# Expression of meshing force of parallel planetary gears

The connection form of parallel coupling planetary gear is shown in Figure 1, and each row has two parts connected. P10 row, P20 row solar wheel input, P10 row, P20 row satellite output, dual input dual output structure.

By adding the rotational stiffness of the connecting parts into the stiffness matrix of the dynamic equation of the double row planetary gear connection, the meshing force becomes the elastic meshing force related to the meshing frequency of the two rows and its multiple frequency

Where, an1 and an2 are the amplitude of meshing force of P10 row and P20 row at corresponding meshing frequency respectively; ω 1、 ω 2 is the meshing frequency of two rows; φ N is the phase angle of the meshing force at the corresponding meshing frequency.

The amplitude frequency diagram of the dynamic response of the external meshing force of the parallel planetary gear system P20 is shown in Figure 2. Among them, the frequencies corresponding to the larger amplitude values are P20 fundamental, 2x, 3x, 6x, 11x, 13X, P10 fundamental and 2x.

The harmonic analysis of the elastic meshing force is carried out to obtain the amplitude of the current tooth frequency and its multiple frequency, and the amplitude of the coupled tooth frequency and its multiple frequency. The meshing force is divided into two parts, TMH and TMB, according to the tooth frequency of two rows and their frequency doubling. Based on the amplitude ratio ∑ FH and ∑ FB of the meshing force of the two parts, the effect of speed and load on the frequency coupling of the meshing force of the double planetary gears is studied. The specific calculation formulas are as follows: 