No abstract

This paper addresses the effect of manufacturing errors such as eccentricity and planet pin positioning errors on the quasi-static behavior of a 3 planet planetary transmission, taking into account different configurations regarding the bearing condition of the sun gear shaft. The aim of the paper is to shed light on some untouched aspects of the load sharing behavior of planetary transmissions, such as the effect of radial positioning errors of the planets when different pressure angles are used, and the impact of the different loadings per planet on the actual load per tooth. A modeling approach is employed, and physical explanations and simplified graphs are provided to help understand the behavior of the transmission when the sun is allowed to float and errors are introduced. The model used, developed by the authors and presented and validated in previous works, hybridizes analytical solutions with finite element models in order to compute the contact forces. The results obtained show that the teeth loads are much lower than expected compared to the planet uneven loads, both in the non-defected and defected transmission, and that radial positioning errors have non-negligible effect on the load sharing ratio under certain operating conditions.

Traditional procedures to calculate efficiency on gear transmissions generally consider sliding friction as the only dissipative effect, and what is more, they are based on the usage of constant friction coefficients. Although this approach gives acceptable efficiency values depending on the transmission application, the utilisation of a variable friction coefficient provides more reliable results of the friction behavior. Within this framework, the influence of the choice of the friction coefficient on the efficiency of shifted spur gears is assessed in this study. The Niemann ′ s friction coefficient formulation, which is constant and commonly applied to traditional approaches, was implemented in this proposal, in order to compare it with two hybrid formulations, which are based on Elastohydrodynamic Lubrication fundamentals and capable to reproduce the friction coefficient in dry contact, boundary, mixed and fluid film conditions of lubrication. These friction coefficient formulations are dependent on the load applied in the conjunction, therefore an enhanced load sharing allows for a better modelling of sliding friction, not only because it depends directly on the normal forces, but due to the friction coefficient load dependence. In this regard, the Load Contact Model previously developed by the authors, which considers the deflections of the adjacent teeth and shifting profile to calculate the load sharing and the friction coefficient, is used, allowing for efficiency values with a high level of accuracy. The efficiency results obtained when hybrid formulations are implemented provides lower values than those determined including Niemann ′ s formulation. Furthermore, there is a shifting profile which makes optimal the efficiency. This shift factor depends on the implemented friction coefficient formulation, concluding the remarkable importance of the friction coefficient choice.

This paper presents a planar spur gear planetary transmission model, describing in great detail aspects such as the geometric definition of geometric overlaps and the contact forces calculation, thus facilitating the reproducibility of results by fellow researchers. The planetary model is based on a mesh model already used by the authors in the study of external gear ordinary transmissions. The model has been improved and extended to allow for the internal meshing simulation, taking into consideration three possible contact scenarios: involute-involute contact, and two types of involute-tip rounding arc contact. The 6 degrees of freedom system solved for a single couple of gears has been expanded to 6 ? 3n degrees of freedom for a planetary transmission with n planets. Furthermore, the coupling of deformations through the gear bodies' flexibility has been also implemented and assessed. A step-by-step integration of the planetary is presented, using two typical configurations, demonstrating the model capability for transmission simulation of a planetary with distinct pressure angles on each mesh. The model is also put to the test with the simulation of the transmission error of a real transmission system, including the effect of different levels of external torque. The model is assessed by means of quasi-static analyses, and the meshing stiffness values are compared with those provided by the literature.

A model for the assessment of the energy efficiency of spur gears is presented in this study, which considers a shifting profile under different operating conditions (40 -600 Nm and 1500 -6000 rpm). Three factors affect the power losses resulting from friction forces in a lubricated spur gear pair, namely, the friction coefficient, sliding velocity and load sharing ratio. Friction forces were implemented using a Coulomb ′ s model with a constant friction coefficient which is the wellknown Niemann formulation. Three different scenarios were developed to assess the effect of the shifting profile on the efficiency under different operating conditions. The first kept the exterior radii constant, the second maintained the theoretical contact ratio whilst in the third the exterior radii is defined by the shifting coefficient. The numerical results were compared with a traditional approach to assess the results.