Figure 1 shows the gear pair meshing coordinate system, S1 and S2 are the dynamic coordinate system of small wheel and large wheel respectively, SS is the gear tooth meshing coordinate system, and H is the axial assembly distance, Σ Is the axle intersection angle, and V is the offset distance of the small wheel.
In the meshing coordinate system SS, two meshing tooth surfaces Σ 1、 Σ The position vector and unit normal vector of 2 are respectively:
Where:
u( 1)、 β ( 1)、 Φ (1) 1 – pinion tooth surface parameters
u( 2)、 β ( 2)、 Φ (2) 1 – gear tooth surface parameters
φ 1 — meshing angle of small wheel
φ 2 — engagement angle of large wheel
Mij — coordinate system transformation matrix
Lij — mij removes the rotation matrix of the last row and last column
As shown in Fig. 2, according to the meshing principle, the two tooth surfaces are in continuous tangent contact during tooth meshing. Therefore, in the fixed coordinate system SS, both tooth surfaces have common contact points at any time, and there are common normal lines at the common contact points, that is, the basic equations of TCA are:
Since | n (1) s | = | n (2) s | = 1, five equations can be obtained from the equations, plus two meshing equations during two tooth surface machining, which have independent scalars. The unknowns are u (1), u (2) β ( 1)、 β ( 2)、 Φ ( 1)1 、 Φ ( 2)1 、 φ 1、 φ 2, 8 in total. At this time, it is advisable to φ 1 is the input quantity, and U (1), u (2) β ( 1)、 β ( 2)、 Φ ( 1)1 、 Φ ( 2)1 、 φ 2, so as to obtain a contact point between the two tooth surfaces. Then change in steps φ 1, continue to solve until the calculated contact point exceeds the effective boundary of the tooth surface. These obtained instantaneous contact points of the tooth surface constitute the tooth surface contact path, and the transmission error of the tooth surface can be obtained as follows:
Where:
φ 10、 φ 20 — initial rotation angle when tooth surface 1 and tooth surface 2 mesh
Z1, Z2 — number of teeth of gear 1 and gear 2
When the elastic deformation D of a given tooth surface is ≤ 0.006 35 mm, the size and direction of the instantaneous contact ellipse can be obtained at each contact point. This series of contact ellipses constitute the tooth surface contact mark.