Main mathematical model of non-NC ball-milling cutter

1 Introduction Ball-end milling cutters are important tools for CNC machining of complex surfaces (especially free-form surfaces). The market demand is large. Due to the complex shape of ball-end milling cutters, most of the processing of ball-end cutters at home and abroad are currently required to be implemented on multi-axis CNC machine tools. Due to the expensive equipment (up to millions of US dollars), the cost of one-piece tool processing is higher. high. In order to reduce the cost of tool processing, the author began to study non-numerical machining methods for ball-end milling cutters in 1991. Together with co-investigators, he proposed the mathematical model of the rake face and flank face of the ball-end milling cutter. Related machining model for ball end mills and serial production of ball end mills. In the in-depth research and development of ball-end cutters, in order to solve the more complex problems of the flank face grinding mechanism in the original machining plan, the use of a planar curved profile instead of a space curve profile to process and succeed, further reducing milling Knife processing costs. In order to have a general concept of the principles and methods of non-numerical machining of ball-end milling cutters by peers, this paper presents a summary of the machining principles described in several literatures and the related backbone mathematical models used to form the final products.

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Fig.1 Principle of rake face machining

2 The rake face machining principle and the trunk model are conducive to cutting. The edge curve of the ball-end milling cutter should be an "S"-shaped spherical curve, that is, the intersection between the rake face and the spherical surface formed by machining should be an "S"-shaped curve. . The principle of machining the rake face of a ball-end milling cutter designed for this purpose is shown in Fig. 1. During machining, the cone wheel rotates around the fixed axis O1, and the workpiece (machined milling cutter) rotates around its axis Oz. The enveloping surface of the sand profile family formed by the cone-faced grinding wheel relative to the workpiece is the rake face. To establish the mathematical model of the rake face machining, select the right-handed right-angle coordinate system s=[O;x,y,z] and s1=[O1;x1,y1,z1] respectively fixedly connected to the workpiece and the grinding wheel (see Figure 1), where the y and y1 axes are parallel and point upwards. If the radius of the large end of the grinding wheel is R2 and the semi-angular angle of the grinding wheel is g, the distance from the big end of the grinding wheel to O1 is p, and the angle between the z and z1 axes is f, then when the radius of the ball head is R, the coordinate system The conversion formula of s1 to coordinate system s is: r={x,y,z}={x1cosf+z1sinf-psinf,y1+R2,x1sinf+z1cosf+R-pcosf} (1) The wheel profile in coordinate system s1 The surface equation is r1={x1,y1,z1}={ucosv,usinv,p+(R2-u)cotg} (2) Let the profile of the grinding wheel revolve around the y1 axis at the angular velocity w1 and the resulting moment at time t The equation is transformed into the coordinate system s, and then it is rotated relative to the Oz axis at the angular velocity w2 to obtain the envelope surface (ie, the rake face) equation of the relative motion lower sand profile surface family.

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(3) 3 blade curve and parameter optimization The spherical surface of the first type in equation (3) is x*2+y*2+z*2=R2 (4) The intersection between the sphere and the rake face For the edge curve. In order to make the edge curve an ideal "S" curve, design variables should be optimized. The design variable is X=[X1,X2,X3]=[p,q,R2] (5) where q=w2/w1. The specific optimization mathematical model can be found in Shi Peiling, Wang Wei and Tang Yuyong's "Study on the Mathematical Model of Spherical Milling Cutter Manufacturing" (published in Journal of Mechanical Engineering, 1994, Issue 5), which is abbreviated here.

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Figure 2 flank machining principle

4 The flank machining principle and the stem model are set by the formula (3), (4) The spherical edge curve obtained is r2={x2, y2, z2} (6) The flank face milled must pass the edge Curve, with enough relief angle. The flank machining principle designed for this purpose is shown in Fig. 2. In the figure, P0 is the fixed point (0, y0, 0) on the y-axis, P2 is any point on the edge curve, P is the corresponding point on the plane curve, and the P-point trajectory is used to ensure the sand profile along the sphere Curve grinding. Set P2 point and perpendicular to the straight bus line axis P0P P point, you only need to find the direction vector P2P to find the P point coordinates, and then you can find the intersection P Pp and y = y0 plane P ( The trajectory of x, y0, z) (ie, the modulus curve). If we set the unit vector on the vector P2P to t={ , , }, cylindrical radius of the wheel is R1, there is

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(7) Find the unit vector t={ , , }, then the P point trajectory equation is rP = r2 + R1t (8) then the straight line POP equation is rp = rp0 + l (rp-rp0) / (rp-rp0) (9) the intersection of this straight line and the plane y = y0 The trajectory is the model curve. Due to limited space, this article no longer lists the specific expressions of the above-mentioned mathematical models. It should be pointed out that the mathematical models given in this section are more concise than the corresponding mathematical models in other documents, and the corresponding model curves are easier to manufacture and easier to replace (in order to meet the needs of tools with different specifications), and thus have obvious advantages. Sex. When processing the flank, there is the problem of optimizing the selection of y0, R1 values. The principle of selection is to make the tool have a certain angle, but also to ensure sufficient strength at the edge. The specific treatment method is omitted here. 5 Conclusion The principle of non-numerical machining of the ball nose cutter and the mathematical model of the backbone have been successfully verified in the production practice. The ball-end milling cutter series developed in cooperation with Harbin Institute of Technology High-tech Park has been put on the market. In addition to the guiding significance of the low-cost mass production of ball-end milling cutters, this paper also has reference value for the mass production of other special rotary milling cutters mentioned in “Some Explanations on Spindle Milling Cutters with Two Axis Linkages”. . However, it should be noted that the content of this paper only reflects the basic framework of the principle of non-numerical machining of ball-end milling cutters. If it is necessary to realize the processing method in the process, the corresponding non-backbone model needs to be filled according to other documents and the framework of this paper, and then be used again. Guide production.

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