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flute intersection determination
ARTICLE IN PRESS
International Journal of Machine Tools & Manufacture 48 (2008) 666–676 www.elsevier.com/locate/ijmactool
Drill flank measurement and flank/flute intersection determination W. Zhang, Y. Peng, F. He, D. Xiong School of Mechanical Engineering and Automation, Dalian Polytechnic University, Dalian 116034, China Received 18 May 2007; received in revised form 26 October 2007; accepted 30 October 2007 Available online 6 November 2007
Abstract Twist drill is a cutting tool of large-batch and diversified varieties. Geometric measurement is indispensable to drill design, manufacture, and regrinding. To meet the demand of advance in manufacturing technology, a three-dimensional (3-D) drill measurement instrument was developed. Method of drill flank and flute measurement by using this device is presented in this work. Cutting angles evaluation is essential to cutting performance appraisal. But, there is no suitable instrument yet to inspect them directly. If the flank/flute intersection is determined, cutting angles calculation would be a routine work of numerical computation. Considering the peculiarity of the drill point configuration, a straightforward method for flank/flute intersection is derived. Procedure of cutting angles calculation is also detailed. The procedure is illustrated by an example of flank and flute inspection, flank/flute intersection, and cutting angles calculation for a standard twist drill. The measurement precision is appraised by using a tool-room microscopy. The maximum error is in the third decimal place. It verifies the validity of the developed 3-D drill meter and the proposed measurement methods. r 2007 Elsevier Ltd. All rights reserved. Keywords: Drill; Measurement; Surface/surface intersection; Cutting angle
1. Introduction Geometric measurement is vital to drill design, manufacture, and regrinding. Drill measurement instrument becomes indispensable to manufacturing shop. Devices for drill point inspection have been reported [1–3] and such products are available in the market. But, they are confined to two-dimensional (2-D). Employing laser sensor, a cost effective and high precision three-dimensional (3-D) drill measurement apparatus was developed and reported [4]. Method for drill flank and flute measurement by using the device is described in this paper. Cutting angles evaluation is essential to performance assessment. Currently, drill-cutting angles could not be measured directly. However, drill-cutting angles can be calculated accurately via flank/flute intersection. Surface/surface intersection has attracted intensive study in the last decades [5–7]. Its start and terminal points are of special interest [8]. Researchers solved the drill flank/flute intersection by polynomial interpolation [9,10]. The results are not Corresponding author.
E-mail address: [email protected] (W. Zhang). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.10.020
accurate enough. An approach of higher precision is presented in this study. Contents of the next sections are as follows: method for flank and flute measurement and data processing; a straightforward solution for flank/flute intersection; calculation of flute surface normal using curve parametrization and B-spline approximation; procedure for cutting angles evaluation. Pseudo-code for the flank/flute intersection is attached in Appendix A. Cutting angle calculations are explained succinctly in Appendix B. Validity of the proposed methods is verified by an example. The flute cross-section measured and the flank/ flute intersection calculated are in good agreement with those inspected by using a tool-room microscopy.
2. Drill flank measurement and grinding parameters determination Drill flanks are usually quadratic surfaces, ellipsoidal, hyperbolic, or conic. The conic ones are prevalent. Using a 2-D meter, the flank surface measurement can be only processed by numerical exploration [3]. Employing a 3-D
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Nomenclature {O: X, Y, Z} global system with drill point center as origin, Z-axis coinciding with the drill axis at the initial position, Y-axis upward, X-axis to the left {A: x, y, z} local system parallel to the global system with an arbitrary point A on the flank as origin {Oc: xc, yc, zc} local system with the cone-apex Oc as origin, zc-axis coinciding with the cone-axis
667
(cx, cy, cz) coordinates of the cone-apex in the global system s intersection angle of the cone-axis with the Zaxis y semi-cone-angle af relief angle gn rake angle r semi-point-angle p helix parameter (p ¼ L/2p, L is the pitch)
laser sensor, the precision can be improved significantly. Principle, configuration, and calibration of this measurement apparatus have been detailed in a published paper [4]. Method for flank measurement and data processing using this device is described in this section.
they determine a spatial line. This is the generatrix passing point A.
2.1. Generatrices allocation
To determine the second generatrix, another point B on the flank is selected as the origin with coordinate system {B: x1, y1, z1} parallel to {O: X, Y, Z}. The second generatrix is determined by the same process shown above. Data in system {B: x1, y1, z1} are transformed into {A: x, y, z} to find the intersecting point, the cone-apex, of these two genratrices. A spatial line is represented by linear equations projected on both planes, associatively. For example (Fig. 3), the
The conic surface is characterized in linear generatrix. If three generatrices are determined, grinding parameters of the flank can be deduced. A global coordinate system {O: X, Y, Z} is established at the initial position, with origin O located at the drill tip. The Z-axis coincides with the drill axis and the laser beam direction (Fig. 1). The drill is chucked at the position that the starting points of both cutting lip are located on the X-axis (Fig. 2). To find a generatrix, an arbitrary point A on the flank is selected as the origin. A local coordinate system {A: x, y, z} is set up parallel to the global system {O: X, Y, Z} as shown in Fig. 2. The laser beam, controlled by joint movement of X- and Y-slide, (Fig. 1) scans along different linear paths passing through point A. The measured curves projected on each plane are likely to be quadratic, for example, y ¼ az2+bz on the y–A–z plane (Fig. 2) and x ¼ c2z þ dz on the x–A–z plane. Among the quadratics on each projection plane, there is one and only one degenerated to be a straight line. If both projections are generated to be straight lines,
Fig. 1. Drill measurement system using laser sensor.
2.3. Intersection of two generatrices
Fig. 2. Generatrix on the flank.
Fig. 3. Intersection of two generatrices.
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generatrix A1A2, passing point A, is expressed as y ¼ a1 z þ b1 ,
(1)
x ¼ c1 z þ d 1 .
(2)
The genertrix B1B2, passing point B, is expressed as
Normal-point formula: Nxðxc xc1 Þ þ Nyðyc yc1 Þ þ Nzðzc zc1 Þ ¼ 0.
(6)
y ¼ a2 z þ b2 ,
(3)
The normal components Nx, Ny, and Nz in Eq. (6) are calculated from Eq. (5). The normal vector is given by
x ¼ c2 z þ d 2 .
(4)
N ¼ Nxi þ Nyj þ Nzk.
The intersecting point, i.e., the apex, can be solved from a set of equations either Eqs. (1)–(3); or (1)–(4). Due to inevitable calculation errors, these spatial lines may be nonintersecting (Fig. 3). However, their projections on the x–A–y plane should intersect at a definite point O1. The corresponding points of point O1 on line A1A2 and B1B2 are Oa and Ob, respectively. The intersecting point Oc can be determined by one-dimensional (1-D) search between Oa and Ob. 2.4. Semi-cone-angle and cone-axis orientation The third generatrix is determined in accordance with the procedure mentioned above, starting from the apex Oc. The coordinate system {A: x, y, z} is converted into another parallel system {Oc: xc, yc, zc} (Fig. 4). A point on each generatrix is selected. The points P1, P2, and P3 with coordinates (xci, yci, zci) i ¼ 1,2,3 are equidistant to the apex Oc. They determine a circular plane perpendicular to the cone-axis. These data provide enough information to determine the flank surface. The circular plane can be expressed in either three-points formula (Eq. (5)) or normal-point formula (Eq. (6)), respectively.
Three-points formula: xc xc1 x x c1 c2 xc3 xc1
yc yc1 yc2 yc1 yc3 yc1
zc zc1 zc2 zc1 ¼ 0. zc3 zc1
(5)
(7)
The semi-cone-angle y is the intersecting angle between the normal N and any generatrix. The conic surface is expressed in the coordinate system {Oc: xc, yc, zc} by the following equation: x2c þ y2c z2c tan2 y ¼ 0.
(8)
Coordinates of the apex Oc in the global system is given by (cx, cy, cz) as shown in Fig. 5. Angle s is the intersection angle between the Z-axis and cone-axis projected on the X–O–Z plane. It is evaluated in accordance with Eqs. (5) and (6): s ¼ tan1 ðNx=NzÞ.
(9)
2.5. Flank surface equation and grinding parameters The local system {Oc: xc, yc, zc} is transformed to the global system {O: X, Y, Z} via the following matrix operation: 32 32 3 2 3 2 1 0 0 cx cos s 0 sin s 0 X xc 76 0 1 0 cy 76 Y 7 6y 7 6 0 1 0 0 76 76 7 6 c7 6 76 76 7. 6 7¼6 4 cc 5 4 sin s 0 cos s 0 54 0 0 1 cz 54 Z 5 0 0 0 1 1 1 0 0 0 1 Then, the conic surface (Eq. (8)) is transformed to the following equation: ððX cx Þ cos s ðZ cz Þ sin sÞ2 þ ðY cy Þ2 ððX cx Þ sin s þ ðZ cz Þ cos sÞ2 tan2 y ¼ 0.
ð10Þ
At the apex, coordinates X, Y, and Z equal to zero. Thus, the grinding parameters cx, cy, cz, s, and y are constrained by the following condition: ðcx cos s þ cz sin sÞ2 þ c2y ðcx sin s cz cos sÞ2 tan2 y ¼ 0.
ð11Þ
2.6. Comparison between measured parameters and known values
Fig. 4. Conic surface.
To verify the procedure mentioned above, a standard twist drill with diameter D ¼ 8 mm, helix angle ¼ 321 and known grinding parameters is measured. The parameters measured are compared with the known values as shown in Table 1.
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Fig. 5. Flank and grinding parameters.
3.1. Comparison of cross- and normal-sectional scanning
Table 1 Grinding parameters comparison cx (mm) cy (mm) cz (mm) s (deg) o (deg) y (deg) Known values 3.302 Measured results 3.182 Deviations 0.012
1.506 1.493 0.013
2.494 2.512 0.018
22.188 14.603 22.578 14.378 0.394 0.227
37.868 37.405 0.263
Maximum deviations are no more than 0.018 mm in length and 0.4 in angle.
To ensure precision of reflection, incident angle between the laser beam and the surface normal should be limited (smaller than 301). The normal vectors on the crosssectional curve are spatial and dispersed in direction. Then, the incident angles are uncontrollable and the shape of the laser reflection spot may be distorted. These defects can be avoided by normal-sectional scanning, because the normal vectors on the normal-sectional profile lay on the same plane. 3.2. Normal sectional measurement
Fig. 6. Drill flute measurement.
3. Measurement of the flute A global coordinate system is set up as shown in Fig. 6. The origin is located at the center of a cross-section to be scanned. X-axis aligns with the laser beam direction. Y-axis is upward. Z-axis coincides with the drill axis at the initial position. The laser sensor is particularly suitable to measure drill flute with 3-D concave surfaces. But, directly scanning on the cross-section is erroneous and normalsectional scanning is imperative. This is to be explained in the sub-section to follow.
If scanning on the normal section directly (Fig. 8a), serious errors occur near both ends of the profile (Fig. 8b), due to out-off limit of the incident angles. Scanning by segments is the remedy (Fig. 9). For normal-sectional scanning, the rotary Table 2 (Fig. 6) should be rotated around the Y-axis to locate the normal section perpendicular to the laser beam, i.e. line n–n coinciding the Y-axis (Fig. 7). For segmental scanning (Fig. 9) on both sides, the rotary Table 1 should be rotated around Z-axis with different angles (for example, +451 or 451). Under these rotations, the section center stands still, because of the configuration of the rotary tables (Fig. 6). The joint rotation of Tables 1 and 2 is to be detailed in the next sub-section. 3.3. Relationship among rotation angles Let OA1 be the unit vector perpendicular to the normal section. Angle b is equal to the helical angle. It is moved to OB1 by rotating Table 1 (Fig. 6) around Z-axis with an angle d: OA1 ¼ ½ 0 sin b
cos b T ,
(12a)
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Fig. 7. Normal section of the drill flute.
Z-component sin b sin y divided by the X-component cos b: tan a ¼ tan b sin d. OC1 is the final 2 cos a 6 OC 1 ¼ 6 4 0 sin a 2 6 OB1 ¼ 6 4
(13)
position of OA1: 3 0 sin a 7 1 0 7 5, 0 cos a 0
3
sin b cos d sin a sin b sin d þ cos a cos b
7 7. 5
ð14Þ
The directional angle b1 of OC1 is evaluated by its components on the YOZ plane: sin b1 ¼ sin b cos d.
(15)
The angle b1 defines the path of laser beam scanning along Y and Z complementarily. The vector OA attached to the normal section (Fig. 10), is rotated via OB to the final position OC. The angle j between OA and OC defines the rotation of the normal section pertinent to the laser beam scanning. The movements are calculated in the same procedure shown above. The results are as follows:
Fig. 8. Scanning the normal section directly.
OA ¼ ½ 0 2
cos b
sin b T ,
sin d cos b
(16)
3
6 7 OB ¼ 4 cos d cos b 5, sin d Fig. 9. Scanning by segments.
2
cos d 6 OB1 ¼ 4 sin d 0
2
32 3 2 3 0 sin d sin b sin d 0 76 7 6 7 cos d 0 54 sin b 5 ¼ 4 cos d sin b 5. cos b cos b 0 1 (12b)
The vector OA1 lies on the YOZ plane (Fig. 10), while OB1 is 3-D with the components shown in Eq. (12b). The laser sensor should keep constant distance with respect to the target surface in X direction. Thus, the vector OB1 should be moved back to the YOZ plane by rotating Table 2 around Y-axis with an angle a. Then, tan a is given by the
6 OC ¼ 4
cos a sin d= cos b
(17)
3
7 cos b cos d 5. cos a sin b cos d cos d
(18)
Then, the value of cos j is given by cos j ¼ OA OC ¼ cos d cos2 b þ sin bðsin a sin d cos b þ cos a sin bÞ. ð19Þ For the given helical angle b and required rotation angles j, d, and a for the rotary Table 1 and 2, respectively, are determined by Eqs. (19) and (13).
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Table 2 Measurement results comparison between laser sensor and microscope (excerpt) No.
1
2
3
4
5
6
7
8
9
10
Abscissa Laser sensor Microscope Deviations
0.6156 2.3141 2.3079 0.0061
0.6099 2.3167 2.3106 0.0009
0.6043 2.3193 2.3132 0.0027
0.5986 2.3219 2.3158 0.0033
0.5931 2.3245 2.3183 0.0061
0.5875 2.3270 2.3208 0.0061
0.5820 2.3295 2.3241 0.0055
0.5766 2.3319 2.3258 0.0038
0.5711 2.3344 2.3283 0.0061
0.5657 2.3368 2.3307 0.0061
The same in abscissas and different in ordinates for each point.
Fig. 10. Vector OA1 perpendicular to the normal section rotated via OB1 to OC1.
3.4. Example of flute measurement Data of a standard twist drill measured: diameter, D=8 mm; helical angle, b=31.9791. Angle of rotation for the left and right side segment: s=7451. Rotation angle for the rotary Table 1: y ¼ 7401. Rotation angle for the rotary Table 2: a ¼ 718.811. Angle of laser beam scanning path: b1 ¼ 25.061. Each segment is scanned 20 times, recording the averaged data and assuring some overlap between consecutive segments (Fig. 11). All these curves are rotated back to their original positions. Before connection, each curve is parameterized using B-spline. The conjunction points are determined in accordance with the condition of agreement in coordinates and tangents (Fig. 12). As a pre-processing for flank/flute intersection, the measured data of the normal-section are converted to the cross-section. Let the coordinates of the points on the normal section be denoted by the arrays X(1–n), Y(1–n), Z(1–n), where Z(n) is greater than Z(1). Let the coordinates of the points on the cross section be denoted by the arrays X0 (1–n), Y0 (1–n) with the same ordinate Z0. The operation is as follows: 32 3 2 2 3 2 3 X ðiÞ 0 X 0 ðiÞ cos oi sin oi 0 6 Y ðiÞ 7 6 sin oi cos oi 0 76 Y ðiÞ 7 6 7 0 54 54 4 0 5 ¼ 4 5, ZðiÞ ZðiÞ Z 0 Z0 0 0 1 (20)
where oi ¼ (Z(i)Z0)tan b/D. D is the drill diameter and b, the helical angle. For the purpose of comparison, another drill of the same kind is cut off along the normal section and the profile is inspected by using KEYENCE VE-9800 microscope. The results measured by the laser sensor are in good agreement with that inspected by the tool-room microscope. The maximum deviation, no more than 0.0061, exhibits at some point shown in Table 2. 4. Flank/flute surfaces intersection There is a substantial body of literature published in the field of surfaces intersection [5–8]. The start and terminal points are the most involved interrogations [8]. Considering the peculiarities of the drill point configuration, this problem can be solved easily. 4.1. Start point To facilitate flank/flute intersection, the global system {O: X, Y, Z} is converted to another system {O: X1, Y1, Z1} by rotating around Z-axis with an angle l, such that the start point A of the intersection is located at a distant r0 to the X1-axis, where r0 is the half-web-thickness (Fig. 13). Transferring system {O: X, Y, Z} to {O: X1, Y1, Z1}, the flank surface is converted from Eq. (10) to Eq. (21) as
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procedure: 1. Determining the Z-ordinate of the measured crosssection The first point (X0(1), Y0(1)) of the measured crosssection is transferred to (Xa, Ya) according to the following equations: X a ¼ X 0 ð1Þ cosððZa Z 1 Þ=pÞ Y 0 ð1Þ sinððZa Z1 Þ=pÞ, (24) Y a ¼ X 0 ð1Þ sinððZa Z 1 Þ=pÞ þ Y 0 ð1Þ cosððZa Z1 Þ=pÞ, (25) where p is the helix parameter (p ¼ L/2p, L is the pitch). Distance Z1 is the Z-ordinate of the measured crosssection. Thus, Z1 is determined by either Eq. (24) or Eq. (25). 2. Transferring the cross-section Points on the measured cross-section, stored in (X0(i), Y0(i), i ¼ 1, n) with ordinate Z1, are transferred to the final position, stored in (X1(i), Y1(i), i ¼ 1, n) with ordinate Za: X 1 ðiÞ ¼ X 0 ðiÞ cosððZ a Z 1 Þ=pÞ Y 0 ðiÞ sinððZ a Z 1 Þ=pÞ, (26) Y 1 ðiÞ ¼ X 0 ðiÞ sinððZ a Z 1 Þ=pÞ þ Y 0 ðiÞ cosððZ a Z 1 Þ=pÞ. (27) 3. Determining all other intersection points All other intersection points corresponding to the points in the array (X1(i), Y1(i), i ¼ 2–i0) are determined by 1-D search, as shown in Appendix A. Fig. 11. Scanning of different segments: (a) central segment, (b) left side segment rotated, and (c) right side segment rotated.
follows: ððX 1 cos l Y 1 sin l cx Þ cos s ðZ 1 cz Þ sin sÞ2 þ ðX 1 sin l þ Y 1 cos l cy Þ2 ððX 1 cos l Y 1 sin l cx Þ cos s þ ðZ 1 cz Þ cos sÞ2 tan2 y ¼ 0.
ð21Þ
The angle l is evaluated by l ¼ sin1 (r0/R), where R is the drill radius. The flank/flute intersection starts at point A with coordinates projected on the X1–O–Y1 plane as Y a ¼ r0 ,
(22)
X a ¼ ðR2 Y 2a Þ0:5 .
(23)
The ordinate Za of the starting point is obtained by substituting Eq. (22) and (23) into Eq. (21).
4.3. Example of flank/flute intersection The flank/flute surface intersection of the standard twist drill is determined with the procedure shown above. Part of the coordinates of the measured cross-sectional profile is listed in Table 3. Part of the calculated coordinates of the intersection points is listed in Table 4, starting at No. 1 and terminates at No. 135. The cutting lip can be measured directly via the laser sensor. Comparison of the measured and calculated data is listed in Table 5. It is impossible to compare point to point accordingly. Only several adjacent points are selected. The cross-sectional profile and intersection curve projected on the X1–O–Y1 plane are shown in the same picture (Fig. 14). It reveals that the flank/flute intersection data calculated are in good agreement with that measured by using the tool-room microscopy. The maximum deviation is no more than 0.003 mm. 5. Flute surface normal and cutting angles calculation
4.2. All other intersection points After determining the start point, all other intersection points are determined in accordance with the following
Determination of the cutting lip paves the way to cutting angles evaluation. The next step is to calculate the flute surface normal.
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Fig. 12. Normal section of the drill flute profile measured: (a) right side segment, (b) left side segment, and (c) all segments connected.
The helical surface equation is expressed in vector form: rðs; yÞ ¼ ðX 0 ðiÞ cos y Y 0 ðiÞsin yÞi þ ðX 0 ðiÞ sin y þ Y 0 ðiÞ cos yÞj þ pyk.
ð28Þ
5.2. Flute surface normal The unit normal at a point on the flute is given by Nr ¼ qr=qs qr=qy=jqr=qs qr=qyj. (29) Fig. 13. Start point of flank/flute intersection.
The partial derivatives qr/qs and qr/qy are derived as follows: qr=qs ¼ ðqX 0 ðiÞ=qs cos q qY 0 ðiÞ=qs sin yÞi þ ðqX 0 ðiÞ=qs sin y þ qY 0 ðiÞ=qs cos yÞj,
5.1. Curvilinear coordinates and flute equation Since the flute surface is represented in discrete data, the surface normal can be calculated only by numerical approximation. Researches tackled it through polynomial interpolation [7,8], but it is defective in precision due to the oscillatory behavior characterized in high-degree polynomial interpolation. The method of curve parametrization and B-spline approximation has proven useful for such data fitting. The flute surface is generated by helical movement of all the points on the cross-sectional profile around the Z-axis. The Cartesian system is parametrized to a curvilinear system (s, y) with point (s(i), y(i)) corresponding to (X0(i), Y0(i)). Arc length s is approximated by the sum of segments ds ¼ ððX 0 ði þ 1Þ X 0 ðiÞÞ2 þ ðY 0 ði þ 1Þ Y 0 ðiÞÞ2 Þ0:5 .
ð30Þ
qr=qy ¼ ðX 0 ðiÞ sin y Y 0 ðiÞ cos yÞi þ ðX 0 ðiÞ cos y Y 0 ðiÞ sin yÞj þ pk.
ð31Þ
Substituting Eqs. (30) and (31) into Eq. (29), the unit normal vector on the flute surface is represented by Nr ¼ Nrx i þ Nry j þ Nrz k, (32) where Nrx ¼ pððqX 0 ðiÞ=qs sin y þ qY 0 ðiÞ=qs cos yÞ=R, Nry ¼ pðqX 0 ðiÞ=qs cos y qY 0 ðiÞ=qs sin yÞ=R, Nrz ¼ pðX 0 ðiÞ X 0 ðiÞ=qs þ Y 0 ðiÞ qY 0 ðiÞ=qsÞ=R
ð33Þ
and R ¼ jqr=qs qr=qyj ¼ ðp2 ðqX 0 ðiÞ=qsÞ2 þ ðqY 0 ðiÞ=qsÞ2 þ ðX 0 qX 0 ðiÞ=qs þ Y 0 qY 0 ðiÞÞ2 Þ0:5 .
ð34Þ
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Table 3 Part of the cross-sectional profile measured No.
1
2
3
4
5
6
7
8
293
294
295
296
297
298
299
300
X Y
3.754 1.376
3.728 1.373
3.702 1.369
3.675 1.365
3.649 1.361
3.623 1.356
3.596 1.352
3.754 1.376
.819 3.748
.829 3.766
.839 3.785
.850 3.804
.860 3.822
.870 3.841
.880 3.860
.891 3.878
Table 4 Part of the coordinates of the flank/flute intersection calculated No.
1
2
3
4
5
6
7
8
129
130
131
132
133
134
135
X Y Z
3.945 .6557 2.163
3.944 .6556 2.162
3.919 .6556 2.147
3.892 .6556 2.131
3.866 .6555 2.115
3.839 .6556 2.099
3.813 .6556 2.083
3.786 .6556 2.067
.5671 .6554 .1135
.5406 .6554 .1175
.5139 .6554 .1015
.4873 .6554 .0855
.4607 .6554 .0695
.4341 .6554 0.0536
.4075 .6554. .0376
Table 5 Comparison of flank/flute intersection data No. 1 X Y Z
21
39
58
77
96
115
3.9459 3.9459 3.4404 3.4407 2.9616 2.9360 2.4561 2.4554 1.9506 1.9503 1.4451 1.4451 0.93964 0.9399 0.6557 0.6557 0.6554 0.6537 0.6553 0.6523 0.6552 0.6576 0.6551 0.6561 0.6552 0.6552 0.6553 0.6548 2.1629 2.1629 1.8593 1.8629 1.5717 1.5629 1.2681 1.2629 0.9645 0.9629 0.6609 0.6629 0.3572 0.3629
In each column, calculated data are on the left and measured, on the right.
where aij and bij are coefficients given by the software. Thus, the partial derivatives of X or Y with respect to s are calculated by using Eq. (35) and (36), respectively. At the node s(i), the former two terms in Eq. (35) and (36) vanish. The partial derivatives are denoted by qX0(i)/qs=ai3 and qY0(i)/qs=bi3, respectively. 5.4. Cutting angles calculation
Fig. 14. Flute cross-sectional profile and flank/flute intersection.
5.3. Partial differentiation In the curvilinear coordinate system (s, y), the variables X and Y are considered as different functions of the same variable s and y. They can be interpolated and differentiated, respectively, by using B-spline functions. The mathematic software Matlab implemented it in a toolbox. For example, the cubic spline interpolation provides piecewise second-order differentiated polynomials within the interval s(i)psps(i+1): qX 2 ðiÞ=qs ¼ ai1 ðs sðiÞÞ2 þ ai2 ðs sðiÞÞ þ ai3 ,
(35)
qY 2 ðiÞ=qs ¼ bi1 ðs sðiÞÞ2 þ bi2 ðs sðiÞÞ þ bi3 ,
(36)
Generally, drill-cutting angles can be only estimated, because no effective measurement apparatus is yet available. For a twist drill, there is rigorous mathematical relationship between the cutting angles and geometric configurations of the flank and flute surfaces. Based on the surface normal and flank/flute intersection, cutting angles can be calculated accurately with maximum error less than 103 in numerical differentiation. For reference, the cutting angles’ definition and calculation procedure are cited [10] and explained more concisely in terms of differential geometry in Appendix B. 5.5. Example of cutting angles calculation Cutting angles of the standard drill are calculated using the methods and procedure shown above. Part of the results is listed in Table 6. Currently, there is almost no instrument available to measure the rake and relief angles accurately. The latter can be estimated by using an angle meter. The relief angles estimated well approach to that calculated.
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Table 6 Part of the calculated cutting angles (in degree) No.
1
2
3
4
5
6
7
8
129
130
131
132
133
134
135
Rake Relief
33.28 15.69
32.56 15.72
32.21 15.75
32.04 15.78
31.81 15.82
31.58 15.85
31.38 15.88
31.18 15.92
25.94 29.92
27.44 30.17
29.03 30.42
30.71 30.65
32.49 30.88
34.39 31.10
36.43 31.30
6. Conclusions This paper associated with another paper [4] presents a procedural method for 3-D measurement of drill flank and flute. Methods for flank/flute intersection and cutting angles evaluation are deduced. Comparison of the measurement data and the inspection results by using a toolroom microscopy are as follows: maximum deviation of the flute cross-sectional profile is 0.0061 mm; deviations of the flank/flute intersection projected on the cross-sectional plane are less than 0.003 mm. Most of the errors are due to the machine parts’ inaccuracy, which are to be improved in the ascertainable production. However, these results verify that the 3-D drill meter and methods are valuable in engineering applications. It is simple in theory and easy to use and can be employed in the design, machining, and reconditioning of drill and other cutting tools. Conventional 2-D drill meters are incapable of performing these tasks, while general-purpose 3-D measurement machines are inconvenient and too expensive to do this everyday work. Appendix A. . Pseudo-code for flank/flute intersection exploration For i ¼ 2–300 (search begins at the point next to the start) x1 ¼ x0(i): y1 ¼ y0(i) (Search interval is initially assumed to be (z1, z2)) z1 ¼ 2.2: x ¼ x1 Cos((z1za)/p)y1 Sin((z1za)/ p): y ¼ x1 Sin((z1za)/p)+y1 Cos((z1za)/p) f1 ¼ f(z1) (Function f(z) is the flank surface equation) z2 ¼ 0: x ¼ x1 Cos((z2za)/p)y1 Sin((z2za)/p): y ¼ x1 Sin((z2za)/p)+y1 Cos((z2za)/p) f2 ¼ f(z2) start: i1 ¼ 0 i1 ¼ i1+1 z0 ¼ (z1+z2)/2 x ¼ x1 Cos((z0za)/p)y1 Sin((z0za)/p): y ¼ x1 Sin((z0za)/p)+y1 Cos((z0za)/p) f0 ¼ f(z0) If Abs(f0)p0.000001 Then GoTo continue : if i14100 GoTo finish (The search fails when iterations exceed 100) If f0 f1o0 Then f2 ¼ f0: z2 ¼ z0: GoTo start Else f1 ¼ f0: z1 ¼ z0: GoTo start End If
continue: zz(i) ¼ z0: xx(i) ¼ x: yy(i) ¼ y (Output (xx(i),yy(i),zz(i)), the intersection point of series number i) Next i Finish: i0 ¼ i (Output the terminal series number i0)
Appendix B. . Cutting angles calculation B.1. Flank surface normal For a flank surface with known equation F(x, y, z) ¼ 0 (Eq. (4)), the unit normal Nf is given by Nf ¼ ðqF =qx i þ qF =qy j þ qF =qz kÞ=ððqF =qxÞ2 þ ðqF =qyÞ2 þ ðqF =qzÞ2 Þ0:5
ðB:1Þ
or in components Nf ¼ Nfx i þ Nfy j þ Nfz k.
(B.2)
B.2. Cutting lip tangent The cutting lip tangent T perpendiculars to both the flute normal Nr (Eq. (21)) and flank normal Nf. Thus, it is given by the vector product: T ¼ Nr Nf =jNrjjN f j or in components T ¼ ðTx i þ Ty j þ Tz kÞ=ððTxÞ2 þ ðTyÞ2 þ ðTzÞ2 Þ0:5 ,
(B.3)
where Tx ¼ Nfy Nrz Nfz Nry, Ty ¼ Nfz Nrx Nfx Nrz, Tz ¼ ðNfx Nry Nfy NrxÞ.
ðB:4Þ
B.3. Bi-normal of flute surface Bi-normal Br of the flute is needed in rake angle calculation. It is normal to both the flute normal Nr and intersection tangent T: Br ¼ Nr T=jNr jjTj. Using Eq. (B.3), it is expressed in components as follows: Br ¼ ðTy Nrz Tz NryÞ=B i þ ðTz Nrx Tx NrzÞ=B j ðB:5Þ þ ðTz Nrx Tx NrzÞ=B k,
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where
The vector Tc is expressed in components as follows:
B ¼ ððTy Nrz Tz NryÞ2 þ ðTz Nrx Tx NrzÞ2 þ ðTz Nrx Tx NrzÞ2 Þ0:5 .
ðB:6Þ
Tc ¼ ðY Nfz i X Nfy j þ ðY Nfx X NfyÞkÞ=jNfjjRj. (B.14) Thus, the releif angle af is given by af ¼ cos1 ðTc VÞ
B.4. Rake angle
(B.15)
or
The rake angle of a point on the lip is defined as the intersection angle between the flute bi-normal Br and the vector Na. Na is normal to both the cutting velocity V and lip tangent T at that point: (B.7)
Na ¼ T V=jTjjVj.
cos af ¼ ððX 2 þ Y 2 Þ0:5 NfzÞ=ððX 2 þ Y 2 ÞNfz þ YNfx X NfyÞ0:5 .
(B.16) B.6. Semi-point angle
The vector V of an arbitrary point (X, Y, Z) on the lip is denoted by
Semi-point angle r of an arbitrary point on the cutting lip is the intersecting angle between the lip tangent and drill axis, Z-axis. The cos(r) is given by
V ¼ Y i X j þ 0 k.
cosðrÞ ¼ ðTxi þ Tyj þ TzkÞð0i þ 0j þ kÞ ¼ Tz
(B.8)
Thus, the unit vector Na is deduced as follows:
or
Na ¼ X Tz=Na i Y Tz=Na j ðX Tx þ Y TyÞ=Na k,
r ¼ cos1 ðTzÞ.
(B.17)
(B.9) where
References 2
2
2 0:5
Na ¼ ððX TzÞ þ ðY TzÞ þ ðX Tx þ Y TyÞ Þ . (B.10) Associating with Eq. (B.5), the rake angle gn formula is derived: gn ¼ cos1 ðNa Br Þ.
(B.11)
B.5. Relief angle The so-called relief angle is the abbreviation of ‘‘circumferential relief angle’’. It is an angle defined in the circumferential plane, the intersection angle between the velocity and circumferential surface tangent at the point. Obviously, the latter is a vector perpendicular to both the flank surface normal and the radius vector. Let Tc denotes this tangent vector, while R the radius vector: Tc ¼ Nf R=jNfjjRj,
(B.12)
where R ¼ X i þ Y j þ 0 k.
(B.13)
[1] B. Ramamoorthy, V. Radhakrishnan, Computer-aided inspection of cutting tool geometry, Precision Engineering 14 (1992) 28–34. [2] Lakshman Hazra, Hideo Kato, Takaharu Kuroda, Practical inspection system of drill point geometry by using simple measurement jig and image processing, Precision Engineering 25 (2001) 206–211. [3] J. Ni, D. Fan, B. Sun, Development of drill measurement system, Technical Report, University of Michigan, 1998. [4] Y. Peng, K. Hiroyuki, W. Zhang, Measurement of drill point geometry using laser sensor, International Journal of Machine Tool and Manufacture 47 (2007) 682–688. [5] C. Asteasu, Intersection of arbitrary surfaces, Computer-aided Design 20 (1988) 533–538. [6] C.L. Bajaj, C.M. Haffmann, R.E. Lynch, Tracing surface intersections, Computer-aided Geometric Design 5 (1988) 285–307. [7] P. Markot, R.L. Msgedson, Procedural method for evaluating the intersection curves of two parametric surfaces, Computer-aided Design 23 (6) (1991) 395–404. [8] Gregor Mullenheim, On determining start points for a surface/ surface intersection algorithm, Computer-aided Geometric Design 8 (1991) 401–408. [9] H.T. Huang, C.I. Weng, C.K. Chan, An analysis of clearance and rake angles along cutting edge for multifacet drill, Journal of Engineering for Industry 116 (1994) 8–16. [10] Kaichun Ren, Jun Ni, Analyses of drill flute and cutting angles, International Journal of Advanced Manufacturing Technology 15 (1999) 546–553.
International Journal of Machine Tools & Manufacture 48 (2008) 666–676 www.elsevier.com/locate/ijmactool
Drill flank measurement and flank/flute intersection determination W. Zhang, Y. Peng, F. He, D. Xiong School of Mechanical Engineering and Automation, Dalian Polytechnic University, Dalian 116034, China Received 18 May 2007; received in revised form 26 October 2007; accepted 30 October 2007 Available online 6 November 2007
Abstract Twist drill is a cutting tool of large-batch and diversified varieties. Geometric measurement is indispensable to drill design, manufacture, and regrinding. To meet the demand of advance in manufacturing technology, a three-dimensional (3-D) drill measurement instrument was developed. Method of drill flank and flute measurement by using this device is presented in this work. Cutting angles evaluation is essential to cutting performance appraisal. But, there is no suitable instrument yet to inspect them directly. If the flank/flute intersection is determined, cutting angles calculation would be a routine work of numerical computation. Considering the peculiarity of the drill point configuration, a straightforward method for flank/flute intersection is derived. Procedure of cutting angles calculation is also detailed. The procedure is illustrated by an example of flank and flute inspection, flank/flute intersection, and cutting angles calculation for a standard twist drill. The measurement precision is appraised by using a tool-room microscopy. The maximum error is in the third decimal place. It verifies the validity of the developed 3-D drill meter and the proposed measurement methods. r 2007 Elsevier Ltd. All rights reserved. Keywords: Drill; Measurement; Surface/surface intersection; Cutting angle
1. Introduction Geometric measurement is vital to drill design, manufacture, and regrinding. Drill measurement instrument becomes indispensable to manufacturing shop. Devices for drill point inspection have been reported [1–3] and such products are available in the market. But, they are confined to two-dimensional (2-D). Employing laser sensor, a cost effective and high precision three-dimensional (3-D) drill measurement apparatus was developed and reported [4]. Method for drill flank and flute measurement by using the device is described in this paper. Cutting angles evaluation is essential to performance assessment. Currently, drill-cutting angles could not be measured directly. However, drill-cutting angles can be calculated accurately via flank/flute intersection. Surface/surface intersection has attracted intensive study in the last decades [5–7]. Its start and terminal points are of special interest [8]. Researchers solved the drill flank/flute intersection by polynomial interpolation [9,10]. The results are not Corresponding author.
E-mail address: [email protected] (W. Zhang). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.10.020
accurate enough. An approach of higher precision is presented in this study. Contents of the next sections are as follows: method for flank and flute measurement and data processing; a straightforward solution for flank/flute intersection; calculation of flute surface normal using curve parametrization and B-spline approximation; procedure for cutting angles evaluation. Pseudo-code for the flank/flute intersection is attached in Appendix A. Cutting angle calculations are explained succinctly in Appendix B. Validity of the proposed methods is verified by an example. The flute cross-section measured and the flank/ flute intersection calculated are in good agreement with those inspected by using a tool-room microscopy.
2. Drill flank measurement and grinding parameters determination Drill flanks are usually quadratic surfaces, ellipsoidal, hyperbolic, or conic. The conic ones are prevalent. Using a 2-D meter, the flank surface measurement can be only processed by numerical exploration [3]. Employing a 3-D
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Nomenclature {O: X, Y, Z} global system with drill point center as origin, Z-axis coinciding with the drill axis at the initial position, Y-axis upward, X-axis to the left {A: x, y, z} local system parallel to the global system with an arbitrary point A on the flank as origin {Oc: xc, yc, zc} local system with the cone-apex Oc as origin, zc-axis coinciding with the cone-axis
667
(cx, cy, cz) coordinates of the cone-apex in the global system s intersection angle of the cone-axis with the Zaxis y semi-cone-angle af relief angle gn rake angle r semi-point-angle p helix parameter (p ¼ L/2p, L is the pitch)
laser sensor, the precision can be improved significantly. Principle, configuration, and calibration of this measurement apparatus have been detailed in a published paper [4]. Method for flank measurement and data processing using this device is described in this section.
they determine a spatial line. This is the generatrix passing point A.
2.1. Generatrices allocation
To determine the second generatrix, another point B on the flank is selected as the origin with coordinate system {B: x1, y1, z1} parallel to {O: X, Y, Z}. The second generatrix is determined by the same process shown above. Data in system {B: x1, y1, z1} are transformed into {A: x, y, z} to find the intersecting point, the cone-apex, of these two genratrices. A spatial line is represented by linear equations projected on both planes, associatively. For example (Fig. 3), the
The conic surface is characterized in linear generatrix. If three generatrices are determined, grinding parameters of the flank can be deduced. A global coordinate system {O: X, Y, Z} is established at the initial position, with origin O located at the drill tip. The Z-axis coincides with the drill axis and the laser beam direction (Fig. 1). The drill is chucked at the position that the starting points of both cutting lip are located on the X-axis (Fig. 2). To find a generatrix, an arbitrary point A on the flank is selected as the origin. A local coordinate system {A: x, y, z} is set up parallel to the global system {O: X, Y, Z} as shown in Fig. 2. The laser beam, controlled by joint movement of X- and Y-slide, (Fig. 1) scans along different linear paths passing through point A. The measured curves projected on each plane are likely to be quadratic, for example, y ¼ az2+bz on the y–A–z plane (Fig. 2) and x ¼ c2z þ dz on the x–A–z plane. Among the quadratics on each projection plane, there is one and only one degenerated to be a straight line. If both projections are generated to be straight lines,
Fig. 1. Drill measurement system using laser sensor.
2.3. Intersection of two generatrices
Fig. 2. Generatrix on the flank.
Fig. 3. Intersection of two generatrices.
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generatrix A1A2, passing point A, is expressed as y ¼ a1 z þ b1 ,
(1)
x ¼ c1 z þ d 1 .
(2)
The genertrix B1B2, passing point B, is expressed as
Normal-point formula: Nxðxc xc1 Þ þ Nyðyc yc1 Þ þ Nzðzc zc1 Þ ¼ 0.
(6)
y ¼ a2 z þ b2 ,
(3)
The normal components Nx, Ny, and Nz in Eq. (6) are calculated from Eq. (5). The normal vector is given by
x ¼ c2 z þ d 2 .
(4)
N ¼ Nxi þ Nyj þ Nzk.
The intersecting point, i.e., the apex, can be solved from a set of equations either Eqs. (1)–(3); or (1)–(4). Due to inevitable calculation errors, these spatial lines may be nonintersecting (Fig. 3). However, their projections on the x–A–y plane should intersect at a definite point O1. The corresponding points of point O1 on line A1A2 and B1B2 are Oa and Ob, respectively. The intersecting point Oc can be determined by one-dimensional (1-D) search between Oa and Ob. 2.4. Semi-cone-angle and cone-axis orientation The third generatrix is determined in accordance with the procedure mentioned above, starting from the apex Oc. The coordinate system {A: x, y, z} is converted into another parallel system {Oc: xc, yc, zc} (Fig. 4). A point on each generatrix is selected. The points P1, P2, and P3 with coordinates (xci, yci, zci) i ¼ 1,2,3 are equidistant to the apex Oc. They determine a circular plane perpendicular to the cone-axis. These data provide enough information to determine the flank surface. The circular plane can be expressed in either three-points formula (Eq. (5)) or normal-point formula (Eq. (6)), respectively.
Three-points formula: xc xc1 x x c1 c2 xc3 xc1
yc yc1 yc2 yc1 yc3 yc1
zc zc1 zc2 zc1 ¼ 0. zc3 zc1
(5)
(7)
The semi-cone-angle y is the intersecting angle between the normal N and any generatrix. The conic surface is expressed in the coordinate system {Oc: xc, yc, zc} by the following equation: x2c þ y2c z2c tan2 y ¼ 0.
(8)
Coordinates of the apex Oc in the global system is given by (cx, cy, cz) as shown in Fig. 5. Angle s is the intersection angle between the Z-axis and cone-axis projected on the X–O–Z plane. It is evaluated in accordance with Eqs. (5) and (6): s ¼ tan1 ðNx=NzÞ.
(9)
2.5. Flank surface equation and grinding parameters The local system {Oc: xc, yc, zc} is transformed to the global system {O: X, Y, Z} via the following matrix operation: 32 32 3 2 3 2 1 0 0 cx cos s 0 sin s 0 X xc 76 0 1 0 cy 76 Y 7 6y 7 6 0 1 0 0 76 76 7 6 c7 6 76 76 7. 6 7¼6 4 cc 5 4 sin s 0 cos s 0 54 0 0 1 cz 54 Z 5 0 0 0 1 1 1 0 0 0 1 Then, the conic surface (Eq. (8)) is transformed to the following equation: ððX cx Þ cos s ðZ cz Þ sin sÞ2 þ ðY cy Þ2 ððX cx Þ sin s þ ðZ cz Þ cos sÞ2 tan2 y ¼ 0.
ð10Þ
At the apex, coordinates X, Y, and Z equal to zero. Thus, the grinding parameters cx, cy, cz, s, and y are constrained by the following condition: ðcx cos s þ cz sin sÞ2 þ c2y ðcx sin s cz cos sÞ2 tan2 y ¼ 0.
ð11Þ
2.6. Comparison between measured parameters and known values
Fig. 4. Conic surface.
To verify the procedure mentioned above, a standard twist drill with diameter D ¼ 8 mm, helix angle ¼ 321 and known grinding parameters is measured. The parameters measured are compared with the known values as shown in Table 1.
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Fig. 5. Flank and grinding parameters.
3.1. Comparison of cross- and normal-sectional scanning
Table 1 Grinding parameters comparison cx (mm) cy (mm) cz (mm) s (deg) o (deg) y (deg) Known values 3.302 Measured results 3.182 Deviations 0.012
1.506 1.493 0.013
2.494 2.512 0.018
22.188 14.603 22.578 14.378 0.394 0.227
37.868 37.405 0.263
Maximum deviations are no more than 0.018 mm in length and 0.4 in angle.
To ensure precision of reflection, incident angle between the laser beam and the surface normal should be limited (smaller than 301). The normal vectors on the crosssectional curve are spatial and dispersed in direction. Then, the incident angles are uncontrollable and the shape of the laser reflection spot may be distorted. These defects can be avoided by normal-sectional scanning, because the normal vectors on the normal-sectional profile lay on the same plane. 3.2. Normal sectional measurement
Fig. 6. Drill flute measurement.
3. Measurement of the flute A global coordinate system is set up as shown in Fig. 6. The origin is located at the center of a cross-section to be scanned. X-axis aligns with the laser beam direction. Y-axis is upward. Z-axis coincides with the drill axis at the initial position. The laser sensor is particularly suitable to measure drill flute with 3-D concave surfaces. But, directly scanning on the cross-section is erroneous and normalsectional scanning is imperative. This is to be explained in the sub-section to follow.
If scanning on the normal section directly (Fig. 8a), serious errors occur near both ends of the profile (Fig. 8b), due to out-off limit of the incident angles. Scanning by segments is the remedy (Fig. 9). For normal-sectional scanning, the rotary Table 2 (Fig. 6) should be rotated around the Y-axis to locate the normal section perpendicular to the laser beam, i.e. line n–n coinciding the Y-axis (Fig. 7). For segmental scanning (Fig. 9) on both sides, the rotary Table 1 should be rotated around Z-axis with different angles (for example, +451 or 451). Under these rotations, the section center stands still, because of the configuration of the rotary tables (Fig. 6). The joint rotation of Tables 1 and 2 is to be detailed in the next sub-section. 3.3. Relationship among rotation angles Let OA1 be the unit vector perpendicular to the normal section. Angle b is equal to the helical angle. It is moved to OB1 by rotating Table 1 (Fig. 6) around Z-axis with an angle d: OA1 ¼ ½ 0 sin b
cos b T ,
(12a)
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Fig. 7. Normal section of the drill flute.
Z-component sin b sin y divided by the X-component cos b: tan a ¼ tan b sin d. OC1 is the final 2 cos a 6 OC 1 ¼ 6 4 0 sin a 2 6 OB1 ¼ 6 4
(13)
position of OA1: 3 0 sin a 7 1 0 7 5, 0 cos a 0
3
sin b cos d sin a sin b sin d þ cos a cos b
7 7. 5
ð14Þ
The directional angle b1 of OC1 is evaluated by its components on the YOZ plane: sin b1 ¼ sin b cos d.
(15)
The angle b1 defines the path of laser beam scanning along Y and Z complementarily. The vector OA attached to the normal section (Fig. 10), is rotated via OB to the final position OC. The angle j between OA and OC defines the rotation of the normal section pertinent to the laser beam scanning. The movements are calculated in the same procedure shown above. The results are as follows:
Fig. 8. Scanning the normal section directly.
OA ¼ ½ 0 2
cos b
sin b T ,
sin d cos b
(16)
3
6 7 OB ¼ 4 cos d cos b 5, sin d Fig. 9. Scanning by segments.
2
cos d 6 OB1 ¼ 4 sin d 0
2
32 3 2 3 0 sin d sin b sin d 0 76 7 6 7 cos d 0 54 sin b 5 ¼ 4 cos d sin b 5. cos b cos b 0 1 (12b)
The vector OA1 lies on the YOZ plane (Fig. 10), while OB1 is 3-D with the components shown in Eq. (12b). The laser sensor should keep constant distance with respect to the target surface in X direction. Thus, the vector OB1 should be moved back to the YOZ plane by rotating Table 2 around Y-axis with an angle a. Then, tan a is given by the
6 OC ¼ 4
cos a sin d= cos b
(17)
3
7 cos b cos d 5. cos a sin b cos d cos d
(18)
Then, the value of cos j is given by cos j ¼ OA OC ¼ cos d cos2 b þ sin bðsin a sin d cos b þ cos a sin bÞ. ð19Þ For the given helical angle b and required rotation angles j, d, and a for the rotary Table 1 and 2, respectively, are determined by Eqs. (19) and (13).
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Table 2 Measurement results comparison between laser sensor and microscope (excerpt) No.
1
2
3
4
5
6
7
8
9
10
Abscissa Laser sensor Microscope Deviations
0.6156 2.3141 2.3079 0.0061
0.6099 2.3167 2.3106 0.0009
0.6043 2.3193 2.3132 0.0027
0.5986 2.3219 2.3158 0.0033
0.5931 2.3245 2.3183 0.0061
0.5875 2.3270 2.3208 0.0061
0.5820 2.3295 2.3241 0.0055
0.5766 2.3319 2.3258 0.0038
0.5711 2.3344 2.3283 0.0061
0.5657 2.3368 2.3307 0.0061
The same in abscissas and different in ordinates for each point.
Fig. 10. Vector OA1 perpendicular to the normal section rotated via OB1 to OC1.
3.4. Example of flute measurement Data of a standard twist drill measured: diameter, D=8 mm; helical angle, b=31.9791. Angle of rotation for the left and right side segment: s=7451. Rotation angle for the rotary Table 1: y ¼ 7401. Rotation angle for the rotary Table 2: a ¼ 718.811. Angle of laser beam scanning path: b1 ¼ 25.061. Each segment is scanned 20 times, recording the averaged data and assuring some overlap between consecutive segments (Fig. 11). All these curves are rotated back to their original positions. Before connection, each curve is parameterized using B-spline. The conjunction points are determined in accordance with the condition of agreement in coordinates and tangents (Fig. 12). As a pre-processing for flank/flute intersection, the measured data of the normal-section are converted to the cross-section. Let the coordinates of the points on the normal section be denoted by the arrays X(1–n), Y(1–n), Z(1–n), where Z(n) is greater than Z(1). Let the coordinates of the points on the cross section be denoted by the arrays X0 (1–n), Y0 (1–n) with the same ordinate Z0. The operation is as follows: 32 3 2 2 3 2 3 X ðiÞ 0 X 0 ðiÞ cos oi sin oi 0 6 Y ðiÞ 7 6 sin oi cos oi 0 76 Y ðiÞ 7 6 7 0 54 54 4 0 5 ¼ 4 5, ZðiÞ ZðiÞ Z 0 Z0 0 0 1 (20)
where oi ¼ (Z(i)Z0)tan b/D. D is the drill diameter and b, the helical angle. For the purpose of comparison, another drill of the same kind is cut off along the normal section and the profile is inspected by using KEYENCE VE-9800 microscope. The results measured by the laser sensor are in good agreement with that inspected by the tool-room microscope. The maximum deviation, no more than 0.0061, exhibits at some point shown in Table 2. 4. Flank/flute surfaces intersection There is a substantial body of literature published in the field of surfaces intersection [5–8]. The start and terminal points are the most involved interrogations [8]. Considering the peculiarities of the drill point configuration, this problem can be solved easily. 4.1. Start point To facilitate flank/flute intersection, the global system {O: X, Y, Z} is converted to another system {O: X1, Y1, Z1} by rotating around Z-axis with an angle l, such that the start point A of the intersection is located at a distant r0 to the X1-axis, where r0 is the half-web-thickness (Fig. 13). Transferring system {O: X, Y, Z} to {O: X1, Y1, Z1}, the flank surface is converted from Eq. (10) to Eq. (21) as
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procedure: 1. Determining the Z-ordinate of the measured crosssection The first point (X0(1), Y0(1)) of the measured crosssection is transferred to (Xa, Ya) according to the following equations: X a ¼ X 0 ð1Þ cosððZa Z 1 Þ=pÞ Y 0 ð1Þ sinððZa Z1 Þ=pÞ, (24) Y a ¼ X 0 ð1Þ sinððZa Z 1 Þ=pÞ þ Y 0 ð1Þ cosððZa Z1 Þ=pÞ, (25) where p is the helix parameter (p ¼ L/2p, L is the pitch). Distance Z1 is the Z-ordinate of the measured crosssection. Thus, Z1 is determined by either Eq. (24) or Eq. (25). 2. Transferring the cross-section Points on the measured cross-section, stored in (X0(i), Y0(i), i ¼ 1, n) with ordinate Z1, are transferred to the final position, stored in (X1(i), Y1(i), i ¼ 1, n) with ordinate Za: X 1 ðiÞ ¼ X 0 ðiÞ cosððZ a Z 1 Þ=pÞ Y 0 ðiÞ sinððZ a Z 1 Þ=pÞ, (26) Y 1 ðiÞ ¼ X 0 ðiÞ sinððZ a Z 1 Þ=pÞ þ Y 0 ðiÞ cosððZ a Z 1 Þ=pÞ. (27) 3. Determining all other intersection points All other intersection points corresponding to the points in the array (X1(i), Y1(i), i ¼ 2–i0) are determined by 1-D search, as shown in Appendix A. Fig. 11. Scanning of different segments: (a) central segment, (b) left side segment rotated, and (c) right side segment rotated.
follows: ððX 1 cos l Y 1 sin l cx Þ cos s ðZ 1 cz Þ sin sÞ2 þ ðX 1 sin l þ Y 1 cos l cy Þ2 ððX 1 cos l Y 1 sin l cx Þ cos s þ ðZ 1 cz Þ cos sÞ2 tan2 y ¼ 0.
ð21Þ
The angle l is evaluated by l ¼ sin1 (r0/R), where R is the drill radius. The flank/flute intersection starts at point A with coordinates projected on the X1–O–Y1 plane as Y a ¼ r0 ,
(22)
X a ¼ ðR2 Y 2a Þ0:5 .
(23)
The ordinate Za of the starting point is obtained by substituting Eq. (22) and (23) into Eq. (21).
4.3. Example of flank/flute intersection The flank/flute surface intersection of the standard twist drill is determined with the procedure shown above. Part of the coordinates of the measured cross-sectional profile is listed in Table 3. Part of the calculated coordinates of the intersection points is listed in Table 4, starting at No. 1 and terminates at No. 135. The cutting lip can be measured directly via the laser sensor. Comparison of the measured and calculated data is listed in Table 5. It is impossible to compare point to point accordingly. Only several adjacent points are selected. The cross-sectional profile and intersection curve projected on the X1–O–Y1 plane are shown in the same picture (Fig. 14). It reveals that the flank/flute intersection data calculated are in good agreement with that measured by using the tool-room microscopy. The maximum deviation is no more than 0.003 mm. 5. Flute surface normal and cutting angles calculation
4.2. All other intersection points After determining the start point, all other intersection points are determined in accordance with the following
Determination of the cutting lip paves the way to cutting angles evaluation. The next step is to calculate the flute surface normal.
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Fig. 12. Normal section of the drill flute profile measured: (a) right side segment, (b) left side segment, and (c) all segments connected.
The helical surface equation is expressed in vector form: rðs; yÞ ¼ ðX 0 ðiÞ cos y Y 0 ðiÞsin yÞi þ ðX 0 ðiÞ sin y þ Y 0 ðiÞ cos yÞj þ pyk.
ð28Þ
5.2. Flute surface normal The unit normal at a point on the flute is given by Nr ¼ qr=qs qr=qy=jqr=qs qr=qyj. (29) Fig. 13. Start point of flank/flute intersection.
The partial derivatives qr/qs and qr/qy are derived as follows: qr=qs ¼ ðqX 0 ðiÞ=qs cos q qY 0 ðiÞ=qs sin yÞi þ ðqX 0 ðiÞ=qs sin y þ qY 0 ðiÞ=qs cos yÞj,
5.1. Curvilinear coordinates and flute equation Since the flute surface is represented in discrete data, the surface normal can be calculated only by numerical approximation. Researches tackled it through polynomial interpolation [7,8], but it is defective in precision due to the oscillatory behavior characterized in high-degree polynomial interpolation. The method of curve parametrization and B-spline approximation has proven useful for such data fitting. The flute surface is generated by helical movement of all the points on the cross-sectional profile around the Z-axis. The Cartesian system is parametrized to a curvilinear system (s, y) with point (s(i), y(i)) corresponding to (X0(i), Y0(i)). Arc length s is approximated by the sum of segments ds ¼ ððX 0 ði þ 1Þ X 0 ðiÞÞ2 þ ðY 0 ði þ 1Þ Y 0 ðiÞÞ2 Þ0:5 .
ð30Þ
qr=qy ¼ ðX 0 ðiÞ sin y Y 0 ðiÞ cos yÞi þ ðX 0 ðiÞ cos y Y 0 ðiÞ sin yÞj þ pk.
ð31Þ
Substituting Eqs. (30) and (31) into Eq. (29), the unit normal vector on the flute surface is represented by Nr ¼ Nrx i þ Nry j þ Nrz k, (32) where Nrx ¼ pððqX 0 ðiÞ=qs sin y þ qY 0 ðiÞ=qs cos yÞ=R, Nry ¼ pðqX 0 ðiÞ=qs cos y qY 0 ðiÞ=qs sin yÞ=R, Nrz ¼ pðX 0 ðiÞ X 0 ðiÞ=qs þ Y 0 ðiÞ qY 0 ðiÞ=qsÞ=R
ð33Þ
and R ¼ jqr=qs qr=qyj ¼ ðp2 ðqX 0 ðiÞ=qsÞ2 þ ðqY 0 ðiÞ=qsÞ2 þ ðX 0 qX 0 ðiÞ=qs þ Y 0 qY 0 ðiÞÞ2 Þ0:5 .
ð34Þ
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Table 3 Part of the cross-sectional profile measured No.
1
2
3
4
5
6
7
8
293
294
295
296
297
298
299
300
X Y
3.754 1.376
3.728 1.373
3.702 1.369
3.675 1.365
3.649 1.361
3.623 1.356
3.596 1.352
3.754 1.376
.819 3.748
.829 3.766
.839 3.785
.850 3.804
.860 3.822
.870 3.841
.880 3.860
.891 3.878
Table 4 Part of the coordinates of the flank/flute intersection calculated No.
1
2
3
4
5
6
7
8
129
130
131
132
133
134
135
X Y Z
3.945 .6557 2.163
3.944 .6556 2.162
3.919 .6556 2.147
3.892 .6556 2.131
3.866 .6555 2.115
3.839 .6556 2.099
3.813 .6556 2.083
3.786 .6556 2.067
.5671 .6554 .1135
.5406 .6554 .1175
.5139 .6554 .1015
.4873 .6554 .0855
.4607 .6554 .0695
.4341 .6554 0.0536
.4075 .6554. .0376
Table 5 Comparison of flank/flute intersection data No. 1 X Y Z
21
39
58
77
96
115
3.9459 3.9459 3.4404 3.4407 2.9616 2.9360 2.4561 2.4554 1.9506 1.9503 1.4451 1.4451 0.93964 0.9399 0.6557 0.6557 0.6554 0.6537 0.6553 0.6523 0.6552 0.6576 0.6551 0.6561 0.6552 0.6552 0.6553 0.6548 2.1629 2.1629 1.8593 1.8629 1.5717 1.5629 1.2681 1.2629 0.9645 0.9629 0.6609 0.6629 0.3572 0.3629
In each column, calculated data are on the left and measured, on the right.
where aij and bij are coefficients given by the software. Thus, the partial derivatives of X or Y with respect to s are calculated by using Eq. (35) and (36), respectively. At the node s(i), the former two terms in Eq. (35) and (36) vanish. The partial derivatives are denoted by qX0(i)/qs=ai3 and qY0(i)/qs=bi3, respectively. 5.4. Cutting angles calculation
Fig. 14. Flute cross-sectional profile and flank/flute intersection.
5.3. Partial differentiation In the curvilinear coordinate system (s, y), the variables X and Y are considered as different functions of the same variable s and y. They can be interpolated and differentiated, respectively, by using B-spline functions. The mathematic software Matlab implemented it in a toolbox. For example, the cubic spline interpolation provides piecewise second-order differentiated polynomials within the interval s(i)psps(i+1): qX 2 ðiÞ=qs ¼ ai1 ðs sðiÞÞ2 þ ai2 ðs sðiÞÞ þ ai3 ,
(35)
qY 2 ðiÞ=qs ¼ bi1 ðs sðiÞÞ2 þ bi2 ðs sðiÞÞ þ bi3 ,
(36)
Generally, drill-cutting angles can be only estimated, because no effective measurement apparatus is yet available. For a twist drill, there is rigorous mathematical relationship between the cutting angles and geometric configurations of the flank and flute surfaces. Based on the surface normal and flank/flute intersection, cutting angles can be calculated accurately with maximum error less than 103 in numerical differentiation. For reference, the cutting angles’ definition and calculation procedure are cited [10] and explained more concisely in terms of differential geometry in Appendix B. 5.5. Example of cutting angles calculation Cutting angles of the standard drill are calculated using the methods and procedure shown above. Part of the results is listed in Table 6. Currently, there is almost no instrument available to measure the rake and relief angles accurately. The latter can be estimated by using an angle meter. The relief angles estimated well approach to that calculated.
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Table 6 Part of the calculated cutting angles (in degree) No.
1
2
3
4
5
6
7
8
129
130
131
132
133
134
135
Rake Relief
33.28 15.69
32.56 15.72
32.21 15.75
32.04 15.78
31.81 15.82
31.58 15.85
31.38 15.88
31.18 15.92
25.94 29.92
27.44 30.17
29.03 30.42
30.71 30.65
32.49 30.88
34.39 31.10
36.43 31.30
6. Conclusions This paper associated with another paper [4] presents a procedural method for 3-D measurement of drill flank and flute. Methods for flank/flute intersection and cutting angles evaluation are deduced. Comparison of the measurement data and the inspection results by using a toolroom microscopy are as follows: maximum deviation of the flute cross-sectional profile is 0.0061 mm; deviations of the flank/flute intersection projected on the cross-sectional plane are less than 0.003 mm. Most of the errors are due to the machine parts’ inaccuracy, which are to be improved in the ascertainable production. However, these results verify that the 3-D drill meter and methods are valuable in engineering applications. It is simple in theory and easy to use and can be employed in the design, machining, and reconditioning of drill and other cutting tools. Conventional 2-D drill meters are incapable of performing these tasks, while general-purpose 3-D measurement machines are inconvenient and too expensive to do this everyday work. Appendix A. . Pseudo-code for flank/flute intersection exploration For i ¼ 2–300 (search begins at the point next to the start) x1 ¼ x0(i): y1 ¼ y0(i) (Search interval is initially assumed to be (z1, z2)) z1 ¼ 2.2: x ¼ x1 Cos((z1za)/p)y1 Sin((z1za)/ p): y ¼ x1 Sin((z1za)/p)+y1 Cos((z1za)/p) f1 ¼ f(z1) (Function f(z) is the flank surface equation) z2 ¼ 0: x ¼ x1 Cos((z2za)/p)y1 Sin((z2za)/p): y ¼ x1 Sin((z2za)/p)+y1 Cos((z2za)/p) f2 ¼ f(z2) start: i1 ¼ 0 i1 ¼ i1+1 z0 ¼ (z1+z2)/2 x ¼ x1 Cos((z0za)/p)y1 Sin((z0za)/p): y ¼ x1 Sin((z0za)/p)+y1 Cos((z0za)/p) f0 ¼ f(z0) If Abs(f0)p0.000001 Then GoTo continue : if i14100 GoTo finish (The search fails when iterations exceed 100) If f0 f1o0 Then f2 ¼ f0: z2 ¼ z0: GoTo start Else f1 ¼ f0: z1 ¼ z0: GoTo start End If
continue: zz(i) ¼ z0: xx(i) ¼ x: yy(i) ¼ y (Output (xx(i),yy(i),zz(i)), the intersection point of series number i) Next i Finish: i0 ¼ i (Output the terminal series number i0)
Appendix B. . Cutting angles calculation B.1. Flank surface normal For a flank surface with known equation F(x, y, z) ¼ 0 (Eq. (4)), the unit normal Nf is given by Nf ¼ ðqF =qx i þ qF =qy j þ qF =qz kÞ=ððqF =qxÞ2 þ ðqF =qyÞ2 þ ðqF =qzÞ2 Þ0:5
ðB:1Þ
or in components Nf ¼ Nfx i þ Nfy j þ Nfz k.
(B.2)
B.2. Cutting lip tangent The cutting lip tangent T perpendiculars to both the flute normal Nr (Eq. (21)) and flank normal Nf. Thus, it is given by the vector product: T ¼ Nr Nf =jNrjjN f j or in components T ¼ ðTx i þ Ty j þ Tz kÞ=ððTxÞ2 þ ðTyÞ2 þ ðTzÞ2 Þ0:5 ,
(B.3)
where Tx ¼ Nfy Nrz Nfz Nry, Ty ¼ Nfz Nrx Nfx Nrz, Tz ¼ ðNfx Nry Nfy NrxÞ.
ðB:4Þ
B.3. Bi-normal of flute surface Bi-normal Br of the flute is needed in rake angle calculation. It is normal to both the flute normal Nr and intersection tangent T: Br ¼ Nr T=jNr jjTj. Using Eq. (B.3), it is expressed in components as follows: Br ¼ ðTy Nrz Tz NryÞ=B i þ ðTz Nrx Tx NrzÞ=B j ðB:5Þ þ ðTz Nrx Tx NrzÞ=B k,
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where
The vector Tc is expressed in components as follows:
B ¼ ððTy Nrz Tz NryÞ2 þ ðTz Nrx Tx NrzÞ2 þ ðTz Nrx Tx NrzÞ2 Þ0:5 .
ðB:6Þ
Tc ¼ ðY Nfz i X Nfy j þ ðY Nfx X NfyÞkÞ=jNfjjRj. (B.14) Thus, the releif angle af is given by af ¼ cos1 ðTc VÞ
B.4. Rake angle
(B.15)
or
The rake angle of a point on the lip is defined as the intersection angle between the flute bi-normal Br and the vector Na. Na is normal to both the cutting velocity V and lip tangent T at that point: (B.7)
Na ¼ T V=jTjjVj.
cos af ¼ ððX 2 þ Y 2 Þ0:5 NfzÞ=ððX 2 þ Y 2 ÞNfz þ YNfx X NfyÞ0:5 .
(B.16) B.6. Semi-point angle
The vector V of an arbitrary point (X, Y, Z) on the lip is denoted by
Semi-point angle r of an arbitrary point on the cutting lip is the intersecting angle between the lip tangent and drill axis, Z-axis. The cos(r) is given by
V ¼ Y i X j þ 0 k.
cosðrÞ ¼ ðTxi þ Tyj þ TzkÞð0i þ 0j þ kÞ ¼ Tz
(B.8)
Thus, the unit vector Na is deduced as follows:
or
Na ¼ X Tz=Na i Y Tz=Na j ðX Tx þ Y TyÞ=Na k,
r ¼ cos1 ðTzÞ.
(B.17)
(B.9) where
References 2
2
2 0:5
Na ¼ ððX TzÞ þ ðY TzÞ þ ðX Tx þ Y TyÞ Þ . (B.10) Associating with Eq. (B.5), the rake angle gn formula is derived: gn ¼ cos1 ðNa Br Þ.
(B.11)
B.5. Relief angle The so-called relief angle is the abbreviation of ‘‘circumferential relief angle’’. It is an angle defined in the circumferential plane, the intersection angle between the velocity and circumferential surface tangent at the point. Obviously, the latter is a vector perpendicular to both the flank surface normal and the radius vector. Let Tc denotes this tangent vector, while R the radius vector: Tc ¼ Nf R=jNfjjRj,
(B.12)
where R ¼ X i þ Y j þ 0 k.
(B.13)
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