4.3 Measurement method with Gear tooth vernier
Measurement method based on Tip circle, measure Chordal tooth thickness upon Pitch cylinder. Refer to Fig. 9 formeasurement method based upon Gear tooth vernier calipers. Fix gear tooth vernier calipers at theoretical value ofTooth depth h and measure deviation between actual Chordal tooth thickness s and its theoretical value. This is atime-honored measurement method with low accuracy due to influence from inconsistent measurement and Jaw
conditions.
For Spur gear, calculation is by following formula :h=mz ­§ π + 2 xtanα ·½ + da − d¸¾®1 − cos¨2 ¯z ¹¿2© 2z

π + 2 xtanα ·s = mzsin §¨¸z ¹© 2zHerebyh : Chordal addendum
m : Modulesz: Chordal tooth thickness

α : Reference pressure angle

x: Rack shift coefficientda : Tip (outside) diameterd: Reference diameter

Refer to Table 2. Below chart shows
and Rack shift coefficient x=0.

h: Number of teeth

s: Chordal addendum and : Chordal tooth thickness for gear with module 1.0

For Helical gear, use Normal surface to measure module, Pressure angle and Rack shift coefficient using value of Normal. Number of teeth z uses (1)Virtual number of teeth for Spur gear.

Table 2. Chordal tooth thickness for standard gearz

h mms mmz

h mms mm12

1.0513

1.566335

1.01761.570313

1.0474

1.567040

1.01541.570414

1.04401.567545

1.01371.5705

Tooth depth slider

15

1.0411

1.567950

1.01231.5705

Vernier

16

1.03851.568360

1.01031.570617

1.0363

1.568670

1.00881.570618

1.0342

1.568880

1.00771.570719

1.03241.569090

1.00691.570720

1.03081.5692100

1.00621.570722

1.02801.5695120

1.00511.570824

1.02571.5697150

1.00411.570826

1.02371.5698200

1.00311.570828

1.02191.5700∞1.00001.570830

1.02061.5701

Tooth depth
scale

Tooth thickness scale
Main scale

VernierTooth depth slider

dodk

TongueShJaw

Fig. 9 Measurement with gear tooth vernier calipers

Values of h and s . (m=1, x=0)

Note: (1) Adopted the old Standard term.

69

＜ Reference ＞ Gear analysis method

How to obtain module, Pressure angle and Rack shiftcoefficient for Involute spur gear,
There are various methods on how to obtain moduleand Pressure angle for Involute spur gear. Method introduced here is by Base pitch measurement.

mα01
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
3.5
3.75
4
4.5
5
5.5
6
6.5
78
910

There is a method of using Sector span to measure
the Base pitch.
For Sector span, assuming n of Number of teeth is En.
Reduce one tooth from n is En-1. Therefore Base pitchPb is by following formula.
pb = En − En − 1 (1)		 = π m cosα

Table 3. Base pitch
14.5°
3.042
3.802
4.562
5.323
6.083
6.843
7.604
8.364
9.125
9.885
10.64511.40612.16613.68715.20816.72818.24919.77021.29124.33227.37430.415Compare Base pitch calculation result by formula (1)with Base pitch in Table 3.
Example for Spur gear:	C alculate module, Pressure
angle and Rack shift coefficient for Number of teeth =12 and Tip (outside) diameter=29.9mm.
Result of measurement for Sector span (Zm) was as follows
Zm= Two (2) teeth of Sector span E2 = 9.855 mm
Zm= Three (3) teeth of Sector span E3 = 15.758 mm
Therefore calculate Pb by following formula (1)

20°
2.952
3.690
4.428
5.166
5.904
6.642
7.380
8.118
8.856
9.594
10.332
11.07011.80913.28514.761
16.23717.71319.18920.66523.61726.56929.521

22.5°
2.902
3.628
4.354
5.079
5.805
6.531
7.256
7.982
8.707
9.433
10.15910.884
11.61013.06114.51215.96317.41518.86620.31723.22026.12229.025Calculating Sector span W’’ for Standard spur gearwith pressure angle 20° is by following formula:
W ' = mcosα {π ( Zm − 0.5) + zinvα }= m(0.01400554 z + 2.95213 zm − 1.47606)		
Calculating Sector span W’” for Rack shifted spur gearwith pressure angle 20° is by following formula:
W " = W ' [standard] + 2 xmsinα
W " = W ' [standard] + 0.68404 xm (2)[Standard] is abbreviation of Standard spur gear.
From above formula (2), calculation for Rack shift coefficient x is as follows:

pb = E 3 − E 2		 = 15.758 − 9.855		 = 5.903mm
With reference to Base pitch chart in Table 3, moduleis 2.0 mm and Pressure angle is 20°.
Calculation formula for Rack shift coefficient
Calculation for Sector span W is by following formulaW = mcosα {π ( Zm − 0.5) + zinvα } + 2 xmsinα

7025°
2.847
3.559
4.271
4.983
5.695
6.406
7.118
7.830
8.542
9.254
9.965
10.67711.38912.81314.23615.6.60
17.0.84
18.50719.93122.77825.62528.473x=

W " −W '0.68404m(3)

Therefore, results are W’”=9.855, W’’ = 9.193. Rack
shift coefficient x is 0-.484.
x=		

9.855 − 9.193
2 × 0.68404= 0.484