An analytical study on the combination of profile relief and lead crown minimizing transmission error in narrow-faced helical gears

Abstract Some original analytical developments and results on the formulation of transmission error in solid spur and helical gears are presented, which lead to closed-form formulae defining the optimum combinations of profile relief and lead crown minimising the time-variation amplitudes of transmission error. Extensive comparisons with the results of benchmark software codes prove that the analytical findings are sound and can be used at the initial design stage to define optimum modifications with minimum effort. Some qualitative general trends can also be derived. In particular, it is shown that optimum tooth modifications depend on a limited number of dimensionless parameters, namely the profile and face contact ratios, the normalised depth and extent of profile modifications along with the normalised lead crown amplitude. The influence of the latter is found to strongly depend on a specific function of face contact ratio, which controls the displacement of optimum profile relief either towards the smaller or larger depths of modification at tooth tips. Finally, the analytical formulae (and the numerical simulations) indicate that the optimum modifications are approximately located along a line segment in the relief versus crown amplitude plane.