Influence of Planet Phasing on Quasi-Static Transmission Error in Planetary Spur Gears with Profile Modifications

In a planetary gear system, the planet phasing depends on the number of teeth in the sun and the ring and the number of planets. When the tooth numbers are both multiples of the number of planets, all planets mesh at the same relative position—which is called synchronous configuration—and the input torque is shared evenly among them. Otherwise, the configuration is asynchronous, or sequentially phased, and the torque sharing is uneven. This directly influences the instantaneous load sharing between the external planet–sun and internal planet–ring meshes, consequently altering both load-induced tooth deflections and the resulting transmission error. The profile relief, frequently used to avoid the mesh-in impact, influences the teeth contact along the interval of relief, which also affects the load distribution, mesh stiffness, and transmission error. Since the transmission error is a source of dynamic load, noise, and vibrations, its peak-to-peak amplitude should be controlled, and the geometry of the profile modification provides an efficient tool. In this paper, the transmission error of spur planetary gears is studied with an analytical model previously developed, based on the minimum elastic potential energy. The study also assesses the influence of the depth and length of the tip relief and compares the behavior of synchronous and asynchronous configurations. As a result of this analysis, it has been found that the variation in the amplitude of transmission error is significantly lower in sequentially phased configurations and reaches the minimum variation for the adjusted depth of relief and medium length of relief. Furthermore, an odd number of teeth on the planets results in a higher mesh stiffness than an even number, which induces a slightly lower peak-to-peak transmission error.