Systematic study on cutting force modelling methods for peripheral milling

ARTICLE IN PRESS International Journal of Machine Tools & Manufacture 49 (2009) 424–432

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Short Communication

Systematic study on cutting force modelling methods for peripheral milling Min Wan , Wei-Hong Zhang The Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, Ministry of Education, Northwestern Polytechnical University, P.O. Box 552, 710072 Xi’An, Shaanxi, People’s Republic of China

a r t i c l e in f o

a b s t r a c t

Article history: Received 25 July 2008 Received in revised form 27 November 2008 Accepted 1 December 2008 Available online 24 December 2008

This paper systematically studies the cutting force modelling methods in peripheral milling process in the presence of cutter runout. Emphasis is put on how to efficiently calibrate the cutting force coefficients and cutter runout. Mathematical derivations and implementation procedures are carried out based on the measured cutting force or its harmonics from Fourier transformation. Five methods are presented in detail. In the first three methods the cutting force coefficients are assumed to be constants whereas in the last two they are taken as functions of instantaneous uncut chip thickness. The first method and the fifth one are taken from literatures for comparison. The second, the third and the fourth methods are original contributions, which are carried out with optimization ideas. The second method proceeds using the first and Nkth harmonic forces as the source signal while the third and the fourth are derived based on the measured cutting forces and its first harmonics. The engagement of the cutter with the workpiece is considered in these three new calibration procedures without the requirement of a prior knowledge of the actual cutter runout. Comparisons among the calibrated results from different methods are made to study the limitations and consistency of the presented methods. Experiments are also conducted to show the prediction ability of all methods. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Peripheral milling Cutting force model Cutter runout Calibration Harmonics

1. Introduction Peripheral milling is a widely used manufacturing process in aerospace, automobile and die-mold industries. The increased complexity of parts and the extreme properties of workpiece materials are continuing to challenge the current manufacturing capability. The development of cutting force models is essential for simulating the critical milling attributes such as machinability, surface quality, cutter wear/fracture and chatter and then for optimizing the cutting parameters [1–8]. Thus, cutting force modelling has been the focus of many studies. The modelling procedure is generally carried out by developing mathematical relationships between the cutting forces and the instantaneous uncut chip thickness (IUCT) through cutting force coefficients. The key of this field is how to efficiently develop the models of cutting force coefficients. A review of the literature shows that there exist two typical categories of cutting force modelling methods according to whether the influence of cutter runout on the cutting force is considered. By ignoring the cutter runout, several researchers have tried to model the cutting forces of milling process [9–16]. The presence of cutter runout causes IUCT to vary over the

 Corresponding author. Tel./fax: +86 29 88495774.

E-mail addresses: [email protected] (M. Wan), [email protected] (W.-H. Zhang). 0890-6955/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2008.12.004

rotation of a multi-tooth cutter. This varying IUCT alters cutting force distributions in different tooth periods [17,18]. Assuming that the runout is known in advance, some researchers studied the cutting force modelling method [17,19–21]. It is well recognized that the cutter offset can be measured off-line with a dial indicator or other electronic means. However, due to the complex in-process interaction and the unbalancing dynamics of the rotating assembly, the actual cutter runout (i.e., the dynamic cutter runout) during milling process could be different from the static cutter runout measured off-line [18]. Generally, it is very difficult to directly measure the dynamic cutter runout. Thus, the problem also arises about how to figure out the dynamic runout parameters. To this end, dynamic cutter runout calibrating has been the focus of many studies [22–28]. However, it is interesting to remark that the above works were carried out based on the assumption that the cutting force coefficients can be treated as constants. To improve the prediction ability, the concept of instantaneous cutting force coefficients has been introduced by many researchers [29–33]. It is also worth noting that the above works are performed for each single method individually. In this paper, five different methods are studied to calibrate the cutting force coefficients and the cutter runout in peripheral milling. Method 1 is taken from Refs. [23,24]. Methods 2–4 are originally developed in this paper. Method 5 is taken from the authors’ previous work [32,33]. The cutting force coefficients are assumed as constants in Methods 1–3 while in Methods 4–5 the cutting force coefficients are

ARTICLE IN PRESS M. Wan, W.-H. Zhang / International Journal of Machine Tools & Manufacture 49 (2009) 424–432

Nomenclature

j

cutter rotation angle instantaneous uncut chip thickness (IUCT) related to the jth disk element of the ith flute at j yi,j(j) angular position of the jth disk element of the ith flute at j f feed per tooth N tooth number of the cutter i0, R helix angle and nominal radii of the cutter z, zi,j axial coordinate and length of the jth disk element of the ith flute c(z) radial lag angle at z FT,i,j(j), FR,i,j(j) tangential and radial cutting force components related to the jth disk element of the ith flute at j FX,i,j(j), FY,i,j(j) X- and Y-components of the cutting forces related to the jth disk element of the ith flute at j FX(j), FY(j) X- and Y-components of the total predicted cutting forces related to the whole cutter at j M FM X ðjÞ, F Y ðjÞ X- and Y-components of the total measured cutting forces related to the whole cutter at j

hi,j(j)

treated as a function of IUCT. Comparisons are made among the results obtained using different methods to ensure the effectiveness of the presented methods. Experimental studies are also discussed to illustrate the implementation procedure and to verify the predicted results. Through this work, it further clearly turns out that the instantaneous cutting force model has better prediction ability than the model with constant coefficients. The main difference of this work from the most related works, i.e., Refs. [23,24,32–34], can be listed as follows: (1) The differences of this paper from Wang’s works [23,24] are twofold. First, the proposed calibration procedures are carried out with optimization idea in this paper, but those in Refs. [23,24] are not. Second, the effect of cutter runout on the entry and exit angles is considered iteratively in this paper whereas in Ref. [23,24] entry and exit angles are treated as nominal values. (2) The main difference of this paper from Ko’s Method [34] is that optimization algorithms are adopted to perform the calibration procedures in this paper, whereas Ko’s Method selects the best-fit one. This difference makes our methods very efficient. (3) The differences of this paper from the author’s previous works [32,33] are as follows. First, the cutting force coefficients and cutter runout are calibrated simultaneously in this paper; while in Refs. [32,33] the cutter runout can be identified only if the cutting force coefficients are calibrated in advance. Second, the measured cutting force is directly used for calibration in this paper, whereas in Refs. [32,33] a separation is required before calibration. (4) Different calibrating methods and different cutting force models are simultaneously adopted to study the difference and the effectiveness of the methods and models.

K T;i;j ½hi;j ðjÞ, K R;i;j ½hi;j ðjÞ tangential and radial instantaneous cutting force coefficients corresponding to hi;j ðjÞ kq, mq (q ¼ T, R) medial parameters used to establish the relationship between K q;i;j ½hi;j ðjÞ and hi;j ðjÞ KT, KR tangential and radial constant-cutting-force coefficients Kq,i,j (q ¼ T, R) abbreviated form of K q;i;j ½hi;j ðjÞ mi number indicating that the current tooth i is removing the material left by the mith previous tooth Ri,j(z), R(z) actual and ideal cutting radii of the jth disk element of the ith flute at z r, l runout offset and its orientation angle Rz, Rr the nominal axial and radial depth of cut yen,yex entry and exit angles jk cutter rotation angle corresponding to the kth sampling point of measured cutting forces Nsap total number of sampling points of measured cutting forces J unit of imaginary number RPM cutter spindle rotation speed (rotation per minute)

as shown in Fig. 1. The tangential and radial cutting force components acting on the jth disk element of the ith flute at an arbitrary j can be expressed as [31–33] F T;i;j ðjÞ ¼ K T;i;j ½hi;j ðjÞhi;j ðjÞzi;j , F R;i;j ðjÞ ¼ K R;i;j ½hi;j ðjÞhi;j ðjÞzi;j

(1)

with hi;j ðjÞ ¼ mi f sin yi;j ðjÞ þ Ri;j ðzÞ  Rimi ;j ðzÞ

(2)

  2ði  1Þp Ri;j ðzÞ ¼ RðzÞ þ r cos l  cðzÞ  N

(3)

end mill collet The jth disk element of the ith flute

zi,j B

Pj

The ith flute

2. Mechanistic cutting force model

Z O

zi,1

X The (i+1)th flute

tooth number i+2

B-B

ω

Y 94 O70

θ (ϕ )

X i+1 F

i-1

i workpiece

The end milling cutter, as usual, is divided into a finite number of disk elements with equivalent axial length along the cutter axis,

L

B z

i+3

In summary, new procedures have been proposed to model the cutting forces in peripheral milling process. Because optimization is introduced into the calibration procedure, the proposed methods are very efficient.

425

F

(ϕ )

Rr

(ϕ )

Fig. 1. Modelling of the peripheral milling process.

ARTICLE IN PRESS 426

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A review of the literature shows that K T;i;j ½hi;j ðjÞ and K R;i;j ½hi;j ðjÞ can be treated either as constants [9,11] or as functions of hi,j(j) [31–34]. FT,i,j(j) and FR,i,j(j) can be further transformed into X- and Ydirections by " # " # " # F X;i;j ðjÞ F T;i;j ðjÞ K T;i;j ¼ Ti;j ðjÞ ¼ hi;j ðjÞzi;j Ti;j ðjÞ (4) K R;i;j F Y;i;j ðjÞ F R;i;j ðjÞ " Ti;j ðjÞ ¼

cos yi;j ðjÞ

sin yi;j ðjÞ

sin yi;j ðjÞ

cos yi;j ðjÞ

# (5)

A summation over all disk elements engaged in the cut yields the total X- and Y-force components: X F s;i;j ðjÞ; s ¼ X; Y (6) F s ðjÞ ¼ i;j

Note that because the axial cutting force is relatively small, it is neglected in this study.

which are obtained from the nominal cutting parameters. For instance, yen ¼ arccosððRr  RÞ=RÞ; yex ¼ p in down milling. Issue (II): The satisfaction of Eq. (7) is related to mi ¼ 1. However, the occurrence of runout will greatly affect the actual cutting radius as well as yen and yex. This means that the above two conditions are not strictly satisfied when runout occurs. As a result, some accuracy will be lost if Method 1 is directly used. Especially, the larger the value of r, the greater the inaccuracy. To improve the calibration accuracy of Method 1, a new method will be proposed here. The details are explained and listed as follows. By reviewing Eqs. (4)–(6) under the assumption of K q;i;j ðjÞ ¼ K q (q ¼ T, R), we can obtain " # " #" # H1 ðjÞ H2 ðjÞ F X ð jÞ KT ¼ (10) F Y ð jÞ H2 ðjÞ H1 ðjÞ KR with H1 ðjÞ ¼ 

X

zi;j hi;j ðjÞcos yi;j ðjÞ,

i;j

3. Identification of the cutting force coefficients and the cutter runout In order to efficiently predict the cutting forces, it is of great importance to calibrate the values of cutting force coefficients and the cutter runout parameters a priori. To systematically understand the calibration procedure, five different calibration methods will be presented in this section. 3.1. Method 1 This is an existing method proposed by Liang and Wang [23] and Wang et al. [24]. The key ideas are presented in brief for the sake of completeness of this study. The cutting forces given in Eq. (1) are in the angle domain. Under the assumption that the cutting force coefficients KT,i,j and KR,i,j can be treated as constants independent of IUCT (symbolized as KT and KR) and mi ¼ 1, the cutting forces can be expanded as follows in frequency domain through convolution analysis [23]: "

F X ðjÞ

#

F Y ðjÞ

¼

þ1 X

("

k¼1

AX ½Nk AY ½Nk

# eJNkj þ

"

AXO ½Nk þ 1 AYO ½Nk þ 1

#

"

eJðNkþ1Þj þ

AXO ½Nk  1 AYO ½Nk  1

)

# eJðNk1Þj

(7) where As[Nk], AsO[Nk+1] and AsO[Nk1] are the parameters related to the harmonics of the predicted cutting forces, as illustrated in Eq. (1). Once the measured cutting forces are obtained, KT, KR, r and l can be immediately calibrated by #1 " M # " # " 1 AX ½0 P 1 ð0Þ P2 ð0Þ KT N CWDð0Þ (8) ¼ M P 2 ð0Þ P1 ð0Þ KR AY ½0 2p

r¼ l¼

jAM YO ½1j , sinðp=NÞðN=2pÞRz jK T P4 ð1Þ  K R P 3 ð1Þj

p 2



p

N

 ffAM YO ½1 þ ff½K T P 4 ð1Þ  K R P 3 ð1Þ

H2 ðjÞ ¼ 

M where AM s ½0 and AsO ½1 are the parameters related to harmonics of the measured cutting forces. For more details of this method and the calculation expressions of P1(0), P2(0), CWD(0), P3(1) and P4(1), one can refer to Refs. [18,23,24].

3.2. Method 2 The accuracy of Method 1 relies on the following two issues: Issue (I): The medial parameters involved in Eq. (7) (e.g., As[Nk], AsO[Nk+1], AsO[Nk1], etc.) depend on yen and yex,

zi;j hi;j ðjÞsin yi;j ðjÞ

(11)

i;j

"

The Fourier series expansion of Eq. (10) can be written as # ( )" # " # þ1 X F X ð jÞ Q 1 ½o Q 2 ½o Joj KT ¼ (12) e F Y ð jÞ Q 2 ½o Q 1 ½o KR o¼1

where Q1[o] and Q2[o] are the Fourier transforms of H1(j) and H2(j), respectively. By combining Eq. (12) with the Nkth harmonics of F M s ðjÞ at o ¼ Nk, the following relationships can be obtained: D½K T ; K R T ¼ b

(13)

with h iT M M M b ¼ ReðAM X ½NkÞ; ImðAX ½NkÞ; ReðAY ½NkÞ; ImðAY ½NkÞ , 2 3 ReðQ 2 ½NkÞ ReðQ 1 ½NkÞ 6 7 6 ImðQ 1 ½NkÞ ImðQ 2 ½NkÞ 7 6 7 D¼6 7 6 ReðQ 2 ½NkÞ ReðQ 1 ½NkÞ 7 4 5 ImðQ 2 ½NkÞ ImðQ 1 ½NkÞ

(14)

where Re(*) and Im(*) indicate the real part and the imaginary part of a complex number. Now, with the aid of Eq. (13), an optimal selection procedure can be used to determine Kq (q ¼ T, R), r and l according to the following steps. Step (2.1): Set r ¼ r0 and l ¼ l0; r0 and l0 are the initially selected values. Practically, r0 and l0 can be obtained using Method 1 so that they are close to the actual case. Step (2.2): Calculate hi,j(j) by hi;j ðjÞ ¼

(9)

X

min fhi;j ðjÞ ¼ mf sin yi;j ðjÞ þ Ri;j ðzÞ  Rim;j ðzÞg

m¼1 to N

(15)

where Ri,j(z) and Rim,j(z) are obtained by Eq. (3). Step (2.3): Calculate Q1[o] and Q2[o] based on Eqs. (15) and (11). Step (2.4): Calculate D using Eq. (14). Then, by using the linear least square method, KT and KR can be immediately obtained by ½K T ; K R T ¼ ðDT DÞ1 DT b

(16)

Step (2.5): Substitute KT and KR obtained from Eq. (16) into Eq. (10). Then, calculate the minimum square deviation d(r, l) between F M s ðjÞ and Fs(j) (s ¼ X, Y) by

dðr; lÞ ¼

2p  X

j¼0

M 2 2 jF M X ðjÞ  F X ðjÞj þ jF Y ðjÞ  F Y ðjÞj



(17)

ARTICLE IN PRESS M. Wan, W.-H. Zhang / International Journal of Machine Tools & Manufacture 49 (2009) 424–432

Step (2.6): If d(r, l) achieves the level of minimum among all cases of different r and l, set KT and KR, r and l as the final results of cutting force coefficients and runout parameters. Otherwise, repeat the above Steps (2.2)–(2.6) by setting r and l to other values rn and ln. The key issue of the above steps is to ascertain the values of rn and ln. According to Ko’s method [34], one can do this by using the so-called parametric study to optimally select rn and ln. That is, for every possible pairs of rn and ln with rmin prn prmax and lmin pln plmax , Steps 2.1–2.6 will be performed. Here, rmax and rmin denote the maximum and minimum possible values of rn, and lmax and lmin denote the maximum and minimum possible values of ln, respectively. The case which has the minimum d(r, l) corresponds to the final results. Obviously, parametric study must sweep all cases in the feasible domain. This will certainly lead to a relatively large computing time. To increase the computing efficiency, an automatic searching procedure will be described here. To do this, the key is to approximately develop the explicit expressions that relate r and l with F M s ðjÞ and Fs(j). For this reason, it is interesting to study the following test case based on the parametric study method. The cutting forces measured from Test 1 (see Table 1) are used. For more details about the experimental conditions, one can refer to Section 4. Here, the distributions of jAY[1]j and +AY[1] vs. r and l are considered. AY[1] is obtained from FY(j) using Fourier Transformation. FY(j) should be calculated from Step (2.5) for every selected set of r and l. It is worth noting that although Method 1 relies on Issues (I) and (II), the calibrated results of r and l using Method 1 are not far from the actual cutter runout. Thus, one can select the values that are close to the calibrated results of cutter runout with Method 1 as the varying range of r and l. In this paper, the calibrated results of r and l from Test 1 are 16.117 mm and 53.0621, respectively. In this study, r and l are chosen to vary from 1e-6 mm (i.e., rmin) to 35 mm (i.e., rmax) and from 401

427

(i.e., lmin) to 601 (i.e., lmax), respectively. The simulation results are shown in Fig. 2. It can be found that both jAY[1]j and +AY[1] are approximately distributed in a planar surface over the considered region. This phenomenon indicates that jAY[1]j and +AY[1] can be locally treated as linear functions of r and l. The same observations can also be made in other cutting conditions and regions of r and l. Thus, the following relations hold: jAY ½1j ¼ E11 r þ E12 l þ E13 ffAY ½1 ¼ E21 r þ E22 l þ E23

(18)

where Euv (u ¼ 1, 2, v ¼ 1, 2, 3) are unknown coefficients that can be determined using the finite-difference scheme in the following way: E11 ¼ ðjAY ½1j3  jAY ½1j1 Þ=Dr1 , E12 ¼ ðjAY ½1j2  jAY ½1j1 Þ=Dl1 , E13 ¼ jAY ½1j1  ðE11 r1 þ E12 l1 Þ, E21 ¼ ½ðffAY ½1Þ3  ðffAY ½1Þ1 =Dr1 , E22 ¼ ½ðffAY ½1Þ2  ðffAY ½1Þ1 =Dl1 , E23 ¼ ðffAY ½1Þ1  ðE21 r1 þ E22 l1 Þ

(19)

where r1, l1 is a set of selected values satisfying rmin pr1 ; r2 prmax and lmin pl1 ; l2 plmax . Assume that r2 ¼ r1 þ Dr1 , l2 ¼ l1 þ Dl1 . With the aid of Steps (2.1)–(2.5), we can obtain jAY[1]j1 and ðffAY ½1Þ1 are related to r1 and l1; jAY[1]j2 and (+AY[1])2 to r1 and l2, jAY[1]j3 and (+AY[1])3 to r2 and l1. Now, rn and ln can be easily obtained by relating Eq. (18) to the M experimental values of jAM Y ½1j and ffAY ½1 through "

rn ln

#

" ¼

E11

E12

E21

E22

#1 "

AM Y ½1  E13 ffAM Y ½1  E23

# (20)

It is worth noting that as jAY[1]j and +AY[1] are linearly approximated over a local design region, it is necessary to update

Table 1 Cutting conditions and measured cutting forces for Test 1. Cutting conditions

The measured cutting forces

Milling type

Down milling

Rz (mm) Rr (mm) f (mm/tooth) Rotation per minute (RPM) Sampling frequency (Hz) Workpiece material Cutter parameters

2 8 0.05 1000 9000 AL 7050 Three-fluted carbide flat end mill with a 301 helix angle and a 16 mm diameter

ARTICLE IN PRESS M. Wan, W.-H. Zhang / International Journal of Machine Tools & Manufacture 49 (2009) 424–432

AY [1]

428

35 30 25 20 15 10 5 0 60 40

55 30 (

50 De g.)

20 45

m (

10 40

)

0

-135

∠(AY [1])

-140 -145 -150 -155 -160 60 40

55 30 (

50 De

g.)

20 45 40

)

m (

10 0

Fig. 2. Distributions of jAY[1]j and +AY[1] vs. q and k: (a) the distribution of jAY[1]j vs. q and k and (b) the distributions of +AY[1] vs. q and k.

the approximation on the new design point in an iterative way. Now, the whole procedure for calibrating Kq (q ¼ T, R), r and l is summarized in Fig. 3.

2 D1 ¼ 4

H1 ðj1 Þ

H1 ðj2 Þ

L

H1 ðjNsap Þ

H2 ðj1 Þ

H2 ðj2 Þ

L

H2 ðjNsap Þ

H2 ðj1 Þ

H2 ðj2 Þ

L

H2 ðjNsap Þ

H1 ðj1 Þ

H1 ðj2 Þ

L

H1 ðjNsap Þ

3T 5

(22)

3.3. Method 3

The other calibrating procedure is similar to that of Method 2.

In Method 2, Kq (q ¼ T, R) is calibrated using the Nkth harmonics of the measured cutting forces, as shown in Eqs. (14) and (16). If the measured cutting forces are considered directly, one can have another calibrating scheme. This can be carried out by only changing the calibrating scheme, i.e., Eq. (16), as follows:

3.4. Method 4

½K T ; K R T ¼ ðDT1 D1 Þ1 DT1 b1 with h

iT b1 ¼ F X ðj1 Þ; F X ðj2 Þ; L; F X ðjNsap Þ; F Y ðj1 Þ; F Y ðj2 Þ; L; F Y ðjNsap Þ ,

(21)

Obviously, Methods 1–3 are all conducted under the assumption that the cutting force coefficients KT,i,j and KR,i,j can be treated as constants KT and KR. In this method, to reflect the size effect, Kq,i,j (q ¼ T, R) will be expressed as the following exponential function [31,35]: K T;i;j ¼ kT ½hi;j ðjÞmT , K R;i;j ¼ kR ½hi;j ðjÞmR

(23)

ARTICLE IN PRESS M. Wan, W.-H. Zhang / International Journal of Machine Tools & Manufacture 49 (2009) 424–432

429

with

Calculate ρ0 and λ0 based on Eqs. (8) and (9).

2

1 61 6 6 6 .. 6. 6 6 61 6 D2 ¼ 6 60 6 60 6 6. 6. 6. 4 0 2

Ln½hi;1 ðj1 Þ

0

Ln½hi;1 ðj2 Þ .. .

0 .. .

3

0

7 7 7 7 7 7 7 7 Ln½hi;1 ðjNsap Þ 0 0 7 7 0 1 Ln½hi;1 ðj1 Þ 7 7 0 1 Ln½hi;1 ðj2 Þ 7 7 7 .. .. .. 7 7 . . . 5 0 1 Ln½hi;1 ðjNsap Þ 3 Ln½F M T;i;1 ðj1 Þ  Ln½hi;1 ðj1 Þ  LnðRz Þ 6 7 6 Ln½F M ðj Þ  Ln½h ðj Þ  LnðR Þ 7 z i;1 6 7 2 2 T;i;1 6 7 6 7 .. 6 7 . 6 7 6 7 M 6 Ln½F T;i;1 ðjNsap Þ  Ln½hi;1 ðjNsap Þ  LnðRz Þ 7 6 7 b2 ¼ 6 7 6 Ln½F M 7 ð j Þ  Ln½h ð j Þ  LnðR Þ z i;1 R;i;1 1 1 6 7 6 7 M 6 Ln½F R;i;1 ðj2 Þ  Ln½hi;1 ðj2 Þ  LnðRz Þ 7 6 7 6 7 .. 6 7 . 6 7 4 5 Ln½F M R;i;1 ðjN sap Þ  Ln½hi;1 ðjNsap Þ  LnðRz Þ

Set ρ1 = ρ0 and λ1= λ0 . Calculate Euvusing Eq. (19) and then calculate ρn and λn using Eq. (20) Set ρ = ρn and λ = λn . Calculate hi,j(φ) using Eq. (15). Then, calculate Q1[ω] and Q2[ω] with the aids of Eq. (11). Determine KT and KR using Eq. (16). Calculate δ(ρ, λ) using Eq. (17).

Is the error between two iterative results of δ(ρ, λ) less than the given tolerance?

0 .. .

where Ln(*) indicates the natural logarithm operation. It is worth noting that if hi;1 ðjk Þ ¼ 0 or F M q;i;1 ðjk Þo0, the corresponding line

yes no The spectrum

60 AY []

(N)

ρ1 = ρn, λ1 = λn

Export the results of KT, KR, ρ and λ. Fig. 3. Whole procedure for Method 2.

=0

40

=3

20 0 -1

0

1

2

3

4

5

6

7

8

9

10

To derive the key schemes of this method, the following two assumptions should hold:

∠(AY [] (Deg.)

200

(I) zi,j is set to be Rz and (II) only one tooth is in contact with the workpiece at any j.

"

F T;i;1 ðjÞ F R;i;1 ðjÞ

#

" ¼ Rz

kT ½hi;1 ðjÞ1mT kR ½hi;1 ðjÞ1mR

#

"

 1 F X ðjÞ ¼ Ti;1 ðjÞ F Y ð jÞ

# (24)

The substitution of F M s ðjÞ into the right hand side of Eq. (24) gives the measured tangential and radial cutting force components, which is symbolized as F M q;i;1 ðjÞ (q ¼ T, R). Now, with the aids of Eq. (24) and F M s ðjÞ, kq, mq (q ¼ T, R), r and l can be determined by ½kt ; mT ; kr ; mR T ¼ ðDT2 D2 Þ1 DT2 b2 , kT ¼ ekt , kR ¼ ekr

(25)

=2

=0

0 -100 The phase

-200 -1

Cutting force coefficients (N/mm2)

Under these two assumptions, F s;i;1 ðjÞ (s ¼ X, Y) constitutes the total cutting force Fs(j). By reviewing Eqs. (1)–(6) and Eq. (23), we can have

100

0

1

2

3 4 5 6 7 The order  of harmonics

1800 1600

8

9

10

Method 1 Method 2 Method 3 Method 4 Method 5

KT,i,j

1400 1200 1000 800

KR,i,j

600 400 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Instantaneous uncut chip thickness (mm)

Fig. 4. Some analyzed results for Test 1 (Rr ¼ 8 mm, Rz ¼ 2 mm, f ¼ 0.05 mm/tooth, RPM ¼ 1000): (a) frequency spectrum and phase of the measured cutting forces and (b) calibrated results of cutting force coefficients from different methods.

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coolant. The workpiece material is Al 7050. Dynamometer 9255B is used to measure the cutting forces. Fig. 4(a) shows the spectrums and phases of the cutting forces measured from Test 1 after Fourier transformation. It can be seen that the amplitude of the Nkth harmonics decreases as k increases. Thus, to ensure calibration accuracy, k ¼ 1 is used in Method 1. That is, the third harmonic is substituted into Eq. (16) to calibrate KT and KR. Fig. 4(b) shows the relationships between the calibrated cutting force coefficients and the IUCT using Methods 1–5. The calibrated results of cutter runout are as follows: (1) for method 1: r ¼ 16:117 mm; l ¼ 53062 ; (2) for method 2: r ¼ 19:022 mm; l ¼ 58:001 ; (3) for method 3: r ¼ 18:634 mm; l ¼ 57:438 ; (4) for method 4: r ¼ 19:885 mm; l ¼ 57:471 ; (5) for method 5: r ¼ 18:839 mm; l ¼ 56:996 . It can be seen that the values of r and l obtained from Methods 2–5 are very close to each other. This means that Methods 2–5 have very good consistency for calibration of runout. By comparison, the results from Method 1 are lesser than those from Methods 2–5. The reason for this is that the values of yex and mi are artificially assumed to be constant at p and 1 in Method 1, respectively, whereas in Methods 2–5, yex and mi are iteratively updated according to actual engagement of the cutter with the workpiece. It is worth noting that the calibrating procedures of Methods 2–4 converge to the final values within 25 iteration steps. If we just use the step lengths in Ref. [34] for parametric investigation, it requires more than 360 iteration steps. It turns out that the proposed methods in this paper are efficient.

should be canceled from D2 and b2. By replacing Eq. (16) with Eq. (25), one can complete this method according to the similar procedure as shown in Fig. 3. 3.5. Method 5 This method is an existing one proposed in Refs. [32,33], where KT,i,j and KR,i,j are expressed as alternative functions of IUCT: K q;i;j ¼ W q1 þ W q2 e½W q3 hi;j ðjÞ ;

q ¼ T; R

(26)

Wq1, Wq2 and Wq3 being constants to be determined from calibration test. The cutting force coefficients are calibrated by the nominal component separated from the measured cutting forces whereas the runout parameters are determined from the perturbation component. For more details of this method, one can refer to Refs. [32,33].

4. Experimental verification Three down milling tests are conducted to verify the above methods. The cutting conditions are listed in the title of Fig. 4 and 5. A three-fluted carbide flat end mill with a 301 helix angle and 16 mm diameter is used. Test 1 is used to calibrate the cutting force coefficients and cutter runout using the proposed methods. Tests 2 and 3 are used to validate the cutting force model developed based on Test 1. All cutting tests are conducted without

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Fig. 5. Measured and predicted cutting forces: (a) Test 2 (Rr ¼ 8 mm, Rz ¼ 1.5 mm, f ¼ 0.05 mm/tooth, RPM ¼ 2000) and (b) Test 3 (Rr ¼ 8 mm, Rz ¼ 3 mm, f ¼ 0.01167 mm/ tooth, RPM ¼ 1000).

ARTICLE IN PRESS M. Wan, W.-H. Zhang / International Journal of Machine Tools & Manufacture 49 (2009) 424–432

An interesting and meaningful verification of the cutting force coefficients and runout calibrated by the above-mentioned methods is achieved by comparison of the predicted and measured cutting forces under different cutting conditions, as shown in Figs. 5(a) and (b). Obviously, in some tooth periods, the predicted cutting forces are far from the measured ones if the cutting force model established with Method 1 is used. This is due to the assumption of yex ¼ p and mi ¼ 1. Under this assumption, accurate calibration can be obtained only when Rz 40:2pR= ðN tanði0 ÞÞ, as stated in Ref. [18]. However, in Test 1, Rz ¼ 2o 2:9  0:2pR=ðN tanði0 ÞÞ. As a result, the cutting force model obtained from Test 1 with Method 1 will have inaccurate predictions. This relation further shows that the accuracy of Method 1 is conditionally ensured. From Fig. 5(a), it can be seen that the cutting forces predicted by the models from Methods 2–5 are very close to each other and also close to the measured ones. From a synthetic analysis of the predicted cutting forces in different tooth periods in Fig. 5(b) it can be seen that the results predicted by both Methods 4 and 5 are close to the measured ones; while the results from Methods 1–3 are relatively far from the experimental results. The reasons for the above results are as follows. In Test 2, relatively large feed per tooth is adopted. This leads to relatively large IUCT in most cutting instants. Correspondingly, Kq,i,j obtained from different methods are close to each other in these cutting instants. For instance, KT,i,j is close to 1200 N/mm2 when hi,j(j) is from 0.02 to 0.08 mm, as shown in Fig. 4(b). As a result, the forces predicted by Methods 2–5 will be close to each other. However, in Test 3, a small feed per tooth is adopted. This will produce relatively small IUCT in most cutting instants, whereas Kq,i,j calibrated by Methods 4 and 5 have dramatic variations due to the size effect. However, Kq,i,j from Methods 1–3 are still assumed to be constants at these instants. This leads to the deviations between the results predicted by Methods 2–5.

5. Conclusions The cutting force modelling procedure for peripheral milling has been systematically investigated. Five different methods for calibrating the cutting force coefficients and cutter runout have been analyzed in detail. Both constant-cutting-force model (see Methods 1–3) and instantaneous cutting force model (see Methods 4 and 5) have been considered. Method 2 is carried out based on the first and the Nkth harmonic forces whereas Methods 3 and 4 proceed by using the measured cutting force and the first harmonics. Methods 1 and 5 are taken from literatures for comparison. The actual engagement of cutters with workpieces is considered in the proposed procedures (i.e., Methods 2–4) iteratively. Experiments are also conducted to demonstrate the validity of the presented methods. It is found that: (1) Because the effects of cutter runout on entry and exit angles are considered in Methods 2–4, they have higher calibration accuracy than Method 1. (2) Methods 2–4 become efficient by using optimization algorithms. (3) When f is relatively large, the five methods have good consistency for cutting force prediction. On the other hand, the constant-cutting-force model (i.e., Methods 1–3) will fail in highly accurate predictions in the case of small f. (4) Because Methods 4–5 consider the size effect of IUCT, they always have good prediction ability even if f is small. That is, instantaneous cutting force model is a recommendable one for highly accurate prediction of cutting force.

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Acknowledgement This work is supported by the NPU Foundation for Fundamental Research (Grant No. JC200810), the National Key Technology R&D Program (Grant No. 2008BAF32B04), the Aeronautical Science Foundation of China (Grant No. 2008ZE53038), the Ao-Xiang Star Program of NPU, the Doctorate Creation Foundation of Northwestern Polytechnical University (Grant No. CX200411), the Youth for NPU Teachers Scientific and Technological Innovation Foundation and the 111 Project (Grant No. B07050).

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