Tài liệu Gears P2 pptx

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, B LA , along the line of action, is: B LA = 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 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: B LA (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) a B = 3440 ____ (arc minutes) R 1 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 TABLE 1.7 SUMMARY OF FUNDAMENTALS SPUR GEARS To Obtain From Known Symbol and Formula Pitch diameter Number of teeth and pitch D = N = N·P c P d π Circular Pitch Diametral pitch or number of teeth and pitch diameter P c = π =  π D P d N Diametral pitch Circular pitch or number of teeth and pitch diameter P d = π = N P c D Number of teeth Pitch and pitch diameter N =DP d = D P c Outside diameter Pitch and pitch diameter or pitch and number of teeth D o =D + 2 = N+2 P d P d Root diameter Pitch diameter and dedendum D R = D - 2b Base circle diameter Pitch diameter and pressure angle D b =D cos φ Base pitch Circular pitch and pressure angle P b = P c cos φ Tooth thickness at standard pitch diameter Circular pitch T std = P c = πD 2 2N Addendum Diametral pitch a = 1 P d Center distance Pitch diameters Or number of teeth and pitch C=D 1 +D 2 =N 1 +N 2 =P c (N1+N2) 2 2P d 2π Contact ratio Outside radii, base radii, center distance and pressure angle mp = (R o ²-R b ²) ½ +(r o ²-r b ²) ½ -C sin φ P c 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) along line of acvon Linear backlash along pitch cirde B LA = B cos φ Backlash, angular Linear backlash a B = 6880 B (arc minutes) D Minimum number of teeth for no undercutting Pressure angle N = 2 sin² φ Dedendum Pitch diameter and root diameter ( D R ) b = ½(D-D R ) Clearance Addendum and dedendum c = b - a Working depth Addendum h k = 2a Pressure angle ( standard ) Base circle diameter and pitch diameter φ =cos -1 D b /D Operating pressure angle Actual operating pitch diameter and base circle diameter φ =cos -1 D b /D' T49 TABLE 1.7 CONT. - SUMMARY OF FUNDAMENTALS HELICAL GEARING To Obtain From Known Symbol and Formula Normal circular pitch Transverse circular pitch P cn = P c cos ψ Normal diametral pitch Transverse diametral pitch P dn = P d cos ψ Axial pitch Circular pitches P a = P c cot ψ = P cn sin ψ Normal pressure angle Transverse pressure angle tan φ n = tan φ cos ψ Pitch diameter Number of teeth and pitch D = N = N P d P dn cos ψ Center distance (parallel shafts) Number of teeth and pitch C = N 1 + N 2 2 P dn cos ψ Center distance (crossed shafts) Number of teeth and pitch C = 1 ( N 1 + N 2 ) 2 P dn cos ψ 1  cos ψ 2 Shaft angle (Crssed shafts) Helix angles of 2 mated gears θ = ψ 1 + ψ 2 Addendum Pitch; or outside and pitch diameters a = 0.5 ( D o - D ) = 1 P d Dedendum Pitch diameter and root diameter (D R ) b = 0.5 ( D - D R ) Clearance Addendum and dedendum c = b-a Working depth Addendum h k = 2a Transverse pressure angle Base circle diameter and pitch circle diameter cos φ t = D b / D Pitch helix angle Number of teeth, normal diametral pitch and pitch diameter cos ψ = N P n D Lead Pitch diameter and pitch helix angle L = π D cos ψ INVOLUTE GEAR PAIRS To Obtain Symbols Spur or Helical Gears ( g gear; p = pinion) Length of action Z A Z A = (C² - (R b +r b )²) ½ (maximum) Z A = (R o ²-R b ²) ½ (r o ²-r b ²-C sin φ r ) ½ Start of active profile SAP SAP p = -(R o ²-R b ²) ½ SAP g = Zmax-(r o ²-r b ²) ½ Contact ratio R c Rcg = ((SAP)² + Rb²) ½ ; R cp = ((SAP)² + r b ²) ½ T50 TABLE 1.7 CONT. - SUMMARY OF FUNDAMENTALS WORM MESHES To Obtain From Known Symbol and Formula Pitch diameter of worm Number of teeth and pitch d w = n w P cn p sin λ Pitch diameter of worm gear Number of teeth and pitch D g = N g P cn π cos λ Lead angle Pitch, diameter, teeth λ = tan -1 n w = sin -1 n w P cn P d d w pd w Lead of worm Number of teeth and pitch L = n w p c = n w p cn cos λ Normal circular pitch Transverse pitch and lead angle P cn = P c cos λ Center distance Pitch diameters C = d w + D g 2 Center distance Pitch, lead angle, teeth C = P cn [ N g + n w ] 2π cos λ sin λ Velocity ratio Number of teeth Z = Ng n w BEVEL GEARING To Obtain From Known Symbol and Formula Velocity ratio Number of teeth Z = N 1 N 2 Velocity ratio Pitch diameters Z = D 1 D 2 Velocity ratio Pitch angles Z = sin γ 1 sin γ 2 Shaft angle Pitch angles Σ = γ 1 + γ 2 T51 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 A o B o . 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 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: P cn = P c 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: P dn = P d = 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: P a = P c cot ψ = P cn = axial pitch (27) sin ψ T53 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: N V = 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 φ n (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 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) P d P dn cos ψ 5.8.3 Center distance — Utilizing equation 30, the center distance of a helical gear mesh is: C = ( N 1 +N 2 ) (31) 2 P dn 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. (m p ) total = (m p ) trans + (m p ) axial (32) where T55 (m p ) trans = value per equation 18 (m p ) axial = F = F tan ψ = F sin ψ P a P c P cn 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: ψ 1 + ψ 2 = θ (33) where: ψ 1 , ψ 2 = 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 New Page 4 file:///C|/A3/D190/HTML/D190T56.htm [9/30/2000 10:10:44 AM] [...]... 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... in worm Ng (43) 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... 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. .. see Figure 1.44 This shape gives rise to the name "octoid" for the tooth form of modem bevel gears T64 T65 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... 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 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:... 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 6.0 RACKS Gear racks (Figure 1.31) are important components in that they are a means of converting rotational motion into linear... 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... 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 7.2 Tooth Parts of Internal Gear Because the... a throated shape to wrap around the Worm T61 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... 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 Tooth thickness of the internal gear can be calculated with equations 9 and 20, but one . 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. 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