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When the rack and the workpiece are engaged for grinding, the intermediate plane of the rack is tangent to the cylindrical surface of the workpiece. In order to form the exhibition movement, the gear to be ground rotates on its own axis, on the one hand, and on the other hand, the grinding wheel is stationary in the horizontal plane. For the convenience of analysis, it is assumed that the gear to be ground is only rotated and not flat, and the rack is used for translation. The analysis results are the same.
Establish a corresponding coordinate system: a fixed coordinate system (P, x, y), a coordinate system (o1, x1, y1) that is fixed with the rack and moves with it, and is fixed with the machined gear and rotates therewith. The coordinate system (o2, x2, y2), o2 is the center of rotation of the gear. At the starting position, (o1, x1, y1) coincides with (P, x, y), the y2 axis coincides with the y direction, and the x2 axis is parallel with x [1].
Let the distance between the intersection of the rack and the pitch line at the starting position and the y-axis be l0, then the equation of the rack tooth profile in (o1, x1, y1) is y1=(x1-l0) cotAt1(1) where: At1 For the end face pressure angle of the rack, At1=arctantanAn1cosB1.
Let the coordinates of any point M on the rack be M(x1, y1), and the intersection of the rack profile normal line passing through point M and the x1 axis is Mc. To make M become the meshing point, the rack should be translated from the starting position. l Distance, ie l=x1 y1cotAt1 (2) At the same time, the gear 2 is rotated from the starting position by the U2 angle, U2=lr2.
When l is a positive value, it means that in order to make M become the meshing point, the rack should move from the starting position to the left, and the gear rotates counterclockwise accordingly; when l is negative, it means that the rack should be from the starting position to the right. Move, the gear rotates clockwise.
According to the principle of gear meshing, the trajectory of the meshing point in the fixed plane is the meshing line [1]. The meshing line can be obtained by the following coordinate transformation formula and coordinate transformation matrix: xyt=M01x1y1t1M01=10-r2U2010001(3) Let t=t1=1, the meshing line equation is x=x1-ly=y1(4) ie xy= - cotAt1, which is a straight line passing through the fixed point P and perpendicular to the tooth profile of the rack, and the angle with the x-axis is -At1.
Therefore, it is known that the coordinate equation of point B is equivalent to the tooth profile of gear 2. Since the point B coordinate equation (9) is obtained when the curve AB is a standard involute, and is any point, the tooth profile equation (6) of the gear 2 is also a standard involute equation. This proves that the gear end section profile of the standard rack and gear meshed on the indexing circle is a standard involute profile.
3 Verification of the involute spiral surface grinding wheel grinding machine grinding helical gear is realized by the principle of helical rack and helical gear meshing. As shown, when the straight rack tilt angle is B1, it meshes with the gear to be ground on the indexing surface, and the truncated shape in the plane of the joint is equivalent to a helical rack with a tooth inclination angle of B1, which is obtained by grinding. The gear tooth profile should be an involute helix in the axial direction.
In the formation of the involute helicoid, in the Fig. 2Formingofinvolutehelicoid, the sub-cylinder of the beveled gear is tangent to the median plane of the helical rack for pure rolling. During the grinding process, the toothed line of the rack on the intermediate plane (inclination angle is B1) is printed on the pitch cylinder of the gear to form a cylindrical spiral of the gear, and its helix angle is also B1.
Let m2m be the cross section of either end of the rack, and the end section of the distance z from the m2m is n2n. As can be seen from the foregoing, the tooth shape formed by the gear in either end section is an involute. The distance between the cross-sections m2m and n2n at both ends of the rack in the horizontal direction is ztanB1, and the normal distance along the rack is ztanB1cosAt1. To make the meshing point of the rack and the gear move from point M to point N, the angle at which the gear rotates is U=ztanB1cosAt1rb2(14) such that the tooth profile of the rack encloses the helical gear of the helix inclination angle B1. From the formula (14), there is zU=r2tanB1 (constant) (15). The above formula shows that when the end section of the gear is involute, the tooth moves in the axial direction around the axis, and the distance moved in the axial direction is z, which is exactly the involute The definition of the helical surface spiral parameter is formed, and the ground gear helix angle B2 is equal to B1. Therefore, when the grinding wheel of the tapered grinding wheel grinding machine is inclined at the B1 angle, the grinding helical gear tooth surface is an end surface pressure angle and is equal to At1, and the sub-circular spiral angle is equal to the involute spiral surface of B1.
Conclusion The taper grinding wheel grinding machine uses the principle of rack and pinion meshing to intermittently grind the involute tooth surface by the forming method. During the grinding process, the machined gear rotates and moves relative to the grinding wheel at a certain speed ratio, and the involute tooth shape of the gear is ground. The conjugate tooth profile of the known gear can be obtained by the tooth normal method. In this paper, the tapered grinding wheel grinding machine correctly grinds the involute spiral surface of the helical gear from the tooth profile equation and the involute spiral surface. Theoretical verification. The verification method and the derivation process are helpful to understand the principle of grinding the gear by the rack-shaped tool, and have theoretical guiding significance for improving the grinding precision.
Principle test for expanding the cutting-by-wire component layer of the cutting gear