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HELICAL GEARS 10.16
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GEARING
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FIGURE 10.8 Pinion proportion factor Cpf. (From Ref. [10.1].)
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become a matched set). Only two values are permissible for Ce either 1.0 or 0.8, as defined by the following requirements: 0.80 Ce = 0.80 1.0 when the compatibility of the gearing is improved by lapping, grinding, or skiving after trial assembly to improve contact when gearing is adjusted at assembly by shimming support bearings and/or housing to yield uniform contact for all other conditions
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If enough detailed information is available, a better estimate of the loaddistribution factor may be obtained by using a more analytical approach. This method, however, requires that the total alignment error et be calculated or estimated. Depending on the contact conditions, one of two expressions is used to calculate the load-distribution factor.
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FIGURE 10.9 Definition of distances S and S1. Bearing span is distance S; pinion offset from midspan is S1. (From Ref. [10.1].)
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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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HELICAL GEARS 10.17
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FIGURE 10.10 Mesh alignment factor Cma. For analytical method for determination of Cma , see Eq. (10.21). (From Ref. [10.1].)
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If the tooth contact pattern at normal operating load essentially covers the entire available tooth face, Eq. (10.22) should be used. If the tooth contact pattern does not cover the entire available tooth face (as would be the case for poorly aligned or high-crowned gears) at normal operating loads, then Eq. (10.23) must be used: Cm = 1.0 + and Cm = where GetZF Wt pb (10.23) GetZF 4Wt pb (10.22)
Wt = tangential tooth load, pounds (lb) G = tooth stiffness constant, (lb/in)/in of face Z = length of line of contact in transverse plane et = total effective alignment error, in/in pb = transverse base pitch, in F = net face width of narrowest member, in
The value of G will vary with tooth proportions, tooth thickness, and material. For steel gears of standard or close to standard proportions, it is normally in the range of 1.5 106 to 2.0 106 psi. The higher value should be used for higher-pressure-angle teeth, which are normally stiffer, while the lower value is representative of more flexible teeth. The most conservative approach is to use the higher value in all cases.
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HELICAL GEARS 10.18
GEARING
For double-helix gears, each half should be analyzed separately by using the appropriate values of F and et and by assuming that half of the tangential tooth load is transmitted by each half (the values for pb, Z, and G remain unchanged). Geometry Factor I. The geometry factor I evaluates the radii of curvature of the contacting tooth profiles based on the pressure angle, helix, and gear ratio. Effects of modified tooth proportions and load sharing are considered. The I factor is defined as follows: I= where Cc = Cx = C = mN =
2 CcCxC mN
(10.24)
curvature factor at operating pitch line contact height factor helical overlap factor load-sharing ratio NG cos o sin o 2 NG + NP cos o sin o NG 2 NG NP
The curvature factor is Cc = (10.25)
for external mesh; for internal mesh, Cc = (10.26)
The contact height factor Cx adjusts the location on the tooth profile at which the critical contact stress occurs (i.e., face-contact ratio > 1.0). The stress is calculated at the mean diameter or the middle of the tooth profile. For low-contact-ratio helical gears (that is, face-contact ratio 1.0), the stress is calculated at the lowest point of single-tooth contact in the transverse plane and Cx is given by Eq. (10.27): Cx = where R1R2 RPRG (10.27)
RP = pinion curvature radius at operating pitch point, in RG = gear curvature radius at operating pitch point, in R1 = pinion curvature radius at critical contact point, in R2 = gear curvature radius at critical contact point, in
The required radii are given by RP = where d sin o 2 RG = D sin o 2 (10.28)
d = pinion operating pitch diameter, in D = gear operating pitch diameter, in o = operating pressure angle in transverse plane, deg R1 = RP Zc (10.29)
and R2 = RG + Zc (10.30)
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