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ScienceDirect Procedia CIRP 58 (2017) 210 – 215
16th CIRP Conference on Modelling of Machining Operations
Tool orientation optimization for reduction of vibration and deformation in ball-end milling of thin-walled impeller blades Tao Huang, Xiao-Ming Zhang*, Han Ding School of Mechanical Science and Engineering, State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China * Corresponding author. E-mail address:
[email protected]
Abstract Due to the flexibility of one-side-supported thin impeller blades, machining defects such as overcut or undercut frequently occur, especially at the front and rear edge connecting the suction surface and the pressure surface, where the thickness are even less than 0.2 millimeter. Vibration and deformation are great challenges for engineers and should be mitigated for better quality. To this end, in the paper a method of tool orientation determination for vibration and deformation reduction during ball-end milling of flexible impeller blades is proposed. In order to reduce vibration level, carefully at the front and rear edge, the direction of resultant cutting force is attempted to be adjusted in the plane in which the stiffness of blade is relatively high. Thus, the cutting force component in the direction of high flexibility is small enough. As a result of modeling, two parts are included, i.e. modeling of tool-workpiece engagement and cutting force in the ball-end finish milling, FEM modeling of structure deformation of flexible workpiece. Machining experiments on 5-axis machining center Mikron UCP 800 verify that the optimized tool orientation could be used for deformation and vibration reduction in milling of flexible parts. ©2017 2017The TheAuthors. Authors. Published by Elsevier B.V.is an open access article under the CC BY-NC-ND license © Published by Elsevier B.V. This Peer-review under responsibility of the scientific committee of The 16th CIRP Conference on Modelling of Machining Operations, in the (http://creativecommons.org/licenses/by-nc-nd/4.0/). person of the Conference Chairs J.C. Outeiro and Prof. Poulachon. under responsibility of theProf. scientifi c committee of The 16thG.CIRP Conference on Modelling of Machining Operations Peer-review Keywords: 5-axis machine; flexible workpiece; tool orientation optimization
1. Introduction Generally, 5-axis milling tool paths as well as tool orientations are generated and optimized in geometrical terms by commercial CAD/CAM software. However, the mechanics of the process has always been little included yet. The neglect of process mechanics may lead to unacceptable high cutting force, excessive defections, vibrations or chatters, poor surface quality and tool wear etc. This motivates researchers to consider mechanical constraints of the process in the tool paths and tool orientations planning. There have been many studies on the prediction of deformations and avoidance of machining errors in flexible part machining. Ratchev et al. [1] reported an integrated methodology for modeling and prediction of surface errors caused by deflection during machining of low-rigidity components. Wan et al. [2,3] developed a general approach with an enhancement of the robustness of the numerical procedure and iterative algorithms for the prediction of static
form errors in peripheral milling of thin-walled workpieces. Both deflections of the cutter and of the workpiece are taken into account. Zhang et al. [4] developed a model to reduce the workpiece vibration aiming to avoid the interference between tool shaft and workpiece with dynamic responses. Lee et al. [5] reduced the workpiece displacement by changing the feed direction and cutter orientation in ball-end milling of cantilever-shaped thin plates. Smith et al. [6] proposed a concept where the uncut portions are proactively designed as sacrificial structures. A simulation system is developed by Biermann et al. [7] for the modeling regenerative workpiece vibrations during the 5-axis milling of turbine blades. Koike et al. [8] presented an algorithm with tool orientation control, which has higher potential to reduce the workpiece displacement significantly for 5-axis milling. Ehsan et al. [9] investigated the effects of lead and tilt angles on cutting force, torque, deflection and surface quality in 5-axis ball-end milling of flexible parts such as turbine blades. They attempted to present a mechanics-based reference map that
2212-8271 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of The 16th CIRP Conference on Modelling of Machining Operations doi:10.1016/j.procir.2017.03.211
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suggests the appropriate tool orientation considering parameters mentioned above. But the dynamic performance of the flexible parts may not be mentioned. In the above literatures, the thin-walled workpieces are always simplified as cantilever-shaped thin plane plates. The stiffness in the normal direction of the local surface is assumed to be weakest, i.e. the most flexible direction, under this situation. However, this assumption is not always true such as on the front and rear arc surfaces which connect the suction surface and the pressure surface of impeller blades. In this paper, the flexibility ellipsoid is introduced to determine the stiffness direction. Then, variation of most flexible directions with respect to the local process coordinate system (PCS) is analyzed along the cutter location points. Cutting force prediction is a key issue and inevitable in the deformation evaluation. Authors have proposed a 5-axis cutting force prediction model based on decoupled chip thickness calculation method [10] which is appropriate for investigating the effects of lead and tilt angles on cutting forces. Since the cutting forces are nonlinear and nondifferentiable functions with respect to lead and tilt angles, the sequential linear programming algorithm is developed for tool orientation determination. 2. Problem statement Manufacturing of flexible part with complex surfaces is of great challenge for engineers. Due to the fickle dynamic characteristic of flexible part and complicated motion of the multi-axis machining process, determining tool path and orientation, cutting depth, spindle speed, and other stabilityrelated factors is not an easy task. To provide a basis for tool orientation optimization in milling of thin impeller blades, this section describes the problem for overview. To distinguish the difference between the cantilevershaped plane plate and the real blade, the geometry as well as the flexibility distribution of the blade, especially near the front and rear arc surfaces, is analyzed. Generally, the suction and pressure surfaces of impeller blades are of free-form surfaces which are blended by arc surfaces. The curvature at the middle part which is far from the front and rear arc surfaces is negligible, thus the dynamic characteristic is similar to that of plane plate and the most flexible direction is in the surface normal direction. However, the curvature near the front and rear arc surfaces is significant. Meanwhile, the most flexible direction is no longer in but even perpendicular to the surface normal direction. Fig.1 gives a cross section of impeller blade. The PCS is presented by FCN, where F denotes the feed direction of the tool, N the surface normal direction, and C cross product of N and F. At location 2, it’s easy to understand that the most flexible direction is in N direction. At location 1 on the front edge, the most flexible direction changes a little from that at location 2, but the PCS rotates 90 degrees counterclockwise from that at location 2; Also, the PCS rotates 90 degrees clockwise from location 2 to location 3. While in machining of cantilever-shaped plane plates, the most flexible direction is always assumed to be in the surface normal direction. The machining process of
impeller blades is rather different from that of plane plates from this point of view.
Fig. 1. Presentation of flexibility ellipsoid in PCS.
On the other hand, milling of such flexible workpiece causes distributed deformation and vibration due to the variation of resultant cutting force vectors including both values and directions. 5-axis cutting force prediction remains a research highlight in the past decades [11,12,13]. Results showed that the 5-axis cutting force is highly influenced by the tool axis inclination angles with respect to PCS. In most instances, tool orientation optimization strategies are carried out for either geometric concern such as collision avoidance or tool deflection or wear concern. In the former problem, the focus is on the workpiece geometric structures, the dynamics of workpiece is out of scope. In the later one, the focus is on reduction in the value of cutting forces, while the direction of the resultant cutting force is rarely involved. However, in the milling of flexible blades, the aim is to minimize the cutting force component in the most flexible direction, thus both value and direction of the 5-axis cutting force need to be taken into account. As the most flexible directions with respect to the PCS are quite different along the tool path, then the tool orientation which changes value and direction of cutting force should be adjusted and optimized point by point. Fig.2 shows the defects appeared frequently at the front and rear edges in practice, where Fig.2(a) and 2(b) are borrowed from the Ref. [14], where two possible reasons related to the geometry were given but the flexible characteristics were not discussed.
Fig. 2. Defects appeared frequently in milling of impeller blades.
A reasonable interpretation is shown in Fig.3(a). The deformation takes place in the most flexible direction which results in intersection between the desired surface and the cutter tooth. In an ideal condition, the tool orientation could be adjusted to achieve that the cutting force component in the most flexible direction turns to zero, as shown in Fig.3(b).
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However, since the tool orientation should meet requirements such as non-interference and non-singularities [15], the ideal condition is always unreachable. Thus we turn to optimize the orientation to minimize the force component in the most flexible direction.
where d f T f means the displacement under unit acting force and S (P1 )T P1 . The matrix S is a real symmetric matrix and eigenvalues are all positive. Therefore, formula is a canonical positive definite quadratic form, whose geometric meaning is an ellipsoid surface, i.e. flexibility ellipsoid. From property of the real symmetric matrix, the spectra decomposition of S can be given as,
S QΛQT
(3)
where Λ diag (O1 , O2 , O3 ), O1 O2 O3 are the eigenvalues of S , and Q is orthogonal matrix. Thus the direction of the major axis of the flexibility ellipsoid can be identified as v f Q[1,0,0]T in the ICS XYZ. Let R denote the rotational matrix of coordinate system FCN with respect to XYZ. Then the major axis of the flexibility ellipsoid in FCN is derived as
vp
RT v f
(4)
Fig. 3. Tool orientation adjustment near edges of blade.
3. Tool orientation optimization procedures Tool orientation optimization is applied at varying positions on a programmed tool path, i.e. tool path is preset. To minimize the related force component, the major preparations are the static flexibility analysis of workpiece and modeling of 5-axis cutting force which are given as follows. 3.1. Static flexibility analysis of thin-walled impeller blades In order to specify the axis directions of flexibility ellipsoid in PCS, static deformation analysis of the blade is performed based on the finite element model (FEM) where commercial software NX Nastran is adopted. Fig.4 shows the static displacement in Z axis of the inertial coordinate system (ICS) XYZ under a predefined load at the corner. In general, a formula between the acting force f and the deformation d at a point can be given as, d
Pf .
(1)
where P is a static flexibility matrix. To determine matrix P , let acting force f [1,0,0]T , then deformation can be calculated by FEM, i.e. d [d x1 , d x 2 , d x3 ]T . Substitute f and d into Eq.(1), we get [P11 , P21 , P31 ]T [d x1 , d x 2 , d x3 ]T . Similarly, let acting force f [0,1,0]T and [0, 0,1]T , then and [P12 , P22 , P32 ]T [d y1 , d y 2 , d y 3 ]T [P13 , P23 , P33 ]T [d z1 , d z 2 , d z 3 ]T . While matrix P is completed, an equation of quadratic form can be derived as, dT T
f f
S
d fTf
1
(2)
Fig. 4. Static deformation analysis based on the finite element model.
3.2. Cutting force modeling in 5-axis milling For cutting force analysis, the tool position and orientation with respect to the blade surface is detailed in Fig.5. The cutter is sliced into discs along the axial direction. For each disc, a linear expressions for cutting forces prediction presented by Lee [16] are adopted,
dFw (t )
Kwc h(t )db K we dS , w t , r , a.
(5)
where, dFt , dFr , dFa are the tangential, radial and axial components of the cutting force act on the disc. dS is length of a cutting edge and db is width of a discrete disc. Cutting force coefficients K wc and K we are obtained from a set of standard calibration tests. The resultant cutting force denoted by f p is the summation of element cutting forces on all of the discs which engage in cutting. Tool coordinate system (TCS) xyz and PCS FCN follow the relationship:
ªF º «C » « » «¬ N »¼
ª sin J « cos J « «¬ 0
cos J cos D sin J cos D sin D
cos J sin D º ª x º sin J sin D »» «« y »» cos D »¼ «¬ z »¼
(6)
The main problem encountered in 5-axis cutting force prediction is how to compute the instantaneous uncut chip thickness (IUCT) and the engagement boundaries. We had developed a decoupled chip thickness model [10], which provides a semi-analytical method, where the IUCT can be expressed as the sum of that contributed from two decomposed motions. The engagement boundaries of cutter
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with respect to workpiece can be deduced by applying IUCT t 0 . It needs to be noted that when tool inclination angles vary, the number of engaged discs, start and exit angels of a disc and the engagement boundaries are all changed. Also, cutting force coefficients of ball-end mill varies along the cutting edge. Based on these facts, the resultant cutting force is significantly influenced by the variation of inclination angles. Simulations and experimental results in Ref.[10] have verified the conclusion.
Fig. 5. Coordinate representation of milling process: inertial coordinate system XYZ, process coordinate system FCN and tool coordinate system xyz. The orientation of cutter is defined by inclination angles D and J .
3.3. Tool orientation planning
min | f (D ,J )
vp fp |
s.t. g d g (D , J ) d g U L
(7)
where v p given in Eq.4 is an unit vector indicating the most flexible direction at the cutting point. For a given point, v p is fixed. f p is the average resultant cutting force which is a function of D and J . Both v p and f p are under FCN. g L d g (D , J ) d gU indicates the feasible region of the tool orientation considering tool-workpiece interference. Suppose | f | is a convex function, since the nonlinearity of cutting force with respect to inclination angles, the method of sequential linear programming [17] can be used to solve P1 . The basic idea is to proceed iteratively by linearizing the objective function and the constraints about the current candidate solutions, thereby reducing the nonlinear problem to a sequence of linear programming problems. Let (D k , J k ) be a candidate solution to problem P1 and consider perturbations of the form (D k 1 , J k 1 ) , we have the first-order differential increment of the force:
f k 1
max
( 'D , 'J )
v p f p (D k 1 , J k 1 ) wf p § wf p · v p f p (D k , J k ) v p ¨ 'D 'J ¸ w D w J © ¹ f k 'f k .
(8)
As the partial derivatives in Eq.8 are not analytic, the difference scheme could be used instead.
[
[ t0 ° k [ 'f d f k f k , ° s.t. ® n[ 'f k d f k f k , ° ° L k k k k U ¯ g d g (D , J ) 'g (D , J ) d g
LP
(9)
The detailed implementation of the above optimizations are given as follows, Algorithm (Tool orientation optimization procedures) Input: Initial tool orientation (D 1 , J 1 ) ; threshold e specifying desired accuracy of the algorithm. Output: Optimal tool orientation (D * , J * ) , and minimum force component value f * Step: 1. Set k 1 ; 2. Compute the cutting force f p (D k , J k ) , function f k , and differences
However, the cutting force is nonanalytic and nonlinear with respect to the inclination angles and. Thus the tool orientation optimization manifests a nonlinear programming problem. To minimize the force component in the most flexible direction, tool orientation planning can be formulated as:
P1
To handle the objective function in P1 which is absolute value of a function, an extra variable [ which satisfies | f k 1 | [ d| f k | , is introduced. The corresponding linear programming problem can be obtained:
'f p (D k , J k )
,
'f p (D k , J k )
; 'D 'J 3. Solve the linear programming problem LP to determine [ k and the increment of ('D k , 'J k ) that appear in 'f k ; 4. Update (D k 1 , J k 1 ) (D k , J k ) ('D k , 'J k ) ; 5. Compute f k 1 v p f p (D k 1 , J k 1 ) ; 6. If | f k | | f k 1 |! e , set k k 1 and go to Step:2; Else exit and report (D * , J * ) (D k 1 , J k 1 ) , and f * f k 1 . 4. Simulation and experiments To identify the cutting coefficients, tests are conducted for specific combination of tool and part material. For each toolworkpiece pair, 18 characterization tests with half-immersion milling are carried out. Then the linear regression analysis is employed to obtain the cutting coefficients K wc and K we (w t , r , a) appearing in Eq.5. Follows are the identified cutting coefficients for the 8 mm ball-end cutter and AL7075 workpiece.
K tc
531.3 306.5T 127.7T 2 53.1T 3
K rc
126 54T 17.4T 2 0.4T 3
K ac
182.9 45.4T 82.2T 2 219.9T 3
K te
0.3, K re
5.6, K ae
(10)
9.3
where T is the position angle of the discs engaged in milling process [10]. The units of K wc and K we are N / mm2 and N / mm , respectively. 4.1. Simulation of cutting force Process parameters for cutting force simulation are: slot milling, feedrate f z 0.1 mm per teeth, the depth of cut b 1.5 mm. The average resultant cutting force vectors in xyz with respect to inclination angles D and J are shown in
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Fig.6. The norms of vectors are relative values to make the figure clearer. The directions and values of resultant cutting forces are quite different while inclination angles change. 4.2. Tool orientation optimization To illustrate the tool orientation optimization algorithm, set the most flexible direction in FCN to be v p [1,1,1]T / 3 and e 0.01 , and inclination angles satisfy 5q D 40q , 90q J 90q . As can be seen from Table 1, after 6 times iterations (each iteration represents a linear programming computation), the objective function f 2 u105 N with an optimal couple of inclination angles D 6.6q , J 64.3q . From Table 1, we can see that the objective function f is becoming smaller with the iterations going on. The initial angles set Iteration 1: (30q ,30q ) result in high f 25.7 N. Using the proposed strategy, we get the optimal angles under which the force component in the flexible direction is nearly zero. It needs to be noted that this is just a special case, not all process can get zero results.
4.3. Experiments The experiment is conducted on Mikron UCP 800 5-axis machining center. The workpiece, tool and feed direction (F) are illustrated in Fig.7. To simulate the cutting condition at the front or rear edges of a blade, The cutting zone is on the flank face of a thin plane plate with size 120mm u 80mm u 3mm . It's obvious that the most flexible direction at the cutting point is in F-direction, i.e. v p [1,0,0]T . The cutting parameters are: two flutes, slot milling, depth of cut b 0.5 mm, spindle speed : 600 rpm and feedrate f r 120 rpm. The inclination angles are set as combination of D 10q , 25q , 40q and J r80q , r40q ,0q . The experimental apparatus and the measured workpiece deformation are shown in Fig.8 and 9, respectively.
Fig. 7. Schematic diagram of the cutting zone and feed direction
Fig. 8. Experiment apparatus and machined workpiece
Fig. 6. Average resultant cutting force vectors (relative value) with respect toinclination angles D and J .
The direction of resultant cutting force in ball-end milling is affected significantly by the cutting depth and cutting width as well as the tool inclination angles. Thus, for different cutting depth or cutting width, the optimized tool inclination angles are quite different. Tab. 1. Optimization results.
Iterations D( ) J( ) f (N)
0 30 30 25.7
1 40 50 17.6
2 40 70 11.9
3 20 70 4.7
4 8.0 67.4 1.2
5 7.0 64.7 0.01
6 6.6 64.3 2e-5
In the experiment, the limit of laser displacement sensor is r2 mm. The deformation is over range while J 80q where most of the cutter element discs are in up milling. And poor surface quality is imprinted. As J turns from positive to negative, some element discs in up milling turns to down milling and the deformation decreases. But when J crosses a certain value, the displacement reverses direction and increases in the opposite direction. It needs to be noted that in milling of rigid workpiece, up milling may not be so bad [10]. But the disadvantage of up milling performs more prominently in the flexible workpiece machining. The simulation in Tab.1 takes only the cutting force into account while the disadvantage of up milling are neglected. From this point of view, the preferred range of J should be of J 0 . while setting 5q D 40q ,0q J 90q , the iteration results are shown in Tab.2 where optimized tool orientation is D 40q , J 42.5q . Since only the ball part of the tool is assumed to be engaged in the slot milling, angle D is set to be less than 40q . In experiments, the deformation as well as
Tao Huang et al. / Procedia CIRP 58 (2017) 210 – 215
vibration are lowest at D Fig.9.
40q and J
40q as seen from
that the optimized tool orientations can reduce machining deformation and improve surface quality. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No. 91648113. References [1]
[2]
[3]
[4]
Fig. 9. Measured displacement with different inclination angles [5] Tab. 2. Iteration results in experiment.
Iterations D( ) J( ) f (N)
0 10 -10 14.2
1 30 0 11.7
2 40 -20 11
3 40 -40 10.9
4 40 -42.5 10.87
As cutting process may destroy the flexible workpiece, exploratory tests are experimented by cutting the flank face of a 3 mm thickness plate. The experimental results verify that the optimized tool orientation could be used for deformation and vibration reduction in edge milling of flexible parts. However, the cutting conditions in this experiment especially the cutting in and out processes are still different from that of edge cutting of a blade. A real impeller blade must be processed for experiment in the future work. It should be noted that the orientation angles can evolve during the machining operation. This paper provides a meted to optimize tool orientation at a specific location on tool path. The orientations will be different along the tool path since the most flexible directions vary at different locations. One can calculate tool orientations at some typical locations on the tool path, and then obtain tool orientations at other locations by vector interpolation.
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13] [14]
5. Conclusion Since the most flexible direction changes along the tool path in ball-end milling of thin-wall blades, which may cause defects especially at the front and rear edges, this paper presents a tool orientation optimization method for reduction of vibration and deformation in the process. The aim is to minimize the cutting force component in the most flexible direction at the cutting point by changing tool inclination angles. The optimization procedure is formulated as a sequential linear programming problem. Experiments indicate
[15]
[16]
[17]
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