Towards general performance diagrams to define optimum profile and lead modifications with regard to transmission error in spur and helical gears

•An original analytical approach to the simulations of mesh elasticity and transmission error is presented.•Numerous comparisons with software code results prove the validity of the methodology.•General performance diagrams are derived, which help define optimum tooth modifications.•The specific contributions of gear macro-geometry and tooth modification parameters are highlighted and quantified. Some theoretical results on mesh elasticity and contact patterns in narrow-faced spur and helical gears are presented, which lead to original semi-analytical definitions of optimum combinations of profile relief and lead crown minimising the time-variation amplitudes of transmission error. Extensive comparisons with results from a benchmark code prove that the proposed methodology is sound and can be used to define optimum modifications with minimum effort and computational time. Dimensionless performance diagrams are derived from systematic simulations over a broad range of gear macro- and micro-geometries. Although no closed-form solutions can be found, general trends are clearly identified, making it possible to define optimum tooth modifications with reasonable precision. The importance of face contact ratio is highlighted as it can lead to highly contrasted results with either one or two solutions minimising the time-variations of transmission error. For a given gear geometry and set tooth profile and lead modifications, it is shown that the specific influence of loading can also be derived from the proposed performance diagrams.