LIBRARY OF CONGRESS. CliapXJ.l&.^^opy right No. Shelf..:.a33 UNITED STATES OF AMERICA. FORMULAS IN GEARING I' 33 WITH PRACTICAL SUGGESTIONS. Ui, >?n«1'^'' PROVIDENCE, R. I. BROWN & SHARPE MANUFACTURING COMPANY, 1896. "x •■■■ Entered according to Act of Congress, in the year 1896 by BROWN & SHARPE MFG. CO., In the Office of the Librarian of Congress at "Washington. Registered at Stationers' Hall, London, Eng. All rights reserved. .v1 Ofo -3f Ss"? PREFACE It is the aim, in the following pages, to condense as much as possible the solution of all problems in gearing which in the ordinary practice may be met with, to the exclusion of prob- lems dealing with transmission of power and strength of gearing. The simplest and briefest being the symbolical expression, it has, whenever available, been resorted to. The mathematics employed are of a simple kind, and will present no difficulty to anyone familiar with ordinary Algebra and the elements of Trigonometry. . ' CONTENTS, FORMULAS IN GEARING. CHAPTER I. Page Systems of Gearing . . . . i CHAPTER n. Spur Gearing — Formulas — Table of Tooth Parts — Comparative Sizes of Gear Teeth 4. CHAPTER HI. Bevel Gears, Axes at Right Angles — Formulas — Bevel Gears, Axes at any Angle — Formulas — Undercut in Bevel Gears — Diameter Incre- ment — Tables for Angles of Edge and Angles of Face — Tables of Natural Lines 11 CHAPTER IV. Worm and Worm Wheel, Formulas — Undercut in Worm Wheels — Table for gashing' Worm Wheels 34 CHAPTER V. Spiral or Screw Gearing — Axes Parallel — Axes at Right Angles — Axes at any Angle — General Formulas— Table of Prime Num- bers and Factors 39 CHAPTER VI. Internal Gearing — Internal Spur Gearing — Internal Bevel Gears cy CHAPTER VII. Gear Patterns 6^ CHAPTER VIII. Dimensions and Form for Bevel Gear Cutters 66 CHAPTER IX. Directions for cutting Bevel Gears with Rotary Cutter 69 CHAPTER X. The Indexing of any Whole or Fractional Number 72 CHAPTER XI. The Gearing of Lathes for Screw Cutting — Simple Gearing — Compound Gearing — Cutting a Multiple Screw 76 FORMULAS IN GEARING. Ch:af»te;r I. SYSTEMS OF GEARING. (Figs. I, 2.) There are in common use two systems of gearing, viz.: the involute and the epicycloidal. In the involute system the-outlines of the working parts of a tooth are single curves, which may be traced by a point in a flexible, inextensible cord being unwound from a circular disk the circumference of which is called the base circle^ the disk being concentric with the pitch circle of the gear. Fl(j, 1. In Fig. I the two base circles are represented as tangent to the line P P. This line (P P) is variously called " the line of pressure," " the line of contact," or '' the line of action." 2 BROWN & SHARPE MFG. CO. In our practice this is drawn so as to make with a normal to the center line (O O') 14%°, or with the center line 75}^°. The rack of this system has teeth with straight sides, the two sides of a tooth making, together, an angle of 29° (twice 14)^°). This applies to gears having 30 teeth or more. For gears having less than 30 teeth special rules are followed, which are explained in our " Practical Treatise on Gearing." Fig, 2, In epicycloidal^ or double-curve teeth, the formation of the curve changes at the pitch circle. The outline of the faces of epicycloidal teeth may be traced by a point in a circle rolling on the outside of pitch circle of a gear, and the flanks by a point in a circle rolling on the inside of the pitch circle. The faces of one gear must be traced by the same circle that traces the flanks of the engaging gear. In our practice the diameter of the rolling or describing circle is equal to the radius of a 15-tooth gear of the pitch required ; this is the base of the system. The same describing circle being used for all gears of the same pitch. The teeth of the rack of this system have double curves, which may be traced by the base circle rolling alternately on each side of the pitch line. An advantage of the involute over the epicycloidal tooth is, that in action gears having involute teeth may be separated a little from their normal positions without interfering with the angular velocity, which is not possible in any other kind of tooth. The obliquity of action is sometimes urged as an objection to involute teeth, but a full consideration of the subject will show that the importance of this has been greatly over-esti- mated. The tooth dimensions for both the involute and epicycloidal gears may be calculated from the formulas in Chapter II. BROWN & SHARPE MFG. CO. CHAPTKR II. SPUR GEARING. (Figs. 3, 4.) Two spur gears in action are comparable to two correspond- ing plain rollers whose surfaces are in contact, these surfaces representing the pitch circles of the gears. Pitch of Gears. For convenience of expression the pitch of gears may be stated as follows : Circular pitch is the distance from the center of one tooth to the center of the next tooth, measured on the pitch line. Diametral pitch is the number of teeth in a gear per inch of pitch diameter. That is, a gear that has, say, six teeth for each inch in pitch diameter is six diameiial pitch, or, as the expres- sion is universally abbreviated, it is " six pitch." This is by far the most convenient way of expressing the relation of diameter to number of teeth. Chordal pitch is a term but little employed. It is the dis- tance from center to center of two adjacent teeth measured in a straight line. PROVIDENCE, R. I. FORMULAS. N = number of teeth. s = addendum. t = thickness of tooth on pitch line. /= clearance at bottom of tooth. D" = working depth of tooth. D" + / = whole depth of tocch. d= pitch diameter. d' = outside diameter. P' = circular pitch. P^ = chord pitch. P = diametral pitch, C = center distance. p_ N + 2 p z= -ZL P' p' = 5 p f=^ = .3x83P' __d ^ N N -f 2 2 2 P lO i8o° lO s D" = 2S P^ — . ^sin N 360° d d P" P' = dTt where sin S = — d = .^ d' = d -h 2 s . NP' BROWN & SHARPE MFG. CO. GEAR WHEELS. TABLE OF TOOTH PARTS CIRCULAR PITCH IN FIRST COLUMN. p' 2 Threads or Teeth per inch Linear. 1 . 11 Thickness of Tooth on Pitch Line. h r Depth of Space below Pitch Line. OH Width of Thread-Tool at End. f P t s D" s+f 1.3732 P'x.31 P'x.335 i 1.5708 1.0000 .6366 1.2732 .7366 .6200 .6700 1^ A 1.6755 .9375 .5968 1.1937 .6906 1.2874 .5813 .6281 If i 1.7952 .8750 .5570 1.1141 .6445 1.2016 .5425 .5863 If A 1.9333 .8125 .5173 1.0345 .5985 1.1158 .5038 .5444 li 1 2.0944 .7500 .4775 .9549 .5525 1.0299 .4650 .5025 h\ a 2.1855 .7187 .4576 .9151 .5294 .9870 .4456 .4816 ^ t\ 2.2848 .6875 .4377 .8754 .5064 .9441 .4262 .4606 lA ii 2.3936 .6562 .4178 .8356 .4834 .9012 .4069 .4397 li i 2.5133 .6250 .3979 .7958 .4604 .8583 .3875 .4188 ItV n 2.6456 .5937 .3780 .7560 .4374 .8156 .3681 .3978 14 f 2.7925 .5625 .3581 .7162 .4143 .7724 .3488 3769 lA H 2.9568 .5312 .3382 .6764 .3913 .7295 .3294 .3559 1 1 3.1416 .5000 .3183 .6366 .3683 .6866 .3100 .3350 n lA 3.3510 .4687 .2984 .5968 .3453 .6437 .2906 .3141 1 H 3.5904 .4375 .2785 .5570 .3223 .6007 .2713 .2931 a 1^ 3.8666 .4062 .2586 .5173 .2993 .5579 .2519 .2722 f 1* 4.1888 .3750 .2387 .4775 .2762 .5150 .2325 .2513 n lA 4.5696 .3437 .2189 .4377 .2532 .4720 .2131 .2303 i li 4.7124 .3333 .2122 .4244 .2455 .4577 .2066 .2233 PROVIDENCE, R. L TABLE OF TOOTH TARTS.— Continued. CIRCULAR PITCH IN FIRST COLUMN. a 55 Threads or Teeth per inch Linear. Diametral Pitch. Thickness of Tooth on Pitch Line. -k id < s M 1 Working Depth ^ 1 of Tooth. Depth of Space beloAV Pitch Line. Whole Depth of Tooth. Width of Thread-Tool at End. Width of Thread at Top. p' 1" P t ^^/ D"+/. P'x.31 P'x.835 1} 5.0265 .3125 .1989 .3979 .2301 .4291 .1938 .2094 A n 5.5851 .2812 .1790 .3581 .2071 .3862 .1744 .1884 i 2 6.2832 .2500 .1592 .3183 .1842 .3433 .1550 .1675 iV 2* 7.1808 .2187 .1393 .2785 .1611 .3003 .1356 .1466 1 2i 7.8540 .2000 .1273 .2546 .1473 .2746 .1240 .1340 f 2| 8.3776 .1875 .1194 .2387 .1381 .2575 .1163 .1256 4 3 9.4248 .1666 .1061 .2122 .1228 .2289 .1033 .1117 iV H 10.0531 .1562 .0995 .1989 .1151 .2146 .0969 .1047 .2 3} 10.9956 .1429 .0909 .1819 .1052 .1962 .0886 .0957 i 4 12.5664 .1250 .0796 .1591 .0921 .1716 .0775 .0838 2 y 4i 14.1372 .1111 .0707 .1415 .0818 .1526 .0689 .0744 i 5 15.7080 .1000 .0637 .1273 .0737 .1373 .0620 .0670 3 GJ 16.7552 .0937 .0597 .1194 .0690 .1287 .0581 .0628 J, 6 6 18.8496 .0833 .0531 .1061 .0614 .1144 .0517 .0558 i 7 21.9911 .0714 .0455 .0910 .0526 .0981 .0443 .0479 J 8 25.1327 .0625 .0398 .0796 .0460 .0858 .0388 .0419 1 9 28.2743 .0555 .0354 .0707 .0409 .0763 .0344 .0372 I'o 10 31.4159 .0500 .0318 .0637 .0368 .0687 .0310 .0335 1 . 1 C IG 50.2655 .0312 .0199 .0398 .0230 .0429 .0194 .0209 BROWN & SHARPE MFG. CO. GEAR WHEELS. TABLE OF TOOTH PAKTS DIAMETKAL PITCH IN FIRST COLUMN. 1 Diametral Pitch. Thickness of Tooth on i Pitch Line. II o ° Depth of Space below Pitch Line. Whole Depth of Tooth. P P' t s D" s+f. D"+/. i 6.2832 3.1416 2.0000 4.0000 2.3142 4.3142 f 4.1888 2.0944 1.3333 2.6666 1.5428 2.8761 1 3.1416 1.5708 1.0000 2.0000 1.1571 2.1571 li 2.5133 1.2566 .8000 1.6000 .9257 1.7257 li 2.0944 1.0472 .6666 1.3333 .7714 1.4381 If 1.7952 .8976 .5714 1.1429 .6612 1.2326 2 1.5708 .7854 .5000 1.0000 .5785 1.0785 2i 1.3963 .6981 .4444 .8888 .5143 .9587 2i 1.2566 .6283 .4000 .8000 .4628 .8628 2J 1.1424 .5712 .3636 .7273 .4208 .7844 3 1.0472 .5236 .3333 .6666 .3857 .7190 3i .8976 .4488 .2857 .5714 .3306 .6163 4 .7854 .3927 .2500 .5000 .2893 .5393 5 .6283 .3142 .2000 .4000 .2314 .4314 6 .5236 .2618 .1666 .3333 .1928 .3595 7 .4488 .2244 .1429 .2857 .1653 .3081 8 .3927 .1963 .1250 .2500 .1446 .2696 9 .3491 .1745 .1111 .2222 .1286 .2397 10 .3142 .1571 .1000 .2000 .1157 .2157 11 .2856 .1428 .0909 .1818 .1052 .1961 12 .2618 .1309 .0833 .1666 .0964 .1798 13 .2417 .1208 .0769 .1538 .0890 .1659 14 .2244 .1122 .0714 .1429 .0826 .1541 PROVIDENCE, R. I. TABLE OF TOOTH TA'RTS— Continued. DIAMETRAL PITCH IN FIRST COLUMN. ■g . II r 5^ Thickness of Tooth on Pitch Line. IS Working Deptli of Tooth. Depth of Space below Pitch Line. p. P'. t. s. D". .0771 D"-^/. 15 .2094 .1047 .0666 .1333 .1438 16 .1963 .0982 .0625 .1250 .0723 .1348 17 .1848 .0924 .0588 .1176 .0681 .1269 18 .1745 .0873 .0555 .1111 .0643 .1198 19 .1653 .0827 .0526 .1053 .0609 .1135 20 .1571 .0785 .0500 .1000 .0579 .1079 22 .1428 .0714 .0455 .0909 .0526 .0980 24 .1309 .0654 .0417 .0833 .0482 .0898 26 .1208 .0604 .0385 .0769 .0445 .0829 28 .1122 .0561 .0357 .0714 .0413 .0770 30 .1047 .0524 .0333 .0666 .0386 .0719 32 .0982 .0491 .0312 .0625 .0362 .0674 34 .0924 .0462 .0294 .0588 .0340 .0634 36 .0873 .0436 .0278 .0555 .0321 .0599 38 .0827 .0413 .0263 .0526 .0304 .0568 40 .0785 .0393 .0250 .0500 .0289 .0539 42 .0748 .0374 .0238 .0476 .0275 .0514 44 .0714 .0357 .0227 .0455 .0263 .0490 46 .0683 .0341 .0217 .0435 .0252 .0469 48 .0654 .0327 .0208 .0417 .0241 .0449 50 .0628 .0314 .0200 .0400 .0231 .0431 56 .0561 .0280 .0178 .0357 .0207 .0385 60 .0524 .0262 .0166 .0333 .0193 .0360 lO BROWN & SHARPE MFG. CO. Comparative Sizes of Gear Teeth. Involute. 8 p Fiij* 4, PROVIDENCE, R. I. I I CHAPTER III. BEVEL GEARS.— AXES AT RIGHT ANGLES. (Fig. 5.) 12 BROWN & SHARPE MFG. CO. FORMULAS. 5^« ^ I Number of teeth \ ^^^r* N5 = j ( pinion P = diametral pitch. P' = circular pitch. a^ = \ center angle = angle of edge j gear. a^j =z ) or pitch angle ( pinion. /3 = angle of top. /?' = angle of bottom. ^« = I angle of face \ ^^^^' gb= \ ^ I pmion. j"'~ !- cutting angle K ^. .' /lb = ) & & ^ pmion. A = apex distance from pitch circle. A' = apex distance from large bottom of tooth. d = pitch diameter. ^' = outside diameter. s = addendum. / = thickness of tooth at pitch line. / = clearance at bottom of tooth. D" = working depth of tooth. D" + /= whole depth of tooth. 2 a = diameter increment. /^ = distance from top of tooth to plane of pitch circle. F = Vi^idth of face. PROVIDENCE, R. I. 13 N N, tan a„ = — — - — : tan cxj ^ 2 sin a 4. z? i" tan p = — — — ; or tan p = — • t^^fy = ^'^no^(^ + Td ^^-3^4sina tan // = £±/ ; N N ^ ■ A ' h = a— ff [See Note, pages 2.) 2 '^'A'^'^m A= N 2 P sin a A'= _^ A' = N COS p' 2 P sin a cos /:^' A = i^ cos ^ sin {a + /^) p= N 2 A sin a V N N P' .... a = — or = a = a -\- 2 a P Tt 2 a =^ 2 s cos a (Seepage 20.) b = ais.na i'^f°'-g^a.>- =f for pinion ( a for pinion = for gear P = —- P'= i^ P' P s = 1 z=z^= .3183 P' ^ = Atan/? P TT ^ ^ ^ j- + / = . 3685 P' ^ + / = A tan /5' 2 2 P F = i + ^- or=: 2P' to3P' P 7 Note. —Formulas containinj^ notations without the designating: letters a and b apply equally to either g:(ar or pinion. If wanted for one or the other, the respective letters are simply attached. BROWN v«t bHAKPE MFG. CO. BEVEL GEARS WITH AXES AT ANY ANGLE. Fifi. a. PROVIDENCE, R. 1. 15 FORMULAS. C = angle formed by axes of gears. t^t" ~ !■ number of teeth -] ^. . ' N5 = j ( pinion. P = diametral pitch, P' = circular" pitch. " Z f angle of edge = pitch angle -j ^9^^- fd = angle of top. fi' = angle of bottom. I angle of face -I ^?^}'' ] ^ { pinio /' ~ >- cutting: ansfle < ^. . /lb = ) & & ( pinic ion. A — apex distance from pitch circle. A' = apex distance from large bottom of tooth. d = pitch diameter. d' = outside diameter. 2a = diameter increment. d = distance from top of tooth to plane of pitch circle. Note. — The formulas for tooth parts as given on page 5 apply equally to thee cases. l6 BROWN & SHARPE MFG. CO, sin C , Nft , , ^ tan a^— — ; or cot a^ = — — A_^ + cot C 1^^-hcosC ^«'^^^ tan a^ = ^^ ^'"^ ^ ; or cot a^ = ^^ "^^ ^ + cot C N„ , ., Nft 3in C — 2: + cos S^ ^ Note. — These formulas are correct only for values of C less than 90°. If C greater than 90°, consult the following: page. ^ 2 sm a ^ JD s tan p = — - — ; or tan /? = _ ; N A ga=9o" - K+ /?) ; g^= 90° - («'5 + /?) h = a — p {See Note, page ^2.) N A 2 i-N P sin or A' A cos/5' ^/ = N " P or = NF 7t d' •=zd -\- 2 a 2 a =^ 2 s cos a a for gear z=z b for pinion. a for pinion ■= b for gear. Note. — See Foot Note on page 13. PROVIDENCE, R. I. 17 l8 BROWN & SHARPE MFG. CO. The formulas given for a^^ and a^ (when C, N^ and N^ are known) undergo some modifications for values of C greater than 90°. For bevel gears at any angle but 90° we may distinguish four cases ; C, N^, N^ being given. /. Case. See pages 14 and 16. //. Case. C is greater than 90°. tan «„= sin (180 -C) ^^^ ^ ^\r^ (iZo - C ^-cos(i8o-C) — "-cos(i8o-C; tan o'j, N, -' N ///. Case. (Xa — 90° ; ^'^ = C — 90° IV. Case. sin E ^ sin E tan (Xa = ; tan a^, = COS E — - -'' _" — cos E N. N For an example to apply to Case III., the following condi- tion must be fulfilled : N„ sin (C - 90°) = N, To distinguish whether a given example belongs to Case II. or case IV., we are guided by the following condition : Is • N sin (C - 00°) \ ^^^^^^^ ^^^^ ^b, we have Case II. " ^ ( l<^i"ger than Nft, we have case IV. PROVIDENCE, R. I. I9 UNDERCUT IN BEVEL GEARS. By undercut in gears is understood a special formation of the tooth, which may be explained by saying that the elements of the tooth below the pitch line are nearer the center line of the tooth than those on the pitch line. Such a tooth outline is to be found only in gears with few teeth. In a pair of bevel gears where the pinion is low-numbered and the ratio high, we are apt to have undercut. For a pair of running gears this condition presents no objection. Should, however, these gears be intended as patterns to cast from, they would be found use- less, from the fact that tliey would not draw out of the sand. We have stated on page 2 (see Fig. i) that the base of our involute system is the 14^° pressure angle. If a pair of bevel gears with teeth constructed on this basis have undercut, we can nearly eliminate the undercut — and for the practical work- ing this is quite sufficient— by taking as a basis for the con- struction of the tooth outline a pressure angle of 20°. The question now is : When do we, and when do we not have undercut ? Let there be : N = number of teeth in gear. n = number of teeth in pinion. nV^ N' + ^' _ N where we have undercut for/ less than 30. This formula is strictly correct for epicycloidal gears only. It is, however, used as a safe and efficient approximation for the involute system. 20 BROWN & SHARPE MFG. CO. DIAMETER INCREMENT. Rule. — The ratio being given or determined, to find the outside diameter divide figures given in table for large and small gear by pitch (P; and add quotient to pitch diameter. GEARS. GEARS. GEARS. RATTH RATJ'^ RATIO. Large Small Large Small Large SmalJ 1 00 1:1 1.41 1.41 1.65 1.05 1.70 4.40 .45 1.94 1.05 1.37 1 42 1.67 5:3 1.03 1.72 4.50 9:2 .44 1.95 1.07 1.36 1.43 1.70' 1.01 1.73 4.60 .42 1 95 1.10 1.35 1.44 1.75 7:4 .99 1.74 4.80 .41 1.96 1.11 10:9 1.34 1.46 1.80 9:5 .97 1.75 5.00 5:1 .39 1.96 1.12 1.33 1.46 1.85 .95 1.76 5.20 .38 1.96 1.18 9:8 1.33 1.47 1.90 .93 1 77 5.40 .37 1.96 1.14 8:7 1.32 1.49 1.95 .91 1.78 5.60 .36 1.97 1.15 1.31 1.50 2.00 2:1 .89 1.79 5.80 .34 1.97 1.16 1.30 1.51 2.10 .87 1.80 6.00 6:1 .33 1.97 1.17 7:6 1.30 1.52 2.20 .84 1.81 6.20 .32 1.97 1.18 1.29 1.53 2.25 9:4 .82 1.82 6.40 .31 1.97 1.19 1.28 1.53 2.30 .80 1.83 6.60 .30 1 97 1.20 6:5 1.28 1.54 2.33 7:3 .78 1.84 6.80 .29 1.98 1.23 1.27 1.55 2.40 .76 1.85 7.00 7:1 .28 1.98 1 25 5:4 1.25 1.56 2.50 5:2 .75 1.86 7.20 .27 1.98 1.27 1.25 1.57 2.60 .73 1.86 7.40 .27 1 98 1.29 9:7 1.24 1.58 2.67 8:3 .71 1.87 7.60 .26 1.98 1.30 1.22 1.59 2.70 .69 1.87 7.80 .26 1.98 1.33 4:3 1.20 1.60 2.80 .67 1.88 8.00 8:1 .25 1.98 1.35 1.18 1.61 2.90 .65 1.89 8.20 .24 1.98 1 37 1.17 1.61 3.00 3:1 .63 1.91 8.40 .24 1.98 1.40 7:5 1.16 1.62 3.20 .60 1.92 8.60 .23 1.98 1.43 10:7 1.15 1.63 3.33 .58 1.92 8.80 .23 1.98 1.45 1.13 1.65 3.40 .56 1 92 9.00 9:1 .22 1.99 1.50 3:2 1.11 1.66 3.50 7:2 .54 1.93 9.20 .22 1.99 1.53 1.10 1.67 3.60 .52 1.93 9.40 .21 1.99 1.55 1.09 1.67 3.80 .50 1.94 9.60 .21 2.00 1.58 1.08 1.68 4.00 4:1 .49 1.94 9.80 .20 2.00 1.60 8:5 1.07 1.68 4.20 .47 1.94 10.00 10:1 .20 2.00 Note. — To be used only for bevel gears with axes at right angle. PROVIDENCE, R. I. 21 TABLES FOR ANGLES OF EDGE AND ANGLES OF FACE. The following four tables have been computed for the convenience in calculating datas for bevel gears with axes at right angle. They do not hold good for bevel gears with axes at any other angle. To use the tables the number of teeth in gear and pinion must be known. Having located the number of teeth in the gear on the horizontal line of figures at the top of the table, and the num- ber of teeth in the pinion on the vertical line of figures on the left-hand side, we follow the two columns to the square formed by their intersections. The two angles found in the same square are the respective angles for gear and pinion. The tables are so arranged that the angle belonging to the gear is always placed above the angle for the pinion. 22 BROWN & SHARPE MFG. CO. TABLE I Angle of Edge. GEAR. 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 12 7341 i6;i9 73*18 16*42' 72*54 17V 72*26 17*32 72*2 17*58 7I-3V 18*26; 71*5' 18*55 70*34 19*26 70* r 19*69' 69*26 20*34 68*50' 2»*io' 68*12 21*48 67*3. 22*29 66*48 23*12 66*2- 23)8 13 ltd 17*35 7.W 18*1' 71*34 18*26 71*7 18*53' 70W 19*8.' 70*9' I9>.' 69V 20*23' 69*5 20>s 68'30' 21*30 67*53 22*7' 67*15' 22*45 66*34 23*86 65*si- 24*9; 65*6' 24;S4;- 64*17' 2543; 14 71 V 70*43 J9*I7' 70*15 I3%s 69*46 eo*M.' 69*.6 2044 68*45 21*5 68'i2' 2(*4« 67*i? 22*23 67'o' 23*0' 66*23 23*87 65*42 24*.6; 64*» 64*H»' 25*4« 63*26' 26*34 62y 27*84 l5 69V eoV 68*2^ 20*34 68'se 68*28 21*32 67's6 22*4 67*n 22*37 66*48 2312 66-12 23*48 65*33' 24*27 64y 25*7- 64',o' 25*50 63*26' 26*34- 62^9 27*21 61 '4» 60*5^ 29*» 16 68V 21*3 68'» 81*48 67V ezV ^7*.o 22*50 66*37 23*23' 6e'2' 23*58 65*26' 24*34: 64*48 25*12 64*8' 25*58 63*26 26*34 62*42 27*»' 61*56' 28*4 61*7 28*53 60*15 29*45 59'er 30V 17 67« 2e*3i' 66*« 23*2 66*27 23*33 65*54 24*6' 65*13' 844.' 64*43' 25*17' 64*6 25*54 63*26 26*34 62*45 27*is 62* r 27*59 61*15 28*45 60*28 29*32 59*37 30*n 58*44 31*6 57*48' 32*12 18 234e 6S'4« 24*M esv 24*4«; €4's3 25*2. 64V 25*56 63'W 26*34 62*47 27*13 62*6 27*54 6l'25' 28*57' 60*36 29*22 59"5r 30*9 59*2' 30« 58*10 3V*M 57'',6 32*4^ 56*.9' 334.' 19 65*e 2**« 64*M. 25*« 64*2 25*58 63» 26*84' 62*49 27*11 62*10 27*56 61*30 28*33 3957 43*24 40*36 48*2^ 41*36 25 58V 3lV 58*0' 32*0' 5720 32*40 56*40 33*eo' 55*57 34*3 ssV 34*47' 54*28 35*32' 53*40' 36*20 52*51' 37*9' 52*g- 38*0' Si"? 38*S' 50*12 39*48- 49*K. 40*46 48*.4 41*46 47*12 44*48 '4. O 26 5737' 32*23' 56*S8 33*e' 56*« 33*41 55*37 34*23' 54*S4 35*6 54*.d 35*sd 53*24 36*M 5296 37Vv 5.*4^ 38*14 5054 39*6 50*,- 39*59 49*5 40*55 48*7 41*53 47*7' 4e*ra 46*5 43*55 1<^ SI 27 56*38 33*« ssW 34* r S5*i8- 34*42 S4V 35*M 53*53 36*7 53*7- 36*53 52^21 37*39' 51^3 38*27 50V 33*17 49"5. 40*9' 48"57 41*3 48''o' 42*0 47*3' 42*57 46'2 43*56 4-5* 28 SS4*; 34*20 55*0 35*0' S4*« 35*4. 53*37 36*23 52*53 37*7' 52*8 37*52 5'> 38*40 50*32 39*28 49«4r 40*13 48*46' 41*12 47''55 42*5 46>6 43 2 46*0 44*0 45* 29 54W 35*6 35*S7 53*22 36*38- S2'^3S 37*er 51*55 38-S 51*9' sa'si' 50*21 39*39 49*32' 40*28 48-4.' 41*19 47*49 42*11 46V 43*6- 45^ 44*2' 45* 30 S3*4i 36*«' 53*7- 36*53 52*26 37*34 5.*42' 38*8 50*56 38*t 50*ie 39*46 49*24 40*36 48*3S 41*25 47*43 42*17 46*5. 43*9 45-56 44*4- 45* 31 37V 58*.3 3747 5l*3»' 33'w 50*48- 39*2 50*2 39'w 48*.6' 4044 48*26 41*32 47*33 42*21 46*47 43*«3 45*54 44*6 45* 32 52*2' 37*« 51*20 38*40 50*38 39*22 49*54 40*6 49*9 40*51 48*22 4i'3»' 47*34 42*2t' 46*44 43*16 45*53 44*7' 45* 33 51*10 38*50 50*2^ 39*31 49''46 40*4 40*58 48*16 41*44 47*2» 42*21 46''4.' 43-.0 45*5. 44*9 45' 34 50*20 39*40 49*3« 40*« 48^Si 41*5' 48'..- 4t*4a' 47*25 42*35 46''38 43*22 45^0 44*10' 45* 35 43*3. 40*29 48*^ 41*18 46*s 41*55 47*« 42*39 46*35 43*25 45*48 44*tt 45' • 36 48« 41*17 48^0 42*0 47''a 42*43 46*33 43*27 4547 44* iW 45* 37 47'k» 42*4' 47'',4 42*4* 46*30 43*30 45!U 44*»4 4S* 3a 4^10 42*S9 46'2» 43*3e 4«^4S 44*15 45' 39 46*2. 43*31 45*43 44*17 45- 40 4542 44*,B 45' 41 45- PROVIDENCE, R. I. 23 TABLE \— (Continued) Angle of Edge. GEAR. 26 25 24 23 22 21 2U 19 18 17 16 15 14 13 12 12 65-.. 64'2Z 25°38 63*ZG 26a4 62*27 27°33 6i'e3 28*37 60*15 29*45 59-2 30*58 57*44 32*16 56 '19 33*4i' 54*47 35*13 53*7' 36*53 51*20 38*40 49*24 40*36 47"i7' 42*43 45* 13 63*26 26V 62*31 27*29 61 '33 2827 eoSr 2^*29 59'2S 30*35 5e*»4 31*46 56-58 33*2 55'37 34*23 54*10 35*50 52*36 37*24 50*54 ^9*6 49' 5 4055 47*7 42*53' 45* 14 6l'4E 28*iB 604S 29*15 594S 30*15 5840 31 '20 57*32 32*28 56*19 33*41' 55*0 35"o 53*37 36*2i 52*6 37m 50*32 39*28 48*48' 4ri2' 46 58 43'e' 45- 15 60*.- 29*53 59- 2 30*5* S8"o 32*0' 56V 33*7 5543 34'|7' 54-28 35*^2 53*7' 36*53' 5lV 38*18 50"i2 38*48' 48°i5' 41*25 46*51 43"9' 45' 16 58*21 31 V 57'e3 32*57' 5fc*l9 33** 1 55*11' 3**49 53-sB 36*2' 52*42' 37*18' 51 M 384,- 11*19 78*36 11*36 78*19 1 1*4.' 78*7 11*53 13 79V 10*23 79*2» I0*3l" 79*20 10*40 79*..' 10*49 79;. 10*59 18*5. 11*9' 78*4.' 11*19 78*3.' 11*29 78*20 11 V 7^9 11*51 77*58 12*2 77V 12*14 77V 12*26 77*22 12*38 77*9 12*51 14 79-<, 78-5. 11*9 784.' II* « 76''32 11*26 78*22 M*3i 76*.. 11*49 78* .' 11*59 77*51- 12*9 77*40 12*26 77*28 I2*3i 77*,7 12*43 77*5 12*55 76*5i 13*8 76-» 13*2.' 76*26 13'3i 76*12 13*48 15 78'l4 78%' 11*56 77*9V 12*6' 77V 12*16 77*34 IE*i6 77*23 12*37 77'.2 12*48 77*o- r3*o' 76*48 13*K>' 76*36 13*»* 76V 13*3^ 76*11' 13*49 75*59 \4°e' 75*4^ 14*16 7^36 14 36 7i*,5 14*45 16 77-26 77-S 12V 12 53 76*57 13*3' 76*45 13*15 76*3» 13*26 76*22 13*3^ 76*10 13*50 75*58 14*2 75V 14*15 7S*3i 14*28 75*« 14*42 75%' 14*56 74V 15*11 74V 15*25 74-,9 15*41 17 76*43 13V 7^K I3W 76*2.' I3*M 76*10 13*50 75*58 14*2 75V I4*ti 75*33 14*27 75*2. 14°m' 75°e' 14*62 74*54 15*6 74^40 15*20 74*25 15*35 74*1.' 15*49 73*56 16*4 73*40 16*20 16 36 18 75^58 14° t 754S 14° KV 75*3S I4*2S 75*23 I4°37 75°.6 14*50 74*59 15*2 74*45 15*15 74*3, 15*29 74*17 15*43 74*3 I5*5i 73*49 16*..' 73*33 16*27 73*.8 I6*4i 73*2 16*58 7245 17*15' 72*29 17*31 19 7S^« »4%7 7S'.- »4'59 74'-« 15*1 1 74^3* 15*24 74*23 15*37 74*Ki 15*56 73V 16*4 73V 16° 18 73*28 16*32 73V 16*47 72*5S 17*2' 72*42 17*18 72*20 17*34 72*9 I7°5l' 71*52 18*8 7l'3» 18*26 20 74-« 74'.6 I5*W IS 57 73*50 16*10 73*37 16*23 73*23 16*4;; 73*9 16*51 72V 17*6 72*39 17*21 72V 17*37 72*7 17'53 71*5.' 18*9' 71V 18*26 71*16 18*44 70''59 19* r 7(f46 19*20 21 73-45 73*32 73*.8 16*42 73*4 16*56 72*50 17*10 72*36 17*14 72*2. 17*39 72V' 17*5f 71*^50 l8*.o 71-34 18*26 71*17 18*43 71*0' l9*o' 7043 19*17 70*24 19*36 70*6- 19*54 69V 20*24 22 73- ,• 16*59 72t,7 I7''l3' 72-« 17*27 72*,,' 17*4.' 72*/ 17*56 71*49 18*11 71*34 18*26 71*18 !e*42 71*2 I8*5R 70*45 .9*15 70V 19*3i 70°.6 19*50 69*52 20*8 69*33 20*27 63''l3 20*47 68*» 21*6 23 72*^' »7%3 72°3 17 $7 71*40 18*11' 71*34 18*26 7I'«' 18*41' 7J*3 18*5^ 70*47 i9*a 70*30 I9*3«i 70*.i 19*46 69*57' 20*3 69>9 20*21 69-20 20*44 2I*6 68*38 21*22' 66*2^ 21*40 68*3 21*57 67*45 22*15 €7V 22*3i 67*8 22*52 66V 23*12 66*28 23*32 66-7' 23*si 65*46 24*14 65V 24*35 65*2 24*58 64*39 25*«.' 28 68%i 21* 15 b8'» 21*31 6S-12 2(*4e 67*55 22*5 67*37- 22*a 67'.9 22*41 67*,' 22» 66*42 23*18 66*22 23*38 66* 2; 23*5i 6S*4i 24*18 6tf2i 24*39 €4*59 25*.' 64*37 25*23 64* w 25*46 63-S6 26*10 29 68V 2l'56 67^47 22*13 22*^ 67*12 22*48 66V 23*6 66*36 23*24 66*.7 23*43 6^57 24*3 Og'37 a4-a 65*,6 24*4^ 64V 25*5 64V 25*26 64V 25*4i 63V 26*.o' 6^26 26*3* 63*2 26*58 30 22*37 6r6 22*54 23*12 66*30 23*90 6^,i 2348 6^52 24*8 65*» 24*27 6S*h; 24*46 25*7 64*32 25*28 64*.o' 25*50 63*4,- 26*11' 63*26 26*34 63-3* 26*57' 62*3,' 27*21 62*14 27*4« 31 66V 23*»8 66'2S 23*3i 66V 23*si 65*46 24*12 65*2^ 24*3. 65*10 24*s« 64*S« 25*10 64*30 2S''30 64*9 25*5. 63V 26*«; 63'2« 26*3i 63V 26*57 62*40 27*20 62*.ri Zf\i 6IV 28*7 6i:28 28*3* 32 66V 23'h« 65*44 24*.e 6S-20 24*34 6S-7' 24*53 64*4^ 25V 64*28 25*32 W*8 25*S2 61^V 26*13 63-2. 26*>i 63-^ 26*56 62V 27*« 62*,9 27*4,' 61*56 28V 6rn 28>8 61*7' 28-53 6(^4. 29*« 33 6523 24*37 65V 24*56 64*45 25*.S 64*2i 2S*S^ 64*7' 25*53 63*47 26*13 63*26 26*34 26*55 62*43 27*17 27*« 61*58 28*4 6I'3^ 28*ti 6l*ii' 60>7 29*3 60-2,' 29*39 59*56 30V 34 644J Z5*.7 2535 64*.' 25*55 26*14 6326 26*34 26*55 62*45 27*is' 62V 27*37 62*.' 27*5? 61*38 61*15 28*45 60V 29*8 60*28 29*32 60*3 29*57 59V 30*23 59*.r 30*49 35 es'ss w;4^ 26*15 63*2fc 26°3* 63; 6 26*54 62*46 27*14 62*25 27*1$ 62*^' 27*56 61*42 28*18 6.*„' 28*41 tO*S7 29*3 60V 39*27 60*9 29*51 59*45 30*15 59*19 30*41 58V 3lV 58*27 31*39 36 63*26 26'>i 2653 62V 27*13 62*27 27*33 6<< 27 54 61*45 28*45 6)V 28*37 61*,' 28*59 60>8 29 2i 60*.5 29°4i 59*51' 30*9' 59*27 30*33 59*2 30*58 58V 31*23 58*10 31*50 57*43 32*17 37 27-.2 62-2S 27» 62^8 27*52 6l*4i 28*12' 6»*27 28*33 6»*b' 28*55 6OV 29*16 60*t,' 29*3i S9*si 30*2 59V 30°2i 59*10 30*50 58*4; 3I*|A 58V 31*46 57*54 32*6 67*2« 32*31 57* .• 32*5» 38 2749 &l*5.' 28*,' fel*36 57*2^ 32*36 56*58 33*2 56*3i 5^6 33'9» 55*,7 34*23 40 60's7 z4i 60^56 29*4.- 60^5 2945' 30*7 59*32 30*28 59*10 30*50 5tf47 31*13 58*« 31*3i 58*0- 32*0' 57*35 32*2fi S7*.6 32*50 56*44 33*16 56V 33*4..' 55*52 34*8 55*24 34*35 54*57- 35*3 41 60*26 29*40 60-0- 30*0' 59*3, 30*21 59:.7; 30 43 58*55 31*5 58*3i 31*28 58*9 31*51 57V 32*5 57*z.- 32*39 56*57 33*3 56V 33*?* 56;6' 33*54 55*39 34*2 1 55*.i 34*48 54*V 35*16 54*6 35*44 4^ S9-4i 30.S 59*» 30*3^ 59V 30*57 58V 3l*ao 58^,8 57*SS 32*5 57V 32*28 57*8 32*si 56*43 33*17 56*19 33*4i' 55*53 34*7 55*27 34*33 55*0' 35*0' 54*33 35*27 54*5' 35*55 53*37 PROVIDENCE, R. 1. 25 TABLE 2.— {Continued.) Angle of Edge. GEAR. 5655545352 SI 50 494847 4645 44 [43 42 as 7? 1? 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 77 m 12° 6 77« 12*32 I2%5 7^4€ 13" 14 76*30 76V I3*s6 134* 75;se 7S"4l 14* f9 l^ii l^A 74"4« 75'tt 7S»e 14*34 14*52 35 15*57 IC^s 17*12 7642 I3'» 76'l3 13*47 75' 14*^2 7^44 14*18 74-51 »5*9' 74« 15*28 74*13 15*47 7363 7S'S6 14*2 7?43 »4'l7 75" 14*32 75^ 14*48 74*56 15*^' 74*21 7^ IS'm »*»7' 73*44| 73*25 73^ 16*56 72 I7V 72*21 7»^7I''34 fix te« >*46 70\\ 75V l5*o M44 15*t6 74» IS*3i 74 12 tB'49 73S5 W*5' 73] 16*231 3I7-73 ^18 78*59 16*42 17 72 39 «7"2I 72j8 174Z 7I°66| 18 71^34 I8*8(i 71 to 18*50 l» »4 ^26 19' 74;3 15*57 16 13 73' 1^ 72V4 72' 35 17*25 72" si 71*55 .e"*5' 71*13 70*52 e*47 19*8 7134 18*26 7 1" 12 1848 70^49 19*11 70*30 70V 19*30 19*53 70" 19*591 C93S 20* 20*5) 737 16*53 7249 17'„ 72*31 17 29 72-13 17*47 7154 18" 6 7l''34 19*26 6943 20"l7 e9"l7 68*52 6686 ^7M 1749 71*53 18*7 18*26 71-15 18*45 7o*»i; 19*6 70-33 19*27 70*2 69*50 19*48 20*10 69 3 20*57 £8 2l'^2« 67 22*^15 66*48 22 43 67-33 30*47 58^ 31*20 31 53 32*28 34*87 29 62*37' 27*23 6^12 27*48 61* 28*15 6119 28*41 60~S1 29*9 60 2J 29*37 59*53 59 30 7 30 37 58 52 31*8 58 31*41 19 57*46 57 12 3248 33 56*37 n 34* o 30 61*49' u' 6r» 28*37 60 29*3' 60*29 2931 29*99 59m 30*28 59*2 S^ 3058 3128 580 32*0' 57*27 3233 56^ 33*7 55^.9 33*41 55-45 sy 34*0 35*32 31 32 61' I 28*18 6036 29*** 60 6 29*S4 59 ta 30*4^ S8*42 31*.8 58*12 574i' 31*48 32*19 S7'8 32*52 57*2i 56*52 32*37 33*8 56*3^ 33*24 34*15 5^ 33*69 SS*26 35*10 35*48 53*34 36*26 59-48 90*12 B9aj 30*39 56 a 31*8 58 31*26 56*3* 56*2 33*26 33*58 56-19 33*41 34*49 54"3s 35' 53*5»| 36*2 52^ 36'm 37*1© 33 59 29 30*31 59*2 30*58 34 58' 26 31* 57'36 32*a» 57-6 32*54 30 54-56 54*2 35*39(36 15 53 8 36*52 37*3. Sl'so 38%' 34 58*44 Se 16 57 46 57 19 31-16 31 44 32 12 3241 56 49 33*11' 5619 33*41 55*5 34*13 34*45 5S*o' 54*» 35*0 3B*3i 54-41 35*19 54 7 3S*S3 53-32 36*28 5252 a 52*18 37*42 51*40 38° 20 50^ 390' SoTit' 35 58 32*0 57*3i 32*28 32 57 56'33 33*27 56 3 33*57' 55J3? 34*28 53*54] 36*8 53*26 36*40 52*44 37°i6 52*8 37''b2 STso 38*30 39 50*4 39'^ 36 S64S 56*19 33*41' 55*49 34*11 55*18 5447 35*13 33*e8|33'S6|34' 54°iB 53*42 35*45 36*18 53*30 Sl'tt 53*6 36*52 5233 37*27 5167 38*3 61-20 38*40 5cr43 39*17 5(7S4«*5«|49' 3646139*25 40%' S6 48 36 'n 4eV 40*49 44*23 37 55» » 55s 34*56 54-34 35*26 54-2 52*»3 37*37 51'47 38*13 48 li 47 §8 'zi 42 4^47* 38 55si 34*9 34*39 15^52 35*8 5423 3^37' 53*51' 36*9 3»42 52W^ 37 3748 51*38 3e°2« 38*57 50^27 39*33 4949 40' 49 II Ao'49 39 SV'm 3451 5?ib 35*5fi 3621 53 36*53 52^3*1 37 •3' 51": 57 38°: 50 54| 39*6 SO 19 394 494« 40*18 S 40"65 46 t7 4I*3J I2J42S3 40 54 28 srse 35*32 36 2 53 28 5258 36%2 37*2 5226 37*34 5I°M 38*6 20 SOV 38*40 39*14 47^ 47»& 4? 3948 40 24-41 I »3»42 k; L°ss' 43*38 m43;I 41 3612 17 3643 5248 37*12 5216 37*44 51 45 15 50 384« 39 39 50 2139 49 40'3b 30 46 54 48 r? 47 40 42*20 42 59 3$ 4419 44*20 ^^ 42 53*6' 36* 51*4 50*3«49|s8 491 S2|37'22i37'S2i38>4 38*86 39*28 40*1 40* 92*38 52-e' 5r3« 4649 H 36 41 II 4< 4.7 42 84 43 I 46jzo 43*40 26 BROWN & SHARPE MYG. CO, TABLE 3. Angle of Face. GEAR. 41 40393837363534 333231 30292827 12 13 S7 70' 13 57 TOail 70*6' 14 39' 69*3 15 ^69* 5' \St4 fe8*3« 15^9- 67'<59 16 1^ 672 16 43 66*-i^ 7l8 fefc*5 I7a3 65i3 18 .9 18 5 63'4d 27 20 * 63 3' 62' 9 49*7 JSVo 13 Wss 69V« 5'. 17 I5'4B 66' I6°a 67°43 6°si 67*9 17 6633 175^ 6556 18* 64*S4| I9°57 63*5 20j 62'i4 14 16 34 68° 16 59 6 729 I7\4 66^6 1750 66°«£ 65-47 6|6S4« 21 -3 63j7 23sl 60*14 25^0 58 6 2 24*4 2 5t\ 17 I9V4 64' 2024 64'io 62 22 24 61 ffo 6I°9 24 594-0 2 759 5'6*|9 27^7 18 2i¥ 63*3 22^ 62341 22%8 23 • 61*17 2343 60V 24 „ 59"5« 25 58 20 26)5 ^7 31 26 57 S6°39 Z7%x 28*9 5449 £350 3043 28 32'<» 19 22 |6g.'3«| 23 20 61*24 23 52 6044] 24i^ 60*" Ii5 58 54 2537 5837 26°5» 57% 2 7 38 56' 28 22 5521 as"* 54'2« 2 9 56 53 5(24 33'« 50*4 20 23' 6130 24 60*53 2432 60*14 2^ 59*34J 26°)6 58* 26 55 5725 27 34 5 6 S3 28(5 5549 28°58 54*6 29°44 54"* 30 31 53*9 51 9 21 24 6025 25 (o 59°46 59S 26 58-26 26"53 574 27 30 5-/o 28(0 56*/4 29'8t 54°36 30(7 53*8 52*50 3(52 51*3 2 3244 50*53 33*3* 34* 49*50 484 7 22 2546 5920 26-9 56' 2653 58° 27^7 57*8 28*5 56*3< 28°43 55*5 29'i>i. 554. 30 5 54*7 304B 53*26 3l°.34 Sfc*32 32 4 2 5/*a 49*4 35*54 47l2 35*20 3?IT3ir^ 46^0 3a'28 23 26°«2 58 (6 27°n 573B 28* 56*56 29(4 5530 2953 35»5 53*57 3(*.B 538 32 52-,s 324« 5/°a4 3336 50*28 34-27 49*29 37% 46* a 45 II 2A 27s7 57t5 28 3 56*35| 29 5553 294; 55*( 54*2« 3I°2 3I°45 52*5 r 32*28 52*2 33 V SI 34°. 50*5 3450 49*20 35-41 4a~2i 30°3 47l, 25 29°34 5534 50" 54*52 30*3 54*» 3I"29 53*23 32°3 2 51 33 37 50*57 34"2j 50 35' 3b'e 4B*i* 36*52 47*6 37^7 -46*. 5 384} 39^ 4**5 ^9*5 J 40 52 A 3*2 26 31 53«| 31*54 53°8 32*34 5 2°;tt 33 6(35 33'58 50 344S 494S 353. 49% 35 "(9 4 8 37io 47* a 38"a 46°. 2 36 56 45*0 27 3I"3 54% 31 39 53*37 32 S7 52''9 333 34 50341 35^ 49%7| 35"3 4945 55 48*5i| 36 3t 4*\ 37a5 A7*7 40 4 44''.o| 42' 28 32''2 53*22 32°39 52*39 33°ie 5IS6 3357] 5 3439 50*25 36^7 48l, 36°52 47^6 3740 47 39\ 40*(i 4r» 43"'9 29 30 31 32 33 34 35 36::, 37 38 39 40 41 3259 3338 52*27 5 3458 so"/* 3539 49 i9 36°23 48*4 33' 5(*33 34'36 50 50 35*5 50*7 37 8 4750 37l^ 46 3556 49*2 o 37l 38^7 36*36 4834 474S|46"55|46 e46. 4f.3 3653 3 394i|4032|4l IS 45*9 3+53 50' 35^ 4950 353 49°57| 36i 49* 3| 36 48"»8 373 4.739 3820 4652 39 5 46*1 3952 4.5 4i°at 43*20 42.M 36V 37' 76 6' 10*47 75 55 10:57 7543 11*6 75^30 iri6 7516 75 3 II* 37 74*49 11*41 11*59 74 35 74 2» 12* U 74*7 I2.*Z2 73*56 16 lO'ssi 75*55 K7'. 7543 75 31 11*26 75 20 Jr37 75 7 7454 11° 56 12:7 74*4*7427 tt' 17 74*13 12*40 12*52 73447^*28 13*5 73*ii' 13*18 72 ii J4'26i4^ 71*45 7r2« 17 11*44 75*10 n:54 745a 12*4 744« 12*13 74'33 12*24 74*26 12*34 74'S' 12*4^ 73°52 I2*S^ 7338 13*7' 7323 13^ 73*9 I3*3t 7252 13*45 72*35 13*59 7?*2I 14' l< 72 3' 18 12*29 I2*4fl: 7425 74*12 12*50 74* f3 ^'46 73'3Z 13-23 7J*I9 I3'34 73*4 IV47 7243 l3*5Sf 7233 14*24 72Z 14*38 7144 14*52 71*28 15*6 71*10 7D*5i' 78i4 69'S9 63*85 17*8' Wat 69*6' 68*46 18* 2' 1 18*30 Id 13*14 73*40 13:25 7327 »:36 73 14 1^48 73* 14* 72*46 7t3i WZ4 72*16 14*36 72 I4*4i 71*45 28 i?5? 7838 15*15' 71*11 15*56 15*44 705470 30 I5;s9 70 ri 20 21 13*59 72*57 14; II' 7243 14:23 7^29 14:34 7214 14*46 72^ 15*4' 7r45 15*11' 71*29 15*25 7ri3 I5»39 7056 16*7 7021 le'ei 70*3 J6*.37 6945 16*53 69*2S 14*43 72*13 K'SS 71*59 15* a' 7/44 15'ai 7/*29 15*33 7/*ia'' is;a6 7058 i5;ss( 7041' J6*83 70*25 16*58 70*8 69' 16*58 69*35!; I7*?ll 69*11 IT*28 I7°46 68*54 6**34 6%" 14 67 52 18° 56; 19* li 67'2£ 67° I I9"4a:20^ 22 15*27 15*4<> 15*53' 7129 7114 70 59 16**; 70*44 e6°20 70*28 16*33 70*11 16*47 17*2' 6955 69 38 17*16' 69*20 17*31 69*1 17*49 68*4^' 16*3 68*2) 18*20 68*4' 18*37 67*43 17*50 68*so' 23 70*46 16*24 70'30 70 16 16*51' 69*59 17; 5' 6943' I7;2b 692669 17*34 8 18*5' 68*33 18*20' 68*14 67*54 18*54 67*34 19* 10 67*14^ 119^ 66*52 66*32 eeTgr 20^ «»^ 6^41 65*<8 21*29 2?^ 64*51 64 a 2?Sq22^ 64*2 63*39 23*10 23*31 24 16*55' 3' 17*9 69*47 17*22 69r3z 17*37 69*15 17*51' 68*59 18*6 68*48 18*21 68*23 18*37 5 IV53 67*45 19*9' 67*27 19*26 67*6^ 44 28* 19* 66*46 6625 20*19 66*3' 25 I7*39|l7*5i 69*21 18*6' 68*48 18*21 68*31 18*36' 68*14 18*52 6756 19*7 67*37 19:2^ 19 67/8 67 19*57 66*39 2^14 20* «L 6620 65 58 20*51 65*37' 21*10 65*14 26 1^2 68*39 18*36 68*22 18*51 68*5 19*6 67*48' 19*22' 67*?d 19 57 67*13 19* Sa 66*53 20"10 66*34 20*26: 66*14 6S*53 21*2 6$*32 21*21 65*11 21*41 64*49 20*38' 66*8' 22' 6426 27 19' 3' 67*57 19*19 67*39' 19*34 6/22 19*49 67*5 20* «■ 66*46 20*2£ 66*28 20*56 6S48 21*13' 65*29 21*3^ 6S*8' 21*50' 64*46 10 22" 24 64* 22*49 63*38' 63*14 62*49 2arsa'a4^ 62*27 62*1' 24*48' 28*10 6I*4« 61*14 25^3^ 60*54 60*27 28 »*4«' 67*16 20*1' 66*59 20^ 66*41' 20*32 66*22 20f>50 66' 21*6 65*44 21*2^ 6^25 21*41 65*5' 22 64*44 22*18' 64*^22 64* r 22*56' 69*38' 2^7 63*15' 23*38 62*52 29 2flr27 66*35 2043 66*17 20*59' 21*1^ 65*59 65*40 2I<'33' 65*21 21*50 65*2' 22-8 64*42 64*21 22*45 63*59 23* 5' 6^37 23r25 63*15 2^4^ 62*52 24-*4 62*28 24*2i 62*s 30 21*9 21*25 65*55 6!r37' 2lW 65*18' 21*58' 6458 22^ 64*39 22*34 22" 6418 52 63' S8 23*«tf 63*38 23*3d esTie' 23^ 62*54 24*10' 62* ad a4'M' 62*7 24*51' 6I'45' 25*12 6i*l8' 24*14' 24*34 62*10 31 21*50' 6S*I4 22*6' 6456 22*2^ 64*36 64*17 2259 63*57 2i*n 63*37 23*35 63*1!; 23*5^ 62* 55' Si^^ 24*54 61*46' 25*17 61*23 25*3S 60*58' 25*8? 26' 60 34 68' 8' 5342 27*9' 2ri4 59*23 58*56 2fi?2fi9 58*38 58*11 28*4<<29*? ST54 57*27 29«2?« 32 6435 22*48' 23*4 6ys6' 23*23 23*40 63*.I6' 2^59' 62*55 2418 62*34 6^12 24P58' 61*50' 25*18' 61*26 2r3si 61*3' 60*3^ 26*23' 26*45^ 60* l!> 59*49 33 S- 23*10' 2r28' 63*36 23*46 63*16 24?4' 62* Z^zi 24*41 ZS"! 56' 62*36 62 15 6I'S3 2^21 6I*3»' 25*42 61*8 26-^ I' 68*44 26;24 60*20 2F4s: 55 27*9' 59*31' 27*31 59' 5' 34 6317 24*8 62*sil 24-*27 62*37 24*4425 62*16 4 61*56 25*23 61*33 25*42 26*3 61 12 25^ 6I-I6' 6049 t26°4^ 60*2 27*7' 59*37 27*29 59*13 2fs2 58*48' 28*16 S8*22 35 24*29 6f39 24*48' 6218' 25*6' 61*58 25*25 61*37 26:4 6054 26*24 6032' 26*45 60*9' 27*6 27Vi 59*44 59* tr^ 27*50 58*56 28*13 58*31' S7S0 28*36' 58*6 5724 29- 1 57*39 5711 5644 36 25*9' 62* i: 25*27 61*41 25-45 61*20 26*5 60*59 26*24 60*36 26*45' 60*15 27:5 5951 2r26' 5928 594 28*10 ;58" 40 58 28:33 15 28*56 29' 5657 30*9' 30^ 56*29 56*1'. 3015 3Pli 5548 552 3l*3rf3ri 37 25*47 61*" 26*6' 61* 60*47 26*25 60*41 26*44 60*20 27>' 5958 2^ 5935 27*45 59*13' 28*7 58*49 25 28*51' 58 ^9^ 57*35 23rM 30*2 57 lb 56*42 38*27 56*15 38 26*44 60*26 27:4 604' 5942 5920 28;4; 58*56 5834 28*4-7 58*9' 29* 5745 ►' 29:3^ 5 57*21' 29*5i 56*55' 30*20 56*30 30*44 55*2' 31*9' 5^35 55 7' 5439 3?l6 32*4< 54 28 53 51 32*57 33*24 53*46 5318 33* 3i 34*1 53* 7' 5238 A9 34V 33 27*3' 60*9' 27*22 594^ 27/42 59 28 2»*2' 59*4' 28 22' 58*42 28;4^ 58*19 29*5 5755 29*27 57*31 29^ 57*7' 30*13 56*41 30*36 31*1' 56 16 5549 31* 2i 5522 31*50 54*54 27*4<< 59*34 2n»3 5932 40 4i 42 28'2rf 58*5«i 28*41 58*27 29*1' 58*5 57*42 57* 29*43' 17' 30*6' 56*54 30*52 56 2' 31* 17 55*37 31*42 55*16 32*6' 54*44 32;3i 54*15' 2rir 58*57 2^37 5837 28*«' 58*22 29'ia; 58*1 28*57 IS' 29*18 57*52 29*39' 57*29' 3crT 57* 3Q*22 4 5640 30*45 56*15 31* 9' 3l*si' 55* St' 55* 25 '4r3rior 13* 54*48 31*55 54*59 3?io!??vi 5432 54 4' 33*12 53*36 29' 57 39 33 29' 55 57" IS* 30*16 56*52 30*38 5^28 31* 56*4' 31*23 55*39 32*35 54*21' 33' 53*54 3^261 53*26 33* 52*58 ^34? 52 29 52 TABLE 4. — (Contimted.) Angle of Face. — Gear. 29 56 55 S4 53 52 51 504948474645444342 12 75 SA 10* le' 7540 lO'.ZB 7524 75 9' 10* sz 74 si 74 $7 7415 ir 30 7358 11*43 73*39 II* «' 73° 20 It" m 7i5B 1755 72 37 I2*4J^ 72 15 irr 71*51 13*19 71*25 j3 14 7456 1116 744« h;28 7424J IfAZ 74 8 11*54: 73*50 12*8 73*32 IZ'zd 73° IZ 12*37 7253 12; 51 72*33 13' 7 72 n' 13* Z3' 71*49 1340 71*26 I3*S« 71-2' K 70 38 14*35 70"li KTz' irib 73*58 7342 12*2912' 7325 43 73 7 12*57 7i49 1 3* 2B 13*43' 72*29 »3*26 728 71 A^ 71*17 71-5 33 70'4» I4*si l(fl7 15" 10 IS'36 69 si 69*26 15'SI' 68*59 15 13' 1 7/1' I3*I6" 72*44 59 A-i 9 14' »4 7r2fl 14*30 71*6' 14*47 70*45 15*5 7tf23 15^3 69*59 15*42 69*34 16' r 69 9 16*22 6&42 16*43 6iiS 67*47 J6 17 •3*59 72*5' 14*13 71*47 14*28 71*2%' I4*44' 71*8' 15" I' 70*49 7027 l6i 1^52 69*42 16* u' 69*19' 16*30 68*54 50 17*10 68*2* I7*3t 67*34 irsei 67*6" 18* »• 66*36 I4°57 71*9' 15-u' 70*49 I5*2«i 70*30 15*44 7016 16* r 69*49 »*I8 69*26 16° 37 69*3' 16*55 17*15 68 39 68 15 17^36 I7;57' 67*50 67*23' 18*20 66*54 I8*4» 66*27 19*6' 6S*S8 19^31 6^27 19 I5°5i 7014 16*7 69*53 16*26 6934 16*42 69*12 17' r 68*43 17*20 26 17*39' 68*3 n*58 67*38 18*20 67*12 18*44' 66*47 19*3' 66* 19 19*27 65*si 19*50 65*20 20 18 64*5« 2tf« 64*18 I6*4«f 69*19 17*2 68*58 »7*23 17*41 68*37 68*15 18* 67*52 18*21 67*29 18*40 67*4 19*1 66*37 19*22 66*12 i9*4c; 65*44 20° 8' 65*16 20*34 64*44 20*59 IS 21*24 63*44 21*52 63*10' 20 17*44 68*26 18" I' 18* 19' 6«r3' 6/41' 18*40' 67*18 19*20 19*41' 66*54 66*30 h€ 5 20*2' 65*38 20*25 65*11' 20*43 64*43 21* 13' 64* li 21*33 63*43 227 63*11 22*32 6^38 23' 162*4" 21 18*39 6l*3r 18*57 67*9' 19*16' 19*37 664666 23 19*58 6S*S8 20*19 65*33 20*41 65*7 21* y 64^39164 21*27 11' 21*52 63*42 22*17 63*13' 2?4S 6241 23*|d 6£8' 23*38 61*34 2<8' 61 22 19*32' 66 38 2^ 6616 65*52 I9*5i 20*33 65*27 20'S5 6^3' 21*17 6437 22r 13 634t 21*40 6410 22*5' 63*41 22*ri 63*13 22*53 6243 2319 62* ii' 23*46 61*40 24 61*7' 24*44*8*14 60*3; S9S6 23 zo'zi 65*47 20*4i 65 23 21*8 6458 21*29' 6433 2r52' 64 8' 22*37 6513 2af2' 62*44 23' 27 62*15 23*54 61*44 24*21 61* 13 25*2» 60* IS' 24^49 6044 25*18' 60*6 59*3l' 26*18 58*54 24 21* 19 64*55 21*39 64* ai 22* r 64*5' 2r24 63* 22*46 23jtf 63 14 6246 23*36 62* 19 24* 6I*4«' 24*26 61*18 24*53 60*47 2sr4a 5941 IZ^Zi 59 6 26*51 58*31' 27;23 57*53' 25 22*11' 22*33' 6339 2fS6' 63*14 23*18' 62*48 2^ 62*^ 24*13 61*56 23°4t' 62*21 24*7 61*53* 24*3i 61*24 24*57 60*53 25*24 60*22 25*52 59*50 2^ 58*55 26*20 59*18 26'50 58*44 27*21' Si'd' 2r52 57*32 28*26 56*S4 26 23*3' 63^5 23*25 6249 6128 25" I 60*59 25*28' 6030 25*53 59 59 26*21 27 27*19' 58*21 27S»^ 57*47 28*2i S/ll W 2927 56*34 SS*5S 27 2F5? 62* 2S6l 24> 58 2440 25' 6152 1*5' 2?5? 6043 25-29 60*37 25'55 60*7 26*22 59*38 26*48' 59*5' 27*17 58*33 27*46 SB- 57*8 28*16; 57*26 28*4f 56* 5l' 29*19 56* IS 29*si 55*38 30^ 54*42 lcr«7 54*59 5r25 54* 2| 29 24*441 61*36 61*9' 25*56' 6014 59*46 26*4« 59*16 27*15 58*45 27*43' 58*13 28*1? 57*42 29* li 56* si ^9^^^| 55*5^ 3flric; 55*2* 25*33' 25*57 60*47 60*21 26*22 si 26*47 59*25 2ri4 58*56 27*40 58*26 28*8 57*5^ 28* 5/22 29*5 S649 29^ 5^15 30*8' 55*40 5i'24 544^° + 4^ or •937 N + 4>f This formula increases the throat diameter, and conse- quently the center distance. The amount of the increase can be found by comparing this value of T with the one as obtained by formula on page ^6. To keep the original center distance, the outside diameter of the worm must be reduced by the same amount the throat diameter is increased. Second Method. — Without changing any of the dimensions we found by the formulas given on page 36, we can avoid the interference to be found in worm wheels of less than 30 teeth by simply increasing the angle of worm thread. We find the value of this angle by the following formula : Let there be 2 y = angle of worm threaa. N = number of teeth in worm wheel. cos ;/ = i/ I _ i_ N From this formula we obtain the following values : N 29 28 27 26 25 24 23 22 21 2y 30X 31 3i>^ 32X 32^ izV^ 34K 35 36 N 19 18 17 16 15 14 13 12 2 y Z^ 39 40 41/^ 42M 44 >^ 46X 48 20 37 As this latter formula involves the making of new hobs in many cases, on account of change of angle, we prefer to reduce the diameter of worm as indicated by first method, if the dis- tance of centers must be absolute. PROVIDENCE, R. 39 CHAPTKR V. SPIRAL OR SCREW GEARING. (Figs. 9, 10, II.) Fig, 9, RIGHT HAND SPIRAL GEARS. In spiral gearing the wheels have cylindrical pitch surfaces, but the teeth are not parallel to the axis. The line in which the pitch surface intersects the face of a tooth is part of a screw line, or helix, drawn at the pitch surface. A screw wheel may have one or any number of teeth. A one-toothed wheel corresponds to a one-threaded screw, a many-toothed wheel to a many-threaded screw. The axes may be placed at any angle. Consider spiral gears with : I. Axes parallel. II. Axes at right angles. III. Axes any angle. 40 BROWN & SHARPE MFG. CO. Fig, 10. LEFT HAND SPIRAL GEAR. Let there be : n"= >• number of teeth in gears •! 7 C = center distance. P = diametral pitch P' = circular pitch. P" = normal diametral pitch. P'" = normal circular pitch. y = angle of axes. Lj = exact lead of spiral on pitch surface. L^ = approximate lead of spiral on pitch surface. T = number of teeth marked on cutter to be used when teeth are to be cut on milling machine. D = pitch diameter. B = blank diameter. " Z !- angle of teeth with axis /= thickness of tooth. s = addendum. D" + / = whole depth of tooth. Note. — Letters a and d occurring at bottom of notations refer to gears a and 3. I. — Axes Parallel. Gears of this class are called twisted gears. The angle of teeth with axes in both gears must be equal and the spirals run in opposite directions. The angles are generally chosen small (seldom over 20°) to avoid excessive end thrust. End thrust may, however, be entirely avoided by combining two pairs of wheels with right and left-hand obliquity. Gears of this class are known as Herringbone gears. They are com- paratively noiseless running at high speed. PROVIDENCE, R. I. 4I II. — Axes at Right Angles. Here we must always have : 1. The teeth of same hand spiral ; 2. The normal pitches equal in both gears ; and 3. The sum of the angles of teeth with axes = 90°. Choosing Angle of Teeth with Axes. 1. If in a pair of gears the ratio of the number of teeth is equal to the direct ratio of the diameters, /. ^., if the number of teeth in the two gears are to each other as their pitch diame- ters, then the angles of the spirals will be 45° and 45° ; for, this condition being fulfilled, the circular pitches of the two gears must be alike, which is only possible with angles of 45°. In such a combination either gear may be the driver, 2. If the ratio of the diameters determined upon is larger or smaller than the ratio of the number of teeth, then the angles are : tan a^ = ^-^ tan ^^ = --±--^ In such gears the velocity ratio is measured by the number of teeth, and not by the diameters. 3. Given N^, N^, and C : If Pa is made = P,,', then we have case '' i " and But if Pa is assumed, then : ^, C7r-y2NaPJ and P ' P ' tan aa = -^ tan a^, = -^^ The gear whose P' or a is larger will ordinarily be the driver, on account of the greater obliquity of the teeth. 4. Given N„, Nj, and C or D. See case " 7 " under III., considering ;/ = 90°. III. — x\xis AT ANY Angle {y). 5. Given case " i," under II., then angles of spirals — }4 y, for the same reason. 6. Analogous cases to "2" and "3," under II., may be worked out, when angles of axes = y, but they have been 42 BROWN & SHARPE MFG. CO. omitted, partly because the formulas are too cumbersome, and partly because they are to some extent covered by cases "5 " and "7." 7. Given N„, N^ and C, or one of the pitch diameters. We find the angles by a graphic method, which for all practical purposes is accurate enough ; ro and v are the axes of gears forming angle / (see diagram, Fig. 11.) On these axes we lay off lines r and v representing the ratio of the number of teeth (velocity ratio), so that N^ : N,, : : r .? : .f z;, and Fig, 11. construct parallelogram r s v. Then, according to Mc- Cord,* the angles formed by the tangent s oxn the pitch con- tact with the axes of the gears insures the least amount of sliding. In bisecting angle y by tangent u and using angles produced in this manner we equally distribute the end thrust on both shafts. Both methods have their advantages ; to profit by both we select angles a^ and a^^ produced by tangent x, bisecting angle u s. Thus we have when angles are found and C given, 2 C TT cos a^ cos al p/ji _ and when D^^ given N„ cos ajj -\- Nj, cos a„ p/)i D„ = D„ 7t cos a^ K 7t COS ai, and * McCord, Kinematics, page 378. providence, r. i. General Formulas. 43 D = FN TV or = TT cos a B = D + 2^ or D pw p = or = p'?i N cos a P'" = P' cos a 7t P" = ~ TT 2 T = Lx = L = (Pitch of cutter.) or 2X + lO N cos a N P^ tan a lo WG„ (6*^^ iV"^/^ 7.) or _N7^ Ptan a or N P' SG. tan o' cos a (6'^f Tf^f^th _____ Outside Diameter - - Angle of Teeth with Axis ----- Normal Circular Pitch ------ Pil-rVi nf r'ntfpr- ________ Thickness of Tooth t _-__-_ Whole Depth D'' + f - - ~ - - - - Exact Lead of Spiral Approximate Lead of Spiral - - - - Gears on Milling Machine to Cut Spiral Gear on Worm 1st Gear on Stud 2nd Gear on Stud - - - Gear on Screw - - - - If the exact lead Li can be obtained by the gears at hand, Li will equal Lo and we shall have from the formula _ 10 W G, (for B. & S. Milling Machine.) S Gi U _ W_G2 10 S Gi Example I. Required the gears for cutting a spiral of 2^" lead. 2^ I . . -- = - factoring, m the most simple way, we have i _ ^ ^ ^ _ I X 28 _ 32 X 28 W G., • 4"^2X2~"56x2'^56 Y64 ^ S Gi PROVIDENCE, R. I. 45 Thus the gearing will be 32 T. on worm, 64 T. ist, on stud, 28 T. 2nd on stud, and 56 T. on screw. Trying these gears on the Milling Machine we find that they cannot be used, and as we have no other regular gears in the ratio of 2 to i that can be used we must try, by factor- ing, to get such ratios for the two pairs of gears as to be able to use the gears at hand, bearing in mind that the combined ratio must be J. T 18 3x6 24 X 6 24 X 48 4 ~ 72 ~ 9T~8 ~ 9 X 64 ■"" 72 X 64 These gears are at hand and the combination can be used on the machine, giving the exact lead of 2^". Example II, Required the gears for cutting a spiral of 8.639" l^^d. 8.639 = 8yVo%; reducing, by continued fractions, to a smaller fraction of approximately the same value, as described on pages 73 and 74 639 ) 1000 ( I 639 361 )639( I 361 278 ) 361 ( 278 83 ) 278 ( 3 249 29 ) 83 ( 2 5« 25 ) 29 ( I 25 4)25(6 24 1)4(4 4 46 BROWN & SHARPE MFG. CO. i 1 2. 7 16 2 3 154 63 S) 1 ? 3 Tf 2^5^ "3^ 2TT TDTJTJ- Selecting if as an approximation near enough for our purpose, and in fact as near as we are likely to find gears for, we have for our lead 8if . Applying the formula as in Ex- .mple 1. W G, 10 S G, m 216 108 10 250 125 9 X 12 9 X 48 factoring we have 72 X 48 ,, . , 25 X 5 100 X 5 = i"^^T^ '^^ ^^^'' required, these being regular gears furnished with the Milling Machine. Proof: 72 X 48 X 10 _ .. -. = 8.640 = L9 100 X 40 o ^ T ' 8-639 =Li .001'' error in lead. In shops where much work is done in milling spirals it is desirable to have a full set of gears for the milling machine, from the smallest to the largest numbers of teeth that can be used. This makes it possible, in most cases, to get closer approximations than could be otherwise obtained, and often saves a great deal of figuring. When the use of continued fractions does not bring a close enough approximation, one method to secure a closer result is to add to or substract from the numerator and de- nominator of the fraction to be reduced, any numbers nearly in proportion to the given fraction, seeing that the numbers added or substracted are such as to make the fraction reduc- able to lower terms. By a little ingenuity and patience ex- tremely close approximations can generally be reached in this way. Take, as an illustration, the fraction in Example II. -6-3-9- 8639 'TOOO" 10 lOOOO Adding 9 to the numerator and 10 to the denominator, these PROVIDENCE, R. I. 47 being in about the same ratio to each other as the numerator and denominator of the fraction, we have 86394-9 = 8648 _ 4324 _ 47 X 92 loooo-f 10 = looio 5005 55 X 91 All of the gears in this case are special. Applying the same proof as in Example II. we find that this train of gears will give a lead of 8.6393+, making an error of .0003'' in the lead. No doubt a much closer approximation than even this could be obtained by further trial. Another method is to multiply both terms of the fraction by some number which will make one term of the fraction easily reducible, and adding one to or subtracting it from the ether term to make it possible to reduce that also. There is an element of uncertainty in both of these methods, as we never feel sure that we have obtained the best combination; practical work, however, rarely requires accuracy beyond a point that can readily be reached. The accompanying list of prime numbers and factors will be found useful in reducing and factoring fractions. 48 BROWN & SHARPE MFG. CO. PRIME NUMBERS AND FACTORS. 1 TO lOOO. i 1 26 2x13 : 51 3x17 76 1 2^x19 2 27 3' 52 2^X13 77 7x11 3 28 2^X7 53 78 2x3x13 4 2- 29 54 2x3-^ 79 30 2x3x5 55 5x11 80 2-*x5 6 2x3 31 56 2'^x7 81 3^ 7 32 2"' 57 3x19 82 2x41 8 2-3 33 3x11 58 2x29 83 9 3- 34 2x17 59 84 2^x3x7 10 2x5 35 5x7 60 2^ X 3 X 5 85 5x17 11 36 2=-'x3-' 61 86 2x43 12 2^x3 37 62 2x31 87 3x29 13 38 2x19 63 3^X7 88 2^X11 14 2x7 39 3x13 64 2'^ 89 15 3x5 40 2V.5 65 5x13 90 2x3^x5 16 2^ 41 ^^ 2x3x11 91 7x13 17 42 2x3x7 67 92 2^x23 18 2x32 43 68 2^x17 93 3x31 19 44 2^x11 69 3x23 94 2x47 20 2^x5 45 3^X5 70 2x5x7 95 5x19 21 3x7 46 2x23 71 96 2^X3 22 2x11 47 72 2^x32 97 23 48 2^X3 73 98 2x7^ 24 2^X3 49 72 74 2x37 99 3^x11 25 52 50 2x52 75 3x52 100 2^x5^ PROVIDENCE, R. I. 49 101 131 .161 7x23 191 102 2x3x17 132 2^x3x11 162 2x3^ 192 2«x3 103 133 7x19 163 193 104 2^X13 134 2x67 164 2^x41 194 2x97 105 3x5x7 135 3^X5 165 3x5x11 195 3x5x13 106 2x53 136 2^x17 166 2x83 196 2^X7- 107 137 167 197 108 22x33 138 2 X 3 X 23 168 2'x3x7 198 2x3^x11 109 139 109 13- 199 110 2x5X11 140 2^x5 X 7 170 2x5x 17 200 2'^X52 111 3x37 141 3x47 171 3^x19 201 3x67 112 2V7 142 2x71 172 2-X43 202 2x101 113 143 11x13 173 203 7x29 114 2x3x19 144 2^X3-^ 174 2x3x29 204 2-x3xl7 115 5x23 145 5x29 175 5^x7 205 5x41 116 2=^X29 146 2x73 176 2^X11 206 2x103 117 3^x13 147 3x7- 177 3x59 207 3-'x23 118 2x59 148 2^x37 178 2x89 208 2^x13 119 7x17 149 179 209 11x19 120 2"^X3X5 150 2 X 3 X 5^ 180 22x3-x5 210 2x3x5x7 121 11^ 151 181 211 1 122 2x61 152 2'^Xl9 182 2x7x13 212 2^X53 123 3x41 153 3-X17 183 3x61 213 3x71 124 2-^x31 154 2x7x11 184 2V23 214 2x107 125 5' 155 5x31 185 5x37 215 5x43 126 2x3^x7 156 2^X3X13 186 2x3x31 216 2=^X3-^ 127 157 187 11 X17 217 7x31 1 128 2-7 158 2x79 188 2^x47 218 2x109 129 3x43 159 3x53 189 3^X7 219 3x73 130 1 2x5x13 160 2-^x5 190 2x5x19 220 2V5X11 1 50 BROWN & SHARPE MFG. CO. 221 - ■■ 13x17 251 281 311 222 2x3x37 252 2^x3^x7 282 2x3x47 312 2^x3x13 223 253 11x23 283 313 224 2^X7 254 2X127 284 2^x71 314 2X157 225 3^x52 255 3x5x17 285 3x5x19 315 3'x5x7 226 2x113 256 2« 286 2x11x13 316 2^x79 227 257 287 7x41 317 228 2^x3x19 258 2x3x43 288 2^X3=^ 318 2x3x53 229 259 7x37 289 17' 319 11x29 230 2x5x23 260 2^x5x13 290 2x5x29 320 2«x5 231 3X7X11 261 3^x29 291 3x97 321 3x107 232 2=^X29 262 2x131 292 2^x73 322 2 X 7 X 23 233 263 293 323 17x19 234 2x3^x13 264 2^x3x11 294 2x3x7' 324 2^X3* 235 5x47 265 5x53 295 5x59 325 5^X13 236 2^x59 266 2x7x19 296 2-^x37 326 2x163 237 3x79 267 3x89 297 3^X11 327 3x109 238 2x7x17 268 2^X67 298 2x149 328 2^X41 239 269 299 13x23 329 7x47 240 2^x3x5 270 2x3^x5 300 2^x3x5' 330 2X3X5X11 241 271 301 7x43 331 242 2x11' 272 2^X17 302 2x151 332 22x83 243 i\' 273 3x7x13 303 3x101 333 3'x37 244 2^x61 274 2x137 304 2^X19 334 2x167 245 5X7' 275 5^x11 305 5x61 335 5x67 246 2x3x41 276 2^x3x23 306 2x3^x17 336 2^x3x7 247 13x19 277 307 337 248 2^X31 278 2X139 308 2'x7xll 338 2x13' 249 3x83 279 3^x31 309 3X103 339 3x113 250 2x5^ 280 2^ X 5 X 7 310 2x5x31 340 2'x5xl7 PROVIDENCE, R. 51 341 11x31 371 7x53 401 431 342 2x3^X19 372 2^x3x31 402 2x3x67 432 2^X3'^ 343 V 373 403 13x31 433 344 2^X43 374 2X11X17 404 2-xlOl 434 2x7x31 345 3x5x23 375 3x5^ 405 3^X5 435 3 X 5 X 29 346 2x173 376 2-^x47 406 2x7x29 436 2^x109 347 377 13x29 407 11X37 437 19x23 348 2^x3x29 378 2x3'^X7 408 2=^x3x17 438 2x3x73 349 879 409 439 350 2x5^x7 380 2-X5X19 410 2x5x41 440 2^^X5x11 351 3^X13 381 3x127 411 3x137 441 3^X7^ 352 2^X11 382 2x191 412 2^x103 412 2x13x17 353 383 413 7x59 443 354 2x3x59 384 2'x3 414 2x3^x23 444 2^x3x37 355 5x71 385 5x7x11 415 5x83 445 5x89 356 2^x89 386 2X193 416 2^X13 446 2x223 357 3x7x17 387 3-X43 417 3x139 447 3x149 358 2X179 388 2-X97 418 2x11x19 448 2«x7 359 389 419 449 360 2^x3^x5 390 2X3X5X13 420 2^X3X5X7 450 2x3^x5' 361 192 391 17x23 421 451 11X41 362 2x181 392 2'^X7- 422 2x211 452 2-X113 363 3x11^ 393 3X131 423 3^X47 453 3x151 364 2^x7x13 394 2x197 424 2^X53 454 2x227 365 5x73 395 5x79 425 5-X17 455 5x7x13 366 2x3x61 396 2^x32x11 426 2x3x71 456 2^x3x19 367 397 427 7x61 457 368 2^X23 398 2x199 428 2-X107 458 2x229 369 3^x41 399 3x7x19 429 3X11X13 459 3'^X17 370 2x5x37 400 2*X52 430 2X5X43 460 2^x5x23 52 BROWN & SHARPE MFG. CO. 461 491 521 551 19x29 462 2X3X7X11 492 2'^x3x41 522 2x3^x29 552 2^x3x23 463 493 17x29 523 553 7x79 464 2^X29 494 2x13x19 524 2^x131 554 2x277 465 3x5x31 495 3^x5x11 525 3x5^x7 555 3x5x37 466 2x233 496 2*X31 526 2x263 556 2^X139 467 497 7x71 527 17x31 557 468 2^x32x13 498 2x3x83 528 2*x3xll 558 2x3^x31 469 7x67 499 529 232 559 13x43 470 2x5x47 500 22x5-^" 530 2x5x53 560 2^X5X7 471 3x157 501 3x167 531 3-X59 561 3x11x17 472 2^X59 502 2X251 532 2-X7X19 562 2x281 473 11X43 503 533 13x41 563 474 2 X 3 X 79 504 23x3^x7 534 2x3x89 564 2^x3x47 475 5^x19 505 5x101 535 5x107 565 5x113 476 2^x7x17 506 2 X 1 1 X 23 536 2'^X67 566 2x283 477 3^X53 507 3x13^' 537 3x179 567 3*X7 478 2x239 508 2^x127 538 2x269 568 2^X71 479 509 539 7=^X11 569 480 2'5x3x5 510 2X3X5X17 540 22x3'^x5 570 2x3X5X19 481 13x37 511 7x 73 541 571 482 2x241 512 29 542 2x271 572 2^x11x13 483 3 X 7 X 23 513 3-^x19 543 3x181 573 3x191 484 2^x11^ 514 2x257 544 2^X17 574 2x7x41 485 5x97 515 5x103 545 5x109 575 5^X23 486 2x3^ 516 2-x3x43 546 2X3X7X13 576 2«X32 487 517 11x47 547 577 488 2^x61 518 2x7x37 548 2^x137 578 2x172 489 3x163 519 3x173 549 3^X61 579 3x193 490 2 X 5 X 72 520 2^x5x13 550 2X5=^X11 580 2^x5x29 PROVIDENCE, R. I. 53 581 7x83 611 13x47 641 671 1 11X61 582 2x3x97 612 2^x3^x17 642 2x3x107 672 2^x3x7 583 11x53 613 643 673 584 2'^X73 614 2x307 644 2^x7x23 674 2x337 585 3^x5x13 615 3x5x41 645 3 X 5 X 43 675 3'^X52 586 2x293 616 2^x7x11 646 2x17x19 676 2^x132 i 587 617 647 677 588 2-^x3x7^ 618 2x3x103 648 2->x3* 678 2x3x113 589 19x31 619 649 11X59 679 7x97 590 2x5x59 620 2^x5x31 650 2X5^X13 680 2'x5xl7 591 3x197 621 3'^X23 651 3x7x31 681 3x227 592 2^X37 622 2x311 652 2-X163 682 2x11x31 593 623 7x89 653 683 594 2x3''xll 624 2^x3x13 654 2x3x109 684 2-^x32x19 595 5x7x17 625 b' 655 5x131 685 5x137 596 2=^X149 626 2x313 656 2^X41 686 2x7^ 597 3x199 627 3x11x19 657 3-^x73 687 3x229 598 2x13x23 62S 2-^X157 658 2x7x47 688 2^X43 599 629 17x37 659 689 600 2^X3X5^ 630 2X3-X5X7 660 2^X3X5X11 690 2X3X5X23' 601 631 661 691 602 2x7x43 632 2'X79 662 2x331 692 2^x173 603 3^x67 633 3x211 663 3x13x17 693 3^X7X11 604 2^x151 634 2x317 664 2'^X83 694 2x347 605 5xlP 635 5x127 665 5x7x19 695 5x139 606 2x3x101 636 2-x3x53 666 2X3-X37 696 2''X3X29 607 637 7^X13 j 667 23 X 29 697 17x41 608 2^X19 638 2x11x29 668 2^X167 698 2x349 609 3 X 7 X 29 639 3^'X71 669 3 X 223 699 3 X 233 610 2x5x61 640 2^X5 670 2x5x67 700 2- X ir X 7 54 BROWN & SHARPE MFG. CO. f 701 731 17x43 761 791 7x113 702 2x3^x13 732 2^x3x61 762 2x3x127 792 2^x3^x11 703 19x37 733 763 7x109 793 13x61 704 2«Xll 734 2x367 764 2-X191 794 2x397 705 3x5x47 735 3 X 5 X 7^ 765 3^x5x17 795 3x5x53 706 2x353 736 2^X23 766 2x383 796 2^x199 707 7x101 737 11X67 767 13x59 797 708 2^x3x59 738 2x3^x41 768 2«x3 798 2X3X7X19 709 739 769 799 17x47 710 2x5x71 740 2^x5x37 770 2X5X7X11 800 2^X5^ 711 3^x79 741 3x13x19 771 3x257 801 3^x89 712 2«X89 742 2x7x53 772 2^x193 802 2x401 713 23x31 743 773 803 11 X73 714 2X3X7X17 744 2'^x3x31 774 2x3^x43 804 2^x3x67 715 5x11x13 745 5x149 775 5^X31 805 5x7x23 716 2^X179 746 2x373 776 2^X97 806 2x13x31 717 3x239 747 3=^X83 777 3x7x37 807 3x269 718 2x359 748 22x11x17 778 2x389 808 2=^X101 719 749 7x107 779 19x41 809 720 2^X3-'X5 750 2x3x5^ 780 2^X3X5X13 810 2x3^x5 721 7x103 751 781 11X71 811 722 2x192 752 2^X47 782 2x17x23 812 2-X7X29 723 3x241 753 3x251 783 3-5x29 813 3x271 724 2^x181 754 2x13x29 784 2^X7^ 814 2x11x37 725 5-'x29 755 5x151 785 5x157 815 5x163 726 2x3xlP 756 2^x3^x7 786 2x3x131 816 2^x3x17 727 757 787 817 19x43 728 2^x7x13 758 2x379 788 2-^x197 818 2x409 729 3« 759 3x11x23 789 3x263 819 3^x7x13 730 L 2x5x73 760 2^x5x19 790 2x5x79 820 2^x5x41 PROVIDENCE, R. I. 55 821 ., 851 23x37 881 911 822 2x3x137 852 2^x3x71 882 2x3^x72 912 2^x3x19 823 853 883 913 11x83 824 2^X103 854 2x7x61 884 2^x13x17 914 2x457 825 3x5^x11 855 3^x5x19 885 3x5x59 915 3x5x61 826 2x7x59 856 2^X107 886 2x443 916 2^x229 827 857 887 917 7x131 828 2^x3^x23 858 2x3x11x13 888 2^^x3x37 918 2x3^x17 829 859 889 7x127 919 830 2x5x83 860 2^x5x43 890 2x5x89 920 2^X5X23 831 3x277 861 3x7x41 891 3^X11 921 3x307 832 2«xl3 862 2x431 892 2^x223 922 2x461 833 7^x17 863 893 19x47 923 13x71 834 2x3x 139 864 2^X33 894 2x3x149 924 2^x3X7X11 835 5x167 865 5x173 895 5x179 925 5^x37 836 2^x11x19 866 2x433 896 2^X7 926 2x463 837 3^X31 867 3x172 897 3x13x23 927 3^x103 838 2x419 868 2^x7x31 898 2x449 928 2^X29 839 869 11x79 899 29x31 929 840 2^X3X5X7 870 2X3X5X29 900 2^x32x52 930 2X3X5X31 841 292 871 13x67 901 17x53 931 7^x19 842 2x421 872 2^x109 902 2x11x41 932 2^x233 843 3x281 873 3^x97 903 3x7x43 933 3x311 844 2^x211 874 2x19x23 904 2^X113 934 2x467 845 5x132 875 5^X7 905 5x181 935 5x11x17 846 2x3^x47 876 2^x3x73 906 2x3x151 936 2=^x3^x13 847 7xlP 877 907 937 848 2^X53 878 2x439 908 2^x227 938 2x7x67 849 3x283 879 3 X 293 909 3^x101 939 3x313 850 2x5^x17 880 2^x5x11 910 2X5X7X13 940 2^x5x47 56 BROWN & SHARPE MFG. CO. 941 956 2^x239 971 986 ^ 2x17x29 942 2x3x157 957 3x11x29 972 2^x3^ 987 3x7x47 943 23x41 958 2x479 973 7x139 988 2^x13x19 944 2^X59 959 7x137 974 2x487 989 23x43 945 3^x5x7 960 2«x3x5 975 3x5-xl3 990 2x32X5X11 946 2x11x43 961 31^ 976 2^X61 991 947 962 2x13x37 977 992 2'x31 948 2-x3x79 963 3^x107 978 2x3x163 993 3x331 949 13x73 964 2^x241 979 11x89 994 2x7x71 950 2X5-X19 965 5x193 980 2^X5X7=^ 995 5x199 951 3x317 966 2X3X7X23 981 3-X109 996 2^x3x83 952 2^x7x17 967 982 2x491 997 953 968 2^X11-^ 983 998 2x499 954 2x3^x53 969 3x17x19 984 2'^x3x41 999 3"x37 955 5x191 970 2x5x97 985 5x197 1000 2'^X5'^ 1 PROVIDENCE, R. I. S7 CHAPTER VX. INTERNAL GEARING. PART A.— INTERNAL SPUR GEARiNG. (Figs. 12, 13, 14, 15, 16.) A little consideration will show that a tooth of an internal or annular gear is the same as the space of a spur — external gear. We prefer the epicycloidal form of tooth in this class of gearing to the involute form, for the reason that the difficulties in overcoming the interference of gear teeth in the involute system are considerable. Special constructions are required when the difference between the number of teeth in gear and pinion is small. In using the system of epicycloidal form of tooth in which the gear of 15 teeth has radial flanks, this difference must be at least 15 teeth, if the teeth have both faces and flanks. Gears fulfilling this condition present no difficulties. Their pitch diameters are found as in regular spur gears, and the inside diameter is equal to the pitch diameter, less twice the adden- dum. If, however, this difference is less than 15, say 6, or 2, or i, then we may construct the tooth outline (based on the epicy- cloidal system) in two different ways. First Method. — To explain this method better, let us sup- pose the case as in Fig. 12, in which the difference between gear and pinion is more than 15 teeth. Here the point o of the describing circle B (the diameter of which in the best practice of the present day is equal to the pitch radius of a 15 tooth gear, of the same pitch as the gears in question) gene- rates the cycloid o, o', o^, o^, etc., when rolling on pitch circle L L of gear, forming the face of tooth ; and when rolling on the outside of L L the flank of the tooth. In like manner is the face and flank of the pinion tooth produced by B rolling out- side and inside of E E (pitch circle of pinion). A little study 58 BROWN & SHARPE MFG. CO. of Fig. 12 (in which the face and flank of a gear tooth are produced) will show the describing circle B divided into 12 equal parts and circles laid through these points (i, 2, 3, etc.), concentric with L L. We now lay off on L L the distances o-i, 1-2, 2-3, etc., of the circumference of B, and obtain points PROVIDENCE, R. I. 59 i^, 2\ 3^, etc. [Ordinarily it is sufficient to use the chord.] It will now readily be seen that B in rolling on L L will success- ively come in contact with i', 2', 3', etc., c meanwhile moving to c\ r', r^, etc. (points on radii through i\ 2\ ^\ etc.), and the generating point o advancing to o\ o^, o^, etc., being the inter- sections of B with c^j c^^ r', etc., as centers and the circles laid through I, 2, 3, etc. Points o, o\ o^ o^, etc., connected with a curve give the face of the tooth ; in like manner the flank is obtained. In this manner the form of tooth is obtained, when the difference of teeth in gear and pinion is less than 15, with the exception that the diameter of describing circle B -y-<^-^) where P = diametral pitch, N and n number of teeth in gears. The distances of the tooth above and below the pitch line as well as the thickness t are determined as in regular spur gears by the pitch, except when the difference in gear and pinion is very small, where we obtain a short tooth, as in Figs. 13 and 14. In such a case the height of tooth is arbitrary and only conditioned by the curve. In internal gears it is best to allow more clearance at bottom of tooth than in ordinary spur gears. 29 TeetJi 42 T. 8 P. 30 TeetJi Fig. 13. In a construction of this kind it is suggested to draw the tooth outline many times full size and reduce by photography. An equally multiplied line A B will help in reducing. 6o BROWN & SHARPE MFG. CO. PROVIDENCE, R. I. 6l Second Method. — The difference between gear and pinion being very small, it is sometimes desirable to obtain a smooth action by avoiding what is termed the " friction of approach- ing action."* This is done, the pinion drivings by giving gear only flanks, Fig. 15, and the gear driving., by giving gear only faces, Fig. 16. In both these cases we have but one describ- ing circle, whose diameter is equal to the difference of the two pitch diameters. The construction of the curve is precisely the same as described under A. The describing circle has been divided into 24 parts simply for the sake of greater accuracy. PART B.-INTERNAL BEVEL GEARS. (Fig. 17.) The pitch surfaces of bevel gears are cones whose apexes are at a common point, rolling upon each other. The tooth forms for any given pair of bevel gears are the same as for a pair of spur gears (of same pitch) whose pitch radii are equal to the respective apex distances of the normal cones (/. f., cones whose elements are perpendicular upon the elements of the bevel gear pitch cones). (Compare Fig 19, page 67.) The same is true of internal bevel gears, with the modifica- tion that here one of the pitch cones rolls inside of the other. The spur gears to whose tooth forms the forms of the bevel gear teeth correspond, resolve themselves into internal spur gears (Fig. 17). The problem is now to be solved as indicated in the first part of this chapter. * McCord, Kinematics, pages 107, io8. 62 BROWN & SHARPE MFG. CO. 8 P. Gear 40 Teeth JPinion 20 Teeth Fig. 17. PROVIDENCE, R. L 63 CMAPTER VII. GEAR PATTERNS. (Fig. 18.) To place in bevel gears the best iron where it belongs, the tooth side of the pattern should always be in the nowel, no matter of what shape the hubs are. Hubs, if short, may be left solid on web ; if long they should be made loose. A long hub should go on a tapering arbor, to prevent tipping in the sand. 1° taper for draft on hubs when loose, and 3° when solid is considered sufficient. Coreprints as a rule are made separate, partly to allow the pattern to be turned on an arbor, partly for convenience, should it be desirable to use different sizes. Put rap- and draw-holes as near to center as possible. Referring to Fig. 18, make L = D for D from y^" to i>^", or even more, should hubs be very long. Otherwise if D is more than I >^" leave L= i>^^ Iron pattern before using should be marked, rusted and waxed. Shrinkage — For cast-iron, yi" per foot. For brass, yV' " Cast-iron gears, especially arm gears, do not always shrink Yi^^ per foot. In making iron patterns the following allow- ances have been found useful : Up to \2" diameter allow no shrink. From 12" to 18" " " Yi regular shrink. " 18" to 24" " '' >^ " " 24" to 48" " " yi " Above 48" " " .10" " for cast-iron. 64 BROWN & SHARPE MFG. CO. PROVIDENCE, R. I. 65 If in gears the teeth are to be cast, the tooth thickness t in the pattern is made smaller than called for by the pitch, to avoid binding of the teeth when cast. No definite rule can be given, as the practice varies on this point. For the different diam- etral pitches we would advise making / smaller by an amount expressed in inches, as given in the following table : DiAM Pitch. Amount t IS Smaller. DiAMo Pitch, Amount t IS Smaller. 16 .010" 5 .020'' 12 .012" 4 .022'' 10 .014" 3 .026'^ 8 .016" 2 .030'' 6 .018" I .040'' 66 BROWN & SHARPE MFG. CO. CHAPTER VIII. DIMENSIONS AND FORM FOR BEVEL GEAR CUTTERS. (Fig. 19.) The data needed to determine the form and thickness of a bevel gear cutter are the following : P = pitch. N= number of teeth in large gear. n = number of teeth in small gear. F = length of face of tooth, measured on pitch line. After having laid out a diagram of the pitch cones a d c and ad/, and laid off the width of face, the problem resolves itself into two parts : Part I. — Determine Proper Curve for Cutter. It will be remembered that in the involute system of cutters (the only one used for bevel gears that are cut with rotary cutter), a set of eight different cutters is made for each pitch, numbering from No. i to No. 8, and cutting from a rack to 12 teeth. Each number represents the form of a cutter suitable to cut the indicated number of teeth. For instance, No. 4 cutter (No. 4 curve) will cut 26 to 34 teeth. In order to find the curve to be used for gear and pinion we simply construct the normal pitch cones by erecting the perpendicular p q through b, Fig. 19. We now measure the lines b q and b p, and taking them as radii, multiplying each by 2 and P we obtain a number of teeth for which cutters of proper curves may be selected. From example we have : Gear : b q — 9^" ; 2 X P X 9.75 = 97 T No. 2 curve. Pinion: b p = 3>^" ; 2 X P X 3-5 = 35 T No. 3 curve. The eight cutters which are made in the involute system for each pitch are as follows : No. I will cut wheels from 135 teeth to a rack. " 2 (■(, 55 134 teeth " 3 « 35 54 " " 4 a 26 34 " " 5 (( 21 25 '' " 6 (( 17 20 " " 7 (( 14 16 " '' 8 (( u 12 13 " PROVIDENCE, R. L 67 68 BROWN & SHARPE MFG. CO. Part II.— Determine Thickness of Cutter. It is very evident that a bevel gear cutter cannot be thicker than the width of the space at small end of tooth ; the practice is to make cutter .005" thinner. Theoretically the cutting angle (/i) is equal to pitch angle less angle of bottom (or /i = a — fi'). Practically, however, better results are obtained by making h = a — f3 (substituting angle of top for angle of bottom), and in calculating the depth at small end, to add the full clearance (/) to the obtained working depth, giving equal amount of clearance at large and small end. This is done to obtain a tooth thinner at the top and more curved. As the small end of tooth determines the thickness of cutter, we shall have to find the tooth part values at small end. From the diagram it will be seen that the values at large end are to those at small end as their respective apex distances {a b and a I). The numerical values of these can be taken from the diagram and the quotient of the larger in the smaller is the constant where- with to multiply the tooth values at large end, to obtain those at small end. In our example we find : ^ 7 ""—= .6s S = constant ir - -d u ab=c,^ "^^ For 5 P we have : /=.3i4i /' = .2057 s = .2000 / = .1310 /=.o3i4 / = :£3i4 s +/=.23i4 / +/=.i624 D" + /= .4314. ^' - -^310 D"' +/=.2934 From the foregoing it is evident that a spur gear cutter could not be used, since a bevel gear cutter must be thinnen If in gears of more than 30 teeth the faces are proportion- ately long, we select a cutter whose curve corresponds to the midway section of the tooth. The curve of the cutter is found by the method explained in Part I. of this Chapter. PROVIDENCE, R. L 69 CHAPITER IX. DIRECTIONS FOR CUTTING BEVEL GEARS WITH ROTARY CUTTER. (Fig. 20.) In order to obtain good results, the gear blanks must be of the right size and form. The following sizes for each end of the tooth must be given the workman : Total depth of tooth. Thickness of tooth at pitch line. Height of tooth above pitch line. These sizes are obtained as explained in Chapter VIII. The workman must further know the cutting angle (see formula on page 13 and compare Chapter VIII.), and be pro- vided with the proper tools with which to measure teeth, etc. In cutting a gear on a universal milling machine the opera- tions and adjustments of the machine are as follows : 1. Set spiral bed to zero line. 2. Set cutter central with spiral head spindle. 3. Set spiral head to the proper cutting angle. 4. Set the index on head for the number of teeth to be cut, leaving the sector on the straight or numbered row of holes, and set the pointer (or in some machines the dial) on cross-feed screw of millmg machine to zero line. 5. As a matter of precaution, mark the depth to be cut for large and small end of tooth on their respective places. 6. Cut two or three teeth in blank to conform with these marks in depth. The teeth will now be too thick on both their pitch circles. 7. Set the cutter off the center by moving the saddle to or from the frame of the machine by means of the cross-feed screw, measuring the advance on dial of same. The saddle must not be moved further than what to good judgment 70 BROWN & SHARPE MFG. CO. Eig. 20. PROVIDENCE, R. I. Jl appears as not excessive ; at the same time bearing in mind that an equal amount of stock is to be taken off each side of tooth. 8. Rotate the gear in the opposite direction from which the saddle is moved off the center, and trim the sides of teeth (A) (Fig. 20.) 9. Then move the saddle the same distance on the opposite side of center and rotate the gear an equal amount in the opposite direction and trim the other sides of teeth (C). 10. If the teeth are still too thick at large end E, move the saddle further off the center and repeat the operation, bearing in mind that the gear must be rotated and the saddle moved an equal amount each way from their respective zero settings. It is generally necessary to file the sides of teeth above the pitch line more or less on the small ends of teeth, as indicated by dotted lines F F. This applies to pinions of less than 30 teeth. For gears of coarser pitch than 5 diametral it is best to make one cut around before attempting to obtain the tooth thickness. The formulas for obtaining the dimensions and angles of gear blanks are given in Chapter III. 72 BROWN & SHARPE MFG. CO. CHAPTER X. THE INDEXING OF ANY WHOLE OR FRAC- TIONAL NUMBER. (Fig. 21 ) Change Gear Fig. 21, In indexing on a machine the question simply is : How- many divisions of the machine index have to be advanced to advance a unit division of the number required. To which is the divisions of machine index answer = number to be indexed Suppose the number of divisions in index wheel of machine to be 2i6. Example I. — Index 72. Answer: 216 72 = 3 (3 turns of v/orm). PROVIDENCE, R. I. 73 Example II. — Index 123. — =i + -93 123 123 If now we should put on worm shaft a change gear having 123 teeth, give the worm shaft, Fig. 21, one turn, and in addi- tion thereto advance 93 teeth of the change gear (to give the fractional turn), we would have indexed correctly one unit of the given number, and so solved the problem. Should we not have change gear 123 we may try those on hand. The ques- tion then is : How many teeth (x) of the gear on hand (for instance 82) must we advance to obtain a result equal to the one when advancing 93 teeth of the 123 tooth gear? We have : -^ = -- where x = ^^ 123 82 Example III. — Index 365, change gear 147. — = -i where ;^ = 87 — -^ 365 147 365 Here 147 is the change gear on hand. In indexing for a unit of 365 we advanceS^teeth of our 147 tooth gear. It is evident that in so doing we advance too fast and will have indexed three teeth of our change gear too many when the circle is completed. To avoid having this error show in its total amount between the last and the first division, we can distribute the error by dropping one tooth at a time at three even intervals. Example IV. — Index igo. 216 _ .26 7^ — ^ "•" — ^ Change gear on hand 88 T — = -^ where j = 12 + 190 88 190 To distribute the error in this case we advance one additional tcoth ot a time of the change gear at eight even intervals. Example V. — Index 117.3913. 216 _ 986087 117.3913 1173913 This example is in nowise different from the preceding ones, except that the fraction is expressed in large numbers. This fraction we can reduce to lower approximate values, which for practical purposes are accurate enough. This is done by the method of continued fractions. [For an explana- 74 BROWN & SHARPE MFG. CO. tion of this method we refer to our " Practical Treatise on Gearing."] 986087 II739I3 986087) II739I3 (I 986087 187826) 986087 (5 939130 46957) 187826 (3 140871 46955) 46957 (^ 46955 2) 46955 (23477 46954 1)2(2 2 . 0" 986087 _ , "V39M , ^ I 5+1 3 + 1 i + i 23477 + 1 2 I 5 c=s I 23477 2 ^1=1 ^ = 5 d = 16 21 493033 986087 a^ = 1 If^ = 6 d^ = ig 25 586944 1173913 ^ ■ the Note. — Find the first two fractions by reduction = - and — , — - , •^ II I + I 6 5 others are then found by the rule j <^ ^ + ^ — " The fraction W is a good approximation; putting therefore a change gear of 25 teeth on worm shaft, we advance (beside the one full turn) 21 teeth to index our unit. Of course, in using any but the correct fraction we have an error every time we index a division ; so that when indexed around the whole circle, we have multiplied this error by the number of divisions. In the present example this error is evidently equal to the difference between the correct and the approximate fraction used. Reducing both common fractions to decimal fractions we have : -5 — = .84000006 1173913 21 — =.84000000 .00000006 = error in each division. PROVIDENCE, R. I. 75 .00000006 X 1 17.3913 = .00000704348 total error in complete circle. This error is expressed in parts of a unit division. (To find this error expressed in inches, multiply it by the distance between two divisions, measured on the circle.) In this case the approximate fraction being smaller than the correct one, in indexing the whole circle we fall short .00000704348 of a division. Example VI.- 15708 983) 1309(1 983 326) 983 (3 978 5) 326 (65 30 26 25 1)5(5 5 o 983 1309 + 1 3+1 65 + 1 5 3 65 I 3 196 983 I 4 261 1309 In using the approximation |^f ^ the error for each division (found as above) will be .000002927, for the whole circle .0000460. In this case, the approximation being larger than the correct fraction, we overreach the circle by the error. /6 BROWN & SHARPE MFG. CO. CHAPTER XI. THE GEARING OF LATHES FOR SCREW CUTTING. (Figs. 22, 23.) The problem of cutting a screw on a lathe resolves itself into connecting the lathe spindle with the lead screw by a train of gears in such a manner that the carriage (which is actuated by Simple a earing. Fig. 22. PROVIDENCE, R. I. 77 the lead screw) advances just one inch, or some definite dis- tance, while the lathe spindle makes a number of revolutions equal to the number of threads to be cut per inch. The lead screw has, with the exception of a very few cases, always a single thread, and to advance the carriage one inch it therefore makes a- number of revolutions equal to its number Compound Gearing- Fig, 23. of threads per inch. Should the lead screw have double thread, it will, to accomplish the same result, make a number of revolutions equal to half its number of threads per inch. It follows that we must know in the first place the number of threads per inch on lead screw. 78 BROWN & SHARPE MFG. CO. It ought to be clearly understood that one or more inter- mediate gears, which simply transmit the motion received from one gear to another, in no wise alter the ultimate ratio of a train of gearing. An even number of intermediate gears simply change the direction of rotation, an odd number do not alter it. The gearing of a lathe to solve a problem in screw cutting can be accomplished by A. Sim. pie gearing. B. Compound gearing. Referring to the diagrams, Figs. 22 and 23, we have in Fig. 22 a case of simple, and in Fig. 23 a case of compound gear- ing. In simple gearing the motion from gear E is transmitted either directly to gear Ron lead screw or through the interme- diate F. In compound gearing the motion of E is transmitted through two gears (G and H) keyed together, revolving on the same stud ;?, by which we can change the velocity ratio of the motion while transmitting it from E to R. With these four variables E, G, H, R, we are enabled to have a wider range of changes than in simple gearing. B and C, being intermediate gears, are not to be considered. If, as is generally the case, gear A equals gear D, we disregard them both, simply remembering that gear E (being fast on same shaft with D) makes as many revolutions as the spindle. Sometimes gear D is twice as large as gear A, then, still con- sidering gear E as making as many revolutions as the spindle, we deal with the lead screw as having twice as many threads per inch as it measures. SIMPLE GEARING. Let there be : the number of teeth in the different gears expressed by their respective letters, as per Fig. 22, and s = threads per inch to be cut, L ■— threads per inch on lead screw ; then I. ^ ^ 5: L E PROVIDENCE, R. I. 79 If now one of the two gears E and R is selected, the other will be : L s 2. The two gears may be found by making -p ""f T f where/ may be any number. 3. The above holds good when a fractional thread is to be cut, but if the fraction is expressed in large numbers, as, for instance, s = 2.833 (2TTHfV)» we first reduce this fraction (y^o^) to lower approximate values by the process of continued fraction (see pages 73 and 74). 833) ICOO (I 833 167) 833 (4 668 165) 167 (I 165 2) 165 (82 16 "^ 1)2(2 2 o I 4 I 82 2 _^ _£ _5_ 414 833 156 497 1000 ± = .S^^ (nearly) and s = 2± 6 6 If in this case L = 4, and we select E = 48, then, since R = ^ R = 34 COMPOUND GEARING. 4. In a lathe geared compound for cutting a screw the product of the drivers (E and H, Fig. 23) multiplied by the num- ber of threads per inch to be cut must equal the product of the driven (G and R) multiplied by the number of threads on lead screw. This is expressed by E.H.^=G.R.Lor ^— i^- = i So BROWN & SHARPE MFG. CO. If three of the gears E, H, G, R have been selected, the fourth one would be either ^ GR L H = ^^ or E s n E H ^ G = or R S=: R L E H^ GL RG L VL.E.H/ E H If a fractional thread is to be cut, as under " 3," we reduce the fraction to lower approximate values. Example.— Gear for 5.2327 threads per inch, lead screw is 6 threads. 2^27 .2327 = -^J- lOOOO — 2327) lOOCO (4 9308 692) 2327 (3 2076 251) 692 (2 502 190) 251 (I 190 "61) 190 (3 183 7) 61 (8 5)7(1 5_ 2j 5 (2 4 i) 2 (2. 2 o 43213 8 I 2 2_ 13 7 10 37 306 343 992 2327 4 13 30 43 159 ^3^5 1474 4263 loooo — = .2327 (nearly) and 5.2327 = 5 — 43 "43 Selecting E = 43, H = 52, R = 50, and ^ E . H . j- , ^ 43 . 1^2 . K^^ G = we have G = ^ — ^ ii-5 = ^g. R . L 50 . 6 '^^ PROVIDENCE, R. I. 8l 5. The examples so far given all deal with single thread. The pitch of a screw is the distance from center of one thread to the center of the next. The lead of a screw is the advance for each complete revolution. In a single thread screw the pitch is equal to the lead, while in a double thread screw the pitch is equal to one-half the lead ; in a triple thread screw equal to one-third the lead, etc. If we have to gear a lathe for a many-threaded screw (double, triple, quadruple, etc.), we simply ascertain the lead, and deal with the lead as we would with the pitch in a single thread screw, /. ^., we divide one inch by it, to obtain the num- ber of threads for which we have to gear our lathe. Example. — Gear for double thread screw, lead = .4654. Number of threads per inch to be geared for is : —1-= -1-=: 2.1487 Lead -4654 Lead screw is four threads per inch. As in previous examples, we reduce the fraction .l4S']=-^Jf-^-^^-Q to lower approximate values by the process of continued frac- tion. From the different values received in the usual way we select : \l = .1487 (nearly) and 2.1487 = 2^ We have therefore : L = 4 (E=74 Selecting ^ G = 30 ( H = 40 G . L 30 . 4 Note, — In using any but the original fraction we commit an error. This error can be found by reducing the approximate fraction used to a decimal fraction, and comparing it with the original fraction. In the above example the original fraction is .1487 and H = . 14864 Error = .00006 inch in lead. In cutting a multiple screw, after having cut one thread, the question arises how to move the thread tool the correct amount for cuttimx the next thread. 82 BROWN & SHARPE MFG. CO. In cutting double, triple, etc., threads, if in simple or com- pound gearing the number of teeth in gear E is divisible by 2, 3, etc., we so divide the teeth ; then leaving the carriage at rest we bring gear E out of mesh and move it forward one division, whereby the spindle will assume the correct position. When E is not divisible we find how many turns (V) of gear R are made to each full turn of the spindle. Dividing this number by 2 for double, by 3 for triple thread, etc., we advance R so many turns and fractions of a turn, being careful to leave the spindle at rest. For compound gearing : G.R When the gear D is twice as large as the gear A (as ex- plained in fifth paragraph, page yS.) the formula would be 2 G. R. If in simple gearing both E and R are not divisible, one remedy would be to gear the lathe compound ; or the face- plate may be accurately divided in two, three or more slots, and all that is then necessary is to move the dog from one slot to another, the carriage remaining stationary.