Modal properties of three-dimensional helical planetary gears

The structured modal properties of single-stage helical planetary gears with equally spaced planets are categorized and mathematically proved. Compared to prior two-dimensional analyses of spur gears, this study examines the three-dimensional motion of the helical gears and shafts. A lumped-parameter model is formulated to obtain the equations of motion. The gear-shaft bodies are modeled as rigid bodies with compliant bearings at arbitrary axial locations on the shafts. A translational and a tilting stiffness account for the force and moment transmission at the gear mesh interface. The derived modal properties generalize those of two-dimensional spur planetary gears; there are twice as many degrees of freedom and natural frequencies due to the added tilting and axial motion. All vibration modes are categorized as rotational–axial, translational–tilting, and planet modes. The modal properties are shown to hold even for configurations that are not symmetric about the gear plane, due to, for example, shaft bearings not being equidistant from the gear plane.