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Mechanics of boring processes—Part I
International Journal of Machine Tools & Manufacture 43 (2003) 463–476
Mechanics of boring processes—Part I F. Atabey a, I. Lazoglu b, Y. Altintas a,∗ a
University of British Columbia, Department of Mechanical Engineering, Manufacturing Automation Laboratory, V6T 1Z4, Vancouver, Canada b Koc University, Department of Mechanical Engineering, 80910 Sariyer, Istanbul, Turkey Received 1 February 2002; received in revised form 28 October 2002; accepted 13 November 2002
Abstract Mechanics of boring operations are presented in the paper. The distribution of chip thickness along the cutting edge is modeled as a function of tool inclination angle, nose radius, depth of cut and feed rate. The cutting mechanics of the process is modeled using both mechanistic and orthogonal to oblique cutting transformation approaches. The forces are separated into tangential and friction directions. The friction force is further projected into the radial and feed directions. The cutting forces are correlated to chip area using mechanistic cutting force coefficients which are expressed as a function of chip-tool edge contact length, chip area and cutting speed. For tools which have uniform rake face, the cutting coefficients are predicted using shear stress, shear angle and friction coefficient of the material. Both approaches are experimentally verified and the cutting forces in three Cartesian directions are predicted satisfactorily. The mechanics model presented in this paper is used in predicting the cutting forces generated by inserted boring heads with runouts and presented in Part II of the article [1]. 2003 Elsevier Science Ltd. All rights reserved. Keywords: Force modeling; Chip load; Single point boring; Orthogonal to oblique transformation
1. Introduction The enlargement of holes is achieved via boring operations. The hole diameter is either enlarged with a single insert attached to a long boring bar, or with a boring head which has a diameter equal to the diameter of the hole to be enlarged. Long boring bars statically and dynamically deform under the cutting forces during boring operations. Excessive static deflections may violate the dimensional tolerance of the hole, and vibrations may lead to poor surface, short tool life and chipping of the tool. Predictions of the force, torque and power are required in order to identify suitable machine tool and fixture set up for a boring operation. A comprehensive engineering model, which allows prediction of cutting forces, torque, power, dimensional surface finish and vibration free cutting conditions, is required in order to plan boring operations in the production floor. Although the other machining processes such as mill-
Corresponding author. Tel.: +1-604-822-5622; fax. 1-604-8222403. E-mail address: [email protected] (Y. Altintas). ∗
ing, turning and drilling have been studied relatively broader and deeper ([2], [3]), there were only a few attempts to model the cutting forces and stability in boring ([4–10]). However, the mechanics and dynamics of the boring process have not been sufficiently modeled for an effective prediction of boring process performance. There are fundamental issues which make the boring process somewhat difficult to model for a reliable prediction of process performance. The boring inserts have nose radius, and may have either uniform or irregular rake face. The distributions of chip thickness, therefore the cutting pressure amplitude and direction, vary as a function of tool nose radius, radial depth of cut and inclination angle. Since the boring bar is long and flexible, it is not possible to remove much larger depths of cuts than the nose radius of the tool, unlike the case of turning and face milling operations. This leads to a non-linear, complex relationship between the cutting force distribution, tool geometry, feedrate and depth of cut. Furthermore, the presence of static and dynamic deflections may influence the engagement conditions, leading to variations in chip load distribution and the cutting pressure. The cutting forces are usually predicted as a function
0890-6955/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0890-6955(02)00276-6
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of uncut chip area that changes in a complex manner in boring due to nose radius and geometry of the tool, and cutting conditions. If the tool rake face has an irregular geometry due to chip breaking grooves and chip-tool contact restriction features, the cutting coefficients are identified using mechanistic models. A series of cutting tests are conducted with the specific tool at different speeds, radial depth of cuts and feedrates. The coefficients are evaluated by curve fitting the force expressions to the measured cutting forces and chip geometry. If the rake face of the tool is smooth and uniform, it is possible to model the cutting edge as an assembly of oblique cutting edges [11]. The cutting pressure at each discrete oblique cutting edge element is modeled by applying the orthogonal to oblique transformation method proposed by Armarego [12,13]. Both approaches have been considered in this article. The paper is organized as follows; first the complex geometry of the chip is modeled analytically for various cutting conditions and tool geometry. The cutting forces are modeled as friction and tangential cutting forces. The friction force is resolved in the radial and feed directions, and the direction of the friction force, i.e. the effective lead angle, is experimentally evaluated from the ratio of the two. The effective lead angle is predicted from the geometry of the chip and tool, and the prediction is improved by a mechanistic model based on regression analysis applied to the model and measurements. The cutting coefficients are modeled as empirical functions of cutting speed, cutting edge contact length and uncut chip area. The forces are modeled using orthogonal to oblique cutting transformation when the rake face of the tool is smooth. This method requires only the tool geometry, shear stress, shear angle and average friction coefficient of the orthogonal cutting process for a specific work material. The paper contains experimental verification of proposed models which are used to predict the cutting forces in all three directions.
2. Chip geometry cut by single insert The fundamental geometry of a boring insert is characterized by a corner radius (R), side cutting edge angle (g) and end cutting edge angle (gc) (Fig. 1). The rake face of the tool may have either flat face or irregular chip breaking and chip contact reduction grooves which affect the cutting mechanics. The cutting edge does not usually have chamfer or curvature unless the workpiece material is not hardened steel. The chip area, and therefore the cutting force distribution, vary as a function of the depth of cut (a), feedrate (c), side cutting edge angle (g) and end cutting edge angle (gc). Fig. 2 also illustrates the relative positions of the insert at successive revolutions of the workpiece in four different configurations. Nine various tool-workpiece inter-
Fig. 1. Friction force distribution along the cutting edge contact length Lc.
ferences can be also defined with respect to the corner radius (R), depth of cut (a), feedrate (c), side cutting edge angle (g) and end cutting edge angle (gc) Notice that the material left behind (uncut material) depends on the feedrate (c) and corner radius (R), and is expected to be large when the feedrate is considerably greater. The uncut material also determines the surface finish quality. While large corner radius and small feedrate create good surface, small corner radius and large feedrate leaves more material (i.e., feed marks) behind causing a rougher surface finish. The most common case encountered in boring applications is when the feedrate is less than the nose radius of the insert. Therefore, only five common interferences have been considered (Figs. 3 and 4). The uncut chip area is evaluated by discretizing the chip into small differential elements in three separate regions as shown in Fig. 3. In the following, the calculation of the uncut chip area (A) and cutting edge contact length (Lc) are explained for only the first configuration. (The uncut chip areas and cutting edge contact lengths are calculated with the same manner for the other configurations shown in Fig. 4). For Region 1 in Fig. 3, the uncut chip area of each differential element is approximated by subtracting the area of the triangle AoBB⬘ from the area of the circular ring sector AODD⬘; 1 AOBB⬘,i ⫽ |OB|i|OB⬘|isin(qi), 2 1 AODD⬘,i ⫽ qiR2, A1,i⬵AODD⬘,i⫺AOBB⬘,i 2
(1)
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Fig. 2.
465
Illustration of four different uncut chip area configurations defined with depth of cut (a), feedrate (c), and corner radius of the tool (R).
Region 2 is considered to be a rectangle, although one side of it (i.e., side KE) has a slight curvature caused by the corner radius of the previous tool position. Area of Region 2 can be approximated as, A2⬵|MG||KM|
(3)
Region 3 is a simple triangle and its area is calculated as, 1 A3 ⫽ |KM||LM|sin(gL) 2
(4)
Finally, total uncut chip area is found by adding the areas for each region, A ⫽ A1 ⫹ A2 ⫹ A3 Fig. 3. Uncut chip area calculations for Configuration #1 (gL ⬎ 0, a ⬎ R(1 + singL)) and definitions of the regions.
The total chip area in Region 1 is evaluated by discrete summation of all differential elements in the curved region bounded by JEG.
冘
The total cutting edge contact with the work material is, Lc ⫽ Lc1 ⫹ Lc2
i⫽1
(6)
where Lc1, Lc2 are the contact lengths in Regions 1 and 2, respectively. If the number of discrete elements in Region 1 is given as n, the contact length in Region 1 will be equal to summation of the discrete contact
n
A1⫺
(5)
冘 n
A1,i
(2)
lengths (i.e., Lc1 =
i=1
冘 n
|DD⬘| = R
θi). Similarly, Lc2
i=1
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Fig. 4.
Uncut chip area configurations (#2…#5) considered in the model process.
is the contact length of Region 2 and equal to the length of MG. Hence the chip area can be identified geometrically as a function of radial depth of cut (a), feedrate (c) and insert geometry (R, g, gc). 3. Modeling of cutting forces in boring The cutting forces are represented by the tangential force (Ft) and friction force (Ffr). Later, friction force is resolved into the feed (Ff) and radial directions (Fr) (Figs. 1, and 5). Since the chip thickness distribution at each point along the cutting edge contact point is different and dependent on the insert geometry (R, g, gc) feedrate (c) and radial depth of cut (a), the distribution of the force along the cutting edge-chip contact zone also varies. At any contact point, the differential cutting forces are modeled as a function of local chip area (dA) and chip-cutting edge contact length (dLc), dFt ⫽ dFtc ⫹ dFte
⫽ Ktc.dA ⫹ Kte.dLc
dFfr ⫽ dFfrc ⫹ dFfre ⫽ Kfrc.dA ⫹ Kfre.dLc
(7)
where dFtc, dFfrc, are contributed by the removal of the chip and dFte, dFfre are due to cutting edge-finish surface rubbing. The edge contact constants (Kte, Kfre) are dependent on the cutting edge condition and preparation. The cutting coefficients (Ktc, Kfrc) are dependent on the local rake, inclination, chip flow angles, cutting conditions and work-tool material properties. They can be determined
Fig. 5.
Illustration of force directions in boring process.
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either mechanistically by conducting cutting tests with each insert, and curve fitting the force measurements against the chip geometry, or using classical laws of cutting mechanics such as orthogonal to oblique cutting transformation proposed by Armarego [13]. Both approaches are presented here for boring operations. 3.1. Mechanistic modeling of cutting forces 3.1.1. Identifications of the cutting force coefficients The cutting force coefficients are determined for two sets of selected inserts (i.e., Kennametal CPMT-32.52 K720 and Valenite CCGT432-FH) used in boring A1 6061-T6 workpiece material. The procedure is general and applicable to other insert geometries as well. The tangential (Ft) and friction (Ffr) forces are modeled as follows, Ft ⫽ Ftc ⫹ Fte
⫽ Ktc.A ⫹ Kte.Lc
Ffr ⫽ Ffrc1 ⫹ Ffrc2 ⫹ Ffre ⫽ Kfrc1.A1 ⫹ Kfrc2.A2 ⫹ Kfre.Lc
(8)
Unlike Ft, Ffr is considered as having two components. The uncut chip area, A1 covers Region 1 and A2 covers Regions 2 and 3 as illustrated in Fig. 3. The direction of total friction force is defined by effective lead angle (fL), which is the angle between the friction force and feed directions (Fig. 1). For relatively large radial depth of cuts, the effective lead angle (fL) tends to approach the side cutting edge angle (gL) of the insert. In such a case, the magnitude ratio of Fr/Ff decreases. In order to determine the edge cutting force coefficients (Kte, Kfre, Kre and Kfe), twenty four experiments have been conducted with Valenite CCGT 432-FH insert at 0.8 mm corner radius, constant 1.5 mm depth of cut and 150 m/min cutting speed but varying feedrate (c) from 0.025–0.19 mm/rev. The cutting forces are measured in three orthogonal directions, (Ft, Fr, Ff). Performing linear regression leads to the following tangential force equation, Ft ⫽ ⫺4721.4 ⫹ 2438.5Lc[N] Lcⱖ1.9556 [mm]
(9)
where Lc is the cutting edge contact length that is dependent on depth of cut and feedrate. The above equation is valid only if Lc greater than 1.9556 mm which corresponds to zero feedrate for 1.5 mm depth of cut. Zero feedrate refers to the rubbing process that is the source of the edge cutting forces. Therefore, substitution of value of 1.9556 mm in the equation above gives tangential edge cutting force (Fte) for given depth of cut (a = 1.5 mm) and zero feedrate (i.e., Fte = Ft|Lc = 1.9556 = 47.33[N]). The tangential edge cutting force coefficient, Kte, is determined as Kte = Fte / Lc = 24.24 N/mm. Kte, represents the tangential rubbing force per unit cutting edge contact length. Following the same procedure, the radial, feed and friction edge cutting forces (Fre, Ffe and Ffre, where Ffre = √F2re + F2fe) and corresponding edge cut-
467
Table 1 Edge cutting force coefficients for the Valenite CCGT432-FH insert Force model
Ft = 2438.5Lc ⫺ 4721.4 Fr = 417.74Lc ⫺ 800.40 Ff = 933.47Lc ⫺ 1779.9 Ffr = √F2r + F2f
Measured edge Edge cutting force cutting forces Fte, coefficients Kte, Kre, Kfe, Kfre N/mm (Lc ⱖ 1.9556 Fre, Ffe, Ffre N mm) 47.41
24.24
16.53
8.45
45.59
23.31
48.49
24.79
ting force coefficients (Kre, Kfe, and Kfre) are found as seen in Table 1. Another set of tests was conducted with Valenite CCGT432-FH insert at different combinations of the cutting parameters within the ranges of 0.025–0.19 mm/rev feed rate, 75–275 m/min cutting speed and 0.25–3.25 mm depth of cut. After removing edge forces (Fe Ffre) from the measured cutting forces, regression analysis is applied to the cutting forces (Fc, Ffc) which led to the following relationships for the cutting force coefficients, Ktc ⫽ eb0Ab1Vb2, Kfrc1 ⫽ em0Lcm11Vm2, Kfrc2
(10)
⫽ en0Lnc21Vn2 where Lc1, Lc2, V are cutting edge contact lengths (in mm) in Region 1 and 2, and cutting speed (in m/min), respectively. bo, b1, b2, m0, m1, m2, no, n1, and n2 are empirical constants obtained using the least-squares method on the experimental data. Their values are; bο = 7.9477, b1 = ⫺0.0853, b2 = ⫺0.2750, mo = 8.1965, m1 = ⫺0.6737, m2 = ⫺0.4210, no = 9.6152, n1 = ⫺0.0241 and n2 = ⫺0.7597. Ktc, Kfrc1, Kfrc2 are given in N/mm2. Once the friction cutting force coefficients, Kfrc1, and the uncut chip area (A), and the cutting edge contact length (Lc,) are determined, the radial (Fr) and feed forces (Ff) are predicted as components of the friction force (Ffr,). However, this requires that the friction force direction, which is defined by the effective lead angle (fL) to be known. The prediction of the effective lead angle consists of two steps; As mentioned above, the uncut chip area (A) is divided into two regions and the friction cutting force (Ffrc) has been expressed separately (Ffrc1 and Ffrc2) in each region for accurate predictions. In the above equations, the cutting force coefficients change inversely as functions of the uncut chip area (A), the cutting edge contact length (Lc), and the cutting speed (V). The variation of the cutting force coefficients with uncut chip area and cutting edge contact length represents the effects of the tool geometry on the cutting forces. The relationships between the cutting force coefficients and the uncut chip area (A), and cutting edge
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contact length (Lc) are non-linear due to the corner radius of the tool and the chip breaking groove along the cutting edge. On the other hand, decrease of the cutting force coefficients with cutting speed is attributed to the reduction in the maximum shear stress of the material and average friction coefficient at high speeds. As noted, the effect of each parameter in the equations of each cutting force coefficient is different. For the depth of cuts which are larger than the corner radius (R), the effect of the cutting edge contact length on the friction cutting force coefficient (Kfrc2) is not as significant as compared to the case in which the depth of cut is less than R. This may be due to the fact that the straight side of the tool dominates the cutting process for the larger depth of cuts. Thus, the friction cutting force coefficient (Kfrc2) does not change much with the depth of cut due to the uniform distribution of friction force on the straight side of the tool. Since the friction force acts perpendicular to the cutting edge contact length and is proportional to the uncut chip area of each differential element along the cutting edge, it can be predicted by assuming that each component of the friction force passes through the gravity center of each related region (Fig. 6). The friction force component of each region is added up vectorially, and the total friction force (Ffr) is obtained. It should be noted that these vectorial friction force components are
Fig. 7.
Determination of gravity center.
presented with the notation Ffr∗1, Ffr∗2 (Fig. 6) and are not identical to the ones in the Eq. (8). Ffrc1, Ffrc2 and Ffre, are the components contributing from each region to the total friction force, and do not have vectorial meanings. Determination of the gravity center of the uncut chip area is shown in the following equation. The calculation is executed with respect to the origin of the corner radius C2 for a given tool position (Fig. 7). XG ⫽
Σni=1AiXGi Σni=1AiYGi , YG ⫽ AT AT
(11)
where Ai, (XG, YG) and AT are the area of a differential element, coordinate of the gravity center with respect to C2 and the total uncut chip area of Region 1, respectively. Based on the definition of the friction force direction, the regional lead angle (f∗L2) in Region 2 and 3 can be assumed to be equal to the side cutting edge angle (gL) of the tool along the straight line of the cutting edge. The total effective lead angle is determined from the sum of two friction force vectors (Fig. 6). The effective lead angle is evaluated from each cutting test as follows, Fig. 6. Illustration of chip geometry, gravity center and friction force components.
Fr fL ⫽ arctan Ff
(12)
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Fig. 8.
469
Fig. 9. Variation of the effective lead angle modification factor Km with V and Lc.
Effective lead angle variation with Lc and V.
After processing the data, analysis has shown that there were certain discrepancies between ‘measured’ and ‘predicted’ effective lead angles based on the above approach (Fig. 8). These discrepancies are perhaps due to the fact that the friction force (Ffr) may not be exactly acting perpendicular to the cutting edge. The difference between the measured and predicted effective lead angle (fL) has been investigated. This has revealed that the effective lead angle shows linear variation with V and Lc. Hence, it can be tuned in the calculation with a modification factor (Km) that is also a linear function of Lc, and V (Fig. 9). fL ⫽ Km(V,Lc)f∗L ∗ L
(13)
where f is the predicted effective lead angle based on the regular procedure described above and fL is the final modified-predicted effective lead angle (Fig. 10) where
Fig. 10. Illustration of predicted and modified-predicted effective lead angle.
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Km1 ⫽ 1.0743⫺0.3567 × 10⫺3Lc ⫹ 0.9763 × 10⫺4V for a ⬍ R
Frc ⫽ Fr⫺Fre
Km2 ⫽ ⫺0.0163 ⫹ 0.6299Lc ⫹ 0.0013V for aⱖR
Ffc ⫽ Ff⫺Ffe (14)
where Lc, and V are given in mm and m/min. The above identified modification factors show that for a ⬍ R, the effective lead angle (fL) has an almost constant deviation along the cutting edge contact length (i.e., the effects of Lc, and V are negligible). However, for a ⬎ R, the deviation has a strong dependence on Lc and V. This may arise from the nature of the chip flow. For a ⬍ R, the chip attempts to flow towards the center of the corner radius with an almost constant deviation. However, along the straight edge, the chip is forced to flow away towards the outside of the contact in the radial direction and with a continuously diverging friction force, causing the chip to curl. The enforcement of the chip exhibits a continuous increase along the cutting edge and reaches a maximum at the end of the contact length. A schematic representation of the directional variation of the friction force is shown in Fig. 11. It should be noted that the cutting edge contact length in Eq. (14) is the total contact length including Region 1. The radial and feed force predictions are made based on the predicted friction force (Ffr) and effective lead angle (fL) as follows,
and corresponding cutting obtained as Krc ⫽
(16) force
coefficients
Frc Ffc , Kfc ⫽ A A
are
(17)
The mechanistic model presented in this paper can be used for the dynamic cutting force model as the friction force is related to the uncut chip area geometry. The geometric factors are the cutting edge contact lengths (Lc1 and Lc2). And the assumption that the friction forces pass through the gravity center of the regions. Applying the above procedure for the Kennametal CPMT-32.52 K720 insert were resulted in the following cutting force coefficients (Eq. 18) given in N/mm2, Ktc ⫽ e8.0428A⫺0.1696V⫺0.2512 Kfrc1 ⫽ e7.7522L⫺0.6093 V⫺0.2189 c1
(18)
Kfrc2 ⫽ e9.3082L⫺0.0541 V⫺0.5470 c2 and effective lead angle modification factors, Km1 ⫽ 1.2963 ⫹ 0.0604Lc ⫽ 0.0006V for a ⬍ R Km2 ⫽ ⫺0.4138 ⫹ 0.7021Lc ⫽ ⫺0.0025V for aⱖR
Fr ⫽ Ffr.sin(fL)
(15)
Ff ⫽ Ffr.cos(fL) Radial and feed cutting force components become,
(19) Edge cutting force coefficients were determined as,
Fig. 11. Deviation of the effective lead angle along the cutting edge contact length.
F. Atabey et al. / International Journal of Machine Tools & Manufacture 43 (2003) 463–476
Kte ⫽ 13.777 [N / mm], Kre ⫽ 13.036 [N / mm], Kfe ⫽ 19.572 [N / mm],
(20)
Kfre ⫽ 23.516 [N / mm] It should be noted that the trend of the variations of the cutting force coefficients and the effective lead angle modification factor in the Kennametal insert are similar to the ones for the Valenite insert, but only the empirical constants are different due to the differences between the geometries of the two inserts. Some of these differences are as the following; The Valenite tool has 5° side cutting edge angle (g) while the Kennametal insert has 0°, the Valenite tool has 7° relief angle while the Kennametal insert has 11°, the grove geometries are different. All these geometrical differences are implicitly considered in the empirical constants. Fig. 12.
Friction force verification for a ⬍ R.
Fig. 13.
Friction force verification for a ⱖ R.
3.1.2. Experimental validation of the mechanistic model Validation tests have been conducted with a workpiece material of Aluminum 6061-T6. Two different inserts have been used in the experiments: 1. Kennametal CPMT 32.52 K720 coated insert with A12-SCFPR3 steel shank boring bar with 0° side cutting edge angle, 2. Valenite CCGT432-FH Carbide PVD coated diamond insert with A-SCLPR/L boring bar with ⫺5° side cutting edge angle g In order to avoid chatter vibrations, the boring bar was clamped to the tool holder with a short length to diameter ratio (L/D = 2.5). Two sets of experiments for each insert were conducted with different combinations of the cutting parameters within the ranges of 0.05–0.19 mm/rev feedrate (c), 75–275 m/min cutting speed (V), 0.25–3.25 mm depth of cut (a). The first set of experiments was used to determine the empirical constants in the equations for the estimation of the cutting force coefficients. In order to validate the mechanistic model, a second set of experiments was conducted under various cutting conditions in the ranges of calibration. The mechanistic model results in good force predictions having less than 10% of absolute average error both for Valenite and Kennametal inserts. Tangential forces Ft (both for a ⬍ R and a ⱖ R together) have 99.5% of correlation. Friction forces for a ⬍ R and a ⱖ R have 93.5 and 98.4% of correlations, respectively. The results of cutting force prediction for the Valenite insert are presented in Figs. 12–18. 3.2. Orthogonal to oblique transformation If the insert’s rake face is uniformly flat without chip breaking or contact reduction grooves, the boring insert’s
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Fig. 15. Effective lead angle verification for a ⬍ R and a ⱖ R. Fig. 14.
Tangential force verification for a ⬍ R and a ⱖ R.
cutting edge can be considered as an assembly of oblique cutting edge elements. Oblique cutting mechanics laws lead to the prediction of cutting pressure at each discrete cutting edge element, which depend on the discrete chip area, edge geometry, and orthogonal cutting parameters of the work material (i.e., shear stress, shear angle and friction angle) which are mapped using classical mechanics laws proposed by Armarego [13]. The details of the orthogonal to oblique cutting transformation can be found in [11]. 3.2.1. Orthogonal tests and identification of oblique parameter Orthogonal cutting parameters, shear stress (ts), friction angle (ba), chip compression ratio (rc) have been identified by Ren [14] for carbide tools and P20 steel material as functions of feedrate (c) and cutting speed (V) as follows:
Fig. 16.
Feed force verification for a ⬍ R.
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Fig. 18. Radial force verification for a ⬍ R and a ⱖ R.
Fig. 17. Feed force verification for a ⱖ R.
ts[MPa] ⫽ 507 ⫹ 1398.76c ⫹ 0.327V ba[deg] ⫽ 33.69⫺12.16c⫺0.022V
(21)
rc ⫽ 0.227 ⫹ 2.71c ⫹ 0.00045V where c is the feedrate in mm/rev, V is the cutting speed in m/min. The expressions are valid within ±5° rake angle range. Edge cutting coefficients Kte, Kre and Kfe (units are given in N/mm) have been also identified by Ren [14] as the following, Kte ⫽ 0.1199 × 10 V ⫺0.1487V ⫹ 76.85 ⫺3
2
Kfe ⫽ 0.1366 × 10⫺3V2⫺0.2007V ⫹ 97.98
(22)
Kre ⫽ Kte.sin(i) Cutting forces as expressed in the general form of Ft ⫽ Ktc.bh ⫹ Kte.b Fr ⫽ Krc.bh ⫹ Kre.b Ff ⫽ Kfc.bh ⫹ Kfe.b
473
(23)
where b mm is the length of differential cutting edge length, and the oblique cutting force coefficients are defined as, Ktc ⫽
cos(bn⫺an) ⫹ tan(i)tan(h)sin(bn) ts sin(fn) cos2(fn ⫹ bn⫺an) ⫹ tan2(h)sin2(bn)
Kfc ⫽
ts sin(bn⫺an) sin(fn)cos(i) cos2(fn ⫹ bn⫺an) ⫹ tan2(h)sin2(bn)
Krc ⫽
ts cos(bn⫺an)⫺tan(h)sin(bn) sin(fn) cos2(fn ⫹ bn⫺an) ⫹ tan2(h)sin2(bn)
冑
冑
(24)
冑
In general, Kte and Kfe are determined in the evaluation of the orthogonal cutting test results. Because there is no radial force component measured in orthogonal cutting tests, Kre is not known. However, experimental investigations have shown that radial cutting edge force Fre is very small and therefore negligible in the transformation method. Oblique angle (i), normal rake (an), normal shear (fn) are evaluated as a function of differen-
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rake (ar) and side cutting edge (g) angles, hence, the cutting in Region 2 and 3 can be considered to obey orthogonal cutting mechanics laws. Region 1: Uncut chip area, Ai, is calculated based on Eq. (2). Approach angle yr = jθ, where j is the counter of the differential elements and qi is the uniform angular increment of each differential element. The following angles are determined as; Orthogonal rake angle aο equals to arctan(tan(af).cos(yr) + tan(ap).sin(yr)) where af, ar, yr, are the side rake, back rake, and side relief angles, respectively. Oblique angle i is determined from arctan(tan(ap).cos(yr) + tan(af).sin(yr)). Normal rake angle an can be found as arctan(tan(a0). cos(i)) where a0 is the orthogonal rake angle. Chip ratio, friction angles and shear stress are given by Eq. (21). Normal rc.cos(an) where an shear angle fn is equal to arctan 1⫺rcsin(an) is the normal rake angle. Normal friction angle bn is determined as arctan(tan(ba). cos(i)). By substituting the above parameters into orthogonal to oblique transformation given in Eq. (24), tangential, radial, and feed cutting force coefficients for each differential element are determined. As can be noticed, the oblique cutting parameters change around Region 1 due to the variation of the approach angle yr. For each differential element, oblique tangential, radial, and feed forces are determined,
冉
冊
Ft,i ⫽ Ktc,iA1,i ⫹ Kte.Lc,i Fr,i ⫽ Krc,iA1,i ⫹ Kre.Lc,i
(25)
Ff,i ⫽ Kfc,iA1,i ⫹ Kfe.Lc,i Then, by summing all the respective force components, total cutting force values along the corresponding directions can be determined in Region 1 as follows,
冘 冘 冘 N
Fig. 19. Evaluation of the oblique cutting parameters for three regions of the uncut chip area.
Fx1 ⫽
Ft1,i
i=1 N
tial cutting edge geometry [11] and evaluated for the particular inserts used in this study. 3.2.2. Experimental validation of the orthogonal to oblique transformation method Valenite CTPGPL-16-3C insert with a nose radius, sharp cutting edge and a flat rake face is selected to demonstrate the prediction of cutting forces using orthogonal to oblique transformation method. The uncut chip area is divided into three regions (Region 1, 2 and 3, Fig. 19) as presented before. Region 1 is discretized into equal angular segments qi , where each element has a different oblique geometry due to tool corner radius of the tool. However, in Region 2 and 3, the uncut chip areas are uniform, and the oblique cutting parameters do not change with location. The selected insert has zero back
Fy1 ⫽
(Ff1,isin(qi)⫺Fr1,icos(qi))
(26)
i=1 N
Fz1 ⫽
(Ff1,icos(qi)⫺Fr1,isin(qi))
i=1
For Region 2: The same equations are used in the force prediction except that the uncut chip area is determined from Eq. (3) and approach angle yr = ⫺γ, where g is the side cutting edge angle of the tool. Cutting force components in three orthogonal measurement directions of dynamometer can be found similarly, Fx,2 ⫽ Ft2 Fy,2 ⫽ Ff2sin(⫺g)⫺Fr2.cos(⫺g)
(27)
Fz,2 ⫽ Ff2cos(⫺g)⫺Fr2.sin(⫺g) For Region 3: The uncut chip area is given by Eq.
F. Atabey et al. / International Journal of Machine Tools & Manufacture 43 (2003) 463–476
475
(4). The approach angle will be approximated as, g yr⯝ . For this region, the edge cutting force compo2 nents are assumed to be zero due to the zero contact length. Cutting force components are obtained by using the equations presented above, Ft,3 ⫽ KtcA3 Fr,3 ⫽ KrcA3
(28)
Ff,3 ⫽ KfcA3 These oblique cutting forces contributed by Region 3 are obtained as, Fx,3 ⫽ Ft3
冉 冊 冉 冊
冉 冊 冉 冊
gl gl Fy,3 ⫽ Ff3sin ⫺ ⫺Fr3·cos ⫺ 2 2
(29)
gl gl Fz,3 ⫽ Ff3cos ⫺ ⫺Fr3·sin ⫺ 2 2
Fig. 20. Comparison of measured and predicted tangential, radial and feed forces using orthogonal to oblique transformation method.
The total forces in dynamometer directions, X, Y and Z are found as follows, Fx ⫽ Fx,1 ⫹ Fx,2 ⫹ Fx,3 Fy ⫽ Fy,1 ⫹ Fy,2 ⫹ Fy,3
(30)
Fz ⫽ Fz,1 ⫹ Fz,2 ⫹ Fz,3 Cutting tests were conducted with the material P20 mold steel by using the Valenite TPC 322J uncoated VC2 grade CTPGL-16-3 C left-hand tool holder at different cutting speeds and depth of cuts, but with constant 0.05 mm/rev feedrates. The tool had 5° side rake angle, 0° back rake and side cutting angles, 11° side relief angle and 0.8 mm corner radius. The cutting speed was varied from 100–240 m/min , and depth of cut was changed in the range of 0.625–1.750 mm. Seven of the experimental force results and corresponding predictions are shown in Fig. 20. Tangential force is predicted with less than 10% average error; however, prediction error in radial and feed forces rises to 25% in some cases. The variation is attributed to geometric modeling errors, as well as the use of classical mechanics laws in evaluating complex chip formation process in boring with inserts.
4. Conclusions A comprehensive model of single point boring operations has been presented. The chip geometry removed by curved boring inserts is modeled as a function of tool geometry, feedrate and radial depth of cut. Due to irregular distribution of chip load around the insert’s cutting edge, the amplitudes and directions of distributed cutting forces change as a function of tool geometry and cutting conditions. As a result, the cutting forces in boring have
a linear dependency with the chip area, but non-linear dependency with the feedrate and radial depth of cut. The cutting coefficients are evaluated mechanistically by conducting cutting tests at different feeds, speeds and depth of cuts with inserts having irregular rake face geometry. The cutting coefficients are estimated by correlating the chip geometry and forces using regression analysis. The cutting coefficients for inserts having smooth rake faces are modeled using orthogonal to oblique transformation method. The models are experimentally proven for single point boring bars used in industry. The models allow the process engineers to investigate the influence of insert geometry, feed, speed and radial depth of cut, boring forces, torque and power. The model is an essential foundation to study the forced and chatter vibrations in boring operations with single point boring bars and multi-insert boring heads.
Acknowledgements This research is conducted at The University of British Columbia and sponsored by National Science and Engineering Research Council of Canada (NSERC), Milacron, Pratt & Whitney Canada, Caterpillar and Boeing Corporations.
References [1] F. Atabey, I. Lazoglu, Y. Altintas, Mechanics of boring processes-Part II: Multi-insert boring heads, Int. J. Machine Tools and Manufacture, Design, Research and Application 43 (2003) 477–484.
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[2] C.A. Van Luttervelt, T.H.C. Childs, I.S. Jawahir, F. Klocke, P.K. Venuvinod, Present situation and future trends in modeling of machining operations, Annals of CIRP 47 (2) (1998) 578–626. [3] K.F. Ehman, S.G. Kapoor, R.E. DeVor, I. Lazoglu, Machining process modeling: A review, J. of Manufacturing Science and Engineering, Trans. of ASME 119 (1997) 655–663. [4] F. Kuster, Cutting dynamics and stability of boring bars, Annals of CIRP 39 (1) (1990) 361–366. [5] G. Subramani, R. Suvada, S.G. Kapoor, R.G. DeVor, W. Meingast, A model for the prediction of force system for cylinder boring process, Proc. XV. NAMRC (1987) 439–446. [6] G.M. Zhang, S.G. Kapoor, Dynamic modeling and dynamic analysis of the boring machining system, J. of Engineering for Industry 109 (1987) 219–226. [7] S.G. Tewani, K.E. Rouch, B.L. Walcott, Cutting process stability of a boring bar with active dynamic absorber, Vibration AnalysisAnalytical and Computational ASME 37 (1991) 205–213. [8] S.G. Tewani, T.C. Switzer, B.L. Walcott, K.E. Rouch, T.R.
[9] [10]
[11]
[12]
[13] [14]
Massa, Active control of machine tool chatter for a boring bar: experimental results, vibration and control of mechanical systems, ASME 61 (1993) 103–115. S. Jayaram, M. Iyer, An analytical model for prediction of chatter stability in boring, Trans. of NAMRI/SME 28 (2000) 203–208. E.W. Parker, Dynamic stability of a cantilever boring bar with machined flats under regenerative cutting conditions, Journal Mechanical Engineering Science 12 (1970) 104–115. Y. Altintas, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations and CNC Design, Cambridge University Press, 2000. E.J.A. Armarego, M. Uthaichaya, Mechanics of cutting approach for force prediction in turning operations, J. of Engineering Production 1 (1977) 1–18. E.J.A., Armarego, Material removal processes—An intermediate course, University of Melbourn, 1993. H. Ren, Mechanics of machining with chamfered tools, Masters Thesis, University of British Columbia, 1998.
Mechanics of boring processes—Part I F. Atabey a, I. Lazoglu b, Y. Altintas a,∗ a
University of British Columbia, Department of Mechanical Engineering, Manufacturing Automation Laboratory, V6T 1Z4, Vancouver, Canada b Koc University, Department of Mechanical Engineering, 80910 Sariyer, Istanbul, Turkey Received 1 February 2002; received in revised form 28 October 2002; accepted 13 November 2002
Abstract Mechanics of boring operations are presented in the paper. The distribution of chip thickness along the cutting edge is modeled as a function of tool inclination angle, nose radius, depth of cut and feed rate. The cutting mechanics of the process is modeled using both mechanistic and orthogonal to oblique cutting transformation approaches. The forces are separated into tangential and friction directions. The friction force is further projected into the radial and feed directions. The cutting forces are correlated to chip area using mechanistic cutting force coefficients which are expressed as a function of chip-tool edge contact length, chip area and cutting speed. For tools which have uniform rake face, the cutting coefficients are predicted using shear stress, shear angle and friction coefficient of the material. Both approaches are experimentally verified and the cutting forces in three Cartesian directions are predicted satisfactorily. The mechanics model presented in this paper is used in predicting the cutting forces generated by inserted boring heads with runouts and presented in Part II of the article [1]. 2003 Elsevier Science Ltd. All rights reserved. Keywords: Force modeling; Chip load; Single point boring; Orthogonal to oblique transformation
1. Introduction The enlargement of holes is achieved via boring operations. The hole diameter is either enlarged with a single insert attached to a long boring bar, or with a boring head which has a diameter equal to the diameter of the hole to be enlarged. Long boring bars statically and dynamically deform under the cutting forces during boring operations. Excessive static deflections may violate the dimensional tolerance of the hole, and vibrations may lead to poor surface, short tool life and chipping of the tool. Predictions of the force, torque and power are required in order to identify suitable machine tool and fixture set up for a boring operation. A comprehensive engineering model, which allows prediction of cutting forces, torque, power, dimensional surface finish and vibration free cutting conditions, is required in order to plan boring operations in the production floor. Although the other machining processes such as mill-
Corresponding author. Tel.: +1-604-822-5622; fax. 1-604-8222403. E-mail address: [email protected] (Y. Altintas). ∗
ing, turning and drilling have been studied relatively broader and deeper ([2], [3]), there were only a few attempts to model the cutting forces and stability in boring ([4–10]). However, the mechanics and dynamics of the boring process have not been sufficiently modeled for an effective prediction of boring process performance. There are fundamental issues which make the boring process somewhat difficult to model for a reliable prediction of process performance. The boring inserts have nose radius, and may have either uniform or irregular rake face. The distributions of chip thickness, therefore the cutting pressure amplitude and direction, vary as a function of tool nose radius, radial depth of cut and inclination angle. Since the boring bar is long and flexible, it is not possible to remove much larger depths of cuts than the nose radius of the tool, unlike the case of turning and face milling operations. This leads to a non-linear, complex relationship between the cutting force distribution, tool geometry, feedrate and depth of cut. Furthermore, the presence of static and dynamic deflections may influence the engagement conditions, leading to variations in chip load distribution and the cutting pressure. The cutting forces are usually predicted as a function
0890-6955/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0890-6955(02)00276-6
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of uncut chip area that changes in a complex manner in boring due to nose radius and geometry of the tool, and cutting conditions. If the tool rake face has an irregular geometry due to chip breaking grooves and chip-tool contact restriction features, the cutting coefficients are identified using mechanistic models. A series of cutting tests are conducted with the specific tool at different speeds, radial depth of cuts and feedrates. The coefficients are evaluated by curve fitting the force expressions to the measured cutting forces and chip geometry. If the rake face of the tool is smooth and uniform, it is possible to model the cutting edge as an assembly of oblique cutting edges [11]. The cutting pressure at each discrete oblique cutting edge element is modeled by applying the orthogonal to oblique transformation method proposed by Armarego [12,13]. Both approaches have been considered in this article. The paper is organized as follows; first the complex geometry of the chip is modeled analytically for various cutting conditions and tool geometry. The cutting forces are modeled as friction and tangential cutting forces. The friction force is resolved in the radial and feed directions, and the direction of the friction force, i.e. the effective lead angle, is experimentally evaluated from the ratio of the two. The effective lead angle is predicted from the geometry of the chip and tool, and the prediction is improved by a mechanistic model based on regression analysis applied to the model and measurements. The cutting coefficients are modeled as empirical functions of cutting speed, cutting edge contact length and uncut chip area. The forces are modeled using orthogonal to oblique cutting transformation when the rake face of the tool is smooth. This method requires only the tool geometry, shear stress, shear angle and average friction coefficient of the orthogonal cutting process for a specific work material. The paper contains experimental verification of proposed models which are used to predict the cutting forces in all three directions.
2. Chip geometry cut by single insert The fundamental geometry of a boring insert is characterized by a corner radius (R), side cutting edge angle (g) and end cutting edge angle (gc) (Fig. 1). The rake face of the tool may have either flat face or irregular chip breaking and chip contact reduction grooves which affect the cutting mechanics. The cutting edge does not usually have chamfer or curvature unless the workpiece material is not hardened steel. The chip area, and therefore the cutting force distribution, vary as a function of the depth of cut (a), feedrate (c), side cutting edge angle (g) and end cutting edge angle (gc). Fig. 2 also illustrates the relative positions of the insert at successive revolutions of the workpiece in four different configurations. Nine various tool-workpiece inter-
Fig. 1. Friction force distribution along the cutting edge contact length Lc.
ferences can be also defined with respect to the corner radius (R), depth of cut (a), feedrate (c), side cutting edge angle (g) and end cutting edge angle (gc) Notice that the material left behind (uncut material) depends on the feedrate (c) and corner radius (R), and is expected to be large when the feedrate is considerably greater. The uncut material also determines the surface finish quality. While large corner radius and small feedrate create good surface, small corner radius and large feedrate leaves more material (i.e., feed marks) behind causing a rougher surface finish. The most common case encountered in boring applications is when the feedrate is less than the nose radius of the insert. Therefore, only five common interferences have been considered (Figs. 3 and 4). The uncut chip area is evaluated by discretizing the chip into small differential elements in three separate regions as shown in Fig. 3. In the following, the calculation of the uncut chip area (A) and cutting edge contact length (Lc) are explained for only the first configuration. (The uncut chip areas and cutting edge contact lengths are calculated with the same manner for the other configurations shown in Fig. 4). For Region 1 in Fig. 3, the uncut chip area of each differential element is approximated by subtracting the area of the triangle AoBB⬘ from the area of the circular ring sector AODD⬘; 1 AOBB⬘,i ⫽ |OB|i|OB⬘|isin(qi), 2 1 AODD⬘,i ⫽ qiR2, A1,i⬵AODD⬘,i⫺AOBB⬘,i 2
(1)
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Fig. 2.
465
Illustration of four different uncut chip area configurations defined with depth of cut (a), feedrate (c), and corner radius of the tool (R).
Region 2 is considered to be a rectangle, although one side of it (i.e., side KE) has a slight curvature caused by the corner radius of the previous tool position. Area of Region 2 can be approximated as, A2⬵|MG||KM|
(3)
Region 3 is a simple triangle and its area is calculated as, 1 A3 ⫽ |KM||LM|sin(gL) 2
(4)
Finally, total uncut chip area is found by adding the areas for each region, A ⫽ A1 ⫹ A2 ⫹ A3 Fig. 3. Uncut chip area calculations for Configuration #1 (gL ⬎ 0, a ⬎ R(1 + singL)) and definitions of the regions.
The total chip area in Region 1 is evaluated by discrete summation of all differential elements in the curved region bounded by JEG.
冘
The total cutting edge contact with the work material is, Lc ⫽ Lc1 ⫹ Lc2
i⫽1
(6)
where Lc1, Lc2 are the contact lengths in Regions 1 and 2, respectively. If the number of discrete elements in Region 1 is given as n, the contact length in Region 1 will be equal to summation of the discrete contact
n
A1⫺
(5)
冘 n
A1,i
(2)
lengths (i.e., Lc1 =
i=1
冘 n
|DD⬘| = R
θi). Similarly, Lc2
i=1
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Fig. 4.
Uncut chip area configurations (#2…#5) considered in the model process.
is the contact length of Region 2 and equal to the length of MG. Hence the chip area can be identified geometrically as a function of radial depth of cut (a), feedrate (c) and insert geometry (R, g, gc). 3. Modeling of cutting forces in boring The cutting forces are represented by the tangential force (Ft) and friction force (Ffr). Later, friction force is resolved into the feed (Ff) and radial directions (Fr) (Figs. 1, and 5). Since the chip thickness distribution at each point along the cutting edge contact point is different and dependent on the insert geometry (R, g, gc) feedrate (c) and radial depth of cut (a), the distribution of the force along the cutting edge-chip contact zone also varies. At any contact point, the differential cutting forces are modeled as a function of local chip area (dA) and chip-cutting edge contact length (dLc), dFt ⫽ dFtc ⫹ dFte
⫽ Ktc.dA ⫹ Kte.dLc
dFfr ⫽ dFfrc ⫹ dFfre ⫽ Kfrc.dA ⫹ Kfre.dLc
(7)
where dFtc, dFfrc, are contributed by the removal of the chip and dFte, dFfre are due to cutting edge-finish surface rubbing. The edge contact constants (Kte, Kfre) are dependent on the cutting edge condition and preparation. The cutting coefficients (Ktc, Kfrc) are dependent on the local rake, inclination, chip flow angles, cutting conditions and work-tool material properties. They can be determined
Fig. 5.
Illustration of force directions in boring process.
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either mechanistically by conducting cutting tests with each insert, and curve fitting the force measurements against the chip geometry, or using classical laws of cutting mechanics such as orthogonal to oblique cutting transformation proposed by Armarego [13]. Both approaches are presented here for boring operations. 3.1. Mechanistic modeling of cutting forces 3.1.1. Identifications of the cutting force coefficients The cutting force coefficients are determined for two sets of selected inserts (i.e., Kennametal CPMT-32.52 K720 and Valenite CCGT432-FH) used in boring A1 6061-T6 workpiece material. The procedure is general and applicable to other insert geometries as well. The tangential (Ft) and friction (Ffr) forces are modeled as follows, Ft ⫽ Ftc ⫹ Fte
⫽ Ktc.A ⫹ Kte.Lc
Ffr ⫽ Ffrc1 ⫹ Ffrc2 ⫹ Ffre ⫽ Kfrc1.A1 ⫹ Kfrc2.A2 ⫹ Kfre.Lc
(8)
Unlike Ft, Ffr is considered as having two components. The uncut chip area, A1 covers Region 1 and A2 covers Regions 2 and 3 as illustrated in Fig. 3. The direction of total friction force is defined by effective lead angle (fL), which is the angle between the friction force and feed directions (Fig. 1). For relatively large radial depth of cuts, the effective lead angle (fL) tends to approach the side cutting edge angle (gL) of the insert. In such a case, the magnitude ratio of Fr/Ff decreases. In order to determine the edge cutting force coefficients (Kte, Kfre, Kre and Kfe), twenty four experiments have been conducted with Valenite CCGT 432-FH insert at 0.8 mm corner radius, constant 1.5 mm depth of cut and 150 m/min cutting speed but varying feedrate (c) from 0.025–0.19 mm/rev. The cutting forces are measured in three orthogonal directions, (Ft, Fr, Ff). Performing linear regression leads to the following tangential force equation, Ft ⫽ ⫺4721.4 ⫹ 2438.5Lc[N] Lcⱖ1.9556 [mm]
(9)
where Lc is the cutting edge contact length that is dependent on depth of cut and feedrate. The above equation is valid only if Lc greater than 1.9556 mm which corresponds to zero feedrate for 1.5 mm depth of cut. Zero feedrate refers to the rubbing process that is the source of the edge cutting forces. Therefore, substitution of value of 1.9556 mm in the equation above gives tangential edge cutting force (Fte) for given depth of cut (a = 1.5 mm) and zero feedrate (i.e., Fte = Ft|Lc = 1.9556 = 47.33[N]). The tangential edge cutting force coefficient, Kte, is determined as Kte = Fte / Lc = 24.24 N/mm. Kte, represents the tangential rubbing force per unit cutting edge contact length. Following the same procedure, the radial, feed and friction edge cutting forces (Fre, Ffe and Ffre, where Ffre = √F2re + F2fe) and corresponding edge cut-
467
Table 1 Edge cutting force coefficients for the Valenite CCGT432-FH insert Force model
Ft = 2438.5Lc ⫺ 4721.4 Fr = 417.74Lc ⫺ 800.40 Ff = 933.47Lc ⫺ 1779.9 Ffr = √F2r + F2f
Measured edge Edge cutting force cutting forces Fte, coefficients Kte, Kre, Kfe, Kfre N/mm (Lc ⱖ 1.9556 Fre, Ffe, Ffre N mm) 47.41
24.24
16.53
8.45
45.59
23.31
48.49
24.79
ting force coefficients (Kre, Kfe, and Kfre) are found as seen in Table 1. Another set of tests was conducted with Valenite CCGT432-FH insert at different combinations of the cutting parameters within the ranges of 0.025–0.19 mm/rev feed rate, 75–275 m/min cutting speed and 0.25–3.25 mm depth of cut. After removing edge forces (Fe Ffre) from the measured cutting forces, regression analysis is applied to the cutting forces (Fc, Ffc) which led to the following relationships for the cutting force coefficients, Ktc ⫽ eb0Ab1Vb2, Kfrc1 ⫽ em0Lcm11Vm2, Kfrc2
(10)
⫽ en0Lnc21Vn2 where Lc1, Lc2, V are cutting edge contact lengths (in mm) in Region 1 and 2, and cutting speed (in m/min), respectively. bo, b1, b2, m0, m1, m2, no, n1, and n2 are empirical constants obtained using the least-squares method on the experimental data. Their values are; bο = 7.9477, b1 = ⫺0.0853, b2 = ⫺0.2750, mo = 8.1965, m1 = ⫺0.6737, m2 = ⫺0.4210, no = 9.6152, n1 = ⫺0.0241 and n2 = ⫺0.7597. Ktc, Kfrc1, Kfrc2 are given in N/mm2. Once the friction cutting force coefficients, Kfrc1, and the uncut chip area (A), and the cutting edge contact length (Lc,) are determined, the radial (Fr) and feed forces (Ff) are predicted as components of the friction force (Ffr,). However, this requires that the friction force direction, which is defined by the effective lead angle (fL) to be known. The prediction of the effective lead angle consists of two steps; As mentioned above, the uncut chip area (A) is divided into two regions and the friction cutting force (Ffrc) has been expressed separately (Ffrc1 and Ffrc2) in each region for accurate predictions. In the above equations, the cutting force coefficients change inversely as functions of the uncut chip area (A), the cutting edge contact length (Lc), and the cutting speed (V). The variation of the cutting force coefficients with uncut chip area and cutting edge contact length represents the effects of the tool geometry on the cutting forces. The relationships between the cutting force coefficients and the uncut chip area (A), and cutting edge
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contact length (Lc) are non-linear due to the corner radius of the tool and the chip breaking groove along the cutting edge. On the other hand, decrease of the cutting force coefficients with cutting speed is attributed to the reduction in the maximum shear stress of the material and average friction coefficient at high speeds. As noted, the effect of each parameter in the equations of each cutting force coefficient is different. For the depth of cuts which are larger than the corner radius (R), the effect of the cutting edge contact length on the friction cutting force coefficient (Kfrc2) is not as significant as compared to the case in which the depth of cut is less than R. This may be due to the fact that the straight side of the tool dominates the cutting process for the larger depth of cuts. Thus, the friction cutting force coefficient (Kfrc2) does not change much with the depth of cut due to the uniform distribution of friction force on the straight side of the tool. Since the friction force acts perpendicular to the cutting edge contact length and is proportional to the uncut chip area of each differential element along the cutting edge, it can be predicted by assuming that each component of the friction force passes through the gravity center of each related region (Fig. 6). The friction force component of each region is added up vectorially, and the total friction force (Ffr) is obtained. It should be noted that these vectorial friction force components are
Fig. 7.
Determination of gravity center.
presented with the notation Ffr∗1, Ffr∗2 (Fig. 6) and are not identical to the ones in the Eq. (8). Ffrc1, Ffrc2 and Ffre, are the components contributing from each region to the total friction force, and do not have vectorial meanings. Determination of the gravity center of the uncut chip area is shown in the following equation. The calculation is executed with respect to the origin of the corner radius C2 for a given tool position (Fig. 7). XG ⫽
Σni=1AiXGi Σni=1AiYGi , YG ⫽ AT AT
(11)
where Ai, (XG, YG) and AT are the area of a differential element, coordinate of the gravity center with respect to C2 and the total uncut chip area of Region 1, respectively. Based on the definition of the friction force direction, the regional lead angle (f∗L2) in Region 2 and 3 can be assumed to be equal to the side cutting edge angle (gL) of the tool along the straight line of the cutting edge. The total effective lead angle is determined from the sum of two friction force vectors (Fig. 6). The effective lead angle is evaluated from each cutting test as follows, Fig. 6. Illustration of chip geometry, gravity center and friction force components.
Fr fL ⫽ arctan Ff
(12)
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Fig. 8.
469
Fig. 9. Variation of the effective lead angle modification factor Km with V and Lc.
Effective lead angle variation with Lc and V.
After processing the data, analysis has shown that there were certain discrepancies between ‘measured’ and ‘predicted’ effective lead angles based on the above approach (Fig. 8). These discrepancies are perhaps due to the fact that the friction force (Ffr) may not be exactly acting perpendicular to the cutting edge. The difference between the measured and predicted effective lead angle (fL) has been investigated. This has revealed that the effective lead angle shows linear variation with V and Lc. Hence, it can be tuned in the calculation with a modification factor (Km) that is also a linear function of Lc, and V (Fig. 9). fL ⫽ Km(V,Lc)f∗L ∗ L
(13)
where f is the predicted effective lead angle based on the regular procedure described above and fL is the final modified-predicted effective lead angle (Fig. 10) where
Fig. 10. Illustration of predicted and modified-predicted effective lead angle.
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Km1 ⫽ 1.0743⫺0.3567 × 10⫺3Lc ⫹ 0.9763 × 10⫺4V for a ⬍ R
Frc ⫽ Fr⫺Fre
Km2 ⫽ ⫺0.0163 ⫹ 0.6299Lc ⫹ 0.0013V for aⱖR
Ffc ⫽ Ff⫺Ffe (14)
where Lc, and V are given in mm and m/min. The above identified modification factors show that for a ⬍ R, the effective lead angle (fL) has an almost constant deviation along the cutting edge contact length (i.e., the effects of Lc, and V are negligible). However, for a ⬎ R, the deviation has a strong dependence on Lc and V. This may arise from the nature of the chip flow. For a ⬍ R, the chip attempts to flow towards the center of the corner radius with an almost constant deviation. However, along the straight edge, the chip is forced to flow away towards the outside of the contact in the radial direction and with a continuously diverging friction force, causing the chip to curl. The enforcement of the chip exhibits a continuous increase along the cutting edge and reaches a maximum at the end of the contact length. A schematic representation of the directional variation of the friction force is shown in Fig. 11. It should be noted that the cutting edge contact length in Eq. (14) is the total contact length including Region 1. The radial and feed force predictions are made based on the predicted friction force (Ffr) and effective lead angle (fL) as follows,
and corresponding cutting obtained as Krc ⫽
(16) force
coefficients
Frc Ffc , Kfc ⫽ A A
are
(17)
The mechanistic model presented in this paper can be used for the dynamic cutting force model as the friction force is related to the uncut chip area geometry. The geometric factors are the cutting edge contact lengths (Lc1 and Lc2). And the assumption that the friction forces pass through the gravity center of the regions. Applying the above procedure for the Kennametal CPMT-32.52 K720 insert were resulted in the following cutting force coefficients (Eq. 18) given in N/mm2, Ktc ⫽ e8.0428A⫺0.1696V⫺0.2512 Kfrc1 ⫽ e7.7522L⫺0.6093 V⫺0.2189 c1
(18)
Kfrc2 ⫽ e9.3082L⫺0.0541 V⫺0.5470 c2 and effective lead angle modification factors, Km1 ⫽ 1.2963 ⫹ 0.0604Lc ⫽ 0.0006V for a ⬍ R Km2 ⫽ ⫺0.4138 ⫹ 0.7021Lc ⫽ ⫺0.0025V for aⱖR
Fr ⫽ Ffr.sin(fL)
(15)
Ff ⫽ Ffr.cos(fL) Radial and feed cutting force components become,
(19) Edge cutting force coefficients were determined as,
Fig. 11. Deviation of the effective lead angle along the cutting edge contact length.
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Kte ⫽ 13.777 [N / mm], Kre ⫽ 13.036 [N / mm], Kfe ⫽ 19.572 [N / mm],
(20)
Kfre ⫽ 23.516 [N / mm] It should be noted that the trend of the variations of the cutting force coefficients and the effective lead angle modification factor in the Kennametal insert are similar to the ones for the Valenite insert, but only the empirical constants are different due to the differences between the geometries of the two inserts. Some of these differences are as the following; The Valenite tool has 5° side cutting edge angle (g) while the Kennametal insert has 0°, the Valenite tool has 7° relief angle while the Kennametal insert has 11°, the grove geometries are different. All these geometrical differences are implicitly considered in the empirical constants. Fig. 12.
Friction force verification for a ⬍ R.
Fig. 13.
Friction force verification for a ⱖ R.
3.1.2. Experimental validation of the mechanistic model Validation tests have been conducted with a workpiece material of Aluminum 6061-T6. Two different inserts have been used in the experiments: 1. Kennametal CPMT 32.52 K720 coated insert with A12-SCFPR3 steel shank boring bar with 0° side cutting edge angle, 2. Valenite CCGT432-FH Carbide PVD coated diamond insert with A-SCLPR/L boring bar with ⫺5° side cutting edge angle g In order to avoid chatter vibrations, the boring bar was clamped to the tool holder with a short length to diameter ratio (L/D = 2.5). Two sets of experiments for each insert were conducted with different combinations of the cutting parameters within the ranges of 0.05–0.19 mm/rev feedrate (c), 75–275 m/min cutting speed (V), 0.25–3.25 mm depth of cut (a). The first set of experiments was used to determine the empirical constants in the equations for the estimation of the cutting force coefficients. In order to validate the mechanistic model, a second set of experiments was conducted under various cutting conditions in the ranges of calibration. The mechanistic model results in good force predictions having less than 10% of absolute average error both for Valenite and Kennametal inserts. Tangential forces Ft (both for a ⬍ R and a ⱖ R together) have 99.5% of correlation. Friction forces for a ⬍ R and a ⱖ R have 93.5 and 98.4% of correlations, respectively. The results of cutting force prediction for the Valenite insert are presented in Figs. 12–18. 3.2. Orthogonal to oblique transformation If the insert’s rake face is uniformly flat without chip breaking or contact reduction grooves, the boring insert’s
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Fig. 15. Effective lead angle verification for a ⬍ R and a ⱖ R. Fig. 14.
Tangential force verification for a ⬍ R and a ⱖ R.
cutting edge can be considered as an assembly of oblique cutting edge elements. Oblique cutting mechanics laws lead to the prediction of cutting pressure at each discrete cutting edge element, which depend on the discrete chip area, edge geometry, and orthogonal cutting parameters of the work material (i.e., shear stress, shear angle and friction angle) which are mapped using classical mechanics laws proposed by Armarego [13]. The details of the orthogonal to oblique cutting transformation can be found in [11]. 3.2.1. Orthogonal tests and identification of oblique parameter Orthogonal cutting parameters, shear stress (ts), friction angle (ba), chip compression ratio (rc) have been identified by Ren [14] for carbide tools and P20 steel material as functions of feedrate (c) and cutting speed (V) as follows:
Fig. 16.
Feed force verification for a ⬍ R.
F. Atabey et al. / International Journal of Machine Tools & Manufacture 43 (2003) 463–476
Fig. 18. Radial force verification for a ⬍ R and a ⱖ R.
Fig. 17. Feed force verification for a ⱖ R.
ts[MPa] ⫽ 507 ⫹ 1398.76c ⫹ 0.327V ba[deg] ⫽ 33.69⫺12.16c⫺0.022V
(21)
rc ⫽ 0.227 ⫹ 2.71c ⫹ 0.00045V where c is the feedrate in mm/rev, V is the cutting speed in m/min. The expressions are valid within ±5° rake angle range. Edge cutting coefficients Kte, Kre and Kfe (units are given in N/mm) have been also identified by Ren [14] as the following, Kte ⫽ 0.1199 × 10 V ⫺0.1487V ⫹ 76.85 ⫺3
2
Kfe ⫽ 0.1366 × 10⫺3V2⫺0.2007V ⫹ 97.98
(22)
Kre ⫽ Kte.sin(i) Cutting forces as expressed in the general form of Ft ⫽ Ktc.bh ⫹ Kte.b Fr ⫽ Krc.bh ⫹ Kre.b Ff ⫽ Kfc.bh ⫹ Kfe.b
473
(23)
where b mm is the length of differential cutting edge length, and the oblique cutting force coefficients are defined as, Ktc ⫽
cos(bn⫺an) ⫹ tan(i)tan(h)sin(bn) ts sin(fn) cos2(fn ⫹ bn⫺an) ⫹ tan2(h)sin2(bn)
Kfc ⫽
ts sin(bn⫺an) sin(fn)cos(i) cos2(fn ⫹ bn⫺an) ⫹ tan2(h)sin2(bn)
Krc ⫽
ts cos(bn⫺an)⫺tan(h)sin(bn) sin(fn) cos2(fn ⫹ bn⫺an) ⫹ tan2(h)sin2(bn)
冑
冑
(24)
冑
In general, Kte and Kfe are determined in the evaluation of the orthogonal cutting test results. Because there is no radial force component measured in orthogonal cutting tests, Kre is not known. However, experimental investigations have shown that radial cutting edge force Fre is very small and therefore negligible in the transformation method. Oblique angle (i), normal rake (an), normal shear (fn) are evaluated as a function of differen-
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rake (ar) and side cutting edge (g) angles, hence, the cutting in Region 2 and 3 can be considered to obey orthogonal cutting mechanics laws. Region 1: Uncut chip area, Ai, is calculated based on Eq. (2). Approach angle yr = jθ, where j is the counter of the differential elements and qi is the uniform angular increment of each differential element. The following angles are determined as; Orthogonal rake angle aο equals to arctan(tan(af).cos(yr) + tan(ap).sin(yr)) where af, ar, yr, are the side rake, back rake, and side relief angles, respectively. Oblique angle i is determined from arctan(tan(ap).cos(yr) + tan(af).sin(yr)). Normal rake angle an can be found as arctan(tan(a0). cos(i)) where a0 is the orthogonal rake angle. Chip ratio, friction angles and shear stress are given by Eq. (21). Normal rc.cos(an) where an shear angle fn is equal to arctan 1⫺rcsin(an) is the normal rake angle. Normal friction angle bn is determined as arctan(tan(ba). cos(i)). By substituting the above parameters into orthogonal to oblique transformation given in Eq. (24), tangential, radial, and feed cutting force coefficients for each differential element are determined. As can be noticed, the oblique cutting parameters change around Region 1 due to the variation of the approach angle yr. For each differential element, oblique tangential, radial, and feed forces are determined,
冉
冊
Ft,i ⫽ Ktc,iA1,i ⫹ Kte.Lc,i Fr,i ⫽ Krc,iA1,i ⫹ Kre.Lc,i
(25)
Ff,i ⫽ Kfc,iA1,i ⫹ Kfe.Lc,i Then, by summing all the respective force components, total cutting force values along the corresponding directions can be determined in Region 1 as follows,
冘 冘 冘 N
Fig. 19. Evaluation of the oblique cutting parameters for three regions of the uncut chip area.
Fx1 ⫽
Ft1,i
i=1 N
tial cutting edge geometry [11] and evaluated for the particular inserts used in this study. 3.2.2. Experimental validation of the orthogonal to oblique transformation method Valenite CTPGPL-16-3C insert with a nose radius, sharp cutting edge and a flat rake face is selected to demonstrate the prediction of cutting forces using orthogonal to oblique transformation method. The uncut chip area is divided into three regions (Region 1, 2 and 3, Fig. 19) as presented before. Region 1 is discretized into equal angular segments qi , where each element has a different oblique geometry due to tool corner radius of the tool. However, in Region 2 and 3, the uncut chip areas are uniform, and the oblique cutting parameters do not change with location. The selected insert has zero back
Fy1 ⫽
(Ff1,isin(qi)⫺Fr1,icos(qi))
(26)
i=1 N
Fz1 ⫽
(Ff1,icos(qi)⫺Fr1,isin(qi))
i=1
For Region 2: The same equations are used in the force prediction except that the uncut chip area is determined from Eq. (3) and approach angle yr = ⫺γ, where g is the side cutting edge angle of the tool. Cutting force components in three orthogonal measurement directions of dynamometer can be found similarly, Fx,2 ⫽ Ft2 Fy,2 ⫽ Ff2sin(⫺g)⫺Fr2.cos(⫺g)
(27)
Fz,2 ⫽ Ff2cos(⫺g)⫺Fr2.sin(⫺g) For Region 3: The uncut chip area is given by Eq.
F. Atabey et al. / International Journal of Machine Tools & Manufacture 43 (2003) 463–476
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(4). The approach angle will be approximated as, g yr⯝ . For this region, the edge cutting force compo2 nents are assumed to be zero due to the zero contact length. Cutting force components are obtained by using the equations presented above, Ft,3 ⫽ KtcA3 Fr,3 ⫽ KrcA3
(28)
Ff,3 ⫽ KfcA3 These oblique cutting forces contributed by Region 3 are obtained as, Fx,3 ⫽ Ft3
冉 冊 冉 冊
冉 冊 冉 冊
gl gl Fy,3 ⫽ Ff3sin ⫺ ⫺Fr3·cos ⫺ 2 2
(29)
gl gl Fz,3 ⫽ Ff3cos ⫺ ⫺Fr3·sin ⫺ 2 2
Fig. 20. Comparison of measured and predicted tangential, radial and feed forces using orthogonal to oblique transformation method.
The total forces in dynamometer directions, X, Y and Z are found as follows, Fx ⫽ Fx,1 ⫹ Fx,2 ⫹ Fx,3 Fy ⫽ Fy,1 ⫹ Fy,2 ⫹ Fy,3
(30)
Fz ⫽ Fz,1 ⫹ Fz,2 ⫹ Fz,3 Cutting tests were conducted with the material P20 mold steel by using the Valenite TPC 322J uncoated VC2 grade CTPGL-16-3 C left-hand tool holder at different cutting speeds and depth of cuts, but with constant 0.05 mm/rev feedrates. The tool had 5° side rake angle, 0° back rake and side cutting angles, 11° side relief angle and 0.8 mm corner radius. The cutting speed was varied from 100–240 m/min , and depth of cut was changed in the range of 0.625–1.750 mm. Seven of the experimental force results and corresponding predictions are shown in Fig. 20. Tangential force is predicted with less than 10% average error; however, prediction error in radial and feed forces rises to 25% in some cases. The variation is attributed to geometric modeling errors, as well as the use of classical mechanics laws in evaluating complex chip formation process in boring with inserts.
4. Conclusions A comprehensive model of single point boring operations has been presented. The chip geometry removed by curved boring inserts is modeled as a function of tool geometry, feedrate and radial depth of cut. Due to irregular distribution of chip load around the insert’s cutting edge, the amplitudes and directions of distributed cutting forces change as a function of tool geometry and cutting conditions. As a result, the cutting forces in boring have
a linear dependency with the chip area, but non-linear dependency with the feedrate and radial depth of cut. The cutting coefficients are evaluated mechanistically by conducting cutting tests at different feeds, speeds and depth of cuts with inserts having irregular rake face geometry. The cutting coefficients are estimated by correlating the chip geometry and forces using regression analysis. The cutting coefficients for inserts having smooth rake faces are modeled using orthogonal to oblique transformation method. The models are experimentally proven for single point boring bars used in industry. The models allow the process engineers to investigate the influence of insert geometry, feed, speed and radial depth of cut, boring forces, torque and power. The model is an essential foundation to study the forced and chatter vibrations in boring operations with single point boring bars and multi-insert boring heads.
Acknowledgements This research is conducted at The University of British Columbia and sponsored by National Science and Engineering Research Council of Canada (NSERC), Milacron, Pratt & Whitney Canada, Caterpillar and Boeing Corporations.
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