A deformable body dynamic analysis of planetary gears with thin rims Introduction Planetary gear sets are used commonly by automotive and aerospace industries。 Typical applications include jet propulsion systems, rotorcraft transmissions, passenger vehicle automatic transmissions and transfer cases and off-highway vehicle gearboxes。 Their high-power-density design combined with their kinematic flexibility in achieving different speed ratios make planetary gears sets often preferable to counter-shaft gear reduction   systems。76221

As planetary gear sets possess unique kinematic and geometric properties, they require specialized design knowledge [1]。 One type of the key parameters, the rim thickness of the gears, must be defined carefully by the designer in order to meet certain design objectives regarding power density, planet load sharing, noise and durability。 From the power density point of view, the rim of the each gear forming the planetary gear set must be as thin as possible in order to minimize mass。 Besides reducing mass, added gear flexibility through reduced rim thickness was shown to reduce the influence of a number of internal gear and carrier errors, and piloting inaccuracies [2]。 In addition, it was also reported that a flexible internal gear helps improve the load sharing amongst the planets when a number of manufacturing and assembly related gear and carrier errors are present [3–6]。 Many of these effects of flexible gear rims were quantified under quasi-static conditions in the absence of any dynamic   effects。

The effect of rim thickness on gear stresses attracted significant attention in the past。 A number of theoretical studies [7–14] modelled mostly a segment of spur gear with a thin rim。 In these studies, the gear segment was typically constrained using certain boundary conditions at the cut ends and a point load along the line of action was applied to a single tooth in order to simulate the forces imposed on a sun or an internal gear by the mating planet。 This segment of the gear was modelled by using the conventional finite element (FE) method with the same boundary conditions  applied  in  order  to  simulate  the  actual  support  conditions。  These  models  do  not

include the other gears in the planetary gear set as the forces acting on the planet–sun gear and planet–internal gear meshes are represented by a single point force applied mostly on a single tooth along the line of action。 While these models were instrumental in qualitatively describing the influence of the rim thickness on the bending stresses of an internal gear, they were not fully capable of describing the behavior observed in a number of experiments on this subject matter [12,15–18]。 The accuracy of the stress predictions were strongly dependent on the suitability of the conventional FEM meshes to simulate the tooth, the boundary conditions imposed to  represent the actual support conditions, and the assumption that a point load can fully describe the actual loads on planet mesh。 Since large portions of the internal gear and the other gears (planets and the sun gear) are left out of these models, it was not possible to investigate the effect of internal gear rim thickness on the overall behavior of the planetary gear set including its influence on the stresses of planets and the sun gear and the load sharing amongst the planets。 Similarly, an accurate prediction of the shape and the amount of gear deflections was also not possible for the same reasons。

Two recent studies by the first author [2,6] employed a non-linear deformable-body model of an entire planetary gear set to investigate the impact of rim flexibilities, especially of the internal gear, on gear stresses and planet load sharing under static conditions。 These studies indicate that reducing rim thickness of the gears improve functionality of the gear set by minimizing the adverse effects of gear and carrier manufacturing errors and by improving the planet load-sharing characteristics under quasi-static conditions。 However, these benefits come at the expense of increased gear stresses。 The practical design question of how thin gear rim thicknesses can be without any durability problems is not possible to answer based on these static analyses alone。 It is expected that behavior of the planetary gear set changes under dynamic conditions as the system flexibility is increased, potentially increasing gear stresses to a certain   extent。