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The tooth profile equation gives only the coordinates of the end points of the involutes of each micro-segment (ie, NP points), in which the involute segments of each micro-segment are still not represented by a unified equation, and the values ​​of the equation are discrete. Therefore, to accurately generate each involute, it is necessary to know the center circle of each NP point. According to the principle of Logix gear formation, the center of each base circle does not coincide with the center of the gear, and they are distributed on the pitch or pitch circle of the Logix gear.
The base circle center coordinate equation of the NP point on the gear For the Logix gear, its logical point, the equation of the NP point, can be obtained by analytical method.
X2=(x1-b)cosU2-(y1-r2)sinU2y2=(x1-b)sinU2 (y1-r2) cosU2 where r2 is the pitch circle radius of the Logix gear. The equation expression of each involute of each micro-segment at the origin of the gear is taken as the coordinate origin. According to the Logix tooth profile principle [4], the center of curvature of each discrete NP point is on the pitch circle, so for any point mi: (xi-xbi)2 (yi-ybi)2=Qmi2 rbi 12( X-xbi)2 (y-ybi)2=rbi 12Logix gears The mi-point coordinate equations can be obtained by solving the above equations to obtain the coordinate equation of the base circle center of the mi point on the Logix gear.
Where: (x, y) is the coordinate of the point on the pitch circle of the Logix gear, and (xi, yi) is the coordinate of the ith NP point. In this way, the coordinates of the center circle of any point can be obtained. In addition, according to the gear meshing principle, the coordinate equation of the center of each NP point on the Logix gear can be obtained. Because the Logix rack and pinion meshes, the center of curvature of the meshing point coincides, that is, the base circle center of the gear and the base of the rack. The center of the circle coincides, so the method of coordinate transformation [5] is used to change the base circle equation of the rack into the equation under the gear coordinate, and the base circle center of the Logix gear can also be obtained. Where (xbg, ybg) is the base circle center coordinate of the Logix gear NP point, and (xbr, ybr) is the base circle center coordinate of the Logix rack NP point.
Conclusion Based on the formation principle and meshing theory of Logix gears, the base circle center equation of the NP point on the Logix tooth profile is derived, which lays a foundation for the establishment of the equation of the micro-segment involute on the tooth profile and the complete Logix tooth profile mathematical model. basis. This study will help further research on Logix gears and find new processing methods.