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Operation Planning Based on Cutting Process Models
Operation Planning Based on Cutting Process Models M. D. Tsai', S. Takata (2)2,M. Inui', F. Kimura (2)', T. Sata (1)3 The University of Tokyo; * Osaka University; The Institute of Physical and Chemical Research Received on January 18,1991
Abstract: This paper describes model based operation planning for pocket milling operations. The tool path and cutting conditions are determined based on the geometric model of workpieces and the physical models of the cutting process. The tool path is generated by using the Voronoi diagram of a cutting area. Cutting conditions to achieve the maximum metal removal rate are determined by evaluating the physical models of cutting torque, chatter vibration, and machining error. To maintain a favorable cutting state, the radial depth of cut is controlled by modifying the tool path distance at the circle path segment and by adding additional tool path segments at the corner. Examples are shown to demonstrate the effectiveness of the method. process Planning, Milling. K E Y WORDS:
1
Introduction
To achieve the small-batch high-variety production in an efficient way, it is necessary to generate error free NC programs in order to eliminate the need for the program verification and modification on the machine tool. Error free means that the NC program does not have any erroneous cutter motion and moreover that the cutting conditions are properly selected not to result in abnormal cutting during operation. A lot of research has been done to achieve this purpose. Wang proposed the utilization of a cutting simulation system for adjusting the feed rate so as to limit the cutting force acting on the tool [8]. Tlusty et al. studied the determination of proper radial and axial depth of cut in pocket milling for realizing chatter free machining [7]. Iwabe et al. made excellent research about the problem of the change of radial depth of cut at the corner based on the analysis of the cutting mechanism of end milling [2]. They proposed to add a supplement tool path to avoid rapid increase of the actual radial depth of cut at the corner. They showed that machining error can be significantly reduced by this method. These research works suggest that the tool path should be determined in connection with the cutting condition considering the physical parameters of the cutting state. For example, radial depth of cut should be controlled so as to avoid chatter vibration and/or excess tool deflection. However, actual radial depth of cut changes depending on the curvature of the tool path even with a constant tool path distance. For generating quality NC programs for various geometries of parts, we need a -more general framework with the capability of geometric and physical evaluation of cutting processes. In this paper, we propose a model based strategy to determine the cutting condition and tool path for achieving high quality machining. This methodology has been developed for NC program generation of pocket milling utilizing the geometric model of workpieces and the physical models of cutting process. Examples are shown demonstrating the NC program generation by the experimental system in which the methodology is implemented.
2
Strategy Toward Optimal Machining
We focus on the pocket machining problem whose boundary is composed mainly by lines and by some circles. Spiral out machining is adopted as illustrated in Figure 1, because it is suitable for the pocket with arbitrary boundary geometry. Tool path for spiral out machining is formed by offset curves of pocket boundary. The distance between two neighboring path segments is called tool path distance, it is usually kept constant. Our goal here is t o generate a NC program which maximizes the metal removal rate while maintaining stable machining and required machining accuracy.
,&Cucle
tool path segment
come
Among parameters which relate to the metal removal rate in milling operations the axial and radial depth of cut must be determined at the time of tool path generation, while the feed rate and cutting speed can be adjusted afterward. Therefore, we try to maximize the axial and radial depth of cut first. Since the axial depth of cut can not be arbitrarily changed during pocket machining, we principally control the radial depth of cut for achieving maximum metal removal rate. The feed rate is then selected considering the limitations of cutting force and tool deflection. Cutting speed is assumed to be given and not discussed in this paper. There are two factors to be considered in determining the radial depth of cut. One is the physical constraint. The other is the change of actual radial depth of cut in spite of the constant tool path distance. With regard to the physical constraints, the radial depth of cut should be restricted not to generate chatter vibration and excess cutting force applied on flutes in roughing operations. In finishing, on the other hand, the number of simultaneously engaged flutes is a main concern, because it relates to the machining error. If only one flute is engaged, the cutting force can be kept significantly smaller, and thus the machining error can be minimized. To evaluate these physical constraints, we utilize the physical models developed for the cutting simulation system [5][6]. The actual radial depth of cut depends on the curvature of the tool path. It has a larger value at the circle tool path segment than at the line tool path segment. Moreover it increases abruptly at the corner. Since the actual radial depth of cut corresponds to the engage angle of the tool against the workpiece and dominates the cutting force, it is desirable to keep it constant as much as possible for maintaining stable cutting and performing reliable operations. Therefore, the tool path distance at the circle tool path segment is so modified that the engage angle remains constant. At the corner, additional tool paths are generated to suppress the excess value of the engage angle as suggested by Iwabe[2]. These modifications can be systematically manipulated by utilizing the geometric model of the workpiece regardless of the pocket geometry as far as its boundary consists of line and circle segments.
3
Determination of Cutting Condition by Evaluating Physical Models
In this chapter, the methodology for determining cutting condition is discussed. The cutting condition concerned here includes axial depth of cut a, radial depth of cut b, and feed rate S. We take the the following procedure to determine these parameters. At first, the axial depth of cut a is selected according to the geometry of product. Then, the maximum allowable radial depth of cut b, or the maximum allowable engage angle ,&, is calculated. In the case of roughing, the criterion about chatter vibration is used in the calculation. In the case of finishing, the number of simultaneously engaged ,, the tool path flutes is, also considered. Based on the value of b, or & is generated by the method explained in the next chapter. The feed rate is then selected based on considering the cutting torque in the case of roughing and the tool deflection in the case of finishing. In the following, the physical model and the criteria which we used in determining the cutting condition are explained. For evaluating the cutting force, we adopt the model illustrated in Figure 2, which is similar to the one used in our cutting simulation system [5][6]. Take the infinitesimal section of the flute which engages in the workpiece at the distance w from the bottom of the tool and a t the angle of 0 from The cutting force acting on this section is expressed by the tangential force dFt and the radial force dF,. dF, and dF, are taken as proportional to the area of the chip element removed by this section, S . sin(0) . dw. Two kinds of engage angles, 0. and 0b are defined in conjunction with the axial depth of cut a and the radial depth of cut b as shown in the figure. The chatter vibration is evaluated by means of a lumped-parameter
m.
Fig.1 Pocket Machining with Spiral Out Tool Path
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model with two degrees of freedom shown in Figure 3. The block diagram representing the relation between feed U ( s ) and the displacement of the tool Y ( s )can be illustrated as Figure 4 161. According to the criterion proposed by Merritt131, stable operation can be performed, if
(V = J K y . ( l / K m ). G m ( j w ) . Kcl) < 0.5.
(1)
The structural dynamics l / I f m . G m ( j w )is calculated in terms of the spring constants and the damping coefficients of the tool. Ify and h‘c are the Laplace transformation of ky and kc. ky represents the coordinate transformation from z I , z2 axes to z1 axis. kc is the cutting stiffness which can he obtained from the cutting force model. In evaluating V, ky and kc are regarded constant in time and substituted for Icy and I f c for convenience, because it is difficult to obtain K c as functions of s. Then V is obtained depending on a and b regardless of S. While rotating the tool, V is evaluated at every angular position, and the maximum value VmaZis identified. The maximum radial depth of cut is determined within the condition of VmOz< 0.5. With regard to the criterion about the cutting torque in determining the feed rate in roughing operation, we adopted the following parameter.
i = a, L, + pt ,
’
Lp.
(2)
Here L, expresses the average torque acting on the tool during one revolution of the spindle. Lp is the maximum cutting torque in one revolution. at and Pt are the weights for blending the effects of L, and Lp. L, can be calculated by integrating r . dFt from 0 to 0,. L p is obtained at the angular position where the total removed area of chip element by the flutes reaches the largest. For example, if 0. 2 06, then L, and Lp are expressed as ( S / 2 x ) K t a band K!Srb/tan(b) respectively, where 4 is the helix angle of the flute. Substituting L, and Lp into Equation ( 2 ) , i can be represented as a function of a, e b and S. Therefore the proper f y d rate can be calculated corresponding to the maximum allowable value of L. In finishing operations, machining error is the main concern in determining the cutting condition. We evaluate the machining error in terms of the tool deflection, because it is a primal cause in end milling. We assume that the tool deflection is proportional to the cutting force, Ife is the proportional coefficient, and calculate it using the model presented in [5]. For minimizing the cutting force, the number of simultaneously engaged flutes should be maintained bellow 1 . The finished surface is generated only when the flute passes P in Figure 2. If another flute is engaged in another place, the cutting force acting on this flute adds to the one which is generating the finished surface, and increases the tool deflection. The condition can be satisfied by selecting the maximum engage angel as &,, 2 (2*/2-0,) where 2 is the number of flutes of the tool. We use this criterion in determining the maximum allowable radial depth of cut for finishing operation. If the number of simultaneously engaged flutes is equal to 1, the largest machining error, Em.,, is induced when the part of the flute at the bottom of the tool passes P. Ern*= can be calculated by integrating dF,, the cutting force component in the direction perpendicular to the finished surface which is obtained from dFt and dF,. If 0. 5 86 the flute is engaged from the tool angle 0 0.. On the other hand, it is engaged from 0 e b , if 0. > 0,. Em.,= can be expressed as,
-
-
where ’+’ and ’-’ represents down-cut milling and up-cut milling respectively. Equation (3) indicates that Ern,,=is obtained as a function of a, e b and S, which can be used in determining the feed rate within the limit of the certain value of Erne=.
4
Tool Path Generation by Controlling Instantaneous Engage Angle
Tool path for pocket machining is generated in the spiral out machining manner, and the tool path distance is determined to let 0, = eb,, when the ,,, when the tool tool is machining along line path segments. 6, may beover & is at the internal corner of two discontinuous path segments or machining along a circle path segment. To control /?b 5 eb,,,, the tool path distance is changed for circle segments, and additional path segments are generated for internal corners.
4.1
Tool Path Generation by Using Voronoi Diagram
Offset calculation is efficiently achieved by using the Voronoi diagram of cutting regions. This approach was initiated by Persson[l] and modified by Held(l1. The Voronoi diagram yields planar divisions as illustrated in Figure 5(a), where each region corresponds to exactly one boundary element, and is defined such that any point in a region has the corresponding element as the nearest one. A border element between two regions is the locus of the point that is in the same distance from two corresponding boundary elements. Such a border element is generally called bisector of the corresponding elements. Once the Voronoi diagram of the pocket area is calculated, the tool path for spiral out machining is efficiently obtained in the following manner. As illustrated in Figure 5(b), the path is started from the innermost point of the pocket area I, and it moves the tool path distance towards the pocket boundary along an arbitrary bisector. Then it enters one region for the first tool path cycle. The tool is moved along the offset curve of the boundary element corresponding to the region until it intersects another bisector of the region. The path generation is continued in the same manner until it reaches the first entering point. By stepping out towards the boundary, the above process is iterated until the path fills the pocket area as illustrated in Figure 5(c). In practice, there exist isle and strait problems in pocket machining, which make it difficult to apply the above procedure directly. The following solution is based on the work by Heldll]. Some unmachined regions remained when isles exist in the pocket area. This difficulty is due to the fact that such pocket area has several separated contours of the pocket boundary because of isles. By stepping out towards one contour the path becomes nearer to this contour, however, it may not become nearer for other contours, therefore, regions corresponding to these contours remain unmachined (see Figure 6(a)). In order to avoid this problem, Some contour bridges are
Region
(a)
Fig. 2 Model for Computing Cutting Force
(b) (C) Fig.5 Tool Path Generation by thevoronoi Dtagram
Fig. 3 Tool Vibration Model Tool
Contour bridge for connecting outmost contour and isle contour Fig. 6 Isle and Connecting Contour Bridge
(a) Unmachined area cause by an isle Fig. 4 Block Diagram of Tool Vibration System
96
(b)
.................. ,..
................... ....................
.;
Fig. 8 Engage Angle in Line Path Segment
......................
Fig. 7 Unmachined Area Caused by a Strait introduced to connect the isle contours with the outmost boundary contour. Each contour bridge is composed by two parallel lines and two circular arcs as illustrated in Figure 6(b). The presence of straits of the pocket also left unmachined area as shown in Figure 7. This problem is solved by dividing the pocket area into several sub-areas without straits. Such divisions are executed at the straits which can be detected in the following manner. A pair of boundary elements are recognized to compose a strait if they define a bisector which does not originate from the boundary, and there exist a point on this bisector which does not decrease the minimal distance to the boundary when it moves on the bisector for an arbitrary small amount.
4.2
I
CI
Controlling Instantaneous Engage Angle
In this section, tool path modification methods for controlling instantaneous engage angle in machining are discussed. Tool p a t h distance for line p a t h segments In a Voronoi region where path segments are linear, the tool path distance, e l, and the instantaneous engage angle, 06, must satisfy the following equation (see Figure 8). cos(8b) = 1 - q / r . (4) 4.2.1
Therefore, we can get the optimal tool path distance by giving 06, as 66. 4.2.2
P a t h modification for circle pat h segments
In the region where path segments are circular, the tool path distance for an arbitrary circle tool path, c,, and the instantaneous engage angle, 06 must
Fig. 9 Tool Path Modification for Achieving Constant Engage Angle in a Region of Circle Tool path Segements
satisfy, where R is the radius of the circle tool path segment as shown in Figure 9(a). Tool path distance is smaller than the diameter of the tool, therefore c, 5 2r is satisfied. We can calculate the optimal tool path distance by giving 06 as 06, of ( 5 ) . In Equation ( 5 ) , - is used for calculations when the Voronoi region for machining corresponds to a concave circle boundary element, otherwise is used. By comparing the equations (4) and ( 5 ) , we can understand that the path distance for the fan-shaped Voronoi region of the concave circle boundary is smaller than the one for the region of the line boundary. As a result, more circle path segments are generated for the fan-shaped region than its adjacent regions with line segments. It is difficult to properly connect circle path segments and line path segments, because the geometry of the line segment and its corresponding circle segment on the border of the regions are discontinuous as shown in Figure 9(b). We assign a one to one correspondence between circle segments and line segments from the pocket boundary side. Then, we introduce additional small line segments to connect them as shown in Figure 9(c). As a result, the innermost circle segment remains. Instead of machining along this small circle segment, we consider an imaginary corner E as shown in figure 9(d). According to the tool path generation method to be given in the following, this imaginary corner is firstly machined. Then the tool start machining the fan-shaped region according to the modified tool path segments. In finishing, because the tool path distance is usually set far below b,, 06 may still be smaller than eb, even in the region of the concave circle boundary. For this case, the tool path distance will not be changed.
+
Additional p a t h segments for machining corners will increase abruptly when the tool is machining an internal comer. As shown in figure ll ( a ), as the tool approaches the comer along the current path segment, 06 starts increasing after the tool passes C, and it remains larger than Ob, until it passes C.. The tool path length between C, and C. becomes longer as the corner becomes sharper. Based on the idea by Iwabe[2], additional circle tool path segments are added for machining the corner. In our method, the tool approaches the corner along the current path segment. Then, it turns away from the current path before passing C, and it moves along the additional circle segment and machines some material from the corner (see Figure lO(a)). Then it returns to the point where the deviation started and approaches the corner again along the current path.
Fig. 10 Additional Tool Path at Comer Since some of the material from the corner was already removed, C, and C. become closer to the corner, and the tool can approach the corner closer than before. The above process is iterated until the length between C, and C. becomes small enough to ignore. The termination condition is achieved
when the length becomes less than 0.5 * r in roughing and c in finishing. If the tool moves along the n f h circle path, 06 takes its maximum value when the tool center locates on En,where the tool rim passes the intersecting point of the bisector of the corner and the circle produced by the tool rim of the n-1-th circle segment (seeFigure lO(b)). The center of the additional circle segment locates on the bisector of the corner. Therefore, if the radius of the circle is calculated, the center of the circle is easily determined. When the tool locates at B,, the radius of the n-th circle, R,,, and the radius of the n-1-th circle, Rln-l, must satisfy,
4.2.3 06
Here d, means the distance between the centers of the n d h circle and the - ~Rt,). If &,-I is known and 0,- is n - l f h circle, and d, = - c ( ~ / 2 ) ( R I ngiven as 86, a quadratic equation of Rt, is obtained. For the first additional circular path segment, 06 takes its maximum value when the tool center locates on B1as shown in figure 1O(c) where the tool rim passes the foot of the perpendicular from the center of the circular path on the machined boundary of the previous cut. The radius of the first path segment, R 1 , satisfy, (7)
97
Fig. 12 Example of Tool Path Generation at Comer for Finishing
Fig.13 Example of Tool Path Generation at Concave Circle Boundary
(C) (d) Fig. 1 1 Example of Tool Path Generation at Comer for Roughing satisfy COS(ob,) = 1 -el/. for roughing operations. Therefore, e b = &,,, in Equation (7). For calculating Ri we set a criterion that b is equal to 1.2 times of b,,, when the tool locates 011 B I . Substituting 86 corresponding to this b into Equation (7), RI can be calculated. CI is selected to
Rl becomes infinite, if
Fig.14 Example of Pocket with Isles and Straits
1. For generating tool paths in spiral out manner, the Voronoi diagram
5
Examples
Four examples are demonstrated to show the'dectiveness of our method. Two examples show the results at the corner, one for the case of roughing, and the other one for finishing. The third example shows the result in the region of concave circle boundary. The last example shows the result for a pocket where straits and islands are included. Figure 11 shows the additional tool paths for corner machining during roughing operation. The tool approaches the corner from the I axis. The intersectional angle is 60". The radius of tool is lOmm and the tool path distance is 10.5mm. Figure l l ( a ) shows the boundary machined by the previous path and the boundary that will be machined by the current path. Figure l l ( b ) shows the first additional tool path and the resultant machined boundary. The tool will return to the deviation point by the line segment, when the tool rim contacts the boundary which must be machined by the current path. Figure l l ( c ) shows the second additional tool path and the resultant machined boundary. Figure l l ( d ) shows the other additional tool paths and the boundaries which are machined by these paths. Figure 12 shows the tool path for corner machining during finishing operation. The tool path distance is lmm. The radial depth of cut is 4mm. The helix angle of the tool is 30". Two small additional circle segments are calculated and shown in the figure. Figure 13 shows modified tool path distance, and the additional tool path segments for machining the imaginary corner in the region of the concave circle boundary. The cutting conditions for roughing and finishing operations are the same with the ones in the above examples. The radius of this circle concave boundary element is 32mm. The tool approaches this region from -y axis. Figure 14 shows the tool path of a pocket area with our method applied. The broken lines show the additional path segments at corner. With our method, feed rate is kept constant regardless the geometry of the tool path.
6
conclusion
In this study, the model based method of the operation planning for pocket milling is developed to achieve the optimal machining.
98
is adopted. 2. The physical models of the cutting process are evaluated to determine the proper radial depth of cut and the feed rate to avoid abnormal cutting states such as chatter vibration, excess tool deflection and excess cutting torque. 3. The instantaneous engage angle can be controlled to achieve stable machining by modifying the tool path distances at the circle tool path segments and adding the supplement tool paths at the corner.
ACKNOWLEDGEMENTS This research work was partly founded by the Product Realizatioii Project organized by the Japan Society of Precision Engineering.
References [I] Held, M.,f989, Geopocket - A Sophisticated Computational Cromctry Solution of Geometrical and Technological Problems Arising from Pocket Machining, CAPE'S9 pp283-293. (21 Iwabe, H., Fujii, Y., Saito, K., Kisinami, T., 1989 Study on Corner Cut by End Adill - Analysis of Cutting Mechanism and N e w Cutting Method at Inside Comer, in Japanese, Journal of Japan Society of Precisioii Engineering, Vo1.55, No.5, pp. 841-846. [3] Merritt, H. E., 1965, Theory ofself-Exited Machine-Tool Chatter Cow tribution to Machine-Tool Chatter Research-f, ASME Journal of Eiig. for Industry, Vo1.87, No.4, pp.447-454. [4] Persson,H., 1978, NC Machining of Arbitrarily Shaped Pockets, Computer-Aided Design Vol.10 No.3 pp.169-174. [5] Takata, S., Tsai, M.D., Inui, M., Sata, T., 1989, A Cutting Simulation System f o r Machinability using a Workpiece Model, Annals of the CIRP, Vo1.38, No.1, pp.417-420. [6] Tsai, M.D., Takata, S., Inui, M., Kimura, F., Sata, T., 1990 Prediction of Chatter Vibration by Means of a Model Based Cutting Simulation System, Annals of the CIRP, Vo1.39, No.1. pp.447-450 [7] Tlusty, J., Smith, S., Zamiidio, C., New N C Routines for. Qualrty i n Milling, Annals of the CIRP, Vol.39, No.1. pp.517-521 [S] Wang, W.P., 1987 Application of Solid Modeling to Automate Machining Pammeters for Complez Parts, 19th CIRP International Seminar on Manufacturing Systems, pp.33-37.
Abstract: This paper describes model based operation planning for pocket milling operations. The tool path and cutting conditions are determined based on the geometric model of workpieces and the physical models of the cutting process. The tool path is generated by using the Voronoi diagram of a cutting area. Cutting conditions to achieve the maximum metal removal rate are determined by evaluating the physical models of cutting torque, chatter vibration, and machining error. To maintain a favorable cutting state, the radial depth of cut is controlled by modifying the tool path distance at the circle path segment and by adding additional tool path segments at the corner. Examples are shown to demonstrate the effectiveness of the method. process Planning, Milling. K E Y WORDS:
1
Introduction
To achieve the small-batch high-variety production in an efficient way, it is necessary to generate error free NC programs in order to eliminate the need for the program verification and modification on the machine tool. Error free means that the NC program does not have any erroneous cutter motion and moreover that the cutting conditions are properly selected not to result in abnormal cutting during operation. A lot of research has been done to achieve this purpose. Wang proposed the utilization of a cutting simulation system for adjusting the feed rate so as to limit the cutting force acting on the tool [8]. Tlusty et al. studied the determination of proper radial and axial depth of cut in pocket milling for realizing chatter free machining [7]. Iwabe et al. made excellent research about the problem of the change of radial depth of cut at the corner based on the analysis of the cutting mechanism of end milling [2]. They proposed to add a supplement tool path to avoid rapid increase of the actual radial depth of cut at the corner. They showed that machining error can be significantly reduced by this method. These research works suggest that the tool path should be determined in connection with the cutting condition considering the physical parameters of the cutting state. For example, radial depth of cut should be controlled so as to avoid chatter vibration and/or excess tool deflection. However, actual radial depth of cut changes depending on the curvature of the tool path even with a constant tool path distance. For generating quality NC programs for various geometries of parts, we need a -more general framework with the capability of geometric and physical evaluation of cutting processes. In this paper, we propose a model based strategy to determine the cutting condition and tool path for achieving high quality machining. This methodology has been developed for NC program generation of pocket milling utilizing the geometric model of workpieces and the physical models of cutting process. Examples are shown demonstrating the NC program generation by the experimental system in which the methodology is implemented.
2
Strategy Toward Optimal Machining
We focus on the pocket machining problem whose boundary is composed mainly by lines and by some circles. Spiral out machining is adopted as illustrated in Figure 1, because it is suitable for the pocket with arbitrary boundary geometry. Tool path for spiral out machining is formed by offset curves of pocket boundary. The distance between two neighboring path segments is called tool path distance, it is usually kept constant. Our goal here is t o generate a NC program which maximizes the metal removal rate while maintaining stable machining and required machining accuracy.
,&Cucle
tool path segment
come
Among parameters which relate to the metal removal rate in milling operations the axial and radial depth of cut must be determined at the time of tool path generation, while the feed rate and cutting speed can be adjusted afterward. Therefore, we try to maximize the axial and radial depth of cut first. Since the axial depth of cut can not be arbitrarily changed during pocket machining, we principally control the radial depth of cut for achieving maximum metal removal rate. The feed rate is then selected considering the limitations of cutting force and tool deflection. Cutting speed is assumed to be given and not discussed in this paper. There are two factors to be considered in determining the radial depth of cut. One is the physical constraint. The other is the change of actual radial depth of cut in spite of the constant tool path distance. With regard to the physical constraints, the radial depth of cut should be restricted not to generate chatter vibration and excess cutting force applied on flutes in roughing operations. In finishing, on the other hand, the number of simultaneously engaged flutes is a main concern, because it relates to the machining error. If only one flute is engaged, the cutting force can be kept significantly smaller, and thus the machining error can be minimized. To evaluate these physical constraints, we utilize the physical models developed for the cutting simulation system [5][6]. The actual radial depth of cut depends on the curvature of the tool path. It has a larger value at the circle tool path segment than at the line tool path segment. Moreover it increases abruptly at the corner. Since the actual radial depth of cut corresponds to the engage angle of the tool against the workpiece and dominates the cutting force, it is desirable to keep it constant as much as possible for maintaining stable cutting and performing reliable operations. Therefore, the tool path distance at the circle tool path segment is so modified that the engage angle remains constant. At the corner, additional tool paths are generated to suppress the excess value of the engage angle as suggested by Iwabe[2]. These modifications can be systematically manipulated by utilizing the geometric model of the workpiece regardless of the pocket geometry as far as its boundary consists of line and circle segments.
3
Determination of Cutting Condition by Evaluating Physical Models
In this chapter, the methodology for determining cutting condition is discussed. The cutting condition concerned here includes axial depth of cut a, radial depth of cut b, and feed rate S. We take the the following procedure to determine these parameters. At first, the axial depth of cut a is selected according to the geometry of product. Then, the maximum allowable radial depth of cut b, or the maximum allowable engage angle ,&, is calculated. In the case of roughing, the criterion about chatter vibration is used in the calculation. In the case of finishing, the number of simultaneously engaged ,, the tool path flutes is, also considered. Based on the value of b, or & is generated by the method explained in the next chapter. The feed rate is then selected based on considering the cutting torque in the case of roughing and the tool deflection in the case of finishing. In the following, the physical model and the criteria which we used in determining the cutting condition are explained. For evaluating the cutting force, we adopt the model illustrated in Figure 2, which is similar to the one used in our cutting simulation system [5][6]. Take the infinitesimal section of the flute which engages in the workpiece at the distance w from the bottom of the tool and a t the angle of 0 from The cutting force acting on this section is expressed by the tangential force dFt and the radial force dF,. dF, and dF, are taken as proportional to the area of the chip element removed by this section, S . sin(0) . dw. Two kinds of engage angles, 0. and 0b are defined in conjunction with the axial depth of cut a and the radial depth of cut b as shown in the figure. The chatter vibration is evaluated by means of a lumped-parameter
m.
Fig.1 Pocket Machining with Spiral Out Tool Path
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model with two degrees of freedom shown in Figure 3. The block diagram representing the relation between feed U ( s ) and the displacement of the tool Y ( s )can be illustrated as Figure 4 161. According to the criterion proposed by Merritt131, stable operation can be performed, if
(V = J K y . ( l / K m ). G m ( j w ) . Kcl) < 0.5.
(1)
The structural dynamics l / I f m . G m ( j w )is calculated in terms of the spring constants and the damping coefficients of the tool. Ify and h‘c are the Laplace transformation of ky and kc. ky represents the coordinate transformation from z I , z2 axes to z1 axis. kc is the cutting stiffness which can he obtained from the cutting force model. In evaluating V, ky and kc are regarded constant in time and substituted for Icy and I f c for convenience, because it is difficult to obtain K c as functions of s. Then V is obtained depending on a and b regardless of S. While rotating the tool, V is evaluated at every angular position, and the maximum value VmaZis identified. The maximum radial depth of cut is determined within the condition of VmOz< 0.5. With regard to the criterion about the cutting torque in determining the feed rate in roughing operation, we adopted the following parameter.
i = a, L, + pt ,
’
Lp.
(2)
Here L, expresses the average torque acting on the tool during one revolution of the spindle. Lp is the maximum cutting torque in one revolution. at and Pt are the weights for blending the effects of L, and Lp. L, can be calculated by integrating r . dFt from 0 to 0,. L p is obtained at the angular position where the total removed area of chip element by the flutes reaches the largest. For example, if 0. 2 06, then L, and Lp are expressed as ( S / 2 x ) K t a band K!Srb/tan(b) respectively, where 4 is the helix angle of the flute. Substituting L, and Lp into Equation ( 2 ) , i can be represented as a function of a, e b and S. Therefore the proper f y d rate can be calculated corresponding to the maximum allowable value of L. In finishing operations, machining error is the main concern in determining the cutting condition. We evaluate the machining error in terms of the tool deflection, because it is a primal cause in end milling. We assume that the tool deflection is proportional to the cutting force, Ife is the proportional coefficient, and calculate it using the model presented in [5]. For minimizing the cutting force, the number of simultaneously engaged flutes should be maintained bellow 1 . The finished surface is generated only when the flute passes P in Figure 2. If another flute is engaged in another place, the cutting force acting on this flute adds to the one which is generating the finished surface, and increases the tool deflection. The condition can be satisfied by selecting the maximum engage angel as &,, 2 (2*/2-0,) where 2 is the number of flutes of the tool. We use this criterion in determining the maximum allowable radial depth of cut for finishing operation. If the number of simultaneously engaged flutes is equal to 1, the largest machining error, Em.,, is induced when the part of the flute at the bottom of the tool passes P. Ern*= can be calculated by integrating dF,, the cutting force component in the direction perpendicular to the finished surface which is obtained from dFt and dF,. If 0. 5 86 the flute is engaged from the tool angle 0 0.. On the other hand, it is engaged from 0 e b , if 0. > 0,. Em.,= can be expressed as,
-
-
where ’+’ and ’-’ represents down-cut milling and up-cut milling respectively. Equation (3) indicates that Ern,,=is obtained as a function of a, e b and S, which can be used in determining the feed rate within the limit of the certain value of Erne=.
4
Tool Path Generation by Controlling Instantaneous Engage Angle
Tool path for pocket machining is generated in the spiral out machining manner, and the tool path distance is determined to let 0, = eb,, when the ,,, when the tool tool is machining along line path segments. 6, may beover & is at the internal corner of two discontinuous path segments or machining along a circle path segment. To control /?b 5 eb,,,, the tool path distance is changed for circle segments, and additional path segments are generated for internal corners.
4.1
Tool Path Generation by Using Voronoi Diagram
Offset calculation is efficiently achieved by using the Voronoi diagram of cutting regions. This approach was initiated by Persson[l] and modified by Held(l1. The Voronoi diagram yields planar divisions as illustrated in Figure 5(a), where each region corresponds to exactly one boundary element, and is defined such that any point in a region has the corresponding element as the nearest one. A border element between two regions is the locus of the point that is in the same distance from two corresponding boundary elements. Such a border element is generally called bisector of the corresponding elements. Once the Voronoi diagram of the pocket area is calculated, the tool path for spiral out machining is efficiently obtained in the following manner. As illustrated in Figure 5(b), the path is started from the innermost point of the pocket area I, and it moves the tool path distance towards the pocket boundary along an arbitrary bisector. Then it enters one region for the first tool path cycle. The tool is moved along the offset curve of the boundary element corresponding to the region until it intersects another bisector of the region. The path generation is continued in the same manner until it reaches the first entering point. By stepping out towards the boundary, the above process is iterated until the path fills the pocket area as illustrated in Figure 5(c). In practice, there exist isle and strait problems in pocket machining, which make it difficult to apply the above procedure directly. The following solution is based on the work by Heldll]. Some unmachined regions remained when isles exist in the pocket area. This difficulty is due to the fact that such pocket area has several separated contours of the pocket boundary because of isles. By stepping out towards one contour the path becomes nearer to this contour, however, it may not become nearer for other contours, therefore, regions corresponding to these contours remain unmachined (see Figure 6(a)). In order to avoid this problem, Some contour bridges are
Region
(a)
Fig. 2 Model for Computing Cutting Force
(b) (C) Fig.5 Tool Path Generation by thevoronoi Dtagram
Fig. 3 Tool Vibration Model Tool
Contour bridge for connecting outmost contour and isle contour Fig. 6 Isle and Connecting Contour Bridge
(a) Unmachined area cause by an isle Fig. 4 Block Diagram of Tool Vibration System
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(b)
.................. ,..
................... ....................
.;
Fig. 8 Engage Angle in Line Path Segment
......................
Fig. 7 Unmachined Area Caused by a Strait introduced to connect the isle contours with the outmost boundary contour. Each contour bridge is composed by two parallel lines and two circular arcs as illustrated in Figure 6(b). The presence of straits of the pocket also left unmachined area as shown in Figure 7. This problem is solved by dividing the pocket area into several sub-areas without straits. Such divisions are executed at the straits which can be detected in the following manner. A pair of boundary elements are recognized to compose a strait if they define a bisector which does not originate from the boundary, and there exist a point on this bisector which does not decrease the minimal distance to the boundary when it moves on the bisector for an arbitrary small amount.
4.2
I
CI
Controlling Instantaneous Engage Angle
In this section, tool path modification methods for controlling instantaneous engage angle in machining are discussed. Tool p a t h distance for line p a t h segments In a Voronoi region where path segments are linear, the tool path distance, e l, and the instantaneous engage angle, 06, must satisfy the following equation (see Figure 8). cos(8b) = 1 - q / r . (4) 4.2.1
Therefore, we can get the optimal tool path distance by giving 06, as 66. 4.2.2
P a t h modification for circle pat h segments
In the region where path segments are circular, the tool path distance for an arbitrary circle tool path, c,, and the instantaneous engage angle, 06 must
Fig. 9 Tool Path Modification for Achieving Constant Engage Angle in a Region of Circle Tool path Segements
satisfy, where R is the radius of the circle tool path segment as shown in Figure 9(a). Tool path distance is smaller than the diameter of the tool, therefore c, 5 2r is satisfied. We can calculate the optimal tool path distance by giving 06 as 06, of ( 5 ) . In Equation ( 5 ) , - is used for calculations when the Voronoi region for machining corresponds to a concave circle boundary element, otherwise is used. By comparing the equations (4) and ( 5 ) , we can understand that the path distance for the fan-shaped Voronoi region of the concave circle boundary is smaller than the one for the region of the line boundary. As a result, more circle path segments are generated for the fan-shaped region than its adjacent regions with line segments. It is difficult to properly connect circle path segments and line path segments, because the geometry of the line segment and its corresponding circle segment on the border of the regions are discontinuous as shown in Figure 9(b). We assign a one to one correspondence between circle segments and line segments from the pocket boundary side. Then, we introduce additional small line segments to connect them as shown in Figure 9(c). As a result, the innermost circle segment remains. Instead of machining along this small circle segment, we consider an imaginary corner E as shown in figure 9(d). According to the tool path generation method to be given in the following, this imaginary corner is firstly machined. Then the tool start machining the fan-shaped region according to the modified tool path segments. In finishing, because the tool path distance is usually set far below b,, 06 may still be smaller than eb, even in the region of the concave circle boundary. For this case, the tool path distance will not be changed.
+
Additional p a t h segments for machining corners will increase abruptly when the tool is machining an internal comer. As shown in figure ll ( a ), as the tool approaches the comer along the current path segment, 06 starts increasing after the tool passes C, and it remains larger than Ob, until it passes C.. The tool path length between C, and C. becomes longer as the corner becomes sharper. Based on the idea by Iwabe[2], additional circle tool path segments are added for machining the corner. In our method, the tool approaches the corner along the current path segment. Then, it turns away from the current path before passing C, and it moves along the additional circle segment and machines some material from the corner (see Figure lO(a)). Then it returns to the point where the deviation started and approaches the corner again along the current path.
Fig. 10 Additional Tool Path at Comer Since some of the material from the corner was already removed, C, and C. become closer to the corner, and the tool can approach the corner closer than before. The above process is iterated until the length between C, and C. becomes small enough to ignore. The termination condition is achieved
when the length becomes less than 0.5 * r in roughing and c in finishing. If the tool moves along the n f h circle path, 06 takes its maximum value when the tool center locates on En,where the tool rim passes the intersecting point of the bisector of the corner and the circle produced by the tool rim of the n-1-th circle segment (seeFigure lO(b)). The center of the additional circle segment locates on the bisector of the corner. Therefore, if the radius of the circle is calculated, the center of the circle is easily determined. When the tool locates at B,, the radius of the n-th circle, R,,, and the radius of the n-1-th circle, Rln-l, must satisfy,
4.2.3 06
Here d, means the distance between the centers of the n d h circle and the - ~Rt,). If &,-I is known and 0,- is n - l f h circle, and d, = - c ( ~ / 2 ) ( R I ngiven as 86, a quadratic equation of Rt, is obtained. For the first additional circular path segment, 06 takes its maximum value when the tool center locates on B1as shown in figure 1O(c) where the tool rim passes the foot of the perpendicular from the center of the circular path on the machined boundary of the previous cut. The radius of the first path segment, R 1 , satisfy, (7)
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Fig. 12 Example of Tool Path Generation at Comer for Finishing
Fig.13 Example of Tool Path Generation at Concave Circle Boundary
(C) (d) Fig. 1 1 Example of Tool Path Generation at Comer for Roughing satisfy COS(ob,) = 1 -el/. for roughing operations. Therefore, e b = &,,, in Equation (7). For calculating Ri we set a criterion that b is equal to 1.2 times of b,,, when the tool locates 011 B I . Substituting 86 corresponding to this b into Equation (7), RI can be calculated. CI is selected to
Rl becomes infinite, if
Fig.14 Example of Pocket with Isles and Straits
1. For generating tool paths in spiral out manner, the Voronoi diagram
5
Examples
Four examples are demonstrated to show the'dectiveness of our method. Two examples show the results at the corner, one for the case of roughing, and the other one for finishing. The third example shows the result in the region of concave circle boundary. The last example shows the result for a pocket where straits and islands are included. Figure 11 shows the additional tool paths for corner machining during roughing operation. The tool approaches the corner from the I axis. The intersectional angle is 60". The radius of tool is lOmm and the tool path distance is 10.5mm. Figure l l ( a ) shows the boundary machined by the previous path and the boundary that will be machined by the current path. Figure l l ( b ) shows the first additional tool path and the resultant machined boundary. The tool will return to the deviation point by the line segment, when the tool rim contacts the boundary which must be machined by the current path. Figure l l ( c ) shows the second additional tool path and the resultant machined boundary. Figure l l ( d ) shows the other additional tool paths and the boundaries which are machined by these paths. Figure 12 shows the tool path for corner machining during finishing operation. The tool path distance is lmm. The radial depth of cut is 4mm. The helix angle of the tool is 30". Two small additional circle segments are calculated and shown in the figure. Figure 13 shows modified tool path distance, and the additional tool path segments for machining the imaginary corner in the region of the concave circle boundary. The cutting conditions for roughing and finishing operations are the same with the ones in the above examples. The radius of this circle concave boundary element is 32mm. The tool approaches this region from -y axis. Figure 14 shows the tool path of a pocket area with our method applied. The broken lines show the additional path segments at corner. With our method, feed rate is kept constant regardless the geometry of the tool path.
6
conclusion
In this study, the model based method of the operation planning for pocket milling is developed to achieve the optimal machining.
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is adopted. 2. The physical models of the cutting process are evaluated to determine the proper radial depth of cut and the feed rate to avoid abnormal cutting states such as chatter vibration, excess tool deflection and excess cutting torque. 3. The instantaneous engage angle can be controlled to achieve stable machining by modifying the tool path distances at the circle tool path segments and adding the supplement tool paths at the corner.
ACKNOWLEDGEMENTS This research work was partly founded by the Product Realizatioii Project organized by the Japan Society of Precision Engineering.
References [I] Held, M.,f989, Geopocket - A Sophisticated Computational Cromctry Solution of Geometrical and Technological Problems Arising from Pocket Machining, CAPE'S9 pp283-293. (21 Iwabe, H., Fujii, Y., Saito, K., Kisinami, T., 1989 Study on Corner Cut by End Adill - Analysis of Cutting Mechanism and N e w Cutting Method at Inside Comer, in Japanese, Journal of Japan Society of Precisioii Engineering, Vo1.55, No.5, pp. 841-846. [3] Merritt, H. E., 1965, Theory ofself-Exited Machine-Tool Chatter Cow tribution to Machine-Tool Chatter Research-f, ASME Journal of Eiig. for Industry, Vo1.87, No.4, pp.447-454. [4] Persson,H., 1978, NC Machining of Arbitrarily Shaped Pockets, Computer-Aided Design Vol.10 No.3 pp.169-174. [5] Takata, S., Tsai, M.D., Inui, M., Sata, T., 1989, A Cutting Simulation System f o r Machinability using a Workpiece Model, Annals of the CIRP, Vo1.38, No.1, pp.417-420. [6] Tsai, M.D., Takata, S., Inui, M., Kimura, F., Sata, T., 1990 Prediction of Chatter Vibration by Means of a Model Based Cutting Simulation System, Annals of the CIRP, Vo1.39, No.1. pp.447-450 [7] Tlusty, J., Smith, S., Zamiidio, C., New N C Routines for. Qualrty i n Milling, Annals of the CIRP, Vol.39, No.1. pp.517-521 [S] Wang, W.P., 1987 Application of Solid Modeling to Automate Machining Pammeters for Complez Parts, 19th CIRP International Seminar on Manufacturing Systems, pp.33-37.