**Theoretically **the tooth thickness on the pitch circle is, regardless of the size of the pressure angle, half of the pitch.

S_{n} =

**
In practice**, the tooth thickness should be slightly below this value, i.e. one has to provide a certain BACKLASH. Otherwise the meshing would not be possible under the right centre distance: the teeth would be jamming. When during tooth hobbing inevitable slight variations occur (deviations in the pitch and the lead on the profile) and when these deviations are larger than the backlash, also the meshing gets jammed.

The backlash will thus be chosen in function of the accuracy degree that one can achieve when toothing. The following are the indications regarding the level of accuracy of the teeth, the tolerances on centre distance and the backlash.

## Class of accuracy

**
**The usually used standardization (DIN) provides 12 accuracy classes, each with its own tolerance zone, defined in function of the module and of the pitch diameter. This standardisation was defined for a normal pressure angle of 20°.

In these reflections we limit ourselves to the qualities 6 to 12, an area that covers practically all the applications in the mechanical construction.

The accuracy class varies according to:

### The manufacturing process

- 9 to 12 for gears that undergo a heat treatment after the finish
- 6 to 12 for teeth obtained by milling, hobbing or shaving
- 4 to 6 for grinded teeth

### Rotational speed

- 10 to 12 for a circumferential speed of 3 m/sec
- 7 to 10 for a circumferential speed of 3 to 6 m/sec
- 4 to 6 for a circumferential speed of 6 to 20 m/sec

## Tolerance on the centre distance

**
**Seven tolerance fields are applicable: from js 5 to 11. In table 1 you will find the most widely used tolerances: js’s 7 to 10.

Tables 2 and 3 also apply to helical gears provided that the indicated values are multiplied by cos β (helix angle on the reference cylinder).

## Backlash

The sum of the clearance on the pinion and wheel (1 wheel + 1 pinion) and defined according to the table of the accuracy class, is roughly equivalent to the backlash calculated according to the following calculation:

J = minimum clearance in mm a = centre distance in mm m = module

class of accuracy 10 to 12: j = 0,03.( );

class of accuracy 8 to 9: j = 0,02.( );

class of accuracy 7 to 8: j = 0,07 + 0,01.( ) milled or hobbed teeth

class of accuracy 6 to 7: j = 0,04 + 0,0065.( ) grinded teeth

To capture the maximum values from the backlash, it is sufficient to increase the obtained values with the above formulas by 50%.

Tolerances on the centre distance |
||||||||

Class of accuracy of centre distance 5 to 6Class of accuracy of centre distance 7 to 9 Class of accuracy of centre distance 10 to 12 |
||||||||

a = centre distance in mm |
Tolerance zone (deviations in μm) – js |
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5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |

more than 10 tot 18 | ± 4,0 | ± 5,5 | ± 9,0 | ± 13,5 | ± 21,5 | ± 35,0 | ± 55,0 | ± 90,0 |

more than 18 tot 30 | ± 4,5 | ± 6,5 | ± 10,5 | ± 16,5 | ± 26,0 | ± 42,0 | ± 65,0 | ± 105 |

more than 30 tot 50 | ± 5,5 | ± 8,0 | ± 12,5 | ± 19,5 | ± 31,0 | ± 50,0 | ± 80,0 | ± 125 |

more than 50 tot 80 | ± 6,5 | ± 9,5 | ± 15,0 | ± 23,0 | ± 37,0 | ± 60,0 | ± 95,0 | ± 150 |

more than 80 tot 120 | ± 7,5 | ± 11,0 | ± 17,5 | ± 27,0 | ± 43,5 | ± 70,0 | ± 110 | ± 175 |

more than 120 tot 180 | ± 9,0 | ± 12,5 | ± 20,0 | ± 31,5 | ± 50,0 | ± 80,0 | ± 125 | ± 200 |

more than 180 tot 250 | ± 10,0 | ± 14,5 | ± 23,0 | ± 36,0 | ± 57,5 | ± 92,5 | ± 145 | ± 230 |

more than 250 tot 315 | ± 11,5 | ± 16,0 | ± 26,0 | ± 40,5 | ± 65,0 | ± 105 | ± 160 | ± 260 |

more than 315 tot 400 | ± 12,5 | ± 18,0 | ± 28,5 | ± 44,5 | ± 70,0 | ± 115 | ± 180 | ± 285 |

more than 400 tot 500 | ± 13,5 | ± 20,0 | ± 31,5 | ± 48,5 | ± 77,5 | ± 125 | ± 200 | ± 315 |

more than 500 tot 630 | ± 15,5 | ± 23,0 | ± 35,0 | ± 55,0 | ± 87,0 | ± 140 | ± 218 | ± 350 |

more than 630 tot 800 | ± 17,0 | ± 25,0 | ± 40,0 | ± 62,0 | ± 100 | ± 160 | ± 250 | ± 395 |

more than 800 tot 1000 | ± 19,5 | ± 29,0 | ± 45,0 | ± 70,0 | ± 115 | ± 180 | ± 290 | ± 455 |

more than 1000 tot 1250 | ± 23,0 | ± 33,0 | ± 52,0 | ± 82,0 | ± 130 | ± 210 | ± 330 | ± 525 |

more than 1250 tot 1600 | ± 27,0 | ± 38,5 | ± 62,0 | ± 97,0 | ± 155 | ± 250 | ± 385 | ± 625 |

## EXAMPLE:

At the design of a gearbox one designs the following gear pairs:

- a wheel with 101 teeth, module 2
- a pinion with 24 teeth, module 2
- the rotational speed of the pinion is 3000 tr/min

The circumferential speed is thus: = = 7,55 m/sec

The teeth are milled and they are suggested to be of an accuracy class 7.

The centre distance is = 125 mm

The permitted deviation is ± 31.5 µm (see table 1 distance accuracy class 8). For a reduction of the centre distance of 0,0315 mm, the backlash between the teeth will reduce with 2.0,0315.sina (a = 20°) = 0,022 mm

The average backlash will be, according to the formula:

j = 0,07 + 0,01.( ) = 0,07 + 0,01.() = 0,115mm and maximum. 0,18mm

which one should increase with 0,022mm, so:

minimum = 0,142mm of 142µm

maximum = 0,202mm of 202µm

**tolerance:** 202µm – 142µm = 60µm

If one examines the following table of deviations depending on the pitch circle diameter Ø48mm and Ø202mm and compares the calculated minimum backlash 142 µm, we can see that the deviation class “d” indicates a base deviation of 44 µm for a a Ø48mm and 80µm for a Ø202mm.

This will thus present a total base deviation of 124 µm, which is consistent with the calculated 142µm. The same method is then used to define the tolerance class; namely for tolerance class “25”, 30 µm for Ø48 and 50 µm for Ø202 = 80 µm (calculated tolerance: 70 µm).

As accuracy class should be thus indicated: “25” – “d”.

The most used tolerance classes are 24 to 27.

Base deviation and graphic definition

Pitch circle diameter mm |
Deviations in µm |
|||||||||||

from | to | a | ab | b | bc | C | cd | d | e | f | g | f |

– | 10 | -100 | -85 | -70 | -58 | -48 | -40 | -33 | -22 | -10 | -5 | 0 |

10 | 50 | -135 | -110 | -95 | -75 | -65 | -54 | -44 | -30 | -14 | -7 | 0 |

50 | 125 | -180 | -150 | -125 | -105 | -85 | -70 | -60 | -40 | -19 | -9 | 0 |

125 | 280 | -250 | -200 | -170 | -140 | -115 | -95 | -80 | -56 | -26 | -12 | 0 |

280 | 560 | -330 | -280 | -230 | -190 | -155 | -130 | -110 | -75 | -35 | -17 | 0 |

560 | 1000 | -450 | -370 | -310 | -260 | -210 | -175 | -145 | -100 | -48 | -22 | 0 |

1000 | 1600 | -600 | -500 | -420 | -340 | -290 | -240 | -200 | -135 | -64 | -30 | 0 |

1600 | 2500 | -820 | -680 | -560 | -460 | -390 | -320 | -270 | -180 | -85 | -41 | 0 |

2500 | 4000 | -1100 | -920 | -760 | -620 | -520 | -430 | -360 | -250 | -115 | -56 | 0 |

Base tolerances |
||||||||||||

Pitch circle diameter mm |
Tolerances in µm |
|||||||||||

from | to | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | |

– | 10 | 3 | 5 | 8 | 12 | 20 | 30 | 50 | 80 | 130 | 200 | |

10 | 50 | 5 | 8 | 12 | 20 | 30 | 50 | 80 | 130 | 200 | 300 | |

50 | 125 | 6 | 10 | 16 | 25 | 40 | 60 | 100 | 160 | 250 | 400 | |

125 | 280 | 8 | 12 | 20 | 30 | 50 | 80 | 130 | 200 | 300 | 500 | |

280 | 560 | 10 | 16 | 25 | 40 | 60 | 100 | 160 | 250 | 400 | 600 | |

560 | 1000 | 12 | 20 | 30 | 50 | 80 | 130 | 200 | 300 | 500 | 800 | |

1000 | 1600 | 16 | 25 | 40 | 60 | 100 | 250 | 400 | 600 | 600 | 1000 | |

1600 | 2500 | 20 | 30 | 50 | 80 | 130 | 200 | 300 | 500 | 800 | 1300 | |

2500 | 4000 | 25 | 40 | 60 | 100 | 160 | 250 | 400 | 600 | 1000 | 1600 |

This nomogram allows you in a quick way to find the minimum clearance for gear pairs mounted into machine tools according to the formula:

j_{n min} = 0,02 +

This nomogram allows you in a quick way to find the minimum clearance for high speed gears and marine gear pairs according to the formula:

j_{n min} =

The diagram below represents the backlash in function of the module for worm and worm gear gears.

The diagram below represents the backlash in function of the modulus for bevel gear sets.