Measurement over multiple teeth or base tangent length

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.

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

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

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

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‘­.
via tablel or Wk = m.cosa.[(k-).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

 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 an with zv = z. and at =  tan-1 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 an+2.xn.mnsin an