FILENAMES: B_Fig1122.gif B_Fig1123.gif B_Fig1124.gif B_Fig1125.gif B_Fig1126.gif B_Fig1127.gif B_Fig1128.gif B_Fig1130.gif B_Fig1131.gif B_Fig1132.gif DESCRIPTION: These are scans of information about bevel gear cutting. Posted by Errol Groff . Errol provided the following description: ================================================================= Page 368 BEVEL GEARS from Machine Tool Operation, Berghardt, Axelrod and Anderson One should have a working knowledge of the parts and principles of the spur gear before attempting to study the bevel gear. To any-one who has this knowledge of spur gears the subject of bevel gears should prove easy to understand, if studied step by step in logical sequence. This text is presented in the following order: 1. Preliminary discussion and definitions. 2. Instruction for laying out gears. A layout is advisable to fix mind certain necessary knowledge as well worth while for the machinist as for the draftsman. 3. Definitions and rules, bevel-gear elements, and tooth parts. 4. Example showing application of the rules. Size of gear selected is the same as for layout (2). 5. Cutting a bevel gear in a milling machine. Do not rush. Try to understand each step before proceeding to the next. Pitch Cylinders, Pitch Cones, Spur Gears, and Bevel Gears. In the same way, as was explained in spur gearing, that the motion of one cylinder will cause motion in a parallel cylinder that touches it, motion may be communicated from one shaft to another at an angle to it (whose center lines would meet if sufficiently prolonged) by means of cones in close contact. The apex of each cone is Page 369 at a point where the center lines of the shafts would meet. Motion may be communicated in the desired velocity ratio (Fig. 11-22, a and b) and at the desired angle (c, d, and e). In the study of spur gears it was learned that positive motion may be transmitted between parallel shafts by means of spur gears, and that spur gears are, in effect, "pitch cylinders" with teeth built on the cylindrical surfaces. Bevel gears are, in effect, "pitch cones" with teeth built on the conical surfaces. Fig. 11-22. Spur gears communicate motion between parallel shafts. These shafts may be (within limits) any distance apart, and motion may be communicated (within limits) in any desired velocity ratio. Similarly, bevel gears communicate motion between shafts whose axes meet when prolonged. In a pair of gears, spur or bevel, the smaller is often called the pinion. Bevel gears are called miter gears when they are of the same size and transmit motion at right angles. Pitch Cones. The cones which represent in bevel gears the original friction surfaces are called the pitch cones. It is on frustums (portions 370 THE MILLING MACHINE at the large end) of these cones that the teeth of bevel gears are built, similarly to the way the teeth of spur gears are built on the pitch cylinders. It is important that one should have a very clear understanding of the elements of the pitch cone, for the reason that, although the bevel gear itself is only a portion of a cone, in drawings the whole cone is laid out and in calculations the whole cone is considered. In the drawing (Fig. 11-23) the triangle AOB represents the pitch cone with the cone center at 0. The line OY is the center line (axis) of the cone, and the lines OA and OB represent the cone (apex) distances. Either angle AOY or BOY is the pitch angle. The line AB will equal the pitch diameter of the bevel gear built on this cone. Pitch Angle. In any bevel gear, the angle that the cone distance makes with the center line is called the pitch angle. Figure 11-22a represents two shafts running at right angles with equal pitch cones. It will be noted that the pitch angle is one half of 90 deg. or 45 deg. It will be observed in Fig.11-22b that in right-angle bevel gears, other than miter gears, the pitch angle of either the gear or of the pinion cannot be 45 deg. and that the pitch angle depends on the relative pitch diameters of the gear and the pinion. The pitch angle is one of the most important factors in bevel-gear calculations. Pitch Diameter. When speaking of the pitch diameter of a bevel gear, the diameter at the large end of the pitch cone is meant. As a matter of fact, the diameter at any given point in the pitch cone is the pitch diameter of that part of the gear, but in practice no thought or consideration is given to any other pitch diameter than the largest in the gear. Face Width of Bevel Gear. In making a bevel gear, the teeth are not cut on the whole pitch cone but only on a portion of it. The distance equal to the length of the tooth is called the face width of the bevel SPUR GEARS AND BEVEL GEARS 371 gear (Fig. 11-24). The ends of the teeth are spoken of as the "large end" and the "small end." In practice, the face width of the gear tooth is frequently shorter but never longer than one third of the cone distance. The dimension should be in sixteenths of an inch or greater, never in thirty-seconds or sixty-fourths. Cone Distance (Apex Distance), Tooth Size and Shape. In any bevel gear of whatever diameter, angle, or face, the teeth and the spaces are largest at the largest diameter of the gear and decrease in size as the cone center of the original pitch cone is approached. If it can be imagined that teeth were cut along the whole pitch cone, halfway down the cone the teeth and the spaces would be half as large as at the large end and all would vanish at the cone center. According to this reasoning, it will be clear that the sizes of bevel-gear-tooth parts, at the large end and at the small end, for example, are Fig. 11-24.proportional to their respective cone distances. This proportion is used in practice (see rule for size of tooth parts at small end on page 377). It will be understood that when the size of the tooth changes, the curve of the tooth also changes, and consequently it will be impossible to cut a bevel gear accurately with a rotary cutter because the cutting edge has a fixed shape and curvature. However, it is possible to mill a fairly accurate tooth shape by taking more than one cut, thus "trimming" the tooth, as will be explained later. The drawing (Fig. 11-24) will serve to illustrate the principles thus far outlined of a bevel gear built on a cone. Let the triangle AOB represent the original pitch cone with a pitch angle of 45 deg. and a pitch diameter of 3 in. Suppose it is decided to put 30 teeth on this gear and to have the face one third of the cone distance. The module (gear 3-in, pitch diameter, 30 teeth) is 110 in., that is, the gear is 1ODP. The cone distance OC is two thirds of the cone distance OA and consequently the module at the small end of tooth is two-thirds of the module at the large end of the tooth. That is, all the tooth parts, addendum, dedendum, thickness, at the small end of the THE MILLING MACHINE Page 372 .. tooth will be two thirds as large as corresponding parts of the tooth at the large end. Laying Out Bevel Gears, Shafts at Right Angles. Perhaps one of the best methods of becoming acquainted with bevel-gear parts and proportions is to lay out first a miter gear, then a gear and pinion. To get parts of miter gears to scale it is only necessary to draw one; to get the angles, proportions, etc., of the gear and pinion, SPUR GEARS AND BEVEL GEARS Page 373 it is necessary to lay out both. It is usually best to make the drawings to a large scale if accuracy is required in measurements. In any event, in order to lay out bevel gears it is necessary to know angle of shafts, numbers of teeth, and pitch of tooth. Two examples with necessary directions follow (Figs. 11-25 and 11-26). Make these drawings full size or larger before proceeding further. Laying Out Bevel Gears, Shafts Not at Right Angles. Bevel gears with shafts at other than right angles (Fig. 11-22, c and d) may be drawn by the method shown in Fig. 11-26, except that the pitch diameters will not be at right angles to each other. Figure 11-26 Here For example, if the angle of the shafts HO and XO (as in Fig. 11-26) were 80 deg. instead of 90 deg., simply draw pitch diameter AM of pinion at an angle of 100 deg. with the pitch diameter AB of gear and proceed as for regular bevels. (The smaller the shaft angle, the greater the distance OX.) If the shaft angle is 110 deg., draw the pitch diameter AM at an angle of 70 deg. with pitch diameter AB and proceed as before. 374 THE MILLING MACHINE The angle formed by the two pitch diameters will be just as many degrees less than 90 deg. as the shaft angle is over 90 deg., or just as many degrees over 90 deg. as the shaft angle is less than 90 deg. Calculations for Bevel Gears. The outside diameter, the face angle, the root angle, etc., may be obtained fairly closely with a scale and protractor from a good drawing. They may be obtained also from tables of gear-tooth parts in handbooks, etc. These tables have been calculated by the use of rules, and in order to read the tables intelligently one should understand the rules. An example showing the calculations for a miter gear is given on page 377. In order to calculate bevel gears, it is important to have a working knowledge of some of the functions of right triangles. If machinists realized how easily and quickly "shop trig" may be understood, more of them would spend a few hours in getting this understanding. Note: For trigonometric formulas and tables see pages 651 to 666. DEFINITIONS AND RULES: BEVEL-GEAR ELEMENTS AND TOOTH PARTS (Fig. 11-27) Other gear terms common to both spur and bevel gears are defined beginning on page 356. Addendum. Same as for spur gear, equals 1 divided by the diametral pitch. Addendum Angle. The angle between the elements of the pitch cone and The face cone in a plane containing the axis of the gear. RULE: The tangent of the addendum angle equals the addendum divided by the cone distance. Angles. Addendum, back, dedendum, face, front, pitch, root, and shaft- see in alphabetical order according to name. Apex Distance. See cone distance. Back Angle. The angle between the plane of the pitch circle and a plane tangent to the large end of the tooth; equals the pitch angle. Back Cone. The cone generated by revolving the back-cone radius about the axis of the gear. Any one of the following books will give the necessary information: Rupert LeGrand, The New American Machinists' Handbook, McGraw-Hill Book Company, Inc., New York, 1955. A. Axelrod, Machine Shop Mathematics, 2d ed., McGraw-Hill Book Cornpany, Inc., New York, 1951. Treatise on Gearing, Brown & Sharpe Manufacturing Company, Providence, R.I., 1951. SPUR GEARS AND BEVEL GEARS 375 Back-cone Radius. The distance perpendicular to the pitch surface from the pitch circle to the axis; equals cone distance times the tangent of the pitch angle. Backing. The distance parallel to the axis from the pitch circle to the face (end) of the shoulder or hub. (Do not confuse with crown backing.) Bore Diameter. The diameter of the hole in the gear. Clearance. Same as for spur gear. Cone Center. The apex of the pitch cone. Cone Distance. The distance from the cone center to any point on the pitch circle; equals one-half the pitch diameter divided by the sine of the pitch angle. Crown. The circle formed by the intersection of the face cone and the back cone extended. Crown Backing. The distance, parallel to the axis, from the crown to the shoulder or hub end. (Do not confuse with backing.) Crown Height. The distance, parallel to the axis, from the cone center to the crown of the gear; equals the product of the cone distance times the cosine of the pitch angle, minus the product of the addendum times the sine of the pitch angle. Dedendum. Same as for spur gear (usually 1.157 divided by the diametral pitch). Dedendum Angle. The angle between elements of the pitch cone and root cone in a plane containing the axis of the gear. The tangent of the dedendum angle equals the dedendum divided by cone distance. Diameter Increment. The amount added to the pitch diameter to obtain the outside diameter; equals two times the addendum multiplied by the cosine of the pitch angle. (It will be observed in Fig. 11-27 that NC, the outside radius, is greater than XA, the pitch radius, by the length of KA, and not by the length of CA, the addendum. Do not add "two addendum" to the pitch diameter of a bevel gear.) Diametral Pitch. Same as for spur gears; equals number of teeth divided by the pitch diameter. Face angle. The angle between an element of the face cone and its axis equals pitch angle plus addendum angle. Face Cone. The right circular cone whose elements contain the top lands of the gear. Face Width. The width of the pitch surface. Front Angle. The angle between the plane of the pitch circle and a plane tangent to the small end of the tooth; equals the pitch angle. 376 THE MILLING MACHINE Heel. A portion of the tooth at the large end; toe on the small end. Increment Angle. See Addendum Angle. Module. See page 354. Mounting Distance. The distance, parallel to the axis, from the cone center to the shoulder or hub end against which the gear is mounted; equals crown backing plus crown height. Outside Diameter. The diameter of the circle which contains the tops of the teeth; equals pitch diameter plus diameter increment. Pitch Angle. The angle between an element of the pitch cone and it's axis. With shafts at right angles, the tangent of pitch angle of the gear equals number of teeth in gear divided by number of teeth in pinion, and pitch angle of pinion equals 90 deg. minus pitch angle of gear. Pitch Circle. The circle formed by the intersection of the pitch cone and a plane perpendicular to the axis. Circumference of pitch circle equals pitch diameter times 3.1416. Pitch Cone. The cone generated by revolving the cone-distance line about the axis of the gear. Pitch Diameter. The diameter of the pitch circle; equals the number of teeth divided by the diametral pitch. Root Angle. The angle between an element of the root cone and its axis; equals pitch angle minus the dedendum angle. Root Circle. The circle containing the bottoms of the tooth spaces. Root Cone. The right circular cone whose elements contain the bottoms of the tooth spaces. Root Diameter. The diameter of the root circle. Fig. 11-27. Parts of a bevel gear. SPUR GEARS AND BEVEL GEARS 377 Shaft Angle. The included angle between the shafts upon which a pair of mating gears are to operate; equals the sum of the pitch angles of the two gears. Thickness, chordal, circular. Same definitions as for spur gears. Toe. A portion of the small end of the tooth-heel on the larger end. Undercut. See definition for undercut on page 359. Virtual Number of Teeth. The number of teeth of a given pitch which would be contained in the virtual pitch circle whose radius is the back-cone radius. RULE. Number of teeth for which to select the cutter for a bevel gear equals the number of teeth in the bevel gear divided by the cosine of the pitch angle. Size of Tooth Parts at Large Ends. Same as for spur gear of same pitch. Size of Tooth Parts at Small End. Divide the cone distance of small end by the cone distance of large end and multiply the respective tooth parts of large end by the quotient. Example Showing Calculations for Miter Gear. Gear selected, 4DP, 20 teeth (see Fig. 11-25). Use rules (Definitions and Rules, Bevel-gear Elements and Tooth Parts) as indicated by the terms introducing the calculations. Addendum = 1/DP = 1/4 in. Dedendum = 1.157/DP = 1.157.4 = 0.289 Pitch diameter = N/DP = 20/4 = 5 in. Pitch angle = 45 deg. (45 deg. always in miter gears). Cone distance = 1/2 pitch diameter divided by the sine of the pitch angle 2.5 in. ö 0.707 = 3.536 in. Diameter increment = two times the addendum multiplied by the cosine of the pitch angle = 2 X 1/4 X 0.707 = 0.3535 in. Outside diameter = pitch diameter plus diameter increment = 5 in. plus 0.3535 in. = 5.3535 in. Addendum angle: Tangent of addendum angle equals addendum divided by the cone distance, equals 1/4 in. ö 3.536 = 0.0707, which is tangent of angle 4 deg. 3 min. Therefore addendum angle is 4 deg. 3 min. 378 THE MILLING MACHINE Dedendum angle: Tangent of dedendum angle equals dedendum divided by cone distance, equals 0.289 ö 3.536 = 0.0817, which is tangent of angle 4 deg. 40 min.. Therefore dedendum angle is 4 deg. 40 min.. Face angle equals pitch angle plus addendum angle = 45 deg. plus 4 deg. 3min. = 49 deg. 3 min.. Root angle (cutting angle) equals the pitch angle minus the dedendum angle = 45deg. minus 4 deg. 40 min. = 40 deg. 20 min.. Size of tooth parts on large end are same as for spur gear of the same pitch. Size of tooth parts at the small end: As one third of the cone distance (3.536 in.) is 1.179 in., let the face width of the teeth measure 1 1/8 in. Then 3.536 in. minus 1.125 in. = 2.411 in., which is the cone distance of the small end of teeth. 2.411 ö 3.536 = 0.682, therefore multiply the parts of the tooth at the large end by 0.682 to obtain the sizes of the corresponding tooth parts at the small end. Addendum equals 1/4 X 0.682 = 0.170 in. Whole depth equals 2.157/4 x 0.682 = 0.367 in. Circular thickness equals . 1.570/4 X 0.682 = 0.267 in. Number of teeth for which to select the cutter for miter gear equals number of teeth/0.707 = 20/0.707 = 28 (This is the same number of cutter as for a spur gear of 28 teeth, therefore the No. 4 bevel-gear cutter is used. Cutting a Bevel Gear in a Milling Machine. As previously stated, it is impossible to cut an accurate bevel gear in a milling machine. It often happens, however, that a bevel gear may be wanted in a hurry, or that an extremely accurate gear is not required, and it is then convenient to know how to mill one (Fig. 11-29). Selecting the Cutter. In the study of spur gearing it has been learned that the same "form" of gear cutter-that is, the same number of cutter-is not used to cut a gear of 20 teeth that is used to cut a gear of 120 teeth. This is because the pitch surface of the 120-tooth gear SPUR GEARS AND BEVEL GEARS 379 has a longer radius of curvature than the 20-tooth gear. In other words, it is more nearly straight. Bevel-gear cutters are made in sets similar to spur-gear cutters with the same range of numbers of teeth to each cutter (see page 364). Bevel-gear cutters have a curve of cutting edge that is right for the large end of the tooth, but they are thinner than spur- gear cutters because they must pass through the spaces at the small end of the teeth. There is one feature in the selection of a bevel-gear Fig. 11-28. Radius of surface. cutter which at first seems difficult to understand; the cutter is not selected for the number of teeth in the bevel gear itself, but for a spur gear having a pitch radius equal to the back- cone radius of the bevel gear. For example, in Fig. 11-28 the number of cutter to select for the bevel gear would be determined by the number of teeth in the spur gear. The reason for this may be explained as follows: The radius of the pitch surface of a spur gear is the same as the radius of the pitch circle of that gear, but the radius of the rolling pitch surface of a bevel gear is longer than the radius of the pitch circle of that bevel gear(5). Thus in Fig. 11-28 the back-cone radius of the bevel gear is equal to the radius of curvature of the spur gear. (5) The difference between the curvature of the rolling pitch surface of a bevel gear and the curvature of the pitch circle of that gear may be easily seen. Select a bevel gear (as large as 10 or 12 in. in diameter is best) lay a piece of paper along the outer edge of the teeth, in close contact against the edge of five or six teeth, and rub it to make an imprint of the teeth on the paper. Trim this paper to the pitch line of these teeth, and it will be cut to approximately the curvature of the rolling pitch surface of the gear. If, now, the paper is held at right angles to the centerline of shaft, with the curve cut on the paper as nearly coincident with the curve of the pitch circle of the gear as possible, the difference between these curves will be apparent. The nearer the bevel gear approaches the spur gear-that is, the less bevel it has-the less is this difference. Fig. 11-29. 380 THE MILLING MACHINE The back-cone radius is not used directly in calculating the cutter to use (see rule below), but if a good drawing is furnished, it may be practically useful in determining the cutter, or in checking the calculation by the rule. RULE: Number of teeth for which to select the cutter equals the number of teeth in the bevel gear divided by the cosine of the pitch angle. Order of Operations. Let it be required to cut a bevel gear. The following directions in general will apply to any bevel gear; but select, for an example, say, a cast- iron gear of 24 teeth, 4DP (Fig. 11-26). The data necessary to cut this bevel gear should be furnished with the order or on the drawing, but may be calculated, as previously explained. The sizes are as follows: DP = 4; whole depth = 0.539 in.; circular thickness = 0.393 in.; addendum = 0.250 in. For small end of tooth, thickness = 0.267 in.; addendum = 0.170 in. The root angle (cutting angle) is 50 deg.10 min. Use No. 3 cutter. IMPORTANT PRECAUTION: In any milling-machine indexing operation, the backlash or lost motion in the index-head worm and worm wheel, and in the feed screws, is a most serious consideration. Do not neglect the lost motion. 1. Check the measurements of the blank, especially the outside diameter and the face angle. 2. For 24 teeth, set the index pin in a circle divisible by 3. The largest circle is best to permit of finer adjustment for reasons hereafter explained (10). Set the sector to two thirds of one turn. 3. Set the dividing head to the root angle (equals 59 deg 10 min ). U 4. Being careful that spindle, arbor, collars, and cutter are clean and that arbor runs true, set the cutter on the arbor so the direction of the cut will be away from the dividing- head spindle. SPUR GEARS AND BEVEL GEARS 381 Have the cutter as near the machine spindle as practicable and bring the work central under the cutter. 5. Adjust the table until the revolving cutter just touches the gear blank at the outside diameter. Fig. 11-30. 6. Raise the table the whole depth (0.539 in.) and take a cut (perhaps a roughing cut will be advisable). Index for one tooth and take another cut. 7. Measure the thickness of tooth, preferably with gear-tooth caliper (Fig. 11-30), at large end and at small end. (Fig. 11-31 shows the gear-tooth vernier in actual use.) 8. Subtract the finished thickness of tooth (I = 0.393 in.; t' = 0.262) from the thickness as measured, and divide by 2 to know how much must be "trimmed" from each side (a, Fig. 11-31). Take THE MILLING MACHINE 382 both halves away and the finished size will remain; take one half away and the size of tooth when one side of the tooth is finished will remain. NOTE: There now are two spaces with a tooth between; the depth of tooth is established and the curve of the cutter is right for the large Fig. 11-31. end, but since the thickness of the cutter is about right for the finished space at the small end, the thickness of tooth at the large end is altogether too great. Since the curve of the cutter is correct for the large end of the tooth, the shape of the tooth at the small end is not right. The job of cutting a bevel gear in a milling machine is to get the correct thickness of tooth, at the pitch line, at both ends of tooth by SPUR GEARS AND BEVEL GEARS 383 trimming both sides of the tooth and then to file the small end to the correct curve (Fig. 11-32). In any motion of a wheel on its axis a point on the rim passes through a greater arc, a greater distance, than a point on the hub. So, by the same principle, in any movement of a bevel gear or bevel gear blank on its axis, the large end moves farther than the small end. That is, if the gear blank is revolved a very little (if the index pin is removed and advanced one or two holes on the plate), the tooth will be cut thinner, and a little more metal will be removed from the large end than from the small end, but, as it happens, not enough in proportion. If the gear is rotated, say, 5 or 6 holes in the Fig. 11-32. 39 circle, the large end may be right; but the small end will be too thin. To avoid this, offset the blank; that is, move the table of the milling machine crosswise, bringing the gear tooth away from the cutter, and then rotate the gear tooth toward the cutter. This has the effect of taking more in proportion off the large end than off the small end. To continue with the directions: 9. With blue-vitriol or dykem blue, paint the spaces cut; take up the lost motion in the cross-feed screw; set the dial at 0, and, for a trial, offset the table about one-seventh the thickness of the tooth at the large end. 10. Pull out the index pin and rotate the blank until the large end of the tooth touches the cutter, and then very carefully (rotating the blank one hole (see 2) in index plate for each cut) trim the side of the tooth until the blue vitriol is nearly all cut off toward the small end. 384 THE MILLING MACHINE 11. Measure the thickness at the large end, and at the small end for sizes "when one side is finished" (see 8). If there remains more metal to be cut from the large end than from the small end, the blank must be offset a little more and the tooth trimmed again. 12. Having obtained the position of the blank to trim one side of the tooth for finish, note the amount of offset and the number of holes which the blank was rotated. 13. Index for the next tooth, take a cut, and so on, all around the gear blank, and there will be 24 teeth with one side finished. NOTE: In a cast-iron gear of 5 pitch or larger or in a steel gear of 8 pitch or larger, it is usually advisable to take a central roughing cut or "stocking" cut before "trimming" either side. 14. Having finished one side of each tooth, offset the gear the same amount from center, being careful about the backlash, in the opposite direction, rotate the blank as many holes as noted (see 12) in the opposite direction, and carefully checking the measurements of first tooth (to finished size, see 8) proceed to trim the other side of each tooth. 15. File the small ends of teeth to size, as indicated at the lines marked F in Fig. 11-32. NOTE: The amount of offset is from one seventh to one sixth of the thickness of the tooth at large end. In the above gear, 4DP - 24 teeth, it is about 0.060 in. In a 12DP 40-tooth miter gear it is about 0.018 in. In a pair of bevel gears, 8DP, gear 24 teeth, pinion 12 teeth, offset for gear is 0.030 in., and for pinion 0.021 in. Too little offset leaves large end too thick.