CALCULATION METHOD OF MILLING CONTACT AREA FOR BALL-END MILLING TOOL WITH TOOL INCLINATION ANGLE

Transcription

1 U.P.B. Sci. Bull., Series D, Vol. 76, Iss. 3, 214 ISSN CALCULATION METHOD OF MILLING CONTACT AREA FOR BALL-END MILLING TOOL WITH TOOL INCLINATION ANGLE Yishu HAO 1, Guoqing TANG 2, Meng ZHANG 3 Milling contact area of ball-end milling tool is studied when tool axis is not arallel to machining surface normal. Based on micro milling force modeling, cutting edge is divided into cutting edge elements along the axis direction by differential. The boundary equation is derived through analytical method. Line integral along cutting edge is used to obtain the whole milling contact area. According to the roosed calculation rogram of integration interval with tool inclination angle, virtual slot cutting exeriment is done. Comared with ublished exerimental results, the accuracy of roosed integration interval searching algorithm and milling force calculation rogram are verified. Keywords: contact area; ball-end milling tool; tool inclination angle 1. Introduction With the raid develoment of aerosace, recision arts and health care, aluminum alloy with high strength, high toughness, anti-corrosion is becoming a key material. To machine-tool system, the main sources of error in machining rocess are the geometry and kinematics error of CNC machine, the thermal deformation of rocess system and the force deformation of rocess system. Geometry and control system error of machine tool accounts for 18% of the final machining error, error caused by heat accounts for 6%, and surface osition error caused by milling force accounts for 76%, so the error caused by milling force is the main source of error[1]. The rediction of cutting forces in five-axis ball-end milling of free-form surfaces requires not only comrehensive modeling of cutting mechanics in three directions, but also the modeling of feed variation contributed by three linear and two rotary motions of the machine tool[2]. For ball-end milling tools, the cutting seed of each element edge is constantly changing [3], and shear force of differential edge is directly related to the milling seed[4]. In slot cutting exeriment, the tool sindle erendicular to the surface of workiece. Milling 1 Prof., School of Mechanical Engineering, Tongji University, China, yishu@tongji.edu.cn 2 Master, Tongji University, China, tangguoqing11@gmail.com 3 Master, Tongji University, China, zhangmengtongji@sina.com

2 82 Yishu Hao, Guoqing Tang, Meng Zhang tool from the ti osition to the maximum deth of cut is involved in cutting, through changing the axial cut deth of cutting tool, the relationshi between the osition of the cutting unit on the cutting edge and the milling force coefficient can be determined. Some scholars take axial deth as olynomial indeendent variable[5], some take cutting thickness and axial deth as variables to fit milling force coefficient[6], thus to calculate milling force, all of them has acquired high calculation accuracy. Due to variable tool axis orientation, free form surfaces of workiece and tool geometry, engagement regions between the tool and the workiece all along the tool ath are very comlex and irregular [7]. At resent, the main calculation methods of tool-workiece contact area are Z-ma method and analytical method. Z-ma method[8] exresses machined surface through using different height values on the corresonding X-Y grid, then tool-workiece contact area is obtained by intersecting swet volume which is generated on the basis of tool ath and tool geometry. Although Z-ma method can be effectively alied in machining rocess modeling for comlex curved surface, it occuies more comuting time when calculating contact area [9]. Analytical method calculates the tool-workiece contact area through solving workiece surface equation and tool geometrical equation. Based on micro milling force modeling theory, cutting edge of ball-end milling tool is divided into cutting edge element along the axis direction by differential in this aer. The boundary equation of contact area when tool axis is not arallel to the surface normal is deduced through analytical calculation. 2. Calculation of milling contact area for ball-end milling tool 2.1 Model of milling rocess for ball-end milling tool Milling force on the edge line involved in cutting can be obtained by integrating micro milling force on the edge line along the cutting edge. The integration interval is the contact area of milling tool and workiece. Line integral method along the cutting edge, which is a siral line shown in fig.1, is used in this aer to calculate the entire milling force. Only one integral limit is required to calculate at every corner osition, so it has high comutational efficiency. Suose the curvature radius of the machined surface is far greater than the sherical radius of ball-end milling tool, that is the workiece s surface is a lane. Take tool angle as abscissa, the distance along the tool axis away from tool ti Z as ordinate, set u coordinate system,z as shown in fig.2. When the tool axis is not arallel to the surface normal, the machining surface equation is the region between the contact start corner and the end corner of the line arallel to the abscissa line. The boundary of

3 Calculation method of milling contact area for ball-end milling tool with tool inclination angle 83 tool-workiece contact surface in,z coordinate system is the rectangular area ABCD shown in fig.2. Siral line equation shown in fig.1 is a slash in,z coordinate system. The edge line of ball-end milling tool intersects contact boundary at two oints, Z coordinate values of these two oints are the uer and lower limit of integral. Fig.1. Siral edge line st j ex Fig.2. Coordinate system and integral interval For the case when tool axis is arallel to the surface normal, the uer and lower limit of integration interval can be judged according to different ositions between the edge line and contact area boundary. As shown in fig.2, the integral limits of different conditions are calculated as follows. If j z, Case, if j z, Case 1, if z a st, if z gx and z a gx, Case 2, if z a st, Case 3, if z a st, Z, (1 Z, a (2 Z, 1 k j (3 Z, 1 k j (4 Z, a (5

4 84 Yishu Hao, Guoqing Tang, Meng Zhang Z, 1 k j (6 Case 4, if z gx and z a gx, the siral groove will be divorced from the cutting zone. Here is the contact start corner, is the contact end corner, of the oint which is on the jth edge line meets z (7 is tool s siral angle, the sloe of the siral angle meets k (8 R When tool axis is not arallel to the surface normal as is shown in fig.3. The machined surface is no longer a straight line arallel to the abscissa. Fig.3. Tool axis and surface normal is not arallel 2.2 Calculation method of milling contact area with tool inclination Through geometrical derivation, boundary height value of the contact area is exressed as a equation of tool corner. z = f ( α Fig.4. Contact area of tool and workiece with inclination In X-Y-Z system established on the basis of ball-end milling tool micro milling force model, the feed direction of ball-end milling tool is the ositive

5 Calculation method of milling contact area for ball-end milling tool with tool inclination angle 85 direction of X-axis. When ball-end milling tool turns an α angle around the line which inclines to the surface to be rocessed and arallel to Y-axis, and the workiece surface intersects the sherical surface to roduce a round. Then the sherical surface is divided into two arts, the small art is contact area of tool and workiece as the shadow art shown in fig.4. Suose α is ositive when ball-end milling tool inclines to the feed direction. The actual contact area is the half of contact surface at the milling direction side. The rojection of workiece surface on X-Z lane is a straight line. Suose the cutting deth is a, then the equation of the rojection line: R a f ( z = α =tanαx+ R (9 cosα Suose Q is a oint on the uer side of the contact area boundary, namely the oint is on the intersection s circular boundary of workiece surface and sherical surface, the X coordinate of the oint: R a z R + x = cosα (1 tanα Connect sherical surface center oint C and Q, make a vertical through Q to the tool axis, the vertical intersect Z-axis at oint M, and a right-angled triangle CMQ can be gotten, where 2 CP = MP + CM (11 According to fig.4, CM R z, CQ R, MQ= R ( R z (12 and for Q, x = MQsin (13 Substitute equation (1 and (12 into equation (13, and R a z R + R cos ( R z sin = α (14 tanα Solve the quadratic equation of one unknown and round solutions that do not meet the ractical significance, R a R 2 a R tan α sin tan αsin ( + R tan αsin z R cosα cosα (15 = + 1+ tan αsin Similarly, the lower side boundary equation on the X-Z lane meets z = cotα x+ R (16 So for one oint on the line,

6 86 Yishu Hao, Guoqing Tang, Meng Zhang R z x = (17 cotα Connect this oint and the sherical surface center C, make a vertical through it to the tool axis, intersects the tool axis at M, R z R ( R z sin = (18 cotα Solve the quadratic equation of one unknown and round solutions that do not meet the ractical significance, R cotα sin z = R (19 1+ cot α sin Different inclination directions corresond to different tool and workiece contact situations. When a R Rcosα, if the feed direction and the inclination direction are in the same lane, as is shown in fig.5. When α π R ( R a st = arctan, ex 2 ( R a sinα When α, 3π R ( R a st = arctan, ex 2 ( R a sinα π R ( R a = + arctan 2 ( R a sinα 3π R ( R a = + arctan 2 ( R a sinα (2 (21 (a α> (b α < (a α > (b α< Fig.5. When a R Rcosα, feed Fig.6. When a > R Rcosα, feed and inclination direction in the same lane and inclination direction in the same lane When a > R Rcosα, if the feed direction and the inclination direction are in the same lane, as is shown in fig.6. When α π R ( R a π R ( R a st = arctan, ex = + arctan (22 2 ( R a sinα 2 ( R a sinα When α

7 Calculation method of milling contact area for ball-end milling tool with tool inclination angle 87 st =, ex = 2π (23 When a < R Rcosα, if the feed direction is vertical to the inclination direction, as is shown in fig.7. when α R ( R a st =, ex = arctan (24 ( R a sin α When α R ( R a st = π arctan ( R a sin α, ex = π (25 When a > R Rcosα, if the feed direction is vertical to the inclination direction, as is shown in fig.8, π st =, ex = (26 2 (a α> (b α < (a α > (b α< Fig.7. When a < R Rcosα, feed Fig.8. When a > R Rcosα, feed direction is vertical to inclination direction direction is vertical to inclination direction Take the situation of a R R cos α as an examle, the feed direction and the inclination direction are in the same lane, and the feed direction corresonds to the inclination direction. The actual osition of tool-workiece contact area is shown in fig.9. Z is the lower boundary of contact area, Z is the uer boundary of the contact area, the actual contact area is sandwiched between and of the uer and lower contact boundary region. When the tool axis of ball-end milling tool is arallel to the surface normal of the workiece, the integral interval can be judged and calculated by simle conditions. When the tool axis is not arallel to the surface normal, suose the edge line equation intersects the boundary of contact area at,z and,z, z,z is the required integral interval. Because the boundary equation of edge rojection and contract area cannot be calculated directly, a search algorithm method is designed to search the intersections of edge line equation and

8 88 Yishu Hao, Guoqing Tang, Meng Zhang contact area boundary along the rojection curve. Set as tool corner, boundary of tool-workiece contact area is shown in fig.1, in,z coordinate system, z Z is the lower side boundary equation, z Z is the uer side boundary equation, z Z is the edge line equation, z Z intersects z Z at,z, z Z intersects z Z at,z, z is the Z coordinate of the contact area start. st π π 2 ex Fig.9. Integration interval when a R Rcosα and the feed direction is consistent with the inclination direction ( i 2, z2 z= z ( 2 ( i 1, z1 z= z ( 3 z= z( 1 Fig.1. Search algorithm diagram when a R Rcosα and the feed direction is consistent with the inclination direction The algorithm flow chart of range-searching,z is shown in fig.11, here ignores the effect of contact start corner and contact end corner to the π 2 π

9 Calculation method of milling contact area for ball-end milling tool with tool inclination angle 89 osition of,z, where k is the sloe of edge line equation. In the algorithm, judging the value Z and Z with the error limit of 1 mm will exist a certain error. Z and Z are the intersections of edge line and contact area boundary. Fig.11. Searching algorithm flow chart when a R Rcosα and the feed direction is consistent with the inclination direction 3. Verification of milling force results for ball-end milling tool Altintas and Lee[1] used integral method to calculate the micro milling force of ball-end milling tool, and did exerimental verification for ball-end milling force model they roosed on a single-edged ball-end milling tool. The workiece material was titanium alloy (Ti6Al4V, the cutting force coefficient of the workiece and cutting tool obtained in the exeriment was exlained. Calculation rogram considering integral limit of inclination is used in this aer, and virtual groove cut simulation exeriment is done according to the same cutting arameters and machining conditions, comared with the ublished exerimental results, the calculation accuracy of the roosed integral interval searching algorithm and milling force calculation rogram can be verified. The smooth lines are the milling force results use the calculation algorithm roosed in this aer. The corrugated lines are the exerimental results in Altintas and Lee s aer. Fig.12 shows the milling force of R =9.5 mm ball-end milling tool when sindle seed n=269 rmin, feed rate er tooth f =.5 mmtooth, axial cutting deth a =1.27 mm and 6.35 mm.

10 9 Yishu Hao, Guoqing Tang, Meng Zhang (a a =1.27 mm (b a =6.35 mm Fig.12. When α, Sindle seed n=269 rmin, R =9.5 mm, β =3, f =.5 mmtooth Fig.13(a shows the milling force of R =9.5 mm ball-end milling tool when sindle seed n=269 rmin, axial cutting deth a =3.81 mm, feed rate er tooth f =.1 mmtooth. Fig.13(b shows the milling force of R =6.5 mm ball-end milling tool when sindle seed n=269 rmin, axial cutting deth a =3.4 mm, feed rate er tooth f =.8 mmtooth. Fig. 13(a shows the maximum Y direction milling force in the exeriment is about 9 N, the maximum Z direction milling force is about 4 N, and in this aer the maximum error of calculated milling force in Y direction is about 12.5%, maximum error of milling force in Z direction is about 3%. (a a =3.81 mm, R =9.5 mm, (b a =3.48 mm, R =6.5 mm, f =.1 mmtooth f =.8 mmtooth Fig.13. Sindle seed n=269 rmin, β =3 Fig.14 shows results of machining exeriment when contact angle is -9, and the milling force of ball-end milling tool when the sindle seed n=269 rmin, R =9.5 mm, axial cutting deth a =6.35 mm, feed rate er tooth f =.5 mmtooth and f =.1 mmtooth. The calculated milling force in this aer has obvious errors after 6

11 Calculation method of milling contact area for ball-end milling tool with tool inclination angle 91 corner in Y and Z direction, and milling force in Y and Z direction aear shar angle at 9. The ossible shar angle osition by numerical calculation in slot cutting is the corner of 18, in semi-slot cutting is 9, they are all in the osition of milling edges begin to leave the milling areas. Using the error limit of 1 mm as a judgment in the rogram, when imrove judgment accuracy, shar angle can be significantly reduced. (a f =.5 mmtooth (b f =.1 mmtooth Fig.14. Semi-slot cutting, sindle seed n=269 rmin, a =6.35 mm, β =3 Using the roosed milling force calculation rogram, which takes tool inclination into consideration, to calculate the milling force corresonding to different milling arameters when α =. The error is relatively large during cutting-in and cutting-out rocess. When the maximum milling force is very large, the error of the milling force is obvious. 4. Conclusions The calculation of milling force is based on micro milling force modeling theory. Integrating the milling force of micro cutting edges along the contact edge line between tool and workiece, milling force in secific contact conditions can be gotten. Geometric analytical method is used to calculate boundary equations of contact area when tool axis of ball-end milling tool is not arallel to the machining surface normal. The calculation results of milling force through numerical otimization are comared with foreign research results roosed calculation method has higher accuracy in finishing rocess. R E F E R E N C E S [1]. Tony L.Schmitz, John C. Ziegert, J.Suzanne Canning, Raul Zaata. Case study: A comarison of errors sources in high-seed milling, Precision Engineering, vol.32, , 28. [2]. Oguzhan Tuysuz,Yusuf Altintas, Hsi-Yung Feng. Prediction of cutting forces in three and

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