The measuring of the tooth thickness

For measuring of gears there are four possibilities:

  • Measuring with the tooth thickness at an adjusted tooth depth ha (see left figure).
  • Measuring with the tooth depth at an adjusted tooth thickness and from this one calculates the tooth thickness (see right figure); this method we leave aside.
  • Measuring the base tangent length across multiple teeth.
  • The diametrically measuring about 2 pins or 2 balls.

Measuring with double toothcaliper or “Brown & Sharpe”

1.1 Straight gear

This special measuring instrument consists of a combination of two calipers, one for setting the tooth height measuring distance, one for measuring the tooth thickness (see sketch). The theoretical tooth thickness on the pitch circle is equal to the pitch (Sn) of the pitch circle arc enclosed by the two flanks of one tooth. The arc length (Sn) represents a half pitch. In practice the tooth thickness must take into account the backlash, so: practical tooth thickness= length of cord – 1/2 of backlash (see chapter backlash).

Calculating of the tooth thickness

Berekenen tanddikteIn the next picture and with the following abbreviations:
mn = module (= ha = addendum)
Image251= chordal height
pn = normal pitch = mn.p
r = pitch circle radius
d = pitch circle diameter
z = number of teeth
Image252= ab =2.r.sinψ = d.sinψ and because d = z.m
and c’b = Image253 of the pitch

Image254=Image255thus
Image252= ab= z.m.sinImage255

Calculating of chordal height

cc’ = m, due to ratio of reference profile, and  Image251 = cc’ + c’e
c’e = r – r.cosψ and as r = Image115Image257 results in:
c’e = Image257.(1
– cosψ) = m.z.(Image258)
dus Image251 = m + m.z.(Image258)
= m.Image259 or
Image251= m.Image260
In the table the values of  Image252 and Image251 are calculated for the modules 1 and for the different tooth numbers. It suffices to multiply these numbers with the number of the used
modules. It is necessary to ensure that for the size Image252 the theoretical value is specified and for the practical value it is necessary to reduce this number thus with ± half of the backlash.

 

Z Image252 Image251 Z Image252 Image251
10 1,564 1,062 24 1,57 1,026
11 1,565 1,056 25 1,57 1,025
12 1,566 1,051 26 1,57 1,024
13 1,567 1,047 27 1,57 1,023
14 1,567 1,044 28-29 1,57 1,022
15 1,568 1,041 30-31 1,57 1,021
16 1,568 1,038 32-33 1,57 1,02
17 1,569 1,036 34-35 1,57 1,019
18 1,569 1,034 36-37 1,57 1,018
19 1,569 1,032 38-39 1,57 1,017
20 1,569 1,031 40-42 1,57 1,016
21 1,569 1,029 43-44 1,57 1,015
22 1,569 1,028 -45 1,571 1,014
23 1,57 1,027 à 1,000

Berekening inwendige vertanding

For the calculation of gears with internal teeth, it is necessary to subtract the arrow pitch  Image252 instead of adding. For gears with correction the half tooth angle Image262 should be increased with +Image263

Special case

Calculation of the tooth thickness at a given radius:

If we take as a basis the tooth thickness S at the pitch circle then it is possible to calculate the tooth thickness on any radius (rp).

from rb = r.cosa = rp cosap we can calculate ap.

The tooth thickness is then

Sp = rp.Image264

1.2 Helical gears:

The same procedure can be applied here by calculating with the virtual number of teeth:

zv =Image265
Image252= zv.mn.sinImage266Image251 = mn. Image267
for the apparently module mn = mt.cosβ. For gears
with correction the half tooth angle Image262 should be increased with +Image268

1.3 Straight helical teeth:

In this case one needs to measure along the side of the large module on the extreme side of the tooth.
The dimensions are the same as those of cylindrical gears with the same module but with a virtual number of teeth:

zv =Image269
Image252= zv.m.sinImage266Image251 = m.Image270

1.4 spiral bevel gears:

Given the helix angle and the arch of the gear, the measuring of the toothcaliper is not properly performed at the ends of the teeth, because at this spot one has to deal with an apparent module. Specialists measure the tooth thickness according to the real module in a specific place of the tooth ends.

1.5 Worm and wormwheels:

The same procedure can be applied here by calculating with the virtual number of teeth:

zv2Image17

Transverse module:

Worm Image18 = Image19.p.mx.cosg &Image20  = Image21
Wormwheel Image22 = zv2.mx.cosg.sinImage23Image24 = mx.Image25
For wormwheels with correction the half tooth angle Image271 should be increased with +Image272

Normal module:

Worm Image18 = Image19.p.mnImage20Image273

Wormwheel Image22 = zv.mn.sinImage23Image24 = mn.Image274
For wormwheels with correction the half tooth angle Image271should be increased with +Image275