When the non-circular gear tooth profile is formed by the shape cutter, if the non-circular gear does not move, the pitch circle of the pinion cutter will roll purely on the pitch curve of the non-circular gear. In the middle, the coordinate system of the non-circular gear is taken as the fixed coordinate Oxy. The coordinate system O1xsys and the pinion cutter are fixed. When the shaper cutter rolls along the non-circular gear joint curve r=r(Υ) to K point rK=r(ΥK), the shape cutter rotates through ΓK, and the coordinates of the center O1 of the shaper in the fixed coordinate system Oxy are (xO1, K, yO1, K), the center line of the K point and the shaper cutter O1K is the normal of the non-circular gear section curve at point K [5].
Since the tooth profile of the pinion cutter and the tooth profile of the non-circular gear are in tangential contact at each instant, mathematically, the tooth profile of the non-circular gear should be the package of the curve family corresponding to the tooth profile of the gear shaping cutter. Network [6]. The enlarged view in the figure is the first slot diagram of the first tooth enveloping non-circular gear of the shaper. It can be clearly seen from the figure that the profile of the shaper is enveloped during the development process. Non-circular contour. The boundary line (envelope) of the tooth profile of the pinion cutter in the non-circular gear body is the non-circular gear tooth profile. Therefore, the non-circular gear tooth profile can be obtained by finding a series of points on these boundary lines.
The tip curve of the non-circular gear is a normal equidistant line [3] of the pitch curve, which has only 2 intersections with each tooth of the non-circular gear. The other normal equidistant lines and non-circular gears of the non-circular gear section curve have at most 2 intersection points, and the two intersection points are respectively on both sides of the tooth gap. If according to certain rules, from the non-circular gear tooth tip to the tooth root, calculate the equidistant line of a certain number of pitch curves, and solve the intersection with each tooth profile (boundary line) (as shown in the magnified view of m1) , m2, m3 point), the tooth profile of each tooth of the non-circular gear can be obtained.
If the number of equidistant lines of the non-circular gear section curve is k in the full height range of the non-circular gear teeth, and the distance between the adjacent equidistant lines is equal, the distance of the t-th equidistant line to the section curve is Ht=ha-tha ha0k(2) where ha—non-circular gear tooth tip height ha0—pinning cutter tip height t=0,1,2,...,k, indicating normal equidistant line from non-circular Gear tip to the root of the tooth. The sign before the outer non-circular gear brackets is positive, and the sign before the inner non-circular gear brackets is negative.
The equation of the t-th normal equidistance line of the knot curve in the gear coordinate system Oxy is [7]xt(Υ)=r(Υ)cosΥ ht(r'(Υ)sinΥ r(Υ)cosΥ)(r2( Υ) r'2(Υ))-12yt(Υ)=r(Υ)sinΥ-ht(r'(Υ)cosΥ-r(Υ)sinΥ)(r2(Υ) r'2(Υ))-12 (3) where r(Υ)——the non-circular gear section curve r′(Υ)—the first derivative of the section curve [4] gives the normal equidistance line of the non-circular gear section curve and The specific method of solving the intersection mt of the boundary line (the point on the non-circular gear tooth profile), while the point on the tooth profile of the pinion corresponding to the point mt is represented by its distance Rt to the center of the shaper.
2 The mechanism of generating the undercut of the non-circular gear According to the tooth profile engagement principle of the pinion cutter and the non-circular gear, under normal meshing conditions, the following relationship exists between them [8] (): ab is an involute pinion cutter The tooth profile F1, b is the addendum corner point, and cd(e)f is the tooth profile F2 of the non-circular gear that it envelops. Wherein, the involute ab segment encloses the cd portion of the non-circular gear tooth profile from point a to point b.
At the same time, the trajectory of point b in the envelope process forms a transition curve ef of the non-circular gear, the point e coincides with the envelope point d, and the point e is the tangent point of the envelope tooth surface and the transition curve.
If the curvature of a point on the pinion F1 and the point of the conjugated non-circular gear F2 is infinite, this indicates that a sharp point has appeared on the tooth profile F2. In the middle, aJb is the tooth profile F1 of the pinion cutter, cJd is the tooth profile F2 conjugated thereto, the aJ segment of the tooth profile F1 meshes with the segment of the cJ of the tooth profile F2, and J is a sharp point on the tooth profile F2. If the tooth body of the tooth profile F1 is on the left side of the aJb, then Jd is inside the body of the tooth, which of course is not achievable. Moreover, if the tooth profile F1 is extended from J to b, when the tooth profile Jb is conjugated with Jd, the trajectory of the point b also intersects with the cJ segment of the tooth profile F2 (the intersection point is e), so that interference occurs, so In the actual machining, the tooth profile F1 of the shaper cutter cuts off the portion below the e point on the non-circular gear tooth profile F2. Therefore, the resulting non-circular gear tooth profile is the middle fec segment. This phenomenon is called The undercut of the non-circular gear [8].
Therefore, theoretically, if it can be calculated whether there is an intersection point e of the envelope profile on the tooth profile, it can be judged whether the tooth profile is undercut, in fact, due to the discontinuity of the numerical calculation, e The point may not be on the set normal equidistance line, and it is difficult to calculate the point e. For this reason, another feature at the point e is used as a criterion for judging whether or not an undercut is generated: when the tooth profile of the pinion is known to enclose the normal tooth profile ce, the radius Rt of the point on the tooth profile of the corresponding pinion cutter It also changes continuously, but the transition curve is the trajectory of the tooth top b of the pinion cutter, and Rt is always equal to the tooth tip radius Ra of the pinion cutter. Due to the undercut, the Rt at the point e is abrupt. Jumping, it is therefore possible to use the Rt on the tooth surface to have a sudden jump as the most obvious and practical criterion for whether or not rooting occurs.
In the middle, let mk be the point on the eccentric ce segment closest to the e point on the equidistant line of the non-circular gear. The thick solid line in the figure indicates that the t point on the tooth profile of the spigot is out of the circle. The mk point on the gear tooth profile, mk-1 is the point on the envelope tooth profile ce of the non-circular gear normal equidistance line closest to mk.
Mk 1 is the point on the transition curve fe closest to the e point on the equidistant line of the non-circular gear. The dotted line indicates that the apex b of the pinion tooth forms the mk 1 point of the transition curve.
From the value calculation model of the gear teeth, the mk-1, mk, mk 1 points and their corresponding pinch cutter point radii are Rk-1, Rk, Rk 1. Considering the error in numerical calculation, the point on the tooth profile of the pinion cutter corresponding to mk 1 point may not be exactly b point, but there will be Rk 1≈Ra in the region very close to point b. Meanwhile, since mk is on the envelope tooth profile, RkRk>Rk-1, Rk 1-RkμRk-Rk-1. According to formula (5), μ1 is calculated. It can be seen that according to μ1, Rt on the tooth surface can be jumped. The point e between mk and mk 1 is the intersection of the envelope profile and the transition curve. The roots have been cut. In order to illustrate the degree of undercut, the tooth height HC of the starting point of the root can be used to illustrate. From the above analysis, the value is HC=ha0 0.5(hk hk 1) (6) where hk and hk 1 are mk. The mk 1 point corresponds to the equidistant line distance, and its value can be calculated according to the formula (2).
In the case where no undercut occurs, the tooth height of the boundary point between the transition curve and the working tooth line is the fixed gap hc. If the HC value exceeds hc, the closer the intersection point e of the envelope profile and the transition curve is to the top of the gear tooth, the more severe the undercut is.
If there is no sudden jump on Rt on the tooth surface, there is ≈1 or <1 according to formula (5), indicating that no undercut occurs, and the point e between mk and mk 1 is the tangent point of the envelope profile and the transition curve. . Therefore, after calculating the tooth profile of each tooth using the pinion numerical calculation model, the transition point and the boundary point equidistance line number k of the envelope tooth surface are first obtained by the equation (4). Then, the root-cutting judgment factor is calculated by the formula (5), and the root-cut analysis of each tooth profile of the non-circular gear is determined based on the value. If a tooth has an undercut, calculate HC and use the HC value to determine the degree of undercut. Finally, based on the obtained undercut conditions of all the teeth, it is determined whether the non-circular gear can be used in production practice.
In the example below, an elliptical gear is taken as an example to illustrate the specific implementation process of determining the undercut based on the numerical model of the inserted tooth. Ellipse long axis radius A=100mm, eccentricity k1=0192531, modulus m=5mm, pressure angle Α=20°, minimum radius of curvature Θmin=14136215mm (at the long axis of the ellipse), number of non-circular gear teeth Z2=29, insert The number of teeth of the cutter is z0=20, the height of the tooth of the gear insert is ha0=6mm, and the radius of the tip of the gear insert is Ra=56mm.
According to formula (1), the maximum modulus mmax of the non-circular gear that does not undergo root cutting should satisfy the condition mmax ≤ 0.117 Θ min = 1.6804 mm (7) Obviously, the m=5 mm of the gear is much larger than the maximum modulus without rooting. The gear will be undercut, but it is not known which roots are undercut and the degree of undercutting, which needs to be judged as described above.
Firstly, according to the elliptical gear parameters and the shaper parameters given above, the calculation model of the gear shaping is solved. If the non-circular gear takes 30 equidistant lines from the crest to the root, the tooth profile of the non-circular gear can be solved. As shown. The coordinates (x2t, y2t) of the left tooth surface of the first tooth of the non-circular gear on the t-normal equidistant line and the radius Rt of the point on the tooth profile of the pinion forming the point are given. And make a radius relationship diagram from the non-circular gear tooth tip to the root point and the corresponding point of the shaper cutter, as shown. According to the above example, it can be seen that it is correct and feasible to use the useful information to solve the tooth profile to determine whether rooting has occurred during the tooth cutting process.
Starting from the formation principle of the non-circular gear root cutting, combined with the digital calculation model of the non-circular gear shaping, the accurate method of the non-circular gear undercut verification is given. By using this method, not only can the teeth of the non-circular gear be undercut, but also the degree of undercut can be estimated and evaluated, which provides a powerful verification method for the design and use of the non-circular gear.
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