In order to facilitate the establishment of the real tooth surface model, the theoretical tooth surface model is established first. In the hypoid gear coordinate system SG, the theoretical tooth surface model of hypoid gear is expressed as:
Where, RG and ng are the position vector and unit normal vector of the theoretical tooth surface of hypoid gear respectively; U and θ Is the tool coordinate system parameter, as shown in Figure 1.
The theoretical tooth surface Σ g of hypoid gear is formed by the envelope of tool cutting edge trajectory surface Σ T, so the theoretical hypoid tooth surface model can be expressed as:
Where Mgt and LGT are the transformation matrices of vectors RG and ng from hypoid gear coordinate system SG to tool coordinate system st respectively; RT and NT are the position vector and unit normal vector of tool cutting edge path surface respectively; UT and θ T is the tool coordinate system parameter; Φ Is the angle of hypoid gear:
Where, NT is the common normal vector of the contact point between the theoretical tooth surface Σ g of hypoid gear and the tool cutting edge trajectory surface Σ T; VTG is the relative velocity of the contact point between the tool cutting edge path surface Σ T and the theoretical tooth surface Σ g of the gear.
The relative position relationship between gear coordinate system SG and CMM installation coordinate system SM is shown in Figure 2, ζ Is the distance between the gear coordinate system SG and the measuring coordinate system SM; δ Is the installation angle of the gear coordinate system SG in the measurement coordinate system SM.
The theoretical gear model is expressed in the measurement coordinate system SM as:
Where, MMG is the coordinate conversion matrix from gear coordinate system SG to measuring coordinate system SM.