# Theoretical tooth surface equation of spiral bevel gear pinion

Theoretical tooth surface equation of spiral bevel gear pinion spiral bevel gear is processed on milling machine. The shaking table mechanism simulates the hypothetical profile gear. The cutting surface of the cutter head is equivalent to the tooth surface of the hypothetical profile gear. The cutting process is the meshing process between the hypothetical profile gear and the processed gear.

The cutting surface of the tool (taking the concave surface of the small wheel as an example) is shown in Figure 1, cutting the conical surface Σ P is expressed in the coordinate system SP as:

Where, u and θ Is the surface coordinate parameter; RP is the tool tip radius; α 1 is the tooth profile angle of the tool.

Figure 2 shows the coordinate system of the small wheel. OM is the center of the shaking table; O1 is the pitch cone vertex of the processed small wheel; SC and S2 are fixed on the shaking table and small wheel respectively, Φ P and Φ 1 is the angle of shaking table and small wheel respectively; SR is the radial tool position and Q1 is the angular tool position; The auxiliary coordinate systems SH and Sb are fixed on the machine tool to determine the installation position of the small wheel. Or is the apex of the root cone of the small wheel, γ 1 is the installation angle of small wheel; XB is the horizontal bed, EM is the vertical wheel position, and XG is the axial wheel position. If the small wheel is processed by denaturing method, the angle of the processed small wheel will be changed Φ 1. Angle with shaking table Φ P shall meet the following relationship:

Where, M1P is the roll ratio of the small wheel; C. D is the second-order denaturation coefficient and

Third order variability coefficient.

Convert the tool equation from the shaking table coordinate system to the workpiece coordinate system:

According to the envelope principle of differential geometry, the meshing equation is:

Where, R1, u, R1, θ、 ｒ１， Φ P is the partial derivative of the tooth surface equation to the coordinate parameters. The accurate expression of small gear tooth surface in S1 can be obtained by simultaneous formula.

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