The International Journal of Advanced Manufacturing Technology (2018) 98:579–591 https://doi.org/10.1007/s00170-018-2181-9
ORIGINAL ARTICLE
Modeling of integral cutting tool grooves using envelope theory and numerical methods Guochao Li 1,2 & Honggen Zhou 1,2 & Xuwen Jing 1,2 & Guizhong Tian 1,2 & Lei Li 1,2 Received: 14 November 2017 / Accepted: 13 May 2018 / Published online: 14 June 2018 # Springer-Verlag London Ltd., part of Springer Nature 2018
Abstract A groove is a key component of the structure of end mills, drills, and other integrated cutting tools. Machining a groove is one of the most difficult, time-consuming, and costly manufacturing process; therefore, for the sake of reduction of the machining cost and meeting the environmental regulations, the modeling of a machined groove with known wheel geometry and position is necessary for cutting tool manufacture. In order to reveal the process more clear and precise, the envelope theory and numerical methods are used. First, the basic calculation procedure for groove section points is built using a meshing equation. Accordingly, four universal problems for the simulation of groove manufacturing process are analyzed by four typical examples. Namely, the wheel side surface may interfere the machined tool edge and lead to an incorrect simulation; the wheel revolution surface may overcut the machined tool edge and produce fake points that would disturb the results; the tip point might not be precise enough; and the groove section points might be distributed unevenly and result in an imprecise groove section. The conditions to solve these problems are established by mathematical formulas and calculated by numerical methods. In addition, an integral procedure is built to simulate the machined groove with correct, precise, and even distribution points. Finally, groove simulation software is developed using MATLAB GUI, and the results are verified. Keywords Cutting tool groove . Manufacturing . Numerical method
Nomenclature OG-XGYGZG OT-XTYTZT gr1, gr2, gr3 ga1, ga2 gb Rg Q m1
Grinding wheel coordinate system Drill coordinate system Fillet radius of end-face of wheel (mm) Wheel angle (deg.) Thickness of grinding wheel (mm) Radius of grinding wheel (mm) A point on the revolution surface of wheel Distance between Q and XG-YG plane (mm)
* Guochao Li
[email protected]
m2 Rt rQ αx (Δx, Δy, Δz) P t n gro_pti con_pti pt1, pt2 dcon_pt
1
2
School of Mechanical Engineering, Jiangsu University of Science and Technology, Zhenjiang, China Jiangsu Provincial Key Laboratory of Advanced Manufacturing for Marine Mechanical Equipment, Jiangsu University of Science and Technology, Zhenjiang, China
Δe′ acc_d
Angle between line QOG and XG-axis (deg.) Radius of drill (mm) Coordinates of Q in wheel coordinate system Rotation angle around the XG-axis (deg.) Coordinates of OG in the drill coordinate system Groove lead (mm) Angle of the wheel rotated around the ZT-axis (deg.) Number of m1 Groove section points, i is its serial number Contact points, i is its serial number Intersections of the wheel edge and blank cylinder Distance between contact points and ZT-axis (mm) The allowed tip precision (mm) Distribution of the groove section points (mm)
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1 Introduction Integral cutting tools, which are manufactured by grinding a raw hard metal rod, obtaining an endmill, a ball-endmill, or a drilling tool, have been widely used in the modern aerospace, automobile, mold, and medical equipment industries [1]. A groove is one of the crucial structures of integral cutting tools, and it has great impact on the tool stiffness, chip removal ability, actual rake angle, cutting edge strength, and dynamics behaviors. It is generally machined by the costly and time-consuming grinding processes. Furthermore, as complex special surfaces, many new design grooves have to use the trial and error approach to develop their grinding processes. Thus, as an alternative of practical trials, and for the sake of reduction of the machining cost and meeting the environmental regulations in the cutting tool groove manufacturing processes, the simulation of the machined groove by using computer technology is necessary. Currently, the cutting tool grooves are mainly simulated by three methods: Boolean, graphically, and analysis. In the Boolean method, the groove manufacturing process was treated as a series of Boolean subtraction operations, where the blank was the “objective” and the wheel was the “tool.” With the secondary development technology, the groove simulation model was obtained in AutoCAD by Mohan [2] and in NX UG by Li [3] and Kim [4]. The Boolean method agrees well with practice so that the results could be used as evidences for other simulation method. However, it is time-consuming and the important groove parameters, such as rake angle, groove width, and core radius, cannot be deduced simultaneously [4]. The graphical method focused on calculating and identifying groove section points by dispersing the wheel and the blank. The blank and the wheel were dispersed into successive discs and shells with different radius, and the points on the groove could be calculated by solving the intersections of each disc and shell. Ko [5, 6] and Uhlmann and Hübert [7] obtained the groove section points by dispersing the wheel and blank into a series of discs and tubes. Alternatively, Beju et al. [8], Li et al. [9], and Karpuschewski et al. [10] calculated the points set on the blank section plane left by the wheel surface family during the manufacturing process. The groove section curve was then identified by a boundary searching method. The graphic method has no wheel geometry limitation. It has strong robust to simulate the groove profile and parameters. However, there is a conflict between the precision and speed for this method. Moreover, it just provides a final result. As a result, the manufacturing process cannot be revealed and analyzed [10]. The main aspect of the analysis method is to establish the implicit or explicit expressions of the machined groove, including the envelope groove and the trajectory groove. As basis of analysis method, the groove machining mechanism has been studied by many researchers. Kang et al. [11] established a relationship between the wheel origin, wheel
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direction vector, and contact points based on the envelope theory that the common normal of the wheel and groove must pass through the wheel axis. Armarego [12] and Kang [13] analyzed the groove manufacturing process graphically and found that the groove was partially enveloped by wheel revolution surface and partially generated by the trajectory of the single point on the wheel generatrix. Accordingly, Zhang et al. [14] determined that the groove may be produced by three different approaches: envelope only, trajectory only, and envelope together with trajectory. Furthermore, Li et al. [15] indicated that the groove surface normal at any contact point must intersect with the wheel axis, whether it was produced using envelope or trajectory. As discussed, the groove machining theory had been illustrated extensively. Accordingly, Hsieh [16] established the groove expression with the contact line formula derived by the envelope theory. Kang et al. [13] built a groove expression with the trajectory of the singular point located on the wheel generatrix. Additionally, Zhang et al. [17] solved for the point connecting the envelope and trajectory groove by translating the surface envelope to the curve envelope. Alternatively, Nguyen and Ko [18] built a universal groove simulation model by substituting the singular points with a series of normal vectors. Thus, the model was simplified and could be solved using envelope theory. Wang and Chen [19] deduced a simulation model for the 1V1 type wheel by combining the analysis and graphical methods. Xiao et al. [20] derived the explicit expressions of groove geometries including rake angle, core radius, and groove width. The analysis method has a high calculation precision and speed, as well as a clear exhibition of the process. However, a complex transcendental equation must be solved when the wheel generatrix consists of a quadratic curve. In this study, in order to improve the accuracy and the efficiency of integral cutting tool groove prediction model compared to traditional methods, as well as to reveal and analyze process problems for groove machining, the groove manufacturing process was modeled by using the envelope theory and the numerical method. This paper is organized as follows. First, the basic calculation procedure for contact points and groove section points is established. Second, four typical examples are carried out to analyze the four deficiencies, including side interference, fake points, tip imprecision, and uneven point distribution. Third, the conditions and methods to solve these problems are discussed, and the groove simulation procedure is established. Lastly, the procedure is programed using MATLAB GUI, and the results are verified.
2 Basic numerical procedure for groove simulation According to the space analytical geometric theory, the contact curve between the wheel and the blank in the groove
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grinding process can be considered as a generatrix for the machined groove. Thus, the groove was simulative as long as the contact curve was calculated. As the curve was almost impossible to define with analytic expressions, the numerical method was employed to express the curve with sufficient discrete points.
2.1 Basic coordinates for the groove manufacturing process The wheel geometry is one of the decisive factors for the machined groove. Rababah et al. [21] and Chen et al. [22] reported that the cutting tool groove is usually machined by standard 1A1 or 1V1 type wheels. Hsieh [23] and Pham [24] described that a nonstandard wheel with two oblique surfaces was also used. For the sake of generality, a wheel with two oblique surfaces was discussed in the study. As presented in Ren et al. [25], the wheel fixed coordinate frame is established in Fig. 1a. The XG-YG plane coincided with one of the wheel side surfaces and the coordinate origin located at the center of the wheel edge (the boundary circle of the wheel side surface). As the groove was mainly machined by the wheel revolution surface, the wheel was defined by eight parameters: Rg, gb, gb1, ga1, ga2, gr1, gr2, and gr3. Obviously, the 1A1 or 1V1 type wheels can be easily derived by substituting 0 for some of the parameters. The wheel revolution surface can be written in the wheel coordinates as reported by Li et al. [3]: G
rQ ¼
G
xQ ; G yQ ; G z Q
T
h iT ¼ f g cosðm2 Þ; f g sinðm2 Þ; m1
2.2 Basic groove simulation procedure As shown in Fig. 1b, the wheel revolution surface at the beginning of the groove manufacturing process was expressed in the blank coordinates as 3 2 1 0 0 Δx 6 0 cosðαx Þ −sinðαx Þ Δy 7 G G G T 7⋅ xQ yQ zQ 1 ð2Þ rQ ¼ 6 4 0 sinðαx Þ cosðαx Þ 0 5 0 0 0 1 During the manufacturing process, the wheel moved in a screw path with a certain lead P. Thus, a family of the wheel revolution surfaces was generated and could be deduced as T rQ ðm1 ; m2 ; t Þ ¼ T xQ T yQ T zQ 1 2 3 cosðt Þ −sinðt Þ 0 0 6 sinðt Þ cosðt Þ 0 7T 0 7⋅ r ¼6 ð3Þ 4 0 0 1 ðPtÞ=ð2πÞ 5 Q 0 0 0 1 where P was the groove lead and t was the angle of the wheel rotated around the ZT-axis. As the groove was an envelope surface generated by the family of wheel surfaces, the meshing equation can be obtained in the form described by Bogale et al. [28]: 0 T 1 ∂ rQ T ∂ r QA T @ ∂m1 ⋅∂ rQ ∂m2 ð4Þ
∂t ¼ 0 ð1Þ
where fg ∈ {fg1, fg2, fg3, fg4, fg5} The wheel position relative to the blank was the other decisive factor for the machined groove. According to the space analytical geometric theory, the wheel position in space consisted of at least six parameters, i.e., three location parameters and three posture parameters. As the wheel and the blank were both revolution solids, and considering that the position parameter along the ZT-axis had no effect on the machined groove geometry, three parameters (αx, Δx and Δy) were selected to define the wheel position, as reported by Nguyen et al. [26] and Wang et al. [27]. As presented in Fig. 1b, the OG-XGYGZG was first concentric with the blank fixed coordinate frame OT-XTYTZT, then rotated around the ZT-axis with αx and moved to the point (Δx, Δy, 0). According to practice, the wheel positions were limited by αx ∈ (0°,90°), Δx ∈ (Rg, Rt + Rg), and Δy ∈ (0, Rg), where Rt is the cutting tool radius and Rg is the wheel radius.
where m1 was the distance between the point on the wheel revolution surface (namely the point Q in Fig. 1a) and the XGYG plane, and m2 was the angle between the line OGXG and OGQ, as illustrated in Fig. 1a. Namely, the contact point should meet the condition: 1 0 T ∂ rQ T ; ∂ rQ C B ð5Þ f contactpoint ðm1 ; m2 ; t Þ ¼ @ ∂m1 T A¼0 ∂m2 ; ∂ rQ ∂t
Theoretically, the contact points could be calculated by solving Eqs. (3) and (5) with a certain t value. However, the obstacle encountered was that an analytical solution for Eq. (5) was nearly impossible unless the wheel generatrix was a simple straight line. On the one hand, Eq. (5) could not be built if there was a singular point on the wheel generatrix. On the other hand, Eq. (5) would be a complex transcendental equation when the wheel generatrix consisted of quadratic curves. To solve these problems, the numerical method and reasonable assumptions were introduced. For the first problem, the singular point was replaced by a transition round corner with a
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Fig. 1 Basic coordinates for the groove manufacturing process. (a) Wheel coordinates; (b) Groove manufacturing process coordinates
(a)
very small radius, which conformed with reality. For the second problem, Eq. (5) was solved to obtain a certain m2 value by using function “fzero()” in MATLAB software with the known m1 and t. After this, the contact points could be calculated by Eqs. (3) and (5) in MATLAB software with the calculation procedure provided in Fig. 2. For convenience, t was set to 0 and m1 = [m11, m12, m13,…, m1n]T was divided equally between 0 and gb. Transforming the calculated contact points to the XT-YT plane, the points on the groove section were obtained and the groove was simulated.
3 Examples and analysis As the groove manufacturing process could be treated as a sequence of Boolean subtraction operations with the wheel as the “cutter” and the blank as the “objective,” a groove grinding simulation system was established by applying NX 8.0 secondary development technology. As reported by Li et al. [29] and Tang et al. [30], the simulated model based on Boolean operations in NX, which was entirely consistent with reality, was used to check the calculation results. Four examples were conducted with different machining parameters provided in Table 1. The main differences were Rt, ga1, and αx. The results of the first three examples are illustrated in Figs. 3, 4, and 5. The contact points and groove section points were first calculated in MATLAB and then imported into NX. The number of m1 was 100 (namely, n = 100). To display clearly, some of the points were represented with red balls. Alternatively, the points in Figs. 3d and 4a were just the calculation results. According to Figs. 3, 4, and 5, some conclusions and problems can be summarized as follows:
(b)
1. The calculated groove section points agree well with the simulated model based on Boolean operations in NX. Thus, the method introduced in Fig. 2 was verified, as shown in Figs. 3c and 4b. 2. There was a one-to-one correspondence between the contact points, the groove section points, and the values of m1. 3. For Ex1 and Ex2, some calculated points were not located on the practical groove section curve if the blank radius was larger than 11 mm (or smaller). Examples of these points include gro_pt12, gro_pt13, gro_pt14, gro_pt15, and gro_pt16; they are also called as fake points and are presented in Figs. 3d and 4. According to Fig. 3a, it can be seen that these fake points were first generated by con_pt12 ~con_pt16 and then overcut by points con_pt10 and con_pt11. This problem, known as the fake points problem, should be solved and supplemented to the calculation procedure because these fake points can disturb the simulation result. 4. The cutting tip was defined imprecisely (known as the tip imprecision problem). The deviation, which was the distance of the desired and calculated tip points, reached 0.3 mm for Ex1, as shown in Fig. 3d. Although the tip precision might be improved by a larger discretization number based on Fig. 2, it was still difficult to ensure the desired precision. Therefore, considering that the cutting tip was crucial for a precision groove model, the tip calculation procedure should be supplemented. 5. The cutting tip might be defined incorrectly. Alternatively, from Ex1 and Ex2, Ex3 provided an accidental case where there were no calculated points located on the practical groove section curve near the tip. As shown in Fig. 5c, gro_pt1 was not located on the groove and there were no calculated points on the groove near the
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Fig. 2 Basic numerical procedure for groove simulation
tip (denoted as gro_fault). Analyzing Fig. 5a, b, this problem can be explained by two factors. One is that the groove was first produced by the wheel edge and then re-machined by the wheel side face. The con_fault illustrated in Fig. 5b was just the part of the wheel side surface Table 1
that produced the gro_fault in Fig. 5c. The other is that the calculation method only considered the wheel revolution surface and ignored its side surface. As a result, no calculation was carried out on the side wheel surface and the gro_fault was not expressed. Considering that the wheel
Groove machining parameters
Examples Cutting tool geometries Wheel parameters Geometries
Ex1 Ex2 Ex3 Ex4
Positions
Rt (mm)
P (mm)
Rg (mm) gb (mm) ga1 (°) ga2 (°) gb1 (mm) gr1 (mm) gr2 (mm) gr3 (mm) Δx (mm) Δy (mm) αx (°)
10 12 10 10
60 60 60 60
75 75 75 75
Note: The entries in italic are variables
20 20 20 20
10 10 90 90
70 70 70 70
5 5 5 5
1 1 1 1
1 1 1 1
1 1 1 1
75 75 75 75
− 15 − 15 − 15 − 15
38 38 50 38
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Fig. 3 Simulation and calculation results for Ex1 (gro_pti denotes groove section points and i is its serial number. con_pti denotes contact points and i is its serial number). (a) Contact points between wheel and groove; (b) Contact points on the grinding wheel; (c) Points on the groove section line; (d) Enlarged view near the tool edge
side was not designed for machining, this problem (called the side interference problem) should be simulated and prevented. 6. The calculated groove section points were distributed unevenly with the even values of m1, as shown in Figs. 3c, 4b, and 5c. The calculated groove section points near the tip were sparser than others. As a result, the accuracy of the important rake surface was low and Fig. 4 Simulation and calculation results for Ex2 (gro_pti denotes groove section points and i is its serial number. con_pti denotes contact points and i is its serial number). (a) Detail View; (b) Global view
could not be improved efficiently by increasing the discretization number. In conclusion, the basic procedure introduced in Fig. 2 was correct, and four problems had to be solved to improve its accuracy and efficiency, namely, the side interference, fake points, tip imprecision, and uneven distribution problems.
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Fig. 5 Simulation and calculation results for Ex3 (gro_pti denotes groove section points and i is its serial number. con_pti denotes contact points and i is its serial number). (a) Contact points (or line) between wheel and groove; (b) Contact points (or line) on the grinding wheel; (c) Points on the groove section line
4 Numerical procedure for groove simulation
2. Definition of ideal cutting tool edge
4.1 Noninterference condition
The helical curve with certain lead P can be written in the general formula Trtooledge = [Txtooledge, Tytooledge, Tztooledge]T = [Rt c o s ( t ) , R t s i n ( t ) , P t / ( 2 π ) ] T. S o l v i n g e q u a t i o n s T ytooledge = Typt1 andTytooledge = Typt2, the values of tpt1 and tpt2 can be derived, respectively. Thus, the ideal cutting tool edge passing through pt1 and pt2 was deduced as
Figure 6 shows the side interference problem. The ellipse line is the intersection of the XG-YG plane and the blank cylinder. pt1 and pt2 are intersections of the wheel edge and blank cylinder and pt1 has a larger value of Tz than pt2. The ideal cutting tool edge is a helical curve pass through pt1 with a certain lead P. The figure shows that Ex4 had no side interference problem compared to Ex3 by reducing αx from 50° to 38° and that the side interference problem would occur only when the ellipse line located below the ideal cutting tool edge in the ZT-axis direction. Therefore, the noninterference condition could be established with the following steps. 1. Definition of pt1 and pt2 Substituting fg = Rg1 into Eq. (3), the wheel edge can be expressed in the blank coordinates as T
rwheeledge ¼
T
xwheeledge T ywheeledge T zwheeledge 2 3 Δx þ Rg1 cosðm2 Þ ¼ 4 Δy þ Rg1 cosðαx Þsinðm2 Þ 5 Rg1 sinðαx Þsinðm2 Þ
xwheeledge 2 þT ywheeledge 2 ¼R2t
xwheeledge T ywheeledge T zwheeledge 2 3 Rt ⋅cos t pt1 −t ¼ 4 Rt ⋅sin t pt1 −t −t⋅p=ð2⋅πÞ5; t∈ 0; t pt1 −t pt2 ð8Þ T zpt1
rwheeledge ¼
As the intersection of the XG-YG plane and blank cylinder, the ellipse line could be given by the vector
Substituting Eq. (6) into Eq. (7), two values of m2 were deduced. The larger value was denoted as m2_max and the smaller value was denoted as m2_min. According to the machining coordinates presented in Fig. 1b, pt1 (Txpt1, Typt1, Tzpt1) and pt2 (Txpt2, Typt2, Tzpt2) can be calculated by substituting m2_min and m2_max into Eq. (6) separately.
xellipseline T yellipseline T zellipseline 2 3 Rt cos t pt1 −t 5 ¼4 R t sin tpt1 −t ; t∈ 0; t pt1 −t pt2 tanðαx Þ Rt sin t pt1 −t −Δy
rellipseline ¼
ð6Þ
ð7Þ
T
3. Definition of ellipse line
T
where Rg1 was the radius of the wheel edge presented in Fig. 1a. The condition for the points located on the blank cylinder was T
T
T
ð9Þ 4. Definition of noninterference condition Considering that the side interference could be prevented as long as the ellipse line had a larger TZ value than that of the ideal cutting tool edge between pt1 and pt2, the noninterference conditions were obtained in the form f condition
non sideinterference
¼ T zellipseline −T zideatooledge ≥ 0; t∈
0; t pt1 −t pt2
ð10Þ
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Fig. 6 Side interference problem analysis. (a) Example3; (b) Example4
Equation (10) could be easily solved using the numerical method. First, a solution set of fcondition_non_sideinterference was calculated by substituting a number for t, divided equally between 0 and tpt1−tpt2, into Eq. (10). Then, Eq. (10) would be satisfied as long as the minimum solution was not less than 0.
In practice, the cutting tool edge could be only machined by the fg1, fg2, or fg3 parts of the wheel provided in Fig. 1a. Thus, the potential fake contact points were obtained by substituting m1 = [0,Gzwheel_pt7], where Gzwheel_pt7 was the Gz value of point 7 in Fig. 1a, into Eqs. (3) and (5). The contact points were written as
4.2 Non-fake point condition To illustrate the fake point problem more clearly, Ex2 was recalculated by increasing n from 100 to 400. The calculated contact points and groove section points were projected onto the XT-YT plane, as shown in Fig. 7. As mentioned, the groove section points gro_pt51~ gro_pt66 were first obtained and then overcut by con_pt 37~ con_pt51 . As a result, the points gro_pt37~ gro_pt51 meet the practical groove section while the fake points gro_pt51~ gro_pt66 must be deleted. The fake points were produced by an overcut. As seen in Fig. 7, the overcut problem can be prevented only by limiting the cutting tool radius to a value smaller than 10.43 mm. In other words, the overcut problem occurred when there was a circle with a smaller radius than Rt, which passed through the calculated groove section curve twice. From the view of mathematical modeling, the non-fake condition indicated that dcon_pt (the distances between contact points and ZT-axis) should monotonically decrease with m1.
2
T
rcon
pt
xcon pt1 6 xcon pt2 ¼6 4 ⋮ xcon ptn
3 zcon pt1 zcon pt2 7 7 ⋮ 5 zcon ptn
ycon pt1 ycon pt2 ⋮ ycon ptn
ð11Þ
The distances between contact points and ZT-axis were 2
dcon
pt
x2con 6 x2 con ¼6 4 x2con
þ y2con 2 pt2 þ ycon ⋮ 2 ptn þ ycon pt1
3 pt1
7 5
pt2 7
ð12Þ
ptn
The non-fake condition can be established as f condition
non fake
¼ dcon pt ðiÞ−dcon pt ði þ 1Þ≥ 0;
i ¼ f1; 2; ⋯; n−1g dcon pt ðiÞ < R2t
ð13Þ
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Fig. 7 Overcut problem (nonfake condition) analysis
Then, if the overcut problem occurred, the produced fake groove section points should be identified and deleted in three steps. First, the critical point (gro_pt51 in Fig. 7) separating the fake and practical points are calculated by Eq. (13). Second, two points (gro_pt50 and gro_pt52 in Fig. 7) nearest to the critical point are deduced. Third, the angles between the two nearest points and the XT-axis are calculated. Thus, the points belonging to the smaller angle side were fake points for a dextral groove.
4.4 Homogenizing of groove section points As presented in Fig. 8, the distribution of the groove section points was evaluated by acc_d. Supposing that there was a point moved from p1 to pn through p2, p3, …, pn-1 sequentially, the path it had traveled from p1 to pi was denoted as acc_di. Obviously, the groove section points would distribute evenly if acc_d = [acc_d2, acc_d3, acc_d4,…, acc_dn] was an arithmetic progression.
4.3 Tip precision condition The tip precision in this study was defined as the subtraction of the cutting tool radius by dgro_pt (distance between groove section point and ZT-axis, as shown in Fig. 7). Therefore, the tip precision condition could be established as 0 ð14Þ f condition tip ðm1 Þ ¼ d gro pt −Rt ≤ Δe where Δe′ was the allowed tip precision. Using the numerical method, the tip precision could be controlled in the allowed error Δe′ in two steps. First, two points located inside and outside of the tip circle were selected and their corresponding values of m1 were obtained. As shown in Fig. 3d, con_pt17 and con_pt11 were the two selected points and their m1 values were denoted as m1_ptin and m1_ptout. Then, the desired tip points could be searched using the dichotomy method. The process could be easily programmed in MATLAB using the pseudo code given in the appendix.
Fig. 8 Definition of acc_d for groove section points distribution
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Taking Ex1, Ex3, and Ex4 for examples, where the relationship between m 1 and acc_d is provided in Fig. 9. Due to the one-to-one correspondence between groove section points and m1 values, the abscissa was the values of m1 dispersing from 0 to gb and the ordinate was the values of acc_d. It was shown that acc_di changed nonlinearly with m1 and changed violently when the wheel generatrix curvature was large. Therefore, the piecewise cubic interpolation method was employed to update m1 with an arithmetic progression acc_d′. It was programmed in MATLAB as 0 0 m1 ¼ interp1 acc d; m1 ; acc d ; ‘ pchip’
where
h i 8 0 > accd 2 ; accd 3 ; accd 4 ; …; accd n > < accd¼ m1 m21 ; mh31 ; m41 ; …; mn1 ; mi1 ¼ iðgb=nÞ i > > : accd0 ¼ accd 0 ; accd 0 ; accd 0 ; …; accd 0 ; accd 0 ¼ iðaccd =nÞ 2 3 4 n i n
the basic procedure. Third, the non-fake points condition was identified, and the fake points were deleted. Next, the desired tip point was searched. Finally, the groove section points were homogenized and the groove geometry parameters were solved with the same method introduced in Li et al. [3].
5 Software development and verifications Cutting tool groove simulation software was developed using MATLAB GUI with the calculation procedure introduced in Fig. 10. As presented in Fig. 11, the input parameters included 15 parameters: the basic groove geometries Rt and P; the wheel positions αx, Δx, and Δy; the wheel geometries Rg, gb, gb1, gr1, gr2, gr3, ga1, and ga2; and the operate parameters Δe′ and n. The groove geometries were defined and exhibited
Updating m1 with m1′ in Fig. 2, the groove section points could be recalculated with a more uniform distribution. The procedure should be repeated until the desired result is obtained.
4.5 Calculation procedure The conditions of noninterference, non-fake points, and tip precision have been established. The methods for the identification and the homogenization of fake points and groove section points were introduced as well. Accordingly, a calculation procedure was built based on the basic procedure provided in Fig. 2, as shown in Fig. 10. First, the noninterference condition was checked. The software would provide a warning and stop the calculation if the side interference problem was detected. Second, the primal groove points were calculated by 90
Ex1
Ex3
Ex4
80
acc_d(mm)
70
m1=1.0
m1=5.0
m1=19.0
60 50
40 30 20 10 0
m1(mm) Fig. 9 Relationships between acc_d and m1 for Ex1, Ex3, and Ex4
Fig. 10 Calculation procedure of groove section points
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Fig. 11 Cutting tool groove simulation software
with both data and graphics. The coordinates of the groove section points were written in a TXT file and the side interference problem was informed. Figure 11 provides a calculation example for Ex2 listed in Table 1.
Using the numerical method and the developed software, Ex1 and Ex2 were recalculated with the operating parameters Δe′ = 0.001 and n = 44 on a PC with the following components: processor, Intel(R) Core(TM) i7-6700HQ; CPU,
Fig. 12 Results of Ex1 and Ex2 with the numerical method. (a) Example 1; (b) Eample 2
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2.6 GHz; and 8 GB of RAM. The groove section points homogenizing procedure was executed twice. Ex1 required 2.31 s and Ex2 required 2.67 s. The results are presented in Fig. 12, which shows that the fake points were identified and deleted (Fig. 12b), the tip points meet the desired precision, and the groove section points are distributed evenly.
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Appendix The pseudo code of the MATLAB program to search the desired tip points. Read m1_ptin, m1_ptout, and Δe' If | fcondition_tip (m1_ptin) |Δe' then if fcondition_tip (m1_middle) • fcondition_tip (m1_middle)
