# Nonlinear dynamic backlash and meshing stiffness of sun gear

The analysis process of the dynamic backlash of the outer meshing pair composed of the sun wheel and the star wheel is the same. The dynamic backlash between the sun gear and the i-th planetary gear is:

In the formula, 2bspi is the tooth side clearance caused by common normal adjustment; RBS and rBPI are the base circle radius of the sun wheel and the i-th planetary wheel respectively, and α SPI (T) is the dynamic engagement angle, which is calculated by the formula; α SPI is the theoretical engagement angle, which is generally 20 °; inv x is the involute function, inv x = Tan X-X.

It is assumed that the center of the sun wheel and the i-th planet wheel are rigid supports. In the planetary frame rotation coordinate system, the relationship between the pressure angle of the sun gear engagement point and the rotation speed is shown in the figure. The coordinate system osxy is at the theoretical center of the sun wheel, and the coordinate axis always points to the theoretical center of the planet wheel. The center of the rotation coordinate system is at the theoretical center of the sun wheel and rotates with the sun wheel. The pressure angle of the engagement point of the sun wheel in a single engagement period can be obtained through the similar analysis of knots

Where, α SPIO is the pressure angle of the engagement point at the beginning of the engagement period, T1 is the beginning of the engagement period, T2 is the ending of the engagement period, and ω s (T) is the relative speed of the sun wheel in the rotation coordinate system of the planet carrier, which is defined as positive for counterclockwise and negative for clockwise.

The phase difference between the sun wheel and the outer meshing pairs of each planet wheel results in the different positions of the initial meshing points between the sun wheel and different planet wheels. Therefore, for the sun wheel and the different planet wheel, the influence of the phase difference can be considered by giving the different initial pressure angle of the meshing point in the formula.

The calculation methods of the pressure angle of the engagement point in the back engagement state of the sun gear and the planetary gear teeth, the pressure angle of the engagement point participating in the engagement of the gear teeth at any time, the dynamic boundary conditions of the single and double teeth engagement and the modification area, and the dynamic engagement stiffness are the same, which will not be discussed in detail.

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