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Systematic simulation procedure of peripheral milling process of thin-walled workpiece
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 7 ( 2 0 0 8 ) 122–131
journal homepage: www.elsevier.com/locate/jmatprotec
Systematic simulation procedure of peripheral milling process of thin-walled workpiece M. Wan ∗ , W.H. Zhang, G. Tan, G.H. Qin Sino-French Laboratory of Concurrent Engineering, The Key Laboratory of Contemporary Design & Integrated Manufacturing Technology, School of Mechatronic Engineering, Northwestern Polytechnical University, P.O. Box 552, 710072 Xi’An, Shaanxi, China
a r t i c l e
i n f o
a b s t r a c t
Article history:
A systematic procedure is developed to simulate the peripheral milling process of
Received 20 October 2006
thin-walled workpiece. The procedure integrates the cutting force module consisting of
Received in revised form
calculating the instantaneous uncut chip thickness (IUCT), calibrating the instantaneous
2 April 2007
cutting force coefficients (ICFC) and the cutting process module consisting of calculating
Accepted 3 June 2007
the cutting configuration and static form errors. It can be used to check the process reasonability and to optimize the process parameters for high precision milling. Key issues such as the theoretical calculation of the IUCT, efficient calibration scheme of ICFC, iterative correc-
Keywords:
tion of IUCT, the radial depth of cut and material removal are studied in detail. Meanwhile,
Cutting force model
the regeneration mechanism in flexible static end milling is investigated both theoretically
Instantaneous uncut chip thickness
and numerically. Comparisons of the cutting forces and form errors obtained numerically
Thin-walled workpiece
and experimentally confirm the validity of the proposed simulation procedure. © 2007 Elsevier B.V. All rights reserved.
Static form error Peripheral milling Regeneration mechanism
1.
Introduction
The peripheral milling of thin-walled workpieces such as entire girders and aero-engine blades is widely used in aerospace and aeronautic industries. A perpetual and everincreasing emphasis is to achieve high machining quality of components. However, the geometric complexity of parts and the extreme machining properties of materials are continuing to challenge the current manufacturing capability. Therefore, reliable machining technologies must be employed to obtain consistent part shapes. Conventionally, trial-and-error methods are adopted to select cutting parameters. It is well known that such an approach will lead to a long setup time and large machining costs. Recently, the advances of computing techniques make it possible to perform machining operations in a comprehensive simulation environment. Before actual
∗
cutting, one can simulate the critical machining characteristics such as cutting forces, tool/workpiece deflection, form errors, cutter wear and vibration (Budak and Altintas, 1992, 1995; Li and Guan, 2004; Tsai and Liao, 1999; Ratchev et al., 2004a,b; Wan et al., 2005; Semercigil and Chen, 2002) firstly by modelling the cutting process and the attributes of the tool/workpiece, etc. Form error compensation and cutting parameters optimization (Ratchev et al., 2006; Depince and Hascoet, 2006; By Chen et al., 2005; Milfelner et al., 2005) are then carried out to ensure the machining quality. In fact, one fundamental and significant basis of all process simulations is the determination of the cutting forces, the accuracy of which depends on how the cutting force models and the corresponding cutting force coefficients (Koenigsberger and Sabberwal, 1961; Altintas and Spence, 1991; Budak et al., 1996; Ko et al., 2002) are established. At
Corresponding author. Tel.: +86 29 88495774; fax: +86 29 88495774. E-mail addresses: [email protected] (M. Wan), [email protected] (W.H. Zhang). 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.06.005
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 7 ( 2 0 0 8 ) 122–131
123
Fig. 1 – Framework of the simulation procedure.
present, three types of cutting force models are widely used: Koenigsberger and Sabberwal (Koenigsberger and Sabberwal, 1961) and Altintas and Spence (Altintas and Spence, 1991) adopted a single coefficient to characterize the effects of the shearing on the rake face and the rubbing at the cutting edge; Budak et al. (Budak et al., 1996) separated the shearing and rubbing effects with two independent coefficients; Cho and co-workers (Ko et al., 2002) used the normal force coefficient Kn , the frictional force coefficient Kf and the chip flow angle c to characterize the cutting force model. It is well known that all above cutting force models rely on the IUCT, which will be influenced by the cutter runout and the cutter/workpiece deflections. Kline and DeVor (Kline and DeVor, 1983) studied the influence of the radial runout on the calculations of IUCT, while the cutter/workpiece deflections are ignored. Sutherland and DeVor (Sutherland and DeVor, 1986) proposed a regeneration model to calculate the IUCT by accounting for both the cutter runout and the cutter deflection. Later, Budak and Altintas (1992) proposed a force-based iteration scheme to study the influence of cutter deflection on the IUCT. It was shown that in few tooth periods, the cutting forces deduced from the regeneration model converge to those produced by using the nominal chip thickness. In the vibration free milling process of thin-walled workpiece, a significant deviation between the planned and machined part profiles is due to the form error induced mainly by the deflection of the tool and the workpiece. To avoid this, the form error prediction becomes an essential issue for machining process control. Ratchev et al. (2004a,b) studied the form error distributions of low-rigidity parts influenced by the workpiece deflection, while the cutter deflection is neglected. Coupling models involving the cutter/workpiece deflection and the workpiece rigidity diminution were proposed in Refs. (Budak and Altintas, 1995; Tsai and Liao, 1999), in which iterative schemes were used to predict the cutting forces and the
form error distributions on a thin-walled rectangular part. The workpiece model was developed by use of one-layer volume elements along the thickness direction so that the material removal can be taken into account by varying the nodal coordinates. It is important to note that the above work was focused on each individual technique, e.g., the development of cutting force model, the calculations of IUCT, the prediction of the cutter/workpiece deflections, the modelling method of material removal, etc. The motivation of this paper is to develop a systematic procedure for simulating the peripheral milling process of thin-walled workpiece. The integrated framework is illustrated in Fig. 1. Key issues associated with the calibration scheme of ICFC, calculation of IUCT, coupling model of the cutter/workpiece system and the material removal model are addressed in detail. Both theoretical and numerical analyses are used to investigate the regeneration mechanism. It is shown that the regeneration phenomenon always disappears after several tooth periods even in flexible static end milling process. Comparisons of simulations and experiment cutting tests are illustrated to show the validity of the proposed procedure.
2.
Cutting force module
2.1.
Basic cutting force model
The peripheral end mill is equivalently divided into a finite number of disk elements along the cutter axis. The total X-, Yand Z-cutting forces acting on the cutter at a particular instant are acquired by numerically summing the force components acting on each individual disk element. Here, the first type of single coefficient cutting force model is considered. For the convenience of study, the jth axial disk element of the ith flute
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is noted as {i, j}. The two extreme points of disk element {i, j} are named as cutter nodes which are symbolized by (i, j) and (i, j + 1), respectively. Note that (i, j) is the cutter node nearer to the cutter free end than node (i, j + 1). For disk element {i, j}, the corresponding cutting forces in the tangential, radial and axial directions can be given as (Budak and Altintas, 1995; Tsai and Liao, 1999; Koenigsberger and Sabberwal, 1961; Altintas and Spence, 1991):
⎧ ⎪ ⎨ Fi,j,T = Ki,j,T hi,j zi,j F
=K
h z
(1)
i,j,R i,j,R i,j i,j ⎪ ⎩F = K h z i,j,z
i,j,z i,j i,j
where zi,j and hi,j are the axial length and the IUCT of {i, j}, respectively. Parameters Ki,j,T , Ki,j,R and Ki,j,z are the instantaneous cutting force coefficients corresponding to hi,j . Assume that the positive directions of axes Y and X in Cartesian coordinate are aligned with the external normal direction of the machined surface and the feed direction of the cutter, respectively. Once three force components are obtained from Eq. (1), they can be mapped along X, Y and Z directions:
⎡
Fi,j,X
⎤
⎡
Fi,j,T
⎤
⎡
Ki,j,T
⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ Fi,j,Y ⎦ = Ti,j ⎣ Fi,j,R ⎦ = hi,j zi,j Ti,j ⎣ Ki,j,R ⎦ Fi,j,Z
Fi,j,z
(2.1)
Ki,j,z
with
⎡
−cos i,j
⎢ Ti,j = ⎣ sin i,j 0
−sin i,j
0
⎤
−cos i,j
⎥ 0⎦
0
(2.2)
1
where i,j is the angular position which is defined as the clockwise angle determined from Y to the disk element {i, j}. Subsequently, the total cutting force components in X-, Yand Z-directions at an arbitrary cutting instant can be evaluated by summing the forces acting on all flutes and disk elements: Fs =
Fi,j,s ,
s = X, Y, Z
(2.3)
Fig. 2 – Geometric illustration of the instantaneous uncut chip thickness (IUCT). (a) The definition of the instantaneous uncut chip thickness (IUCT) and (b) the local amplified sketch map.
where
a=
ıx = mi f + ıt,X − ıt,X , i,j i−m ,j
ı2x + ı2y ,
i
ıy = ıt,Y − ıt,Y , i,j i−m ,j i
ˇ = − i,j + arccos
ı y
a
i,j
2.2.
Calculations of IUCT
By definition, hi,j refers to the radial distance between the tooth path to be generated by the disk element {i, j} and the workpiece surface left by the previous disk element {i − mi , j} with the same angular angle position i,j . mi means that the current tooth i is removing the material left by the mi th previous tooth. The occurrence of cutter runout will lead to mi = 1. As shown in Fig. 2, hi,j is the IUCT associated with i,j and it will change in function of i,j . Physically, due to the cutter/workpiece deflections, the axis of the cutter will shift from its nominal position. Correspondingly, the two adjacent tool paths will deviate from their nominal positions, too. By taking into account the effect of cutter/workpiece deflections, hi,j can be calculated as:
hi,j = −a cos ˇ + Radi,j −
Rad2i−m ,j − a2 sin2 ˇ i
(3)
Radi,j is the radius of the tooth path generated by disk element {i, j}. (ıt,X , ıt,Y ) and (ıt,X , ıt,Y ) denote the deviation values i,j i,j i−mi ,j i−mi ,j between the path centres of disk elements {i, j} and {i − mi , j}. f is the feed per tooth. Due to the presence of cutter runout, the rotation radius of an arbitrary cutter element {i, j} can be obtained as (Wang and Liang, 1996; Azeem et al., 2004):
Radi,j = R + cos −
2(i − 1) z tan i0 − R N
(4)
Here, R and i0 are the nominal geometry radius and helical angle of the cutter, respectively. and are the runout parameters. N is the tooth number of the cutter. As Radi−mi ,j >> ıx , Radi−mi ,j >> ıy and Radi,k−mi >> a sin ˇ, Eq. (3) can be finally approximated as: hi,j = −a cos ˇ + Radi,j − Radi−mi ,j
(5)
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Alternatively, if the influence of the cutter/workpiece deflections is negligible, hi,j is expressed as: hi,j = mi f sin i,j + Radi,j − Radi−mi ,j
(6)
In fact, for a vibration free cutting process, the cutting force must be stabilized after a transient period. Here, the satiability implies that the cutting forces in the current tooth period converge to those obtained in the previous tooth period, i.e., mi = 1 and Radi,j = Radi−1,j . The equivalent forces will induce the equivalent deflections so that following relations exist: ıt,X = ıt,X , i,j i−1,j
ıt,Y = ıt,Y i−1,j i,j
(7)
With these conditions, Eq. (6) can be simplified as: hi,j = f sin i,j
⎧ ⎪ ⎨ Fj,X (ϕ) = [−KT (ϕ) cos i,j − KR (ϕ) sin i,j ]hj (ϕ)zi,j F
(ϕ) = [KT (ϕ) sin − KR (ϕ) cos i,j ]hj (ϕ)zi,j
j,Y i,j ⎪ ⎩ F (ϕ) = K (ϕ)h (ϕ)z z j,Z j i,j
To calibrate the cutting force coefficients experimentally, the workpiece has to be a relative large block with sufficient rigidity. Meanwhile, the cutter is assumed to be a rigid one. Therefore, the influence of the cutter/workpiece deflections on the IUCT can be negligible so that Eq. (6) can be directly used to calculate the IUCT. In the calibration procedure, the cutting force coefficients are conventionally determined using measured average cutting forces. These techniques are useful, but in most cases the identified coefficients are only valid in the range of test conditions. Especially, to obtain robust application range of the related cutting force coefficients, a great number of cutting tests must be performed by changing feeds, depths of cut, etc. To save the test costs and to improve the calibration accuracy, we propose here an alternative method to calibrate the instantaneous cutting force coefficients Ki,j,T , Ki,j,R and Ki,j,z using instantaneous values of the measured cutting forces, instead of the average ones. It will be shown that just one cutting test will be needed to calibrate Ki,j,T , Ki,j,R and Ki,j,z . If the values of Fi,j,X , Fi,j,Y and Fi,j,Z can be measured in advance, Ki,j,q (q = T, R, z) can be obtained from Eq. (2.1). However, the measured cutting forces using dynamometer are available only in the form of total cutting forces Fs (s = X, Y, Z) and cannot be decomposed into force components Fi,j,X , Fi,j,Y and Fi,j,Z . These make it practically impossible to establish ‘strict’ relationship between Ki,j,q and hi,j . Therefore, an approximate approach is proposed below. As we know, i,j refers to the angular position of disk element {i, j} at the cutter rotation angle ϕ, as shown in Fig. 1. After the cutter rotation with an angle of (2k)/N (k = 1, 2, . . ., N − 1), the angular position of disk
N
hi,j
(10)
i=1
(8)
Calibration of ICFC
(9)
with
hj (ϕ) =
This is the so-called nominal value of hi,j in static milling. It means that hi,j converges to its nominal value of f sin i,j . This conclusion deduced from the basic definition of hi,j agrees completely with that from the force-based iteration of Budak (Budak and Altintas, 1992). Especially, it gives the theoretical explanation why nominal IUCT value, f sin i,j , is widely used by many researchers to study the cutting force model.
2.3.
element {i + k, j} equals i,j , too. If the cutting force coefficients are assumed to have the same values Kq (ϕ) (q = T, R, z) at ϕ + (2k)/N (k = 0, 1, 2, . . ., N − 1), the sum of each X-, Y- and Z-force component of the disk elements located at the angular position i,j over one rotation period corresponds to:
As the cutter/workpiece deflections can be ignored in the calibration procedure, Eq. (6) will be used to evaluate hj (ϕ) so that Eq. (10) can be rewritten as:
hj (ϕ) =
N
(mi f sin i,j + Radi,j − Radi−mi ,j ) = Nf sin i,j
(11)
i=1
Note that the effect of cutter radial runout on hj (ϕ) vanishes. It turns out that the sum of three force components of the cutting edge elements with the same angular position i,j is independent of the runout. The substitution of Eq. (11) into Eq. (9) and then the summation along the cutter axis produces:
⎡
FX (ϕ)
⎡
⎤
A
⎢B ⎢ ⎢ ⎥ ⎣ FY (ϕ) ⎦ = Nfzi,j ⎢ ⎣0
−B A 0
FZ (ϕ)
⎤
⎡ ⎤ KT (ϕ) ⎥ 0 ⎥⎢ ⎥ ⎥ ⎣ KR (ϕ) ⎦ ⎦ sin i,j 0
Kz (ϕ)
i,j
where
A=−
(cos i,j sin i,j )
and
(12)
B=
i,j
(sin i,j sin i,j )
i,j
Here, Fs (ϕ) (s = X, Y, Z) only depend upon cutting force coefficients and the feed per tooth no matter what the runout is. Denote FsM (ϕ) (s = X, Y, Z) to be measured cutting forces at rotation angle ϕ. Fs (ϕ) can be obtained by averaging measured cutting forces: 1 M Fs N N
Fs (ϕ) =
i=1
ϕ+
2(i − 1) N
(13)
With the known Fs (ϕ), Kq (ϕ) (q = T, R, z) can be obtained immediately from Eq. (12). This ICFC model reveals the relationship between Kq (ϕ) and ϕ. The interesting thing is that with Eqs. (12) and (13), the influence of the cutter runout is eliminated in the calibration procedure of the cutting force coefficients. The discrete values of Kq (ϕ) obtained from Eq. (12) can be fully exploited to give rise to a spectrum of cutting force coefficients for their accurate fitting.
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(k+1)
the cutter node (i, j). ϕi,j
denotes the variation of the immer(k+1)
sion angle which can be evaluated in terms of Rr
(k+1)
ϕi,j
= arccos
(k+1)
1−
Rr (i, j) Radi,j
(i, j) by:
(16)
The IUCT is iterated as: (k+1)
= −a(k) cos ˇ(k) + Radi,j − Radi−mi ,j
hi,j
(17)
with
(k)
=
a
t,X(k)
(mi f + ıi,j
Fig. 3 – Illustration of immersion angle. ˇ
3.
Calculations of cutting configuration
3.1.1. Correction of cutting parameters and numerical simulation of the regeneration phenomenon As shown in Fig. 3, due to the existence of deflections caused by the cutting forces, the ideal contact curve AB will be shortened to QB. Meanwhile, the nominal intersection line DF between the cutter and the workpiece will be shifted to EH. As a result, for any engaged cutter node, the radial depth of cut, IUCT and the immersion angle will be deviated from their nominal values. From this viewpoint, for disk element {i, j} at any cutting position, the radial depth of cut is corrected by: (k+1)
(k)
t,Y(k)
(i, j) = Rr (i, j) − [ıi,j
w,Y(k)
+ı
]
i,j (k+1) (k) (i, j) − Rr (i, j) ≤ ε, k = 0, 1, . . . , n s.t. Rr
= − i,j + arccos
(14)
t,Y(k)
(k+1)
= ui,j
zi,j
(k+1)
where ui,j denotes the correction factor in the (k + 1)th iteration defined as: (k+1)
(k+1)
and
i
a(k)
(k+1) (k+1) (k) (k) h˜ i,j = (hi,j − h˜ i,j ) + h˜ i,j ,
=
(ϕi,j
+
i,j
− )D
2zi,j tan i0
(15)
where D is the diameter of the cutter. i,j designates the angle determined clockwise from the positive direction of axis Y to
0<≤1
(18)
(k+1)
where h˜ i,j is the corrected value of the IUCT. is the weighted parameter given a priori. (2) Based on the correction factor defined in Eq. (15), the divergence during the iteration process of Eq. (14) can be avoided if the following scheme is adopted: (k+1)
u˜ i,j
(k+1)
= (ui,j
(k)
(k)
k = 0, 1, . . . , n ˜ (k+1) (k+1) − ui,j ≤ ε s.t. 0 < ≤ 1, ui,j
w,Y(k)
of cut. ıi,j and ıi,j denote the normal deflections of disk element {i, j} and the workpiece at the cutting position after the kth iteration, respectively. ε is the prescribed tolerance to control the iteration process. As shown in Fig. 3, due to the deflection, a certain number of cutter disk elements that should be completely in cut becomes now partially engaged, e.g., {i, j}. The actual contact (k+1) axial length zi,j can be written as:
ui,j
t,Y(k)
− ıi−m ,j
i
(1) A relative small change of chip thickness is used:
(k+1)
(k+1)
t,Y(k)
ıi,j
t,Y(k) 2
− ıi−m ,j )
Eq. (17) is the so-called numerical scheme for regeneration phenomenon simulation. During the iteration process of Eqs. (14) and (17), numerical oscillations may however occur due to the relative large change of radial depth of cut and the chip thickness. To ensure the convergence, the below sub-iteration algorithms are adopted:
where Rr (i, j) is the corrected radial depth of cut in the (k + 1)th iteration. Rr (i, j) is the nominal values of radial depth
zi,j
t,Y(k)
i
Cutting process module
3.1.
Rr
t,X(k) 2
− ıi−m ,j ) + (ıi,j
− u˜ i,j ) + u˜ i,j ,
(19)
(k+1)
where u˜ i,j is the corrected value of the correction factor. is the weighted parameter given a priori.
3.1.2.
Modelling of material removal
To consider the rigidity change of the workpiece due to the material removal in the machining process, the idea of softening materials is implemented to correct the element stiffness matrix in terms of its volume variation so that: Ki = i Ki
(20)
where Ki is the nominal stiffness matrix of element i. i denotes the ratio of volume variation of element i after sweeping with:
i =
Vi Vi
(10−6 = ε ≤ i ≤ 1)
(21)
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Table 1 – Cutting conditions Cutter no. A B B A A A
Test no.
Milling type
Radial depth of cut, Rr (mm)
Axial depth of cut, Rz (mm)
Feed per tooth, f (mm/tooth)
RPM
1 2 3 4 5 6
Down milling Down milling Down milling Down milling Down milling Up milling
10 10 10 10 0.95 0.95
1 1 1 1.2 29 29
0.05 0.05 0.05 0.05 0.04 0.0533
1000 300 180 300 200 300
in which Vi and Vi designate the remaining and nominal volume of element i before and after cutting, respectively. Here, a lower bound ε is used to prevent the singularity of the element stiffness matrix when the material is completely removed in milling. This treatment is more efficient than remeshing.
3.2.
Prediction of form error
The form error can be defined as the deviation of the machined surface from its nominal position. In the peripheral milling process, the cutter/workpiece deflections in the direction normal to the machined surface constitute the form errors. At a certain position P, the form error ep is evaluated by: eP = ıt,Y + ıw,Y i,j i,j
sion of 102 mm × 29.4 mm × 2.21 mm. The plate of Test 6 is also flexible with a dimension of 102 mm × 29.4 mm × 3.26 mm. 3D finite element meshes are used to discretize the workpiece. Cutter A has a tool gauge length of 61 mm and is discretized into 61 axial elements. Especially, Cutter A is designed as a single flute in order to eliminate the runout effects on form error distributions in Test 5 and Test 6. The Young’s moduli of the cutter and the workpiece are 600 GPa and 71.7 GPa, respectively. The flexibility of the plates is considerable when the cutting is carried out with a radial depth of 0.95 mm and axial depth of cut of 29 mm. Cutting forces are measured with a Kistler 9255B dynamometer.
4.1.
Calibration of the cutting force coefficients
(22)
and ıw,Y are the normal projections of the tool where ıt,Y i,j i,j deflection and the workpiece deflection corresponding to P, respectively. ıt,Y is calculated according to the method i,j presented in Ref. (Budak and Altintas, 1995), while ıw,Y is cali,j culated using FEA method. It is well recognized that the cutting forces are strongly influenced by the cutter/workpiece deflections as well as the rigidity variation of the workpiece due to material removal. In different situations, two available models are selected to calculate the form errors: (1) Rigid model. It is a simplified model in which the cutter/workpiece deflections and the workpiece rigidity change caused by the material removal are ignored. The form errors will be calculated directly based on the nominal cutting parameters. (2) Flexible model. Both the cutter/workpiece deflections and the workpiece rigidity change induced by material removal are taken into account. At any cutting instant, Eqs. (18) and (19) are used to correct the radial depth of cut and the IUCT.
The purpose of this section is twofold. On one hand, the verification will be made to show whether the runout effect is able to be eliminated by averaging the measured data with Eq. (13) if a multiple teeth cutter, i.e., Cutter B in our case is used to calibrate the cutting force coefficients. On the other hand, it will be shown how the cutting force coefficients are practically calibrated with the above procedure. To do these, the cutting force data associated with Cutter A are firstly measured and will be considered to be the authentic nominal values of Fs (ϕ) (s = X, Y, Z) as it is a single flute one without runout effect. These nominal values will be then used as reference data to check the validity of Eq. (13) applied to Cutter B. By virtue of Eq. (13), obtained averaged cutting force curves associated with Test 2 and Test 3 are shown in Fig. 4. They are very close to reference curves of Test 1 related to Cutter A. Therefore Eq. (13) is efficient to eliminate the runout effect. Now, Test 4 will be selected to calibrate the cutting force coefficients for Cutter A. Calibrated Kq (ϕ) (q = T, R, z) are shown in Fig. 5 versus the instantaneous average chip thickness
4. Example and the experimental verification Two carbide flat end mills with identical diameter of 20 mm and a helix angle of 30◦ are adopted to cut aluminum alloy 7050. The only difference is that Cutter A is a single-fluted flat end mill and Cutter B is a two-fluted one. Table 1 shows the cutting conditions. Tests 1–4 are used to cut a thick block for calibration of the cutting force coefficients. Test 5 and Test 6 are selected cantilever plates used to validate the proposed cutting force model. The plate of Test 5 is flexible with a dimen-
Fig. 4 – Comparison of averaged cutting forces.
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Fig. 5 – Cutting force coefficients.
(IACT), h(ϕ) that is calculated at ϕ by:
N i,U h(ϕ) =
i=1
N
i,L
i=1
f sin d
(23)
(i,L − i,U )
where i,L and i,U are the minimum and maximum angular positions of cutting edge i. As an exponent-like relation exists between Kq (ϕ) and h(ϕ), the following nonlinear fitting function is used: Kq (ϕ) = Wq1 + Wq2 e[Wq3 h(ϕ)]
(q = T, R, z)
Fig. 6 – Cutting forces FY of Test 5. (a) Iteration results and (b) comparison of measured and predicted cutting forces.
(24)
with Ws1 , Ws2 and Ws3 being constants to be determined. The fitted coefficients are also illustrated in Fig. 5. The overall trend of the fitted results corresponds well to the discrete values calculated. To characterize the relationship between Ki,j,q and the instantaneous uncut chip thickness hi,j of every
4.3.
Prediction of the form errors
Figs. 9 and 10 show the form errors along the cutter axial direction at two feed stations of Test 5 and Test 6 predicted by the
disk element, h(ϕ) is replaced by hi,j in Eq. (24) for calculation of Ki,j,q . This relationship is used for the numerical simulations through all cutting tests given below.
4.2.
The distribution of IUCT
With the aid of the ICFC obtained above and the iterative algorithms presented in Section 3.1.1, the distributions of the IUCT and the radial depth of cut are calculated and plotted for Test 5 and Test 6. Figs. 6 and 7 show that the cutting force based on the iterated chip thickness convergences to that using nominal chip thickness in both magnitude and shape after few tooth periods. Take an example. The convergence histories of the instantaneous uncut chip thickness associated with cutter element {1, 22} in down milling and cutter element {1, 12} in up milling versus the cutter angular position during the milling process are illustrated in Fig. 8. The iteration value converges very quickly to the nominal value after several rotation angles whatever it is concerned with down milling or up milling. All these curves confirm that the regeneration mechanism is short lived and the cutting forces can be well predicted by using the nominal chip thickness, as analysed from the theoretical viewpoint in Section 2.2. Therefore, the nominal values of the IUCT defined in Eq. (8) can be directly used to predict the cutting forces and the form errors.
Fig. 7 – Cutting forces FX of Test 6. (a) Iteration results and (b) comparison of measured and predicted cutting forces.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 7 ( 2 0 0 8 ) 122–131
129
Fig. 8 – Evolution of the chip thickness vs. cutter rotation angle: (a) Test 5 and (b) Test 6. Fig. 10 – Form errors of Test 6. Form errors in the position where the cutter feeds (a) 12 mm and (b) 84 mm.
present simulation model and measured by experiment. It can be seen that the results predicted by Rigid model and Flexible model have a good coherence in the variation tendency when compared with the experimental data. Especially, the Flexible model is shown to be more reliable because the material removal and the coupling effect are considered. The iteration history of correction factor u1,16 in down milling is shown in Fig. 10 when the cutter has a rotation of 26.65◦ from the feed position 42 mm. The iteration scheme is very efficient because it is stabilized after only a few steps. In summary, all these curves demonstrate the validity of the cutting force model developed in this paper.
5.
Fig. 9 – Form errors of Test 5. Form errors in the position where the cutter feeds (a) 42 mm and (b) 90 mm.
Discussion
In the vibration free milling process of low-rigidity thin-walled parts, deflections induced by cutting forces are inevitable to cause surface form errors that will deteriorate extremely the accuracy and quality of the parts. If the form errors violate seriously the dimensional tolerance, the machining process will lead to waste products. To reduce the errors, one of the most efficient strategies is to simulate the milling process combining the finite element method with cutting mechanism and furthermore to optimize the cutting parameters to improve the cutting process. The proposed study is focused on the simulation techniques of cutting force modelling and form error prediction. The simulation procedure is developed by considering the chip thickness variation (or regeneration) both
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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 7 ( 2 0 0 8 ) 122–131
surface form errors. All these show that our simulation procedure is reliable and has good robustness. The good consistency between the predicted and measured results also shows that the relationship between the instantaneous cutting force coefficients and the instantaneous uncut chip thickness can be established by fitting the discretized values of the coefficients that are calibrated directly from the measured data as function of the IACT of the whole cutter. Then, instantaneous uncut chip thickness of every separate disk element can be substituted into the fitted function for calculating the instantaneous cutting force coefficients associated with every single disk element. Fig. 11 – Test 5. Iteration history of correction factor u1,16 .
theoretically and numerically. Down milling and up milling are used for model verification. Numerical results show that the value of instantaneous uncut chip thickness hi,j will converge to its nominal value of f sin i,j in few tooth periods (Figs. 6 and 7) even in flexible static milling process. These results just give a corresponding numerical explanation of the conclusions deuced from the theoretical derivation in last paragraph in Section 2.2. Especially, the conclusion that is based on the basic definition of hi,j agrees completely with that from the force-based iteration of Budak and Altintas (1992). This gives the convictive explanation why nominal IUCT value, f sin i,j , is widely used by many researchers to study the cutting force model in static milling process. The surface form errors are predicted by modelling the mill deflection using cantilevered beam theory and modelling the workpiece deflection using finite element analysis. The coupling equilibrium state between the cutter and the workpiece is obtained using the improved iterative scheme that takes into account the recursive correction between cutting force, cutter/workpiece deflection and the immersion angle. From Fig. 9, it can be seen that the form errors are smallest at the cantilevered bottom of the workpiece, but they are not zero because of the cutter deflections. As the surface generation point shifts to the free end of the workpiece (Z-axis), the form errors increase due to the increase of the cutter stiffness and the workpiece flexibility. By comparing Fig. 9(a) with (b), it is found that both predicted and measured results indicate increasing trend in the form error amplitudes in the feed direction. This is due to the decreasing stiffness of the workpiece as a result of material removal. All these phenomena are in satisfactory agreement with the available referenced ones (Budak and Altintas, 1995; Tsai and Liao, 1999). It is worth mentioning that all existing results associated with the milling process of thin-walled workpiece are focused on the down milling. Alternatively, Fig. 10 gives an inspection of form error distribution in up milling. Compared with the experiment results (Figs. 9 and 10), it indicates that the proposed methodology can be successfully used to simulate both down milling process and up milling process. Especially, from Figs. 6–11, it can be seen that although the cutting force coefficients are calibrated only by one test with a considerable saving of experiment costs, the proposed approach has good accuracy in prediction of cutting force and
6.
Conclusions
A systematic procedure has been constructed to simulate the peripheral milling process of thin-walled workpiece. The developed procedure is able to predict the milling precision before actual cutting. It comprises of two major modules: cutting force module and cutting process module. Some significant improvements have been conducted about calculations of IUCT, calibrations of ICFC, corrections of cutting parameters and the material removal modelling. Especially, both theoretical and numerical analyses prove that the regeneration phenomenon is short lived in flexible static end milling. This indicates that the nominal value of the IUCT can be directly adopted in the simulation process. The form errors in flat end milling of thin-walled workpiece are studied numerically and experimentally to show the validity of the proposed procedure. It turns out that the developed procedure is an essential part of subsequent research on developing efficient methodology for optimum selection of cutting process parameters and tool path parameters in high-accurate machining of complex low-rigidity parts.
Acknowledgements This work is supported by the Doctorate Creation Foundation of Northwestern Polytechnical University (Grant no. CX200411), the Youth for NPU Teachers Scientific and Technological Innovation Foundation and the Natural Science Foundation of Shaanxi Province (Grant no. 2004E2 17).
references
Altintas, Y., Spence, A., 1991. End milling force algorithms for CAD systems. CIRP Ann. 40 (1), 31–34. Azeem, A., Feng, H.Y., Wang, L.H., 2004. Simplified and efficient calibration of a mechanistic cutting force model for ball-end milling. Int. J. Mach. Tools Manuf. 44, 291–298. Budak, E., Altintas, Y., 1992. Flexible milling force model for improved surface error predictions. In: Proceedings of Engineering System Design and Analysis, vol. 47(1), ASME, Istanbul, Turkey, pp. 89–94. Budak, E., Altintas, Y., 1995. Modeling and avoidance of static form errors in peripheral milling of plates. Int. J. Mach. Tools Manuf. 35, 459–476. Budak, E., Altintas, Y., Armarego, E.J.A., 1996. Prediction of milling force coefficients from orthogonal cutting data. ASME J. Manuf. Sci. Eng. 118, 216–224.
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By Chen, J.-S., Huang, Y.-K., Chen, M.O.-S., 2005. Feedrate optimization and tool profile modification for the high-efficiency ball-end milling process. Int. J. Mach. Tools Manuf. 45, 1070–1076. Depince, P., Hascoet, J.-Y., 2006. Active integration of tool deflection effects in end milling. Part 2. Compensation of tool deflection. Int. J. Mach. Tools Manuf. 46, 945–956. Kline, W.A., DeVor, R.E., 1983. The effect of runout on cutting geometry and forces in end milling. Int. J. Mach. Tool Des. Res. 23 (2/3), 123–140. Ko, J.H., Yun, W.S., Cho, D.W., Ehmann, K.F., 2002. Development of a virtual machining system. Part 1. Approximation of the size effect for cutting force prediction. Int. J. Mach. Tools Manuf. 42, 1595–1605. Koenigsberger, F., Sabberwal, A.J.P., 1961. An investigation into the cutting force pulsations during milling operations. Int. J. Mach. Tool Des. Res. 1, 15–33. Li, X.L., Guan, X.P., 2004. Time-frequency-analysis-based minor cutting edge fracture detection during end milling. Mech. Syst. Signal Process. 18 (6), 1485–1496. Milfelner, M., Cus, F., Balic, J., 2005. An overview of data acquisition system for cutting force measuring and optimization in milling. J. Mater. Process. Technol. 164/165, 1281–1288.
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Ratchev, S., Liu, S., Huang, W., Becker, A.A., 2004a. A flexible force model for end milling of low-rigidity parts. J. Mater. Process. Technol. 153/154, 134–138. Ratchev, S., Liu, S., Huang, W., Becker, A.A., 2004b. Milling error prediction and compensation in machining of low-rigidity parts. Int. J. Mach. Tools Manuf. 44, 1629–1641. Ratchev, S., Liu, S., Huang, W., Becker, A.A., 2006. An advanced FEA based force induced error compensation strategy in milling. Int. J. Mach. Tools Manuf. 46, 542–551. Semercigil, S.E., Chen, L.A., 2002. Preliminary computations for chatter control in end milling. J. Sound Vib. 249, 622–633. Sutherland, J.W., DeVor, R.E., 1986. An improved method for cutting force and surface error prediction in flexible end milling systems. ASME J. Eng. Ind. 108, 269–279. Tsai, J.S., Liao, C.L., 1999. Finite-element modeling of static surface errors in the peripheral milling of thin-walled workpieces. J. Mater. Process. Technol. 94, 235–246. Wan, M., Zhang, W.H., Qiu, K.P., Gao, T., Yang, Y.H., 2005. Numerical prediction of static form errors in peripheral milling of thin-walled workpieces with irregular meshes. ASME J. Manuf. Sci. Eng. 127, 13–22. Wang, J.-J.J., Liang, S.Y., 1996. Chip load kinematics in milling with radial cutter runout. ASME J. Eng. Ind. 118, 111–116.
journal homepage: www.elsevier.com/locate/jmatprotec
Systematic simulation procedure of peripheral milling process of thin-walled workpiece M. Wan ∗ , W.H. Zhang, G. Tan, G.H. Qin Sino-French Laboratory of Concurrent Engineering, The Key Laboratory of Contemporary Design & Integrated Manufacturing Technology, School of Mechatronic Engineering, Northwestern Polytechnical University, P.O. Box 552, 710072 Xi’An, Shaanxi, China
a r t i c l e
i n f o
a b s t r a c t
Article history:
A systematic procedure is developed to simulate the peripheral milling process of
Received 20 October 2006
thin-walled workpiece. The procedure integrates the cutting force module consisting of
Received in revised form
calculating the instantaneous uncut chip thickness (IUCT), calibrating the instantaneous
2 April 2007
cutting force coefficients (ICFC) and the cutting process module consisting of calculating
Accepted 3 June 2007
the cutting configuration and static form errors. It can be used to check the process reasonability and to optimize the process parameters for high precision milling. Key issues such as the theoretical calculation of the IUCT, efficient calibration scheme of ICFC, iterative correc-
Keywords:
tion of IUCT, the radial depth of cut and material removal are studied in detail. Meanwhile,
Cutting force model
the regeneration mechanism in flexible static end milling is investigated both theoretically
Instantaneous uncut chip thickness
and numerically. Comparisons of the cutting forces and form errors obtained numerically
Thin-walled workpiece
and experimentally confirm the validity of the proposed simulation procedure. © 2007 Elsevier B.V. All rights reserved.
Static form error Peripheral milling Regeneration mechanism
1.
Introduction
The peripheral milling of thin-walled workpieces such as entire girders and aero-engine blades is widely used in aerospace and aeronautic industries. A perpetual and everincreasing emphasis is to achieve high machining quality of components. However, the geometric complexity of parts and the extreme machining properties of materials are continuing to challenge the current manufacturing capability. Therefore, reliable machining technologies must be employed to obtain consistent part shapes. Conventionally, trial-and-error methods are adopted to select cutting parameters. It is well known that such an approach will lead to a long setup time and large machining costs. Recently, the advances of computing techniques make it possible to perform machining operations in a comprehensive simulation environment. Before actual
∗
cutting, one can simulate the critical machining characteristics such as cutting forces, tool/workpiece deflection, form errors, cutter wear and vibration (Budak and Altintas, 1992, 1995; Li and Guan, 2004; Tsai and Liao, 1999; Ratchev et al., 2004a,b; Wan et al., 2005; Semercigil and Chen, 2002) firstly by modelling the cutting process and the attributes of the tool/workpiece, etc. Form error compensation and cutting parameters optimization (Ratchev et al., 2006; Depince and Hascoet, 2006; By Chen et al., 2005; Milfelner et al., 2005) are then carried out to ensure the machining quality. In fact, one fundamental and significant basis of all process simulations is the determination of the cutting forces, the accuracy of which depends on how the cutting force models and the corresponding cutting force coefficients (Koenigsberger and Sabberwal, 1961; Altintas and Spence, 1991; Budak et al., 1996; Ko et al., 2002) are established. At
Corresponding author. Tel.: +86 29 88495774; fax: +86 29 88495774. E-mail addresses: [email protected] (M. Wan), [email protected] (W.H. Zhang). 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.06.005
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123
Fig. 1 – Framework of the simulation procedure.
present, three types of cutting force models are widely used: Koenigsberger and Sabberwal (Koenigsberger and Sabberwal, 1961) and Altintas and Spence (Altintas and Spence, 1991) adopted a single coefficient to characterize the effects of the shearing on the rake face and the rubbing at the cutting edge; Budak et al. (Budak et al., 1996) separated the shearing and rubbing effects with two independent coefficients; Cho and co-workers (Ko et al., 2002) used the normal force coefficient Kn , the frictional force coefficient Kf and the chip flow angle c to characterize the cutting force model. It is well known that all above cutting force models rely on the IUCT, which will be influenced by the cutter runout and the cutter/workpiece deflections. Kline and DeVor (Kline and DeVor, 1983) studied the influence of the radial runout on the calculations of IUCT, while the cutter/workpiece deflections are ignored. Sutherland and DeVor (Sutherland and DeVor, 1986) proposed a regeneration model to calculate the IUCT by accounting for both the cutter runout and the cutter deflection. Later, Budak and Altintas (1992) proposed a force-based iteration scheme to study the influence of cutter deflection on the IUCT. It was shown that in few tooth periods, the cutting forces deduced from the regeneration model converge to those produced by using the nominal chip thickness. In the vibration free milling process of thin-walled workpiece, a significant deviation between the planned and machined part profiles is due to the form error induced mainly by the deflection of the tool and the workpiece. To avoid this, the form error prediction becomes an essential issue for machining process control. Ratchev et al. (2004a,b) studied the form error distributions of low-rigidity parts influenced by the workpiece deflection, while the cutter deflection is neglected. Coupling models involving the cutter/workpiece deflection and the workpiece rigidity diminution were proposed in Refs. (Budak and Altintas, 1995; Tsai and Liao, 1999), in which iterative schemes were used to predict the cutting forces and the
form error distributions on a thin-walled rectangular part. The workpiece model was developed by use of one-layer volume elements along the thickness direction so that the material removal can be taken into account by varying the nodal coordinates. It is important to note that the above work was focused on each individual technique, e.g., the development of cutting force model, the calculations of IUCT, the prediction of the cutter/workpiece deflections, the modelling method of material removal, etc. The motivation of this paper is to develop a systematic procedure for simulating the peripheral milling process of thin-walled workpiece. The integrated framework is illustrated in Fig. 1. Key issues associated with the calibration scheme of ICFC, calculation of IUCT, coupling model of the cutter/workpiece system and the material removal model are addressed in detail. Both theoretical and numerical analyses are used to investigate the regeneration mechanism. It is shown that the regeneration phenomenon always disappears after several tooth periods even in flexible static end milling process. Comparisons of simulations and experiment cutting tests are illustrated to show the validity of the proposed procedure.
2.
Cutting force module
2.1.
Basic cutting force model
The peripheral end mill is equivalently divided into a finite number of disk elements along the cutter axis. The total X-, Yand Z-cutting forces acting on the cutter at a particular instant are acquired by numerically summing the force components acting on each individual disk element. Here, the first type of single coefficient cutting force model is considered. For the convenience of study, the jth axial disk element of the ith flute
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is noted as {i, j}. The two extreme points of disk element {i, j} are named as cutter nodes which are symbolized by (i, j) and (i, j + 1), respectively. Note that (i, j) is the cutter node nearer to the cutter free end than node (i, j + 1). For disk element {i, j}, the corresponding cutting forces in the tangential, radial and axial directions can be given as (Budak and Altintas, 1995; Tsai and Liao, 1999; Koenigsberger and Sabberwal, 1961; Altintas and Spence, 1991):
⎧ ⎪ ⎨ Fi,j,T = Ki,j,T hi,j zi,j F
=K
h z
(1)
i,j,R i,j,R i,j i,j ⎪ ⎩F = K h z i,j,z
i,j,z i,j i,j
where zi,j and hi,j are the axial length and the IUCT of {i, j}, respectively. Parameters Ki,j,T , Ki,j,R and Ki,j,z are the instantaneous cutting force coefficients corresponding to hi,j . Assume that the positive directions of axes Y and X in Cartesian coordinate are aligned with the external normal direction of the machined surface and the feed direction of the cutter, respectively. Once three force components are obtained from Eq. (1), they can be mapped along X, Y and Z directions:
⎡
Fi,j,X
⎤
⎡
Fi,j,T
⎤
⎡
Ki,j,T
⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ Fi,j,Y ⎦ = Ti,j ⎣ Fi,j,R ⎦ = hi,j zi,j Ti,j ⎣ Ki,j,R ⎦ Fi,j,Z
Fi,j,z
(2.1)
Ki,j,z
with
⎡
−cos i,j
⎢ Ti,j = ⎣ sin i,j 0
−sin i,j
0
⎤
−cos i,j
⎥ 0⎦
0
(2.2)
1
where i,j is the angular position which is defined as the clockwise angle determined from Y to the disk element {i, j}. Subsequently, the total cutting force components in X-, Yand Z-directions at an arbitrary cutting instant can be evaluated by summing the forces acting on all flutes and disk elements: Fs =
Fi,j,s ,
s = X, Y, Z
(2.3)
Fig. 2 – Geometric illustration of the instantaneous uncut chip thickness (IUCT). (a) The definition of the instantaneous uncut chip thickness (IUCT) and (b) the local amplified sketch map.
where
a=
ıx = mi f + ıt,X − ıt,X , i,j i−m ,j
ı2x + ı2y ,
i
ıy = ıt,Y − ıt,Y , i,j i−m ,j i
ˇ = − i,j + arccos
ı y
a
i,j
2.2.
Calculations of IUCT
By definition, hi,j refers to the radial distance between the tooth path to be generated by the disk element {i, j} and the workpiece surface left by the previous disk element {i − mi , j} with the same angular angle position i,j . mi means that the current tooth i is removing the material left by the mi th previous tooth. The occurrence of cutter runout will lead to mi = 1. As shown in Fig. 2, hi,j is the IUCT associated with i,j and it will change in function of i,j . Physically, due to the cutter/workpiece deflections, the axis of the cutter will shift from its nominal position. Correspondingly, the two adjacent tool paths will deviate from their nominal positions, too. By taking into account the effect of cutter/workpiece deflections, hi,j can be calculated as:
hi,j = −a cos ˇ + Radi,j −
Rad2i−m ,j − a2 sin2 ˇ i
(3)
Radi,j is the radius of the tooth path generated by disk element {i, j}. (ıt,X , ıt,Y ) and (ıt,X , ıt,Y ) denote the deviation values i,j i,j i−mi ,j i−mi ,j between the path centres of disk elements {i, j} and {i − mi , j}. f is the feed per tooth. Due to the presence of cutter runout, the rotation radius of an arbitrary cutter element {i, j} can be obtained as (Wang and Liang, 1996; Azeem et al., 2004):
Radi,j = R + cos −
2(i − 1) z tan i0 − R N
(4)
Here, R and i0 are the nominal geometry radius and helical angle of the cutter, respectively. and are the runout parameters. N is the tooth number of the cutter. As Radi−mi ,j >> ıx , Radi−mi ,j >> ıy and Radi,k−mi >> a sin ˇ, Eq. (3) can be finally approximated as: hi,j = −a cos ˇ + Radi,j − Radi−mi ,j
(5)
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Alternatively, if the influence of the cutter/workpiece deflections is negligible, hi,j is expressed as: hi,j = mi f sin i,j + Radi,j − Radi−mi ,j
(6)
In fact, for a vibration free cutting process, the cutting force must be stabilized after a transient period. Here, the satiability implies that the cutting forces in the current tooth period converge to those obtained in the previous tooth period, i.e., mi = 1 and Radi,j = Radi−1,j . The equivalent forces will induce the equivalent deflections so that following relations exist: ıt,X = ıt,X , i,j i−1,j
ıt,Y = ıt,Y i−1,j i,j
(7)
With these conditions, Eq. (6) can be simplified as: hi,j = f sin i,j
⎧ ⎪ ⎨ Fj,X (ϕ) = [−KT (ϕ) cos i,j − KR (ϕ) sin i,j ]hj (ϕ)zi,j F
(ϕ) = [KT (ϕ) sin − KR (ϕ) cos i,j ]hj (ϕ)zi,j
j,Y i,j ⎪ ⎩ F (ϕ) = K (ϕ)h (ϕ)z z j,Z j i,j
To calibrate the cutting force coefficients experimentally, the workpiece has to be a relative large block with sufficient rigidity. Meanwhile, the cutter is assumed to be a rigid one. Therefore, the influence of the cutter/workpiece deflections on the IUCT can be negligible so that Eq. (6) can be directly used to calculate the IUCT. In the calibration procedure, the cutting force coefficients are conventionally determined using measured average cutting forces. These techniques are useful, but in most cases the identified coefficients are only valid in the range of test conditions. Especially, to obtain robust application range of the related cutting force coefficients, a great number of cutting tests must be performed by changing feeds, depths of cut, etc. To save the test costs and to improve the calibration accuracy, we propose here an alternative method to calibrate the instantaneous cutting force coefficients Ki,j,T , Ki,j,R and Ki,j,z using instantaneous values of the measured cutting forces, instead of the average ones. It will be shown that just one cutting test will be needed to calibrate Ki,j,T , Ki,j,R and Ki,j,z . If the values of Fi,j,X , Fi,j,Y and Fi,j,Z can be measured in advance, Ki,j,q (q = T, R, z) can be obtained from Eq. (2.1). However, the measured cutting forces using dynamometer are available only in the form of total cutting forces Fs (s = X, Y, Z) and cannot be decomposed into force components Fi,j,X , Fi,j,Y and Fi,j,Z . These make it practically impossible to establish ‘strict’ relationship between Ki,j,q and hi,j . Therefore, an approximate approach is proposed below. As we know, i,j refers to the angular position of disk element {i, j} at the cutter rotation angle ϕ, as shown in Fig. 1. After the cutter rotation with an angle of (2k)/N (k = 1, 2, . . ., N − 1), the angular position of disk
N
hi,j
(10)
i=1
(8)
Calibration of ICFC
(9)
with
hj (ϕ) =
This is the so-called nominal value of hi,j in static milling. It means that hi,j converges to its nominal value of f sin i,j . This conclusion deduced from the basic definition of hi,j agrees completely with that from the force-based iteration of Budak (Budak and Altintas, 1992). Especially, it gives the theoretical explanation why nominal IUCT value, f sin i,j , is widely used by many researchers to study the cutting force model.
2.3.
element {i + k, j} equals i,j , too. If the cutting force coefficients are assumed to have the same values Kq (ϕ) (q = T, R, z) at ϕ + (2k)/N (k = 0, 1, 2, . . ., N − 1), the sum of each X-, Y- and Z-force component of the disk elements located at the angular position i,j over one rotation period corresponds to:
As the cutter/workpiece deflections can be ignored in the calibration procedure, Eq. (6) will be used to evaluate hj (ϕ) so that Eq. (10) can be rewritten as:
hj (ϕ) =
N
(mi f sin i,j + Radi,j − Radi−mi ,j ) = Nf sin i,j
(11)
i=1
Note that the effect of cutter radial runout on hj (ϕ) vanishes. It turns out that the sum of three force components of the cutting edge elements with the same angular position i,j is independent of the runout. The substitution of Eq. (11) into Eq. (9) and then the summation along the cutter axis produces:
⎡
FX (ϕ)
⎡
⎤
A
⎢B ⎢ ⎢ ⎥ ⎣ FY (ϕ) ⎦ = Nfzi,j ⎢ ⎣0
−B A 0
FZ (ϕ)
⎤
⎡ ⎤ KT (ϕ) ⎥ 0 ⎥⎢ ⎥ ⎥ ⎣ KR (ϕ) ⎦ ⎦ sin i,j 0
Kz (ϕ)
i,j
where
A=−
(cos i,j sin i,j )
and
(12)
B=
i,j
(sin i,j sin i,j )
i,j
Here, Fs (ϕ) (s = X, Y, Z) only depend upon cutting force coefficients and the feed per tooth no matter what the runout is. Denote FsM (ϕ) (s = X, Y, Z) to be measured cutting forces at rotation angle ϕ. Fs (ϕ) can be obtained by averaging measured cutting forces: 1 M Fs N N
Fs (ϕ) =
i=1
ϕ+
2(i − 1) N
(13)
With the known Fs (ϕ), Kq (ϕ) (q = T, R, z) can be obtained immediately from Eq. (12). This ICFC model reveals the relationship between Kq (ϕ) and ϕ. The interesting thing is that with Eqs. (12) and (13), the influence of the cutter runout is eliminated in the calibration procedure of the cutting force coefficients. The discrete values of Kq (ϕ) obtained from Eq. (12) can be fully exploited to give rise to a spectrum of cutting force coefficients for their accurate fitting.
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(k+1)
the cutter node (i, j). ϕi,j
denotes the variation of the immer(k+1)
sion angle which can be evaluated in terms of Rr
(k+1)
ϕi,j
= arccos
(k+1)
1−
Rr (i, j) Radi,j
(i, j) by:
(16)
The IUCT is iterated as: (k+1)
= −a(k) cos ˇ(k) + Radi,j − Radi−mi ,j
hi,j
(17)
with
(k)
=
a
t,X(k)
(mi f + ıi,j
Fig. 3 – Illustration of immersion angle. ˇ
3.
Calculations of cutting configuration
3.1.1. Correction of cutting parameters and numerical simulation of the regeneration phenomenon As shown in Fig. 3, due to the existence of deflections caused by the cutting forces, the ideal contact curve AB will be shortened to QB. Meanwhile, the nominal intersection line DF between the cutter and the workpiece will be shifted to EH. As a result, for any engaged cutter node, the radial depth of cut, IUCT and the immersion angle will be deviated from their nominal values. From this viewpoint, for disk element {i, j} at any cutting position, the radial depth of cut is corrected by: (k+1)
(k)
t,Y(k)
(i, j) = Rr (i, j) − [ıi,j
w,Y(k)
+ı
]
i,j (k+1) (k) (i, j) − Rr (i, j) ≤ ε, k = 0, 1, . . . , n s.t. Rr
= − i,j + arccos
(14)
t,Y(k)
(k+1)
= ui,j
zi,j
(k+1)
where ui,j denotes the correction factor in the (k + 1)th iteration defined as: (k+1)
(k+1)
and
i
a(k)
(k+1) (k+1) (k) (k) h˜ i,j = (hi,j − h˜ i,j ) + h˜ i,j ,
=
(ϕi,j
+
i,j
− )D
2zi,j tan i0
(15)
where D is the diameter of the cutter. i,j designates the angle determined clockwise from the positive direction of axis Y to
0<≤1
(18)
(k+1)
where h˜ i,j is the corrected value of the IUCT. is the weighted parameter given a priori. (2) Based on the correction factor defined in Eq. (15), the divergence during the iteration process of Eq. (14) can be avoided if the following scheme is adopted: (k+1)
u˜ i,j
(k+1)
= (ui,j
(k)
(k)
k = 0, 1, . . . , n ˜ (k+1) (k+1) − ui,j ≤ ε s.t. 0 < ≤ 1, ui,j
w,Y(k)
of cut. ıi,j and ıi,j denote the normal deflections of disk element {i, j} and the workpiece at the cutting position after the kth iteration, respectively. ε is the prescribed tolerance to control the iteration process. As shown in Fig. 3, due to the deflection, a certain number of cutter disk elements that should be completely in cut becomes now partially engaged, e.g., {i, j}. The actual contact (k+1) axial length zi,j can be written as:
ui,j
t,Y(k)
− ıi−m ,j
i
(1) A relative small change of chip thickness is used:
(k+1)
(k+1)
t,Y(k)
ıi,j
t,Y(k) 2
− ıi−m ,j )
Eq. (17) is the so-called numerical scheme for regeneration phenomenon simulation. During the iteration process of Eqs. (14) and (17), numerical oscillations may however occur due to the relative large change of radial depth of cut and the chip thickness. To ensure the convergence, the below sub-iteration algorithms are adopted:
where Rr (i, j) is the corrected radial depth of cut in the (k + 1)th iteration. Rr (i, j) is the nominal values of radial depth
zi,j
t,Y(k)
i
Cutting process module
3.1.
Rr
t,X(k) 2
− ıi−m ,j ) + (ıi,j
− u˜ i,j ) + u˜ i,j ,
(19)
(k+1)
where u˜ i,j is the corrected value of the correction factor. is the weighted parameter given a priori.
3.1.2.
Modelling of material removal
To consider the rigidity change of the workpiece due to the material removal in the machining process, the idea of softening materials is implemented to correct the element stiffness matrix in terms of its volume variation so that: Ki = i Ki
(20)
where Ki is the nominal stiffness matrix of element i. i denotes the ratio of volume variation of element i after sweeping with:
i =
Vi Vi
(10−6 = ε ≤ i ≤ 1)
(21)
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Table 1 – Cutting conditions Cutter no. A B B A A A
Test no.
Milling type
Radial depth of cut, Rr (mm)
Axial depth of cut, Rz (mm)
Feed per tooth, f (mm/tooth)
RPM
1 2 3 4 5 6
Down milling Down milling Down milling Down milling Down milling Up milling
10 10 10 10 0.95 0.95
1 1 1 1.2 29 29
0.05 0.05 0.05 0.05 0.04 0.0533
1000 300 180 300 200 300
in which Vi and Vi designate the remaining and nominal volume of element i before and after cutting, respectively. Here, a lower bound ε is used to prevent the singularity of the element stiffness matrix when the material is completely removed in milling. This treatment is more efficient than remeshing.
3.2.
Prediction of form error
The form error can be defined as the deviation of the machined surface from its nominal position. In the peripheral milling process, the cutter/workpiece deflections in the direction normal to the machined surface constitute the form errors. At a certain position P, the form error ep is evaluated by: eP = ıt,Y + ıw,Y i,j i,j
sion of 102 mm × 29.4 mm × 2.21 mm. The plate of Test 6 is also flexible with a dimension of 102 mm × 29.4 mm × 3.26 mm. 3D finite element meshes are used to discretize the workpiece. Cutter A has a tool gauge length of 61 mm and is discretized into 61 axial elements. Especially, Cutter A is designed as a single flute in order to eliminate the runout effects on form error distributions in Test 5 and Test 6. The Young’s moduli of the cutter and the workpiece are 600 GPa and 71.7 GPa, respectively. The flexibility of the plates is considerable when the cutting is carried out with a radial depth of 0.95 mm and axial depth of cut of 29 mm. Cutting forces are measured with a Kistler 9255B dynamometer.
4.1.
Calibration of the cutting force coefficients
(22)
and ıw,Y are the normal projections of the tool where ıt,Y i,j i,j deflection and the workpiece deflection corresponding to P, respectively. ıt,Y is calculated according to the method i,j presented in Ref. (Budak and Altintas, 1995), while ıw,Y is cali,j culated using FEA method. It is well recognized that the cutting forces are strongly influenced by the cutter/workpiece deflections as well as the rigidity variation of the workpiece due to material removal. In different situations, two available models are selected to calculate the form errors: (1) Rigid model. It is a simplified model in which the cutter/workpiece deflections and the workpiece rigidity change caused by the material removal are ignored. The form errors will be calculated directly based on the nominal cutting parameters. (2) Flexible model. Both the cutter/workpiece deflections and the workpiece rigidity change induced by material removal are taken into account. At any cutting instant, Eqs. (18) and (19) are used to correct the radial depth of cut and the IUCT.
The purpose of this section is twofold. On one hand, the verification will be made to show whether the runout effect is able to be eliminated by averaging the measured data with Eq. (13) if a multiple teeth cutter, i.e., Cutter B in our case is used to calibrate the cutting force coefficients. On the other hand, it will be shown how the cutting force coefficients are practically calibrated with the above procedure. To do these, the cutting force data associated with Cutter A are firstly measured and will be considered to be the authentic nominal values of Fs (ϕ) (s = X, Y, Z) as it is a single flute one without runout effect. These nominal values will be then used as reference data to check the validity of Eq. (13) applied to Cutter B. By virtue of Eq. (13), obtained averaged cutting force curves associated with Test 2 and Test 3 are shown in Fig. 4. They are very close to reference curves of Test 1 related to Cutter A. Therefore Eq. (13) is efficient to eliminate the runout effect. Now, Test 4 will be selected to calibrate the cutting force coefficients for Cutter A. Calibrated Kq (ϕ) (q = T, R, z) are shown in Fig. 5 versus the instantaneous average chip thickness
4. Example and the experimental verification Two carbide flat end mills with identical diameter of 20 mm and a helix angle of 30◦ are adopted to cut aluminum alloy 7050. The only difference is that Cutter A is a single-fluted flat end mill and Cutter B is a two-fluted one. Table 1 shows the cutting conditions. Tests 1–4 are used to cut a thick block for calibration of the cutting force coefficients. Test 5 and Test 6 are selected cantilever plates used to validate the proposed cutting force model. The plate of Test 5 is flexible with a dimen-
Fig. 4 – Comparison of averaged cutting forces.
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Fig. 5 – Cutting force coefficients.
(IACT), h(ϕ) that is calculated at ϕ by:
N i,U h(ϕ) =
i=1
N
i,L
i=1
f sin d
(23)
(i,L − i,U )
where i,L and i,U are the minimum and maximum angular positions of cutting edge i. As an exponent-like relation exists between Kq (ϕ) and h(ϕ), the following nonlinear fitting function is used: Kq (ϕ) = Wq1 + Wq2 e[Wq3 h(ϕ)]
(q = T, R, z)
Fig. 6 – Cutting forces FY of Test 5. (a) Iteration results and (b) comparison of measured and predicted cutting forces.
(24)
with Ws1 , Ws2 and Ws3 being constants to be determined. The fitted coefficients are also illustrated in Fig. 5. The overall trend of the fitted results corresponds well to the discrete values calculated. To characterize the relationship between Ki,j,q and the instantaneous uncut chip thickness hi,j of every
4.3.
Prediction of the form errors
Figs. 9 and 10 show the form errors along the cutter axial direction at two feed stations of Test 5 and Test 6 predicted by the
disk element, h(ϕ) is replaced by hi,j in Eq. (24) for calculation of Ki,j,q . This relationship is used for the numerical simulations through all cutting tests given below.
4.2.
The distribution of IUCT
With the aid of the ICFC obtained above and the iterative algorithms presented in Section 3.1.1, the distributions of the IUCT and the radial depth of cut are calculated and plotted for Test 5 and Test 6. Figs. 6 and 7 show that the cutting force based on the iterated chip thickness convergences to that using nominal chip thickness in both magnitude and shape after few tooth periods. Take an example. The convergence histories of the instantaneous uncut chip thickness associated with cutter element {1, 22} in down milling and cutter element {1, 12} in up milling versus the cutter angular position during the milling process are illustrated in Fig. 8. The iteration value converges very quickly to the nominal value after several rotation angles whatever it is concerned with down milling or up milling. All these curves confirm that the regeneration mechanism is short lived and the cutting forces can be well predicted by using the nominal chip thickness, as analysed from the theoretical viewpoint in Section 2.2. Therefore, the nominal values of the IUCT defined in Eq. (8) can be directly used to predict the cutting forces and the form errors.
Fig. 7 – Cutting forces FX of Test 6. (a) Iteration results and (b) comparison of measured and predicted cutting forces.
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Fig. 8 – Evolution of the chip thickness vs. cutter rotation angle: (a) Test 5 and (b) Test 6. Fig. 10 – Form errors of Test 6. Form errors in the position where the cutter feeds (a) 12 mm and (b) 84 mm.
present simulation model and measured by experiment. It can be seen that the results predicted by Rigid model and Flexible model have a good coherence in the variation tendency when compared with the experimental data. Especially, the Flexible model is shown to be more reliable because the material removal and the coupling effect are considered. The iteration history of correction factor u1,16 in down milling is shown in Fig. 10 when the cutter has a rotation of 26.65◦ from the feed position 42 mm. The iteration scheme is very efficient because it is stabilized after only a few steps. In summary, all these curves demonstrate the validity of the cutting force model developed in this paper.
5.
Fig. 9 – Form errors of Test 5. Form errors in the position where the cutter feeds (a) 42 mm and (b) 90 mm.
Discussion
In the vibration free milling process of low-rigidity thin-walled parts, deflections induced by cutting forces are inevitable to cause surface form errors that will deteriorate extremely the accuracy and quality of the parts. If the form errors violate seriously the dimensional tolerance, the machining process will lead to waste products. To reduce the errors, one of the most efficient strategies is to simulate the milling process combining the finite element method with cutting mechanism and furthermore to optimize the cutting parameters to improve the cutting process. The proposed study is focused on the simulation techniques of cutting force modelling and form error prediction. The simulation procedure is developed by considering the chip thickness variation (or regeneration) both
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surface form errors. All these show that our simulation procedure is reliable and has good robustness. The good consistency between the predicted and measured results also shows that the relationship between the instantaneous cutting force coefficients and the instantaneous uncut chip thickness can be established by fitting the discretized values of the coefficients that are calibrated directly from the measured data as function of the IACT of the whole cutter. Then, instantaneous uncut chip thickness of every separate disk element can be substituted into the fitted function for calculating the instantaneous cutting force coefficients associated with every single disk element. Fig. 11 – Test 5. Iteration history of correction factor u1,16 .
theoretically and numerically. Down milling and up milling are used for model verification. Numerical results show that the value of instantaneous uncut chip thickness hi,j will converge to its nominal value of f sin i,j in few tooth periods (Figs. 6 and 7) even in flexible static milling process. These results just give a corresponding numerical explanation of the conclusions deuced from the theoretical derivation in last paragraph in Section 2.2. Especially, the conclusion that is based on the basic definition of hi,j agrees completely with that from the force-based iteration of Budak and Altintas (1992). This gives the convictive explanation why nominal IUCT value, f sin i,j , is widely used by many researchers to study the cutting force model in static milling process. The surface form errors are predicted by modelling the mill deflection using cantilevered beam theory and modelling the workpiece deflection using finite element analysis. The coupling equilibrium state between the cutter and the workpiece is obtained using the improved iterative scheme that takes into account the recursive correction between cutting force, cutter/workpiece deflection and the immersion angle. From Fig. 9, it can be seen that the form errors are smallest at the cantilevered bottom of the workpiece, but they are not zero because of the cutter deflections. As the surface generation point shifts to the free end of the workpiece (Z-axis), the form errors increase due to the increase of the cutter stiffness and the workpiece flexibility. By comparing Fig. 9(a) with (b), it is found that both predicted and measured results indicate increasing trend in the form error amplitudes in the feed direction. This is due to the decreasing stiffness of the workpiece as a result of material removal. All these phenomena are in satisfactory agreement with the available referenced ones (Budak and Altintas, 1995; Tsai and Liao, 1999). It is worth mentioning that all existing results associated with the milling process of thin-walled workpiece are focused on the down milling. Alternatively, Fig. 10 gives an inspection of form error distribution in up milling. Compared with the experiment results (Figs. 9 and 10), it indicates that the proposed methodology can be successfully used to simulate both down milling process and up milling process. Especially, from Figs. 6–11, it can be seen that although the cutting force coefficients are calibrated only by one test with a considerable saving of experiment costs, the proposed approach has good accuracy in prediction of cutting force and
6.
Conclusions
A systematic procedure has been constructed to simulate the peripheral milling process of thin-walled workpiece. The developed procedure is able to predict the milling precision before actual cutting. It comprises of two major modules: cutting force module and cutting process module. Some significant improvements have been conducted about calculations of IUCT, calibrations of ICFC, corrections of cutting parameters and the material removal modelling. Especially, both theoretical and numerical analyses prove that the regeneration phenomenon is short lived in flexible static end milling. This indicates that the nominal value of the IUCT can be directly adopted in the simulation process. The form errors in flat end milling of thin-walled workpiece are studied numerically and experimentally to show the validity of the proposed procedure. It turns out that the developed procedure is an essential part of subsequent research on developing efficient methodology for optimum selection of cutting process parameters and tool path parameters in high-accurate machining of complex low-rigidity parts.
Acknowledgements This work is supported by the Doctorate Creation Foundation of Northwestern Polytechnical University (Grant no. CX200411), the Youth for NPU Teachers Scientific and Technological Innovation Foundation and the Natural Science Foundation of Shaanxi Province (Grant no. 2004E2 17).
references
Altintas, Y., Spence, A., 1991. End milling force algorithms for CAD systems. CIRP Ann. 40 (1), 31–34. Azeem, A., Feng, H.Y., Wang, L.H., 2004. Simplified and efficient calibration of a mechanistic cutting force model for ball-end milling. Int. J. Mach. Tools Manuf. 44, 291–298. Budak, E., Altintas, Y., 1992. Flexible milling force model for improved surface error predictions. In: Proceedings of Engineering System Design and Analysis, vol. 47(1), ASME, Istanbul, Turkey, pp. 89–94. Budak, E., Altintas, Y., 1995. Modeling and avoidance of static form errors in peripheral milling of plates. Int. J. Mach. Tools Manuf. 35, 459–476. Budak, E., Altintas, Y., Armarego, E.J.A., 1996. Prediction of milling force coefficients from orthogonal cutting data. ASME J. Manuf. Sci. Eng. 118, 216–224.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 7 ( 2 0 0 8 ) 122–131
By Chen, J.-S., Huang, Y.-K., Chen, M.O.-S., 2005. Feedrate optimization and tool profile modification for the high-efficiency ball-end milling process. Int. J. Mach. Tools Manuf. 45, 1070–1076. Depince, P., Hascoet, J.-Y., 2006. Active integration of tool deflection effects in end milling. Part 2. Compensation of tool deflection. Int. J. Mach. Tools Manuf. 46, 945–956. Kline, W.A., DeVor, R.E., 1983. The effect of runout on cutting geometry and forces in end milling. Int. J. Mach. Tool Des. Res. 23 (2/3), 123–140. Ko, J.H., Yun, W.S., Cho, D.W., Ehmann, K.F., 2002. Development of a virtual machining system. Part 1. Approximation of the size effect for cutting force prediction. Int. J. Mach. Tools Manuf. 42, 1595–1605. Koenigsberger, F., Sabberwal, A.J.P., 1961. An investigation into the cutting force pulsations during milling operations. Int. J. Mach. Tool Des. Res. 1, 15–33. Li, X.L., Guan, X.P., 2004. Time-frequency-analysis-based minor cutting edge fracture detection during end milling. Mech. Syst. Signal Process. 18 (6), 1485–1496. Milfelner, M., Cus, F., Balic, J., 2005. An overview of data acquisition system for cutting force measuring and optimization in milling. J. Mater. Process. Technol. 164/165, 1281–1288.
131
Ratchev, S., Liu, S., Huang, W., Becker, A.A., 2004a. A flexible force model for end milling of low-rigidity parts. J. Mater. Process. Technol. 153/154, 134–138. Ratchev, S., Liu, S., Huang, W., Becker, A.A., 2004b. Milling error prediction and compensation in machining of low-rigidity parts. Int. J. Mach. Tools Manuf. 44, 1629–1641. Ratchev, S., Liu, S., Huang, W., Becker, A.A., 2006. An advanced FEA based force induced error compensation strategy in milling. Int. J. Mach. Tools Manuf. 46, 542–551. Semercigil, S.E., Chen, L.A., 2002. Preliminary computations for chatter control in end milling. J. Sound Vib. 249, 622–633. Sutherland, J.W., DeVor, R.E., 1986. An improved method for cutting force and surface error prediction in flexible end milling systems. ASME J. Eng. Ind. 108, 269–279. Tsai, J.S., Liao, C.L., 1999. Finite-element modeling of static surface errors in the peripheral milling of thin-walled workpieces. J. Mater. Process. Technol. 94, 235–246. Wan, M., Zhang, W.H., Qiu, K.P., Gao, T., Yang, Y.H., 2005. Numerical prediction of static form errors in peripheral milling of thin-walled workpieces with irregular meshes. ASME J. Manuf. Sci. Eng. 127, 13–22. Wang, J.-J.J., Liang, S.Y., 1996. Chip load kinematics in milling with radial cutter runout. ASME J. Eng. Ind. 118, 111–116.