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Chinese Journal of Aeronautics 22(2009) 677-684

Chinese Journal of Aeronautics

www.elsevier.com/locate/cj a

Corner-milling of Thin Walled Cavities on Aeronautical Components

Wu Qiong*, Zhang Yidu, Zhang Hongwei

State Key Laboratory of Virtual Reality Technology and Systems, School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

Received 4 November 2008; accepted 2 March 2009

Abstract

This article presents a mathematical model of helical end-milling forces through experimental identification of the cutting coefficients and analyzes the changes of corner-milling forces under different conditions. In allusion to the corner-milling process, the relationship between working parameters and the corner coordinates is investigated by way of combination of tool tracing and cutting geometrodynamics. The milling parameters are optimized by changing the coordinates of tool center and working parameters without altering cutting forces. By applying the optimized parameters to milling practice, a comparison is made to show the improved product quality. Based on these optimized parameters, a finite element method (FEM) program is used to compute deformation values of a workpiece's corner, which evidences few effects that optimized parameters can exert on the corner deformation.

Keywords: machining; mathematical models; trace analysis; milling cutters; parameter estimation; optimization; deformation; finite element method

1. Introduction

As new materials, large structures and complicated designs have found ever more increasing applications in the aeronautic and aerospace industry, and the use of constructions characterized by thin walled cavities is booming year after year. The structure of this type is generally manufactured by high-speed numerical control (NC) milling. However, as most cavities are configured dramatically complicated with pockets and thin walls, adoption of traditional cutting methods would often lead to over cutting, lack cutting and chatter during corner-milling. Some workpieces even have to be ground manually to eliminate defects. This stands not only to shorten tool's service life, but seriously deteriorate the product accuracy and manufacturing efficiency as well. As a result, it is required to develop methods to determine reasonable milling trace paths and optimize cutting parameters to create a highly effective working process.

Recent researches on pocket milling are mainly categorized into two directions. One is to optimize the

Corresponding author. Tel.: +86-10-82317756.

E-mail address: wuqlc@126.com

Foundation item: National Defense Basic Research Program (D0620060433)

1000-9361/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(08)60158-2

milling trace path to ensure a constant cutting capac-ity[1-6] and, meantime, improve the accuracy and stability of milling. The other is to investigate milling deformation, milling forces and longitudinal errors during corner-milling. There are a lot of works pertinent to static deformation of spindles and workpieces caused by cutting forces and thermal stresses during corner-milling. Circumferential errors resulted from run-out of spindles and vibration of the tool and/or workpiece exert baleful effects upon the machining quality. Refs.[7]-[11] were devoted to modifying and optimizing cutting parameters to improve dimensional accuracy. This article is supposed to explore the correlation between tool motion traces and experimental milling forces to attain stable parameters for corner-milling and apply a parameterized finite element method (FEM) to calculating workpiece deformation, which enables us to determine cutting parameters through re-optimization and guarantee the ultimate cutting accuracy.

2. Cutting Forces Per Tooth

The cutting force is one of the factors that decisively affect the cutting process, for an unstable cutting force often results in machining defects in a workpiece. It is for this reason that the cutting force is chosen to be the analytical base in the ongoing study. Apart from the factors, such as tool geometry, machining conditions

and materials, the material hardness, friction factors and stresses also make a significant difference to the cutting force. Since cutting forces at the corners are more complicated, it is necessary to adopt an appropriate end-milling force model as a base to calculate accurate corner-milling forces. The helical end-milling force model used herein is quoted from the analytical study by Y. Altintas, et al.[12]

2.1. Modeling of cutting forces

Given a differential dz along z-axis (see Fig.1, where n is spindle speed), the differential tangential (dFt), radial (dFr) and axial (dFa) cutting forces acting on an infinitesimal segment of the cutting edge are given by Y. Altintas , et al.[13]:

dFt . {fa z) = (Ktc hJ (0j (z)) + Kte)dz

dFr,J {fa z) = (Krchj (^ (z)) + Kre )dz

rc JyrJy

dFa, j (0, z) = (Kachi (^ (z)) + Kae)dz

where (¡) is rotation angle of the milling cutter; hJ (fa ( z ))the chip thickness which is related to the

cutting edge and varies with the position of the cutting point and cutter's rotation, J the number of flute (J = 0, 1, •••, N); the edge cutting force coefficients Kte, Kre and Kae are constants; as the shear force coefficients Ktc, Krc and Kac are complicated to compute by formulas, they are determined by resorting to orthogonal cutting tests using an oblique transformation method[14]. hJ (fa (z)) is evaluated with the kinematics

of milling formula[15],

h. z) = csin(/). (z)

and z directions.

Fig.2 shows the model of cutting forces per tooth.

Fig.2 Model of cutting forces per tooth.

By finding indefinite integral of Eq.(1) from zi to zj (^2), can be obtained

— f Nacr

-[Ktc cos(2^) - Krc(2^- sin(20))] +

(-Kte sin ^ + Kre cos f)

Nac 8K

[ Ktc(2^- sin(20) + Krccos(2^)]-

— (Kte cos ^ + Kre sin f)

Fz ("Kacccos <P + Kae^)

where a is the length of cutting edge, c the cutting output per tooth.

According to the changing of cutting width and by substituting fa = n and fa = 0 into Eqs.(4)-(6), the following equations expressing the average forces per tooth in the three directions can be obtained:

Fx =-—k^

Nac Na

~~TK tc +-K te

F. - ——K. + N-K.

Fig.1 Model of milling forces.

The tangential (dFt), radial (dFr) and axial (dFa) cutting forces are then projected onto the Cartesian coordinate system then we can obtain

" Fx " - cos <f

Fy = sin ^

_ Fz _ 0

where Fx, Fy and Fz are cutting forces projected in x, y

Ref.[7] points out that the edge cutting force coefficients Kre, Kte and Kae are constant with little effect on the resultant forces and thus can be found the following solution, in which, to achieve the resultant cutting forces, the edge cutting force coefficients are actually ignored:

F = yj ( Fx )2 + ( Fy )2 + ( Fz )2 =

(-+ ( + ( N^Kac)2

The shear force coefficients Ktc, Krc and Kac are found through experiments, which are described in Section 2.2.

2.2. Experimental cutting force coefficients

Fig.3 illustrates the experimental setup and Table 1 lists the milling parameters used for identifying the shear force coefficients Ktc, Krc and Kac. The devices used in experiment were as follows: an X50A vertical mill, an INV303A intellectual signal collector and analyzer, a Kistler measuring cell, an electricity amplifier and an interface cable. To simulate the milling process, was used SimuCut experimental software developed by School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics[16-17]. The software for analysis is supplied with a measuring cell.

Fig.3 Experimental setup and site. Table 1 Cutting parameters

Parameter Value Parameter Value

Rotation speed/ (r-min-1) 500 Tool material Hard alloy

Feed speed/ (mm-min-1) 400 Number of tooth 2

Cutting depth/mm 2 Helical angle/(°) 20

Cutting width/mm 6 Length of cantilever/ mm 120

Workpiece material 7075Al Outer diameter/mm 12

Hardness HRA 90 Rake angle/(°) 20

Different cutting forces were achieved by changing the feed speed and recorded (see Fig.4). These forces were then compared to simulated values (see Fig.5) to implement correction of the average cutting forces in simulation. Shear force coefficients were also obtained from simulation in order to reduce material waste and improve the efficiency instead of experiment.

2 3 4 5 6 7 Time/s

Fig.4 Milling forces in experiment.

Fig.5 Comparison of milling forces from experiment and simulation.

The average cutting forces per tooth can easily be determined through simulation and experiments. When a piece of 7075 Al alloy was milled with a cutter of hard alloy, the shear forces coefficients in N/mm2 were

Krc = 893.3, Ktc = 1 880.0, Kac= 157.1 2.3. Average cutting forces during corner-milling

Experiment demonstrated that under the same cutting conditions—cutting width ae, cutting depth ap and feed speed Vf, tool traces would profoundly affect the cutting forces.

Fig.6 shows the average cutting forces produced by three different tool traces, where 6 is the angle corresponding to cutting arc L, R the radius of the milling cutter. The maximal average cutting force is represented by a concave curve, while the minimal a convex curve. Based on experimental results, the relationship of the parameters that affect the average cutting forces is defined by

F = f Vf , L, ß, Ktc, Krc, Kac)

where L is the length of cutting arc for milling, p the helial angle.

S 0.28

■S 0.24

2 0.20

0.2 Time/s

•S 0.24

0.2 Time/s

0.2 Time/s

Fig.6 Milling forces produced by three different traces.

Just as Fig.7 shows, the milling cutter, which is unfolding along an edge, has the parameters described by

nN L tan ß = ae

Substituting Eqs.(10)-(12) into Eq.(8) yields the following equation for the resultant cutting force per tooth:

cos ß Vf

cos ß Vf

( cos ß Vf

Length of cutting edge a

Fig.7 Relationship of edge-milling parameters.

3. Influences of Tool Trace

Many corners can be encountered during highspeed cutting of cavities. A sudden change in the feed direction during cutting would cause a tapering point at the corner of the tool trace. In Fig.8, if the point A, at which the profile of the milling cutter intersects with the roughing profile, moves to the point D, the length

of the milling arc AA' suddenly increases to arc DD'. According to Eq.(13), the length of the milling arc is directly related to the average cutting forces per tooth.

Fig.8 shows the corresponding changes in the cutting force.

Reality force curve in the corner

\ / B

! \ Ideal force curve in the corner

Position of cutting (b)

Fig.8 Change of arc length and cutting force caused by pointed machining trace.

Once the milling cutter's center coincides with the tapering point of the tool trace, the cutting force skyrockets from the maximum to the minimum then followed by a continual cutting process with the constant cutting force. When the length of the milling arc reaches maximum at the position of the corner point, the feed speed in this direction drops to zero. The sudden change in direction not only causes fluctuation in the cutting forces and deteriorates the product quality, but also makes it difficult to control the milling process by using NC system. As a result, it is urgently needed to find an ideal cutting-force curve for corner-cutting that is in position to guarantee machining stability.

In practical milling process, an arc machining trace is generally adopted to avoid that direction is changed suddenly, as shown in Fig.9. Point A moves to point E

and the arc length changes from AA' to EE' . The cutting arc for this process changes rather gradually compared to pointed machining, and the real force curve tends towards the ideal curve. Analysis results demonstrate that the machining trace directly influences the cutting quality.

§ 0.45

^ 0.30

£ 0.15 <

Reality force curve in the corner

Ideal force curve in the corner

Position of cutting (b)

Fig.9 Change of arc length altering cutting forces when

arc-machining.

4. Influences of Feed Speed

At present, corners of a cavity are commonly first machined with a large roughing milling cutter, and then finished with a precision milling cutter (see Fig.10).

Fig.10 Precision machining of a corner.

As experience has suggested, the corner feed speed should decrease somewhat to reduce the cutting forces. After the corner-cutting has been completed, the feed speed increases again to improve the cutting efficiency. If the feed speed remains too low, the efficiency would be significantly reduced if there are too many corners in a cavity to be fabricated. In contrast, an unduly minor feed speed would trigger chatter, deformation and

over-cutting in the milling process, which certainly exacerbates the product quality. In the following, a detailed analysis of a corner with a 1/4 circle will be carried out and effects of changing parameters during cutting investigated.

Fig.11 shows that corner-milling can be divided into six sects along the tool trace according to points A, B, C, D and E to yield: ® (», A), © (A, B), © (B, C), © (C, D), © (D, E) and © (E, »). Note that as sects ©-© are just the inverse of the first three sects, only sects ©-© should be investigated, for the results of the other sects can easily be obtained in the same way. In Fig.11, h is the wall thickness of corner; R1 and R2 are the radii of rough and precision machining profiles.

Milling tool

Workpiece

Fig.11 Separated process of corner-cutting.

The milling cutter's center moves from far away to points A, B, C, and the milling cutter's profile and the precision machining profile T1 have three tangential points, Ai, Bi and Ci (see Fig.12, where a1, a2 and a3 are the angles corresponding to cutting arc A1A2 , B1B2 and QC2 ). The front profile of the milling cutter and the rough machining profile intersects at A2, B2 and C2. The nine points can be denoted in the coordinates by A (xa, yA), A1 (xAl, yAx), A2 (xa2, yA2), B (Xb, ys), B1 (Xbx, yB) B2 (XB2, yB2), C (Xc, yC), C1 (xQ, yCj) and C2 (XC2, yc2).

Fig.12 Changing cutting arc length.

Fig.12 shows that points A, B and C are critically important in the^whole corner-milling process. A1A2 , B1B2 and QC2 are the corresponding cutting arcs.

When the milling cutter's center passes through them, the cutting forces per tooth will be subjected to a change. ® (<», A) means cutting forces remaining stable; ® (A, B) cutting forces soaring at an increasing rate; ® (B, C) cutting forces increasing, however, at a decreasing rate. At the point C, the milling force reaches the maximum but the change rate of cutting force is decreasing to zero.

4.1. Analysis of the (x>, A) process

When the milling cutter moves from far away to point A, both the rough machining profile T2 and the tool machining trace T3 are straight lines.

Since arc A,A2 = Ra,,

R - a„

R - a„

cos a1 =

a1 = arccos-

L = A1A2 = R arccos-

R - a„

According to Eq.(15) and Eq.(13), the cutting arc is constant, so the average cutting forces per tooth will be stable only if the cutting parameters remain unchanged.

4.2. Analysis of the (A, B) process

As the milling cutter moves from A to B, the tool machining trace is a straight line, viz x = R, but the rough machining profile T2 is an arc expressed by

(x - R)2 + (y - R)2 = R2 (x < Rj, y < Rj) (16)

The milling cutter circle is

(x - xb ) + (y - yB) = R

In the machining process, the tool center's coordinates of (xB, yB) can be found at any time from the NC system program.

The Bi coordinates (xB1, yB1) are given by

xB, = xB ~ R, yB1 = yB

The B2 coordinates (xB2, yB2) at any time can easily be obtained by solving Eq.(16) and Eq.(17).

The chord length B1B2 can be obtained from the following formula:

B1B2 = yj(xB, - xB2)2 +(yB, - >B2 )2

The length of

can be expressed as

2 = Ra2

Since then

L = 2R arcsin

a2 = 2arcsin[BjB2/(2R)]

V(XB. -xb2)2 + (y^ -yB2)2 /(2R)

According to Eq.(20) and Eq.(13), the average cutting forces per tooth in the process can easily be obtained.

4.3. Analysis of the (B, C) process

As the milling cutter moves from B to C, the tool machining trace as an arc can be defined by

(x-R2)2 + (y-R2)2 = (R -R)2 (21)

The rough machining profile T2 is also an arc, which is described by

(x - R1)2 + (y - R1)2 = R12 (x < R1, y < R1) (22)

The milling cutter's circle can be expressed by

(x - xc )2 + (y - yC )2 = R2 (23)

Coordinates (xC, yC) of C, the tool's center, can be obtained from the NC system program in the machining process. The coordinates of C1 can be solved in terms of the variable 6 by using

R2 - yC

cos 6 = —-C, sin U = -

R2 - R

R2 - R

xC = xC - R cos 0, yC = yC - R sin 0

The C2 coordinates (xC2, yC2) at any time can easily be obtained by solving Eqs.(22)-(23). Chord length C1C2 can be obtained from Eq.(24) as follows

C1C2 ( XC, - xCzf +( yC, - yC2)2 The length of C^C2 can be expressed by

L = C,C2 = Ra3

then L = 2R arcsin

a3 = 2arcsin[C1C2 /(2R)]

- xc2)2 + (yc. - yc2)2 /(2R)

The average cutting forces per tooth in the process can easily be obtained from Eq.(27) and Eq.(13).

The average cutting forces per tooth can be achieved from Eq.(15), Eq.(20), Eq.(27) and Eq.(13). By symmetric principle, cutting forces of (C, D), (D, E) and (E, ®) can be easily obtained. As the milling process is going on, Vf is modified to keep the cutting force F constant. Vf can be solved at the center coordinates at any time using all of the equations.

5. Calculation and Tests

As an example, the above-introduced formulas are used to find the parameters dependent on feed speed during cutting. Assume the radius of rough machining profile R1 = 20 mm, the radius of rough machining R2 = 25 mm, the cutting depth ap = 5 mm, the milling cut-

ter's radius R = 6 mm, the initial feed speed Vf = 400 mm/min and the cutting width ae=5 mm.

By applying Eq.(15), Eq.(20), Eq.(27) and Eq.(13) to a corner, the coordinates of the cutter's center and the cutting forces can be found as a function of feed speed. If F is set to be constant and the trace in coordinates (x, y) is input, the feed speed can be obtained at any position and any time. Table 2 lists the parameters.

Table 2 Parameters of corner-cutting in milling process

Tool center's coordinates (x, y)/mm Feed speed/ (mm-min-1) Cutting arc length/mm Cutting force/kN Rotation speed/ (r-min-1)

(6.0, 30.0) 400.0 9.42 0.350 8 000

(6.0, 29.0) 400.0 9.42 0.350 8 000

(6.0, 28.0) 400.0 9.42 0.350 8 000

(6.0, 27.0) 400.0 9.42 0.350 8 000

(6.0, 26.0) 400.0 9.42 0.350 8 000

(6.0, 25.0) 392.1 9.51 0.350 8 000

(6.0, 24.0) 381.4 9.76 0.350 8 000

(6.0, 23.0) 364.7 10.38 0.350 8 000

(6.1, 22.0) 350.2 10.91 0.350 8 000

(6.3, 21.0) 337.7 11.38 0.350 8 000

(6.5, 20.0) 314.8 12.22 0.350 8 000

(6.7, 19.0) 302.3 12.69 0.350 8 000

(7.0, 18.0) 293.6 13.01 0.350 8 000

(7.3, 17.0) 280.7 13.49 0.350 8 000

(7.6, 16.0) 275.8 13.67 0.350 8 000

(8.0, 15.0) 270.2 13.88 0.350 8 000

(8.5, 14.0) 268.6 13.93 0.350 8 000

(9.0, 13.0) 266.1 14.03 0.350 8 000

(9.6, 12.0) 265.5 14.13 0.350 8 000

The parameters in Table 2 are applied to milling process and the results before and after optimization are compared in Fig.13. To clarify the results, the corner has been separated into three parts according to surface roughness (see Fig. 14) and Table 3 lists the roughness data thereon.

From Table 3, it is noted that optimized cutting parameters have exerted powerful influences upon the Part 2 of the corner. Comparison of results evidences that modification of the feed speed can not only avoid

(a) Before optimization

(b) After optimization

Fig.14 Three parts separated according to surface roughness.

Table 3 Surface roughness data pertinent to Fig.14

Part 1 2 3

Before optimization 3.2 25.0 3.2

After optimization 3.2 3.2 3.2

chatter and over-cutting, but also improve the cutting efficiency. Furthermore these cutting parameters can also be applied to calculate the deformation of the thin walled corner and further testify optimization and reliability of parameters.

6. Corner Deformation

Owing to the prominent stiffness of the sharp pointed corner, this location is naturally subjected to less deformation than other sites. Now load the optimized parameters calculated in Section 4 onto the geometric model using auto program design language (APDL) and simulate corner deformation using FEM software.

Simulate deformations as the function of thickness in the milling process. Fig.15 shows the instantaneous deformation.

Fig.13 Comparison of results between before and after optimization.

Fig.15 1 mm-thinkness deformation of corner by FEM.

Fig. 16 presents the points and fitting curves that represent several groups of data gathered in milling process. The fitting curves of deflection are parabolas in the case of an unfolded corner. It is observed that minimal deformations occur at sharp pointed corners regardless of the thickness. Specific data in Fig. 16

corroborate that as thickness increases from 1.0 mm to 3.0 mm, the deformation of the pointed corner decreases from 0.043 mm to 0.022 mm while the deformation of the two flanks falls from 0.800 mm to 0.139 mm showing an almost 5.8-fold reduction.

To sum up, thickness and cutting parameters of a thin walled cavity exert few effects upon the deformation of a pointed corner but have strong effects on the two flanks. Optimized parameters can easily be applied to corner-cutting without worrying about obvious workpiece deformation.

Fig.16 Curves showing deformation as function of thickness and position under the same condition.

7. Conclusions

Changes in the cutting force are comparatively complicated during corner-milling. By dividing the process of corner-milling into six sects, this article investigates one by one. This approach is adopted to associate experimental data and the cutting geometrodynamics with FEM to model the milling forces and deformations during corner-milling. Furthermore, it introduces a method to provide proper cutting parameters which are to be put into the NC system to ensure the stability of corner-cutting and greatly improve the processing efficiency.

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Wu Qiong Born in 1980, he received B.S. and M.S. degrees from Shenyang University of Chemical Technology in 2002 and 2005 respectively, and now he is a Ph.D. candidate in Beijing University of Aeronautics and Astronautics. His main research interest includes analysis and simulation of manufacturing and design as well as FEM. E-mail: wuqlc@126.com

Zhang Yidu Born in 1959, as a Ph.D., he is a professor in Beijing University of Aeronautics and Astronautics. His main research interest includes multi-subject optimization technology, transmission and simulation of manufacturing and design, etc. E-mail: ydzhang@buaa.edu.cn

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