side of B B and parallel to it; the distance from D D to E E being as many eighths of an inch if it be 8 pitch as there are to be teeth in the gear. In the example the number of teeth is 24; therefore the distance from D D to E E will be 24, or 14 inches each side of B B. K K and L L are similarly drawn, but there being only 16 teeth in the small gear, the distance from K K to L L will be 16, or 1 inch each side of C C. Then through the intersections of D D and L L, E E and L L, and E E and K K, draw the diagonals F A. These are the pitch lines. Through the same point draw lines as G G at right angles to the pitch lines, forming the backs of the teeth. On these lines lay off of an inch each side of the pitch lines, and draw M A and N A, forming the faces and bottoms of the teeth. The lines H H are drawn parallel to G G, the distance between them being the width of the face. The face of the larger gear should be turned to the lines M A, and the small gear to N A. For other pitches the same rules apply. If 4 pitch, use 4ths instead of 8ths; if 3 pitch, 3ds, and so on. Bevel gears should always be turned to the exact diameters and angles of the drawings and the teeth cut at the correct angle. = LAY out the bevel gears and draw lines A and B at right angles to the center angle line. Extend this to the center lines and measure A and B. The distance A the radius of a spur gear of the same pitch, and finding the number of teeth in such a gear we have the right cutter for the bevel gear in question. Calling the gears 8 pitch and the distance A 4 inches. Then 2 X 4 X 8 teeth, so that a No. 2 cutter is the one to use. For the pinion, if B is 2 inches, then 2 X 2 X 8 = 32 or a No. 4 cutter is the one to use. = 64 USING THE BEVEL GEAR TABLE TAKE a pair of bevel gears 24 and 72 teeth, 8 diametral pitch Divide the pinion by the gear — 24 ÷ 72 =-3333. This is the tangent of the center angles. Look in the seven columns unde center angles for the nearest number to this. The nearest is .3340 in the center column, as all these are decimals to four places. Fol low this out to the left and find 18 in the center angle column. the .3346 is in the column marked .50 the center angle of the pinion is 18.50 degrees. Looking to the right under center angles for gears find 71 and add the .50 making the gear angle 71.50 degrees. This gives Center angle of pinion 18.5 degrees. As In the first column opposite 18 is 36. Divide this by the number of teeth in the pinion, 24, and get 1.5 degrees. This is the angle increase for this pair of gears, and is the amount to be added to the center angle to get the face angle and to be deducted to get the cut angles. This gives Pinion center angle 18.5+ 1.5 = 20 degrees face angle. Pinion center angle 18.5- = 17 degrees cut angle. 73 degrees face angle. 1.5 70 degrees cut angle. = For the outside diameter go to the column of diameter increase and in line with 18 find 1.90. Divide this by the pitch, 8, and get .237, which is the diameter increase for the pinion. same line to the right and find .65 for the gear increase. by the pitch, .8, and get .081 for gear increase. This gives Pinion, 24 teeth, 8 pitch diameter. Gear, 72 teeth, 8 pitch diameter. = = 3 inches + .237 Follow the 3.237 inches outside 9.081 inches outside Another way of selecting the cutter is to divide the number of teeth in the gear by the cosine of the center angle C and the answer is the number of teeth in a spur gear from which to select the cutter. For the pinion the process is the same except the number of teeth in the pinion is divided by the sine of the center angle. Formula Tangent of C Number of teeth to use in selecting cutter for gear Number of teeth to use in selecting cutter for pinion Any pair of gears can be figured out in the same way, bearing in mind that when finding the center angle for the gear, to read the parts of a degree from the decimals at the bottom, and that for the pinion they are at the top. In the example worked out the tangent came in the center column so that it made no difference. If, however, the tangent had been .3476 we read the pinion angle at the top, 19.17 degrees and the gear angle at the bottom, 70.83. By noting that the sum of the two angles is 90 degrees, we can be sure we are right. .0000 I .0029 .0058 .0087 .0116 .0145 0175 0349 88 .07 0524 .0699 .0875 .1022 .1051 .1405 .1584 1763 1944 .2126 2493 3026 .3057 3443 3839 52 1.78 27 5317 5505 5543 5774 7536 .7581 .7627 .7673 7720 .7766 .7813 .7813 7860 .7907 7954 .8002 8050 .8098 .8098 .8146 .8195 .8243 8292 .8342 .8391 .8391 8441 .8491 .8541 .8591 .8642 .8693 .8693 8744 .8796 .8847 .8899 .8952 .9004 77 1.48 42 .9004 9057 .9110 .9163 .9217 .9271 9325 79 1.46 43 9325 .9380 9435 9490 .9545 9601 .9657 .9657 9713 9770 .9827 .9884 .9942 1.0000 1.0000 1.0058 1.0117 1.0176 1.0235 1.0295 1.0355 from 2 to 10, inclusive, omitting 9. A TABLE The arrangement and use of the table needs no explanation. cipal dimensions of miter gears (center angle 45 degrees), the num the possible numbers of teeth from 12 to 60, inclusive, and pitches ber of teeth and the pitch being known. The table covers most of THE accompanying table is of service in determining the prinFOR DIMENSIONS FOR MITER GEARS SPIRAL GEARS THE term spiral gear is usually applied to gears having angular teeth and which do not have their shafts or axis in parallel lines, and usually at right angles. Spiral gears take the place of bevel gears and give a smoother action as well as allowing greater speed ratios in a given space. When gears with angular or skew teeth run on parallel shafts they are usually called helical gears. In considering speed ratios for spiral gears the driving gear can be taken as a worm having as many threads as there are teeth and the driven as the worm wheel with its number of teeth, so that one revolution of the driver will turn a point on the pitch circle of the driven gear as many inches as the product of the lead of the driver and the number of its teeth. Divide this by the circumference of the pitch circle of the driven gear to get the revolutions of the driven. When the spiral angles are 45 degrees, the speed ratio depends entirely on the number of teeth as in bevel gears, but for other angles of spiral the following formula will be useful: Let Ri R2 Di D2 = = = Revs. of Driver. Revs. of Driven. Pitch diameter of Driver. R2 R1 R2 D. = = = = = D D2 X cotangent of helix angle of driver with its axis. D1 X R1 D2 R1 X cotangent of helix angle of driver with its axis. cotangent of helix |