the gears and rack for a particular arrangement. There is not always a unique solution, and sometimes no solution for a given set of values. An example of the graphical output is shown below, along with a motor assembly that was designed using the code.
See Also
Here are some other generators for involute gears:
Parametric Involute Bevel and Spur Gears
Gear template generator by Matthias Wandel
Designing Involute Profile Gear Pairs in Art of Illusion by Forrest Higgs
Other Gear Stuff on the wiki:
RBS/Gears: The RepRap Building System (RBS) Gears
Gear design has a little more information on rackandpinion and alternatives.
Here are some pages with helpful information on terminology and formulas:
The Involute Curve, Drafting a Gear in CAD and Applications by Nick Carter
Introduction to Mechanisms, Chapter 7 Gears by Yi Zhang, with Susan Finger and Stephannie Behrens