Approximate Modeling Method for Large Modulus Hardened Roller Razor Profiles

1 Introduction The development of mechanical transmission technology has increasingly higher performance requirements for gear parts. In order to obtain gear products with high load carrying capacity, strong resistance to pitting corrosion, high transmission accuracy, and good surface quality, hardened tooth surfaces (HRC48-62) Gear machining technology has become a research hotspot in gear manufacturing. Hardened roller razor is a new type of gear cutting tool for hard tooth surface finishing. It not only retains the characteristics of carbide scraping hob, but also has the advantages of disc shaped cutter teeth and edge belt. Circular grinding hob round grinding process. When the tool is used to machine hard tooth surface gears, the cutting method of “rolling with a belt” can be used to obtain higher machining accuracy, cutting efficiency and lower surface roughness value. One of the key technologies in the design and manufacture of hard-tooth roller razors is to obtain a high-precision roller razor tooth profile. Especially for large-modulus roller razors, it is more difficult to achieve higher tooth profile accuracy due to the longer tooth profile arc. Therefore, according to the manufacturing method of the hard tooth surface roller razor tooth profile, by selecting a reasonable tooth shape approximate modeling method to improve the roller razor tooth profile precision, it is an important content of the roller razor design and manufacture. 2 Roller razor tooth manufacturing method Roller razor tooth shape is manufactured by using a circular grinding involute worm, so that the teeth on both sides of the roller razor are located on the involute worm. The shaft section of the grinding wheel rotates with a roundabout spiral angle relative to the axis of the roller razor, so that the section of the grinding wheel shaft coincides with the subsection of the roller razor tooth groove, and the grinding wheel is placed in the roller razor tooth groove, and the grinding wheel and the roller razor Trajectory movement can be circular milled. Rolling razor teeth groove on both sides of the law section of the tooth profile symmetrical, after grinding one side will roll razor U-turn and then grinding the other side. Theoretically, the contact line is obtained according to the known conditions of rolling razor involute worm and contact line, and then the contact line is rotated around the axis of the grinding wheel to obtain the rotating surface of the grinding wheel and its axial cross-sectional shape, but the contact line obtained is A complex space curve makes it very difficult to dress the wheel. Therefore, the approximate method is mostly used in actual operation, that is, the law section curve of the involute worm groove rounding of the rolling razor is first obtained, and then the curve is rotated around the axis of the grinding wheel to obtain the turning surface of the grinding wheel, and the resulting error is very high. Small, negligible. Because the roll section of the roller razor is a complicated plane curve, it is difficult to dress the wheel. Therefore, a simple curve can be used to approximate the section curve of the roll razor method, and the molding error can be controlled within a very small range. This method can not only reduce the dressing difficulty of the grinding wheel, but also achieve higher tooth profile accuracy. 3 Rolling razor involute worm gear profile approximation 1
Figure 1 Linear modeling error

Figure 2 Arc modeling error

Straight line fitting method using a straight line to fit a roller razor involute worm gear section profile curve can obviously reduce the difficulty of grinding wheel dressing, but the resulting approximation of the modeling error can guarantee the required tooth profile accuracy must be calculated to verify. In order to make the straight line and the rolling razor involute worm cross-section profile curve to achieve the optimal state, there must be two intersections between them, and the resulting three extremes of fitting error are shown in Figure 1. E1 and e3 are the maximum fitting errors of the tooth root and the top of the tooth respectively, and e2 is the maximum fitting error of a point in the middle of the tooth. To achieve an optimal fit, the absolute values ​​of the three fitting errors should be equal, ie, e1=-e2=e3, and the maximum fitting error ∆e=max{|e1-e2|,|e3-e2|}. The above requirements can be achieved through optimization calculations. When fitting, first set the linear equation to x=kz+c. After determining the formulae for e1, e2, and e3, establish the objective function F=|e1+e2|+|e2+e3|. The design variable is the linear slope k and The constant c can be reflected by the continuous movement of the two intersections. When F approaches 0, the best fit is achieved, and the k and c values ​​can be determined based on the intersection point, thereby determining the straight line position. Arc Fitting Method Figure 2 shows a section of a tooth profile curve of an involute worm in a roll razor. Involute worm method section profile curve still has involute nature, that is, the closer the radius from the base circle is, the smaller the radius of curvature is, and the larger the distance from the base circle is, the larger the radius of curvature is. In the section profile shown in Fig. 2, a point represents the root of the tooth, close to the base circle, point b represents the top of the tooth, and away from the base circle, the radius of curvature from point a to point b increases from small to large. Because the radius of curvature of each point on the arc is the same, if we select a curve with a radius of curvature greater than a point and less than b point to fit the curve of the normal section profile, there will be three intersections and four extreme error of the fitting. E1, e2, e3, e4. In order to achieve the optimal fitting and improve the modeling accuracy, it is necessary to constantly change the positions of the three intersections so that e1=-e2=e3=-e4, and the optimization calculation is performed accordingly. When fitting, set the fitting arc equation to (Z-Zc)2+(X-Xc)2=R2, then the fitting extreme error is ei=R-[(Zi-Zc)2+(X1-Xc) ]2]1⁄2 (i=1,2,3,4) where R is the radius of the fitted arc Zc, and Xc is the circle center coordinate of the fitted arc Z1, X1—Rolling razor involute worm The coordinates of the extreme point error of the cross-section curve fitting establish the optimal objective function as F=|e1+e2|+|e2+e3|+|e3+e4|+|e1+e4|, and the design variables are Zc, Xc, R, The optimization calculation is reflected in the constant movement of the three intersections. When F approaches 0, the best fit is achieved and the maximum fitting error ∆e=max{|e1-e2|,|e3-e2|,|e1-e4|,e3-e4|} . According to the position of the three intersections in the optimal fitting state, the values ​​of Zc, Xc, and R can be found, and the position and radius of the fitting arc can be determined.

Figure 3 Hyperbolic Fitting

Figure 4 Hyperbolic Modeling Error

Hyperbolic Fitting A line with a distance H from the rotary axis and a rotation angle a rotates around a rotary axis to obtain a hyperboloid with the equation x2/H2+y2/H2-z2/(H/tana)2=1. . Hyperbola can be obtained by cutting in any direction along the rotation axis of the hyperboloid. According to the straight line trajectory dressing, a grinding wheel with a hyperbolic axis can be obtained. Fig. 3 shows the method of fitting a roller razor involute worm gear with a hyperbola. The hyperbola is obtained by rotating about the axis of the grinding wheel. Its coordinate system is O'-Z'X', and the line segment a'b' is a segment on the hyperbola. O-ZX is the coordinate system where the tooth profile curve of the roll razor method is located, ab is the tooth profile curve of the roll razor, a is the tooth root, and b is the tooth tip. When the two curves are fitted, the coordinate system of the O'-Z'X' coordinate system needs to be transformed. After translation and rotation, the OZX coordinate system is re-weighted, and the coordinate systems of the two curves are the same. Since the radius of curvature of the tooth profile curve of the roller razor method gradually increases from point a to point b, the curvature radius of the hyperbola gradually decreases from the point a' to the point b' according to the formula of the radius of curvature. Therefore, appropriate parameters should be selected so that the double The radius of curvature at the a' point on the curve is greater than the radius of curvature at the a point on the tooth profile of the roller razor, and the radius of curvature at the b' point on the hyperbola is smaller than the radius of curvature at the b point on the tooth profile of the razor. When fitting, there will be three intersections and four fitting extreme errors e1, e2, e3, e4 will result. As shown in Figure 4, when the best fit is achieved, there is e1=-e2=e3=-e4, from which an optimized objective function F=|e1+e2|+|e2+e3|+|e3+e4 can be established |+|e1+e4|. The hyperbolic shape of the grinding wheel is determined by the parameters H, a, and the position of the grinding wheel is determined by the parameters Z0, X1, b. Although the shape and position of the grinding wheel are determined by five parameters, the fitting curve has only three intersection points, ie only three of these parameters can be determined. For this reason, two of the parameters (X0, b) can be used as constraints when optimizing the calculation, and the remaining three parameters are used as design variables and are reflected as the constant movement of the three intersection points. When F approaches 0, the best fit is achieved and the maximum fitting error ∆e=max{|e1-e2|,|e3-e2|,|e1-e4|,|e3-e4|} . Three design variables can be determined from the three identified intersection points. Adding two given parameters can determine the shape and position of the grinding wheel. 4 Modeling error calculation example The basic parameters of the large modulus hard tooth roller razor are: normal surface module mn=14mm, normal surface tooth angle an=20°, top circle outer diameter D=290mm, sub-circular helix angle g=3°7', single head, right handed. Using the three approximate modeling methods described above, the approximate styling error simulation of the tooth profile of the roller razor method was performed. The calculation results are shown in the table to the right. Approximate modeling Simulation calculation Error table Modeling method Modeling error (μm) e1 e2 e3 e4 ∆e Linear fitting 5.89 -5.84 5.86 ... 11.73 Arc fitting 0.326 -0.384 0.308 -0.329 0.674 Hyperbolic fitting 0.577 -0.503 0.570 -0.56 1.133 From the table to the right, the straight line fitting has the largest modeling error (∆e=11.73 μm); the hyperbolic fitting has less modeling error (∆e=1.133 μm); the arc fitting has the smallest molding error (∆e=0.674). Μm). Since roll razors have not yet established relevant accuracy standards, the hob accuracy standard can be temporarily applied. The ISO 4468-1982 and GB 6064-85 standards require a tolerance of 10 μm for the AA class hob with a module mn > 10 to 16 mm. Compared with the results of Table 1, we can see that the modeling error of the straight line fitting is larger than that of the standard specified tooth shape tolerance and cannot be used; the modeling error of arc fitting and hyperbolic fitting is far smaller than the standard tolerance, and it is an ideal modeling curve. Since the hyperbola has the feature of dressing the grinding wheel in a straight line, it is the preferred curve for the approximate shape. 5 Conclusions The approximate modelling calculation and analysis of the tooth profile of the involute worm tooth slotting method for large-modulus hard-tooth roller razor has been performed. The results show that the error of the straight line fitting method is large, exceeding the tolerance of the tooth shape specified by the standard. Can not be used; circular arc fitting method error is minimal; hyperbolic fitting method error is also very small, and can be a straight line dressing wheel, is an ideal approximation modeling method.