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Cutting force formulation of taper end-mills using differential geometry
Precision Engineering 23 (1999) 196 –203
Cutting force formulation of taper end-mills using differential geometry T. Huanga, D.J. Whitehouseb,* a
Department of Mechanical Engineering, Tianjin University, Tianjin 300072, People’s Republic of China b Department of Engineering, University of Warwick, Coventry CV4 7AL, United Kingdom Received 18 June 1998; received in revised form 16 December 1998; accepted 29 December 1998.
Abstract In this paper, a mechanistic model to formulate the nonlinear three-dimensional (3-D) cutting forces of taper end-mills by means of differential geometry is presented. The relationship between the tool geometry and the cutting force directions is analyzed. A cutting coefficient estimation procedure is developed. The model is verified by milling carbon steel specimens. For a set of given cutting conditions, the results show close agreement between the measured cutting forces and the model predictions. © 1999 Elsevier Science Inc. All rights reserved. Keywords: Milling force prediction; Taper end-mills; Process parameter identification; Differential geometry
Nomenclature A Fn, Ff F K n, K f, c K 1, K 2, K 3 R t, n, b tc
␣, , ␥n , 1, 2
rotation matrix pressure and friction cutting force vectors on the rake face total cutting force vector pressure coefficient, friction coefficient, chip flow angle on the rake face cutting force coefficients cutter radius at the tool tip unit vectors defining the curvilinear coordinate system on the rake face chip thickness unit vector defining the chip movement direction half-apex angle, helix angle, normal rake angle tool rotation angle, tool position angle integration limits
1. Introduction It has long been acknowledged that cutting force is one of the most sensitive output variables in the milling process. * Corresponding author. Tel.: ⫹44-1203-523154; fax: ⫹44-1203-471457 or ⫹44-1203-473-558. E-mail address: [email protected] (D.J. Whitehouse)
Furthermore, this force can be affected by any variation in the tool/workpiece geometry, the materials used, the cutting conditions and the dynamic characteristics of machine–toolfixture system. Therefore, a fundamental understanding of the cutting force system plays an important role in accurate analysis and prediction of the capability of the machining process. The analysis and subsequent prediction should make it possible to reduce greatly the time required for NC programming verification in test cuts and to enable the adaptive control strategies to be reliably simulated before the real implementation is attempted [1]. Considerable research effort has been devoted to formulating mechanismic models for cutting force prediction in the milling process [1– 4]. Moreover, modeling methodologies together with procedures for estimating the process parameters have been reported by Bayoumi et al. [3,5] and Yucesan and Altintas [6] regarding the three-dimensional (3-D) cutting force prediction of helical end-mills having arbitrary profiles in general and cylindrical configurations in particular. In Bayoumi’s model, it was assumed that the cutting force components could be linearly represented in proportion to the frontal area of the chip being removed and that a Taylor’s series expansion technique could be used for the process parameter approximations in terms of tool rotation angle. Similarly, to cope with the nonlinear problem encountered in parameter identification, the empirical equation employed in the Yucesan’s model was that the specific cutting force coefficients could be expressed as an exponential decaying function of the averaged uncut chip thickness.
0141-6359/99/$ – see front matter © 1999 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 1 - 6 3 5 9 ( 9 9 ) 0 0 0 1 0 - 0
T. Huang, D.J. Whitehouse / Precision Engineering 23 (1999) 196 –203
The main difference between these two models is the question of how to define and determine the unknown cutting coefficients; an old topic that may date back to the early work of Sabberval [7,8]. Taper end-mills have found wide application in the die fabrication for automobile and aerospace components. Depending upon the geometry of the cutting edge, there are two main types of taper end-mills: the straight edge type such as reamer/drill reamer; and the constant helix angle type with a flat/ball nosed end for die sinking. Although the cutting force prediction models for the helical end-mills with cylindrical configuration have been well developed, the corresponding models for the taper end-mills have been less investigated. The cutting force model using end-mills of this kind was studied by Ramaraj and Eleftheriou [9]. Based upon a basic shear plane model in oblique cutting theory, the shear and normal forces were formulated in a local Cartesian system. From this, a series of complicated coordinate transformations was implemented to obtain the total force in conventional Cartesian coordinates. However, in addition to using a rather ambiguous way to define the tool geometry, this method suffers from the problem that the representation of the primary deformation zone by a shear plane may be unrealistic, and the mean angle of friction may be insufficient to define completely the frictional conditions on the tool face, as noted by Bayoumi et al. [5,6]. Consequently, it is necessary to have a set of experimentally determined strain–stress relationships in the deformed zone. This is, however, very difficult, if not impossible. In this paper, an instantaneous rigid force model of end milling operation using taper end-mills with constant helix angle is developed. The formulation procedure involves the representation of cutting edge by means of differential geometry, the definition of the curvilinear coordinate system, the development of the mechanistic cutting force components along the directions of normal pressure and chip movement, the evaluation of the total instantaneous cutting force, and the identification of the process parameters. The model is experimentally verified by cutting middle carbon steel.
Consider a taper end-mill as shown in Fig. 1. The parametric equation of the revolute surface of the cutter in the local cutting edge fixed Cartesian coordinate O ⫺ xyz has the form [Eq. (1)] (1)
where ␣ , R, and denote the half apex angle, the cutter radius, and the position angle measured at the tool tip, respectively. According to differential geometry [10], the fundamental quadratic form of any curve upon a taper surface is given by [Eq. (2)] I ⫽ sec2␣ dz 2 ⫹ 共R ⫹ z tan ␣ 兲 2d 2
Fig. 1. Tool geometry and cutting force components.
Let  be the helix angle defined by the angle between the positive directions of the curve of a cutting edge and the longitude at point P( x, y, z), then [Eq. (3)] results cos2  ⫽
sec2␣ dz 2 共R ⫹ z tan ␣ 兲 2d 2 ⫹ sec2 ␣ dz 2
(3)
or [Eq. (4)] sec2 ␣ dz 2 ⫽ 共R ⫹ z tan ␣ 兲 2 cot2 d 2
(4)
For the cutter having a right-handed helix angle, taking the root square of Eq. (4) leads to [Eq. (5)] 1 dz ⫽ cos ␣ cot  d R ⫹ z tan ␣
(5)
Integrating Eq. (5) with respect to and remembering that z兩 ⫽0 ⫽ 0 gives [Eq. (6)]
冉
1n 1 ⫹ tan ␣
冊
z ⫽ sin ␣ cot  R
(6)
Exponentiating both sides of Eq. (6) and substituting into Eq. (1) leads to the position vector r expressed in terms of the tool position angle [Eq. (7)]
2. Tool geometry
r ⫽ 关共R ⫹ z tan ␣ 兲cos 共R ⫹ z tan ␣)sin z兴
197
(2)
r ⫽ r共 兲 ⫽ Re sin␣cot[cos i ⫹ sin j ⫹ cot ␣ 共1 ⫺ e ⫺sin␣cot)k兴
(7)
This indicates that the generating curve of the cutting edge for taper end-mills with constant helix angle is a 3-D spiral.
3. Curvilinear coordinate system It has been found that the differential cutting force acting on the rake surface at point P can be divided into two orthogonal components in O ⫺ xyz, as shown in Fig. 1 [3].
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One component is referred to as the normal pressure force, which is perpendicular to the rake surface. The other component is called the friction force, which is located on the rake surface, along the direction of chip movement. The procedure to determine the direction cosines of these cutting force components is implemented by manipulating a set of unit vectors defining the curvilinear coordinate of the cutting edge. As shown in Fig. 1, the orthogonal curvilinear coordinate system along the cutting edge is defined by a unit normal vector n, which is perpendicular to the rake surface, a unit vector t tangent to the helix flute, and a unit vector b that is orthogonal to both t and n. The procedure to generate these vectors is described as follows. First, differentiating Eq. (7) with respect to gives the tangent unit vector t of the cutting edge [Eq. (8)]. t⫽
r ⫽ 共sin ␣ cos  cos ⫺ sin  sin )i 兩r 兩
(13)
Obviously, the unit vectors n and defining the differential cutting force components for the taper end-mills are functions of the tool geometry parameters, including halfapex angle ␣, helix angle , rake angles ␥ n , position angle , and chip flow angle c . It has been noticed, however, that despite the chip flow angle c being able to take the value of the inclination angle to a first degree of approximation, the large friction at the tool– chip interface may cause a strong deviation of the former from the latter. In other words, the chip flow angle is heavily affected by such factors as the adhesive property between the tool and the workpiece materials, lubrication conditions, and other relevant characteristics of the process; therefore, this angle must be determined by experimentation. 4. Cutting force formulation
⫹ 共sin ␣ cos  sin ⫹ sin  cos )j ⫹ cos ␣ cos k (8) The primary normal vector a and the auxiliary normal vector c of the cutting edge as shown in Fig. 1 can be derived by [Eqs. (9, 10)]., a⫽
⫽ b cos c ⫹ t sin c
r1 ⫺ r ⫽ ⫺cos ␣ cos i ⫺ cos ␣ sin j ⫹ sin ␣k 兩r 1 ⫺ r兩
(9)
c ⫽ t ⫻ a ⫽ 共sin ␣ sin  cos ⫹ cos  sin )t ⫹ 共sin ␣ sin  sin ⫺ cos  cos )j ⫹ cos ␣ sin k (10) where (r1 ⫺ r) ⫻ t ⫽ 0, r1 ⫽ {0 0 Re sin␣cot[tan ␣ ⫹ cot ␣(1 ⫺ e ⫺sin␣cot)]}. Therefore t, a, and c constitute a Frenet frame at point P. Instead of using the rake angle defined in the x–y plane, as addressed by Bayoumi and Yucesan [3,5], let ␥ n be the normal rake angle defined as an angle oriented from the primary normal vector a in the normal plane perpendicular to the rake surface. The reason for this is that the normal rake angle remains constant along the cutting edge and is the geometric parameter used as the cutter is manufactured. Consequently, the unit vectors b and n can be generated [Eqs. (11, 12)]. b ⫽ cos ␥ na ⫺ sin ␥ nc
(11)
n ⫽ sin ␥ na ⫹ cos ␥ nc
(12)
According to the mechanismic model in the oblique cutting characterized by the nonzero inclination angle, the direction of chip movement as shown in Fig. 1 must locate on the rake surface expanded by the unit vectors b and t. Let c be the chip flow angle defined as an angle oriented from unit vector b, then the unit vector defining the direction of the differential friction force on the rake surface can be given as
According to the kinematics in milling, the uncut chip thickness with zero runout can be approximated as: t c共 , 兲 ⫽ f t sin共 ⫹ 兲
(14)
where ft is the feed rate per tooth, and is the tool rotation angle in down milling. As indicated previously, the differential cutting force vector acting at a certain point on the cutting edge can be divided into two orthogonal components; that is, the pressure force in the opposite direction of the normal of rake face and the friction force in the direction of chip movement. These differential cutting force components acting on an infinitesimal element can be respectively written as: dF n共 , 兲 ⫽ ⫺A共 兲 K n共 兲n共 兲⌫共 , 兲
dz d d
dF f共 , 兲 ⫽ A共 兲 K f共 兲 K n共 兲 共 兲⌫共 , 兲
dz d d
(15)
In the above expressions, dz/d ⫽ R cos ␣ cot  e sin␣cot, K n and K f are the unknown pressure and friction coefficients associated with the cutting force components acting on the rake surface and are assumed to be the function of tool position angle ; ⌫( , ) ⫽ f[t c ( , )] represents the function introduced to describe the contribution of the uncut chip thickness to the cutting forces, and A( ) denotes the rotation matrix from the local to the global Cartesian coordinates. Taking the summation and integrating along the cutting edge leads to the resultant cutting force vector in the global Cartesian system [Eq. (16)]. F共 兲 ⫽ A共 兲
冕
2
K n共 兲兵⫺n共 兲 ⫹ K f共 兲关b共 兲cos c共 兲
1
⫹ t共 兲sin c共 兲兴其⌫共 , 兲 Define [Eq. (17)]
dz d d
(16)
T. Huang, D.J. Whitehouse / Precision Engineering 23 (1999) 196 –203
再
K 1共 兲 ⫽ K n共 兲 K 2共 兲 ⫽ K n共 兲 K f共 兲cos c共 兲 K 3共 兲 ⫽ K n共 兲 K f共 兲sin c共 兲
(17)
as the cutting force coefficients, then Eq. (16) can be rewritten as follows [Eq. (18)]: F共 兲 ⫽ A共 兲
冕
2
关⫺K 1共 兲n共 兲 ⫹ K 2共 兲b共 兲
1
⫹ K 3共 兲t共 兲兴⌫共 , 兲
dz d d
(18)
In the above expressions, 1 and 2 are the integration limits representing the angular locations of the starting and end points of the cutting engagement of the flute and can be determined by the method presented in the Appendix. After taking the integral, K 1 , K 2 , and K 3 can be determined by means of the least-squares estimation using the cutting force measurements. The procedure to identify these coefficients is described in the following section. Thus, the unknown pressure and friction coefficients K n , K f as well as the chip flow angle c can be determined directly by the following inverse transformations [Eq. (19)]
冦
K n共 兲 ⫽ K 1共 兲
冋 册
c共 兲 ⫽ arctan
K 3共 兲 K 2共 兲
K 2共 兲 sec c共 兲 K f共 兲 ⫽ K 1共 兲
(19)
Strictly speaking, it is impossible to obtain the closed form expression because of the unknown parameters and complexity of the integrand, even for the cutters with simple geometrical profile. As a result, a numerical integration procedure must be performed to calculate the total cutting force. It should be pointed out that the above formulations are developed for a single fluted taper end-mill. The derivations can be slightly extended for the multiple cutter by summing the force acting on each flute at a specific tool rotation angle.
199
functional capabilities described above. These approaches generally are based upon the widely accepted empirical experience that cutting force coefficients in milling are influenced by changes in the chip thickness, the cutter geometry, tool/workpiece material properties, lubrication conditions, and so forth. The most commonly used form for parametric relationships is an exponential decaying function of the chip thickness [7,8]. This results in nonlinear identification procedures in cutting force coefficient estimation. Three effective approaches, among others, have been developed by Altintas [2], Bayoumi [3,5], and Yucesan [6]. In Altintas’ [2], it is proposed that the chip thickness can be replaced by its average value within the range of engagement. Therefore, the process parameters (the tangential and radial cutting coefficients expressed as the exponential decaying function of the average chip thickness) were able to be moved before the integral in both parameter identification and cutting force evaluation (in effect a use of the mean value theorem). However, use of the average chip thickness may cause inaccurate cutting force prediction for smaller actual chip thickness [3]. In the further improvements made by Bayoumi [3,5], the process parameters (the pressure and friction coefficients K n , K f and the chip flow angle c ), which are directly expressed as a function of both the rotation angle and the position angle, are replaced by the effective moving average parameters that are functions of the tool rotation angle. With a similar starting point to Bayoumi’s method, the first step in the approach proposed by Yucesan and Altintas is to transform K n , K f , and c into the cutting force coefficients K 1 , K 2 , and K 3 as given in Eq. (17). The second step is to approximate K 1 , K 2 , and K 3 by a series expansion in such a way that the coefficients of the polynomials can be easily estimated by linear regression. This is followed by fitting K 1 , K 2 , and K 3 as exponential functions of chip thickness that change with the tool position angle and the rotation angle. Finally, the chip thickness is averaged within the range of the flute engagement so that the average cutting coefficients are obtained as a function of the tool rotation angle. Therefore, the average cutting coefficients can be brought before the integral operator in the cutting force prediction. Hereafter, we propose another method to identify the
5. Parameter Identification Because of the complexity of the cutting process, analytical prediction of the process parameters remains to be well developed. Therefore, it is common practice to determine these unknown parameters by experiment. In principle, a good identification procedure should be beneficial to: predicting the instantaneous cutting force with sufficient accuracy; investigating the influence of tool geometry and cutting conditions, such as feed rate, axial and radial depths of cut on the cutting force; implementing effective and reliable algorithms for real-time simulation; and building up a cutting database with ease for the later use. Several approaches have been developed to achieve the
Table 1 Cutting parameters No.
Feed rate (mm/tooth)
Depth of cut (mm)
1 2 3 4 5 6 7 8 9
0.01992 0.03983 0.05975 0.01992 0.03983 0.05975 0.01992 0.03983 0.05975
10 10 10 15 15 15 20 20 20
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Fig. 2. The layout of instrumentation in the cutting test.
process parameters, which involves solving a set of bilinear equations. Suppose that a group of discrete cutting force data F1, F2, F3, . . . , Fn has been measured through cutting test under the specific cutting condition, then the following error functional in a quadratic form can be formulated as follows (Eq. 20).
冘 储F ⫺ F共 兲储 n
J⫽
j
j
2 2
(20)
j⫽1
According to Sabberval’s [7,8] observations, the instantaneous cutting force is closely related to the uncut chip thickness variation, which is a function of both the tool position angle and tool rotation angle. On the one hand, consider that the cutting force coefficients are formulated in such a way that they are only a function of tool position angle K j can, therefore, be approximated by a power series in terms of the tool position angle [Eq. (21)].
冘 ␣ 共 j ⫽ 1, 2, 3兲 n
K j共 兲 ⫽
k
jk
(21)
k⫽0
On the other hand, account for the effect of the uncut chip thickness on cutting force, a Chebyshev polynomial [Eq. (22)] can be used to approximate the unknown function ⌫(, )
冘 h cos兵共l ⫹ 1兲arccos关t 共, 兲兴其 m
⌫共 , 兲 ⫽
l
c
(22)
l⫽0
where h 0 ⫽ 1 for the independence of the coefficients. The least-squares estimation can be used to implement parameter identification. Thus, taking the derivative of Eq. (20) with respect to the unknown coefficients will lead to a set of bilinear equations [Eq. (23)]
再 DyBx ⫽⫽ CE where x ⫽ 共a 11 a 12, · · · , a 1n, · · · , a 31 a 32, · · · , a 3n兲 T, y ⫽ 共h 1 h 2, · · · , h m兲 T
(23)
The Newton–Raphson method is available to solve this problem. Note here that for a large class of milling cutters, the rake angles are the same. Therefore, it would be helpful at the same time to use a group of cutting force measurements obtained by changing such cutting parameters as feed rate and depth of cut. This usually allows accuracy of the cutting coefficients; hence, the cutting force predictions to be improved.
Fig. 3. The pressure and friction coefficients K n , K f , and the chip flow angle c versus the axial depth of cut and feed rate 1.
T. Huang, D.J. Whitehouse / Precision Engineering 23 (1999) 196 –203
201
Fig. 4. Comparison between the measured and predicted cutting forces under different cutting conditions.
6. Experimental verification A group cutting test is used to verify the validity of the model presented in this paper. A Vcenter-65 machining center with FANUCOM controller is used in the experiment. A KISTLER three-component dynamometer, type 9525A is used to measure the cutting forces. A photoelectric sensor is used to pick up the rotation speed of the spindle and to measure the phase of the cutting force. The mill is a three fluted taper end-mill made of HSS, 16-mm diameter at the tool tip, normal rake angle of 13°, apex half angle of 4°, and helix angle of 30°. The material of the workpiece is 45 middle carbon steel. The tests are carried out in a semifinished down milling on a flat surface with spindle speed 318 rpm, radial depth of cut 6 mm, and an oil coolant is used. The changeable cutting parameters as given in Table 1 are the feed rate and axial depth of cut. The feed rate is chosen
to cover a range of chip thickness. The axial depth of cut is varied to verify the validity of cutting parametric relationships. No variation in the radial depth of cut and cutting speed are considered, because it has been observed that they do not have significant effects on the process parameters in end milling. In the tests, the cutting forces are first recorded by a SONY 8-channel high-fidelity tape recorder, and then are sampled at a rate of 400 samples per second through an AST 386/25 PC with KH8345A data acquisition board. The layout of the test is shown in Fig. 2. Note that the normal rake angle and the helix angle are constants along the cutting edge, and the cutting speed changes linearly along the axial depth of cut. Because the cutting speed has little bearing upon the magnitude of cutting force, it is reasonable to assume that the cutting coefficients K 1 , K 2 , and K 3 are constant. Meanwhile, a fourth order Chebyshev polynomial is used to approximate the
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unknown function ⌫(, ). Fig. 3 shows the variations of the identified pressure and friction coefficients K n , K f , and the chip flow angle c versus the depth of cut and feed rate. It can be observed from these figures that utilizing the identification model presented in this paper allows the above parameters to have little correlation with the changes in the cutting conditions. The Chebyshev polynomial corresponding to the first cutting condition is identified and given by ⌫( , ) ⫽ t c ( , ) ⫹ 0.02296 cos{2 arccos[t c ( , )]} ⫹ 0.07043 cos{3 arccos[t c ( , )]} ⫹ 0.02939 cos{4 arccos[t c ( , )]}. Fig. 4 (a– d) shows the measured and predicted cutting forces under different cutting conditions. The data in Fig. 4 (a) are used to identify the cutting coefficients, which are then employed to predict the cutting forces under the next three cutting conditions. It can be observed from these figures that close agreement between the measured cutting forces and the model predictions is obtained.
7. Conclusions A mechanismic model to formulate the nonlinear, 3-D cutting forces of taper end-mills is presented in this article. For the given tool geometry and workpiece material, the experimental results show that the correlation between the feed rate and cutting coefficients is very small; therefore, little experimental effort has to be made to build up a cutting force library. In addition, the formulation technique proposed is so general that it can be expanded with little effort to deal with cutting force predictions using other tool geometries.
Fig. A. Cutting geometry for die-sinking with taper end-mills.
2 ⫽ cos⫺1 Acknowledgments This research work is supported by the Natural Science Foundations of China and Tianjin Municipality.
冉
R ⫹ h tan ␣ ⫺ t R ⫹ h tan ␣
Application of taper end-mills to die-sinking may include two possible cutting formats: (1) profile milling of plenary surfaces with a fixed axial depth of cut for semifinish cutting [see Fig. A(a)]; and (2) parallel milling of the same profile with a fixed axial depth of cut for finish cutting [see Fig. A(b)]. Fig. A.3 shows the geometric situation for semifinish milling. In the figure 1, 2 are the immersion angles at the tool tip with a radial depth of cut t and at the other extreme end with a radial depth of cut t ⫹ h tan ␣, where h is the axial depth of cut.
1 ⫽ cos⫺1
冉 冊
冉
R⫺t R⫺t , 2 ⫽ cos⫺1 R R ⫹ h tan ␣
冊
(A.1)
Whereas, for the finish milling, 2 (See Fig. A.4) is given by
(A.2)
The maximum tool position angle associated with the given axial engagement is given by
冋 冉
max ⫽ csc ␣ tan  1n Appendix
冊
h tan ␣ ⫹1 R
冊册
(A.3)
Thus, the integration limits for the both cutting geometries, as shown in Fig. A(e), (f) can be determined by means of the following algorithm. Lower limits SELECT CASE ; CASE 2 ⫺ 1 ⱕ ⬍ 2 (cases A and B) 1 ⫽ 0 Case 2 ⫺ (2 ⫹ max) ⬍ ⬍ 2 ⫺ 1 (cases C and D) 1 ⫽ 2 ⫺ ⫺ cos⫺1[(1 ⫺ t/R)e ⫺ 1sin␣cot] CASE 0 ⱕ ⱕ 2 ⫺ (2 ⫹ max) (case E) disengagement END SELECT Upper limit SELECT CASE
T. Huang, D.J. Whitehouse / Precision Engineering 23 (1999) 196 –203
CASE 2 ⫺ max ⱕ ⬍ 2 (cases A and C) 2 ⫽ 2 ⫺ CASE 2 ⫺ (2 ⫹ max) ⬍ ⬍ 2 ⫺ max (cases B and D) 2 ⫽ max CASE 0 ⬍ ⬍ 2 ⫺ (2 ⫹ max) (case E) disengagement END SELECT
References [1] Smith S, Tlusty J. An overview of modeling and simulation of the milling process. J Eng Ind 1991;113(2):169 –75. [2] Altintas Y, Spence A. End-milling force algorithm for CAD system. Ann CIRP 1991;40(1):31– 4.
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[3] Bayoumi AE, Yucesan G, Kendall LA. Analytic mechanistic cutting force model for milling operations: A theory and methodology. J Eng Ind 1994a;116(3):324 –330. [4] Wang J-JJ, Liang SY, Book WJ. Convolution analysis of milling force pulsation. J Eng Ind 1994;116(1):17–25. [5] Bayoumi AE, Yucesan G, Kendall LA. Analytic mechanistic cutting force model for milling operations: A case study of helical milling operation. J Eng Ind 1994b;116(3):331–39. [6] Yucesan G, Altintas Y. Improved modeling of cutting force coefficients in peripheral milling. Int J Machine Tools Manufact 1994; 34(4):473– 487. [7] Sabberval AJP. Chip section and cutting force during milling operation. Ann CIRP 1961;10:197–203. [8] Sabberval, AJP. Cutting force in down milling. Int J Machine Tools Design Res 1962;2(1):27– 41. [9] Ramaraj TC, Eleftheriou E. Analysis of the mechanics of machining with tapered end-milling cutters. J Eng Ind 1994;116(3):398 – 404. [10] Eisenhart LP. A treatise on the differential geometry of curves and surfaces. New York: Dover, 1960.
Cutting force formulation of taper end-mills using differential geometry T. Huanga, D.J. Whitehouseb,* a
Department of Mechanical Engineering, Tianjin University, Tianjin 300072, People’s Republic of China b Department of Engineering, University of Warwick, Coventry CV4 7AL, United Kingdom Received 18 June 1998; received in revised form 16 December 1998; accepted 29 December 1998.
Abstract In this paper, a mechanistic model to formulate the nonlinear three-dimensional (3-D) cutting forces of taper end-mills by means of differential geometry is presented. The relationship between the tool geometry and the cutting force directions is analyzed. A cutting coefficient estimation procedure is developed. The model is verified by milling carbon steel specimens. For a set of given cutting conditions, the results show close agreement between the measured cutting forces and the model predictions. © 1999 Elsevier Science Inc. All rights reserved. Keywords: Milling force prediction; Taper end-mills; Process parameter identification; Differential geometry
Nomenclature A Fn, Ff F K n, K f, c K 1, K 2, K 3 R t, n, b tc
␣, , ␥n , 1, 2
rotation matrix pressure and friction cutting force vectors on the rake face total cutting force vector pressure coefficient, friction coefficient, chip flow angle on the rake face cutting force coefficients cutter radius at the tool tip unit vectors defining the curvilinear coordinate system on the rake face chip thickness unit vector defining the chip movement direction half-apex angle, helix angle, normal rake angle tool rotation angle, tool position angle integration limits
1. Introduction It has long been acknowledged that cutting force is one of the most sensitive output variables in the milling process. * Corresponding author. Tel.: ⫹44-1203-523154; fax: ⫹44-1203-471457 or ⫹44-1203-473-558. E-mail address: [email protected] (D.J. Whitehouse)
Furthermore, this force can be affected by any variation in the tool/workpiece geometry, the materials used, the cutting conditions and the dynamic characteristics of machine–toolfixture system. Therefore, a fundamental understanding of the cutting force system plays an important role in accurate analysis and prediction of the capability of the machining process. The analysis and subsequent prediction should make it possible to reduce greatly the time required for NC programming verification in test cuts and to enable the adaptive control strategies to be reliably simulated before the real implementation is attempted [1]. Considerable research effort has been devoted to formulating mechanismic models for cutting force prediction in the milling process [1– 4]. Moreover, modeling methodologies together with procedures for estimating the process parameters have been reported by Bayoumi et al. [3,5] and Yucesan and Altintas [6] regarding the three-dimensional (3-D) cutting force prediction of helical end-mills having arbitrary profiles in general and cylindrical configurations in particular. In Bayoumi’s model, it was assumed that the cutting force components could be linearly represented in proportion to the frontal area of the chip being removed and that a Taylor’s series expansion technique could be used for the process parameter approximations in terms of tool rotation angle. Similarly, to cope with the nonlinear problem encountered in parameter identification, the empirical equation employed in the Yucesan’s model was that the specific cutting force coefficients could be expressed as an exponential decaying function of the averaged uncut chip thickness.
0141-6359/99/$ – see front matter © 1999 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 1 - 6 3 5 9 ( 9 9 ) 0 0 0 1 0 - 0
T. Huang, D.J. Whitehouse / Precision Engineering 23 (1999) 196 –203
The main difference between these two models is the question of how to define and determine the unknown cutting coefficients; an old topic that may date back to the early work of Sabberval [7,8]. Taper end-mills have found wide application in the die fabrication for automobile and aerospace components. Depending upon the geometry of the cutting edge, there are two main types of taper end-mills: the straight edge type such as reamer/drill reamer; and the constant helix angle type with a flat/ball nosed end for die sinking. Although the cutting force prediction models for the helical end-mills with cylindrical configuration have been well developed, the corresponding models for the taper end-mills have been less investigated. The cutting force model using end-mills of this kind was studied by Ramaraj and Eleftheriou [9]. Based upon a basic shear plane model in oblique cutting theory, the shear and normal forces were formulated in a local Cartesian system. From this, a series of complicated coordinate transformations was implemented to obtain the total force in conventional Cartesian coordinates. However, in addition to using a rather ambiguous way to define the tool geometry, this method suffers from the problem that the representation of the primary deformation zone by a shear plane may be unrealistic, and the mean angle of friction may be insufficient to define completely the frictional conditions on the tool face, as noted by Bayoumi et al. [5,6]. Consequently, it is necessary to have a set of experimentally determined strain–stress relationships in the deformed zone. This is, however, very difficult, if not impossible. In this paper, an instantaneous rigid force model of end milling operation using taper end-mills with constant helix angle is developed. The formulation procedure involves the representation of cutting edge by means of differential geometry, the definition of the curvilinear coordinate system, the development of the mechanistic cutting force components along the directions of normal pressure and chip movement, the evaluation of the total instantaneous cutting force, and the identification of the process parameters. The model is experimentally verified by cutting middle carbon steel.
Consider a taper end-mill as shown in Fig. 1. The parametric equation of the revolute surface of the cutter in the local cutting edge fixed Cartesian coordinate O ⫺ xyz has the form [Eq. (1)] (1)
where ␣ , R, and denote the half apex angle, the cutter radius, and the position angle measured at the tool tip, respectively. According to differential geometry [10], the fundamental quadratic form of any curve upon a taper surface is given by [Eq. (2)] I ⫽ sec2␣ dz 2 ⫹ 共R ⫹ z tan ␣ 兲 2d 2
Fig. 1. Tool geometry and cutting force components.
Let  be the helix angle defined by the angle between the positive directions of the curve of a cutting edge and the longitude at point P( x, y, z), then [Eq. (3)] results cos2  ⫽
sec2␣ dz 2 共R ⫹ z tan ␣ 兲 2d 2 ⫹ sec2 ␣ dz 2
(3)
or [Eq. (4)] sec2 ␣ dz 2 ⫽ 共R ⫹ z tan ␣ 兲 2 cot2 d 2
(4)
For the cutter having a right-handed helix angle, taking the root square of Eq. (4) leads to [Eq. (5)] 1 dz ⫽ cos ␣ cot  d R ⫹ z tan ␣
(5)
Integrating Eq. (5) with respect to and remembering that z兩 ⫽0 ⫽ 0 gives [Eq. (6)]
冉
1n 1 ⫹ tan ␣
冊
z ⫽ sin ␣ cot  R
(6)
Exponentiating both sides of Eq. (6) and substituting into Eq. (1) leads to the position vector r expressed in terms of the tool position angle [Eq. (7)]
2. Tool geometry
r ⫽ 关共R ⫹ z tan ␣ 兲cos 共R ⫹ z tan ␣)sin z兴
197
(2)
r ⫽ r共 兲 ⫽ Re sin␣cot[cos i ⫹ sin j ⫹ cot ␣ 共1 ⫺ e ⫺sin␣cot)k兴
(7)
This indicates that the generating curve of the cutting edge for taper end-mills with constant helix angle is a 3-D spiral.
3. Curvilinear coordinate system It has been found that the differential cutting force acting on the rake surface at point P can be divided into two orthogonal components in O ⫺ xyz, as shown in Fig. 1 [3].
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One component is referred to as the normal pressure force, which is perpendicular to the rake surface. The other component is called the friction force, which is located on the rake surface, along the direction of chip movement. The procedure to determine the direction cosines of these cutting force components is implemented by manipulating a set of unit vectors defining the curvilinear coordinate of the cutting edge. As shown in Fig. 1, the orthogonal curvilinear coordinate system along the cutting edge is defined by a unit normal vector n, which is perpendicular to the rake surface, a unit vector t tangent to the helix flute, and a unit vector b that is orthogonal to both t and n. The procedure to generate these vectors is described as follows. First, differentiating Eq. (7) with respect to gives the tangent unit vector t of the cutting edge [Eq. (8)]. t⫽
r ⫽ 共sin ␣ cos  cos ⫺ sin  sin )i 兩r 兩
(13)
Obviously, the unit vectors n and defining the differential cutting force components for the taper end-mills are functions of the tool geometry parameters, including halfapex angle ␣, helix angle , rake angles ␥ n , position angle , and chip flow angle c . It has been noticed, however, that despite the chip flow angle c being able to take the value of the inclination angle to a first degree of approximation, the large friction at the tool– chip interface may cause a strong deviation of the former from the latter. In other words, the chip flow angle is heavily affected by such factors as the adhesive property between the tool and the workpiece materials, lubrication conditions, and other relevant characteristics of the process; therefore, this angle must be determined by experimentation. 4. Cutting force formulation
⫹ 共sin ␣ cos  sin ⫹ sin  cos )j ⫹ cos ␣ cos k (8) The primary normal vector a and the auxiliary normal vector c of the cutting edge as shown in Fig. 1 can be derived by [Eqs. (9, 10)]., a⫽
⫽ b cos c ⫹ t sin c
r1 ⫺ r ⫽ ⫺cos ␣ cos i ⫺ cos ␣ sin j ⫹ sin ␣k 兩r 1 ⫺ r兩
(9)
c ⫽ t ⫻ a ⫽ 共sin ␣ sin  cos ⫹ cos  sin )t ⫹ 共sin ␣ sin  sin ⫺ cos  cos )j ⫹ cos ␣ sin k (10) where (r1 ⫺ r) ⫻ t ⫽ 0, r1 ⫽ {0 0 Re sin␣cot[tan ␣ ⫹ cot ␣(1 ⫺ e ⫺sin␣cot)]}. Therefore t, a, and c constitute a Frenet frame at point P. Instead of using the rake angle defined in the x–y plane, as addressed by Bayoumi and Yucesan [3,5], let ␥ n be the normal rake angle defined as an angle oriented from the primary normal vector a in the normal plane perpendicular to the rake surface. The reason for this is that the normal rake angle remains constant along the cutting edge and is the geometric parameter used as the cutter is manufactured. Consequently, the unit vectors b and n can be generated [Eqs. (11, 12)]. b ⫽ cos ␥ na ⫺ sin ␥ nc
(11)
n ⫽ sin ␥ na ⫹ cos ␥ nc
(12)
According to the mechanismic model in the oblique cutting characterized by the nonzero inclination angle, the direction of chip movement as shown in Fig. 1 must locate on the rake surface expanded by the unit vectors b and t. Let c be the chip flow angle defined as an angle oriented from unit vector b, then the unit vector defining the direction of the differential friction force on the rake surface can be given as
According to the kinematics in milling, the uncut chip thickness with zero runout can be approximated as: t c共 , 兲 ⫽ f t sin共 ⫹ 兲
(14)
where ft is the feed rate per tooth, and is the tool rotation angle in down milling. As indicated previously, the differential cutting force vector acting at a certain point on the cutting edge can be divided into two orthogonal components; that is, the pressure force in the opposite direction of the normal of rake face and the friction force in the direction of chip movement. These differential cutting force components acting on an infinitesimal element can be respectively written as: dF n共 , 兲 ⫽ ⫺A共 兲 K n共 兲n共 兲⌫共 , 兲
dz d d
dF f共 , 兲 ⫽ A共 兲 K f共 兲 K n共 兲 共 兲⌫共 , 兲
dz d d
(15)
In the above expressions, dz/d ⫽ R cos ␣ cot  e sin␣cot, K n and K f are the unknown pressure and friction coefficients associated with the cutting force components acting on the rake surface and are assumed to be the function of tool position angle ; ⌫( , ) ⫽ f[t c ( , )] represents the function introduced to describe the contribution of the uncut chip thickness to the cutting forces, and A( ) denotes the rotation matrix from the local to the global Cartesian coordinates. Taking the summation and integrating along the cutting edge leads to the resultant cutting force vector in the global Cartesian system [Eq. (16)]. F共 兲 ⫽ A共 兲
冕
2
K n共 兲兵⫺n共 兲 ⫹ K f共 兲关b共 兲cos c共 兲
1
⫹ t共 兲sin c共 兲兴其⌫共 , 兲 Define [Eq. (17)]
dz d d
(16)
T. Huang, D.J. Whitehouse / Precision Engineering 23 (1999) 196 –203
再
K 1共 兲 ⫽ K n共 兲 K 2共 兲 ⫽ K n共 兲 K f共 兲cos c共 兲 K 3共 兲 ⫽ K n共 兲 K f共 兲sin c共 兲
(17)
as the cutting force coefficients, then Eq. (16) can be rewritten as follows [Eq. (18)]: F共 兲 ⫽ A共 兲
冕
2
关⫺K 1共 兲n共 兲 ⫹ K 2共 兲b共 兲
1
⫹ K 3共 兲t共 兲兴⌫共 , 兲
dz d d
(18)
In the above expressions, 1 and 2 are the integration limits representing the angular locations of the starting and end points of the cutting engagement of the flute and can be determined by the method presented in the Appendix. After taking the integral, K 1 , K 2 , and K 3 can be determined by means of the least-squares estimation using the cutting force measurements. The procedure to identify these coefficients is described in the following section. Thus, the unknown pressure and friction coefficients K n , K f as well as the chip flow angle c can be determined directly by the following inverse transformations [Eq. (19)]
冦
K n共 兲 ⫽ K 1共 兲
冋 册
c共 兲 ⫽ arctan
K 3共 兲 K 2共 兲
K 2共 兲 sec c共 兲 K f共 兲 ⫽ K 1共 兲
(19)
Strictly speaking, it is impossible to obtain the closed form expression because of the unknown parameters and complexity of the integrand, even for the cutters with simple geometrical profile. As a result, a numerical integration procedure must be performed to calculate the total cutting force. It should be pointed out that the above formulations are developed for a single fluted taper end-mill. The derivations can be slightly extended for the multiple cutter by summing the force acting on each flute at a specific tool rotation angle.
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functional capabilities described above. These approaches generally are based upon the widely accepted empirical experience that cutting force coefficients in milling are influenced by changes in the chip thickness, the cutter geometry, tool/workpiece material properties, lubrication conditions, and so forth. The most commonly used form for parametric relationships is an exponential decaying function of the chip thickness [7,8]. This results in nonlinear identification procedures in cutting force coefficient estimation. Three effective approaches, among others, have been developed by Altintas [2], Bayoumi [3,5], and Yucesan [6]. In Altintas’ [2], it is proposed that the chip thickness can be replaced by its average value within the range of engagement. Therefore, the process parameters (the tangential and radial cutting coefficients expressed as the exponential decaying function of the average chip thickness) were able to be moved before the integral in both parameter identification and cutting force evaluation (in effect a use of the mean value theorem). However, use of the average chip thickness may cause inaccurate cutting force prediction for smaller actual chip thickness [3]. In the further improvements made by Bayoumi [3,5], the process parameters (the pressure and friction coefficients K n , K f and the chip flow angle c ), which are directly expressed as a function of both the rotation angle and the position angle, are replaced by the effective moving average parameters that are functions of the tool rotation angle. With a similar starting point to Bayoumi’s method, the first step in the approach proposed by Yucesan and Altintas is to transform K n , K f , and c into the cutting force coefficients K 1 , K 2 , and K 3 as given in Eq. (17). The second step is to approximate K 1 , K 2 , and K 3 by a series expansion in such a way that the coefficients of the polynomials can be easily estimated by linear regression. This is followed by fitting K 1 , K 2 , and K 3 as exponential functions of chip thickness that change with the tool position angle and the rotation angle. Finally, the chip thickness is averaged within the range of the flute engagement so that the average cutting coefficients are obtained as a function of the tool rotation angle. Therefore, the average cutting coefficients can be brought before the integral operator in the cutting force prediction. Hereafter, we propose another method to identify the
5. Parameter Identification Because of the complexity of the cutting process, analytical prediction of the process parameters remains to be well developed. Therefore, it is common practice to determine these unknown parameters by experiment. In principle, a good identification procedure should be beneficial to: predicting the instantaneous cutting force with sufficient accuracy; investigating the influence of tool geometry and cutting conditions, such as feed rate, axial and radial depths of cut on the cutting force; implementing effective and reliable algorithms for real-time simulation; and building up a cutting database with ease for the later use. Several approaches have been developed to achieve the
Table 1 Cutting parameters No.
Feed rate (mm/tooth)
Depth of cut (mm)
1 2 3 4 5 6 7 8 9
0.01992 0.03983 0.05975 0.01992 0.03983 0.05975 0.01992 0.03983 0.05975
10 10 10 15 15 15 20 20 20
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Fig. 2. The layout of instrumentation in the cutting test.
process parameters, which involves solving a set of bilinear equations. Suppose that a group of discrete cutting force data F1, F2, F3, . . . , Fn has been measured through cutting test under the specific cutting condition, then the following error functional in a quadratic form can be formulated as follows (Eq. 20).
冘 储F ⫺ F共 兲储 n
J⫽
j
j
2 2
(20)
j⫽1
According to Sabberval’s [7,8] observations, the instantaneous cutting force is closely related to the uncut chip thickness variation, which is a function of both the tool position angle and tool rotation angle. On the one hand, consider that the cutting force coefficients are formulated in such a way that they are only a function of tool position angle K j can, therefore, be approximated by a power series in terms of the tool position angle [Eq. (21)].
冘 ␣ 共 j ⫽ 1, 2, 3兲 n
K j共 兲 ⫽
k
jk
(21)
k⫽0
On the other hand, account for the effect of the uncut chip thickness on cutting force, a Chebyshev polynomial [Eq. (22)] can be used to approximate the unknown function ⌫(, )
冘 h cos兵共l ⫹ 1兲arccos关t 共, 兲兴其 m
⌫共 , 兲 ⫽
l
c
(22)
l⫽0
where h 0 ⫽ 1 for the independence of the coefficients. The least-squares estimation can be used to implement parameter identification. Thus, taking the derivative of Eq. (20) with respect to the unknown coefficients will lead to a set of bilinear equations [Eq. (23)]
再 DyBx ⫽⫽ CE where x ⫽ 共a 11 a 12, · · · , a 1n, · · · , a 31 a 32, · · · , a 3n兲 T, y ⫽ 共h 1 h 2, · · · , h m兲 T
(23)
The Newton–Raphson method is available to solve this problem. Note here that for a large class of milling cutters, the rake angles are the same. Therefore, it would be helpful at the same time to use a group of cutting force measurements obtained by changing such cutting parameters as feed rate and depth of cut. This usually allows accuracy of the cutting coefficients; hence, the cutting force predictions to be improved.
Fig. 3. The pressure and friction coefficients K n , K f , and the chip flow angle c versus the axial depth of cut and feed rate 1.
T. Huang, D.J. Whitehouse / Precision Engineering 23 (1999) 196 –203
201
Fig. 4. Comparison between the measured and predicted cutting forces under different cutting conditions.
6. Experimental verification A group cutting test is used to verify the validity of the model presented in this paper. A Vcenter-65 machining center with FANUCOM controller is used in the experiment. A KISTLER three-component dynamometer, type 9525A is used to measure the cutting forces. A photoelectric sensor is used to pick up the rotation speed of the spindle and to measure the phase of the cutting force. The mill is a three fluted taper end-mill made of HSS, 16-mm diameter at the tool tip, normal rake angle of 13°, apex half angle of 4°, and helix angle of 30°. The material of the workpiece is 45 middle carbon steel. The tests are carried out in a semifinished down milling on a flat surface with spindle speed 318 rpm, radial depth of cut 6 mm, and an oil coolant is used. The changeable cutting parameters as given in Table 1 are the feed rate and axial depth of cut. The feed rate is chosen
to cover a range of chip thickness. The axial depth of cut is varied to verify the validity of cutting parametric relationships. No variation in the radial depth of cut and cutting speed are considered, because it has been observed that they do not have significant effects on the process parameters in end milling. In the tests, the cutting forces are first recorded by a SONY 8-channel high-fidelity tape recorder, and then are sampled at a rate of 400 samples per second through an AST 386/25 PC with KH8345A data acquisition board. The layout of the test is shown in Fig. 2. Note that the normal rake angle and the helix angle are constants along the cutting edge, and the cutting speed changes linearly along the axial depth of cut. Because the cutting speed has little bearing upon the magnitude of cutting force, it is reasonable to assume that the cutting coefficients K 1 , K 2 , and K 3 are constant. Meanwhile, a fourth order Chebyshev polynomial is used to approximate the
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unknown function ⌫(, ). Fig. 3 shows the variations of the identified pressure and friction coefficients K n , K f , and the chip flow angle c versus the depth of cut and feed rate. It can be observed from these figures that utilizing the identification model presented in this paper allows the above parameters to have little correlation with the changes in the cutting conditions. The Chebyshev polynomial corresponding to the first cutting condition is identified and given by ⌫( , ) ⫽ t c ( , ) ⫹ 0.02296 cos{2 arccos[t c ( , )]} ⫹ 0.07043 cos{3 arccos[t c ( , )]} ⫹ 0.02939 cos{4 arccos[t c ( , )]}. Fig. 4 (a– d) shows the measured and predicted cutting forces under different cutting conditions. The data in Fig. 4 (a) are used to identify the cutting coefficients, which are then employed to predict the cutting forces under the next three cutting conditions. It can be observed from these figures that close agreement between the measured cutting forces and the model predictions is obtained.
7. Conclusions A mechanismic model to formulate the nonlinear, 3-D cutting forces of taper end-mills is presented in this article. For the given tool geometry and workpiece material, the experimental results show that the correlation between the feed rate and cutting coefficients is very small; therefore, little experimental effort has to be made to build up a cutting force library. In addition, the formulation technique proposed is so general that it can be expanded with little effort to deal with cutting force predictions using other tool geometries.
Fig. A. Cutting geometry for die-sinking with taper end-mills.
2 ⫽ cos⫺1 Acknowledgments This research work is supported by the Natural Science Foundations of China and Tianjin Municipality.
冉
R ⫹ h tan ␣ ⫺ t R ⫹ h tan ␣
Application of taper end-mills to die-sinking may include two possible cutting formats: (1) profile milling of plenary surfaces with a fixed axial depth of cut for semifinish cutting [see Fig. A(a)]; and (2) parallel milling of the same profile with a fixed axial depth of cut for finish cutting [see Fig. A(b)]. Fig. A.3 shows the geometric situation for semifinish milling. In the figure 1, 2 are the immersion angles at the tool tip with a radial depth of cut t and at the other extreme end with a radial depth of cut t ⫹ h tan ␣, where h is the axial depth of cut.
1 ⫽ cos⫺1
冉 冊
冉
R⫺t R⫺t , 2 ⫽ cos⫺1 R R ⫹ h tan ␣
冊
(A.1)
Whereas, for the finish milling, 2 (See Fig. A.4) is given by
(A.2)
The maximum tool position angle associated with the given axial engagement is given by
冋 冉
max ⫽ csc ␣ tan  1n Appendix
冊
h tan ␣ ⫹1 R
冊册
(A.3)
Thus, the integration limits for the both cutting geometries, as shown in Fig. A(e), (f) can be determined by means of the following algorithm. Lower limits SELECT CASE ; CASE 2 ⫺ 1 ⱕ ⬍ 2 (cases A and B) 1 ⫽ 0 Case 2 ⫺ (2 ⫹ max) ⬍ ⬍ 2 ⫺ 1 (cases C and D) 1 ⫽ 2 ⫺ ⫺ cos⫺1[(1 ⫺ t/R)e ⫺ 1sin␣cot] CASE 0 ⱕ ⱕ 2 ⫺ (2 ⫹ max) (case E) disengagement END SELECT Upper limit SELECT CASE
T. Huang, D.J. Whitehouse / Precision Engineering 23 (1999) 196 –203
CASE 2 ⫺ max ⱕ ⬍ 2 (cases A and C) 2 ⫽ 2 ⫺ CASE 2 ⫺ (2 ⫹ max) ⬍ ⬍ 2 ⫺ max (cases B and D) 2 ⫽ max CASE 0 ⬍ ⬍ 2 ⫺ (2 ⫹ max) (case E) disengagement END SELECT
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[3] Bayoumi AE, Yucesan G, Kendall LA. Analytic mechanistic cutting force model for milling operations: A theory and methodology. J Eng Ind 1994a;116(3):324 –330. [4] Wang J-JJ, Liang SY, Book WJ. Convolution analysis of milling force pulsation. J Eng Ind 1994;116(1):17–25. [5] Bayoumi AE, Yucesan G, Kendall LA. Analytic mechanistic cutting force model for milling operations: A case study of helical milling operation. J Eng Ind 1994b;116(3):331–39. [6] Yucesan G, Altintas Y. Improved modeling of cutting force coefficients in peripheral milling. Int J Machine Tools Manufact 1994; 34(4):473– 487. [7] Sabberval AJP. Chip section and cutting force during milling operation. Ann CIRP 1961;10:197–203. [8] Sabberval, AJP. Cutting force in down milling. Int J Machine Tools Design Res 1962;2(1):27– 41. [9] Ramaraj TC, Eleftheriou E. Analysis of the mechanics of machining with tapered end-milling cutters. J Eng Ind 1994;116(3):398 – 404. [10] Eisenhart LP. A treatise on the differential geometry of curves and surfaces. New York: Dover, 1960.