Measurement over multiple teeth or base tangent length

Maat over meerdere tanden of tandwijdte

STRAIGHT NOT CORRECTED GEARS

The principle consists in measuring the distance between flanks of together lying teeth (measurement of Wildhaber), this provides accurate indications about the tooth thickness, is easy to apply and requires, outside a regular but precise caliper, no special instruments. This measurement is based on the characteristics of the involute gear where anti-homologous tooth flanks on the tangent lines to the base circle determine the segments of equal length regardless of the location of the tangent point (see picture).

Since the segments are equal to the length of an arc of the base circle, one measures thus a distance that is a multiple of the basic pitch + a tooth thickness on the base circle. Given the fact there is a relation between this distance and the tooth thickness on the pitch circle, it is possible to measure the latter indirectly but accurately and easily.

Note:
In the formulas and in the tables the theoretical tooth thickness is always indicated; the foreseeable backlash should to be deducted from that value (see chapter: backlash).
The following formula indicates the theoretical measurement across a certain number of teeth for the involute gear, in function of the gear characteristics of the gear.
Wk = m.cos(α).[(k-0,5).π + z.inv(α)]
In which:
Wk = distance between the lips of the caliper in mm or base tangent length is sufficient
m = module
α = pressure angle
k = number of teeth between the lips of the caliper
z = number of teeth of the gear to be checked
inv = involute function, either (tan(α)-α)
In order not to have to calculate the value of Wk for every case, tables were formatted for all ordinary tooth numbers, module and for pressure angles 14⁰30 ‘ and 20⁰. Since the dimension Wk is directly proportional to the module it is sufficient to multiply the found value with the module, subtracting half of the backlash from there and obtaining in this way the required control of the tooth thickness with the caliper.

Rechte niet gecorrigeerde vertanding

Straight not corrected gear

Pressure angle 14⁰30’
z= number of teeth of the wheel
k= number of teeth between the lips of the caliper
Wk= theoretical dimension with module 1

z k Wk z k Wk z k Wk
5 2 4,589129 49 4 10,908382 93 8 23,310686
6 2 4,594498 50 4 10,913751 94 8 23,316055
7 2 4,599866 51 5 13,960644 95 8 23,321423
8 2 4,605234 52 5 13,966013 96 9 26,368317
9 2 4,610602 53 5 13,971381 97 9 26,373685
10 2 4,615971 54 5 13,976749 98 9 26,379053
11 2 4,621339 55 5 13,982117 99 9 26,384421
12 2 4,626707 56 5 13,987485 100 9 26,389790
13 2 4,632075 57 5 13,992854 101 9 26,395158
14 2 4,637443 58 5 13,998222 102 9 26,400526
15 2 4,642812 59 5 14,003590 103 9 26,405894
16 2 4,648180 60 5 14,008958 104 9 26,411262
17 2 4,653548 61 6 17,055852 105 9 26,416631
18 2 4,658916 62 6 17,061220 106 9 26,421999
19 2 4,664285 63 6 17,066589 107 10 29,468893
20 2 4,669653 64 6 17,071957 108 10 29,474261
21 2 4,675021 65 6 17,077325 109 10 29,479629
22 2 4,680389 66 6 17,082693 110 10 29,484997
23 2 4,685757 67 6 17,088062 111 10 29,490366
24 2 4,691126 68 6 17,093430 112 10 29,495734
25 2 4,696494 69 6 17,098798 113 10 29,501102
26 3 7,743388 70 6 17,104166 114 10 29,506470
27 3 7,748756 71 6 17,109534 115 10 29,511838
28 3 7,754124 72 6 17,114903 116 10 29,517207
29 3 7,759492 73 7 20,161796 117 10 29,522575
30 3 7,764861 74 7 20,167165 118 10 29,527943
31 3 7,770229 75 7 20,172533 119 10 29,533311
32 3 7,775597 76 7 20,177901 120 11 32,580205
33 3 7,780965 77 7 20,183269 121 11 32,585573
34 3 7,786333 78 7 20,188638 122 11 32,590942
35 3 7,791702 79 7 20,194006 123 11 32,596310
36 3 7,797070 80 7 20,199374 124 11 32,601678
37 3 7,802438 81 7 20,204742 125 11 32,607046
38 4 10,849332 82 7 20,210110 126 11 32,612414
39 4 10,854700 83 7 20,215479 127 11 32,617783
40 4 10,860068 84 8 23,262372 128 11 32,623151
41 4 10,865437 85 8 23,267741 129 11 32,628519
42 4 10,870805 86 8 23,273109 130 11 32,633887
43 4 10,876173 87 8 23,278477 131 11 32,639256
44 4 10,881541 88 8 23,283845 132 12 35,686149
45 4 10,886909 89 8 23,289214 133 12 35,691518
46 4 10,892278 90 8 23,294582 134 12 35,696886
47 4 10,897646 91 8 23,299950 135 12 35,702254
48 4 10,903014 92 8 23,305318 136 12 35,707622
Rechte niet gecorrigeerde vertanding

Straight not corrected gear

Pressure angle 14°30′
z= number of teeth of the wheel
k= number of teeth between the lips of the caliper
Wk= theoretical dimension with module 1

z k Wk z k Wk z k Wk
137 12 35,712990 172 15 45,025455 207 18 54,337919
138 12 35,718359 173 15 45,030823 208 18 54,343288
139 12 35,723727 174 15 45,036191 209 18 54,348656
140 12 35,729095 175 15 45,041560 210 18 54,354024
141 12 35,734463 176 15 45,046928 211 18 54,359392
142 12 35,739832 177 15 45,052296 212 18 54,364761
143 12 35,745200 178 15 45,057664 213 18 54,370129
144 13 38,792094 179 16 48,104558 214 18 54,375497
145 13 38,797462 180 16 48,109926 215 19 57,422391
146 13 38,802830 181 16 48,115294 216 19 57,427759
147 13 38,808198 182 16 48,120663 217 19 57,433127
148 13 38,813566 183 16 48,126031 218 19 57,438495
149 13 38,818935 184 16 48,131399 219 19 57,443864
150 13 38,824303 185 16 48,136767 220 19 57,449232
151 13 38,829671 186 16 48,142136 221 19 57,454600
152 13 38,835039 187 16 48,147504 222 19 57,459968
153 13 38,840408 188 16 48,152872 223 19 57,465337
154 13 38,845776 189 16 48,158240 224 19 57,470705
155 13 38,851144 190 16 48,163609 225 19 57,476073
156 14 41,898038 191 17 51,210502 226 19 57,481441
157 14 41,903406 192 17 51,215870 227 20 60,528335
158 14 41,908774 193 17 51,221239 228 20 60,533703
159 14 41,914142 194 17 51,226607 229 20 60,539071
160 14 41,919511 195 17 51,231975 230 20 60,544440
161 14 41,924879 196 17 51,237343 231 20 60,549808
162 14 41,930247 197 17 51,242712 232 20 60,555176
163 14 41,935615 198 17 51,248080 233 20 60,560544
164 14 41,940984 199 17 51,253448 234 20 60,565913
165 14 41,946352 200 17 51,258816 235 20 60,571281
166 14 41,951720 201 17 51,264185 236 20 60,576649
167 14 41,957088 202 17 51,269553 237 20 60,582017
168 15 45,003982 203 18 54,316446 238 20 60,587385
169 15 45,009350 204 18 54,321815 239 20 60,592754
170 15 45,014718 205 18 54,327183 240 20 60,598122
171 15 45,020087 206 18 54,332551 241 21 63,645016
Rechte niet gecorrigeerde vertanding

Straight not corrected gear

Pressure angle 20°00′
z= number of teeth of the wheel
k= number of teeth between the lips of the caliper
Wk= theoretical dimension with module 1

z k Wk z k Wk z k Wk
8 2 4,540241 44 6 16,852967 80 10 29,165692
9 2 4,554247 45 6 16,866972 81 10 29,179697
10 2 4,568253 46 6 16,880978 82 10 29,193703
11 2 4,582258 47 6 16,894983 83 11 32,159840
12 2 4,596264 48 6 16,908989 84 11 32,173845
13 2 4,610269 49 6 16,922994 85 11 32,187851
14 2 4,624275 50 7 19,889131 86 11 32,201856
15 3 7,590412 51 7 19,903137 87 11 32,215862
16 3 7,604417 52 7 19,917142 88 11 32,229868
17 3 7,618423 53 7 19,931148 89 11 32,243873
18 3 7,632428 54 7 19,945153 90 11 32,257879
19 3 7,646434 55 7 19,959159 91 11 32,271884
20 3 7,660439 56 7 19,973165 92 12 35,238021
21 3 7,674445 57 7 19,987170 93 12 35,252027
22 3 7,688450 58 8 22,953307 94 12 35,266032
23 3 7,702456 59 8 22,967313 95 12 35,280038
24 4 10,668593 60 8 22,981318 96 12 35,294043
25 4 10,682599 61 8 22,995324 97 12 35,308049
26 4 10,696604 62 8 23,009329 98 12 35,322054
27 4 10,710610 63 8 23,023335 99 12 35,336060
28 4 10,724615 64 8 23,037340 100 12 35,350065
29 4 10,738621 65 8 23,051346 101 13 38,316202
30 4 10,752626 66 8 23,065351 102 13 38,330208
31 4 10,766632 67 9 26,031488 103 13 38,344213
32 4 10,780637 68 9 26,045494 104 13 38,358219
33 5 13,746774 69 9 26,059499 105 13 38,372225
34 5 13,760780 70 9 26,073505 106 13 38,386230
35 5 13,774785 71 9 26,087510 107 13 38,400236
36 5 13,788791 72 9 26,101516 108 13 38,414241
37 5 13,802796 73 9 26,115522 109 13 38,428247
38 5 13,816802 74 9 26,129527 110 14 41,394384
39 5 13,830807 75 10 29,095664 111 14 41,408389
40 5 13,844813 76 10 29,109670 112 14 41,422395
41 6 16,810950 77 10 29,123675 113 14 41,436400
42 6 16,824956 78 10 29,137681 114 14 41,450406
43 6 16,838961 79 10 29,151686 115 14 41,464411
Rechte niet gecorrigeerde vertanding

Straight not corrected gear

Pressure angle 14°30′
z= number of teeth of the wheel
k= number of teeth between the lips of the caliper
Wk= theoretical dimension with module 1

z k Wk z k Wk z k Wk
116 14 41,478417 158 19 56,827307 200 24 72,176197
117 14 41,492422 159 19 56,841312 201 24 72,190202
118 15 44,458559 160 19 56,855318 202 24 72,204208
119 15 44,472565 161 20 59,821455 203 24 72,218213
120 15 44,486571 162 20 59,835460 204 25 75,184350
121 15 44,500576 163 20 59,849466 205 25 75,198356
122 15 44,514582 164 20 59,863471 206 25 75,212361
123 15 44,528587 165 20 59,877477 207 25 75,226367
124 15 44,542593 166 20 59,891483 208 25 75,240372
125 15 44,556598 167 20 59,905488 209 25 75,254378
126 15 44,570604 168 20 59,919494 210 25 75,268383
127 16 47,536741 169 20 59,933499 211 25 75,282389
128 16 47,550746 170 21 62,899636 212 25 75,296395
129 16 47,564752 171 21 62,913642 213 26 78,262531
130 16 47,578757 172 21 62,927647 214 26 78,276537
131 16 47,592763 173 21 62,941653 215 26 78,290543
132 16 47,606768 174 21 62,955658 216 26 78,304548
133 16 47,620774 175 21 62,969664 217 26 78,318554
134 16 47,634780 176 21 62,983669 218 26 78,332559
135 17 50,600917 177 21 62,997675 219 26 78,346565
136 17 50,614922 178 22 65,963812 220 26 78,360570
137 17 50,628928 179 22 65,977817 221 26 78,374576
138 17 50,642933 180 22 65,991823 222 27 81,340713
139 17 50,656939 181 22 66,005828 223 27 81,354718
140 17 50,670944 182 22 66,019834 224 27 81,368724
141 17 50,684950 183 22 66,033840 225 27 81,382729
142 17 50,698955 184 22 66,047845 226 27 81,396735
143 17 50,712961 185 22 66,061851 227 27 81,410740
144 18 53,679098 186 22 66,075856 228 27 81,424746
145 18 53,693103 187 23 69,041993 229 27 81,438752
146 18 53,707109 188 23 69,055999 230 28 84,404889
147 18 53,721114 189 23 69,070004 231 28 84,418894
148 18 53,735120 190 23 69,084010 232 28 84,432900
149 18 53,749125 191 23 69,098015 233 28 84,446905
150 18 53,763131 192 23 69,112021 234 28 84,460911
151 18 53,777137 193 23 69,126026 235 28 84,474916
152 18 53,791142 194 23 69,140032 236 28 84,488922
153 19 56,757279 195 23 69,154037 237 28 84,502927
154 19 56,771285 196 24 72,120174 238 28 84,516933
155 19 56,785290 197 24 72,134180 239 29 87,483070
156 19 56,799296 198 24 72,148186 240 29 87,497075
157 19 56,813301 199 24 72,162191 241 29 87,511081

Straight corrected gear

The measurement across the teeth of a corrected wheel or pinion with straight gear= Wk‘­.
Image278via tablel or Wk = m.cosa.[(k-Image19).p + z.inva] + 2.x.m.sina
with factors as:

  • Wk= width according to previous tables
  • x= coefficient of correction
  • x.m= correction in mm (positive or negative), measured at the radius
  • m= module
  • K= coefficient from the table below
Module K when
AP = 14°30′
K when
AP = 20°
Module K when
AP = 14°30′
K when
AP = 20°
1 0,50076 0,68404 8 4,00608 5,47232
1,5 0,75114 1,02606 9 4,50684 6,15636
2 1,00152 1,36808 10 5,00760 6,84040
2,5 1,25190 1,71010 11 5,50836 7,52444
3 1,50228 2,05212 12 6,00912 8,20848
3,5 1,75266 2,39414 13 6,50836 8,89252
4 2,00304 2,73616 14 6,00912 9,57656
4,5 2,25342 3,07818 15 6,50988 10,26060
5 2,50380 3,42020 16 7,01064 10,94464
5,5 2,75418 3,76222 17 7,51140 11,62868
6 3,00456 4,10424 18 9,01216 12,31272
6,5 3,25494 4,44626 19 9,51292 12,99676
7 3,50632 4,78828 20 10,01520 13,68080
7,5 3,75570 5,13030

Helical teeth without correction

meetmethode The measurement method on a different number of teeth is also applicable for helical gears, provided that one takes into account the apparent modules mt, the apparent pressure angle at and of the virtual number of teeth zn. The formula is then consequently: Wnk = mn.cos anImage280
with zv = z.Image281 and at =  tan-1Image282
This method of measurement is often limited by the width of the gear and by the angle. The following table lists some values from the involute function.

Helical teeth with correction

The following formula can be applied for this:
Wnk= mn.cos anImage280+2.xn.mnsin an