T32. Great Circle Tooth Profile

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1 SECTION BEVEL GEARING For intersecting shafts, evel gears offer a good means of transmitting motion and power. Most transmissions occur at right angles, Figure -1, ut the shaft angle can e any value. Ratios up to 4:1 are common, although higher ratios are possile as well..1 Development And Geometry Of Bevel Gears Fig. -1 Typical Right Bevel Gear Bevel gears have tapered elements ecause they are generatend operate, in theory, on the surface of a sphere. Pitch diameters of mating evel gears elong to frusta of cones, as shown in Figure -2a. In the full development on the surface of a sphere, a pair of meshed evel gears are in conjugate engagement as shown in Figure -2. O P Great Circle Tooth Profile Line of Action Pitch Line Common Apex of Cone Frusta (a) Pitch Cone Frusta Fig. -2 Trace of Spherical Surface O P' Pitch Cones of Bevel Gears γ 1 γ 2 ω 2 ω1 O' O" () Pitch Cones and the Development Sphere Fig Bevel Gear Tooth Proportions Spherical Basis of Octoid Bevel Crown Gear Bevel gear teeth are proportionen accordance with the standard system of tooth proportions used for spur gears. However, the pressure angle of all standard design evel gears is limited to. Pinions with a small numer of teeth are enlargeutomatically when the design follows the Gleason system. Since evel-tooth elements are tapered, tooth dimensions and pitch diameter are referenced to the outer end (heel). Since the narrow end of the teeth (toe) vanishes at the pitch apex (center of reference generating sphere), there is a practical limit to the length (face) of a evel gear. The geometry andentification of evel gear parts is given in Figure -5. The crown gear, which is a evel gear having the largest possile pitch angle (definen Figure -), is analogous to the rack of spur gearing, and is the asic tool for generating evel gears. However, for practical reasons, the tootorm is not that of a spherical involute, annstead, the crown gear profile assumes a slightly simplified form. Although the deviation from a true spherical involute is minor, it results in a line-of-action having a figure- trace in its extreme extension; see Figure -4. This shape gives rise to the name "octoid" for the tootorm of modern evel gears. Root Face Cone Dist. Face Pitch Apex to Back Pitch Apex to Crown Shaft Pitch Apex Pitch Pitch Crown to Back Whole Depth O 2 O 2 Pitch Dia. P O P O.D. Back Cone Dist. O 1 O 1 Fig. - Meshing Bevel Gear Pair with Conjugate Crown Gear Fig. -5 Bevel Gear Pair Design Parameters T2

2 . Velocity Ratio The velocity ratio, i, can e derived from the ratio of several parameters: d 1 sin i = = = (-1) d 2 sin where: δ = pitch angle (see Figure -5).4 Forms Of Bevel Teeth * In the simplest design, the tooth elements are straight radial, converging at the cone apex. However, it is possile to have the teeth curve along a spiral as they converge on the cone apex, resulting in greater tooth overlap, analogous to the overlapping action of helical teeth. The result is a spiral evel tooth. In addition, there are other possile variations. One is the zerol evel, which is a curved tooth having elements that start and end on the same radial line. Straight evel gears come in two variations depending upon the farication equipment. All current Gleason straight evel generators are of the Coniflex form which gives an almost imperceptile convexity to the tooth surfaces. Older machines produce true straight elements. See Figure -a. Straight evel gears are the simplest and most widely used type of evel gears (a) Straight Teeth for the transmission of power and/or motion etween intersecting shafts. Straight evel gears are recommended: 1. When speeds are less than 00 meters/min (00 feet/min) at higher speeds, straight evel gears may e noisy. 2. When loads are light, or for high static loads when surface wear is not a critical factor.. When space, gear weight, and mountings are a premium. This includes planetary gear sets, where space does not permit the inclusion of rolling-element earings. Other forms of evel gearing include the following: Coniflex gears (Figure -) are produced y current Gleason straight evel gear generating machines that crown the sides of the teeth in their lengthwise direction. The teeth, therefore, tolerate small amounts of misalignment in the assemly of the gears and some displacement of the gears under load without concentrating the tooth contact at the ends of the teeth. Thus, for the operating conditions, Coniflex gears are capale of transmitting larger loads than the predecessor Gleason straight evel gears. Spiral evels (Figure -c) have curved olique teeth which contact each () Coniflex Teeth (Exaggerated Tooth Curving) Fig. - (c) Spiral Teeth (d) Zerol Teeth other gradually and smoothly from one end to the other. Imagine cutting a straight evel into an infinite numer of short face width sections, angularly displace one relative to the other, and one has a spiral evel gear. Welldesigned spiral evels have two or more teeth in contact at all times. The overlapping tooth action transmits motion more smoothly and quietly than with straight evel gears. R Forms of Bevel Gear Teeth Zerol evels (Figure -d) have curved teeth similar to those of the spiral evels, ut with zero spiral angle at the middle of the face width; and they have little end thrust. Both spiral and Zerol gears can e cut on the same machines with the same circular face-mill cutters or ground on the same grinding machines. Both are produced with localized tooth contact which can e controlled for length, width, and shape. Functionally, however, Zerol evels are similar to the straight evels and thus carry the same ratings. In fact, Zerols can e usen the place of straight evels without mounting changes. Zerol evels are widely employen the aircraft industry, where groundtooth precision gears are generally required. Most hypoid cutting machines can cut spiral evel, Zerol or hypoid gears..5 Bevel Gear Calculations Let and e pinion and gear tooth numers; shaft angle ; and pitch cone angles and ; then: tan = + cos (-2) tan = + cos Generally, shaft angle = 0 is most used. Other angles (Figure -) are sometimes used. Then, it is called evel gear in nonright angle drive. The 0 case is called evel gear in right angle drive. When = 0, Equation (-2) ecomes: = tan ( ) 1 (-) = tan ( ) 1 Fig. - m The Pitch Cone of Bevel Gear Miter gears are evel gears with = 0 and =. Their speed ratio / = 1. They only change the direction of the shaft, ut do not change the speed. Figure - depicts the meshing of evel gears. The meshing must e consideren pairs. It is ecause the pitch cone angles and are restricted y the gear ratio /. In the facial view, which is normal to the contact line of pitch cones, the meshing of evel gears appears to e similar to the meshing of spur gears. m * The material in this section has een reprinted with the permission of McGraw Hill Book Co., Inc., New York, N.Y. from "Design of Bevel Gears" y W. Coleman, Gear Design and Applications, N. Chironis, Editor, McGraw Hill, New York, N.Y. 1, p. 5. T

3 .5.1 Gleason Straight Bevel Gears The straight evel gear has straight teetlanks which are along the surface of the pitch cone from the ottom to the apex. Straight evel gears can e groupento the Gleason type and the standard type. d d 2 R 2 0 δ R 1 h a h d 1 θ a Fig. - The Meshing of Bevel Gears δ In this section, we discuss the Gleason straight evel gear. The Gleason Company defined the tooth profile as: whole depth h =2.1m; top clearance c a = 0.1m; and working depth h w = 2.000m. The characteristics are: Design specified profile shifted gears: In the Gleason system, the pinion is positive shiftend the gear is negative shifted. The reason is to distriute the proper strength etween the two gears. Miter gears, thus, do not neeny shifted tooth profile. The top clearance is designed to e parallel The outer cone elements of two paired evel gears are parallel. That is to ensure that the top clearance along the whole tooth is the same. For the standard evel gears, top clearance is variale. It is smaller at the toe angger at the heel. Tale -1 shows the minimum numer of teeth to prevent undercut in the Gleason system at the shaft angle = 0. Tale -2 presents equations for designing straight evel gears in the Gleason system. The meanings of the dimensions anngles are shown in Figure -. All the equations in Tale -2 can also e applied to evel gears with any shaft angle. The straight evel gear with crowning in the Gleason system is calle Coniflex gear. It is manufactured y a special Gleason Coniflex machine. It can successfully eliminate poor tooth wear due to improper mounting and assemly. The first characteristic of a Gleason straight evel gear is its profile shifted tooth. From Figure -, we can see the positive tooth profile shift in the pinion. The tooth thickness at the root diameter of a Gleason pinion is larger than that of a standard straight evel gear. Tale -1 The Minimum Numers of Teeth to Prevent Undercut Pressure z Comination of Numers of Teeth 1 z2 (14.5 ) 2 / Over 2 2 / Over 2 2 / Over 1 2 / Over 5 25 / Over / Over 5 (25 ) 1 / Over 1 15 / Over 1 14 / Over 20 1 / Over 0 1 / Over 1 Fig Standard Straight Bevel Gears A evel gear with no profile shifted tooth is a standard straight evel gear. The applicale equations are in Tale -. These equations can also e applied to evel gear sets with other than 0 shaft angle..5. Gleason Spiral Bevel Gears Dimensions and s of Bevel Gears A spiral evel gear is one with a spiral tootlank as in Figure -. The spiral is generally consistent with the curve of a cutter with the diameter d c. The spiral angle β is the angle etween a generatrix element of the pitch cone and the tootlank. The spiral angle just at the tootlank center is called central spiral angle β m. In practice, spiral angle means central spiral angle. All equations in Tale - are dedicated for the manufacturing method of Spread Blade or of Single Side from Gleason. If a gear is not cut per the Gleason system, the equations will e different from these. The tooth profile of a Gleason spiral evel gear shown here has the whole depth h = 1.m; top clearance c a = 0.1m; and working depth h w = 1.00m. These Gleason spiral evel gears elong to a stu gear system. This is applicale to gears with modules m > 2.1. Tale -4 shows the minimum numer of teeth to avoid undercut in the Gleason system with shaft angle = 0 and pressure angle α n =. If the numer of teeth is less than 12, Tale -5 is used to determine the gear sizes. All equations in Tale - are also applicale to Gleason evel gears with any shaft angle. A spiral evel gear set requires matching of hands; left-hannd right-hans a pair. T4

4 No Tale -2 The Calculations of Straight Bevel Gears of the Gleason System Item Shaft Module Pressure Numer of Teeth Pitch Diameter Pitch Cone Symol Formula m α, d zm δ 1 tan 1 ( ) + cos Example Pinion Gear Cone Distance d 2 2sin.0204 Face Width It should e less than / or m 22 h a1 h a m h a2 0.40m 0.540m + cosδ () 1 cos 2.1m h a tan 1 ( / ) Outer Cone Root Cone Outside Diameter Pitch Apex to Crown θ a1 θ a2 2 1 δ + θ a δ d + 2h a cosδ cosδ h a sinδ Axial Face Width Inner Outside Diameter cos 2 sin No Tale - Item Shaft Module Pressure Numer of Teeth Pitch Diameter Pitch Cone Calculation of a Standard Straight Bevel Gears Example Symol Formula Pinion Gear m α, d zm tan ( 1 ) +cos Cone Distance d 2 2sin Face Width Outer Cone Root Cone Outside Diameter h a θ a It should e less than / or m 1.00 m 1.25 m tan 1 ( / ) tan 1 (h a / ) δ + θ a δ d + 2h a cosδ Pitch Apex to Crown cosδ h a sinδ Axial Face Width Inner Outside Diameter cos 2 sin T5

5 Gleason Straight Bevel Gear Pinion Gear Standard Straight Bevel Gear Pinion Gear Fig. - The Tooth Profile of Straight Bevel Gears d c β m 2 2 δ R v Fig. - Spiral Bevel Gear (Left-Hand) Tale -4 The Minimum Numers of Teeth to Prevent Undercut β m = 5 Pressure Comination of Numers of Teeth z2 1 / Over 1 1 / Over 1 15 / Over 1 14 / Over 20 1 / Over / Over 2 Tale -5 Dimensions for Pinions with Numers of Teeth Less than 12 Numer of Teeth in Pinion Numer of Teeth in Gear Working Depth Whole Depth Gear Pinion Circular Tooth Thickness of Gear Pressure Spiral Shaft NOTE: All values in the tale are ased on m = 1. s 2 h w h h a2 h a α n β m Over Over Over Over Over Over T

6 .5.4 Gleason Zerol Spiral Bevel Gears When the spiral angle β m = 0, the evel gear is calle Zerol evel gear. The calculation equations of Tale -2 for Gleason straight evel gears are applicale. They also should take care again of the rule of hands; left and right of a pair must e matched. Figure -12 is a left-hand Zerol evel gear. Fig. -12 Left-Hand Zerol Bevel Gear No Shaft Tale - Outside Radial Module Normal Pressure Spiral Numer of Teeth and Spiral Hand Radial Pressure Pitch Diameter Item Pitch Cone The Calculations of Spiral Bevel Gears of the Gleason System Symol m α n β m, α t d Formula tanα tan 1() n cosβ m zm tan ( 1 ) +cos Pinion Example 0 5 Gear 20 (L) 40 (R) Cone Distance Face Width d 2 2sin It should e less than / or m h a1 h a2 1.00m h a2 0.0m 0.40m + cosδ () 1 cos m h a tan 1 ( / ) Outer Cone Root Cone Outside Diameter Pitch Apex to Crown θ a1 θ a2 2 1 δ + θ a δ d + 2h a cosδ cosδ h a sinδ Axial Face Width Inner Outside Diameter cos 2 sin SECTION WORM MESH The worm mesh is another gear type used for connecting skew shafts, usually 0. See Figure -1. Worm meshes are characterized y high velocity ratios. Also, they offer the advantage of higher load capacity associated with their line contact in contrast to the point contact of the crossed-helical mesh. Fig. -1 Typical Worm Mesh T