- Email: [email protected]
Note on a novel method for machining parameters optimization in a chatter-free milling process
International Journal of Machine Tools & Manufacture 72 (2013) 11–15
Contents lists available at SciVerse ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Short Communication
Note on a novel method for machining parameters optimization in a chatter-free milling process Xiaoming Zhang n, Han Ding State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China
art ic l e i nf o
a b s t r a c t
Article history: Received 14 April 2013 Accepted 25 April 2013 Available online 9 May 2013
Machining parameters optimization in a chatter-free milling process is a common issue for practical engineers at shop floor. The detailed and easily applicable optimization algorithm taking the milling mechanism into account is absent to date. This communication deals with the procedures of optimization of machining parameters in the chatter-free milling process. First, the model of machining parameters optimization is given based on the milling dynamics. The optimization objects are to minimize the tool vibration response and maximize the material removal rate, and the constraint conditions are to keep the milling process chatter-free or stable and let the spindle speed below a predefined one. Emphasis is put on how to select the spindle speed for reducing the tool vibration response and maximization of the material removal rate simultaneously. Augmented Lagrangian function method is employed to solve the nonlinear optimization problem. A numerical example is provided to show the effectiveness of the developed results. Besides, improvements on the calculation scheme of the tool vibration response have been made. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Milling dynamics Machining parameters optimization Tool vibration response
1. Introduction Machining parameters optimization in a milling process plays an important role in the practical manufacturing engineering. The aims are to improve the part quality and maximize the material removal rate (MRR). Traditionally, trial-and-error and heuristic approaches are employed to obtain the optimal machining parameters. It is well recognized that these methods are timeconsuming and will lead to long machining periods and large machining costs. The process of searching for an optimal machining parameters cannot be effectively carried out without reliable optimization procedures. Providing accurate and efficient optimization procedures to obtain the optimal machining parameters is therefore an important activity. In this communication, we employ the augmented Lagrangian function method for solving the machining parameters optimization formulation. The emphasis is how to select the optimal spindle speed for reducing tool vibration response and maximization of MRR simultaneously in an effective and a reliable way. The problem of machining parameters optimization has been tackled by many authors and many works have been reported in the literature. Among them we quote the works in [1–3] and the references therein. These references have focused on mathematical
n
Corresponding author. Tel.: +86 2787559842. E-mail address: [email protected] (X. Zhang).
0890-6955/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmachtools.2013.04.006
modeling approaches to determine optimal machining parameters with regard to various objective functions. But the machining parameters optimization with chatter-free condition constraint has been addressed little. Budak and Tekeli [4] proposed a method to determine the optimal combination of axial and radial depths of cut so that chatter-free MRR is maximized. The optimal machining parameters are derived based on the two Lobe diagrams, which express the relations of stable axial depth of cut verse spindle speed and stable radial depth of cut verse spindle speed, respectively. But the tool vibration response was not taken into account, so the part quality cannot be guaranteed. Insperger et al. [5] discussed the selection of optimal spindle speed considering both the stability diagram and the surface location error diagram. But the method adopted there is heuristic and the detailed mathematical programming model is absent. Wan et al. [6] developed a linear programming model to increase the cutting efficiency and to satisfy the tolerance in the meantime. The work is successful based on the fact that the explicit linear relationship expressions of the part error in terms of machining parameters, i.e. feed per tooth and radial depth of cut could be derived. But there is no linear relationship between the part error in terms of spindle speed. So the method proposed in Ref. [6] cannot be adopted directly to tackle the machining parameters optimization problem taking the spindle speed into account. To the best of our knowledge the methodology we are using in this communication to tackle the optimization of machining parameters in a chatter-free milling process has never been used before. The optimization of machining parameters in the
12
X. Zhang, H. Ding / International Journal of Machine Tools & Manufacture 72 (2013) 11–15
M N t v x yp v Φ Ω ζ φ
Nomenclature a b C fz g K Kn Kt LA
radial depth of cut axial depth of cut damping matrix of tool feed per tooth switching function stiffness matrix of tool normal cutting force coefficients tangential cutting force coefficients augmented Lagrangian function
chatter-free milling process is modeled as a nonlinear optimization formulation, in which the stability diagram as a constraint condition is approximated by a polynomial curve. The nonlinear optimization problem is solved by the augmented Lagrangian function method. Besides, improvements on the calculation scheme of the tool vibration response have been made, which is not addressed in Refs. [5,3]. The rest of this communication is organized as follows. In Section 2, a description of the milling dynamics model is given. And also calculation scheme of the tool vibration response is presented. Section 3 presents the model of machining parameters optimization for chatter-free milling process, and detailed optimization procedures. Numerical example is given to show the effectiveness of the proposed methodology to handle the machining parameters optimization problem. Conclusions are reached in Section 4.
2. Milling model 2.1. Milling dynamics The standard two degrees of freedom milling process is shown in Fig. 1. The tool is assumed to be compliant relative to the rigid workpiece. The vibration is excited by the summation of cutting force. The governing equation of motion has the following form: € þ CxðtÞ _ þ KxðtÞ ¼ HðtÞb½xðtÞ−xðt−τÞ þ GðtÞ MxðtÞ with " M¼
mx
0
0
my
# ;
" C¼
cx
0
0
cy
#
" ;
K¼
kx
0
0
ky
ð1Þ
#
where M, C, K, and F are the modal mass, damping, stiffness, and cutting force matrices, respectively. The terms mx;y , cx;y , kx;y and F x;y are the corresponding components in the flexible directions of the system. b is the axial depth of cut. τ ¼ 60=NΩ is the tooth passing period in seconds, in which N is the number of teeth on the cutting tool and Ω the spindle speed in rpm. xðtÞ ¼ ½xðtÞ yðtÞT is the dynamic response vector and xðtÞ−xðt−τÞ the dynamic chip thickness, see Fig. 1.
mass matrix of tool number of tool tooth time vector of machining parameters dynamic response vector tool vibration response along the Y-direction vector of machining parameters transition matrix of milling system spindle speed damping coefficient angular position of the cutting edge
HðtÞ is the matrix given as follows, which represents the component of cutting forces that depend on the position vector " # N −K t sc−K n s2 −K t c2 −K n sc HðtÞ ¼ ∑ g½φj ðtÞ ð2Þ K t sc−K n c2 K t s2 −K n sc j¼1 where Kt and Kn are the tangential and normal cutting force coefficient components, respectively. For a tool with N evenly spaced teeth, φj is the angular position of the jth cutting edge and expressed as φj ðtÞ ¼
2πΩ 2πðj−1Þ tþ ; 60 N
j ¼ 1; 2; …; N
s ¼ sin φj ðtÞ, c ¼ cos φj ðtÞ and the function g½φj ðtÞ acts as a switching function, which is equal to 1 if the tooth is active and 0 if it is not cutting 1; φe o φj ðtÞ oφa g½φj ðtÞ ¼ 0 otherwise where φe and φa are the angles where the jth tooth enters and exits the cut, respectively. For down-milling operation, φa ¼ π, for up-milling, φe ¼ 0. Note that the entry and exit angles may vary due to heavy vibrations of the tool. This effect is neglected here, and the angles φe and φa are approximated by constant values as it is usually done in the literature. GðtÞ is the vector that represents the components of the cutting force which are independent of the position vector " # N K t sc þ K n s2 GðtÞ ¼ ∑ g½φj ðtÞbf z ð3Þ −K t s2 þ K n sc j¼1 where f z ¼ vf τ denotes the feed per tooth and vf is the feed speed. 2.2. Milling stability prediction and calculation of tool vibration response To simplify the milling dynamics, the motion of the tool is decomposed in the following form [5]: xðtÞ ¼ xp ðtÞ þ ξðtÞ
ð4Þ
where xp ðt þ τÞ ¼ xp ðtÞ is the forced periodic chatter-free motion of the tool, and xp ðtÞ ¼ ½xp ðtÞ; yp ðtÞT , and ξðtÞ is a perturbation corresponding to the self-excited vibration of the system. Substituting Eq. (4) into Eq. (1) results in € þ CξðtÞ _ þ KξðtÞ þ Mx€ p ðtÞ þ Cx_ p ðtÞ þ Kxp ðtÞ MξðtÞ ¼ HðtÞb½ξðtÞ−ξðt−τÞ þ GðtÞ
ð5Þ
For the ideal case, when no chatter arises, i.e., ξðtÞ≡0, and the motion is described by xðtÞ ¼ xp ðtÞ, the corresponding equations of motion are the ordinary differential ones Fig. 1. Dynamic chip thickness.
Mx€ p ðtÞ þ Cx_ p ðtÞ þ Kxp ðtÞ ¼ GðtÞ
ð6Þ
X. Zhang, H. Ding / International Journal of Machine Tools & Manufacture 72 (2013) 11–15
The assumption in Eq. (4) is appropriate, if Eq. (6) has a τperiodic solution. Since the excitation GðtÞ is τperiodic, the stationary (particular) solution of Eq. (6) is also τperiodic. This validates assumption (4). The detailed analysis of validation of assumption in Eq. (4) can be found in Ref. [5]. Eqs. (5) and (6) imply the equations € þ CξðtÞ _ þ KξðtÞ ¼ HðtÞb½ξðtÞ−ξðt−τÞ MξðtÞ
ð7Þ
Eq. (7) determines the stability of the milling system. Milling stability has been investigated by several researchers. The stability analysis of the milling system can be performed only by applying approximated numerical methods since there is no closed solution to time-delay differential equation of milling dynamics. Alternatively it can be carried out by means of time-domain simulations [7], in the frequency domain [8] or by applying methods based on delay differential equation theory, such as the semi-discretization method [9] and the time finite element approach (TFEA) [10]. Here we adopt the semi-discretization method [9] to obtain the Lobe diagram, where the combinations of cutting depth and spindle speed below the Lobe diagram curve state the stability of milling process. The detailed stability analysis is omitted here. When the milling process is stable, the tool vibration response along the Y-direction affects greatly on the part quality. From Eq. (6), we can get my y€ p ðtÞ þ cy y_ p ðtÞ þ ky yp ðtÞ N
s:t:
MRR
for series of
2
LA ðv; s; λÞ ¼ f ðvÞ
8 1 2 > > < −λi ci ðvÞ þ si ci ðvÞ 2 þ ∑ > 1 2 i ¼ 1;2;3> : − λi =si 2
j¼1
ð9Þ
where the coefficients A1;2;3 , ω1;2;3 , ϕj1;j2;j3 , ζ, and ωn are given in the Appendix.
ð13Þ
otherwise
where v ¼ ½b; ΩT f ðvÞ ¼ maxjyp ðvÞj c1 ¼ lðvÞ c2 ¼ Ω−Ω0 j ¼ 1; 2; …; m
where lðvÞ is the cubic polynomial curve to interpolate the obtained Lobe diagram λmax ðAðb; ΩÞÞ−1 ≤0. We give the algorithmic framework as follows:
Step 1: Given initial points v1 and λ1i ; s1i 4 0 , tolerance ε≥0, k≔1. Step 2: Find an approximate minimizer vkþ1 of LA ðv; sk ; λk Þ, i.e.
3. Nonlinear optimization of machining parameters in the chatter-free milling process
vkþ1 ¼ arg min LA ðv; sk ; λk Þ. If ∥c− ðvkþ1 Þ∥∞ ≤ ε, STOP and report the minimizer. Step 3: Let 8 < ski if c−i ðvkþ1 Þ ≤ 14jc−i ðvk Þj : maxf10ski ; k2 g
3.1. Model of optimization of machining parameters In general, a machining parameters optimization problem can be stated as follows: min
ð10Þ
where the optimization objects are to minimize the tool vibration response along the Y-direction, denoted as maxjyp ðb; ΩÞj and maximize MRR, i.e. f MRR ðb; ΩÞ, and constraint conditions are to keep the milling process stable, which means that the maximal eigenvalue of transition matrix Φ is less than or equal to 1 [9]; and the spindle speed is less than the predefined one, which is set by the spindle system and tool suppliers. Here MRR is defined as
otherwise
set k≔k þ 1 and update Lagrange multipliers maxfλki −ski ci ðvkþ1 Þ; 0g to obtain λkþ1 and go to Step 2.
¼ λkþ1 i
To obtain the minimizer vkþ1 ¼ arg min LA ðv; sk ; λk Þ in Algorithm 1, we need the differentiation of the augmented Lagrangian function with respect to the machining parameters ∇v LA ðv; s; λÞ. The detailed expressions are given as follows: ∇v LA ðv; s; λÞ ¼ ∇v f ðvÞ −λi ∇v ci ðvÞ þ si ci ðvÞ∇v ci ðvÞ þ ∑ i ¼ 1;2 0
if ci ðvÞ−μi λi o 0 otherwise ð14Þ
With the expressions of yp in Eq. (9), ∇v f ðvÞ can be easily obtained as follows: ∇v f ðvÞ ¼ ½maxj∇b yp ðb; ΩÞj; maxj∇Ω yp ðb; ΩÞjT
ð15Þ
ð11Þ
where a is the axial depth of cut. The optimization formulation presented in Eqs. (10) can be transformed into the sequential optimization ones, that is maxjyp ðb; ΩÞj
if ci ðvÞ−μi λi o 0
Algorithm 1 (The augmented Lagrangian function method). Define the constraint penalty function c−i ðvÞ ¼ minf0; ci ðvÞg; i ¼ 1; 2
N
yp ðtÞ ¼ ∑ g½φj ðtÞ½A1 cosðω1 t−ϕj1 Þ
fmaxjyp ðb; ΩÞj; f MRR ðb; ΩÞg ( λmax ðΦðb; ΩÞÞ ≤1 s:t: Ω ≤Ω0
ð12Þ
ð8Þ
Note that the right side of Eq. (8) has the sinusoid form, so we can obtain the following solution:
þA2 cosðω2 t−ϕj2 Þ þ A3 expð−ζωn tÞ cosðω3 t−ϕj3 Þ
j ¼ 1; 2; …; m
The optimization problem given in Eqs. (12) is solved by the augmented Lagrangian function method. The basic idea of this method is that it transforms the nonlinear optimization problem into an unconstrained optimization one by introducing a penalty function, named augmented Lagrangian function [11]. The augmented Lagrangian function achieves these goals by including an explicit estimate of the Lagrange multipliers in the objective. The augmented Lagrangian function can be defined as
j
j¼1
min
MRR
j f MRR
c3 ¼ f MRR ðb; ΩÞ−f MRR ;
¼ bf z ∑ g½φj ðtÞð−K t sin φj ðtÞ þ K n sin φj ðtÞ cos φj ðtÞÞ
f MRR ðb; ΩÞ ¼ a b N Ω f z
8 λ ðΦðb; ΩÞÞ ≤1 > < max Ω ≤Ω0 > j :f ðb; ΩÞ ≤f ;
13
3.2. Numerical example In order to demonstrate the usage and effectiveness of the proposed optimization scheme, a numerical example of machining parameters optimization is given. The input parameters to the
14
X. Zhang, H. Ding / International Journal of Machine Tools & Manufacture 72 (2013) 11–15
optimization model include (1) modal matrices of spindle-tool; (2) cutting coefficients; (3) feed per tooth; and (4) initial machining parameters, i.e. cutting depth and spindle speed. The output parameters are the optimal machining parameters and tool vibration response. The modal matrices are 0:02 0 M¼ kg 0 0:02 1:50 0 C¼ Ns=m 0 1:50 400 000 0 K¼ N=m 0 400 000
Fig. 2. Diagram lobe.
4
tool vibration response (m)
The cutting coefficients are K t ¼ 7:0 108 N=m2 and K n ¼ 2:0 108 N=m2 . The feed per tooth was f z ¼ 0:16 mm. The simulation is performed using an D ¼8 mm diameter end mill with a single cutting edge (N ¼1), down-milling and the radial depth of cut a ¼ 13 D. Normally, we pay attention to the MRR at a particular interval, because the MRR at that interval is adopted for the practical milling for the tradeoff between the cutting load limits and milling efficiency. In our simulation, the radial depth of cut a, number of tool tooth N, and feed per tooth fz are fixed, so MRR is determined by the combination of spindle speed Ω and axial depth of cut b, see Eq. (11). The combination of spindle speed interval and axial cutting depth interval [1.5e04 rpm 2.5e04 rpm]*[2.5e −04 m, 3.0e−03 m] is selected as the preferred one. We solve the sequential optimization formulations as shown in Eqs. (12) by i setting f MRR ði ¼ 1; 2; …; 100Þ, which can be obtained by setting the combinations bnΩ in the third constraint as
x 10−4
2 0 −2 X: 0.002934 Y: −0.0004204
−4 −6
0
0.002
0.004
ΩðiÞ ¼ 1:5e04 : 1:0e04=9 : 2:5e04
for for
0.006
0.008
0.01
time (t) Fig. 3. Optimization results: case 1.
bðjÞ ¼ 2:5e−04 : 2:75e−03=9 : 3:0e−03
Single optimization with upper limit bðjÞnΩðiÞ end The stability of the milling process can be determined by the semi-discretization method [9], which is adopted to obtain the Lobe diagram as shown in Fig. 2. In Fig. 2 the upper one describes the relationships among λmax ðΦÞ, b, and Ω, and the below one is the contour curve at λmax ðΦÞ ¼ 1. The contour curve is approximated by a cubic polynomial, which is treated as a constraint condition in Eqs. (12) (Fig. 3). Using Algorithm 1, we can obtain the optimized machining parameters, given in Table 1. In Table 1, the first optimization parameters ðb; ΩÞ ¼ ð6:49e−4 m; 1:61e4 rpmÞ lead to the minimized tool vibration response jyp j ¼ 0:4204e−3 m among the 100 times optimization procedures, presented in Eqs. (12). And the second one ðb; ΩÞ ¼ ð2:33e−3 m; 2:45e4 rpmÞ leads to jyp j ¼ 0:2259e−2 m, which is the optimization results given the initial values ðb; ΩÞ ¼ ð3:00e−3 m; 2:69e4 rpmÞ. Case 1 implies a desired tool vibration response with low MRR. But the latter case gives a great MRR with a large tool vibration response. The two cases are helpful for the decision making in a practical milling process. The tool vibration responses with initial and optimal machining parameters are given in Fig. 4.
4. Conclusion This communication deals with the machining parameters optimization in a chatter-free milling process. The problem has been solved using the augmented Lagrangian function method.
Table 1 Simulation results of machining parameters optimization.
Case 1 Case 2
2 tool vibration response (m)
end
x 10
Optimal parameters ðb; ΩÞ
Optimization value jyp j
(6.49e−4 m, 1.61e4 rpm) (2.33e−3 m, 2.45e4 rpm)
0.4204e−3 m 0.2259e−2 m
−3
1 0 −1 X: 0.003636 Y: −0.002259
−2 −3
0
0.002
0.004
0.006
0.008
0.01
time (t)
Fig. 4. Optimization results: case 2.
The detailed optimization procedures have been given with a numerical example verification. Besides, improvements on the calculation scheme of the tool vibration response have been made. Our studies take the Lobe diagram as a nonlinear constraint condition in optimization formulation, differing from the trialand-error and heuristic approaches in traditional parameters selection. The procedures presented can be employed to solve a
X. Zhang, H. Ding / International Journal of Machine Tools & Manufacture 72 (2013) 11–15
general MRR optimization problem in a milling process in an applicable and effective way.
15
ϕj3 : depends on the initial conditions qffiffiffiffiffiffiffiffiffiffiffiffiffiffi cy where ωn ¼ ky =my and ζ ¼ 2my ωn
Acknowledgments This work was partially supported by the National Basic Research Program of China (2011CB706804), the National Natural Science Foundation of China (51005087, 51121002, 51120155001) and NCET-12-0222. Prof. Xiaoming Zhang acknowledges the support of the Alexander von Humboldt Foundation. Appendix A The coefficients A1;2;3 , ω1;2;3 and ϕj1;j2;j3 in Eq. (9) are given as follows:
2ky
bf z K t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ½1−ðω1 =ωn Þ2 2 þ ð2ζω1 =ωn Þ2
2ky
bf z K n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ½1−ðω2 =ωn Þ2 2 þ ð2ζω2 =ωn Þ2
A1 ¼ A2 ¼
A3 : depends on the initial conditions πΩ ω 1 ¼ ω2 ¼ 15 qffiffiffiffiffiffiffiffiffiffiffi ω 3 ¼ ωn
1−ζ 2
4πðj−1Þ 2ζω1 =ωn −arctan N 1−ðω1 =ωn Þ2 4πðj−1Þ π 2ζω2 =ωn − −arctan ϕj2 ¼ N 2 1−ðω2 =ωn Þ2
ϕj1 ¼
References [1] A.I. Sönmez, A. Baykasoǧ lu, T. Dereli, İ.H. Filiz, Dynamic optimization of multipass milling operations via geometric programming, International Journal of Machine Tools and Manufacture 39 (2) (1999) 297–320. [2] J. Wang, E.J.A. Armarego, Computer-aided optimization of multiple constraint single pass face milling operations, Machining Science and Technology 5 (1) (2001) 77–99. [3] M.H. Kurdi, Robust Multicriteria Optimization of Surface Location Error and Material Removal Rate in High-Speed Milling Under Uncertainty, Ph.D. Thesis, University of Florida, 2006. [4] E. Budak, A. Tekeli, Maximizing chatter free material removal rate in milling through optimal selection of axial and radial depth of cut pairs, CIRP Annals— Manufacturing Technology 54 (1) (2005) 353–356. [5] T. Insperger, J. Gradišek, M. Kalveram, G. Stépán, K. Winert, E. Govekar, Machine tool chatter and surface location error in milling processes, Journal of Manufacturing Science and Engineering 128 (2006) 913. [6] M. Wan, W.H. Zhang, G.H. Qin, Z.P. Wang, Strategies for error prediction and error control in peripheral milling of thin-walled workpiece, International Journal of Machine Tools and Manufacture 48 (12–13) (2008) 1366–1374. [7] M.L. Campomanes, An improved time domain simulation for dynamic milling at small radial immersions, Journal of Manufacturing Science and Engineering 125 (2003) 416. [8] Y. Altintaş, E. Budak, Analytical prediction of stability lobes in milling, CIRP Annals—Manufacturing Technology 44 (1) (1995) 357–362. [9] T. Insperger, G. Stépán, Updated semi-discretization method for periodic delay-differential equations with discrete delay, International Journal for Numerical Methods in Engineering 61 (1) (2004) 117–141. [10] B.P. Mann, Dynamics of Milling Process, Ph.D. Thesis, Washington University, St. Louis, MO, 2003. [11] J. Nocedal, S.J. Wright, Numerical Optimization, Springer, 1999.
Contents lists available at SciVerse ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Short Communication
Note on a novel method for machining parameters optimization in a chatter-free milling process Xiaoming Zhang n, Han Ding State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China
art ic l e i nf o
a b s t r a c t
Article history: Received 14 April 2013 Accepted 25 April 2013 Available online 9 May 2013
Machining parameters optimization in a chatter-free milling process is a common issue for practical engineers at shop floor. The detailed and easily applicable optimization algorithm taking the milling mechanism into account is absent to date. This communication deals with the procedures of optimization of machining parameters in the chatter-free milling process. First, the model of machining parameters optimization is given based on the milling dynamics. The optimization objects are to minimize the tool vibration response and maximize the material removal rate, and the constraint conditions are to keep the milling process chatter-free or stable and let the spindle speed below a predefined one. Emphasis is put on how to select the spindle speed for reducing the tool vibration response and maximization of the material removal rate simultaneously. Augmented Lagrangian function method is employed to solve the nonlinear optimization problem. A numerical example is provided to show the effectiveness of the developed results. Besides, improvements on the calculation scheme of the tool vibration response have been made. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Milling dynamics Machining parameters optimization Tool vibration response
1. Introduction Machining parameters optimization in a milling process plays an important role in the practical manufacturing engineering. The aims are to improve the part quality and maximize the material removal rate (MRR). Traditionally, trial-and-error and heuristic approaches are employed to obtain the optimal machining parameters. It is well recognized that these methods are timeconsuming and will lead to long machining periods and large machining costs. The process of searching for an optimal machining parameters cannot be effectively carried out without reliable optimization procedures. Providing accurate and efficient optimization procedures to obtain the optimal machining parameters is therefore an important activity. In this communication, we employ the augmented Lagrangian function method for solving the machining parameters optimization formulation. The emphasis is how to select the optimal spindle speed for reducing tool vibration response and maximization of MRR simultaneously in an effective and a reliable way. The problem of machining parameters optimization has been tackled by many authors and many works have been reported in the literature. Among them we quote the works in [1–3] and the references therein. These references have focused on mathematical
n
Corresponding author. Tel.: +86 2787559842. E-mail address: [email protected] (X. Zhang).
0890-6955/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmachtools.2013.04.006
modeling approaches to determine optimal machining parameters with regard to various objective functions. But the machining parameters optimization with chatter-free condition constraint has been addressed little. Budak and Tekeli [4] proposed a method to determine the optimal combination of axial and radial depths of cut so that chatter-free MRR is maximized. The optimal machining parameters are derived based on the two Lobe diagrams, which express the relations of stable axial depth of cut verse spindle speed and stable radial depth of cut verse spindle speed, respectively. But the tool vibration response was not taken into account, so the part quality cannot be guaranteed. Insperger et al. [5] discussed the selection of optimal spindle speed considering both the stability diagram and the surface location error diagram. But the method adopted there is heuristic and the detailed mathematical programming model is absent. Wan et al. [6] developed a linear programming model to increase the cutting efficiency and to satisfy the tolerance in the meantime. The work is successful based on the fact that the explicit linear relationship expressions of the part error in terms of machining parameters, i.e. feed per tooth and radial depth of cut could be derived. But there is no linear relationship between the part error in terms of spindle speed. So the method proposed in Ref. [6] cannot be adopted directly to tackle the machining parameters optimization problem taking the spindle speed into account. To the best of our knowledge the methodology we are using in this communication to tackle the optimization of machining parameters in a chatter-free milling process has never been used before. The optimization of machining parameters in the
12
X. Zhang, H. Ding / International Journal of Machine Tools & Manufacture 72 (2013) 11–15
M N t v x yp v Φ Ω ζ φ
Nomenclature a b C fz g K Kn Kt LA
radial depth of cut axial depth of cut damping matrix of tool feed per tooth switching function stiffness matrix of tool normal cutting force coefficients tangential cutting force coefficients augmented Lagrangian function
chatter-free milling process is modeled as a nonlinear optimization formulation, in which the stability diagram as a constraint condition is approximated by a polynomial curve. The nonlinear optimization problem is solved by the augmented Lagrangian function method. Besides, improvements on the calculation scheme of the tool vibration response have been made, which is not addressed in Refs. [5,3]. The rest of this communication is organized as follows. In Section 2, a description of the milling dynamics model is given. And also calculation scheme of the tool vibration response is presented. Section 3 presents the model of machining parameters optimization for chatter-free milling process, and detailed optimization procedures. Numerical example is given to show the effectiveness of the proposed methodology to handle the machining parameters optimization problem. Conclusions are reached in Section 4.
2. Milling model 2.1. Milling dynamics The standard two degrees of freedom milling process is shown in Fig. 1. The tool is assumed to be compliant relative to the rigid workpiece. The vibration is excited by the summation of cutting force. The governing equation of motion has the following form: € þ CxðtÞ _ þ KxðtÞ ¼ HðtÞb½xðtÞ−xðt−τÞ þ GðtÞ MxðtÞ with " M¼
mx
0
0
my
# ;
" C¼
cx
0
0
cy
#
" ;
K¼
kx
0
0
ky
ð1Þ
#
where M, C, K, and F are the modal mass, damping, stiffness, and cutting force matrices, respectively. The terms mx;y , cx;y , kx;y and F x;y are the corresponding components in the flexible directions of the system. b is the axial depth of cut. τ ¼ 60=NΩ is the tooth passing period in seconds, in which N is the number of teeth on the cutting tool and Ω the spindle speed in rpm. xðtÞ ¼ ½xðtÞ yðtÞT is the dynamic response vector and xðtÞ−xðt−τÞ the dynamic chip thickness, see Fig. 1.
mass matrix of tool number of tool tooth time vector of machining parameters dynamic response vector tool vibration response along the Y-direction vector of machining parameters transition matrix of milling system spindle speed damping coefficient angular position of the cutting edge
HðtÞ is the matrix given as follows, which represents the component of cutting forces that depend on the position vector " # N −K t sc−K n s2 −K t c2 −K n sc HðtÞ ¼ ∑ g½φj ðtÞ ð2Þ K t sc−K n c2 K t s2 −K n sc j¼1 where Kt and Kn are the tangential and normal cutting force coefficient components, respectively. For a tool with N evenly spaced teeth, φj is the angular position of the jth cutting edge and expressed as φj ðtÞ ¼
2πΩ 2πðj−1Þ tþ ; 60 N
j ¼ 1; 2; …; N
s ¼ sin φj ðtÞ, c ¼ cos φj ðtÞ and the function g½φj ðtÞ acts as a switching function, which is equal to 1 if the tooth is active and 0 if it is not cutting 1; φe o φj ðtÞ oφa g½φj ðtÞ ¼ 0 otherwise where φe and φa are the angles where the jth tooth enters and exits the cut, respectively. For down-milling operation, φa ¼ π, for up-milling, φe ¼ 0. Note that the entry and exit angles may vary due to heavy vibrations of the tool. This effect is neglected here, and the angles φe and φa are approximated by constant values as it is usually done in the literature. GðtÞ is the vector that represents the components of the cutting force which are independent of the position vector " # N K t sc þ K n s2 GðtÞ ¼ ∑ g½φj ðtÞbf z ð3Þ −K t s2 þ K n sc j¼1 where f z ¼ vf τ denotes the feed per tooth and vf is the feed speed. 2.2. Milling stability prediction and calculation of tool vibration response To simplify the milling dynamics, the motion of the tool is decomposed in the following form [5]: xðtÞ ¼ xp ðtÞ þ ξðtÞ
ð4Þ
where xp ðt þ τÞ ¼ xp ðtÞ is the forced periodic chatter-free motion of the tool, and xp ðtÞ ¼ ½xp ðtÞ; yp ðtÞT , and ξðtÞ is a perturbation corresponding to the self-excited vibration of the system. Substituting Eq. (4) into Eq. (1) results in € þ CξðtÞ _ þ KξðtÞ þ Mx€ p ðtÞ þ Cx_ p ðtÞ þ Kxp ðtÞ MξðtÞ ¼ HðtÞb½ξðtÞ−ξðt−τÞ þ GðtÞ
ð5Þ
For the ideal case, when no chatter arises, i.e., ξðtÞ≡0, and the motion is described by xðtÞ ¼ xp ðtÞ, the corresponding equations of motion are the ordinary differential ones Fig. 1. Dynamic chip thickness.
Mx€ p ðtÞ þ Cx_ p ðtÞ þ Kxp ðtÞ ¼ GðtÞ
ð6Þ
X. Zhang, H. Ding / International Journal of Machine Tools & Manufacture 72 (2013) 11–15
The assumption in Eq. (4) is appropriate, if Eq. (6) has a τperiodic solution. Since the excitation GðtÞ is τperiodic, the stationary (particular) solution of Eq. (6) is also τperiodic. This validates assumption (4). The detailed analysis of validation of assumption in Eq. (4) can be found in Ref. [5]. Eqs. (5) and (6) imply the equations € þ CξðtÞ _ þ KξðtÞ ¼ HðtÞb½ξðtÞ−ξðt−τÞ MξðtÞ
ð7Þ
Eq. (7) determines the stability of the milling system. Milling stability has been investigated by several researchers. The stability analysis of the milling system can be performed only by applying approximated numerical methods since there is no closed solution to time-delay differential equation of milling dynamics. Alternatively it can be carried out by means of time-domain simulations [7], in the frequency domain [8] or by applying methods based on delay differential equation theory, such as the semi-discretization method [9] and the time finite element approach (TFEA) [10]. Here we adopt the semi-discretization method [9] to obtain the Lobe diagram, where the combinations of cutting depth and spindle speed below the Lobe diagram curve state the stability of milling process. The detailed stability analysis is omitted here. When the milling process is stable, the tool vibration response along the Y-direction affects greatly on the part quality. From Eq. (6), we can get my y€ p ðtÞ þ cy y_ p ðtÞ þ ky yp ðtÞ N
s:t:
MRR
for series of
2
LA ðv; s; λÞ ¼ f ðvÞ
8 1 2 > > < −λi ci ðvÞ þ si ci ðvÞ 2 þ ∑ > 1 2 i ¼ 1;2;3> : − λi =si 2
j¼1
ð9Þ
where the coefficients A1;2;3 , ω1;2;3 , ϕj1;j2;j3 , ζ, and ωn are given in the Appendix.
ð13Þ
otherwise
where v ¼ ½b; ΩT f ðvÞ ¼ maxjyp ðvÞj c1 ¼ lðvÞ c2 ¼ Ω−Ω0 j ¼ 1; 2; …; m
where lðvÞ is the cubic polynomial curve to interpolate the obtained Lobe diagram λmax ðAðb; ΩÞÞ−1 ≤0. We give the algorithmic framework as follows:
Step 1: Given initial points v1 and λ1i ; s1i 4 0 , tolerance ε≥0, k≔1. Step 2: Find an approximate minimizer vkþ1 of LA ðv; sk ; λk Þ, i.e.
3. Nonlinear optimization of machining parameters in the chatter-free milling process
vkþ1 ¼ arg min LA ðv; sk ; λk Þ. If ∥c− ðvkþ1 Þ∥∞ ≤ ε, STOP and report the minimizer. Step 3: Let 8 < ski if c−i ðvkþ1 Þ ≤ 14jc−i ðvk Þj : maxf10ski ; k2 g
3.1. Model of optimization of machining parameters In general, a machining parameters optimization problem can be stated as follows: min
ð10Þ
where the optimization objects are to minimize the tool vibration response along the Y-direction, denoted as maxjyp ðb; ΩÞj and maximize MRR, i.e. f MRR ðb; ΩÞ, and constraint conditions are to keep the milling process stable, which means that the maximal eigenvalue of transition matrix Φ is less than or equal to 1 [9]; and the spindle speed is less than the predefined one, which is set by the spindle system and tool suppliers. Here MRR is defined as
otherwise
set k≔k þ 1 and update Lagrange multipliers maxfλki −ski ci ðvkþ1 Þ; 0g to obtain λkþ1 and go to Step 2.
¼ λkþ1 i
To obtain the minimizer vkþ1 ¼ arg min LA ðv; sk ; λk Þ in Algorithm 1, we need the differentiation of the augmented Lagrangian function with respect to the machining parameters ∇v LA ðv; s; λÞ. The detailed expressions are given as follows: ∇v LA ðv; s; λÞ ¼ ∇v f ðvÞ −λi ∇v ci ðvÞ þ si ci ðvÞ∇v ci ðvÞ þ ∑ i ¼ 1;2 0
if ci ðvÞ−μi λi o 0 otherwise ð14Þ
With the expressions of yp in Eq. (9), ∇v f ðvÞ can be easily obtained as follows: ∇v f ðvÞ ¼ ½maxj∇b yp ðb; ΩÞj; maxj∇Ω yp ðb; ΩÞjT
ð15Þ
ð11Þ
where a is the axial depth of cut. The optimization formulation presented in Eqs. (10) can be transformed into the sequential optimization ones, that is maxjyp ðb; ΩÞj
if ci ðvÞ−μi λi o 0
Algorithm 1 (The augmented Lagrangian function method). Define the constraint penalty function c−i ðvÞ ¼ minf0; ci ðvÞg; i ¼ 1; 2
N
yp ðtÞ ¼ ∑ g½φj ðtÞ½A1 cosðω1 t−ϕj1 Þ
fmaxjyp ðb; ΩÞj; f MRR ðb; ΩÞg ( λmax ðΦðb; ΩÞÞ ≤1 s:t: Ω ≤Ω0
ð12Þ
ð8Þ
Note that the right side of Eq. (8) has the sinusoid form, so we can obtain the following solution:
þA2 cosðω2 t−ϕj2 Þ þ A3 expð−ζωn tÞ cosðω3 t−ϕj3 Þ
j ¼ 1; 2; …; m
The optimization problem given in Eqs. (12) is solved by the augmented Lagrangian function method. The basic idea of this method is that it transforms the nonlinear optimization problem into an unconstrained optimization one by introducing a penalty function, named augmented Lagrangian function [11]. The augmented Lagrangian function achieves these goals by including an explicit estimate of the Lagrange multipliers in the objective. The augmented Lagrangian function can be defined as
j
j¼1
min
MRR
j f MRR
c3 ¼ f MRR ðb; ΩÞ−f MRR ;
¼ bf z ∑ g½φj ðtÞð−K t sin φj ðtÞ þ K n sin φj ðtÞ cos φj ðtÞÞ
f MRR ðb; ΩÞ ¼ a b N Ω f z
8 λ ðΦðb; ΩÞÞ ≤1 > < max Ω ≤Ω0 > j :f ðb; ΩÞ ≤f ;
13
3.2. Numerical example In order to demonstrate the usage and effectiveness of the proposed optimization scheme, a numerical example of machining parameters optimization is given. The input parameters to the
14
X. Zhang, H. Ding / International Journal of Machine Tools & Manufacture 72 (2013) 11–15
optimization model include (1) modal matrices of spindle-tool; (2) cutting coefficients; (3) feed per tooth; and (4) initial machining parameters, i.e. cutting depth and spindle speed. The output parameters are the optimal machining parameters and tool vibration response. The modal matrices are 0:02 0 M¼ kg 0 0:02 1:50 0 C¼ Ns=m 0 1:50 400 000 0 K¼ N=m 0 400 000
Fig. 2. Diagram lobe.
4
tool vibration response (m)
The cutting coefficients are K t ¼ 7:0 108 N=m2 and K n ¼ 2:0 108 N=m2 . The feed per tooth was f z ¼ 0:16 mm. The simulation is performed using an D ¼8 mm diameter end mill with a single cutting edge (N ¼1), down-milling and the radial depth of cut a ¼ 13 D. Normally, we pay attention to the MRR at a particular interval, because the MRR at that interval is adopted for the practical milling for the tradeoff between the cutting load limits and milling efficiency. In our simulation, the radial depth of cut a, number of tool tooth N, and feed per tooth fz are fixed, so MRR is determined by the combination of spindle speed Ω and axial depth of cut b, see Eq. (11). The combination of spindle speed interval and axial cutting depth interval [1.5e04 rpm 2.5e04 rpm]*[2.5e −04 m, 3.0e−03 m] is selected as the preferred one. We solve the sequential optimization formulations as shown in Eqs. (12) by i setting f MRR ði ¼ 1; 2; …; 100Þ, which can be obtained by setting the combinations bnΩ in the third constraint as
x 10−4
2 0 −2 X: 0.002934 Y: −0.0004204
−4 −6
0
0.002
0.004
ΩðiÞ ¼ 1:5e04 : 1:0e04=9 : 2:5e04
for for
0.006
0.008
0.01
time (t) Fig. 3. Optimization results: case 1.
bðjÞ ¼ 2:5e−04 : 2:75e−03=9 : 3:0e−03
Single optimization with upper limit bðjÞnΩðiÞ end The stability of the milling process can be determined by the semi-discretization method [9], which is adopted to obtain the Lobe diagram as shown in Fig. 2. In Fig. 2 the upper one describes the relationships among λmax ðΦÞ, b, and Ω, and the below one is the contour curve at λmax ðΦÞ ¼ 1. The contour curve is approximated by a cubic polynomial, which is treated as a constraint condition in Eqs. (12) (Fig. 3). Using Algorithm 1, we can obtain the optimized machining parameters, given in Table 1. In Table 1, the first optimization parameters ðb; ΩÞ ¼ ð6:49e−4 m; 1:61e4 rpmÞ lead to the minimized tool vibration response jyp j ¼ 0:4204e−3 m among the 100 times optimization procedures, presented in Eqs. (12). And the second one ðb; ΩÞ ¼ ð2:33e−3 m; 2:45e4 rpmÞ leads to jyp j ¼ 0:2259e−2 m, which is the optimization results given the initial values ðb; ΩÞ ¼ ð3:00e−3 m; 2:69e4 rpmÞ. Case 1 implies a desired tool vibration response with low MRR. But the latter case gives a great MRR with a large tool vibration response. The two cases are helpful for the decision making in a practical milling process. The tool vibration responses with initial and optimal machining parameters are given in Fig. 4.
4. Conclusion This communication deals with the machining parameters optimization in a chatter-free milling process. The problem has been solved using the augmented Lagrangian function method.
Table 1 Simulation results of machining parameters optimization.
Case 1 Case 2
2 tool vibration response (m)
end
x 10
Optimal parameters ðb; ΩÞ
Optimization value jyp j
(6.49e−4 m, 1.61e4 rpm) (2.33e−3 m, 2.45e4 rpm)
0.4204e−3 m 0.2259e−2 m
−3
1 0 −1 X: 0.003636 Y: −0.002259
−2 −3
0
0.002
0.004
0.006
0.008
0.01
time (t)
Fig. 4. Optimization results: case 2.
The detailed optimization procedures have been given with a numerical example verification. Besides, improvements on the calculation scheme of the tool vibration response have been made. Our studies take the Lobe diagram as a nonlinear constraint condition in optimization formulation, differing from the trialand-error and heuristic approaches in traditional parameters selection. The procedures presented can be employed to solve a
X. Zhang, H. Ding / International Journal of Machine Tools & Manufacture 72 (2013) 11–15
general MRR optimization problem in a milling process in an applicable and effective way.
15
ϕj3 : depends on the initial conditions qffiffiffiffiffiffiffiffiffiffiffiffiffiffi cy where ωn ¼ ky =my and ζ ¼ 2my ωn
Acknowledgments This work was partially supported by the National Basic Research Program of China (2011CB706804), the National Natural Science Foundation of China (51005087, 51121002, 51120155001) and NCET-12-0222. Prof. Xiaoming Zhang acknowledges the support of the Alexander von Humboldt Foundation. Appendix A The coefficients A1;2;3 , ω1;2;3 and ϕj1;j2;j3 in Eq. (9) are given as follows:
2ky
bf z K t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ½1−ðω1 =ωn Þ2 2 þ ð2ζω1 =ωn Þ2
2ky
bf z K n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ½1−ðω2 =ωn Þ2 2 þ ð2ζω2 =ωn Þ2
A1 ¼ A2 ¼
A3 : depends on the initial conditions πΩ ω 1 ¼ ω2 ¼ 15 qffiffiffiffiffiffiffiffiffiffiffi ω 3 ¼ ωn
1−ζ 2
4πðj−1Þ 2ζω1 =ωn −arctan N 1−ðω1 =ωn Þ2 4πðj−1Þ π 2ζω2 =ωn − −arctan ϕj2 ¼ N 2 1−ðω2 =ωn Þ2
ϕj1 ¼
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