# Gears P2

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## Gears P2

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When the center distance is increased by a relatively small amount, ∆C, a backlash space develops between mating teeth, as in Figure 1.21. The relationship between center distance increase and linear backlash, BLA, along the line of action, is: BLA

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1. When the center distance is increased by a relatively small amount, ∆C, a backlash space develops between mating teeth, as in Figure 1.21. The relationship between center distance increase and linear backlash, BLA, along the line of action, is: BLA = 2(∆C)sin φ (21) This measure along the line-of-action is useful when inserting a feeler gage between teeth to measure backlash. The equivalent linear backlash measured along the pitch circle is given by: B = 2(∆C) tan φ (22a) where: ∆C = change in center distance φ = pressure angle Hence, an approximate relationship between center distance change and change in backlash is: ∆C= 1.933 ∆B for 14½° pressure-angle gears (22b) ∆C= 1.374 ∆B for 20° pressure-angle gears (22c) T47
2. Although these are approximate relationships they are adequate for most uses. Their derivation, limitations, and correction factors are detailed in Reference 5. Note that backlash due to center distance opening is dependent upon the tangent function of the pressure angle. Thus, 20° gears have 41% more backlash than 14½º gears, and this constitutes one of the few advantages of the lower pressure angle. Equations 22 are a useful relationship, particularly for converting to angular backlash. Also for fine-pitch gears the use of feeler gages for measurement is impractical, whereas an indicator at the pitch line gives a direct measure. The two linear backlashes are related by: BLA (23) B = _____ cos φ The angular backlash at the gear shaft is usually the critical factor in the gear application. As seen from Figure 1.20a this is related to the gear’s pitch radius as follows: B (24) ____ (arc minutes) aB = 3440 R1 Obviously, angular backlash is inversely proportional to gear radius. Also, since the two meshing gears are usually of different pitch diameters, the linear backlash of the measure converts to different angular values for each gear. Thus, an angular backlash must be specified with reference to a particular shaft or gear center. 4.11 Summary of Gear Mesh Fundamentals The basic geometric relationships of gears and meshed pairs given in the above sections are summarized in Table 1.7. T48
3. TABLE 1.7 SUMMARY OF FUNDAMENTALS SPUR GEARS To Obtain From Known Symbol and Formula D = N = N·Pc Pitch diameter Number of teeth and pitch Pd π Diametral pitch or number of Pc =π= π D Circular Pitch teeth and pitch diameter Pd N Circular pitch or number of Pd = π=N Diametral pitch teeth and pitch diameter Pc D N =DPd = D Number of teeth Pitch and pitch diameter Pc Pitch and pitch diameter or Do =D + 2 = N+2 Outside diameter pitch and number of teeth Pd Pd Root diameter Pitch diameter and dedendum DR = D - 2b Base circle diameter Pitch diameter and pressure angle Db=D cos φ Base pitch Circular pitch and pressure angle Pb = Pc cos φ Tooth thickness at Tstd = Pc = πD Circular pitch standard pitch diameter 2 2N a= 1 Addendum Diametral pitch Pd Pitch diameters Or number C=D1+D2=N1+N2=Pc(N1+N2) Center distance of teeth and pitch 2 2Pd 2π Outside radii, base radii, center mp = (Ro²-Rb²)½+(ro²-rb²)½-C sin φ Contact ratio distance and pressure angle Pc cos φ Backlash (linear) From change in center distance B = 2 (∆C) tan φ Backlash (linear) From change in tooth thickness B = ∆T Backlash (linear) BLA = B cos φ Linear backlash along pitch cirde along line of acvon Backlash, angular Linear backlash aB = 6880 B (arc minutes) D Minimum number of N= 2 Pressure angle teeth for no undercutting sin² φ Pitch diameter and Dedendum b = ½(D-DR) root diameter ( DR ) Clearance Addendum and dedendum c=b-a Working depth Addendum hk = 2a Pressure angle Base circle diameter and pitch φ =cos-1 Db/D ( standard ) diameter Operating pressure Actual operating pitch diameter φ =cos-1 Db/D' angle and base circle diameter T49
4. TABLE 1.7 CONT. - SUMMARY OF FUNDAMENTALS HELICAL GEARING To Obtain From Known Symbol and Formula Normal circular pitch Transverse circular pitch Pcn = Pc cos ψ Pdn = Pd Normal diametral pitch Transverse diametral pitch cos ψ Pa = Pc cot ψ = Pcn Axial pitch Circular pitches sin ψ Normal pressure angle Transverse pressure angle tan φn = tan φ cos ψ D = N = N Pitch diameter Number of teeth and pitch Pd Pdn cos ψ Center distance C = N1 + N2 Number of teeth and pitch (parallel shafts) 2 Pdn cos ψ Center distance C= 1 ( N1 + N2 ) Number of teeth and pitch (crossed shafts) 2 Pdn cos ψ1 cos ψ2 Shaft angle θ = ψ1 + ψ2 Helix angles of 2 mated gears (Crssed shafts) Pitch; or outside and pitch a = 0.5 ( Do - D ) = 1 Addendum diameters Pd Pitch diameter and root Dedendum b = 0.5 ( D - DR ) diameter (DR) Clearance Addendum and dedendum c = b-a Working depth Addendum hk = 2a Transverse pressure Base circle diameter and cos φt = Db / D angle pitch circle diameter Number of teeth, cos ψ = N Pitch helix angle normal diametral pitch and Pn D pitch diameter Pitch diameter and Lead L = π D cos ψ pitch helix angle INVOLUTE GEAR PAIRS To Obtain Symbols Spur or Helical Gears ( g gear; p = pinion) ZA = (C² - (Rb+rb)²)½ (maximum) Length of action ZA ZA = (Ro²-Rb²)½ (ro²-rb²-C sin φr)½ SAPp = -(Ro²-Rb²)½ Start of active profile SAP SAPg = Zmax-(ro²-rb²)½ Contact ratio Rc Rcg = ((SAP)² + Rb²)½; Rcp = ((SAP)² + rb²)½ T50
5. TABLE 1.7 CONT. - SUMMARY OF FUNDAMENTALS WORM MESHES To Obtain From Known Symbol and Formula dw = nw Pcn Pitch diameter of worm Number of teeth and pitch p sin λ Pitch diameter of Dg = Ng Pcn Number of teeth and pitch worm gear π cos λ λ = tan -1 nw = sin-1 nw Pcn Lead angle Pitch, diameter, teeth Pddw pdw L = nwpc = nw pcn Lead of worm Number of teeth and pitch cos λ Normal circular pitch Transverse pitch and lead angle Pcn = Pc cos λ C = dw + Dg Center distance Pitch diameters 2 C = Pcn [ Ng + nw ] Center distance Pitch, lead angle, teeth 2π cos λ sin λ Z = Ng Velocity ratio Number of teeth nw BEVEL GEARING To Obtain From Known Symbol and Formula Z = N1 Velocity ratio Number of teeth N2 Z = D1 Velocity ratio Pitch diameters D2 Z = sin γ1 Velocity ratio Pitch angles sin γ2 Shaft angle Pitch angles Σ = γ1 + γ2 T51
6. 5.0 HELICAL GEARS The helical gear differs from the spur gear in that its teeth are twisted along a helical path in the axial direction. It resembles the spur gear in the plane of rotation, but in the axial direction it is as if there were a series of staggered spur gears. See Figure 1.22. This design brings forth a number of different features relative to the spur gear, two of the most important being as follows: 1. tooth strength is improved because of the elongated helical wrap around tooth base support. 2. contact ratio is increased due to the axial tooth overlap. Helical gears thus tend to have greater load-carrying capactiy than spur gears of the same size. Spur gears, on the other hand, have a somewhat higher efficiency. Helical gears are used in two forms: 1. Parallel shaft applications, which is the largest usage. 2. Crossed-helicals (or spiral gears) for connecting skew shafts, usually at tight angles. 5.1 Generation of the Helical Tooth The helical tooth form is involute in the plane of rotation and can be developed in a manner similar to that of the spur gear. However, unlike the spur gear which can be viewed essentially as two dimensional, the helical gear must be portrayed in three dimensions to show changing axial features. Referring to Figure 1.23, there is a base cylinder from which a taut plane is unwrapped, analogous to the unwinding taut string of the spur gear in Figure 12. On the plane there is a straight line AB, which when wrapped on the base cylinder has a helical trace AoBo. As the taut plane is unwrapped any point on the line AB can be visualized as tracing an involute from the base cylinder. Thus, there is an infinite series of involutes generated by line AB, all alike, but displaced in phase along a helix on the base cylinder. Again a concept analogous to the spur-gear tooth development is to imagine the taut plane being wound from one base cylinder on to another as the base cylinders rotate in opposite directions. The result is the generation of a pair of conjugate helical involutes. If a reverse direction of rotation is assumed and a second tangent plane is arranged so that it crosses the first, a complete involute helicoid tooth is formed. T52
7. 5.2 Fundamental of Helical Teeth In tho piano of rotation the helical gear tooth is involute and all of the relationships govorning spur gears apply to the helical. However, tho axial twist of the teeth introduces a holix anglo. Since the helix angle varies from the base of the tooth to the outside radnjs, the helix angle, w~ is detned as the angle between the tangent to the helicoidal tooth at the intersection of the pitch cylinder and the tooth profile, and an element of the pitch cylinder. See Figure 1.24. The direction of the helical twist is designated as either left or right. The direction is defined by the right-hand rule. 5.3 Helical Gear Relationships For helical gears there are two related pitches: one in the plane of rotation and the other in a plane normal to the tooth. In addition there is an axial pitch. These are defined and related as follows: Referring to Figure 1.25, the two circular pitches are related as follows: Pcn = Pc cos ψ = normal circular pitch (25) The normal circular pitch is less than the transverse or circular pitch in the plane of rotation, the ratio between the two being equal to the cosine of the helix angle. Consistent with this, the normal diametral pitch is greater than the transverse pitch: Pdn = Pd = normal diametral pitch (26) cos ψ The axial pitch of a helical gear is the distance between corresponding points of adjacent teeth measured parallel to the gears axis—see Figure 1.26. Axial pitch, p1. is related to circular pitch by the expressions: Pa = Pc cot ψ = Pcn = axial pitch (27) sin ψ T53
8. 5.4 Equivalent Spur Gear The true involute pitch and involute geometry of a helical gear is that in the plane of rotation. However, in the normal plane, looking at one tooth, there is a resemblance to an involute tooth of a pitch corresponding to the normal pitch. However, the shape of the tooth corresponds to a spur gear of a larger number of teeth, the exact value depending on the magnitude of the helix angle. The geometric basis of deriving the number of teeth in this equivalent tooth form spur gear is given in Figure 1.27. The result of the transposed geometry is an equivalent number of teeth given as: NV = N (28) cos³ψ This equivalent number is also called a virtual number because this spur gear is imaginary. The value of this number is its use in determining helical tooth strength. 5.5 Pressure Angle Although strictly speaking, pressure angle exists only for a gear pair, a nominal pressure angle can be considered for an individual gear. For the helical gear there is a normal pressure angle as well as the usual pressure angle in the plane of rotation. Figure 1.28 shows their relationship, which is expressed as: tan φ = tan φn (29) cos ψ 5.6 Importance of Normal Plane Geometry Because of the nature of tooth generation with a rack-type hob, a single tool can generate helical gears at all helix angles as well as spur gears. However, this means the normal pitch is the common denominator, and usually is taken as a standard value. Since the true involute features are in the transverse plane, they will differ from the standard normal values. Hence, there is a real need for relating parameters in the two reference planes. T54 f
9. 5.7 Helical Tooth Proportions These follow the same standards as those for spur gears. Addendum, dedendum, whole depth and clearance are the same regardless of whothor measured in tho piano of rotation er the normal piano. Pressure angle and pitch are usually specified as standard values in tho normal plane, but there are times when they are specified standard in the transverse plane. 5.8 Parallel Shaft Helical Gear Meshes Fundamental information for the design of gear meshes is as follows: 5.8.1 Helix angle — Both gears of a meshed pair must have the same helix angle. However, the helix directions must be opposite, i.e., a left-hand mates with a right-hand helix. 5.8.2 Pitch dIameter — This is given by the same expression as for spur gears, but if the normal pitch is involved it is a function of the helix angle. The expressions are: D=N= N (30) Pd Pdn cos ψ 5.8.3 Center distance — Utilizing equation 30, the center distance of a helical gear mesh is: C = ( N1+N2 ) (31) 2 Pdn cos ψ Note that for standard parameters in the normal plane, the center distance will not be a standard value compared to standard spur gears. Further, by manipulating the helix angle (ψ) the center distance can be adjusted over a wide range of values. Conversely, it is possible a. to compensate for significant center distance changes (or erçors) without changing the speed ratio between parallel geared shafts; and b. to alter the speed ratio between parallel geared shafts without changing center distance by manipulating helix angle along with tooth numbers. 5.8.4 Contact Ratio — The contact ratio of helical gears is enhanced by the axial overlap of the teeth. Thus, the contact ratio is the sum of the transverse contact ratio, calculated in the same manner as for spur gears (equation 18), and a term involving the axial pitch. (mp)total = (mp)trans + (mp)axial (32) where T55
10. New Page 4 (mp)trans = value per equation 18 (mp)axial = F = F tan ψ = F sin ψ Pa Pc Pcn and F = face width of gear. 5.8.5 Involute interference — Helical gears cut with standard normal pressure angles can have considerably higher pressure angles in the plane of rotation (see equation 29), depending on the helix angle. Therefore, referring to equation 19, the minimum number of teeth without undercutting can be significantly reduced and helical gears having very low tooth numbers without undercutting are feasible. 5.9 Crossed Helical Gear Meshes These are also known as spiral and screw gears. They are used for interconnecting skew shafts, such as in Figure 1.29. They can be designed to connect shafts at any angle, but in most applications the shafts are at right angles. 5.9.1 Helix angle and hands — The helix angles need not be the same. However, their sum must equal the shaft angle: ψ1 + ψ2 = θ (33) where: ψ1, ψ2 = the respective helix angles of the two gears θ = shaft angle (the acute angle between the two shafts when viewed in a direction parallel ing a common perpendicular between the shafts) Except for very small shaft angles, the helix hands are the same. 5.9.2 Pitch — Because of the possibility of ditferent helix angles for the gear pair, the transverse pitches may not be the same. However, the normal pitches must always be identical. T56 file:///C|/A3/D190/HTML/D190T56.htm [9/30/2000 10:10:44 AM]
11. 5.9.3 Center Distance — The pitch diameter of a crossed-helical gear is given by equation 30, and the center distance becomes: C = 1 ( N1 + N2 ) (34) 2Pdn cos ψ1 cos ψ2 Again it is possible to adjust the center distance by manipulating the helix angle. However, both gear helix angles must be altered consistently in accordance with equation 33. 5.9.4 Velocity ratio — Unlike spur and parallel shaft helical meshes the velocity ratio (gear ratio) cannot be determined from the ratio of pitch diameters, since these can be altered by juggling of helix angles. The speed ratio can be determined only from the number of teeth as follows: velocity ratio Z = N1 (35) N2 or if pitch diameters are introduced the relationship is: Z = D1 cos ψ1 (36) D2 cos ψ2 5.10 Axial Thrust of Helical Gears In both parallel-shaft and crossed shaft applications helical gears develop an axial thrust load. This is a useless force that loads gear teeth and bearings and must accordingly be considered in the housing and bearing design. In some special instrument designs this thrust load can be utilized to actuate face clutches, provide a friction drag, or other special purpose. The magnitude of the thrust load depends on the helix angle and is given by the expression: WT =Wt tanψ (37) where: WT = axial thrust load Wt = transmitted load The direction of the thrust load is related to the hand of the gear and the direction of rotation. This is depicted in Figure 1.29. When the helix angle is larger than about 20°, the use of double helical gears with opposite hands (Figure 1 .30b) or herringbone gears (Figure 1.30a) is worth considering. T57
12. 6.0 RACKS Gear racks (Figure 1.31) are important components in that they are a means of converting rotational motion into linear motion. In theory the rack is a gear with infinite pitch diameter, resulting in an involute profile that is essentially a straight line, and the tooth is of simple V form. Racks can be both spur and helical. A rack will mesh with all gears of the same pitch. Backlash is computed by the same formula as for gear pairs, equation 22. However, the pressure angle and the gears pitch radius remain constant regardless of changes in the relative position of the gear and rack. Only the pitch line shifts accordingly as the gear center is altered. See Figure 1.32. 7.0 INTERNAL GEARS A special feature of spur and helical gears is their capability of being made in an internal form, in which an internal gear mates with an ordinary external gear. This offers considerable versatility in the design of planetary gear trains and miscellaneous instrument packages. 7.1 Development of the Internal Gear The gears considered so far can be imagined as equivalent pitch circle friction discs which roll on each other with external contact If instead, one of the pitch circles rolls on the inside of the ether, it forms the basis of internal gearing. In addition, the larger gear must have the material forming the teeth on the convex side of the involute profile, such that the internal gear is an inverse of the common external gear, see Figure 1.33a. The base circles, line of action and development of the involute profiles and action are shown in Figure 1.33b. As with spur gears there is a taut generating string that winds and unwinds between the base circles. However, in this case the string does not cross the line of centers, and actual contact and involute development occurs on an extension of the common tangent. Otherwise, action parallels that for external spur gears. T58
13. 7.2 Tooth Parts of Internal Gear Because the internal gear is reversed relative to the external gear, the tooth parts are also reversed relative to the ordinary (external) gear. This is shown in Figure 1.34. Tooth proportions and standards are the same as for external gears except that the addendum of the gear is reduced to avoid trimming of the teeth in the fabrication process. T59
14. Tooth thickness of the internal gear can be calculated with equations 9 and 20, but one must remember that the tooth and space thicknesses are reversed, (see Figure 1.35). Also, in using equation 10 to calculate tooth thickness at various radii, (see Figure 1.36), it is the tooth space that is calculated and the internal gear tooth thickness is obtained by a subtraction from the circular pitch at that radius, Thus, applying equation 10 to Figure 1.36, 7.3 Teeth Thickness Measurement In a procedure similar to that used for external gears, tooth thickness can be measured indirectly by gaging with pins, but this time the measurement is "under" the pins, as shown in Figure 1.37. Equations 11 thru 13 are modified accordingly to yield: For an even number of teeth: M= 2 ( Rc - dw ) (38) 2 For an odd number of teeth: M = 2(Rc cos 90º - dw ) (39) N 2 inv φ1=inv φ + π - T - dw N D Dcos φ where: Rc = cos φ R cos φ1 T60
15. 7.4 Features of Internal Gears General advantages: 1. Lend to compact design since the center distance is less than for external gears. 2. A high contact ratio is possible. 3. Good surface endurance due to a convex profile surface working against a concave surface. General disadvantages: 1. Housing and bearing supports are more complicated, because the external gear nests within the internal gear. 2. Low velocity ratios are unsuitable and in many cases impossible because of interferences. 3. Fabrication is limited to the shaper generating process, and usually special tooling is required. 8.0 WORM MESH The worm mesh is another gear type used for connecting skew shafts, usually 90º, see Figure 1.38. Worm meshes are characterized by high velocity ratios. Also, they offer the advantage of the higher load capacity associated with their line contact in contrast to the point contact of the crossed-helical mesh 8.1 Worm Mesh Geometry The worm is equivalent to a V-type screw thread, as evident from Figure 1.39. The mating worm-gear teeth have a helical lead. A central section of the mesh, taken through the worm’s axis and perpendicular to the wormgear’s axis, as shown in Figure 1.39, reveals a rack-type tooth for the worm, and a curved involute tooth form for the wormgear. However, the involute features are only true for the central section. Sections on either side of the worm axis reveal non-symmetric and non-involute tooth profiles. Thus, a worm-gear mesh is not a true involute mesh. Also, for conjugate action the center distance of the mesh must be an exact duplicate of that used in generating the wormgear. To increase the length of action the wormgear is made of a throated shape to wrap around the Worm. T61
16. 8.2 Worm Tooth Proportions Worm tooth dimensions, such as addendum, dedendum, pressure angle, etc., follow the same standards as those for spur and helical gears. The standard values apply to the central section of the mesh, (see Figure 1.40a). A high pressure angle is favored and in some applications values as high as 25º and 30° are used. 8.3 Number of Threads The worm can be considered resembling a helical gear with a high helix angle. For extremely high helix angles, there is one continuous tooth or thread. For slightly smaller angles them can be two, three, or even more threads. Thus, a worm is characterized by the number of threads, nw. 8.4 Worm and Wormgear Calculations Referring to Figure 1.40b and recalling the relationships established for normal and transverse pitches in Par.5, the following defines the geometry of worm mesh components. T62
17. 8.4.1 Pitch Diameters, Lead snd Lead Angle Pitch diameter of worm = dw = nw Pcn (40) π sin λ Pitch diameter of wormgear = Dg = Ng Pcn (41) π cos λ where: nw = number of threads of worm L = lead of worm = nwpc = nw Pcn cos λ λ = lead angle = tan-1 nw Pddw = sin-1 nw Pcn πdw Pcn = Pc cos λ 8.4.2 Center Distance of Mesh c = dw + Dg = Pcn [ Ng + nw ] (42) 2 2π cos λ sin λ T63
18. 8.5 Velocity Ratio The gear ratio of a worm mesh cannot be calculated from the ratio of the pitch diameters. It can be determined only from the ratio of tooth numbers: velocity ratio = Z = no. teeth in worm gear = Ng (43) no. threads in worm 9.0 BEVEL GEARING For intersecting shafts, bevel gears offer a good means of transmitting motion and power. Most transmissions occur at right angles (Figure 1.41), but the shaft angle can be any value. Ratios up to 4:1 are common, although higher ratios are possible as well. 9.1 Development and Geometry of Bevel Gears Bevel gears have tapered elements because they can be generated by rolling cones, their pitch surfaces lying on the surface of a sphere. Pitch diameters of mating bevel gears belong to frusta of cones, as shown in Figure 1.42. In the full development on the surface of a sphere, a pair of meshed bevel gears and a crown gear are in conjugate engagement as shown in Figure 1.43. The crown gear, which is a bevel gear having the largest possible pitch angle (defined in Figure 1.43), is analogous to the rack of spur gearing, and is the basic tool for generating bevel gears. However, for practical reasons the tooth form is not that of a spherical involute, and instead, 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-S trace in its extreme extension, see Figure 1.44. This shape gives rise to the name "octoid" for the tooth form of modem bevel gears. T64
19. T65
20. 9.2 Bevel Gear Tooth Proportions Bevel gear teeth are proportioned in accordance with the standard system of tooth proportions used for spur gears. However, the pressure angle of all standard design bevel gears is limited to 200. Pinions with a small number of teeth are enlarged automatically when the design follows the Gleason system. Since bevel-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 bevel gear. The geometry and identification of bevel gear parts is given in Figure 1.45. 9.3 Velocity Ratio The velocity ratio can be derived from the ratio of several parameters: velocity ratio = Z = N1 = D1 = sin γ1 (44) N2 D2 sin γ2 where: γ = pitch angle (Figure 1.45) T66