The indexable ball end mill is an advanced tool for machining free-space surfaces. Compared with the integral ball end mill, the indexable ball end mill not only has all the advantages of the indexable cutting tools, but also can significantly improve the cutting edge due to the use of the processing methods for the overlapping cutting of the overlapping blades. Free-cutting state reduces the non-free coefficient, thereby reducing the cutting force [1]. The literature [2] has studied the basic theory of the geometry of the end edge of the indexable ball end mill. However, there are no relevant reports in the literatures at home and abroad for the method of overlapping peripheral edge and end edge of indexable ball end mills. Based on the theory of [2], this paper establishes a mathematical model of the method of lapping peripheral edge and end edge of the indexable ball end mill. The model can be used to calculate the geometrical error of the machining surface near the overlap. Peripheral shape control parameters.
Figure 1 Coordinate system
Fig. 2 Calculation of the geometrical error of the machined surface of the rake face of blade 1 at the lap of the end edge and peripheral edge The coordinate system shown in Fig. 1 was established. The indexable ball end mill radius is R and the blade inclination is ls. The rake face of the blade is initially set to a square shape (the square blade is easy to process, and all four sides can be used as cutting edges, and the blade utilization rate is high), and the square side length is A (see FIG. 2 ). The tangent vector [2] where the end edge is at the overlap is known as T0 at q=0, ie T0=(0, 1 ,m )T (1+m2)1â„2 (1+m2)1â„2 where m =cotw1 (w1 is the helix angle). From the above equation, T0 is perpendicular to the x axis. Let the cylindrical parameter equation be r = (x, y, z) T = (Rcosq, Rsinq, V) T (1) where: q - the angle variable V - the cylinder height variable (in Figure 2 is negative Value) Let N be the midpoint of the blade's cutting edge and be on the cylindrical surface (see Figure 1). Let the CD direction vector be P. Assume that the tangent at the overlap after the peripheral edge rotates is parallel to T0. Since CD is a straight edge, T0×P=0 must be taken in the same coordinate system. (2) Take P as T0. It must satisfy formula (2). The equation for a straight line CD over a point N (Rcosq, Rsinq, V) is {x=Rcosq y-Rsinq = zV 1/(1+m2)1â„2 m/(1+m2)1â„2 (3) Get { x = Rcosq y = Rsinq + zV m (4) From equation (4), we get x2+y2=(Rcosq)2+(Rsinq+ zV ) m (5) Let x=0 in equation (5), get the straight line After the CD (cutting edge) rotates around the oz axis, the single-leaf hyperboloid axial section equation is y2-(Rsinq+zV)2=(Rcosq)2m (6) For ease of calculation, let point C be on the xoy plane (see Figure 1) ), the projection length of the cutting edge CD on the z-axis is -2V, and -2V = Acosls, then V = - (Acosls) / 2 (7) Substituting equation (7) into equation (6) yields y2- (z+mRsinq+0.5Acosls ) 2=(Rcosq)2 m (8) As shown in FIG. 3, except for N point, C and D are not at the cylinder surface but will be at z=0 or z=2V. Produces the maximum shape error of the machined surface. Let ymax be the maximum positive coordinate value on the y-axis of the single-leaved hyperboloid axis profile after the C-point rotation. Substituting Z=0 into (8) yields ymax=[(Rsinq+A cosls)2+R2cos2q]1â„2 2m ( 9) Obviously, the maximum machining surface geometry error at point C is Dmax = ymax - R (10)
Fig. 3 Axis section of the revolving surface 2 Method of reducing the geometrical error of the peripheral edge machining surface The calculation of the peripheral edge modification angle f The ideal state of the peripheral edge processing is that the three points D, N, and C are intersected by the rake face and the cylinder face. The elliptical line (at this time Dmax = 0), and the peripheral edge machining surface geometry error is due to the blade on the C, D two points are not on the cylindrical surface (see Figure 1), and in Figure 2 CC "on the blade The extra part of the protruding ellipse line will result in "multi-cutting" in the machining. Since the elliptical edge line of the blade is difficult to machine, it is considered to replace the corresponding elliptic curve with a straight line NC". As shown in Figure 2, if the CND segment is chamfered f from both sides of the N point, the shape of the CND segment blade will be a polyline D"NC", which will obviously reduce the error of the machining surface geometry caused by C, D points. This method can partially eliminate machining surface shape errors without increasing the manufacturing difficulty of the blade. Although the CC "line is not necessarily perpendicular to the oz axis, but in order to partially eliminate the error in the machining surface geometry caused by the CC segment, Dmax can still be approximated as the CC" value. Thus, after the chamfered blade NC "segment after rotation" The maximum surface geometry error should be reduced. In FIG. 1 and FIG. 2, the size of the chamfer f can be determined by the right-angled triangle NCC as tanf=2 {[(Rsinq+ A cosls)2+R2sin2q]1â„2}-RA 2m (11) The formula (11) is both a peripheral edge The formula for the modification angle f is also an index for controlling the modification parameters, and its value is related to R, A, ls, and m. C “The calculation of the maximum surface shape error can be calculated after the peripheral edge modification angle f is calculated. The maximum surface shape error at point C (see Figure 2). Connect C"D" with its axis of symmetry at point N'. The D'max generated at point C is equivalent to moving point N to point N'. The maximum surface shape error value produced by C"D" at point C. In this case, C"D" should be considered as an imaginary blade, but the radius of rotation is not R, but (R-Dmax). If used (R- Dmax) In place of R in (9), the maximum coordinate value of the single-leaf hyperboloid axis section after C point rotation is obtained on the y-axis, and the maximum surface shape error value is obtained as D'max= [(Rsinq-Dmaxsinq+A cosls)2+(R-Dmax)2sin2q]1â„2-R 2m (12) After replacing the corresponding arc with the straight lines NC "and ND" (see Fig. 2, Fig. 3), will produce "less Cut phenomenon. Let the negative error generated at this time be Dc0. Since C0 is at the midpoint of the NC, when Dc0 is found, the blade radius R becomes (R-0.5Dmax) and the blade length A becomes 0.5A. The maximum coordinate value on the y-axis of the single-leaved hyperboloid axis section after rotation at C0 is obtained, and let it be Yc0, then Yc0=[(Rsinq-0.5Dmaxsinq+A cosls)2+(R-0.5Dmax)2sin2q] 1â„2 2m (13) Substituting equation (13) into equation (10) yields Dc0=Yc0-R (14) 3 Influence of the cutting edge on the overlapping of the end edge and the peripheral edge According to the above analysis, only the end edge is considered. The peripheral edge of the tangent at the lap is parallel to T0 and will produce the maximum surface shape error, Dmax, which will produce "multiple cuts." After the cutting edge is trimmed, the error will be reduced to D'max, but in two The tangent near the laps of the two blades is not necessarily parallel, resulting in a "less cut" Dc0, and at this point the blade axis needs to be fine-tuned so that C" is still on the xoy plane. Because the cutter has a feed motion, "less cut" is preferable to "multi cut" because the "less cut" part can be removed by the N point on the peripheral edge that does not produce the machining surface geometry error. Reducing the blade length A and increasing the radius R of the cutter can reduce the angle of the tangent of the cross section of the curved surface where the two blades are machined.
Fig. 4 Calculation of the overlap of the peripheral edge and the end edge 4 The angle a' is the angle between the revolution surface machined by the peripheral edge and the axis profile of the revolution surface machined by the end edge near the overlap. As shown in FIG. 4 , the axial profile of the turning surface machined by the end edge is a semicircle whose tangent at the point = is a plumb line. Therefore, it is only necessary to solve the axis profile curves GFE′ and y of the turning surface machined by the peripheral blade. The angle between the intersection point E' of the shaft and the plumb line may be sufficient. First find the tangent slope of the E' point. As shown in Fig. 4, the hyperbola is known to have E', F, and G points, and the coordinates in the yoz coordinate system are E'(R+D'max, 0), F(R+Dc0, -0.25Acosls, respectively). ), G(R+D'max, -0.5Acosls). Examine the hyperbola of the positive part of the Y-axis, refer to equation (6), and let it be in the form y2 - (zt)2 = 1 a2 b2 (15) Substituting the coordinates of E', F, and G in equation (15), Available t=-(Acosls)/4 (16) a2=(R+Dc0)2 (17) b2=(R+Dc0)2A2cos2ls 16(D'max-Dc0)(2R+D'max+Dc0) ( 18) Let a be the angle between the tangent line and the positive direction of the y-axis in the coordinate system, then the slope is tana. From equation (15), the slope of the hyperbola at E' point is tana=- b2+(R+D'max) A2t (19) Substituting equations (16), (17), and (18) into equation (19) yields a=arctan [ Acosls(R+D'max)] 4(D'max-Dc0)(2R+D) 'max+Dc0) (20) Since a' is the angle between the tangent line and the positive z-axis, a'=90°-a (21) Equation (21) is the revolution after machining the peripheral edge f-angle. The formula for solving the angle a' between the surface of the curved surface and the turning surface of the end edge at the overlap. D'max and a' are two indicators for measuring the geometrical error of the machining surface near the lap joint. The smaller the a' value is, the smaller the angle of the curved surface in the axial section of the rotating surface machined near the lap is. The smaller the D'max, the better the fit at the lap. 5 Calculation example Let the blade peripheral radius R = 25mm, blade length A = 12mm, w1 = 20°, ls = 15°. Calculate the values ​​of Dmax, D'max, f, Dc0, and a'. Let q = 45°, available from m = cotw1: m = 2.7475. From formula (10) available: Dmax = 1.5335mm. From formula (11) available: f=14.34°. From Formula (12), D'max=1.45×10-4mm. From formula (14) available: Dc0 = -0.00984mm. From equation (21) we can get: a'=0.395°. The above calculations show that the use of a symmetrical polygonal line-shaped cutting edge instead of a straight line-shaped cutting edge can significantly reduce the geometrical error of the machined surface of the overlapped surface at the peripheral edge of the indexable ball end mill, and enable the peripheral edge and end edge to rotate. An approximately first-order smooth connection is achieved at the lap, and the blade is easy to manufacture. The four sides of the original square on the rake face of the insert can be used as the cutting edge. When used, it can be rotated by 90°, that is, one blade can be equivalent to four linear blades and a single blade. Adopting the overlapping method can not only improve the utilization rate of the blade, but also facilitate the standardization of the blade. Therefore, it has good economic and practical value.
Figure 1 Coordinate system
Fig. 2 Calculation of the geometrical error of the machined surface of the rake face of blade 1 at the lap of the end edge and peripheral edge The coordinate system shown in Fig. 1 was established. The indexable ball end mill radius is R and the blade inclination is ls. The rake face of the blade is initially set to a square shape (the square blade is easy to process, and all four sides can be used as cutting edges, and the blade utilization rate is high), and the square side length is A (see FIG. 2 ). The tangent vector [2] where the end edge is at the overlap is known as T0 at q=0, ie T0=(0, 1 ,m )T (1+m2)1â„2 (1+m2)1â„2 where m =cotw1 (w1 is the helix angle). From the above equation, T0 is perpendicular to the x axis. Let the cylindrical parameter equation be r = (x, y, z) T = (Rcosq, Rsinq, V) T (1) where: q - the angle variable V - the cylinder height variable (in Figure 2 is negative Value) Let N be the midpoint of the blade's cutting edge and be on the cylindrical surface (see Figure 1). Let the CD direction vector be P. Assume that the tangent at the overlap after the peripheral edge rotates is parallel to T0. Since CD is a straight edge, T0×P=0 must be taken in the same coordinate system. (2) Take P as T0. It must satisfy formula (2). The equation for a straight line CD over a point N (Rcosq, Rsinq, V) is {x=Rcosq y-Rsinq = zV 1/(1+m2)1â„2 m/(1+m2)1â„2 (3) Get { x = Rcosq y = Rsinq + zV m (4) From equation (4), we get x2+y2=(Rcosq)2+(Rsinq+ zV ) m (5) Let x=0 in equation (5), get the straight line After the CD (cutting edge) rotates around the oz axis, the single-leaf hyperboloid axial section equation is y2-(Rsinq+zV)2=(Rcosq)2m (6) For ease of calculation, let point C be on the xoy plane (see Figure 1) ), the projection length of the cutting edge CD on the z-axis is -2V, and -2V = Acosls, then V = - (Acosls) / 2 (7) Substituting equation (7) into equation (6) yields y2- (z+mRsinq+0.5Acosls ) 2=(Rcosq)2 m (8) As shown in FIG. 3, except for N point, C and D are not at the cylinder surface but will be at z=0 or z=2V. Produces the maximum shape error of the machined surface. Let ymax be the maximum positive coordinate value on the y-axis of the single-leaved hyperboloid axis profile after the C-point rotation. Substituting Z=0 into (8) yields ymax=[(Rsinq+A cosls)2+R2cos2q]1â„2 2m ( 9) Obviously, the maximum machining surface geometry error at point C is Dmax = ymax - R (10)
Fig. 3 Axis section of the revolving surface 2 Method of reducing the geometrical error of the peripheral edge machining surface The calculation of the peripheral edge modification angle f The ideal state of the peripheral edge processing is that the three points D, N, and C are intersected by the rake face and the cylinder face. The elliptical line (at this time Dmax = 0), and the peripheral edge machining surface geometry error is due to the blade on the C, D two points are not on the cylindrical surface (see Figure 1), and in Figure 2 CC "on the blade The extra part of the protruding ellipse line will result in "multi-cutting" in the machining. Since the elliptical edge line of the blade is difficult to machine, it is considered to replace the corresponding elliptic curve with a straight line NC". As shown in Figure 2, if the CND segment is chamfered f from both sides of the N point, the shape of the CND segment blade will be a polyline D"NC", which will obviously reduce the error of the machining surface geometry caused by C, D points. This method can partially eliminate machining surface shape errors without increasing the manufacturing difficulty of the blade. Although the CC "line is not necessarily perpendicular to the oz axis, but in order to partially eliminate the error in the machining surface geometry caused by the CC segment, Dmax can still be approximated as the CC" value. Thus, after the chamfered blade NC "segment after rotation" The maximum surface geometry error should be reduced. In FIG. 1 and FIG. 2, the size of the chamfer f can be determined by the right-angled triangle NCC as tanf=2 {[(Rsinq+ A cosls)2+R2sin2q]1â„2}-RA 2m (11) The formula (11) is both a peripheral edge The formula for the modification angle f is also an index for controlling the modification parameters, and its value is related to R, A, ls, and m. C “The calculation of the maximum surface shape error can be calculated after the peripheral edge modification angle f is calculated. The maximum surface shape error at point C (see Figure 2). Connect C"D" with its axis of symmetry at point N'. The D'max generated at point C is equivalent to moving point N to point N'. The maximum surface shape error value produced by C"D" at point C. In this case, C"D" should be considered as an imaginary blade, but the radius of rotation is not R, but (R-Dmax). If used (R- Dmax) In place of R in (9), the maximum coordinate value of the single-leaf hyperboloid axis section after C point rotation is obtained on the y-axis, and the maximum surface shape error value is obtained as D'max= [(Rsinq-Dmaxsinq+A cosls)2+(R-Dmax)2sin2q]1â„2-R 2m (12) After replacing the corresponding arc with the straight lines NC "and ND" (see Fig. 2, Fig. 3), will produce "less Cut phenomenon. Let the negative error generated at this time be Dc0. Since C0 is at the midpoint of the NC, when Dc0 is found, the blade radius R becomes (R-0.5Dmax) and the blade length A becomes 0.5A. The maximum coordinate value on the y-axis of the single-leaved hyperboloid axis section after rotation at C0 is obtained, and let it be Yc0, then Yc0=[(Rsinq-0.5Dmaxsinq+A cosls)2+(R-0.5Dmax)2sin2q] 1â„2 2m (13) Substituting equation (13) into equation (10) yields Dc0=Yc0-R (14) 3 Influence of the cutting edge on the overlapping of the end edge and the peripheral edge According to the above analysis, only the end edge is considered. The peripheral edge of the tangent at the lap is parallel to T0 and will produce the maximum surface shape error, Dmax, which will produce "multiple cuts." After the cutting edge is trimmed, the error will be reduced to D'max, but in two The tangent near the laps of the two blades is not necessarily parallel, resulting in a "less cut" Dc0, and at this point the blade axis needs to be fine-tuned so that C" is still on the xoy plane. Because the cutter has a feed motion, "less cut" is preferable to "multi cut" because the "less cut" part can be removed by the N point on the peripheral edge that does not produce the machining surface geometry error. Reducing the blade length A and increasing the radius R of the cutter can reduce the angle of the tangent of the cross section of the curved surface where the two blades are machined.
Fig. 4 Calculation of the overlap of the peripheral edge and the end edge 4 The angle a' is the angle between the revolution surface machined by the peripheral edge and the axis profile of the revolution surface machined by the end edge near the overlap. As shown in FIG. 4 , the axial profile of the turning surface machined by the end edge is a semicircle whose tangent at the point = is a plumb line. Therefore, it is only necessary to solve the axis profile curves GFE′ and y of the turning surface machined by the peripheral blade. The angle between the intersection point E' of the shaft and the plumb line may be sufficient. First find the tangent slope of the E' point. As shown in Fig. 4, the hyperbola is known to have E', F, and G points, and the coordinates in the yoz coordinate system are E'(R+D'max, 0), F(R+Dc0, -0.25Acosls, respectively). ), G(R+D'max, -0.5Acosls). Examine the hyperbola of the positive part of the Y-axis, refer to equation (6), and let it be in the form y2 - (zt)2 = 1 a2 b2 (15) Substituting the coordinates of E', F, and G in equation (15), Available t=-(Acosls)/4 (16) a2=(R+Dc0)2 (17) b2=(R+Dc0)2A2cos2ls 16(D'max-Dc0)(2R+D'max+Dc0) ( 18) Let a be the angle between the tangent line and the positive direction of the y-axis in the coordinate system, then the slope is tana. From equation (15), the slope of the hyperbola at E' point is tana=- b2+(R+D'max) A2t (19) Substituting equations (16), (17), and (18) into equation (19) yields a=arctan [ Acosls(R+D'max)] 4(D'max-Dc0)(2R+D) 'max+Dc0) (20) Since a' is the angle between the tangent line and the positive z-axis, a'=90°-a (21) Equation (21) is the revolution after machining the peripheral edge f-angle. The formula for solving the angle a' between the surface of the curved surface and the turning surface of the end edge at the overlap. D'max and a' are two indicators for measuring the geometrical error of the machining surface near the lap joint. The smaller the a' value is, the smaller the angle of the curved surface in the axial section of the rotating surface machined near the lap is. The smaller the D'max, the better the fit at the lap. 5 Calculation example Let the blade peripheral radius R = 25mm, blade length A = 12mm, w1 = 20°, ls = 15°. Calculate the values ​​of Dmax, D'max, f, Dc0, and a'. Let q = 45°, available from m = cotw1: m = 2.7475. From formula (10) available: Dmax = 1.5335mm. From formula (11) available: f=14.34°. From Formula (12), D'max=1.45×10-4mm. From formula (14) available: Dc0 = -0.00984mm. From equation (21) we can get: a'=0.395°. The above calculations show that the use of a symmetrical polygonal line-shaped cutting edge instead of a straight line-shaped cutting edge can significantly reduce the geometrical error of the machined surface of the overlapped surface at the peripheral edge of the indexable ball end mill, and enable the peripheral edge and end edge to rotate. An approximately first-order smooth connection is achieved at the lap, and the blade is easy to manufacture. The four sides of the original square on the rake face of the insert can be used as the cutting edge. When used, it can be rotated by 90°, that is, one blade can be equivalent to four linear blades and a single blade. Adopting the overlapping method can not only improve the utilization rate of the blade, but also facilitate the standardization of the blade. Therefore, it has good economic and practical value.
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