29-12-2012, 06:45 PM
Modified involute helical gears: computerized design, simulation of meshing and stress analysis
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Abstract
The contents of the paper cover: (i) computerized design, (ii) methods for generation, (iii) simulation of meshing, and
(iv) enhanced stress analysis of modified involute helical gears. The approaches proposed for modification of conventional
involute helical gears are based on conjugation of double-crowned pinion with a conventional helical involute
gear. Double-crowning of the pinion means deviation of cross-profile from an involute one and deviation in longitudinal
direction from a helicoid surface. The pinion-gear tooth surfaces are in point contact, the bearing contact is localized and
oriented longitudinally, edge contact is avoided, the influence of errors of alignment on the shift of bearing contact and
vibration and noise are reduced substantially. The developed theory is illustrated with numerical examples that confirm
the advantages of the gear drives of the modified geometry in comparison with conventional helical involute gears.
2003 Elsevier B.V. All rights reserved.
Profile-crowned pinion and gear tooth surfaces
The profile-crowned pinion and gear tooth surfaces are designated as Rr and R2, respectively, wherein R1
indicates the pinion double-crowned surface.
3.1. Generation of Rr
Profile-crowned pinion tooth surface Rr is generated as the envelope to the pinion rack-cutter surface Rc.
The derivation of Rr is based on the following considerations:
ii(i) Movable coordinate systems Scðxc; ycÞ and Srðxr; yrÞ are rigidly connected to the pinion rack-cutter and
the pinion, respectively (Fig. 9(a)). The fixed coordinate system Sm is rigidly connected to the cutting
machine.
i(ii) The rack-cutter and the pinion perform related motions, as shown in Fig. 9(a), where sc ¼ rp1wr is the
displacement of the rack-cutter in its translational motion, and wr is the angle of rotation of the pinion.
Fig. 9. Generation of profile-crowned tooth surfaces by application of rack-cutters: (a) for pinion generation by rack-cutter Rc; (b) for
gear generation by rack-cutter Rt.
F.L. Litvin et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 3619–3655 3627
(iii) Using coordinate transformation from coordinate system Sc to coordinate system Sr we obtain a family
of generating surfaces Rc represented in Sr by the following matrix equation:
Longitudinal crowning of pinion by plunging disk
We remind that errors of shaft angle and lead angle cause a discontinuous linear function of transmission
errors (see Section 4) and high acceleration and vibration of the gear drive become inevitable.
Longitudinal crowning of the pinion tooth surface, in addition to profile crowning, is provided for
transformation of the shape of the function of transmission errors and reduction of noise and vibration.
The contents of this section cover longitudinal crowning of the pinion by application of a plunging
generating disk. The same goal (double-crowning) may be achieved by application of a generating worm
(see Section 6).