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2 2 Z = (r o r b)1/2 + (R 2 R 2 )1/2 Co sin o o b
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cos T Pd
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(10.2)
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This equation is for an external gear mesh. For an internal gear mesh, the length of the line of action is
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2 Z = (R 2 R 2)1/2 (r 2 r b)1/2 + Co sin o l b o
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where Pd = transverse diametral pitch as manufactured T = transverse pressure angle as manufactured, degrees (deg) ro = effective pinion outside radius, inches (in) Ro = effective gear outside radius, in Rl = effective gear inside radius, in o = operating transverse pressure angle, deg rb = pinion base radius, in Rb = gear base radius, in Co = operating center distance, in The operating transverse pressure angle o is o = cos 1 C cos T Co (10.5)
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The manufactured center distance C is simply C= NP + NG 2Pd (10.6)
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for external mesh; for internal mesh, the relation is C= NG NP 2Pd (10.7)
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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.
HELICAL GEARS 10.8
GEARING
The contact ratio mP in the transverse plane (profile-contact ratio) is defined as the ratio of the total length of the line of action in the transverse plane Z to the base pitch in the transverse plane pb. Thus mP = Z pb (10.8)
The diametral pitch, pitch diameters, helix angle, and normal pressure angle at the operating pitch circle are required in the load-capacity evaluation of helical gears. These terms are given by Pdo = for external mesh; for internal mesh, Pdo = Also, d= NP Pdo D= NG Pdo (10.11) NG NP 2Co (10.10) NP + NG 2Co (10.9)
B = tan 1 (tan cos T) o = tan 1 tan B cos o
(10.12)
(10.13)
No = sin 1 (sin o cos B) where Pdo = operating diametral pitch B = base helix angle, deg o = helix angle at operating pitch point, deg No = operating normal pressure angle, deg d = operating pinion pitch diameter, in D = operating gear pitch diameter, in
(10.14)
10.5 LOAD RATING
Reference [10.1] establishes a coherent method for rating external helical and spur gears. The treatment of strength and durability provided here is derived in large part from this source. Four factors must be considered in the load rating of a helical-gear set: strength, durability, wear resistance, and scoring probability. Although strength and durability must always be considered, wear resistance and scoring evaluations may not be required for every case. We treat each topic in some depth.
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.
HELICAL GEARS 10.9
HELICAL GEARS
10.5.1 Strength and Durability The strength of a gear tooth is evaluated by calculating the bending stress index number at the root by st = where WtKa Pd KbKm Kv FE J (10.15)
st = bending stress index number, pounds per square inch (psi) Ka = bending application factor FE = effective face width, in Km = bending load-distribution factor Kv = bending dynamic factor J = bending geometry factor Pd = transverse operating diametral pitch Kb = rim thickness factor
The calculated bending stress index number st must be within safe operating limits as defined by st where satKL KTKR (10.16)
sat = allowable bending stress index number KL = life factor KT = temperature factor KR = reliability factor
Some of the factors which are used in these equations are similar to those used in the durability equations. Thus we present the basic durability rating equations before discussing the factors: sc = Cp where Wt Ca 1 Cm Cv dFN I (10.17)
sc = contact stress index number Ca = durability application factor Cv = durability dynamic factor d = operating pinion pitch diameter FN = net face width, in Cm = load-distribution factor Cp = elastic coefficient I = durability geometry factor
The calculated contact stress index number must be within safe operating limits as defined by sc where sacCLCH CTCR (10.18)
sac = allowable contact stress index number CL = durability life factor CH = hardness ratio factor CT = temperature factor CR = reliability factor
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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