7。Conclusion

The investigations carried out in relation to the undercutting of teeth of involute gears by a rack-cutter, result in the following conclusions:

1。The undercutting of teeth is done by the rectilinear profile of the rack-cutter (traditional case — type I), as well as by the rack-cutter fillet, on which the teeth crests of the tool's cutting teeth are cut (the non-traditional case — type II)。

2。The undercutting of teeth, caused by the rack-cutter fillet is found in two variants, defined in the present work as “undercutting

— type IIa” and “undercutting — type IIb”。

3。In the presence of the undercutting — type IIа the tooth thickness in their bottom is decreased without cutting their involute profile, and in the presence of undercutting — type IIb a part of the involute profile is additionally cut。

4。If the teeth of a gear, cut by a rack-cutter, are undercut of type IIb, they are also undercut of type IIa。

5。The condition for non-undercutting — type II is defined uniquely by two independent parameters: the number z of the teeth cut and the profile angle α of the rack-cutter。

6。In order to avoid the undercutting of the involute teeth (type I, IIa and IIb), it is necessary to satisfy the traditional boundary condition (2) as well as the boundary condition  (19)。

7。The proposed indices of undercutting δr, δt, λr and λt allow the extent of teeth undercutting in radial and tangential direction to be expressed by the respective quantitative value。

References

[1] F。L。 Litvin, A。 Fuentes, Gear Geometry and Applied Theory, Cambridge University Press, Cambridge, 2004。 [2]   F。L。 Litvin, Theory of Gearing, Nauka, Moscow, 1968。 , (in   Russian)。

[3]  J。R。 Colbourne, The Geometry of Involute Gears, Springer-Verlag, New York, 1987。

[4]  Z。X。 Chen, Proof of the undercutting phenomenon for an involute tooth profile on a cylindrical gear, Mechanism and Machine Theory 27 (1992) 93–95。 [5]  R。T。 Tseng, C。B。 Tsay, Mathematical model and undercutting of cylindrical gears with curvilinear shaped teeth, Mechanism and Machine Theory 36 (2001)

1189–1202。

[6]  L。C。 Chao, C。B。 Tsay, Tooth flank, undercutting and tooth pointing of spherical gears, Mechanism and Machine Theory 46 (2011) 534–543。

[7] W。S。 Wang, Z。H。 Fong, Undercutting and contact characteristics of longitudinal cycloidal spur gears generated by the dual face-hobbing method, Mechanism and Machine  Theory 46 (2011)  399–411。

[8] S。L。 Chang, C。B。 Tsay, L。I。 Wu, Mathematical model and undercutting analysis of elliptical gears generated by rack cutters, Mechanism and Machine Theory 31 (1996)   879–890。

[9]  G。C。 Mimmi, P。E。 Pennacchi, Non-undercutting conditions in internal gears, Mechanism and Machine Theory 35 (2000)  477–490。

[10] Z。 Ye, W。 Zhang, Q。 Huang, C。 Chen, Simple explicit formulae for calculating limit dimensions to avoid undercutting in the rotor of a Cycloid rotor pump, Mechanism and Machine Theory 41 (2006)    405–414。

[11] P。 Pennacchi, Comments on “Simple explicit formulae for calculating limit dimensions to avoid undercutting in the rotor of a Cycloid rotor pump” by Ye, Zhonghe; Zhang, Wei; Huang, Qinghai; Chen, Chuanming [Mech。 Mach。 Theory 41(4), 2006, pp。 405–414], Mechanism and Machine Theory 42 (2007) 1672–1675。

[12] A。 Kapelevich, Y。 Shekhtman, Tooth Fillet Profile Optimization for Gears with Symmetric and Asymmetric Teeth, Gear Technology, September/October (2009)   73–79。

[13] F。L。 Litvin, Q。 Lian, A。L。 Kapelevich, Asymmetric modified gear drives: reduction of noise, localization of contact, simulation of meshing and stress analysis, Computer Methods in Applied Mechanics and Engineering 188 (2000)    363–390。