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Trochoidal machining for the high-speed milling of pockets
Accepted Manuscript Title: Trochoidal machining for the high-speed milling of pockets Author: Wu Shixiong Li Bin Ma Wei Wang Chengyong PII: DOI: Reference:
S0924-0136(16)30031-0 http://dx.doi.org/doi:10.1016/j.jmatprotec.2016.01.033 PROTEC 14714
To appear in:
Journal of Materials Processing Technology
Received date: Revised date: Accepted date:
6-9-2015 3-1-2016 30-1-2016
Please cite this article as: Shixiong, Wu, Bin, Li, Wei, Ma, Chengyong, Wang, Trochoidal machining for the high-speed milling of pockets.Journal of Materials Processing Technology http://dx.doi.org/10.1016/j.jmatprotec.2016.01.033 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Trochoidal machining for the high-speed milling of pockets
Wu Shixiong*, Li Bin, Ma Wei, Wang Chengyong Mechanical and Electrical Engineering Institute, Guangdong University of Technology, Panyu Higher Education Mega Center , P. C. 510006, Guangzhou, China
*
Corresponding author. Tel.: +86-18925114386. E-mail address: [email protected]
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Abstract When machining the pocket of a mould in high-speed milling mode, the tool load at a pocket’s narrow area or corner may sharply increase because of the presence of a higher amount of material to be cut. A trochoidal machining method considering milling force, machining tool, and pocket geometry is proposed in this paper. First, a method for the geometric modelling of the engagement angle in trochoidal machining is proposed. Maximum and mean values of the milling force are analysed; meanwhile, the corresponding relationship between the milling force curve and the engagement angle curve during the trochoidal machining process is analysed. Based on fundamental experiments on trochoidal machining, results for the milling force and tool wear are obtained; then, a proper control strategy for cavity trochoidal milling machining is proposed. Based on this control strategy for trochoidal milling machining, two realizations of cavity trochoidal milling machining are proposed. Finally, comparison experiments on cavity machining are conducted. Compared with the feedrate adjustment method, trochoidal machining provides better control over the milling force and tool wear at corners and narrow slots. The milling force and machining vibrations are smaller, and the tool wear is substantially reduced. Keywords: high-speed milling; trochoidal machining; milling force
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1. Introduction
High-speed milling provides various benefits in terms of improved production efficiency, machining precision, surface quality, etc. It has been successfully applied in the mould industry and promotes the rapid development thereof. Contour-parallel tool paths are a common high-speed milling method for cavity moulds, where the tool path is generally calculated based on the contour offset and intersection. However, corners and narrow areas such as slots may easily appear between contours. If no special treatments are applied, the following issues can frequently arise in high-speed machining: (1) The engagement angle or engagement arc length between the tool and uncut materials can be greatly increased, resulting in a sharp increase in the contact material. (2) The tool load amount can be much higher at a corner or slot, resulting in a greater fatigue or damage to the tool. These problems are particularly serious in the high-speed milling of harder materials. The engagement angle and cutting load variations have generated concern by scholars. Various studies have been conducted, including the analysis and modelling of the engagement angle, milling force, and other aspects. Kline et al. (1982) presented a mechanistic model for the force system in end milling and stated that varying the cutting engagement and chip thickness could cause variations in the cutter load. Choy and Chan (2003) reported that the instantaneous cutter sweep angle (CSA) is a suitable parameter for studying chip load in 2.5D pocket milling and proposed a comprehensive modeling method; experiments showed that the model can accurately predict a cutter load pattern at cornering cut. Wei et al. (2010) proposed an effective milling force model for pocket machining and pointed out that the varying feed direction and cutter
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engagement could affect the milling force for the entire process. These studies show that the close relationship between tool contact and load changes during machining process. This relationship is used as the basis to establish methods for reducing cutting load, such as the adjustment of the cutting parameters or trajectory. Two major approaches, namely, the adaptive control and geometry modification of the tool path, to addressing the cutting load variation problem can be identified. The adaptive control approach focuses on controlling the cutting performance by instantaneously adjusting the cutting parameters when milling the work pieces. Spence and Altintas (1994) developed a simulation system based on constructive solid geometry (CSG) and concluded that the feed rate can be automatically scheduled to satisfy force, torque, and part dimensional error constraints. Tarng and Shyur (1993) argued that cutting stability is greatly dependent on the radial depth of cut and proposed a method for identifying the radial depth of cut in pocket machining; they concluded that the feedrate can be adjusted accordingly to maintain a constant material removal rate. Bae et al. (2003) assumed that the cutting force is a function of two major independent variables, namely, 2D chip-load and the feedrate; a simplified cutting model was proposed to adjust feedrate for pocket machining. Liu et al. (2015) presented a feedrate optimization strategy with multiple constrains including relative chip volume, milling force, and cutter deflection. The experimental results demonstrated that the optimized feedrate can satisfy the requirements for a pocket milling process. The adaptive control approach ， which is used to adjust the instantaneous contact relationship between cutter and the workpiece, exhibits a certain effect on controlling the cutting force and protecting the cutting tools during machining. Because the cutting force and vibration may increase almost instantaneously at a sharp corner or a narrow
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slot, the adaptive control approach could help to reduce to a certain degree but not completely avoid the negative influences. The second approach is the treatment of the trajectory by changing the track form. Zhao et al. (2007) demonstrated that residual materials at the corner can be effectively removed by inserting a biarc curve. Through the technique, the tool may be subjected to high cutting loads because the tool enters the corner in a direct manner. Choy and Chan (2003) inserted bow-like tool path segments at a corner and concluded that the improved tool path can clear the accumulated material and reduce the cutting load at pocket corners; this progressive cutting method can effectively control the milling load. Elber et al. (2004) considered the need for C1 continuous tool paths and presented a pocket tool path containing a series of circular arcs; this method mainly focuses on the geometrical analysis and discussion. Ibaraki and Yamaji (2010) proposed that the materials of the medial axis areas of the pocket should be removed by trochoidal grooving to effectively control the tool load during the late milling stage. Ferreira and Ochoa presented a method to generate trochoidal tool paths for pocket milling process with multiple tools and concluded that the method can avoid the momentary increments in the radial depth of cut. The second method is more effective in decreasing the load variation and protecting the milling tool; however, few papers have comprehensively considered the milling force, cutting tool, and processing. Trochoidal machining for high-speed milling pockets can be categorized as the second method. It is a method of progressively cutting away material and is very suitable for milling narrow areas or sharp corners, where the cutting load changes considerably (Fig. 1). In recent years, certain commercial CAM software has included trochoidal machining methods. Several scholars have also launched corresponding studies on
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trochoidal machining. Through process experiments, Uhlmann et al. (2013) holds that a trochoidal milling strategy for TiAl6V4 workpieces offers considerable potential to improve energy consumption and process time during production. For difficult-tomachine materials such as Nickel-based superalloys, Pleta et al. (2014) provided a comparison of trochoidal milling with a traditional milling technique. They believe that the process of trochoidal machining can result in improved productivity and efficiency. Aiming at the workpiece surface with holes and bosses, Otkur and Lazoglu (2007) proposed an approach to model the milling force for trochoidal milling. The experimental results demonstrated the good agreement between the predicted forces and the measured forces. Rauch et al. (2009) investigated the trochoidal model which infers curvature and tangency continuity, and concluded that the cutting time and tool life can be effectively improved in a high dynamics machining tool. Rauch and Hascoet (2007) developed algorithms that can be used to generate trochoidal and plunge cutting trajectories for pocket milling. The results of this study lead to a better definition of the strengths and weaknesses of trochoidal milling and plunge cutting strategies in rough machining of aluminium alloys with a focus on optimizing the choice of strategies. In these trochoidal models, the trochoidal circles normally present equal diameters. In this paper, the trochoidal definition is expanded to include two types of trochoidal machining: isometric circle trochoid (Fig. 1b) and variable circle trochoid (Fig. 1c). Ibaraki and Yamaji (2010) reported that the medial axis areas of a pocket should be removed by trochoidal tool paths prior to high-speed contour-parallel cutting. The experiments demonstrated improved machining efficiency and tool wear. In the study, the machining parameters, such as the size or distance of trochoidal circles, are specified based on experience. Unsuitable parameters may cause reduced efficiency or
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increased the tool load. In the present work, a trochoidal model and a control strategy for controlling parameters are proposed and used to generate several kinds of pocket trochoidal paths. Attempting to address the processing problems in milling narrow areas or corners in a pocket, this paper will present research on trochoidal machining. The milling forces, tool wear, tool path and cavity geometry will be comprehensively considered. The remainder of the paper is organized as follows. The modelling of the engagement angle and milling force in trochoidal machining is presented in Section 2. The control strategies for trochoidal machining are presented and analysed in Section 3. The realized methods of trochoidal machining in pockets are given in Section 4, followed by experimental identification in Section 5 and conclusions in Section 6.
2. Modelling of the engagement angle and force in trochoidal machining
2.1 Basic concepts The engagement angle represents the contact geometrical relationship between the tool and the uncut material of the workpiece. As the tool travels along the tool path, the engagement angle continuously varies, as shown in Fig. 2. The engagement angle will reach its maximum value in the concave corner area. Accordingly, the amount of material to be cut will reach a maximum at the concave corner, similar to the milling force. As mentioned by Choy and Chan (2003), the cutter load is directly related to the cutter engage arc length or the engagement angle. Therefore, the engagement angle can be used to characterize the change trend of the milling force under certain conditions.
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When the cutter mills the pocket in a contour-parallel cutting manner, the cutter load is very large at sharp corners, slots or other narrow areas. This is due to the large cutting engagement angle, which may even reach 180 degrees in the milling of a slot.
Trochoidal machining can decrease the engagement angle during processing because the material is gradually milled by a series of continuous circles. Therefore, the cutting load can be greatly reduced, and the cutter life can also be improved. In the straight trajectory area as shown in Fig. 2, the engagement angle can be solved according to the tool path contour spacing d and the tool radius r, namely, astr=arcos[(r-d)/r]
(1)
In this paper, the value of astr will be taken as an important reference to control the trochoidal machining process. When we generate the cutting tool path for a mould pocket, the contour spacing d can be determined by experience or by consulting reference handbooks. Thereupon, the engagement angle astr can be easily determined.
2.2 Geometric modelling of engagement angle in trochoidal machining The following is the geometric modelling of the engagement angle change process of the trochoidal trajectory, where the characteristics and rules are mastered and applied to a milling force control. The derivation process adopts a variable circle mode (Figs. 3ac), and an isometric circle mode is regarded as a special case (Fig. 3d).
In Fig. 3a, the trochoidal paths are shown in dashed lines. The dashed-line circle (O2) and circle (O1) represent the current circle and the previous circle, respectively. To facilitate an analysis, each dashed-line circle is offset by a distance of r (tool radius) and is indicated by a solid line. The tool rolls along the dashed line circle, which is
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equivalent to rolling on the inner side of the solid-line offset circle. One cycle of trochoid includes two sections: a connected curve (SE) and a circle (E-G-H-E). A 3D coordinate system with the centre O1 of the previous offset circle as the origin (0, 0) is established. The distance between O1O2 is L. The tool centre C and the tangency point E1 are connected, C and O1 are connected, C and D are connected, and D and O1 are connected. Obviously, ∠DCE1 is the engagement angle. The radius of the previous offset circle is set as R1, the radius of the current offset circle is set as R2, and the tool radius is set as r. Obviously, |CD|=r，|CE1|=r， |O1D|=|O1S|= R1, |O2E|= R2, R1=rtro1+r, R2=rtro2+r. To avoid a tedious calculation at the circular intersection, a geometric modelling method for the engagement angle in trochoidal machining is proposed as follows. It mainly includes two sections (Figs.3 b-c). (1) Geometric modelling on the first section SE Calculating angle ∠E1CO1 Fig. 3b shows that the cutter rolls along the curve SE and that the tool centre trajectory is S’E’. The coordinates of the cutter centre C(xc,yc) can be computed using the tangency point E1 and vector E1C . The length of CO1 is CO1 ( x C x O1 ) 2 ( y C y O1 ) 2
In addition, the length of E1O1 is E1O1 ( x E1 xO1 ) 2 ( y E1 yO1 ) 2
Obviously, from the geometric relationship, ∠E1CO1 can be calculated by the following formula:
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cos E1CO1 2
2
(2)
2
( CO1 CE1 E1O1 ) /( 2 * CO1 * CE1 ) 2
2
( CO1 r 2 E1O1 ) /( 2 * CO1 * r )
Calculating the angle ∠DCO1 ∠DCO1 is an interior angle of the triangle △DCO1, and its range is 0<∠DCO1<π. It can be obtained using the cosine theorem: cos∠DCO1=(|CO1|2+r2–R12)/(2*|CO1|*r)
(3)
Calculating the engagement angle ∠DCE1 Obviously, the engagement angle ∠DCE1 satisfies the following geometric formula: atro=∠DCE1=2π–∠E1CO1–∠DCO1
(4)
When SE1 increases in the variation range, the engagement angle with corresponding change may be calculated using this geometric calculation. (2) Geometric modelling of the second section EGHE Establishing the trajectory equation for tool centre The moving trajectory of the cutter centre C is the arc centred by O2 (xO2, yO2) with radius R2-r. Therefore, the following equations for the variation in C(xc, yc) may be established. xc=xO2 +(R2–r)*cosφ yc=yO2 + (R2–r)*sinφ
(5)
Calculating the angle ∠DCO1 Obviously, ∠DCO1 is an interior angle of the triangle △DCO1, and its range is 0< ∠DCO1<π. From the geometric relationship, |CD|=r，|O1D|=R1 are known, and the length of CO1 is h2=( xc–0)2+( yc–0)2,
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According to the cosine theorem, cos∠DCO1=(h2+r2–R12)/(2*h*r)
(6)
The angle ∠DCO1 can be obtained by the inverse cosine. Calculating the angle ∠E1CO1 The angle ∠E1CO1 may be calculated as the vector dot product of O2 C and CO1 . However, because ∠E1CO1 is likely to be larger than π (see Fig. 3c), the relationship of the vector cross product is required to identify whether ∠E1CO1 is greater than or less than π. The vector quantity CO1 is set as a=[ax，ay，0], and ax = –xO2 – (R2–r)*cosφ ay = –yO2 – (R2–r)*sinφ The vector quantity
O2 C
is set as b=[bx，by，0], and
bx =(R2–r)*cosφ by =(R2–r)*sinφ According to the vector dot product, cos∠E1CO1=a·b/(|a|·|b|) =（ax*bx+ay*by）/[(ax2+bx2)+( ay2+by2)]
(7)
∠E1CO1 can be acquired using the inverse cosine in the range between 0 and π.
Calculating the engagement angle ∠DCE1 Obviously, the engagement angle satisfies the following geometric formula: atro=∠DCE1=2π–∠E1CO1–∠DCO1
(8)
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When the angle φ increases in the variation range, the engagement angle with corresponding change may be calculated using this geometric calculation. (3) Determining processing limitations (Fig. 4) First, the current offset circle (centred by O2) should intersect the previous offset circle (centred by O1). The limitations thereof should be studied; that is, determine whether the current offset circle is internally tangent or externally tangent with the previous offset circle (see Fig. 4a). The distance of O1 and O2 should meet the following condition: R1–R2
(9)
Second, after the cutter travels through the previous offset circle and the current circle, no residual material should remain in their middle area. According to the geometrical relationship, the limitation should be as follows: O1, O2, and the cutter centre are in a straight line, and the tool circle is internally tangent with the current offset circle and externally tangent with the previous offset circle (see Fig. 4b). Therefore, S=(L –R1)+R2<2*r That is, the distance of O1 and O2 should satisfy L < R1–R2+2*r
(10)
In most cases, the cutting width S is less than r to avoid large cutting loads and tool fatigue; therefore, the following inequality can be obtained. L < R1–R2+r
(11)
(4) Calculating and drawing the variation diagram of the engagement angle A trochoidal path normally incorporates a few trip cycles. Each cycle is combined with one connected curve and one circle arc. By calculating the engagement angle for a
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series of positions in a trip cycle, a variation curve diagram of the engagement angle can be drawn. Fig. 5 includes several typical examples. The horizontal coordinate axis is the rolling length, and the vertical axis is the engagement angle (in degrees). When the ratios of r, R1, and R2 vary, similar results can be obtained. Based on the curve diagram analysis, two conclusions can be drawn: (1) Increasing the centre distance of two neighbouring circles can increase the maximum value of the engagement angle of the trochoidal path. (2) Assuming that the cutter radius (r) and the trochoidal circle spacing (L) are certain, the maximum engagement angle will decrease with an increasing trochoidal circle size.
2.3 Modelling of milling force in trochoidal machining Altintas et al. (1991) presented an efficient simulation system for milling mechanics by using tool path generation algorithms based on a solid modeler; they also experimentally validated their models. According to the mechanistic force model, the differential forces are defined in the tangential, radial, and axial directions as in Eq. (12). Then, they are transformed into x, y, and z coordinates as in Eq. (13) dFt ( ) ( K tc h( ) K te )dz dFr ( ) ( K rc h( ) K re )dz dF ( ) ( K h( ) K )dz ac ae a
(12)
dFx ( ) dFt cos dFr sin dFy ( ) dFt sin dFr cos dFz ( ) dFa
(13)
The instantaneous forces can be expressed as follows: ex
ex
ex
st
st
st
Fx dFx , Fy dFy , Fz dFz
(14)
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In addition, the resultant force is 2
2
F Fx Fy Fz
2
(15)
2.4 Maximum and mean milling force or engagement angle in trochoidal machining 2.4.1 Maximum milling force or engagement angle The maximum milling force and engagement angle are denoted as Ftro_max and atro_max. Because the engagement angle and milling force vary in a similar manner, it is only necessary to analyse the engagement angle. Fig. 6a shows that the line O1O2 intersects the circle O1 at point M. First, during the process of the cutter-circle rolling along the curve SE, the engagement angle will gradually increase and obtain a periodical extremum at point E. Then, the cutter-circle continues rolling along the arc EGH, and the engagement angle will gradually increase. When the cutter-circle passes through point M, the angle reaches its maximum (Fig. 6b); then, it gradually decreases. Another particular case is when the point M is located within the cutter circle when the cutter circle is tangent to the point E. In this case, a maximum engagement angle occurs. In the subsequent rolling path of EGH, the engagement angle will decrease step by step. 2.4.2 Computing the variation curve diagram of milling force For specific workpiece materials, if the milling parameters of the cutting tool are given, the method in Chapter 2.3 can be used to calculate the dynamic milling force for trochoidal machining. Corresponding to the EGH path in Fig. 6, Fig. 7 shows the dynamic milling force calculated under some parameters. The milling force first increases to the maximum gradually, and then decreases gradually. In Fig. 7, the two adjacent dots illustrate a rotation of the cutting tool. In this case, the cutting tool has
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four edges, so there are four peak values between the two adjacent dots, respectively corresponding to the peak values obtained when the four edges cut in or out the workpiece. A series of peak points (F1, F2, … Fi, Fi+1 … Fn) for the first edge of the cutting tool during the machining can be connected to form a curve which is defined as the variation curve of milling force in this paper. The maximum of the series of peak points is defined as Ftro_max. In addition, the series of peak points can be averaged, thus to obtain an average value Ftro_ave, as below: Ftro _ ave ( F1dl F2 dl Fi dl Fn dl ) /(ndl ) ( F1 F2 Fi Fn ) /(n) (16)
Similarly, atro_max and atro_ave can be obtained from the variation curve of the engagement angle in Fig. 5. The four values, Ftro_max、Ftro_ave、atro_max and atro_ave, will be used in subsequent calculations and analysis.
2.5 Variation curve diagram of milling force and engagement angle The large axial and small radial depths of cut are often adopted in cavity machining, which can provide improved use the cutting capacity of the cutting edges of the tool and allow more materials to be machined before the cutting tool becomes blunt. Usually, the radial depth of cut is smaller than 1/3 of the tool diameter, or even much smaller; while the axial depth of a cut may be larger than 1/3 of the tool diameter, or even much larger. In this context, carry out a calculation and comparative analysis for the variation curve of the trochoidal milling force and the variation curve of the engagement angle. Fig. 8 shows that for a cycle in trochoidal machining, we can first calculate each peak value of the milling force and then arrange them into a milling force curve. Second, we can calculate each changing engagement angle for trochoidal machining and then arrange
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them into another curve. Finally, we extend the engagement angle curve to make the peaks of the two curves meet. After comparison of the two curves, we find that the changing trend of the engagement angle is very similar to the changing trend of the milling force. Therefore, we can use the engagement angle as a substitute for the milling force for analysis and calculation such as for large axial depths of cut and small radial depths of cut in cavity machining.
3. Control strategy for trochoidal milling
In high-speed cavity milling, a cutting mode with a large axial depth of cut and small radial depth of cut is often used. This is taken as the background; then, we will implement a serial of fundamental experiments (Fig. 9) and subsequently propose the control strategy for trochoidal machining. (1) The milling force and tool wear increase with an increasing centre distance between neighbouring circles. An excessively large centre distance will increase the possibility of wear and damage to the cutting tool. The characteristics of the force change and tool wear are clearly indicated in Fig. 5 and Fig. 8.
(2) If the centre distance is set to equal the contour spacing of contour-parallel cutting, the milling force of trochoidal machining will be particularly high compared with contour-parallel cutting, which is more likely to make the tool fatigue and even become damaged.
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However, the tool exhibits good heat dissipation in the trochoidal machining process; therefore, the ultimate tool wear tends to be low.
According to the above experiments on trochoidal machining, the main problem in trochoidal machining is the huge milling force, which tends to cause fatigue and breakage of the tool. Hence, the primary task of trochoidal machining control is to control the milling force. The following control strategy for trochoidal milling will focus on the milling force. Moreover, the cavity geometry, tool wear, and milling efficiency will also be considered. The following will select the engagement angle as the auxiliary control parameter, not the spacing of the contour-parallel path. This is mainly because the spacing of the contour-parallel path is not an exact expression of milling force changes, but the engagement angle can provide for a very good expression (see 4.2).
3.2 Control strategy for trochoidal machining in cavity milling If milling cavity is performed using the contour-parallel cutting method, the processing parameters, such as path spacing, cutting depth, and feedrate, are generally given by consulting a manual or by experience, which usually consider the bearing capacity of the cutting tool. Taking these basic parameters of contour-parallel cutting as a reference for trochoidal milling, and considering the above trochoidal machining features, a trochoidal machining control strategy is proposed as follows: (1) According to the basic parameters of the contour-parallel cutting, such as the cutting tool radius (r) and path spacing (d), calculate the engagement angle (astr) or the milling force (Fstr) at the straight trajectory area of the contour-parallel cutting.
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(2) According to the cavity geometry features, select an appropriate small circle, determine the area to insert the trochoid, and compute the location of the initial circle of the trochoidal path. (3) Calculate a reasonable centre distance of neighbouring circles to meet the control requirements of the engagement angle or milling force. Assuming that the mean engagement angle in trochoidal machining is atro ave and that the maximum engagement angle is atro_max, calculate the following three parameters.
Llow: Ftro max=Fstr; Calculate a proper centre spacing of trochoidal circles to make
the maximum milling force equal to the milling force Fstr in the contour-parallel cutting; this spacing is defined as Llow.
Lmid: Ftro ave =Fstr; Calculate a proper centre spacing of trochoidal circles to
make the average milling force equal to the milling force astr in the contour-parallel cutting; this spacing is defined as Lmid.
Lhigh: Ftro ave =1.5Fstr; Four areas can be delineated according to the three values above (as shown in Fig.
10): Area 1: The machining is safe, and the cutting tool is subject to relatively small forces and obtains a longer service life, but the milling efficiency is reduced. Area 2: The machining is relatively safe, the tool wear is reduced, and the tool load is greater than that of the machining mode in contour-parallel cutting. Area 3: The cutting tool load is larger, and the possibility of fatigue for cutting tool increases. However, the machining efficiency will also be improved. Area 4: The machining efficiency is higher, but the cutting tool load is larger; moreover, the tool can be easily damaged, and tool wear is more serious.
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Obviously, in general, Area 2 or Area 3 should be chosen in high-speed cavity milling. Area 3 can be chosen to improve the milling efficiency if the strength of the cutting tool permits. To obtain better control, we can set the values of t1 and t2 (for example t1=10% and t2= 50%) and calculate a proper centre spacing of neighbouring trochoidal-circles to make the average milling force in trochoidal milling satisfy Ftro_ave≤Fstr * (1+t1)
(17)
or the maximum engagement angle satisfy Ftro_max≤Fstr* (1+t2)
(18)
Sometimes the engagement angle a (astr、atro、atro_max) can be used as a substitute for the milling force (Fstr、Ftro、Ftro_max) to simplify the calculation. (4) According to the geometry features of the cavity, corner, and narrow slot, compute the trochoidal path in the material gathering area. The corresponding algorithm will be shown in Part 4.
4. Trochoidal machining for cavity
4.1 Geometry description Contour-parallel tool paths represent a common high-speed milling method for cavity moulds. In this paper, the contour-parallel cutting approach adopts the milling mode from inside to outside. The path elements involved are mainly arcs and linear segments. The following analysis will focus on the line-line corner type, and other corner types composed of complex curves can be deduced accordingly.
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Fig. 11 shows the tool path of contour-parallel cutting. Assuming that the current trajectory is U and V, the trajectory formula can be expressed as U: A1 x B1 y C1 F1t1 0 V: A2 x B2 y C 2 F2 t1 0
(19)
where F1 and F2 refer to the offset director and take on the value of 1 or -1; the coefficients of x and y are normalized. The previous trajectory with the offset distance d is PU: A1 x B1 y C1 F1 (t1 d ) 0 PV: A2 x B2 y C 2 F2 (t1 d ) 0
(20)
The bisector of two geometric curves is the trajectory in which each point is equidistant to both curves. As mentioned by the paper of Persson (1978), the bisector of the corner may be a straight line, parabola, or hyperbola. Calculating the bisector is an important step in establishing the trochoidal path. Let the curves U and V be offset by an amount equal to h. The offset curves intersect at a point M, which is a point on the bisector. The intersection point can be calculated by the following formula: A1 x B1 y C1 F1 (t1 h) 0 A2 x B2 y C 2 F2 (t1 h) 0
(21)
Thus, the coordinate of the intersection point M is x m ( B1 C 2 B 2 C1 ) / T (t1 h)( B1 F2 B 2 F1 ) / T y m ( A2 C1 A1C 2 ) / T (t1 h)( A2 F1 A1 F2 ) / T
(22)
By calculating a series of intersection points, the bisector of U and V can be acquired.
4.2 Trochoidal machining for corners of a cavity
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Material is more easily gathered at a cavity corner, especially corners with smaller angles, while milling corners with trochoid realize smaller tool loads. The path elements involved are mainly clockwise and anticlockwise arcs and linear segments, which may be combined to produce a total of nine types of corners as mentioned by the paper of Choy and Chan (2003). Trochoidal machining of cavity corners can be realized in three ways, as shown in Fig. 12, Corner trochoidal machining with variable circles. Material at the formed corner of
each contour ring can be gradually removed by the method of variable trochoidal circles. Corner trochoidal machining with isometric circles. Material at the formed corner of
each contour ring can be gradually removed by the method of isometric circles. The first and second method generate a trochoidal path at the corner of each ring in the contour-parallel path. Perforative corner trochoidal machining along the medial axis. This uses the method
of Voronoi diagrams to calculate the medial axis of the cavity and arrange the perforative trochoid according to the medial axis.
The computation algorithm for trochoidal machining includes the following steps: (1) Calculate the engagement angle or the milling force at the straight line area of the contour-parallel cutting path, which is regarded as a basic reference for the trochoidal calculation. (2) Compute the guiding curve. If the first or second method of trochoidal machining is adopted, analyse the corner type and calculate the corner bisector or the offset curve
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of the corner edge. If the third method of trochoidal machining is adopted, calculate the medial axis of the cavity. (3) Determine the initial circle and calculate the size and location of the trochoidal circle step by step, guided by the curves in Step 2 so that the engagement angle or milling force can satisfy the control requirements presented in Section 3. (4) Add the new trochoidal circle and the middle curve between two adjacent circles into the stored linked list. 4.2.1 Corner trochoidal machining with variable circles As shown in Fig. 12a, a few variable circles may be applied to mill the material of the remaining corner. The circle (c1) on the end position of the corner is very specific and is selected as the initial trochoid circle. Other trochoid circles will be computed one by one until the corner materials are cut out. Finally, all of the trochoid circles are stored in the trochoid list in reverse order. The calculation of the next trochoid circle constitutes the key sub-algorithms. Fig. 13 shows that the current trajectory is U and V, and the trajectory with the offset distance r (tool radius) is UW and VW. The circle centred by N and tangential to UW and VW is the offset circle of the current trochoid circle. The computation on the next trochoid circle involves two steps: (1)Analyse the situation when the cutter circle is internally tangential to the current offset circle at point G on the corner bisector and calculate the other intersection point S between the cutter circle and the bisector curve (see Fig. 13a). Then, calculate the other offset circle externally tangential to the cutter circle at point S and set its centre as the point M. The in-centre point M will be taken as a limit point, and the next trochoid circle will be obtained between M and N.
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(2) Establish a function Ftro_ave=Fstr * (1+t1); Z(P)= Ftro(P)–Ftro_ave≤ε
(23)
where t1 is a specified value (for example t1=15%), Ftro (P) is the average milling force when the cutter rolls within the current offset circle, and ε is a relatively small control value (e.g., ε=1). First, for M and N, find the midpoint P((MX+NX)/2, (MY+NY)/2) on the corner
bisector. Take it as a centre and create a new circle that is tangential to UW and VW (see Fig. 13b). Take the circle centred by P and the circle centred by N as the present analysis
objects of the trochoidal trajectory. Calculate the average milling force when the cutter rolls along the current arc EGH, and set the angle as atro(P). Calculate Z (P) according to formula (23). If Z (P) >0 and Z (P) ≤ε, the point P is
the required trochoidal-circle centre; exit the calculation. Otherwise, if Z (P) <0, then set M=P, or if Z (P) >0, set N=P. Repeat the above three steps until an appropriate P is obtained. Make the circle
centred by P offset inward a distance of r; then, the next required trochoid circle can be obtained. Sometimes the maximum milling force Ftro max is adopted as the analysed object. Therefore, the average engagement angle in formula (23) will be replaced by the maximum milling force Ftro max. If the engagement angle is used as a substitute for the milling force, formula (23) is converted to
23
Z(P)= atro(P)–atro_ave≤ε
(24)
The algorithm is similar to that above. 4.2.2 Corner trochoidal machining with isometric circles The method of trochoidal milling corners with isometric circles can facilitate calculation and improve the continuity of the trajectory. It can also effectively reduce the cutting tool load. According to whether the trochoid circle can coat the corner arc formed in previous machining, two implementation methods can be obtained, as shown in Fig. 14a and Fig. 14b. U-V is the current trajectory in Fig. 14, PU-PV is the previous trajectory, and PSQ is the corner arc formed after the previous machining. The distance from P to UW is |PH1|. Make a circle with |PH1| as diameter; then, internally offset by a distance of r (tool radius). Then,
rh1 PH 1 / 2 r 2
2
| A1 x P B1 y P C1 F1 (t1 r ) | / 2 A1 B1 r
(25)
If the radius of the selected trochoid circle rh satisfies rh>rh1, this indicates that the trochoid circle can coat the former corner arc formed by previous milling. Thus, the method in Fig. 14a can be used. To calculate its machining trochoid, a suitable centre distance (L) of neighbouring circles should be first calculated so that the trochoidal machining can satisfy the limits of the engagement angle and the milling force. Then, the curve U is offset by rh to obtain the curve Up. According to the geometric characteristics of the corner curve PU-PV, we determine the initial circle (O1) on the curve Up and then calculate the trochoid circles guided by the curve Up one by one and store the results in the path list. Assuming that the coordinate of the current trochoid
24
circle centre is (xoi,yoi), the next trochoid circle centre (xoj,yoj) on the curve Up should satisfy the following relation: (26)
( xOi xOj ) 2 ( yOi yOj ) 2 L2
In certain cases, the selected trochoid circle is small (rh
4.3 Trochoidal machining for narrow areas Narrow areas, such as slots, often occur in cavity machining. The width of the narrowest area can be defined as h=k*d (such as k=3~5, where d is the cutter diameter). Trochoidal machining remains a very good method for the machining of narrow areas because it helps reduce the cutting load. However, if the actual width exceeds this value, the contour-parallel cutting approach is preferred for removing the material from the narrow area.
25
There are mainly two types of treatment for the narrow area: (1) calculating the bisector or medial axis of the narrow area and using the trochoidal method with variable circles to mill materials (Fig. 15a) and (2) calculating the bisector or medial axis of the narrow area and using the trochoidal method with isometric circles to first mill the materials; then, the trajectory of contour-parallel cutting is used to remove the residual material (Fig. 15b). The corresponding calculation method is similar to that in 4.2.
5. Experiment
5.1 Test platform The test platform consists of a high-speed machining centre (DMC-60T), a tool workpiece system, and a cutting force measuring system. The force measuring system includes a force platform, a charge amplifier, and data acquisition software. The force measuring device is a Kistler9265B dynamic piezoelectric dynamometer with the sensitivity of 0.05 N. The charge amplifier is a Kistler5019. The test results for the milling force include three component forces: Fx, Fy, and Fz. Their synthesized force can be calculated using the formula
F FX 2 FY 2 FZ 2
.
5.2 Comparative experiment Fig. 16 shows a cavity model by the factory. The sizes of the cavity are 100*70mm. There is a 40 degree corner in B area of the model, and a slot in C area. The material of workpiece is P20, and its hardness is HRC36. The tool path generated from CAM software. Use this tool path for direct machining of workpieces without any auxiliary processing. See Fig. 16b for relevant process parameters and machined cavities. After the machining of 14 workpieces, the tool wear reaches the wear-out failure criterion
26
(VB > 0.3 mm). Test the milling force of a tool path loop and obtain the result (Fig. 16c). Large loads often change abruptly at corners. Check the tool edge, and tipping (Fig. 16d) may be found, which indicates the loads on the tool are very large. In order to improve the durability of the cutting tool, feedrate adjusting machining at the corner is considered firstly. Because the commercial CNC (Computerized Numerical Control) system provides limited user settings, changing the feedrate is more suitable. The following measures can be adopted: (1) Change the feed rate at a point 6 mm before each corner, and then the commercial CNC system will complete the deceleration; after the tool passes through the corner, change the feed rate to the normal speed. (2) at the corner of B area, the feed-rate is slowed down to 1/3 of normal speed. At the corner of D area, the feed-rate is slowed down to 1/2 of normal speed. (3) Area C is a slot. Take 2 times for cutting and the axial depth is 2 mm each time. The feed-rate is slowed down to 1/2 of normal speed. Fig. 17b shows a single cavity that has been machined. The total machining time of a single cavity is 88 seconds. Upon checking the flank face of the cutting tool after machining the 18th cavity (Fig. 17c), the cutting lip has been found to be severely worn. Fig. 17d shows the milling force of one of the tool path loops. Compared to the original method (Fig. 16), the abruptly-changing load at the corner decreases, but the maximum load remains large. After the cutting tool exceeds the wear-out failure criterion (VB > 0.3 mm), 22 cavities can ultimately be machined. The number is higher when compared to the original method.
To obtain a better machining, trochoidal-aided machining could be considered. With the straight line area A as a reference and by controlling the average and maximum
27
engagement angles in trochoidal machining, the calculated results shown in Table 1 are obtained.
A machining method adopting perforative corner trochoid along the medial axis is shown in Fig. 18. The calculation of the parameters of the trochoid is shown in Table 1B. In the machining, the trochoid is first run in the middle of the cavity; then, the contour-parallel cutting path is run. Finally, the narrow slot in area C is processed. The total machining time of a single cavity is 113 seconds. The milling force during the process is shown in Fig. 18d. After the cutting tool exceeds the wear-out failure criterion (VB > 0.3 mm), 30 cavities can ultimately be machined. The number is much higher when compared to the original method. In Fig. 19, a trochoidal path is added at the corner of each ring in the contour-parallel path. The calculation of the parameters of trochoid is shown in Tables 1-B and 1-D. The total machining time of a single cavity is 101 seconds. The milling force during the process is shown in Fig. 19d. The milling force during the process is effectively controlled. After the cutting tool exceeds the wear-out failure criterion (VB > 0.3 mm), 33 cavities can ultimately be machined. The above four machining methods can be defined as M1, M2, M3 and M4, respectively. The number of cavities machined was 14, 22, 30 and 33, respectively. Obviously, by using the deceleration method (M2), tool life can be prolonged when compared with the original path method (M1), which, however, will be much longer by using methods M3 and M4. Considering that the change of loads at the corner is an important factor influencing tool life, a comparative analysis should be made for these methods. The maximum loads and average loads were observed during the machining
28
of Corner B when using each of these methods; Fig. 20a shows the results obtained. With M2, both the maximum and average loads decrease to a certain extent when compared to M1; however, both types of load still remain large. This indicates that because the cutting tool has a large amount of contact with the workpiece material in a unit of time, the loads on the tool are still heavy even if the tool has decelerated. However, with M3 and M4 where load optimization has been conducted, both the maximum and average load decrease significantly when compared to M1 and M2; therefore, there is less tool wear and fatigue and the tool life is extended greatly. Further compare M3 and M4; the result indicates that the tool wear of the method M3 is greater and the machining efficiency of the method M3 is much lower. The main reason for this is that the trochoidal track is longer in larger corner processing (the corners of D1 and D2 are close to 80 degrees). Therefore, by optimizing the trochoidal geometry and reducing the length of the trochoidal path, greater process efficiency can be obtained, and later research will be conducted in this aspect. Additionally, compared to the conventional method, M1, the number of cavities machined has a great increase with the two trochoidal methods (M3, M4). However, there currently is also a limitation for trochoidal methods now - their machining efficiency is lower. Fig. 20b shows the comparison of machining time with all four methods. M3 and M4 require more time to machine the product mainly because of the need for having additional auxiliary trochoidal paths in the tool path. The machining time when using these methods is closely related to the size of cavity. The larger the cavity is, the less the proportion of the corner path in the overall tool path will be, and the narrower the time difference between the former and latter two paths will become.
29
In short, the trochoidal path methods present excellent load control capacity and greatly extend tool life. Further research studies on the optimization of trochoidal tool paths will be carried out in the future. The path geometry will be optimized to reduce the length of the trochoidal path, and the depth of cut, feed rate and other parameters will be increased within a controllable range of tool loads, thus to enhance machining efficiency.
6. Conclusion
(1) A method for the geometric modelling of the engagement angle in trochoidal machining is proposed. The corresponding relationship between the milling force curve and the engagement angle curve is analyzed through simulation. The changing trend of the engagement angle is very similar to the changing trend of the milling force. Therefore, in certain circumstances, the engagement angle can be used as a substitute for the milling force for analysis and calculation such as for large axial depths of cut and small radial depths of cut in cavity machining. (2) In high-speed cavity milling, a cutting mode with a large axial depth of cut and small radial depth of cut are often used. This is taken as the background; then, we have implemented a serial of fundamental experiments of trochoidal machining. The result shows that milling force is the key issue in trochoidal machining, and it should be controlled. Based on the obtained effective conclusions, a proper control strategy for cavity trochoidal machining has been proposed. (3) Based on the trochoidal control strategy, two realizations of cavity trochoidal machining have been proposed, and practical cavity machining experiments are compared. Experiments show that the feedrate adjustment method has weak control on
30
the milling force and tool vibrations, therefore, the tool wear is relatively big, and it is easy for the tool to be fatigued and damaged. However, trochoidal milling method is more effective at milling force control, reduction of tool vibrations, and tool wear. By increasing the axial depth of cut, the milling efficiency and tool wear of trochoidal milling method is better than the feedrate adjustment method. In the future, the research of trochoidal machining should be focused on geometry optimization, milling with high axial depth of cut and high speed-rate, in order to improve the efficiency of trochoidal milling. Acknowledgements
This work reported in this paper is conducted in conjunction with ‘NSFC-Guangdong Collaborative Fund Key Program U12012081’.
31
References
Altintas, Y., Spence, A., Tlusty, J., 1991. End milling force algorithms for CAD systems. CIRP Annals-Manufacturing Technology, 40(1), 31-34. Bae, S. H., Ko, K., Kim, B. H., Choi, B. K., 2003. Automatic feedrate adjustment for pocket machining. Computer-Aided Design, 35(5), 495-500. Choy, H. S., Chan, K. W., 2003. A corner-looping based tool path for pocket milling. Computer-Aided Design, 35(2), 155-166. Choy, H. S., Chan, K. W., 2003. Modeling cutter swept angle at cornering cut. International Journal of CAD/CAM, 3(1), 1-12. Elber, G., Cohen, E., Drake, S., 2005. MATHSM: Medial axis transform toward high speed machining of pockets. Computer-Aided Design, 37(2), 241-250. Ferreira, J. C., Ochoa, D. M., 2013. A method for generating trochoidal tool paths for 2½D pocket milling process planning with multiple tools. Proceedings of the Institution of Mechanical
Engineers,
Part
B:
Journal
of
Engineering
Manufacture,
0954405413487897. Ibaraki, S., Yamaji, I., Matsubara, A., 2010. On the removal of critical cutting regions by trochoidal grooving. Precision Engineering, 34(3), 467-473. Kline, W. A., DeVor, R. E., Lindberg, J. R., 1982. The prediction of cutting forces in end milling with application to cornering cuts. International Journal of Machine Tool Design and Research, 22(1), 7-22. Liu, X., Ding, Y. P., Yue, C., Zhang, R., Tong, X., 2015. Off-line feedrate optimization with multiple constraints for corner milling of a cavity. The International Journal of Advanced Manufacturing Technology, 1-9.
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Otkur, M., Lazoglu, I., 2007. Trochoidal milling. International Journal of Machine Tools and Manufacture, 47(9), 1324-1332. Persson, H., 1978. NC machining of arbitrarily shaped pockets. Computer-Aided Design, 10(3), 169-174. Pleta, A., Ulutan, D., Mears, L., 2014. Investigation of Trochoidal Milling in Nickel-Based Superalloy Inconel 738 and Comparison With End Milling. In ASME 2014 International Manufacturing Science and Engineering Conference collocated with the JSME 2014 International Conference on Materials and Processing and the 42nd North American Manufacturing Research Conference (pp. V002T02A058-V002T02A058). American Society of Mechanical Engineers. Rauch, M., Duc, E., Hascoet, J. Y., 2009. Improving trochoidal tool paths generation and implementation using process constraints modelling.International Journal of Machine Tools and Manufacture, 49(5), 375-383. Rauch, M., Hascoet, J. Y., 2007. Generation of plunging and trochoidal toolpaths for pocket milling. Mecanique & Industries, 8(5), 445-453. Spence, A. D., Altintas, Y., 1994. A solid modeller based milling process simulation and planning system. Journal of Manufacturing Science and Engineering, 116(1), 61-69. Tarng, Y. S., Shyur, Y. Y., 1993. Identification of radial depth of cut in numerical control pocketing routines. International Journal of Machine Tools and Manufacture, 33(1), 111. Uhlmann, E., Fürstmann, P., Rosenau, B., Gebhard, S., Gerstenberger, R., Müller, G., 2013. The potential of reducing the energy consumption for machining TiAl6V4 by using innovative metal cutting processes. 11th Global Conference on Sustainalble Manufactruing, september, berlin.
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Wei, Z. C., Wang, M. J., Han, X. G., 2010. Cutting forces prediction in generalized pocket machining. The International Journal of Advanced Manufacturing Technology, 50(5-8), 449-458. Zhang, M., Liu, W. J., Li, L., 2011. Sharp corner clean-up CPO tool-path generating method for complex pocket. Computer Integrated Manufacturing Systems, 17(11), 2399-2404. Zhao, Z. Y., Wang, C. Y., Zhou, H. M., Qin, Z., 2007. Pocketing toolpath optimization for sharp corners. Journal of materials processing technology,192, 175-180.
34
Figure Captions Fig. 1. Trochoidal machining for a pocket: (a) narrow areas or sharp corners in a cavity;
(b) trochoid with isometric circles; and (c) trochoid with variable circles.
Fig. 2. Engagement angle.
Fig. 3. Engagement angle model of trochoidal trajectory: (a) variable circle mode; (b)
the first section SE of one trochoidal cycle; (c) the second section E-G-H-E of one trochoidal cycle; and (d) isometric circle mode.
Fig. 4. Processing limitations.
Fig. 5. Curves of the engagement angle: (a) variable circle mode (R1>R2 ) and (b)
isometric circle mode (R1=R2).
Fig. 6. Maximum and mean milling force: (a) rolling along SE and (b) rolling along
EGH.
Fig. 7. Simulating the calculation of the maximum force and mean force.
Fig. 8. Comparison of milling force and engagement angle: (a) spacing L=0.5 mm; (b)
spacing L=1.0mm; and (c) spacing L=1.5 mm.
Fig. 9. Comparison of two milling modes: (a) set L=d in pocket machining; (b) flank
wear; and (c) Milling force.
35
Fig. 10. Control strategy for trochoidal machining.
Fig. 11. Geometry description.
Fig. 12. Three techniques of trochoidal machining: (a) generate a trochoidal path at the
corner of each contour ring and (b) generate a trochoidal path at the medial axial of cavity.
Fig. 13. Calculate the next cyclic circle.
Fig. 14. Calculation schematic for the trochoidal path: (a) the first method and (b) the
second method.
Fig. 15. Trochoidal machining for narrow areas: (a) method with variable circles and (b)
method with isometric circles.
Fig. 16. Original machining method: (a) machining path generated from a CAM
software; (b) machined pocket; (c) Flank wear (VB=0.32mm) after machining the 15th cavity; and (d) milling force.
Fig. 17. Feedrate adjusting machining: (a) machining path; (b) machined pocket; (c)
Flank wear (VB=0.256mm) after machining the 18th cavity; and (d) milling force.
Fig. 18. Trochoidal machining 1: (a) machining path; (b) machined pocket; (c) flank
wear (VB=0.215mm) after machining the 18th cavity; and (d) milling force.
36
Fig. 19. Trochoidal machining 2: (a)machining path; (b) machined pocket; (c) flank
wear (VB=0.2mm) after machining the 18th cavity; (d) milling force.
Fig. 20. Comparison of four methods: (a) comparison of milling force (in B area of
the model); and (b) comparison of the machining time of a single cavity.
37
Fig. 1.
Fig. 2.
38
Fig. 3.
Fig. 4.
39
Fig. 5.
40
Fig. 6.
Fig. 7.
41
Fig. 8.
42
Fig. 9.
43
Fig. 10.
44
Fig. 11.
45
Fig. 12.
Fig. 13.
46
Fig. 14.
47
Fig. 15.
Fig. 16.
48
Fig. 17.
Fig. 18.
49
Fig. 19.
50
(a)
(b)
Fig. 20.
51
Table 1. Computed trochoidal parameters.
Area A
r=3.0 mm d=1.0 mm astr= 480 (See chapter 2.1)
Area B
Area D
rtro=1.8 mm L=0.72 mm
52
r1=1.0 mm rtro =2.4 mm L=2.1 mm atro max =690 atro ave =530 t1 =10%, t2 =45% (See chapter 3.2)
S0924-0136(16)30031-0 http://dx.doi.org/doi:10.1016/j.jmatprotec.2016.01.033 PROTEC 14714
To appear in:
Journal of Materials Processing Technology
Received date: Revised date: Accepted date:
6-9-2015 3-1-2016 30-1-2016
Please cite this article as: Shixiong, Wu, Bin, Li, Wei, Ma, Chengyong, Wang, Trochoidal machining for the high-speed milling of pockets.Journal of Materials Processing Technology http://dx.doi.org/10.1016/j.jmatprotec.2016.01.033 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Trochoidal machining for the high-speed milling of pockets
Wu Shixiong*, Li Bin, Ma Wei, Wang Chengyong Mechanical and Electrical Engineering Institute, Guangdong University of Technology, Panyu Higher Education Mega Center , P. C. 510006, Guangzhou, China
*
Corresponding author. Tel.: +86-18925114386. E-mail address: [email protected]
1
Abstract When machining the pocket of a mould in high-speed milling mode, the tool load at a pocket’s narrow area or corner may sharply increase because of the presence of a higher amount of material to be cut. A trochoidal machining method considering milling force, machining tool, and pocket geometry is proposed in this paper. First, a method for the geometric modelling of the engagement angle in trochoidal machining is proposed. Maximum and mean values of the milling force are analysed; meanwhile, the corresponding relationship between the milling force curve and the engagement angle curve during the trochoidal machining process is analysed. Based on fundamental experiments on trochoidal machining, results for the milling force and tool wear are obtained; then, a proper control strategy for cavity trochoidal milling machining is proposed. Based on this control strategy for trochoidal milling machining, two realizations of cavity trochoidal milling machining are proposed. Finally, comparison experiments on cavity machining are conducted. Compared with the feedrate adjustment method, trochoidal machining provides better control over the milling force and tool wear at corners and narrow slots. The milling force and machining vibrations are smaller, and the tool wear is substantially reduced. Keywords: high-speed milling; trochoidal machining; milling force
2
1. Introduction
High-speed milling provides various benefits in terms of improved production efficiency, machining precision, surface quality, etc. It has been successfully applied in the mould industry and promotes the rapid development thereof. Contour-parallel tool paths are a common high-speed milling method for cavity moulds, where the tool path is generally calculated based on the contour offset and intersection. However, corners and narrow areas such as slots may easily appear between contours. If no special treatments are applied, the following issues can frequently arise in high-speed machining: (1) The engagement angle or engagement arc length between the tool and uncut materials can be greatly increased, resulting in a sharp increase in the contact material. (2) The tool load amount can be much higher at a corner or slot, resulting in a greater fatigue or damage to the tool. These problems are particularly serious in the high-speed milling of harder materials. The engagement angle and cutting load variations have generated concern by scholars. Various studies have been conducted, including the analysis and modelling of the engagement angle, milling force, and other aspects. Kline et al. (1982) presented a mechanistic model for the force system in end milling and stated that varying the cutting engagement and chip thickness could cause variations in the cutter load. Choy and Chan (2003) reported that the instantaneous cutter sweep angle (CSA) is a suitable parameter for studying chip load in 2.5D pocket milling and proposed a comprehensive modeling method; experiments showed that the model can accurately predict a cutter load pattern at cornering cut. Wei et al. (2010) proposed an effective milling force model for pocket machining and pointed out that the varying feed direction and cutter
3
engagement could affect the milling force for the entire process. These studies show that the close relationship between tool contact and load changes during machining process. This relationship is used as the basis to establish methods for reducing cutting load, such as the adjustment of the cutting parameters or trajectory. Two major approaches, namely, the adaptive control and geometry modification of the tool path, to addressing the cutting load variation problem can be identified. The adaptive control approach focuses on controlling the cutting performance by instantaneously adjusting the cutting parameters when milling the work pieces. Spence and Altintas (1994) developed a simulation system based on constructive solid geometry (CSG) and concluded that the feed rate can be automatically scheduled to satisfy force, torque, and part dimensional error constraints. Tarng and Shyur (1993) argued that cutting stability is greatly dependent on the radial depth of cut and proposed a method for identifying the radial depth of cut in pocket machining; they concluded that the feedrate can be adjusted accordingly to maintain a constant material removal rate. Bae et al. (2003) assumed that the cutting force is a function of two major independent variables, namely, 2D chip-load and the feedrate; a simplified cutting model was proposed to adjust feedrate for pocket machining. Liu et al. (2015) presented a feedrate optimization strategy with multiple constrains including relative chip volume, milling force, and cutter deflection. The experimental results demonstrated that the optimized feedrate can satisfy the requirements for a pocket milling process. The adaptive control approach ， which is used to adjust the instantaneous contact relationship between cutter and the workpiece, exhibits a certain effect on controlling the cutting force and protecting the cutting tools during machining. Because the cutting force and vibration may increase almost instantaneously at a sharp corner or a narrow
4
slot, the adaptive control approach could help to reduce to a certain degree but not completely avoid the negative influences. The second approach is the treatment of the trajectory by changing the track form. Zhao et al. (2007) demonstrated that residual materials at the corner can be effectively removed by inserting a biarc curve. Through the technique, the tool may be subjected to high cutting loads because the tool enters the corner in a direct manner. Choy and Chan (2003) inserted bow-like tool path segments at a corner and concluded that the improved tool path can clear the accumulated material and reduce the cutting load at pocket corners; this progressive cutting method can effectively control the milling load. Elber et al. (2004) considered the need for C1 continuous tool paths and presented a pocket tool path containing a series of circular arcs; this method mainly focuses on the geometrical analysis and discussion. Ibaraki and Yamaji (2010) proposed that the materials of the medial axis areas of the pocket should be removed by trochoidal grooving to effectively control the tool load during the late milling stage. Ferreira and Ochoa presented a method to generate trochoidal tool paths for pocket milling process with multiple tools and concluded that the method can avoid the momentary increments in the radial depth of cut. The second method is more effective in decreasing the load variation and protecting the milling tool; however, few papers have comprehensively considered the milling force, cutting tool, and processing. Trochoidal machining for high-speed milling pockets can be categorized as the second method. It is a method of progressively cutting away material and is very suitable for milling narrow areas or sharp corners, where the cutting load changes considerably (Fig. 1). In recent years, certain commercial CAM software has included trochoidal machining methods. Several scholars have also launched corresponding studies on
5
trochoidal machining. Through process experiments, Uhlmann et al. (2013) holds that a trochoidal milling strategy for TiAl6V4 workpieces offers considerable potential to improve energy consumption and process time during production. For difficult-tomachine materials such as Nickel-based superalloys, Pleta et al. (2014) provided a comparison of trochoidal milling with a traditional milling technique. They believe that the process of trochoidal machining can result in improved productivity and efficiency. Aiming at the workpiece surface with holes and bosses, Otkur and Lazoglu (2007) proposed an approach to model the milling force for trochoidal milling. The experimental results demonstrated the good agreement between the predicted forces and the measured forces. Rauch et al. (2009) investigated the trochoidal model which infers curvature and tangency continuity, and concluded that the cutting time and tool life can be effectively improved in a high dynamics machining tool. Rauch and Hascoet (2007) developed algorithms that can be used to generate trochoidal and plunge cutting trajectories for pocket milling. The results of this study lead to a better definition of the strengths and weaknesses of trochoidal milling and plunge cutting strategies in rough machining of aluminium alloys with a focus on optimizing the choice of strategies. In these trochoidal models, the trochoidal circles normally present equal diameters. In this paper, the trochoidal definition is expanded to include two types of trochoidal machining: isometric circle trochoid (Fig. 1b) and variable circle trochoid (Fig. 1c). Ibaraki and Yamaji (2010) reported that the medial axis areas of a pocket should be removed by trochoidal tool paths prior to high-speed contour-parallel cutting. The experiments demonstrated improved machining efficiency and tool wear. In the study, the machining parameters, such as the size or distance of trochoidal circles, are specified based on experience. Unsuitable parameters may cause reduced efficiency or
6
increased the tool load. In the present work, a trochoidal model and a control strategy for controlling parameters are proposed and used to generate several kinds of pocket trochoidal paths. Attempting to address the processing problems in milling narrow areas or corners in a pocket, this paper will present research on trochoidal machining. The milling forces, tool wear, tool path and cavity geometry will be comprehensively considered. The remainder of the paper is organized as follows. The modelling of the engagement angle and milling force in trochoidal machining is presented in Section 2. The control strategies for trochoidal machining are presented and analysed in Section 3. The realized methods of trochoidal machining in pockets are given in Section 4, followed by experimental identification in Section 5 and conclusions in Section 6.
2. Modelling of the engagement angle and force in trochoidal machining
2.1 Basic concepts The engagement angle represents the contact geometrical relationship between the tool and the uncut material of the workpiece. As the tool travels along the tool path, the engagement angle continuously varies, as shown in Fig. 2. The engagement angle will reach its maximum value in the concave corner area. Accordingly, the amount of material to be cut will reach a maximum at the concave corner, similar to the milling force. As mentioned by Choy and Chan (2003), the cutter load is directly related to the cutter engage arc length or the engagement angle. Therefore, the engagement angle can be used to characterize the change trend of the milling force under certain conditions.
7
When the cutter mills the pocket in a contour-parallel cutting manner, the cutter load is very large at sharp corners, slots or other narrow areas. This is due to the large cutting engagement angle, which may even reach 180 degrees in the milling of a slot.
Trochoidal machining can decrease the engagement angle during processing because the material is gradually milled by a series of continuous circles. Therefore, the cutting load can be greatly reduced, and the cutter life can also be improved. In the straight trajectory area as shown in Fig. 2, the engagement angle can be solved according to the tool path contour spacing d and the tool radius r, namely, astr=arcos[(r-d)/r]
(1)
In this paper, the value of astr will be taken as an important reference to control the trochoidal machining process. When we generate the cutting tool path for a mould pocket, the contour spacing d can be determined by experience or by consulting reference handbooks. Thereupon, the engagement angle astr can be easily determined.
2.2 Geometric modelling of engagement angle in trochoidal machining The following is the geometric modelling of the engagement angle change process of the trochoidal trajectory, where the characteristics and rules are mastered and applied to a milling force control. The derivation process adopts a variable circle mode (Figs. 3ac), and an isometric circle mode is regarded as a special case (Fig. 3d).
In Fig. 3a, the trochoidal paths are shown in dashed lines. The dashed-line circle (O2) and circle (O1) represent the current circle and the previous circle, respectively. To facilitate an analysis, each dashed-line circle is offset by a distance of r (tool radius) and is indicated by a solid line. The tool rolls along the dashed line circle, which is
8
equivalent to rolling on the inner side of the solid-line offset circle. One cycle of trochoid includes two sections: a connected curve (SE) and a circle (E-G-H-E). A 3D coordinate system with the centre O1 of the previous offset circle as the origin (0, 0) is established. The distance between O1O2 is L. The tool centre C and the tangency point E1 are connected, C and O1 are connected, C and D are connected, and D and O1 are connected. Obviously, ∠DCE1 is the engagement angle. The radius of the previous offset circle is set as R1, the radius of the current offset circle is set as R2, and the tool radius is set as r. Obviously, |CD|=r，|CE1|=r， |O1D|=|O1S|= R1, |O2E|= R2, R1=rtro1+r, R2=rtro2+r. To avoid a tedious calculation at the circular intersection, a geometric modelling method for the engagement angle in trochoidal machining is proposed as follows. It mainly includes two sections (Figs.3 b-c). (1) Geometric modelling on the first section SE Calculating angle ∠E1CO1 Fig. 3b shows that the cutter rolls along the curve SE and that the tool centre trajectory is S’E’. The coordinates of the cutter centre C(xc,yc) can be computed using the tangency point E1 and vector E1C . The length of CO1 is CO1 ( x C x O1 ) 2 ( y C y O1 ) 2
In addition, the length of E1O1 is E1O1 ( x E1 xO1 ) 2 ( y E1 yO1 ) 2
Obviously, from the geometric relationship, ∠E1CO1 can be calculated by the following formula:
9
cos E1CO1 2
2
(2)
2
( CO1 CE1 E1O1 ) /( 2 * CO1 * CE1 ) 2
2
( CO1 r 2 E1O1 ) /( 2 * CO1 * r )
Calculating the angle ∠DCO1 ∠DCO1 is an interior angle of the triangle △DCO1, and its range is 0<∠DCO1<π. It can be obtained using the cosine theorem: cos∠DCO1=(|CO1|2+r2–R12)/(2*|CO1|*r)
(3)
Calculating the engagement angle ∠DCE1 Obviously, the engagement angle ∠DCE1 satisfies the following geometric formula: atro=∠DCE1=2π–∠E1CO1–∠DCO1
(4)
When SE1 increases in the variation range, the engagement angle with corresponding change may be calculated using this geometric calculation. (2) Geometric modelling of the second section EGHE Establishing the trajectory equation for tool centre The moving trajectory of the cutter centre C is the arc centred by O2 (xO2, yO2) with radius R2-r. Therefore, the following equations for the variation in C(xc, yc) may be established. xc=xO2 +(R2–r)*cosφ yc=yO2 + (R2–r)*sinφ
(5)
Calculating the angle ∠DCO1 Obviously, ∠DCO1 is an interior angle of the triangle △DCO1, and its range is 0< ∠DCO1<π. From the geometric relationship, |CD|=r，|O1D|=R1 are known, and the length of CO1 is h2=( xc–0)2+( yc–0)2,
10
According to the cosine theorem, cos∠DCO1=(h2+r2–R12)/(2*h*r)
(6)
The angle ∠DCO1 can be obtained by the inverse cosine. Calculating the angle ∠E1CO1 The angle ∠E1CO1 may be calculated as the vector dot product of O2 C and CO1 . However, because ∠E1CO1 is likely to be larger than π (see Fig. 3c), the relationship of the vector cross product is required to identify whether ∠E1CO1 is greater than or less than π. The vector quantity CO1 is set as a=[ax，ay，0], and ax = –xO2 – (R2–r)*cosφ ay = –yO2 – (R2–r)*sinφ The vector quantity
O2 C
is set as b=[bx，by，0], and
bx =(R2–r)*cosφ by =(R2–r)*sinφ According to the vector dot product, cos∠E1CO1=a·b/(|a|·|b|) =（ax*bx+ay*by）/[(ax2+bx2)+( ay2+by2)]
(7)
∠E1CO1 can be acquired using the inverse cosine in the range between 0 and π.
Calculating the engagement angle ∠DCE1 Obviously, the engagement angle satisfies the following geometric formula: atro=∠DCE1=2π–∠E1CO1–∠DCO1
(8)
11
When the angle φ increases in the variation range, the engagement angle with corresponding change may be calculated using this geometric calculation. (3) Determining processing limitations (Fig. 4) First, the current offset circle (centred by O2) should intersect the previous offset circle (centred by O1). The limitations thereof should be studied; that is, determine whether the current offset circle is internally tangent or externally tangent with the previous offset circle (see Fig. 4a). The distance of O1 and O2 should meet the following condition: R1–R2
(9)
Second, after the cutter travels through the previous offset circle and the current circle, no residual material should remain in their middle area. According to the geometrical relationship, the limitation should be as follows: O1, O2, and the cutter centre are in a straight line, and the tool circle is internally tangent with the current offset circle and externally tangent with the previous offset circle (see Fig. 4b). Therefore, S=(L –R1)+R2<2*r That is, the distance of O1 and O2 should satisfy L < R1–R2+2*r
(10)
In most cases, the cutting width S is less than r to avoid large cutting loads and tool fatigue; therefore, the following inequality can be obtained. L < R1–R2+r
(11)
(4) Calculating and drawing the variation diagram of the engagement angle A trochoidal path normally incorporates a few trip cycles. Each cycle is combined with one connected curve and one circle arc. By calculating the engagement angle for a
12
series of positions in a trip cycle, a variation curve diagram of the engagement angle can be drawn. Fig. 5 includes several typical examples. The horizontal coordinate axis is the rolling length, and the vertical axis is the engagement angle (in degrees). When the ratios of r, R1, and R2 vary, similar results can be obtained. Based on the curve diagram analysis, two conclusions can be drawn: (1) Increasing the centre distance of two neighbouring circles can increase the maximum value of the engagement angle of the trochoidal path. (2) Assuming that the cutter radius (r) and the trochoidal circle spacing (L) are certain, the maximum engagement angle will decrease with an increasing trochoidal circle size.
2.3 Modelling of milling force in trochoidal machining Altintas et al. (1991) presented an efficient simulation system for milling mechanics by using tool path generation algorithms based on a solid modeler; they also experimentally validated their models. According to the mechanistic force model, the differential forces are defined in the tangential, radial, and axial directions as in Eq. (12). Then, they are transformed into x, y, and z coordinates as in Eq. (13) dFt ( ) ( K tc h( ) K te )dz dFr ( ) ( K rc h( ) K re )dz dF ( ) ( K h( ) K )dz ac ae a
(12)
dFx ( ) dFt cos dFr sin dFy ( ) dFt sin dFr cos dFz ( ) dFa
(13)
The instantaneous forces can be expressed as follows: ex
ex
ex
st
st
st
Fx dFx , Fy dFy , Fz dFz
(14)
13
In addition, the resultant force is 2
2
F Fx Fy Fz
2
(15)
2.4 Maximum and mean milling force or engagement angle in trochoidal machining 2.4.1 Maximum milling force or engagement angle The maximum milling force and engagement angle are denoted as Ftro_max and atro_max. Because the engagement angle and milling force vary in a similar manner, it is only necessary to analyse the engagement angle. Fig. 6a shows that the line O1O2 intersects the circle O1 at point M. First, during the process of the cutter-circle rolling along the curve SE, the engagement angle will gradually increase and obtain a periodical extremum at point E. Then, the cutter-circle continues rolling along the arc EGH, and the engagement angle will gradually increase. When the cutter-circle passes through point M, the angle reaches its maximum (Fig. 6b); then, it gradually decreases. Another particular case is when the point M is located within the cutter circle when the cutter circle is tangent to the point E. In this case, a maximum engagement angle occurs. In the subsequent rolling path of EGH, the engagement angle will decrease step by step. 2.4.2 Computing the variation curve diagram of milling force For specific workpiece materials, if the milling parameters of the cutting tool are given, the method in Chapter 2.3 can be used to calculate the dynamic milling force for trochoidal machining. Corresponding to the EGH path in Fig. 6, Fig. 7 shows the dynamic milling force calculated under some parameters. The milling force first increases to the maximum gradually, and then decreases gradually. In Fig. 7, the two adjacent dots illustrate a rotation of the cutting tool. In this case, the cutting tool has
14
four edges, so there are four peak values between the two adjacent dots, respectively corresponding to the peak values obtained when the four edges cut in or out the workpiece. A series of peak points (F1, F2, … Fi, Fi+1 … Fn) for the first edge of the cutting tool during the machining can be connected to form a curve which is defined as the variation curve of milling force in this paper. The maximum of the series of peak points is defined as Ftro_max. In addition, the series of peak points can be averaged, thus to obtain an average value Ftro_ave, as below: Ftro _ ave ( F1dl F2 dl Fi dl Fn dl ) /(ndl ) ( F1 F2 Fi Fn ) /(n) (16)
Similarly, atro_max and atro_ave can be obtained from the variation curve of the engagement angle in Fig. 5. The four values, Ftro_max、Ftro_ave、atro_max and atro_ave, will be used in subsequent calculations and analysis.
2.5 Variation curve diagram of milling force and engagement angle The large axial and small radial depths of cut are often adopted in cavity machining, which can provide improved use the cutting capacity of the cutting edges of the tool and allow more materials to be machined before the cutting tool becomes blunt. Usually, the radial depth of cut is smaller than 1/3 of the tool diameter, or even much smaller; while the axial depth of a cut may be larger than 1/3 of the tool diameter, or even much larger. In this context, carry out a calculation and comparative analysis for the variation curve of the trochoidal milling force and the variation curve of the engagement angle. Fig. 8 shows that for a cycle in trochoidal machining, we can first calculate each peak value of the milling force and then arrange them into a milling force curve. Second, we can calculate each changing engagement angle for trochoidal machining and then arrange
15
them into another curve. Finally, we extend the engagement angle curve to make the peaks of the two curves meet. After comparison of the two curves, we find that the changing trend of the engagement angle is very similar to the changing trend of the milling force. Therefore, we can use the engagement angle as a substitute for the milling force for analysis and calculation such as for large axial depths of cut and small radial depths of cut in cavity machining.
3. Control strategy for trochoidal milling
In high-speed cavity milling, a cutting mode with a large axial depth of cut and small radial depth of cut is often used. This is taken as the background; then, we will implement a serial of fundamental experiments (Fig. 9) and subsequently propose the control strategy for trochoidal machining. (1) The milling force and tool wear increase with an increasing centre distance between neighbouring circles. An excessively large centre distance will increase the possibility of wear and damage to the cutting tool. The characteristics of the force change and tool wear are clearly indicated in Fig. 5 and Fig. 8.
(2) If the centre distance is set to equal the contour spacing of contour-parallel cutting, the milling force of trochoidal machining will be particularly high compared with contour-parallel cutting, which is more likely to make the tool fatigue and even become damaged.
16
However, the tool exhibits good heat dissipation in the trochoidal machining process; therefore, the ultimate tool wear tends to be low.
According to the above experiments on trochoidal machining, the main problem in trochoidal machining is the huge milling force, which tends to cause fatigue and breakage of the tool. Hence, the primary task of trochoidal machining control is to control the milling force. The following control strategy for trochoidal milling will focus on the milling force. Moreover, the cavity geometry, tool wear, and milling efficiency will also be considered. The following will select the engagement angle as the auxiliary control parameter, not the spacing of the contour-parallel path. This is mainly because the spacing of the contour-parallel path is not an exact expression of milling force changes, but the engagement angle can provide for a very good expression (see 4.2).
3.2 Control strategy for trochoidal machining in cavity milling If milling cavity is performed using the contour-parallel cutting method, the processing parameters, such as path spacing, cutting depth, and feedrate, are generally given by consulting a manual or by experience, which usually consider the bearing capacity of the cutting tool. Taking these basic parameters of contour-parallel cutting as a reference for trochoidal milling, and considering the above trochoidal machining features, a trochoidal machining control strategy is proposed as follows: (1) According to the basic parameters of the contour-parallel cutting, such as the cutting tool radius (r) and path spacing (d), calculate the engagement angle (astr) or the milling force (Fstr) at the straight trajectory area of the contour-parallel cutting.
17
(2) According to the cavity geometry features, select an appropriate small circle, determine the area to insert the trochoid, and compute the location of the initial circle of the trochoidal path. (3) Calculate a reasonable centre distance of neighbouring circles to meet the control requirements of the engagement angle or milling force. Assuming that the mean engagement angle in trochoidal machining is atro ave and that the maximum engagement angle is atro_max, calculate the following three parameters.
Llow: Ftro max=Fstr; Calculate a proper centre spacing of trochoidal circles to make
the maximum milling force equal to the milling force Fstr in the contour-parallel cutting; this spacing is defined as Llow.
Lmid: Ftro ave =Fstr; Calculate a proper centre spacing of trochoidal circles to
make the average milling force equal to the milling force astr in the contour-parallel cutting; this spacing is defined as Lmid.
Lhigh: Ftro ave =1.5Fstr; Four areas can be delineated according to the three values above (as shown in Fig.
10): Area 1: The machining is safe, and the cutting tool is subject to relatively small forces and obtains a longer service life, but the milling efficiency is reduced. Area 2: The machining is relatively safe, the tool wear is reduced, and the tool load is greater than that of the machining mode in contour-parallel cutting. Area 3: The cutting tool load is larger, and the possibility of fatigue for cutting tool increases. However, the machining efficiency will also be improved. Area 4: The machining efficiency is higher, but the cutting tool load is larger; moreover, the tool can be easily damaged, and tool wear is more serious.
18
Obviously, in general, Area 2 or Area 3 should be chosen in high-speed cavity milling. Area 3 can be chosen to improve the milling efficiency if the strength of the cutting tool permits. To obtain better control, we can set the values of t1 and t2 (for example t1=10% and t2= 50%) and calculate a proper centre spacing of neighbouring trochoidal-circles to make the average milling force in trochoidal milling satisfy Ftro_ave≤Fstr * (1+t1)
(17)
or the maximum engagement angle satisfy Ftro_max≤Fstr* (1+t2)
(18)
Sometimes the engagement angle a (astr、atro、atro_max) can be used as a substitute for the milling force (Fstr、Ftro、Ftro_max) to simplify the calculation. (4) According to the geometry features of the cavity, corner, and narrow slot, compute the trochoidal path in the material gathering area. The corresponding algorithm will be shown in Part 4.
4. Trochoidal machining for cavity
4.1 Geometry description Contour-parallel tool paths represent a common high-speed milling method for cavity moulds. In this paper, the contour-parallel cutting approach adopts the milling mode from inside to outside. The path elements involved are mainly arcs and linear segments. The following analysis will focus on the line-line corner type, and other corner types composed of complex curves can be deduced accordingly.
19
Fig. 11 shows the tool path of contour-parallel cutting. Assuming that the current trajectory is U and V, the trajectory formula can be expressed as U: A1 x B1 y C1 F1t1 0 V: A2 x B2 y C 2 F2 t1 0
(19)
where F1 and F2 refer to the offset director and take on the value of 1 or -1; the coefficients of x and y are normalized. The previous trajectory with the offset distance d is PU: A1 x B1 y C1 F1 (t1 d ) 0 PV: A2 x B2 y C 2 F2 (t1 d ) 0
(20)
The bisector of two geometric curves is the trajectory in which each point is equidistant to both curves. As mentioned by the paper of Persson (1978), the bisector of the corner may be a straight line, parabola, or hyperbola. Calculating the bisector is an important step in establishing the trochoidal path. Let the curves U and V be offset by an amount equal to h. The offset curves intersect at a point M, which is a point on the bisector. The intersection point can be calculated by the following formula: A1 x B1 y C1 F1 (t1 h) 0 A2 x B2 y C 2 F2 (t1 h) 0
(21)
Thus, the coordinate of the intersection point M is x m ( B1 C 2 B 2 C1 ) / T (t1 h)( B1 F2 B 2 F1 ) / T y m ( A2 C1 A1C 2 ) / T (t1 h)( A2 F1 A1 F2 ) / T
(22)
By calculating a series of intersection points, the bisector of U and V can be acquired.
4.2 Trochoidal machining for corners of a cavity
20
Material is more easily gathered at a cavity corner, especially corners with smaller angles, while milling corners with trochoid realize smaller tool loads. The path elements involved are mainly clockwise and anticlockwise arcs and linear segments, which may be combined to produce a total of nine types of corners as mentioned by the paper of Choy and Chan (2003). Trochoidal machining of cavity corners can be realized in three ways, as shown in Fig. 12, Corner trochoidal machining with variable circles. Material at the formed corner of
each contour ring can be gradually removed by the method of variable trochoidal circles. Corner trochoidal machining with isometric circles. Material at the formed corner of
each contour ring can be gradually removed by the method of isometric circles. The first and second method generate a trochoidal path at the corner of each ring in the contour-parallel path. Perforative corner trochoidal machining along the medial axis. This uses the method
of Voronoi diagrams to calculate the medial axis of the cavity and arrange the perforative trochoid according to the medial axis.
The computation algorithm for trochoidal machining includes the following steps: (1) Calculate the engagement angle or the milling force at the straight line area of the contour-parallel cutting path, which is regarded as a basic reference for the trochoidal calculation. (2) Compute the guiding curve. If the first or second method of trochoidal machining is adopted, analyse the corner type and calculate the corner bisector or the offset curve
21
of the corner edge. If the third method of trochoidal machining is adopted, calculate the medial axis of the cavity. (3) Determine the initial circle and calculate the size and location of the trochoidal circle step by step, guided by the curves in Step 2 so that the engagement angle or milling force can satisfy the control requirements presented in Section 3. (4) Add the new trochoidal circle and the middle curve between two adjacent circles into the stored linked list. 4.2.1 Corner trochoidal machining with variable circles As shown in Fig. 12a, a few variable circles may be applied to mill the material of the remaining corner. The circle (c1) on the end position of the corner is very specific and is selected as the initial trochoid circle. Other trochoid circles will be computed one by one until the corner materials are cut out. Finally, all of the trochoid circles are stored in the trochoid list in reverse order. The calculation of the next trochoid circle constitutes the key sub-algorithms. Fig. 13 shows that the current trajectory is U and V, and the trajectory with the offset distance r (tool radius) is UW and VW. The circle centred by N and tangential to UW and VW is the offset circle of the current trochoid circle. The computation on the next trochoid circle involves two steps: (1)Analyse the situation when the cutter circle is internally tangential to the current offset circle at point G on the corner bisector and calculate the other intersection point S between the cutter circle and the bisector curve (see Fig. 13a). Then, calculate the other offset circle externally tangential to the cutter circle at point S and set its centre as the point M. The in-centre point M will be taken as a limit point, and the next trochoid circle will be obtained between M and N.
22
(2) Establish a function Ftro_ave=Fstr * (1+t1); Z(P)= Ftro(P)–Ftro_ave≤ε
(23)
where t1 is a specified value (for example t1=15%), Ftro (P) is the average milling force when the cutter rolls within the current offset circle, and ε is a relatively small control value (e.g., ε=1). First, for M and N, find the midpoint P((MX+NX)/2, (MY+NY)/2) on the corner
bisector. Take it as a centre and create a new circle that is tangential to UW and VW (see Fig. 13b). Take the circle centred by P and the circle centred by N as the present analysis
objects of the trochoidal trajectory. Calculate the average milling force when the cutter rolls along the current arc EGH, and set the angle as atro(P). Calculate Z (P) according to formula (23). If Z (P) >0 and Z (P) ≤ε, the point P is
the required trochoidal-circle centre; exit the calculation. Otherwise, if Z (P) <0, then set M=P, or if Z (P) >0, set N=P. Repeat the above three steps until an appropriate P is obtained. Make the circle
centred by P offset inward a distance of r; then, the next required trochoid circle can be obtained. Sometimes the maximum milling force Ftro max is adopted as the analysed object. Therefore, the average engagement angle in formula (23) will be replaced by the maximum milling force Ftro max. If the engagement angle is used as a substitute for the milling force, formula (23) is converted to
23
Z(P)= atro(P)–atro_ave≤ε
(24)
The algorithm is similar to that above. 4.2.2 Corner trochoidal machining with isometric circles The method of trochoidal milling corners with isometric circles can facilitate calculation and improve the continuity of the trajectory. It can also effectively reduce the cutting tool load. According to whether the trochoid circle can coat the corner arc formed in previous machining, two implementation methods can be obtained, as shown in Fig. 14a and Fig. 14b. U-V is the current trajectory in Fig. 14, PU-PV is the previous trajectory, and PSQ is the corner arc formed after the previous machining. The distance from P to UW is |PH1|. Make a circle with |PH1| as diameter; then, internally offset by a distance of r (tool radius). Then,
rh1 PH 1 / 2 r 2
2
| A1 x P B1 y P C1 F1 (t1 r ) | / 2 A1 B1 r
(25)
If the radius of the selected trochoid circle rh satisfies rh>rh1, this indicates that the trochoid circle can coat the former corner arc formed by previous milling. Thus, the method in Fig. 14a can be used. To calculate its machining trochoid, a suitable centre distance (L) of neighbouring circles should be first calculated so that the trochoidal machining can satisfy the limits of the engagement angle and the milling force. Then, the curve U is offset by rh to obtain the curve Up. According to the geometric characteristics of the corner curve PU-PV, we determine the initial circle (O1) on the curve Up and then calculate the trochoid circles guided by the curve Up one by one and store the results in the path list. Assuming that the coordinate of the current trochoid
24
circle centre is (xoi,yoi), the next trochoid circle centre (xoj,yoj) on the curve Up should satisfy the following relation: (26)
( xOi xOj ) 2 ( yOi yOj ) 2 L2
In certain cases, the selected trochoid circle is small (rh
4.3 Trochoidal machining for narrow areas Narrow areas, such as slots, often occur in cavity machining. The width of the narrowest area can be defined as h=k*d (such as k=3~5, where d is the cutter diameter). Trochoidal machining remains a very good method for the machining of narrow areas because it helps reduce the cutting load. However, if the actual width exceeds this value, the contour-parallel cutting approach is preferred for removing the material from the narrow area.
25
There are mainly two types of treatment for the narrow area: (1) calculating the bisector or medial axis of the narrow area and using the trochoidal method with variable circles to mill materials (Fig. 15a) and (2) calculating the bisector or medial axis of the narrow area and using the trochoidal method with isometric circles to first mill the materials; then, the trajectory of contour-parallel cutting is used to remove the residual material (Fig. 15b). The corresponding calculation method is similar to that in 4.2.
5. Experiment
5.1 Test platform The test platform consists of a high-speed machining centre (DMC-60T), a tool workpiece system, and a cutting force measuring system. The force measuring system includes a force platform, a charge amplifier, and data acquisition software. The force measuring device is a Kistler9265B dynamic piezoelectric dynamometer with the sensitivity of 0.05 N. The charge amplifier is a Kistler5019. The test results for the milling force include three component forces: Fx, Fy, and Fz. Their synthesized force can be calculated using the formula
F FX 2 FY 2 FZ 2
.
5.2 Comparative experiment Fig. 16 shows a cavity model by the factory. The sizes of the cavity are 100*70mm. There is a 40 degree corner in B area of the model, and a slot in C area. The material of workpiece is P20, and its hardness is HRC36. The tool path generated from CAM software. Use this tool path for direct machining of workpieces without any auxiliary processing. See Fig. 16b for relevant process parameters and machined cavities. After the machining of 14 workpieces, the tool wear reaches the wear-out failure criterion
26
(VB > 0.3 mm). Test the milling force of a tool path loop and obtain the result (Fig. 16c). Large loads often change abruptly at corners. Check the tool edge, and tipping (Fig. 16d) may be found, which indicates the loads on the tool are very large. In order to improve the durability of the cutting tool, feedrate adjusting machining at the corner is considered firstly. Because the commercial CNC (Computerized Numerical Control) system provides limited user settings, changing the feedrate is more suitable. The following measures can be adopted: (1) Change the feed rate at a point 6 mm before each corner, and then the commercial CNC system will complete the deceleration; after the tool passes through the corner, change the feed rate to the normal speed. (2) at the corner of B area, the feed-rate is slowed down to 1/3 of normal speed. At the corner of D area, the feed-rate is slowed down to 1/2 of normal speed. (3) Area C is a slot. Take 2 times for cutting and the axial depth is 2 mm each time. The feed-rate is slowed down to 1/2 of normal speed. Fig. 17b shows a single cavity that has been machined. The total machining time of a single cavity is 88 seconds. Upon checking the flank face of the cutting tool after machining the 18th cavity (Fig. 17c), the cutting lip has been found to be severely worn. Fig. 17d shows the milling force of one of the tool path loops. Compared to the original method (Fig. 16), the abruptly-changing load at the corner decreases, but the maximum load remains large. After the cutting tool exceeds the wear-out failure criterion (VB > 0.3 mm), 22 cavities can ultimately be machined. The number is higher when compared to the original method.
To obtain a better machining, trochoidal-aided machining could be considered. With the straight line area A as a reference and by controlling the average and maximum
27
engagement angles in trochoidal machining, the calculated results shown in Table 1 are obtained.
A machining method adopting perforative corner trochoid along the medial axis is shown in Fig. 18. The calculation of the parameters of the trochoid is shown in Table 1B. In the machining, the trochoid is first run in the middle of the cavity; then, the contour-parallel cutting path is run. Finally, the narrow slot in area C is processed. The total machining time of a single cavity is 113 seconds. The milling force during the process is shown in Fig. 18d. After the cutting tool exceeds the wear-out failure criterion (VB > 0.3 mm), 30 cavities can ultimately be machined. The number is much higher when compared to the original method. In Fig. 19, a trochoidal path is added at the corner of each ring in the contour-parallel path. The calculation of the parameters of trochoid is shown in Tables 1-B and 1-D. The total machining time of a single cavity is 101 seconds. The milling force during the process is shown in Fig. 19d. The milling force during the process is effectively controlled. After the cutting tool exceeds the wear-out failure criterion (VB > 0.3 mm), 33 cavities can ultimately be machined. The above four machining methods can be defined as M1, M2, M3 and M4, respectively. The number of cavities machined was 14, 22, 30 and 33, respectively. Obviously, by using the deceleration method (M2), tool life can be prolonged when compared with the original path method (M1), which, however, will be much longer by using methods M3 and M4. Considering that the change of loads at the corner is an important factor influencing tool life, a comparative analysis should be made for these methods. The maximum loads and average loads were observed during the machining
28
of Corner B when using each of these methods; Fig. 20a shows the results obtained. With M2, both the maximum and average loads decrease to a certain extent when compared to M1; however, both types of load still remain large. This indicates that because the cutting tool has a large amount of contact with the workpiece material in a unit of time, the loads on the tool are still heavy even if the tool has decelerated. However, with M3 and M4 where load optimization has been conducted, both the maximum and average load decrease significantly when compared to M1 and M2; therefore, there is less tool wear and fatigue and the tool life is extended greatly. Further compare M3 and M4; the result indicates that the tool wear of the method M3 is greater and the machining efficiency of the method M3 is much lower. The main reason for this is that the trochoidal track is longer in larger corner processing (the corners of D1 and D2 are close to 80 degrees). Therefore, by optimizing the trochoidal geometry and reducing the length of the trochoidal path, greater process efficiency can be obtained, and later research will be conducted in this aspect. Additionally, compared to the conventional method, M1, the number of cavities machined has a great increase with the two trochoidal methods (M3, M4). However, there currently is also a limitation for trochoidal methods now - their machining efficiency is lower. Fig. 20b shows the comparison of machining time with all four methods. M3 and M4 require more time to machine the product mainly because of the need for having additional auxiliary trochoidal paths in the tool path. The machining time when using these methods is closely related to the size of cavity. The larger the cavity is, the less the proportion of the corner path in the overall tool path will be, and the narrower the time difference between the former and latter two paths will become.
29
In short, the trochoidal path methods present excellent load control capacity and greatly extend tool life. Further research studies on the optimization of trochoidal tool paths will be carried out in the future. The path geometry will be optimized to reduce the length of the trochoidal path, and the depth of cut, feed rate and other parameters will be increased within a controllable range of tool loads, thus to enhance machining efficiency.
6. Conclusion
(1) A method for the geometric modelling of the engagement angle in trochoidal machining is proposed. The corresponding relationship between the milling force curve and the engagement angle curve is analyzed through simulation. The changing trend of the engagement angle is very similar to the changing trend of the milling force. Therefore, in certain circumstances, the engagement angle can be used as a substitute for the milling force for analysis and calculation such as for large axial depths of cut and small radial depths of cut in cavity machining. (2) In high-speed cavity milling, a cutting mode with a large axial depth of cut and small radial depth of cut are often used. This is taken as the background; then, we have implemented a serial of fundamental experiments of trochoidal machining. The result shows that milling force is the key issue in trochoidal machining, and it should be controlled. Based on the obtained effective conclusions, a proper control strategy for cavity trochoidal machining has been proposed. (3) Based on the trochoidal control strategy, two realizations of cavity trochoidal machining have been proposed, and practical cavity machining experiments are compared. Experiments show that the feedrate adjustment method has weak control on
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the milling force and tool vibrations, therefore, the tool wear is relatively big, and it is easy for the tool to be fatigued and damaged. However, trochoidal milling method is more effective at milling force control, reduction of tool vibrations, and tool wear. By increasing the axial depth of cut, the milling efficiency and tool wear of trochoidal milling method is better than the feedrate adjustment method. In the future, the research of trochoidal machining should be focused on geometry optimization, milling with high axial depth of cut and high speed-rate, in order to improve the efficiency of trochoidal milling. Acknowledgements
This work reported in this paper is conducted in conjunction with ‘NSFC-Guangdong Collaborative Fund Key Program U12012081’.
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References
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Figure Captions Fig. 1. Trochoidal machining for a pocket: (a) narrow areas or sharp corners in a cavity;
(b) trochoid with isometric circles; and (c) trochoid with variable circles.
Fig. 2. Engagement angle.
Fig. 3. Engagement angle model of trochoidal trajectory: (a) variable circle mode; (b)
the first section SE of one trochoidal cycle; (c) the second section E-G-H-E of one trochoidal cycle; and (d) isometric circle mode.
Fig. 4. Processing limitations.
Fig. 5. Curves of the engagement angle: (a) variable circle mode (R1>R2 ) and (b)
isometric circle mode (R1=R2).
Fig. 6. Maximum and mean milling force: (a) rolling along SE and (b) rolling along
EGH.
Fig. 7. Simulating the calculation of the maximum force and mean force.
Fig. 8. Comparison of milling force and engagement angle: (a) spacing L=0.5 mm; (b)
spacing L=1.0mm; and (c) spacing L=1.5 mm.
Fig. 9. Comparison of two milling modes: (a) set L=d in pocket machining; (b) flank
wear; and (c) Milling force.
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Fig. 10. Control strategy for trochoidal machining.
Fig. 11. Geometry description.
Fig. 12. Three techniques of trochoidal machining: (a) generate a trochoidal path at the
corner of each contour ring and (b) generate a trochoidal path at the medial axial of cavity.
Fig. 13. Calculate the next cyclic circle.
Fig. 14. Calculation schematic for the trochoidal path: (a) the first method and (b) the
second method.
Fig. 15. Trochoidal machining for narrow areas: (a) method with variable circles and (b)
method with isometric circles.
Fig. 16. Original machining method: (a) machining path generated from a CAM
software; (b) machined pocket; (c) Flank wear (VB=0.32mm) after machining the 15th cavity; and (d) milling force.
Fig. 17. Feedrate adjusting machining: (a) machining path; (b) machined pocket; (c)
Flank wear (VB=0.256mm) after machining the 18th cavity; and (d) milling force.
Fig. 18. Trochoidal machining 1: (a) machining path; (b) machined pocket; (c) flank
wear (VB=0.215mm) after machining the 18th cavity; and (d) milling force.
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Fig. 19. Trochoidal machining 2: (a)machining path; (b) machined pocket; (c) flank
wear (VB=0.2mm) after machining the 18th cavity; (d) milling force.
Fig. 20. Comparison of four methods: (a) comparison of milling force (in B area of
the model); and (b) comparison of the machining time of a single cavity.
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Fig. 1.
Fig. 2.
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Fig. 3.
Fig. 4.
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Fig. 5.
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Fig. 6.
Fig. 7.
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Fig. 8.
42
Fig. 9.
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Fig. 10.
44
Fig. 11.
45
Fig. 12.
Fig. 13.
46
Fig. 14.
47
Fig. 15.
Fig. 16.
48
Fig. 17.
Fig. 18.
49
Fig. 19.
50
(a)
(b)
Fig. 20.
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Table 1. Computed trochoidal parameters.
Area A
r=3.0 mm d=1.0 mm astr= 480 (See chapter 2.1)
Area B
Area D
rtro=1.8 mm L=0.72 mm
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r1=1.0 mm rtro =2.4 mm L=2.1 mm atro max =690 atro ave =530 t1 =10%, t2 =45% (See chapter 3.2)