Calculate Change Gears for Machine Tools

Most machine tools have at least one gearbox, but the ability to set the gear ratio can vary significantly. Simple ratios—for example, switching between “High/Low” or “Inch/Metric”—might be possible by flipping a lever or engaging one gear. However, in gear-making equipment such as hobbers, shapers, and grinders, an exact ratio is required to produce a gear with the correct number of teeth. These ratios use four gears, arranged as (A/B) × (C/D), to give the machine the correct gear ratio. Such change-gear calculations can be found in the Machinery’s Handbook, in OEM tables, or generated with this change-gear calculator. The machine itself (hobber, shaper, or 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 handles the math for you and provides a range of solutions from only the number of teeth and the Machine Constant. The range of solutions grows further when two of the change gears form a common ratio (e.g., 1:1, 1:2, 2:3). Rather than listing every option—20/20, 21/21, and so on—the calculator treats them as 1:1. Likewise, a 2:1 ratio is simply 2:1, and any equivalent gear pair (20:40, 30:60) will work.

 

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 individual gears (e.g. 125:21) also make a proper mesh more challenging. Our calculator prefers 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.

 

📌 Quick Reference: Finding the Machine Constant

  • Install four identical gears (A = B = C = D)

  • Rotate the workpiece once

  • Count hob rotations during that turn

  • The count = Machine Constant

 

Need to find solutions for other equipment? Here are some other options.


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Making "Prime-Numbered Gears"

Making a gear with a prime number of teeth can be an extra challenge. A prime number, by definition, cannot be broken down to a product of any two numbers other than 1 and itself. That is, the only way to multiply to get 73 is 73*1. Practically, this often means that you'll need the exact change gear to make the gear you want to.

And yet, "prime-numbered gears" remain extremely popular in gearbox design. The same attribute that makes prime numbered gears difficult to machine makes them attractive for evenly distributing wear across a pair of meshing gears. Consider two gears with the same number of teeth, i.e. 40 and 40. Each tooth is always in contact with its specific mate. If one gear had 41 teeth, that gear would need to rotate 40 times before finding its original "tooth mate," distributing any non-uniformities (wear, damage, manufacturing defects, etc.) across the entire set of teeth, not just concentrated on one point.

The prime numbers up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Change Gears for Machine Tools

Simplify the process of determining change gears for any machine tool. The general form of the problem involves setting a ratio between an input shaft and an output shaft using either 2, 4, 6, 8, ,..in standard and reverted gearboxes. For the standard 4-gear double reduction gearbox the most common solutions can be found on our free calculators page. Our tools also handle non-standard ratios for helix angles or other machines like Pfauter, Niles, Lorenz, Maag, Jones & Lamson, Barber Coleman, Mikron, Fellows, and many others. Enter your change gears into GearCalculators.com to calculate ratios to 7 significant figures.

References

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

    1965, 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. Sadly this set of handy tables was removed in the 23rd edition)

    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)

Here's a Quick Summary:

› Input your machine constant

› (optionally) Include a list of change gears you have

› Change gear combinations are sorted to prefer common ratios