The relative spatial position of each chain link and the phase angle of engaged tooth are not concerned in traditional sprocket designs. As such, the chain shifting might fail regularly. In this study, an optimized path of one chain for up-shifting is derived. Such path is formed by several bended chain links sustain yaw and roll. Empirically, appropriate bending, yaw and torsion angles for tooth chamfer, could be shorten the up-shifting distance and phase angle required, eventually facilitate not only more ascending points compactly arranged on sprocket but larger chain laterodeviation which curtails sprockets clearance and integrates the sprocket unit. Then again, the optimized shifting path promotes efficiency and decreases required movement yielded by derailleur tappet pressure.

This paper presents an approach for obtaining an optimized geometry for the flank of atooth by minimizing the equivalent contact stress. The stress calculation method is basedon Hertz theory. As the majority of tooth profiles are involute, the geometric variationof the flank of the tooth is achieved variationally relative to the involute profile. Theoptimum profile is obtained by Monte Carlo simulation. During this optimization, apolynomial expression of the tooth geometry is used. The parameters influencing thesimulation are the four characteristic contact points. The approach presented has beenapplied in a representative case. A study of the geometric sensitivity was conducted onthe optimized tooth profile. Two different approaches were considered: by Monte Carlosimulation and analytical propagation. The robust and linear nature of the behavior of thetooth profile was demonstrated when it was subjected to geometric variations.