Change Gear Calculator

Most machine tools have at least one gearbox, but the ability to set the gear ratio may vary wildly. Simple ratios, i.e. changing between "High/Low" or "Inch/Metric" might be possible by flipping a lever, or engaging one gear. But in gear-making equipment, like hobbers, shapers, and grinders, an exact ratio is required to make a gear with the right number of teeth. These ratios use four gears arranged (A/B)*(C/D) to give the machine the correct gear ratio. These change gear calculations can be found in the Machinery's Handbook or tables from the OEM, or generated with this change gear calculator. The machine itself (hobber, shaper, grinder) will also have an internal Machine Constant, so for a given tooth count, one needs to solve: Machine Constant / Number of Teeth = (A/B)*(C/D)

Our change gear calculator for hobbers, shapers, and grinders, takes the math out of equation and allows you to get a range of solutions from just the Number of Teeth and Machine Constant. The range of solutions grows even more when two of the change gears form a common ratio (i.e. 1:1, 1:2, 2:3). Rather than repeat out every option: 20/20, 21/21, .... the calculator will treat this as 1:1. Likewise a 2;1 ratio is just 2:1, and any pair of gears (20:40, 30:60) will work.

How to Find Your Machine Constant

The machine constant for a hobbing machine can be found by installing all four change gears with the same number of teeth, and recording how many times the hob rotates relative to the gear blank.  With the machine off, you can usually rotate it by hand to do this for smaller machines.  For larger machines that must be energized to make this happen, use any modern camera and some indicators to mark the rotation of the hob and the table in the same frame.  Count it by watching your video.  That is, with A, B, C, and D all the same, as the workpiece rotates one time, how many times does the hob rotate? This relative rotation count will determine the Machine Constant for your machine.

Range of Acceptable Gears

Not all gears will fit inside your machine. Large gears may not physically fit in the cabinet, or require too high of a center distance. High gear ratios between individuals gears (e.g. 125:21) also make a proper mesh more challenging. Our calculator preferences common ratios (1:1, 1:2, 2:3) and gears that are close in size to give you the most options that will fit your machine.  We did not constrain any output data, but we prioritize answers with the smallest physical gear size.

Scissor for Pfauter RS00

Scissor mechanism for Pfauter RS00

Change Gear Calculation - The Math

From a math standpoint, the selection of change gears is a problem of continued fractions, and conjugate fraction methods.  But there are many boundary conditions, such as the machine limits on the "Scissor", the change gears the operator actually possesses, and the quantity of each gear in the change gear set.

A tabular solution to a very specific set of ratios is in the Pfauter references below.  They solve for gear ratios between 0.10000 to 1.0000 and tabularize the resulting change gears (a/b)*(c/d) using change gear sets from 18 to 80 teeth.  These books are difficult to find.


1950, Pfauter Wechselräder Tabellen, Becher und Körner (German)

1964, Pfauter Change Gear Tables, Becher and Koerner (English)

1971, ASME, A Computer Algorithm to Design Compound Gear Trains for Arbitrary Ratio, Dil Pare

1981, Machinery's Handbook 21st Edition (1981),  Pages 1435-1465
Includes a nice set of logarithm tables for Tooth Counts Between 15-120  Handbook Editions prior to the 23rd have this set of handy tables.

Machinery's Handbook 29th Edition (2012), Page 11 - Continued Fractions
Page 12 Conjugate Fractions
There is a great example calculation on how to manually iterate to a solution