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Analytical Modelling of Milling Forces for Helical End Milling Based on a Predictive Machining Theory
Available online at www.sciencedirect.com
ScienceDirect Procedia CIRP 31 (2015) 258 – 263
15th CIRP Conference on Modelling of Machining Operations
Analytical Modelling of Milling Forces for Helical End Milling Based on a Predictive Machining Theory Zhongtao Fua, Wenyu Yanga,*, Xuelin Wanga, Jürgen Leopoldb a
State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Technology, Huazhong University of Science and Technology, Wuhan,430074, China. b
Formerly Fraunhofer Institute for Machine Tools and Forming Technology, Chemnitz, 09661, Germany.
* Corresponding author. Tel.: +86-27-875-48180; fax: +86-27-875-59416. E-mail address: [email protected]
Abstract Milling forces play an important role in the milling process and are generally calculated by the mechanistic or numerical methods which are considered time-consuming and impractical for various cutting conditions and workpiece-tool pair. Therefore, this paper proposes an analytical method for modelling the milling forces in helical end milling process based on a predictive machining theory, which regards the workpiece material properties, tool geometry, cutting conditions and types of milling as the input data. In this method, each cutting edge is discretized into a series of infinitesimal elements along the cutter axis and the cutting action of each element is equivalent to the classical oblique cutting process. The three dimensional cutting force components applied on each element are predicted analytically using this predictive oblique cutting model with the effect of cutting edge radius. Finally, the proposed analytical model of milling forces is verified by the published results and the simulation values using the software AdvantEdge FEM.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2015 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The International Scientific Committee of the “15th Conference on Modelling of Machining Operations”. Peer-review under responsibility of the International Scientific Committee of the “15th Conference on Modelling of Machining Operations Keywords: Analytical modelling; Milling forces; Helical end milling; Predictive machining theory
1. Introduction The helical end milling has been used extensively to machine parts with sculptured surfaces, such as propellers, turbines. The cutting force is the basis for predicting the chatter-free cutting parameters, jig and fixture strength and process optimization [1], [2]. Therefore, the accuracy prediction of cutting force model is very crucial to improve the machining performance, the workpiece surface quality and process stability. To this end, considerable studies have been focused on the milling forces model with various cutting conditions and workpiece-tool pair. According to the available studies, there exist three main methods to model milling forces, i.e. mechanistic, numerical, and analytical ones [2]-[19]. The mechanistic methods [2]-[9], which are commonly used to milling forces model, regard the cutting force coefficients as the relation between process variables and milling forces. The coefficients are calibrated from a set of milling [2]-[6] or orthogonal experiments [7]-[9] for a given tool-workpiece pair. However, this approach is
only valid for a certain workpiece-tool material pair and the calibration done by numerous experiments is impractical due to high cost and hardware setup complexity. Currently, the numerical methods [10], [12] are focused on the study of the interaction between the tool and the workpiece and the complex thermo-mechanical phenomenon. An arbitrary or updated Lagrangian formulation is employed in the finite element technology to simulate the machining process and predict the milling forces. Nevertheless, this approach is time-consuming and the commercial simulation software is too costly to be available. For the analytical methods, they try to establish mathematical relations between the milling forces and several mechanical aspects like friction, geometry and material behavior. For instance, Merchant [12] and Armarego [14] are two of the main contributors to this kind of models by the development of orthogonal and oblique cutting mechanics. Oxley [14] developed a predictive machining theory that allows for the high strain rate, high temperature and thermal properties of the work material. Based on Oxley’s predictive
2212-8271 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the International Scientific Committee of the “15th Conference on Modelling of Machining Operations doi:10.1016/j.procir.2015.03.013
Zhongtao Fu et al. / Procedia CIRP 31 (2015) 258 – 263
259
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Zhongtao Fu et al. / Procedia CIRP 31 (2015) 258 – 263
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Zhongtao Fu et al. / Procedia CIRP 31 (2015) 258 – 263 Pn
Finally, the cutting forces contributed by all cutting edges are summed to obtain the total instantaneous forces on the end-milling cutter as follows˖
Dn Chip
t2
Fx (I )
Vc
In
re
-
t1
R
+
K
Workpiece
In Fig. 3, re is the edge radius, U 0 is the prow angle which inclined to the uncut workpiece surface (10 deg in present analysis), R is the radius of the circular fan filed centered at P, the fan field angles J 0 ,- and K are found from the geometric and friction relationship. the plough forces can be determined as follows˖ ª(1 2- 2J 0 sin(2K ))sin(I J 0 K ) º dPc W s « » lPC w ¬cos(2K ) cos(In J 0 K ) ¼ (19) (1 2 2 J sin(2 K )) cos( I J K ) ª º 0 0 dPf W s « l w » PC ¬cos(2K )sin(In J 0 K ) ¼ where W s is shear flow stress of the primary shear plane PH;
the width of cut w is equal to dz / cos Os ; K 0.5arccos P , P is the friction coefficient between tool and workpiece; J 0 K I arcsin( 2 sin U0 sinK ) ; - S / 4 U0 In ; lPC R sinK , ª 2 R sin U0 º S D 2 sin K « re tan( n ) » 2 > R sin U0 @ ; 4 2 tan(S / 2 D n ) ¼» ¬«
ª cos Os « 0 « ¬« sin Os
cos Os sin In º » dFs ½ cos In » ® ¾ dN s ¿ » sin Os sin In ¼ ¯
0º dPc ½ 1 »» ® ¾ dPf 0 ¼» ¯ ¿
(20)
As shown in Fig.1 (a), the components dFt , j , dFr , j , dFa , j in local coordinates are transformed to the global coordinates using the relations: dFx , j ½ ° ° ® dFy , j ¾ ° dF ° ¯ z, j ¿
ª cos I j « sin I j « ¬« 0
sin I j cos I j 0
0 º dFt , j ½ ° ° 0 »» ® dFr , j ¾ ° ° 1¼» ¯ dFa , j ¿
(21)
The differential cutting forces are integrated along the in-cut portion of the j-th cutting edge and the total cutting forces produced by this cutting edge can be obtained as: z (I ) (22) Fq [I j ( z )] ³ dFq [I j ( z )]dz, q x, y, z 2
j 1
Nt
¦F
z, j
[I ( z )]
(23)
j 1
m 0.8, J 0 1 1,, Tm 1933K, Tr 293K and the material density U is 4430kg / m3 , the specific heat c p is 542J / (kgK) , the thermal conductivity k p is 6.7W / (mK) and the Taylor-Quinney coefficient F 0.85 . The carbide milling cutter used in up-milling has fourflute cutting edge, 30deg helix angle, 12deg rake angle and 19.05mm diameter. And the cutter used in down-milling has the same tool geometry as in up-milling except that the rake angle is 00. The cutting conditions are listed in Table 1. Table 1. Cutting conditions used in [8] Case
Due to the edge radius, the effective normal rake angle will vary depending on the edge radius and the depth of cut. The detailed calculation can be referenced to [24]. Therefore, for the infinitesimal element of the j-th cutting edge, the three differential cutting force components dFt , j , dFr , j , dFa , j (tangential, radial, axial), applied to point P in Fig.2 (a), are evaluated from the following matrix form: ª cos Os cos In cosK sh sin Os sin K sh « sin In cosK sh « «¬ sin Os cos In cosK sh cos Os sin Ksh
[I ( z )], Fz (I )
A computer program in MATLAB 7.9 is developed to implement the feasibility of the proposed analytical model of milling forces for helical end milling. The published results [8] obtained from the mechanistic model and experimental data are selected for comparison, in which two tests, i. e. halfimmersion up-milling and half-immersion down-milling, are carried out. The workpiece material is Ti6Al4V whose material parameters in Eq. (14) are given by [24]: A=862.5MPa, B=331.2 Mpa, C 0.0128, n 0.34,
2
dFt , j ½ ° ° ® dFr , j ¾ ° dF ° ¯ a, j ¿
y, j
3.1. Comparison with the published results
C
Fig. 3. Waldrof’s slip-line field for plough forces
R
Nt
¦F
[I ( z )], Fy (I )
3. Model validation
P
J0
V
x, j
j 1
Tool
H
U0
Nt
¦F
j
z1 (I j )
where z1 (I j ) and z2 (I j ) are the lower and upper axial engagement limits of the in-cut portion.
1 2
Cutting velocity V (m/min) 30 30
Feed rate
ft (mm/tooth) 0.050 0.0127
Radial depth of cut d r (mm)
Axial depth of cut a p (mm)
type
9.525 9.525
5.08 5.08
up down
Likewise, the proposed analytical model utilizes the same constants of the workpiece material, cutting conditions, tool geometry. The mean friction angle E n is obtained from Appendix A, where the sliding friction coefficient P s is taken as 0.6 for this tool-workpiece pair. From the orthogonal cutting database [8], the material constant is G 450 and the plough force coefficients are Kte 24 N/mm; Kre 43 N/mm. Fig. 4 shows the comparison of milling forces obtained by the proposed analytical model and the published results [8]. It can be observed that the milling forces calculated in the present work, are in good agreement with the published results obtained by the mechanistic model and experimental data. There exists a small deviation on the amplitude of the maximum milling forces between the proposed analytical model and the published results. This deviation might be attributed to the cutter deflection which is not considered in this proposed model. Furthermore, the mean shear flow stress W s predicted in the primary shear zone is about 592MPa and very close to the value 613MPa of the orthogonal cutting database in [8].
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Zhongtao Fu et al. / Procedia CIRP 31 (2015) 258 – 263
in Table 2 and Table 3. The mean friction angle is obtained from Appendix A, where the sliding friction coefficient is taken as 0.5 for this tool-workpiece pair and the material constant is G 450 . Fy
Fy
Fz
Fz
Fx (a)
Case 1: Half-immersion up-milling
Fx
(a) Case 1: Up-milling Fy
Fz Fy
Fx
Fz
Fx
(b) Case 2: Half-immersion down-milling Fig. 4. Comparison of milling forces between the analytical model and the published results [8]
3.2. Comparison with the results using simulation software To further verify the proposed analytical milling forces model, we compare with the simulation results using the software AdvantEdge FEM which adopts the adaptive mesh technology and ALE (Arbitrary Lagrangian-Eulerian) approach to simulate various cutting processes. In the end milling simulation, the workpiece material is AISI 304 stainless steel and its material constitutive model are described by Eq. (14) and the related parameters of material properties are listed in Table 2. And Table 3 gives the geometry of helical end milling cutter and the cutting conditions. Table 2. Material properties of AISI 304 [25] A(MPa)
B(MPa)
310 Density U (kg/m3)
C
J 0 /s
Tm(K)
Tr (K)
0.65 1.00 1.00 Thermal conductivity
1673
293
n
1000 0.07 Heat capacity
m
c p ( J / (kgK ) )
k p ( W / (mK) )
Taylor-Quinney Coefficient F
440
17.3
0.90
7900
Table 3. Tool geometry and cutting conditions Diameter D (mm)
Helix angle
Rake angle D r (deg)
Number of teeth N t
Edge radius
i0 (deg)
8
30
5
4
0.04
Spindle Case 1 2
speed
nr
(rpm) 2000 4000
Feed rate
re (mm)
ft
Radial depth of cut d r
Axial depth of cut a p
Type
(mm) 0.15 0.10
(mm) 4 4
(mm) 2.0 2.0
up down
The proposed model adopts the same parametric values
(b) Case 2: Down-milling Fig. 5. Comparison of milling forces between the proposed model and simulation results The simulation data of milling forces are compared with the proposed analytical model in one revolution of the cutter under the corresponding cutting conditions in Table 3. It can be seen from Fig. 5 that the waveforms of milling forces obtained in the present work, agrees well with the simulation results using the AdvantEdge FEM in terms of the amplitude, phase and pulsation pattern. The derivations of the average and peak milling forces are evaluated to be less than 15%. The possible reason for the derivations may be the frequent remeshing and the discretization of the cutting zone. Therefore, the simulation results clearly demonstrate the effectiveness of the proposed analytical model of milling forces. 4. Conclusion In this paper, an analytical modelling of milling forces for helical end milling has been developed based on a predictive machining theory. In the model, milling forces are predicted from the input data of workpiece material properties, tool geometry, cutting conditions and types of milling. Each cutting edge of the helical cutter is discretized into a series of infinitesimal elements and the cutting forces of each element are predicted analytically using the classical oblique cutting model. Compared with the published data obtained from the experimental data and mechanistic model, the proposed analytical model has a good agreement for milling forces. The
Zhongtao Fu et al. / Procedia CIRP 31 (2015) 258 – 263
simulation results using the software AdvantEdge FEM shows further that the proposed model is efficient and suitable. In addition, the proposed model has a priority to the mechanistic model which requires many experiments for given toolworkpiece pair and cutting conditions. However, the effect of tool eccentricity and deflection in milling process need to be considered in future work for more accurate prediction of milling forces. Acknowledgements This work was partially supported by the National Basic Research Program of China (2014CB046704), the National Natural Science Foundation of China (51375182), National Science and Technology Support Plan (2014BAB13B01). Appendix A: The mean friction angle E n The friction behavior in metal cutting was analyzed in cutting process by Ozlu et al. [27], who presented an analytical dual zone model of tool and chip interface by sticking and sliding zones. tan E n
Ws ª
1/ [ § § W · «1 [ ¨ 1 ¨ s ¸ ¨ P0 « © Ps P0 ¹ © ¬
·º ¸» ¸ ¹ ¼»
(A1)
where, W s is shear stress of the primary shear plane PH; cos 2 E n [ 1 W is the normal pressure at [ 2 sin[2(In E n D n )] s tool tip; [ is the exponential constant for pressure distribution P0
4
(usually taken as 3); P s is the sliding friction coefficient for the tool workpiece pair. References [1] Altintas Y., Manufacturing automation: metal cutting mechanics, machine tool vibrations, and CNC design. 2012: Cambridge university press. [2] Davim J. P., Machining of Complex Sculptured Surfaces. 2012: Springer. [3] Fu H., DeVor R.E., Kapoor S.G., A mechanistic model for the prediction of the force system in face milling operations. J Manuf Sci E-T ASME, 1984. 106(1): p.81-88. [4] Endres W.J., DeVor R.E., Kapoor S.G., A dual-mechanism approach to the prediction of machining forces, Part 1: model development. J Manuf Sci E-T ASME, 1995. 117(4): p.526-533. [5] Engin S., Altintas Y., Mechanics and dynamics of general milling cutters. Part I: helical end mills. Int J Mach Tool Manu, 2001. 41(15): p.2195-2212.
[6] Gradišek J., Kalveram M., Weinert K., Mechanistic identification of specific force coefficients for a general end mill. Int J Mach Tool Manu, 2004. 44(4): p.401-414. [7] Yang M., Park H., The prediction of cutting force in ball-end milling. Int J Mach Tool Manu, 1991. 31(1): p.45-54. [8] Budak E., Altintaş Y., Armarego E.J.A., Prediction of milling force coefficients from orthogonal cutting data. J Manuf Sci E-T ASME, 1996. 118(2): p.216-224. [9] Lee P., Altintaş Y., Prediction of ball-end milling forces from orthogonal cutting data. Int J Mach Tool Manu, 1996. 36(9): p.1059-1072. [10] Armarego E., Deshpande N.P., Force prediction models and CAD/CAM software for helical tooth milling processes. III. End-milling and slotting operations. Int J Prod Res, 1994. 32(7): p.1715-1738. [11] Bäker M., Finite element simulation of high-speed cutting forces. J Mater Process Tech, 2006. 176 (1-3): p.117-126. [12] Jin X., Altintas Y., Prediction of micro-milling forces with finite element method. J Mater Process Tech, 2012. 212(3): p.542-552. [13] Merchant M.E., Basic mechanics of the metal cutting process. J Appl Mech-T ASME, 1944. 11(A): p.168-175. [14] Armarego E., Deshpande N.P., Force prediction models and CAD/CAM software for helical tooth milling processes. II. Peripheral milling operations. Int J Prod Res, 1993. 31(10): p.2319-2336. [15] Oxley P.L.B., Mechanics of Machining: an Analytical Approach to Assessing Machinability, Ellis Horwood Publisher, 1989. [16] Young H., Mathew P., Oxley P.L.B., Predicting cutting forces in face milling. Int J Mach Tool Manu, 1994. 34(6): p.771-783. [17] Li H.Z., Li X.P., Zhang W.B., Modelling of cutting forces in helical end milling using a predictive machining theory. Int J Mech Sci, 2001. 43(8): p.1711-1730. [18] Moufki A., Dudzinski D., Le Coz G., An analytical thermomechanical modelling of peripheral milling process using a predictive machining theory. Adv Mater Res, 2011. 223: p.93-100. [19] Fontaine, M., Moufki A., Devillez A., Dudzinski D., Predictive force model for ball-end milling and experimental validation with a wavelike form machining test. Int J Mach Tool Manu, 2006. 46(3-4): p.367-380. [20] Li B., Wang X., Hu Y., Analytical prediction of cutting forces in orthogonal cutting using unequal division shear-zone model. Int J Adv Manuf Tech, 2011. 54(5-8): p.431-443. [21] Li B., Hu Y., Wang X., Li C., Li X., An analytical model of oblique cutting with application to end milling. Mach Sci Technol, 2011. 15(4): p.453-484. [22] Zvorykin K.A., Work and stress necessary for separation of metal chips. Proceedings of the Kharkov Technological Institute, Ukraine, 1893. [23] Waldorf D.J., DeVor R.E., Kapoor S.G., A slip-line field for ploughing during orthogonal cutting. J Manuf Sci E-T ASME, 1998. 120(4): p.693-699. [24] Manjunathaiah J., Analysis and a New Model for the Orthogonal Machining Process in the Presence of Edge-Radiused (Non-Sharp) Tools. Ph.D., University of Michigan, 1998. [25] Meyer H. W., Kleponis D. S., Modelling the high strain rate behaviour of titanium undergoing ballistic impact and penetration. Int J Impact Eng, 2001. 26: 509-521. [26] Mori L.F., Lee S., Xue Z.Y., Vaziri A., Queheillalt D.T., et al. Deformation and fracture modes of sandwich structures subjected to underwater impulsive loads. J Mech Mater Struct, 2007. 2(10): p.1981-2006. [27] Ozlu E., Budak E., Molinari A., Analytical and experimental investigation of rake contact and friction behavior in metal cutting. Int J Mach Tool Manu, 2009. 49(11): p.865-875.
263
ScienceDirect Procedia CIRP 31 (2015) 258 – 263
15th CIRP Conference on Modelling of Machining Operations
Analytical Modelling of Milling Forces for Helical End Milling Based on a Predictive Machining Theory Zhongtao Fua, Wenyu Yanga,*, Xuelin Wanga, Jürgen Leopoldb a
State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Technology, Huazhong University of Science and Technology, Wuhan,430074, China. b
Formerly Fraunhofer Institute for Machine Tools and Forming Technology, Chemnitz, 09661, Germany.
* Corresponding author. Tel.: +86-27-875-48180; fax: +86-27-875-59416. E-mail address: [email protected]
Abstract Milling forces play an important role in the milling process and are generally calculated by the mechanistic or numerical methods which are considered time-consuming and impractical for various cutting conditions and workpiece-tool pair. Therefore, this paper proposes an analytical method for modelling the milling forces in helical end milling process based on a predictive machining theory, which regards the workpiece material properties, tool geometry, cutting conditions and types of milling as the input data. In this method, each cutting edge is discretized into a series of infinitesimal elements along the cutter axis and the cutting action of each element is equivalent to the classical oblique cutting process. The three dimensional cutting force components applied on each element are predicted analytically using this predictive oblique cutting model with the effect of cutting edge radius. Finally, the proposed analytical model of milling forces is verified by the published results and the simulation values using the software AdvantEdge FEM.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2015 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The International Scientific Committee of the “15th Conference on Modelling of Machining Operations”. Peer-review under responsibility of the International Scientific Committee of the “15th Conference on Modelling of Machining Operations Keywords: Analytical modelling; Milling forces; Helical end milling; Predictive machining theory
1. Introduction The helical end milling has been used extensively to machine parts with sculptured surfaces, such as propellers, turbines. The cutting force is the basis for predicting the chatter-free cutting parameters, jig and fixture strength and process optimization [1], [2]. Therefore, the accuracy prediction of cutting force model is very crucial to improve the machining performance, the workpiece surface quality and process stability. To this end, considerable studies have been focused on the milling forces model with various cutting conditions and workpiece-tool pair. According to the available studies, there exist three main methods to model milling forces, i.e. mechanistic, numerical, and analytical ones [2]-[19]. The mechanistic methods [2]-[9], which are commonly used to milling forces model, regard the cutting force coefficients as the relation between process variables and milling forces. The coefficients are calibrated from a set of milling [2]-[6] or orthogonal experiments [7]-[9] for a given tool-workpiece pair. However, this approach is
only valid for a certain workpiece-tool material pair and the calibration done by numerous experiments is impractical due to high cost and hardware setup complexity. Currently, the numerical methods [10], [12] are focused on the study of the interaction between the tool and the workpiece and the complex thermo-mechanical phenomenon. An arbitrary or updated Lagrangian formulation is employed in the finite element technology to simulate the machining process and predict the milling forces. Nevertheless, this approach is time-consuming and the commercial simulation software is too costly to be available. For the analytical methods, they try to establish mathematical relations between the milling forces and several mechanical aspects like friction, geometry and material behavior. For instance, Merchant [12] and Armarego [14] are two of the main contributors to this kind of models by the development of orthogonal and oblique cutting mechanics. Oxley [14] developed a predictive machining theory that allows for the high strain rate, high temperature and thermal properties of the work material. Based on Oxley’s predictive
2212-8271 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the International Scientific Committee of the “15th Conference on Modelling of Machining Operations doi:10.1016/j.procir.2015.03.013
Zhongtao Fu et al. / Procedia CIRP 31 (2015) 258 – 263
259
260
Zhongtao Fu et al. / Procedia CIRP 31 (2015) 258 – 263
261
Zhongtao Fu et al. / Procedia CIRP 31 (2015) 258 – 263 Pn
Finally, the cutting forces contributed by all cutting edges are summed to obtain the total instantaneous forces on the end-milling cutter as follows˖
Dn Chip
t2
Fx (I )
Vc
In
re
-
t1
R
+
K
Workpiece
In Fig. 3, re is the edge radius, U 0 is the prow angle which inclined to the uncut workpiece surface (10 deg in present analysis), R is the radius of the circular fan filed centered at P, the fan field angles J 0 ,- and K are found from the geometric and friction relationship. the plough forces can be determined as follows˖ ª(1 2- 2J 0 sin(2K ))sin(I J 0 K ) º dPc W s « » lPC w ¬cos(2K ) cos(In J 0 K ) ¼ (19) (1 2 2 J sin(2 K )) cos( I J K ) ª º 0 0 dPf W s « l w » PC ¬cos(2K )sin(In J 0 K ) ¼ where W s is shear flow stress of the primary shear plane PH;
the width of cut w is equal to dz / cos Os ; K 0.5arccos P , P is the friction coefficient between tool and workpiece; J 0 K I arcsin( 2 sin U0 sinK ) ; - S / 4 U0 In ; lPC R sinK , ª 2 R sin U0 º S D 2 sin K « re tan( n ) » 2 > R sin U0 @ ; 4 2 tan(S / 2 D n ) ¼» ¬«
ª cos Os « 0 « ¬« sin Os
cos Os sin In º » dFs ½ cos In » ® ¾ dN s ¿ » sin Os sin In ¼ ¯
0º dPc ½ 1 »» ® ¾ dPf 0 ¼» ¯ ¿
(20)
As shown in Fig.1 (a), the components dFt , j , dFr , j , dFa , j in local coordinates are transformed to the global coordinates using the relations: dFx , j ½ ° ° ® dFy , j ¾ ° dF ° ¯ z, j ¿
ª cos I j « sin I j « ¬« 0
sin I j cos I j 0
0 º dFt , j ½ ° ° 0 »» ® dFr , j ¾ ° ° 1¼» ¯ dFa , j ¿
(21)
The differential cutting forces are integrated along the in-cut portion of the j-th cutting edge and the total cutting forces produced by this cutting edge can be obtained as: z (I ) (22) Fq [I j ( z )] ³ dFq [I j ( z )]dz, q x, y, z 2
j 1
Nt
¦F
z, j
[I ( z )]
(23)
j 1
m 0.8, J 0 1 1,, Tm 1933K, Tr 293K and the material density U is 4430kg / m3 , the specific heat c p is 542J / (kgK) , the thermal conductivity k p is 6.7W / (mK) and the Taylor-Quinney coefficient F 0.85 . The carbide milling cutter used in up-milling has fourflute cutting edge, 30deg helix angle, 12deg rake angle and 19.05mm diameter. And the cutter used in down-milling has the same tool geometry as in up-milling except that the rake angle is 00. The cutting conditions are listed in Table 1. Table 1. Cutting conditions used in [8] Case
Due to the edge radius, the effective normal rake angle will vary depending on the edge radius and the depth of cut. The detailed calculation can be referenced to [24]. Therefore, for the infinitesimal element of the j-th cutting edge, the three differential cutting force components dFt , j , dFr , j , dFa , j (tangential, radial, axial), applied to point P in Fig.2 (a), are evaluated from the following matrix form: ª cos Os cos In cosK sh sin Os sin K sh « sin In cosK sh « «¬ sin Os cos In cosK sh cos Os sin Ksh
[I ( z )], Fz (I )
A computer program in MATLAB 7.9 is developed to implement the feasibility of the proposed analytical model of milling forces for helical end milling. The published results [8] obtained from the mechanistic model and experimental data are selected for comparison, in which two tests, i. e. halfimmersion up-milling and half-immersion down-milling, are carried out. The workpiece material is Ti6Al4V whose material parameters in Eq. (14) are given by [24]: A=862.5MPa, B=331.2 Mpa, C 0.0128, n 0.34,
2
dFt , j ½ ° ° ® dFr , j ¾ ° dF ° ¯ a, j ¿
y, j
3.1. Comparison with the published results
C
Fig. 3. Waldrof’s slip-line field for plough forces
R
Nt
¦F
[I ( z )], Fy (I )
3. Model validation
P
J0
V
x, j
j 1
Tool
H
U0
Nt
¦F
j
z1 (I j )
where z1 (I j ) and z2 (I j ) are the lower and upper axial engagement limits of the in-cut portion.
1 2
Cutting velocity V (m/min) 30 30
Feed rate
ft (mm/tooth) 0.050 0.0127
Radial depth of cut d r (mm)
Axial depth of cut a p (mm)
type
9.525 9.525
5.08 5.08
up down
Likewise, the proposed analytical model utilizes the same constants of the workpiece material, cutting conditions, tool geometry. The mean friction angle E n is obtained from Appendix A, where the sliding friction coefficient P s is taken as 0.6 for this tool-workpiece pair. From the orthogonal cutting database [8], the material constant is G 450 and the plough force coefficients are Kte 24 N/mm; Kre 43 N/mm. Fig. 4 shows the comparison of milling forces obtained by the proposed analytical model and the published results [8]. It can be observed that the milling forces calculated in the present work, are in good agreement with the published results obtained by the mechanistic model and experimental data. There exists a small deviation on the amplitude of the maximum milling forces between the proposed analytical model and the published results. This deviation might be attributed to the cutter deflection which is not considered in this proposed model. Furthermore, the mean shear flow stress W s predicted in the primary shear zone is about 592MPa and very close to the value 613MPa of the orthogonal cutting database in [8].
262
Zhongtao Fu et al. / Procedia CIRP 31 (2015) 258 – 263
in Table 2 and Table 3. The mean friction angle is obtained from Appendix A, where the sliding friction coefficient is taken as 0.5 for this tool-workpiece pair and the material constant is G 450 . Fy
Fy
Fz
Fz
Fx (a)
Case 1: Half-immersion up-milling
Fx
(a) Case 1: Up-milling Fy
Fz Fy
Fx
Fz
Fx
(b) Case 2: Half-immersion down-milling Fig. 4. Comparison of milling forces between the analytical model and the published results [8]
3.2. Comparison with the results using simulation software To further verify the proposed analytical milling forces model, we compare with the simulation results using the software AdvantEdge FEM which adopts the adaptive mesh technology and ALE (Arbitrary Lagrangian-Eulerian) approach to simulate various cutting processes. In the end milling simulation, the workpiece material is AISI 304 stainless steel and its material constitutive model are described by Eq. (14) and the related parameters of material properties are listed in Table 2. And Table 3 gives the geometry of helical end milling cutter and the cutting conditions. Table 2. Material properties of AISI 304 [25] A(MPa)
B(MPa)
310 Density U (kg/m3)
C
J 0 /s
Tm(K)
Tr (K)
0.65 1.00 1.00 Thermal conductivity
1673
293
n
1000 0.07 Heat capacity
m
c p ( J / (kgK ) )
k p ( W / (mK) )
Taylor-Quinney Coefficient F
440
17.3
0.90
7900
Table 3. Tool geometry and cutting conditions Diameter D (mm)
Helix angle
Rake angle D r (deg)
Number of teeth N t
Edge radius
i0 (deg)
8
30
5
4
0.04
Spindle Case 1 2
speed
nr
(rpm) 2000 4000
Feed rate
re (mm)
ft
Radial depth of cut d r
Axial depth of cut a p
Type
(mm) 0.15 0.10
(mm) 4 4
(mm) 2.0 2.0
up down
The proposed model adopts the same parametric values
(b) Case 2: Down-milling Fig. 5. Comparison of milling forces between the proposed model and simulation results The simulation data of milling forces are compared with the proposed analytical model in one revolution of the cutter under the corresponding cutting conditions in Table 3. It can be seen from Fig. 5 that the waveforms of milling forces obtained in the present work, agrees well with the simulation results using the AdvantEdge FEM in terms of the amplitude, phase and pulsation pattern. The derivations of the average and peak milling forces are evaluated to be less than 15%. The possible reason for the derivations may be the frequent remeshing and the discretization of the cutting zone. Therefore, the simulation results clearly demonstrate the effectiveness of the proposed analytical model of milling forces. 4. Conclusion In this paper, an analytical modelling of milling forces for helical end milling has been developed based on a predictive machining theory. In the model, milling forces are predicted from the input data of workpiece material properties, tool geometry, cutting conditions and types of milling. Each cutting edge of the helical cutter is discretized into a series of infinitesimal elements and the cutting forces of each element are predicted analytically using the classical oblique cutting model. Compared with the published data obtained from the experimental data and mechanistic model, the proposed analytical model has a good agreement for milling forces. The
Zhongtao Fu et al. / Procedia CIRP 31 (2015) 258 – 263
simulation results using the software AdvantEdge FEM shows further that the proposed model is efficient and suitable. In addition, the proposed model has a priority to the mechanistic model which requires many experiments for given toolworkpiece pair and cutting conditions. However, the effect of tool eccentricity and deflection in milling process need to be considered in future work for more accurate prediction of milling forces. Acknowledgements This work was partially supported by the National Basic Research Program of China (2014CB046704), the National Natural Science Foundation of China (51375182), National Science and Technology Support Plan (2014BAB13B01). Appendix A: The mean friction angle E n The friction behavior in metal cutting was analyzed in cutting process by Ozlu et al. [27], who presented an analytical dual zone model of tool and chip interface by sticking and sliding zones. tan E n
Ws ª
1/ [ § § W · «1 [ ¨ 1 ¨ s ¸ ¨ P0 « © Ps P0 ¹ © ¬
·º ¸» ¸ ¹ ¼»
(A1)
where, W s is shear stress of the primary shear plane PH; cos 2 E n [ 1 W is the normal pressure at [ 2 sin[2(In E n D n )] s tool tip; [ is the exponential constant for pressure distribution P0
4
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