# Meshing phase calculation of Spur Planetary Gear System

The meshing phase relationship of planetary gears can be determined according to the position angle of planetary gears and the number of teeth of gears. According to the meshing frequency of gears, there are the following relationships between the meshing phases of each meshing pair in the planetary gear train

Where:

γ Sn — the meshing phase between the sun gear and the nth planetary gear, (n = 1, 2, 3,…, n);

γ RN — meshing phase between the inner gear ring and the nth planetary gear;

ψ n — the angle between the nth planetary gear and the reference planetary gear (planetary gear 1) in the circumferential direction,

In the formula, it is specified that “+” is taken when the planet carrier rotates anticlockwise, and “-” when it is rotated clockwise.

For the planetary gear transmission system studied, the angle between the adjacent planetary gears is 120 ° because three planetary gears are evenly arranged on the planetary carrier, and the planetary gear 1 is taken as the reference gear, and we can know that ψ 1 = 0,

Ψ2=2π/3，Ψ3=4π/3 。 When the planet carrier rotates anticlockwise, the phase relationship among the gears is γ S1 = γ RI = 0, γ S2 = γ R2 = Zr / 3, γ S3 = γ R3 = 2Zr / 3, which means that the meshing phases of the second and third planetary gear relative to the first planetary gear are respectively ahead of Zr / 3 and 2Zr / 3.

In planetary gear transmission system, many pairs of “sun gear planetary gear” and “planetary gear inner gear ring” gear pairs are meshed at the same time. Although the meshing stiffness of each pair of sun gear planetary gear (or planetary gear inner gear ring) is the same, the meshing phase between different tooth pairs is not always the same. According to the relationship between meshing phase and meshing stiffness given in reference, it can be seen that:

Where:

Kspn — meshing stiffness between the nth pair of planetary gear and sun gear;

Krpn — the meshing stiffness between the nth pair of “planetary gear inner gear ring”;

KSP1 — meshing stiffness between the first pair of “planetary gear sun gear” meshing pair;

Krp1 — meshing stiffness between the first pair of “planetary gear inner gear ring” meshing pair;

γ SR — the phase difference between the nth pair of sun gear planetary gear and the nth pair of planetary gear inner ring gear. When the number of planetary gear teeth is even, γ sR = 0, when the number of planetary gear teeth is odd, γ s = 1 / 2;

TM — meshing period. The inner and outer meshing pairs have the same meshing period. 