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The root cut analysis of the LogiX gear is used to form the LogiX gear with the basic rack envelope (ie, the fan method). If the gear has a particularly small number of teeth, the LogiX gear will also undergo root cutting. The root cut of the LogiX gear is essentially the same as the undercut of the involute gear, and the tooth top of the rack cutter cuts off a portion of the profile of the machined gear. In the range of error tolerance, the micro-segment involute on the LogiX rack can be replaced by a micro-segment line, so that the LogiX rack can be regarded as connected by a plurality of involute racks with different pressure angles. Made.
The root-cut schematic diagram of the LogiX gear is shown in the figure. In the figure, mi-1, mi and mi 1 are the adjacent 3 NP points on the LogiX rack, respectively, using the micro-segment line mi-1mi, mimi 1 instead of the micro-segment involute Line mi-1mi, mimi 1, extend the mimi-1 node line at the si point, extend the mi 1mi node line at si 1 point, and then pass mi, mi 1 point as the parallel line miti, mi 1ti 1 . Here, simiti can be thought of as an involute rack, and mi-1miti is part of the involute rack. Similarly, si 1mi 1ti 1 can also be regarded as an involute rack, and mimi 1ti 1 is part of the involute rack. The mi-1mi, mimi 1 two-piece rack will cut out two connected involutes on the gear. For the involute rack simiti, if the cut gear of the cut is undercut, the cut-in portion on the rack should be its crest portion, the portion near the mi. If it can be proved that the gears it cuts are not undercut, then the mi-1mi, which is part of the rack, obviously will not cut into the gears. Therefore, if the gears cut by the involute rack formed by each micro-segment line on the LogiX rack profile do not undergo root cutting, it can be considered that the gears processed by the LogiX rack will not be undercut. phenomenon.
The root cut of the logiX tooth profile non-circular gear is known from the above. When the logiX non-circular gear is machined, the rack tool can also be regarded as a series of involute racks with different pressure angles. For each involute, the straight profile is constant value = 0, d = 0, so the radius of curvature of the tooth profile can be obtained by equation (1), ie = rP P1sin0. (3) shows the geometric meaning of equation (3): when the tool rack and the gear tooth profile are tangent to an arbitrary M, the instantaneous node is P, and the radius of curvature of the node curve at point P is P1, the center of curvature O. From O to MP, the vertical line is N, then MN is the radius of curvature of the M point profile. It is easy to prove this.
MN=MP PN=rP P1sin0=, (4) Since the condition of the root cut limit point on the rack profile is =0, that is, rP=-P1sin0. Its geometric meaning, rP=MP in the figure is negative. This formula indicates that if the absolute value of y3 of the tooth angle is larger than the absolute value of the ordinate of the N point, the undercut limit of the tool rack will be undercut. The condition that does not produce undercut is |y3|P1sin20, (5) When machining standard non-circular gears, y3 is the tooth tip ham of the rack tool. For the logiX tooth profile rack knife, it is known that each micro segment is gradually opened. The ha sum of the line is different, and the values ​​are ha(i)=ham=y3m=y3(i)m, where m is the equivalent modulus of the corresponding tooth of the non-circular gear, and (i) is The pressure angle of the NP point at mi, and y3 is the ordinate of the tooth profile of the logiX rack above the pitch line. From (5) and (6), there are ha(i)mP1sin2(i), (7) For the entire gear, the smallest place of P1 is the easiest to cut, so the conditional formula of the check root cut is ha(i) mP1minsin2(i), (8) where m is the equivalent modulus of the corresponding tooth at the minimum radius of curvature of the pitch curve.
The minimum number of teeth in which the logiX non-circular gear does not occur under the equation (7) is zmin=2y3(i) msin2(i)(9) The formula (9) can be used to calculate the minimum number of teeth corresponding to each micro-segment line. As long as the number of gears of the logiX gear is greater than the largest one, no undercut will occur.
Conclusion By analyzing the root-cutting principle of the LogiX tooth profile non-circular gear, it can be seen that the root cut is most likely to occur at the minimum radius of curvature of the non-circular gear. When designing the LogiX tooth profile non-circular gear, calculate the minimum number of teeth corresponding to each micro-segment line, as long as the LogiX gear number is greater than the largest one, no undercut will occur.