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HELICAL GEARS 10.19
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for external gears; for internal gears, R2 = RG Zc (10.31)
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where Zc is the distance along the line of action in the transverse plane to the critical contact point. The value of Zc is dependent on the transverse contact ratio. For helical gears where the face-contact ratio 1.0, Zc is found by using Eq. (10.32). For normal helical gears where the face-contact ratio is > 1.0, Eq. (10.33) is used: Zc = pb 0.5[(d 2 d 2 )1/2 (d2 d 2 )1/2] o b b and
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2 Zc = 0.5 [(d2 d 2 )1/2 (d m d 2 )1/2] b b
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(10.32)
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mF > 1.0
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(10.33)
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where pb = do = db = dm =
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base pitch, in pinion outside diameter, in pinion base diameter, in pinion mean diameter, in
The pinion mean diameter is defined by Eq. (10.34) or (10.35). For external mesh, dm = Co For internal mesh, dm = DI + do Co 2 (10.35) Do do 2 (10.34)
where Do = external gear outside diameter and DI = internal gear inside diameter. The helical factor C accounts for the partial helical overlap action which occurs in helical gears with a face-contact ratio mF 1.0. For helical gears with a facecontact ratio > 1.0, C is set equal to unity; for low-contact helical gears, it is C = where Z= F= mF = Cx = Cxn = b = 1 mF +
2 Cx n Zm F CxF sin b
(10.36)
total length of line of action in transverse plane, in net minimum face width, in face-contact ratio contact height factor [Eq. (10.27)] contact height factor for equivalent normal helical gears [Eq. (10.37)] base helix angle, deg
The Cxn factor is given by Cxn = R1nR2n RPRG (10.37)
where R1n = curvature radius at critical point for equivalent normal helical pinion, in R2n = curvature radius at critical contact point for equivalent normal helical gear, in
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HELICAL GEARS 10.20
GEARING
The curvature radii are given by R1n = RP Zc R2n = RG + Z R2n = RG Zc external gears (10.39) internal gears (10.38)
where Eq. (10.38) applies to external gears and Eq. (10.39) to either, as appropriate. Also, the term Zc is obtained from Eq. (10.32). The load-sharing ratio mN is the ratio of the face width to the minimum total length of the contact lines: mN = where F Lmin (10.40)
mN = load-sharing ratio F = minimum net face width, in Lmin = minimum total length of contact lines, in
The calculation of Lmin is a rather involved process. For most helical gears which have a face-contact ratio of at least 2.0, a conservative approximation for the loadsharing ratio ratio mN may be obtained from mN = PN 0.95Z (10.41)
where pN = normal circular pitch in inches and Z = length of line of action in the transverse plane in inches. For helical gears with a face-contact ratio of less than 2.0, it is imperative that the actual value of Lmin be calculated and used in Eq. (10.40). The method for doing this is shown in Eqs. (10.42) through (10.45): Lmin = 1 [(P1 Q1) + (P2 Q2) + sin b + (Pi Qi) + + (Pn Qn)] where n = limiting number of lines of contact, as given by n= (Z/tan b) + F px (10.43) (10.42)
Also, Pi = sum of base pitches in inches. The ith term of Pi is the lesser of ipx tan b or Z (10.44)
Finally, Qi = remainder of base pitches in inches. Its value is Qi = 0 if ipx F
But when ipx > F, then Qi is the ith term and is the lesser of (ipx F) tan b or Z (10.45)
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