# Nonlinear backlash of Spur Planetary Gear System

As shown in the figure is a sketch of the gear meshing pair with backlash. Its function is to reserve the side clearance in the process of processing and manufacturing in order to facilitate the lubrication between the two teeth during gear meshing and to avoid the occurrence of gear seizure during gear meshing.

When the gear teeth are meshing, the existence of the backlash between the teeth causes the gear meshing pair to show the changing impact phenomenon of “contact disengagement contact” during meshing, which makes the dynamic response of the gear transmission system present obvious nonlinear characteristics during the transmission of motion and power. For a pair of assembled gear pairs, when one of the gears is fixed, the swing of the other gear in the circumferential direction is defined as the backlash, which is expressed by the arc length on the dividing circle or measured by the linear displacement along the meshing line. In general, the gear dynamic model is analyzed based on the motion on the meshing line. Therefore, the backlash in this paper refers to the backlash measured on the meshing line. If the clearance is assumed to be symmetric, the nonlinear function of the backlash of the external gear meshing pair is expressed as follows:

As shown in the figure, it is the function of backlash of gear meshing pair. Generally, in the meshing pair with backlash, the backlash is usually expressed by F (x, b) to represent its nonlinear function. According to the meaning of the backlash and its concrete forms, the piecewise function is generally used to express the backlash, and the formula is the expression of the backlash. Generally, the piecewise function is used to describe the backlash in mathematical sense, so that the approximate analytical method can be used to solve the dynamic response of the system under the influence of backlash. According to the reference [41], the representation method of the backlash of the internal and external gear pair is the same, so the tooth side clearance of the internal gear pair refers to the external meshing gear pair, and will not be repeated here.

Where:

b — backlash of gear pair.

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