Minimax Optimization Strategy for Process Parameters Planning ...

Journal of Manufacturing Science and Engineering. Received May 26, 2016; Accepted manuscript posted November 9, 2016. doi:10.1115/1.4035184 Copyright (c) 2016 by ASME

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Minimax optimization strategy for process parameters planning: towards interference-free between tool and flexible workpiece in milling process Xiao-Ming Zhanga,∗, Dong Zhanga , Le Caoa , Tao Huanga , J¨ urgen Leopoldb , Han Dinga a State

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Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China b Fraunhofer Institute for Machine Tools and Forming Technology, Chemnitz 09661, Germany

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Abstract

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In milling of flexible workpieces, like axial-flow compressor impellers with thin-wall blades and deep channels, interference occurrence between workpiece and tool shaft is a great adverse issue. Even though interference avoidance plays a mandatory role in tool path generation stage, the generated tool path remains just a nominally interference-free one. This challenge is attributed to the fact that workpiece flexibility and dynamic response cannot be considered in tool path generation stage. This paper presents a strategy in process parameters planning stage, aiming to avoid the interference between tool shaft and flexible workpiece with dynamic response in milling process. The interference problem is formulated as that to evaluate the approaching extent of two surfaces, i.e. the vibrating workpiece and the swept envelope surface generated by the tool undergoing spatial motions. A metric is defined to evaluate quantitatively the approaching extent. Then a minimax optimization model is developed, in which the optimization objective is to maximize the metric, so as the interference-free can be guaranteed while constraints require the milling process to be stable and process parameters to fall into preferred intervals in which material removal rate is satisfactory. Finish milling of impeller using a conical cutter, governed by a nominally interference-free tool path is numerically simulated to illustrate the dynamics responses of the spatially distributed nodal points on the thin-wall blade, and approaching extent of the time-varying vibrating blades to the tool swept envelope surface. Furthermore, the present model results suggest to use an optimal process parameters set in finish milling, as a result improving machining efficiency in addition to ensuring the interference-free requirement. The model results are verified against milling experiments.

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Keywords: Process parameters planning, Minimax optimization strategy, Computer aided manufacturing, Flexible workpiece machining

∗ Corresponding

author, Email: [email protected]

Preprint submitted to J. Manuf. Sci. Eng.-Trans. ASME

October 30, 2016

Downloaded From: http://manufacturingscience.asmedigitalcollection.asme.org/ on 11/08/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use


1. Introduction

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The five-axis milling process is widely adopted in the manufacturing of flexible workpiece, as turbine blades, jet engine compressor and etc. Normally in milling process planning stage, a interference-free numerical control(NC) tool path is programmed by a commercial computer-aided manufacturing(CAM) system, followed by the process parameters planning. The final machined workpiece surface is obtained via graphically subtracting the swept volume generated by the tool moving along the programmed tool path from the current raw stock. So the interference-free is only geometrically avoiding interference between machined workpiece and tool swept volume surface under the programmed tool path. CAM system may give the software manipulator a choice to change the cutter size and fine-tune tool path to avoid interference, unfortunately cannot take the workpiece flexibility and vibration quantitatively into account. It is this fact that makes the process parameters planning towards interference-free between tool and vibrating flexible workpiece in milling process necessary. The milling process of flexible workpiece is complicated due to the distributed deformation and vibration, the time-variant structure parameter caused by material removal, the dynamic interaction between the cutting tool and workpiece. In order to improve the machining quality and surface profile precision, many researchers[1, 2, 3, 4, 5] focused on the force-induced errors during the stable milling process. Based on the finiteelement method, Tsai[1] predicted the quasi-statically equilibrium deformation of the tool and thin-wall workpiece by applying the mechanical force model to simulate the instant cutting force. Instead, Ratchev[2, 3] adopted a theoretical force model to include the material properties and tool geometry variables and presented an integrated adaptive machining planning environment. Recently, Kang[4] proposed an efficient iterative algorithms for error prediction, which showed faster convergence speed in the iteration.For the sake of high profile precision, strategies were developed to reduce the surface location errors. Ratchev[5] introduced a multi-level machining error compensation approach to limit the errors in the permission level by correcting the tool path. Qin[6] considered the workpiece-clamping process and developed a nonlinear mathematical programming formulation to determine the workpiece machining error. Wan[7] optimally selected the appropriate process parameters without tolerance violation by setting up an approximate linear relationship of the dimensional error with respect to the radial depth of cut and the feed. Chaves-Jacob[8] proposed a new optimal strategy for the finishing of impeller blades, considering the three indicators expressing requirements of flow, mechanical and machining. Lartigue[9] presented a kinematics approach-based envelope surface expression and via that gave a flank milling strategy for machining of a sculptured surface, and the positioning errors can be precisely corrected by the optimization of the tool control points, as a result the envelope surface of the tool movement better fitting the surface to be machined. In terms of the machining system dynamic feature, the chatter is another important issues discussed by[10, 11, 12, 13, 14]. Thevenot[10] constructed a 3D lobs diagram by introducing the dynamic characteristics variation with respect to tool position without consideration of the tool vibration. Bravo[11] synthetically considered the dynamic response of the tool and workpiece because of their similar dynamic behavior and presented a group of lobes diagrams at different intermediate stages of the machining process. Contrast to Bravo[11], Ma˜ n´e[12] analytically modeled the dynamic behaviors 2



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of the spindle-bearing system taking the gyroscopic moment of the spindle rotor and speed-dependent bearing stiffness into account. Adetoro[13] predicted the stability lobes including the nonlinearity of cutting force coefficients as well as thin wall dynamics. Seguy[14] examined the link between the chatter and surface roughness evolution and found that the dominate mode migration along with the tool path and the coupled mode interaction strongly influenced the surface topography of the thin wall workpiece. Most of these existing studies, however, have focused on estimating and compensating static and dynamic deflection of the cutter or workpiece, as well as the machining stability. Interference derived from dynamic response of flexible workpiece and the coping strategy/specific system solutions remain underexploited. The remainder of this paper offers the followings:

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(1) The interference between tool and workpiece is modeled as the approaching extent evaluation of the tool swept envelope surface and the vibrating workpiece surface in milling process. A new metric is introduced for quantitatively evaluating the interference, upon which the process parameters planning is formulated as a minimax optimization problem. The constraint minimax optimization problem cannot be solved directly due to the non-differential of optimization object, so it is reformulated as a sequential linear programming one. (2) The surfaces expressions of the tool swept envelope and vibrating workpiece are presented, which are essential for the mathematical equations of the process parameters planning. The former surface is developed based on the two-parameter sphere congruence theory, while the latter can be obtained by invoking the cutting force model, the finite element analysis of workpiece structure, and the milling dynamics analysis. (3) Simulations of five-axis finish milling process are given, in which an impeller with thin-wall blades and deep channels is machined using a conical cutter. An interferencefree tool path is programmed with a CAM system, and the process parameters are adopted from the proposed planning strategy, as well as the optional ones. With the time-varying cutting force, the distributed-parameter flexible workpiece dynamic responses are solved with experimentally determined damping coefficients. The dynamic process of the defined metric can be displayed, which are associated with the time and process parameters in milling process. With the model and simulation results a higher spindle speed is suggested for finish milling than the one recommended by CAM system, so as to improve machining efficiency. (4) Milling of an AL7075 impeller is carried out to verify the proposed planning method. While adopting the optimized process parameters and optional ones, the interference states in milling process are identified. Also the cutting force and impeller blade vibration response are measured in milling process and compared against the model results.

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2. Problem statement and formulation To provide a basis for process parameters planning towards interference avoidance in the milling process, this section describes the problem, presents a new metric for numerically evaluating the extent of the approaching between tool swept envelope surface and the vibrating workpiece, and based on the metric give a mathematical model with optimization algorithm to tackle the interference problem. 3



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2.1. Problem description The nominal interference-free tool path programmed by CAM system, and the widely discussed chatter stability are necessary but not sufficient conditions to guarantee the interference-free between tool and vibrating workpiece while milling a flexible workpiece. The problem is best illustrated in Fig.1 where the programmed and actual tool paths (discrete cutter locations) for milling an impeller blade are compared. With cutting force applied on the contact points of discrete locations, the flexible blade deformation varies from point to point which could result in unwanted interference between the tool and vibrating blade, as shown in Fig.1-(b). The blade vibration response characteristic of time-varying and spatially distribution must be suppressed by appropriate optimization of the process parameters to prevent any potential unwanted interference between tool and vibrating blade, which is achieved without sacrificing machining efficiency. Discrete cutter locations

Vibrated blade (in dark blue) Tool-blade interference

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(a) Programmed cutter locations

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Flexible blade at nominal positions (in light blue)

(b) Unwanted interference between tool and vibrating blade

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Figure 1: Milling of a thin-wall flexible blade

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2.2. Dynamic metric for evaluating approaching extent of tool-workpiece In milling process the flexible workpiece vibration is not only time-varying but also spatially distributed. It is difficult to find an appropriate metric to evaluate the approaching extent between vibrating workpiece and tool. We give a dynamic metric Φ, which is the minimal distance from the vibrating workpiece X to the swept envelope surface S generated by the tool undergoing spatial motions, shown in Fig.2. The metric can be stated as: Φ(t, w) = min dpi ,X(t,w) (1) i

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where pi ∈ S, i = 1, 2, · · · , n are the points distributed on the tool swept envelope surface, and dpi ,X(t,w) , i = 1, 2, · · · , n are the distances from points pi , i = 1, 2, · · · , n to the foot point p0i , i = 1, 2, · · · , n on surface X(t, w), respectively. Here t denotes time and w the process parameters vector. In this paper, considering the generated tool path is given, the spindle speed Ω and feed per tooth f are selected as the process parameters for optimization, i.e. w = (Ω, f )T . The vibrating workpiece surface X(t, w) is associated with process parameters and time-varying. Note that the interference between tool shaft and workpiece is the emphasis and the contact area for cutting between cutter and workpiece should be excluded from the surfaces approaching extent evaluation.

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S

Tool swept envelope surface

X

Workpiece surface

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Figure 2: Dynamic metric

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The geometrical meaning of the dynamic metric Φ(t, w) is straightforward, i.e. if the following condition is satisfied, Φ(t, w) > 0 (2)

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the interference-free between tool and vibrating workpiece is guaranteed. Furthermore, maximization of Φ(t, w) means reducing the possibility of interference occurrence to the greatest extent. This point is important when considering the existence of unforeseeable factors in a practical flexible workpiece machining process. Also we should point out that in Eq.(2) we consider only the interference (collision) between tool shaft and flexible workpiece in the milling process, and the undercut is not taken into account. The undercut is an important adverse issue in the tool path generation for flank milling, however in this paper the point milling is employed for finish milling, interference-free is the predominant issue.

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2.3. Mathematical formulation of the problem The interference-free oriented process parameters planning can be formulated as:

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max min Φ(t, w) = min dpi ,X(t,w) , pi ∈ S, i = 1, 2, · · · , n w t i  U  w ≤ w s.t. M L ≤ M (w) ≤ M U   F(w) ≤ 0

P1

(3)

In Eqs.(3), the function for optimization is Φ(t, w) = min dpi ,X(t,w) , pi ∈ S, i =

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1, 2, · · · , n, which represents the extent of surface S approaching X. The optimization objective max min Φ(t, w) means minimizing the dynamic metric Φ(t, w) regarding the w

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time t in a time interval, i.e. t ∈ [t1 , t2 ] during the milling operations, simultaneously maximizing Φ(t, w) regarding the process parameters w, with feasible interval given by the constraint conditions in Eqs.(3). M is material removal rate, and M U , M L are the corresponding upper and lower bounds respectively; wU is the upper bound of process parameters, normally recommended by the spindle system and tool suppliers. F ≤ 0 states the milling process stable, which is determined by a chatter stability Lobe[15, 16, 17]. The Lobe is usually given in a chart, in which the boundary curve differs stable and unstable milling conditions in the form of axial depth of cut limit versus spindle speed for a specific radial width of cut. It should be noted that the chatter stability Lobe discussed here is calculated for workpiece flexibility and the tool is assumed as rigid. 5



2.4. Optimization implementation The formulation given in Eqs.(3) is a constraint minimax optimization problem, in which optimization object Φ(t, w) = min dpi ,X(t,w) exhibits non differential. We cannot i

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solve it directly by adopting an established optimization procedures. Next, we detail the algorithm for the process parameters optimization problem presented in Eqs.(3). By introducing one extra variable ξ, P1 can be reformulated as the following differentiable constrained optimization problem, stated as follows:

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(4)

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P2

max ξ  ξ − dpi ,X(tj ,w) ≤ 0, i = 1, 2, · · · , n, j = 1, 2, · · · , m      w ≤ wU s.t.  M L ≤ M (w) ≤ M U     F(w) ≤ 0

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where tj (j = 1, 2, · · · , m) are the sampling times during milling process tj ∈ [t1 , t2 ]. P2 can be solved by the method of sequential approximation programming, which has been employed to handle many practical nonlinear constrained optimization problems. The basic idea of this method is to proceed iteratively by linearizing the objective function and the constraint ones about the current candidate solutions, thereby reducing the given nonlinear problem to a sequence of linear programming problems[18]. Let (wk , ξ k ) be a candidate solution to problem P2 and consider perturbations of the form: (wk+1 , ξ k+1 )

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Using the first-order differential increment of the distance[19], we have: dpi ,X(tj ,wk+1 ) = dpi ,X(tj ,wk ) + ∆dpi X(tj ,wk ) 0

0

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= dpi ,X(tj ,wk ) − [npi · Xw1 , · · · , npi · Xwl ]T · ∆wk

(5)

i = 1, 2, · · · , n, j = 1, 2, · · · , m

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where Xwk , k = 1, 2, · · · l are the partial derivatives of surface X to process parameters 0 w; p0i is the footpoint of pi relative to surface X, and npi is the unit outward normal vector of surface X at point p0i (see Fig.3).

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d pi , X

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Figure 3: Point-to-surface distance 6



Then we have the linearized constraint functions: (ξ k + ∆ξ) − (dpi ,X(tj ,wk ) + ∆dpi ,X(tj ,wk ) ) ≤ 0,

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i = 1, 2, · · · , n, j = 1, 2, · · · , m Clearly, the linearized objective function is ∆ξ. Thus, the corresponding linear programming problem can be obtained.

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(6)

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LP

max ∆ξk  k (ξ + ∆ξk ) − (dpi ,X(tj ,wk ) + ∆dpi ,X(tj ,wk ) ) ≤ 0,      wk + ∆wk ≤ wU      L ∂M ∆wk ≤ M U M ≤ M (wk ) + s.t. ∂w    ∂F    F(wk ) + ∆wk ≤ 0   ∂w   − ∆ξk ≤ 0

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The detailed implementation of the above optimizations are given with a flow chart as follows: Algorithm (Process parameters optimization procedures) Input: Initial process parameters w0 ; threshold τ specifying desired accuracy of the algorithm. Output: Optimal process parameters w∗ ; and optimized and desired distance Φ(t∗ , w∗ ) = min dpi ,X(t∗ ,w∗ ) i

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at the time t∗ . Step 0:

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1. Set k = 0; 2. Compute the distance dpi ,X(tj ,w0 ) , i = 1, 2, · · · , n, j = 1, 2, · · · , m; 3. Set ξ 0 = min dpi ,X(tj ,w0 ) , i = 1, 2, · · · , n, j = 1, 2, · · · , m; i,j

Step 1:

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1. Solve the linear programming problem LP to determine the increment of ∆wk that appears in ∆dpi ,X(tj ,wk ) ; 2. Update wk+1 = wk + ∆wk ; 3. Compute min dpi ,X(tj ,wk+1 ) , i = 1, 2, · · · , n, j = 1, 2, · · · , m;

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4. Update ξ k+1

= min dpi ,X(tj ,wk+1 ) , i = 1, 2, · · · , n, j = 1, 2, · · · , m; i,j

k

i

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5. If |ξ /ξ − 1| > τ , set k = k + 1 and go to Step1:(2); Else exit and report w∗ = wk , and Φ(t∗ , w∗ ) = min dpi ,X(t∗ ,w∗ ) at the time t∗ .

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initial process parameters

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k =0

X(t j , w k ) i, j

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x k = min dp , X (t ,w Dw k = arg LP

X(t j , w k +1 ) i

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x k +1 x k - 1 > t

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x k +1 = min dp , X (t

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Optimized process parameters

Figure 4: Flow chart of the optimization procedures 3. Expressions of tool swept envelope surface S and vibrating workpiece surface X in mathematical formulation Equations (3)

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To implement the optimization planning presented in Eqs.(3), we have to know the expressions of tool swept envelope surface S and vibrating workpiece surface X. This section offers the followings: 1. The analytical expression of tool swept envelope surface S based on the twoparameter sphere congruence theory. When the programmed tool path and the tool geometry are given, S can be determined. 2. The expression of vibrating workpiece surface X(t) at any time tj , (j = 1, 2, · · · , m) during milling process with programmed tool path, which is developed with the help of models of workpiece structural dynamics, cutting force acting on the workpiece, and milling dynamics response analysis. 3.1. Expressions of tool swept envelope surface S By using the theory of two-parameter sphere congruence, we can obtain the expression of swept envelope surface of tool undergoing an arbitrary spatial motion. Consider a 8



c(a) = 0

T 0 a ,

r(a) = R0 cos φ + a sin φ

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conical tool used frequently in finish milling, as shown in Fig.5-a. The tool surface can be represented as the envelope surface of a one-parameter family of spheres. The center c and radius r of any sphere in the family can be expressed as functions of the parameter a , the height measured from the bottom circle and along the cutter axis, (7)

H x

a

f

R0

r O

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R0 R0

r (a)

f

º

(a)

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where a ∈ [R0 tan φ, H/ cos2 φ + R0 tan φ], and r(a) can be seen from Fig.5-b.

(b)

(c)

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Figure 5: (a) Conical tool geometry, (b) Radius r as function of parameter a, (c) Tool swept envelope surface

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The rigid body motion of a tool is represented as a form of matrix-vector pair [R(t), p(t)], t ∈ [t0 , t1 ], where R is a 3 × 3 rotation matrix and p is a 1 × 3 translation vector. The tool motion is described by the matrix [R(t), p(t)], which transforms the initial tool position to arbitrary one, see Fig.6. Then the tool swept envelope surface can be expressed as a two-parameter family of spheres, given by: S(a, t) = R(t) 0

T 0 a + p(t) + r(a, t)n(a, t)

(8)

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where r(a, t) = R0 cos φ + a sin φ, (a, t) ∈ [R0 tan φ, H/ cos2 φ + R0 tan φ] × [t0 , t1 ], and n(a, t) is the unit normal vector of the envelope. A surface is generated by a conical tool under spatial motion is presented in Fig.5-c. More detailed descriptions can be referred to our developed work[20].

z (t )

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Figure 6: Reference frame for tool motion

reference frame

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3.2. Expressions of vibrating workpiece surface X The vibrating workpiece surface changes its geometry due to the dynamic cutting force acting on the varying positions determined by the programmed tool path. To get the expressions of the geometry, we need workpiece structural dynamics model based on finite element approach (FEA), cutting force model with chip thickness calculation involved, and integrated milling dynamics. The block diagram in Fig.7 shows how to get the vibration response of workpiece surface δX(t, w) in milling process. End

Cutting parameters

w

No Yes

Tool path

Cutting force

F(t )

Structural matrices

M, C, K Response  X(t j , w)

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Engagement ?

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t j  t j 1  t j  1,2, m

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Figure 7: Diagram of getting the vibration response of workpiece surface in milling process

V

Cei

= αi Kei + βi Mei

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Using FEA the workpiece structural dynamics model can be obtained with structural modal damping knowledge derived from experimental modal tests. The shell or solid element is applied to develop the finite element model of the workpiece. The stiffness, mass and damping matrices of the element are Z e Ki = ET EEdV V Z (9) Mei = NT ρNdV

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where E = LN and L is a differential operation matrix; N is the shape function matrix; E is the elastic matrix; ρ is the mass density matrix. Coefficients αi and βi are calculated with the modal test experiments. The global stiffness, mass and damping matrices of the structure are then obtained from the element matrices, respectively[21]. Also the effects of material removal on the structural modal shapes should be taken into account. Matrix perturbation theory can be adopted to handle this issue, see our developed work[22]. The tool is assumed to be rigid relative to the flexible workpiece. The workpiece vibration is excited by the summation of cutting force. Considering the regenerative effect with the workpiece structural dynamics, the governing equations of milling dynamics 10



response have the following form: ¨ w) + Cδ X(t, ˙ w) + KδX(t, w) = F(t, w) Mδ X(t,

(10)

Z

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where M, C, and K are the modal mass, damping, and stiffness matrices, assembled from ¨ δ X, ˙ and δX are the acceleration, the element matrices given by Eqs.(9), respectively; δ X, velocity, and displacement of the finite element nodes vibration, respectively; F(t, w) is the distributed cutting forces acting on the workpiece, including the average static and the varied dynamic components (detailed in the Appendix, see Eqs.(17)), which cause blade deformation and vibration respectively. For cutting force analysis, the tool position and orientation relative to the blade surface is detailed in Fig.8. The cutter is sliced into discs along the axial direction. For each disc, a linear expressions for cutting forces prediction presented by Lee[23] are used.

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X

a

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Feed

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Blade

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Figure 8: Coordinate representation of milling process: fixed coordinate system XYZ, process coordinate system FCN and tool coordinate system xyz. The orientation of cutter is defined by lead angle α and tilt angle γ.

dFw (t) = Kwc h(t, w)db + Kwe dS,

w = t, r, a.

(11)

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where, dFt , dFr , dFa are the tangential, radial and axial components of the cutting force acting on the disc. dS is length of a cutting edge and db is width of a discrete disc. Cutting force coefficients Kwc and Kwe are obtained from a set of standard calibration tests. The main problem encountered in five-axis cutting force prediction is how to compute the instantaneous uncut chip thickness h(t, w) and the engagement boundaries. We had developed the decoupled chip thickness model [24], which provides a semi-analytical method, where static uncut chip thickness hs (t, w) can be expressed as the sum of which distributed from two decomposed motions, determined by the lead angle α and tilt angle γ (Fig.8). The engagement boundaries of cutter with workpiece can be deduced by applying hs (t, w) = 0. It should be noted that the geometry of ball-end cutter in reference[24] is representative, the method can also be applied to conical milling cutter or flat end cutter. The details of uncut chip thickness and cutting force calculations are given in Appendix. Then we can obtain the expressions of vibrating workpiece surface X(t, w), given as: X(t, w) = X0 + δX(t, w) 11

(12)



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where X0 are the finite element nodes coordinates of initial blade surface and δX(t, w) in Eqs.(10) are the displacement increments of finite element nodes of the blade surface, expressed in the same coordinates system. The resolution of Eqs.(10) to obtain δX(t, w) can resort to the finite element software. In this study we adopt ANSYS for the transient dynamic response δX(t, w) and the simulation results are detailed in the next section. 4. Model results and experimental verification

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For illustrating and validating the process parameters planning, simulations are given to demonstrate its application in finish milling of a flexible workpiece, followed by the experiments in which an impeller with thin-wall blades and deep channels is machined. This section provides the followings: first the system parameters identification is given to obtain cutting force coefficients, and workpiece modal damping parameters; with system parameters the proposed process parameters planning simulations are developed on the basis of the programmed tool path; the simulation results are verified against by experiments of impeller milling with optimal process parameters and other optional ones.

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4.1. Experiment design and system parameters identification We use a Mikron UCP800 machining center to carry out the experiments. As shown in Fig.9, an AL7075 impeller is selected as the flexible workpiece for milling operations, and the cutter is carbide conical ball nose end mill. The cutter parameters are shown in Table 1:

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Table 1: Conical cutter parameters- Sandvik R216.52 Teeth number (N )

Conical degree (φ)

Helix angle (β)

4 mm

2

10◦

40◦

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Diameter (D)

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The vibration responses are measured by an acceleration sensor B&K accelerometer, which is fixed on the impeller blade surface. While the cutting forces are measured by a rotating cutting force dynamoneter type Kistler 9123C, which is fixed between the spindle and the tool holder (see Fig.9a). The acceleration and force signals are collected and processed by a NI signal acquisition instrument at an acquisition frequency of 20 kHz.

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Dynamometer

Impeller

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Accelerometer

Fixture

(b)

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Figure 9: Experiments: (a) Experimental setup (b) Impeller geometry

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The experiments can be divided into three steps E1-E3 given below, in which E1 and E2 are the system parameters identification experiments prior to the impeller machining. E1: Cutting force coefficients calibration. The cutting tests are conducted for specific tool and part material to identify the cutting coefficients. For each tool-workpiece pair, 18 characterization tests with half-immersion cuts are carried out. Then the linear regression analysis is employed to obtain the cutting coefficients Ktc , Krc , Kac , Kte , Kre and Kae appearing in Eq.(11). Follows are the identified cutting coefficients for the 4 mm cutter in Table 1 and AL7075 workpiece. Ktc = 564.3 + 281θ + 106θ2 − 48.9θ3

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Krc = 12.9 + 17.6θ + 58.7θ2 + 13.7θ3 Kac = 29.1 − 31.2θ − 72.4θ2 − 108.8θ3 Kte = −1.12 × 103 Kre = 78.5

(13)

Kae = 4.71 × 103

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where θ is the position angle of the discs engaged in milling process[24]. E2: Identification of workpiece structural modal parameters. By using the modal test experiments with an acceleration sensor, the modal shape parameters are identified by the Rational Fraction Polynomial (RFP) method[25]. The first and second order modal shape parameters, as well as AL7075 material parameters are given in Table 2. E3: Finish milling of impeller, with measurements of blade vibration response and cutting force. With the given tool path generated by a CAM system and optimized process parameters w∗ derived from the proposed planning strategy, vibration response of workpiece in the milling process using a B&K accelerometer can be obtained. In Table 2, f1 , f2 , ζ1 and ζ2 are the first and second order modal natural frequencies and damping coefficients, respectively; E, ρ and ν are the elastic modulus, mass density and Poisson’s ratio of the impeller material AL7075, respectively. 13



Table 2: Parameters of workpiece structure f2 (Hz)

ζ1

ζ2

E (Pa)

ρ (kg/m3 )

ν

3200

5010

1.078%

1.052%

2.7E+7

2.8E+6

0.33

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f1 (Hz)

(a)

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4.2. Finish milling simulation and experimental verification We have uploaded the tool path simulation video to YouTube at https://www. youtube.com/watch?v=PIaE3-Uy2tY. Cutter contact point trajectory and some discrete cutter locations extracted from the programmed NC tool path are displayed in Fig.(10). In the generated tool path, the finish allowance is 0.2mm, and the distance between two passes is 0.357mm corresponding to a scallop/cusp height of 0.008mm. After post-processing associated with machine tool kinematics, the tool path is transformed into G-codes, and drive the machine tool to conduct the milling operations.

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Figure 10: Tool path for NC machining of impeller blade: (a) Cutter contact point trajectory (b) Discrete cutter locations

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A segment of tool path is selected for simulation. During the segment the cutting force and vibration response of impeller blade are calculated under different process parameters, then the dynamic metric is presented to evaluate the interference between tool shaft and blades. The transient vibration response of impeller blade is calculated on ANSYS. We select the element SOLID187 for analysis. The tetrahedral structural solid element is a higher order 3-D, 10-node element. Each node has three degrees of freedom: translations in the nodal x, y, and z directions. The numbers of elements and nodes for one blade are 5847 and 23242, respectively. In the simulations, the spindle speed Ω and feed per tooth f are selected as the process parameters for optimization, i.e. w = (Ω, f )T . To ensure the machining efficiency and considering limitations of spindle the process parameters are constrained in pre-defined intervals, given as follows: Ω ∈ [5000, 10000] rpm or Vc ∈ [62.8, 125.6] m/min f ∈ [0.8, 1.1] mm/tooth

(14)

14



Ma nu

sc r

ip

tN

ot

Co

py ed

ite d

where Vc is the cutting speed, Vc = Ω·π ·D. With the optimization strategy presented in Section 2, we can obtain the optimization results in the milling process, as well as the intermediate numerical iteration results, listed in Table 3. As can be seen from Table 3 that after 6 iterations (each iteration represents a linear programming computation), the dynamic metric are Φ = 53.6µm with an optimal set of process parameters w∗ = (Ω∗ , f ∗ )T , and Ω∗ = 7450rpm and f ∗ = 0.09mm/tooth, respectively; F = N ·Ω·f is feed speed, an alterative for material removal rate representative of machining efficiency. With the optimal set of process parameters, the dynamic changes of the metric Φ with respect to the time t during milling operations are displayed in Fig.11. It can be seen from Fig.11 that with the optimal parameters set w∗ , the metric changes in milling process and approaches the minimal value 53.6µm when t = 1.15s. The spatial distributions of distance between the two surfaces S and X are shown in Fig.12, from which we can find the areas in dark blue, sensitive for interference. From Table 3, we can see that the dynamic metric Φ is becoming greater with the iterations going on. The initial parameters set Iteration 1: (5500rpm, 0.1mm/tooth), conservative ones with low machining efficiency, result in the least metric Φ = 12.3µm, which means the possible interference between tool and flexible impeller blades. It should be noted here that the criteria for interference-free has been denoted as Φ > 0 in Eqs.(2). The metric Φ is calculated from the developed model, which inevitably contains deviations from the real value. On the other hand, the uncertainties normally exist in a milling experiment, as described in our developed work[26]. Therefore, from a practical machining viewpoint we have to adjust the process parameters to increase Φ as great as possible to reduce the possibility of interference. Using the proposed strategy, fortunately we find a better candidate Iteration 2: (7400rpm, 0.105mm/tooth), increasing feed speed by 41.27%, in addition to magnify the dynamic metric from 12.3µm to 45.2µm. With further optimization iterations, the improvements for dynamic metric as well as feed speed go slow. Finally we get the optimal one, Iteration 6: (7450rpm, 0.09mm/tooth) with a greater metric and a satisfying machining efficiency. The improvements lie on the two sides, one is machining efficiency, and the other the possibility of interference is greatly reduced. Table 3: Optimization results

5500 7400 7410 7400 7450 7460 7450

f (mm/tooth) 0.1 0.105 0.1 0.1 0.095 0.09 0.09

F (mm/min) 1100 1554 1482 1480 1415.5 1342.8 1341

Φ (µm) 12.3 45.2 46.5 46.6 52.2 53.3 53.6

Ac

ce

pt

0 1 2 3 4 5 6

Ω (rpm)

ed

Iteration number

15



130 120

ite d

100 90 80 70 60

53.6 50 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Co

Time (s)

py ed

Dynamic metric (µm)

110

tN

ot

Figure 11: Changes of dynamic metric Φ with respect to time t: Φ(w∗ , t), where w∗ = (7450rpm, 0.09mm/tooth)T , Iteration 6 in Table 3. The arrow points to the minimum of the metric Φ(w∗ , t).

μm

sc r

ip

Sensitive area for interference

ed

Ma nu

S

ce

pt

Figure 12: Spatial distributions of distances(displayed by color bar) between S and X in the milling process, with process parameters: Ω = 7450rpm, and f = 0.09mm/tooth, Iteration 6 in Table 3.

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However, when process parameters are given without dynamic response analysis, the interference may occur. For example, one set of process parameters Ω = 6500rpm, and f = 0.1mm/tooth is adopted for milling operations, as a result the interferences between tool shaft and blades appear, see Fig.13-a. The interference areas distinguish themselves from the common vibration marks on the blade surfaces. Using the proposed model we can get dynamic metric Φ = −15.7µm with the given process parameters set, a negative value which explains why interference occurs. As a comparison, the optimal 16



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process parameters Ω = 7450rpm and f = 0.09mm/tooth, Iteration 6 in Table 3 are adopted for milling operations, resulting in interference-free surface shown in Fig.13-b. In this case the dynamic metric Φ = 53.6µm. The video of vibrating blade using the optimal process parameters has been uploaded to YouTube at https://www.youtube. com/watch?v=a-RHhkkKssQ. !"#$%#$#!&#'($#(

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,(-

,.-

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Figure 13: (a) Interference occurrence, with process parameters: Ω = 6500rpm, and f = 0.1mm/tooth, resulting in dynamic metric Φ = −15.7µm, (b) Interference-free surface, with process parameters: Ω = 7450rpm, and f = 0.09mm/tooth, resulting in dynamic metric Φ = 53.6µm, Iteration 6 in Table 3.

Ac

ce

pt

ed

Ma nu

sc r

ip

In addition, for validation of the proposed cutting force model with identified cutting force coefficients, as well as the finite element model of blades with identified damping coefficients, the cutting forces and accelerations from simulation and experimental results are compared, shown in Figs.14-15. The position for measuring accelerations can be seen in Fig.9. It can be seen that the models of cutting force and acceleration supply a satisfying accuracy from Figs.14-15. Here the tool movement and the blade vibration signal should be synchronised. When the cutter enters in the engagement zone, see Fig.10-a, the vibration acceleration will experience a sudden increase. Therefore the tool entrance position is matching the beginning of vibration acceleration signal increasing. With the machining operations going on, the vibration signal is traced by the machining time, e.g., tc , while with the machining time the cutter can be positioned by the tool path length f · N · tc · Ω/60 mm where f , N and Ω are the feed rate per tooth, number of cutter teeth and spindle speed, respectively.

17



50

50 Fx Fy

30

20

20

-10

10 0 -10

-20

-20

-30

-30

-40

-40 0.4

0.6

0.8 Time (s)

1

ite d

0

-50 0.2

1.2

0.4

(a)

py ed

10

-50 0.2

Fx Fy

40

30

Cutting force (N)

Force (N)

40

0.6

0.8 Time (s)

1

1.2

(b)

ot

Co

Figure 14: Comparison of cutting forces: (a) simulation results and (b) experimental results, with process parameters: Ω = 7450rpm, and f = 0.09mm/tooth, Iteration 6 in Table 3

1000

tN

1000

800

800

600

Acceleration (m/s2 )

400

0 -200 -400 -600 -800 0.55

0.6

0.65

0.7

0.75 Time (s)

0.8

0.85

0.9

0.95

1

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-1000 0.5

400 200

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200

sc r

Acceleration (m/s 2 )

600

(a)

0

-200 -400 -600 -800

-1000 0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

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0.95

1

Time (s)

(b)

ed

Figure 15: Comparison of accelerations: (a) simulation results and (b) experimental results, with process parameters: Ω = 7450rpm, and f = 0.09mm/tooth, Iteration 6 in Table 3

Ac

ce

pt

As can be seen from Figs.14, the cutting forces obtained from simulations are less than those from experimental measurement. The reason is that in simulations the cutting force computation supposes the semi-finished surface is smooth, in other words, does not consider the scollop height areas (the height is about 0.05mm), left by the neighbored tool passes in semi-finish process, see Figs.16. The scollop height areas positively contribute to the cutting forces. Therefore, the practical or measured cutting forces are greater than the simulation results.

18



Semi-finished surface Finished surface

Finished surface

Presumed semi-finished surface in simulation

Semi-finished surface

ite d

Scallop height area

Allowance

py ed

Workpiece

Figure 16: Scallop height area in finish milling

Co

5. Conclusions and remarks

ot

In this investigation, the interference-free milling operation of thin-walled workpieces has been studied. According to the model and experimental results reported above, the following conclusions and remarks can be drawn:

Ac

ce

pt

ed

Ma nu

sc r

ip

tN

(a) Model development. The interference-free problem is modeled as the evaluation of approaching extent between tool and flexible workpiece during the milling operations. A dynamic metric for the evaluation has been introduced, and with that as the optimization objective the process parameter planning formulation is developed to guarantee the interference-free requirement in milling process. This model is proposed for the first time in this field to our best knowledge. Minimax optimization strategy is adopted to implement the process parameter planning model. (b) Experiment design and verification. Finish milling of an axial-flow compressor impeller is carried out to verify the proposed model. Comparison results reveal that the proposed model can give an optimal set of process parameters, as as to reduce the blade vibration and guarantee interference-free. Furthermore, the model and simulation results suggest to use a higher spindle speed to improve cutting efficiency, still not cause interference. Experimental results also validate this point. (c) Practical applications. The proposed model and accompanying simulation results are helpful to decision making for process parameters planning in practical flexible workpiece milling operations, because the quantitative information of workpiece vibrations, characteristics of time-varying and spatial-distributed is provided. Note that the proposed process parameters planning model is applicable to milling of impellers with different geometries. The differences lie on the cutting force calculation and impeller structural dynamics. (d) Some remarks. i) The outcome of the proposed planning method is conservative for judging interference occurrence, because the worst case scenario is considered in the defined dynamic metric. ii) The effects of material removal on modal frequencies and shapes of workpieces should be considered. In this study, the finish allowance is 0.2mm, about a tenth part of the finish part. The FEM simulations indicate that the changes of the first five nature frequencies for blades before and after finish milling are less than 10%. However, if the finish allowances are great and cannot be ignored, the variations of modal parameters should be taken into account. Also noted that tool wear or lubrication can affect the precision of cutting force prediction. This 19



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point clearly represents the direct continuation of this study. iii) Milling stability depends on cutter locations because the impeller frequency response functions vary from point to point on blade surface. Fortunately we can off-line preestimate the stability Lobes for diverse machining areas. iv) Last but not least, the selection of process parameters for optimization deserves attention. In five-axis milling the lead and tilt angles affect cutting forces, resulting in workpiece vibration. The process parameters optimization taking into the two angles into account is an interesting work, although the adopted optimization model itself is consistent with the proposed one. Acknowledgement

ot

Co

This work was partially supported by the the National Natural Science Foundation of China (51375005) and the National Basic Research Program of China (2013CB035800). The authors thanks Mr. Jin Yue for the help on numerical control milling operations. The first author thanks the Alexander von Humboldt Foundation for the support during his stays at TU Darmstadt.

tN

Appendix: Cutting force calculation

ip

The decoupled chip thickness model[24] provides a semi-analytical method, where the static uncut chip thickness hs (t, w) can be expressed as the sum of which distributed from two decomposed motions,

sc r

hs (t, w) = hl (t, w) + ht (t, w)

(15)

Ma nu

where hl and ht are the chip thickness distributed from the motions with only lead angle α and tilt angle γ (see Fig.8), respectively. The chip thickness hs (t, w) can be either positive or negative so that the engagement boundary of cutter with workpiece can be deduced by applying hs (t, w) = 0. When regenerative effects are taken into account, the instantaneous uncut chip thickness should be amended, h(t, w) = hs (t, w) + cos(ϕj (t, w)) sin(α)DT [δX(t, w) − δX(t − τ, w)]

(16)

Ac

ce

pt

ed

where D is the cutting force distribution matrix, which denotes the location of the cutting force applied on the workpiece. δX(t, w) is the dynamic response vector and δX(t, w) − δX(t − τ, w) represents the regenerative effect. The regenerative effect results from phase differences between the vibration waves left on both sides of the chip. τ = 60/N Ω 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. For a tool with N evenly spaced teeth, ϕj is the angular position of the jth cutting edge and expressed as: ϕj =

2π(j − 1) 2πΩ t+ 60 N

j = 1, 2, · · · , N

Substituting Eq.(16) into (11), the cutting force matrix in Eq.(10) can be stated as follows: F(t, w) = D{H(t, w)[X(t, w) − X(t − τ, w)] + G(t, w)} (17) 20



H(t, w) is a matrix which represents the component of cutting forces that depend on the position vector, given as follows: gi [ϕj (t, w)] s2 α

Ktc scϕ − Krc c2 ϕ scα DT Kac cϕ

(18)

ite d

H(t, w) =

N X M X j=1 i=1

py ed

where M is the number of discrete discs. The function gi [ϕj (t)] acts as a switching function, which is equal to 1 if the tooth of the ith disc is active and 0 if it is not cutting. ( 1 ϕie < ϕj (t, w) < ϕia gi [ϕj (t, w)] = 0 otherwise

Co

where ϕie and ϕia are the angles where the jth tooth of the ith disc enters and exits the cut, respectively. G(t, w) is the vector that represents the components of the cutting force which are independent of the position vector:

gi [ϕj (t, w)] s2 α

scα

Ktc scϕ − Krc c2 ϕ Kte scϕ − Kre c2 ϕ his (t)db + dS Kac cϕ Kae cϕ

tN

j=1 i=1

ot

G(t, w) =

N X M X

sc r

ip

(19) where his (t, w) is the static uncut chip thickness of the ith disc. The total cutting force is the sum of force components contributed by all the discs engaged in cutting. As a result, the instantaneous cutting force F(t, w) is a function of process parameters w, including variables: inclination angles α and γ, depth of cut b, feed per tooth f and spindle speed Ω .

Ma nu

References

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[1] J.-S. Tsai and C.-L. Liao. Finite-element modeling of static surface errors in the peripheral milling of thin-walled workpieces. Journal of Materials Processing Technology, 94(2):235–246, 1999. [2] S. Ratchev, S. Liu, W. Huang, and A. A Becker. Milling error prediction and compensation in machining of low-rigidity parts. International Journal of Machine Tools and Manufacture, 44(15):1629–1641, 2004. [3] S. Ratchev, S. Liu, W. Huang, and A. A Becker. A flexible force model for end milling of low-rigidity parts. Journal of Materials Processing Technology, 153:134–138, 2004. [4] Y.-G. Kang and Z.-Q. Wang. Two efficient iterative algorithms for error prediction in peripheral milling of thin-walled workpieces considering the in-cutting chip. International Journal of Machine Tools and Manufacture, 73:55–61, 2013. [5] S. Ratchev, S. Liu, W. Huang, and A. A Becker. An advanced fea based force induced error compensation strategy in milling. International Journal of Machine Tools and Manufacture, 46(5):542–551, 2006. [6] G.H. Qin, W.H. Zhang, Z.X. Wu, and M. Wan. Systematic modeling of workpiece-fixture geometric default and compliance for the prediction of workpiece machining error. ASME Journal of Manufacturing Science and Engineering, 129(4):789–801, 2007. [7] M. Wan, W.H. Zhang, G.H. Qin, and 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):1366–1374, 2008. [8] J. Chaves-Jacob, G. Poulachon, and E. Duc. Optimal strategy for finishing impeller blades using 5-axis machining. The International Journal of Advanced Manufacturing Technology, 58(5-8):573– 583, 2012.

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[9] C. Lartigue, E. Duc, and A. Affouard. Tool path deformation in 5-axis flank milling using envelope surface. Computer-Aided Design, 35(4):375–382, 2003. [10] V. Thevenot, L. Arnaud, G. Dessein, and G. Cazenave-Larroche. Integration of dynamic behaviour variations in the stability lobes method: 3d lobes construction and application to thin-walled structure milling. The International Journal of Advanced Manufacturing Technology, 27(7-8):638–644, 2006. [11] U. Bravo, O. Altuzarra, L.N. L´ opez De Lacalle, J.A. S´ anchez, and F.J. Campa. Stability limits of milling considering the flexibility of the workpiece and the machine. International Journal of Machine Tools and Manufacture, 45(15):1669–1680, 2005. [12] I. Ma˜ n´ e, V. Gagnol, B.C. Bouzgarrou, and P. Ray. Stability-based spindle speed control during flexible workpiece high-speed milling. International Journal of Machine Tools and Manufacture, 48(2):184–194, 2008. [13] O.B. Adetoro, W.M. Sim, and P.H. Wen. An improved prediction of stability lobes using nonlinear thin wall dynamics. Journal of Materials Processing Technology, 210(6):969–979, 2010. [14] S. Seguy, G. Dessein, and L. Arnaud. Surface roughness variation of thin wall milling, related to modal interactions. International Journal of Machine Tools and Manufacture, 48(3):261–274, 2008. [15] Y. Altinta¸s and E. Budak. Analytical prediction of stability lobes in milling. CIRP AnnalsManufacturing Technology, 44(1):357–362, 1995. [16] B.P. Mann, T. Insperger, P.V. Bayly, and G. St´ ep´ an. Stability of up-milling and down-milling, part 2: experimental verification. International Journal of Machine Tools and Manufacture, 43(1):35–40, 2003. [17] T. Insperger and G. St´ ep´ an. Updated semi-discretization method for periodic delay-differential equations with discrete delay. International Journal for Numerical Methods in Engineering, 61(1):117–141, 2004. [18] P. Marcotte and J. Dussault. A sequential linear programming algorithm for solving monotone variational inequalities. SIAM Journal on Control and Optimization, 27(6):1260–1278, 1989. [19] L.M. Zhu, Z.H. Xiong, H. Ding, and Y.L. Xiong. A distance function based approach for localization and profile error evaluation of complex surface. ASME Journal of Manufacturing Science and Engineering, 126(3):542–554, 2004. [20] L. Zhu, X.-M. Zhang, G. Zheng, and H. Ding. Analytical expression of the swept surface of a rotary cutter using the envelope theory of sphere congruence. ASME Journal of Manufacturing Science and Engineering, 131(4):041017, 2009. [21] J.T. Oden, C.A.M. Duarte, and O.C. Zienkiewicz. A new cloud-based hp finite element method. Computer Methods in Applied Mechanics and Engineering, 153(1):117–126, 1998. [22] X.-M. Zhang, L.-M. Zhu, and H. Ding. Matrix perturbation method for predicting dynamic modal shapes of the workpiece in high-speed machining. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 224(1):177–183, 2010. [23] P. Lee and Y. Altinta¸s. Prediction of ball-end milling forces from orthogonal cutting data. International Journal of Machine Tools and Manufacture, 36(9):1059–1072, 1996. [24] T. Huang, X.-M. Zhang, and H. Ding. Decoupled chip thickness calculation model for cutting force prediction in five-axis ball-end milling. The International Journal of Advanced Manufacturing Technology, 69(5-8):1203–1217, 2013. [25] A. Carcaterra and W. D’Ambrogio. An iterative rational fraction polynomial technique for modal identification. Meccanica, 30(1):63–75, 1995. [26] X.-M. Zhang, L.M. Zhu, D. Zhang, H. Ding, and Y.L. Xiong. Numerical robust optimization of spindle speed for milling process with uncertainties. International Journal of Machine Tools and Manufacture, 61:9–19, 2012.

List of Figures

Milling of a thin-wall flexible blade . . . . . . . . . . Dynamic metric . . . . . . . . . . . . . . . . . . . . . Point-to-surface distance . . . . . . . . . . . . . . . . Flow chart of the optimization procedures . . . . . . (a) Conical tool geometry, (b) Radius r as function Tool swept envelope surface . . . . . . . . . . . . . . 22

Ac

1 2 3 4 5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of parameter a, (c) . . . . . . . . . . . .

4 5 6 8 9



14

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16

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13

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12

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11

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9 10

List of Tables

Conical cutter parameters- Sandvik R216.52 . . . . . . . . . . . . . . . . . Parameters of workpiece structure . . . . . . . . . . . . . . . . . . . . . . Optimization results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 14 15

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1 2 3

sc r

8

Reference frame for tool motion . . . . . . . . . . . . . . . . . . . . . . . . 9 Diagram of getting the vibration response of workpiece surface in milling process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Coordinate representation of milling process: fixed coordinate system XYZ, process coordinate system FCN and tool coordinate system xyz. The orientation of cutter is defined by lead angle α and tilt angle γ. . . . 11 Experiments: (a) Experimental setup (b) Impeller geometry . . . . . . . . 13 Tool path for NC machining of impeller blade: (a) Cutter contact point trajectory (b) Discrete cutter locations . . . . . . . . . . . . . . . . . . . . 14 Changes of dynamic metric Φ with respect to time t: Φ(w∗ , t), where w∗ = (7450rpm, 0.09mm/tooth)T , Iteration 6 in Table 3. The arrow points to the minimum of the metric Φ(w∗ , t). . . . . . . . . . . . . . . . . . . . 16 Spatial distributions of distances(displayed by color bar) between S and X in the milling process, with process parameters: Ω = 7450rpm, and f = 0.09mm/tooth, Iteration 6 in Table 3. . . . . . . . . . . . . . . . . . . 16 (a) Interference occurrence, with process parameters: Ω = 6500rpm, and f = 0.1mm/tooth, resulting in dynamic metric Φ = −15.7µm, (b) Interferencefree surface, with process parameters: Ω = 7450rpm, and f = 0.09mm/tooth, resulting in dynamic metric Φ = 53.6µm, Iteration 6 in Table 3. . . . . . . 17 Comparison of cutting forces: (a) simulation results and (b) experimental results, with process parameters: Ω = 7450rpm, and f = 0.09mm/tooth, Iteration 6 in Table 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Comparison of accelerations: (a) simulation results and (b) experimental results, with process parameters: Ω = 7450rpm, and f = 0.09mm/tooth, Iteration 6 in Table 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Scallop height area in finish milling . . . . . . . . . . . . . . . . . . . . . . 19

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23


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