Numerical Simulation of Machined Surface ...

Tong Gao Weihong Zhang1 Kepeng Qiu Min Wan The Key Laboratory of Contemporary Design & Integrated Manufacturing Technology, Northwestern Polytechnical University, P.O. Box 552, 710072 Xi’an, Shaanxi, China

1

Numerical Simulation of Machined Surface Topography and Roughness in Milling Process Machined surface topography is very critical since it directly affects the surface quality, especially the surface roughness. Based on the trajectory equations of the cutting edge relative to the workpiece, a new method is developed for the prediction of machined surface topography. This method has the advantage of simplicity and is a meshindependent direct computing method over the traditional interpolation scheme. It is unnecessary to discretize the cutting edge or to mesh the workpiece. The topography value of any point on the machined surface can be calculated directly, and the spindle runout can be taken into account. The simulation of machined surface topography is successfully carried out for both end and ball-end milling processes. In the end milling process, a fast convergence of solving the trajectory equation system by the NewtonRaphson method can be ensured for topography simulation at any node on the machined surface thanks to the appropriate choice of the starting point. In the ball-end milling process, this general algorithm is applicable to any machined surface. Finally, the validity of the method is demonstrated by several simulation examples. Simulation results are compared to experimental ones, and a good agreement is obtained. 关DOI: 10.1115/1.2123047兴

Introduction

Milling is a very common manufacturing process widely used in a variety of sectors, such as the aerospace and automobile industries. In order to ensure the machining quality and to reduce cost and time, it is indispensable to develop research on milling process simulations, including the prediction of cutting forces, form errors, and surface quality. At the process-planning stage, the simulation results will greatly help the process engineers to select appropriate values for the process parameters. Among others, finding the machined surface topography and the geometric shape and texture of the machined surface is essential because the latter directly affects the surface quality, especially the surface roughness. For a ball-end milling process, the surface topography also affects the cutting force and chip load calculations. Simulations of machined surface topography constitute an active research topic in the manufacturing community. Relevant published research work can be summarized as follows. Kline et al. 关1兴 discussed the effects of cutter runout on the shape of the tooth marks in end milling process. Jung et al. 关2,3兴 developed the so-called ridge method to predict the characteristic lines of the cut remainder for a disk tool in the ball-end milling process, and three types of ridges are defined to this end. Imani et al. 关4兴, Imani and Elbestawi 关5兴, and Sadeghi et al. 关6兴 used solid modeling techniques and Boolean operations to deal with the geometric simulation of the ball-end milling operations. In summary, many researchers employed discretization and interpolation techniques to simulate the machined surface topography. Elbestawi et al. 关7兴 and Ismail et al. 关8兴 studied the trochoid path for surface generation of end milling. The tool path is discretized into segments to simulate

1 To whom correspondence should be addressed. Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received June 8, 2004; final manuscript received March 15, 2005. Review conducted by Y. C. Shin.

96 / Vol. 128, FEBRUARY 2006

the surface topography. Based on the concepts of parallel reference section levels and elementary linear sections, Bouzakis et al. 关9兴 modeled the workpiece and cutting edge to simulate the topography in the ball-end milling process. Furthermore, Ehmann and Hong 关10兴, Xu et al. 关11兴 and Xu 关12兴 simulated the topography of the end milling by meshing cutter and workpiece into small elements. Yan et al. 关13兴 and Lazoglu 关14兴 applied a similar method to the ball-end milling process. Li et al. 关15兴 formulated the trajectory equations of the cutting edge relative to the workpiece and simulated the surface topography in the end milling process. Antoniadis et al. 关16兴 determined the machined surface roughness for ball-end milling, based on shape-function interpolation over a number of finite linear segments of the workpiece. In fact, all above methods depend on how the workpiece, cutting edge, and tool path are discretized and the coherence between these discretizations. This paper presents a general numerical method to simulate the machined surface topography and roughness in milling process. First, the trajectory equation system of the cutting edge relative to the workpiece in the milling process is formulated with an illustration of the height of the cut remainder. Then, numerical methods are developed to solve the equation system for end and ballend milling. Finally, some examples are studied to evaluate the topography and roughness. Results are compared to corresponding experimental ones. In addition, the developed algorithm has also the advantage of determining the tool position whenever the machined surface is generated in any desired node. This will be helpful in the prediction of form errors due to machining deformation.

2

Simulation Algorithm

In the milling process, the machined surface quality depends on a variety of factors, e.g., tool geometry, tool path, spindle runout, material properties of the workpiece, vibration of the overall machine system, etc. In this paper, an analytical model is proposed for the prediction of the topography of the generated surface. 2.1 Basic Equations and General Simulation Algorithm. First, consider an arbitrary contour mill and machined surface

Copyright © 2006 by ASME

Transactions of the ASME

Fig. 1 Coordinate systems in the milling process

Fig. 2 Coordinate systems in the end milling process

illustrated in Fig. 1. Under the premise of disregarding the influence of material properties of the workpiece and the tool, and vibration of the machine system, the machined surface topography mainly depends on the tool geometry, tool path, and spindle runout. In order to facilitate the description of the relative motion relationship between the blade of the cutter and workpiece in the milling process, a set of coordinate systems are established a priori, as shown in Fig. 1.

Generally, because solutions of ␪ and t are not unique, multiple values of z can be obtained after substituting ␪ and t into z = z共␪ , t兲. By definition, the topography value that we have to evaluate refers to the minimum of z. In the following sections, discussions will be focused on the topography simulations of end milling and ball-end milling processes, respectively.

1. OWXWY WZW represents a reference coordinate system fixed on the workpiece. 2. OAXAY AZA is the local coordinate system fixed on the main shaft of the milling machine. 3. OCXCY CZC is the local coordinate system fixed on the cutter. The cutter revolves round the spindle, i.e., axis OAZA, with the angle speed ␻. 4. OCX jY jZ j is defined as the local coordinate system attached to the jth cutting edge. 5. NXNY NZN is the local coordinate system whose origin is located at node N on the machined surface with NZN being the normal vector of the machined surface at N.

2.2 Topography Simulation of End Milling Process. Without the loss of generality, the effect of spindle runout is included in the topography simulation. Based on the above rule, the involved set of coordinate systems is established, as shown in Fig. 2. 1. OWXWY WZW represents a reference coordinate system attached to the workpiece. 2. OAXAY AZA is the local coordinate system attached to the main shaft of the mill machine. Axis OAZA is along the main shaft. Axes OAXA and OAY A are parallel to OWXW and OWY W, respectively. T关xT , y T , zT兴 is the initial position of the origin OA. 3. OCXCY CZC is the local coordinate system fixed on the cutter. Axis OCY C and the vector of spindle runout e are superposed. The cutter revolves with the angle speed ␻ round the spindle, i.e., axis OAZA. The angle between axis OCY C and OAY A is ␻t + ␮, with ␮ being the initial angle. 4. OCX jY jZ j denotes the local coordinate system attached to the jth cutting edge. Axis OCX j intersects with the jth cutting edge and the angle between axes OCX j and OCXC is ␾ j. 5. NXNY NZN is the local coordinate system of the given node N共xN , y N , zN兲 on the machined surface. Axis NY N is along the feed direction.

In addition, f is the feed vector. ␾ j is the angle between the axis OCX j and axis OCXC. In the coordinate system OCX jY jZ j, let ␪ and z be curvilinear parameters, then the coordinates of a given point P on the jth cutting edge can be expressed by 关11兴 g关z共␪兲兴cos ␪,

g关z共␪兲兴sin ␪,

z共␪兲

共1兲

where g共z兲 is parametric equation of the generator. Based on these coordinate systems and the expression in Eq. 共1兲, the trajectory equations of point P in the local coordinate system NXNY NZN can be obtained through a series of coordinate transformations as follows:

冦

x = x共␪,t兲 y = y共␪,t兲 z = z共␪,t兲

冧

共2兲

Based on the geometrical characteristics of the cutting edge and tool path, the topography corresponds to the z value with such ␪ and t satisfying the following equation system:

再

x共␪,t兲 = 0 y共␪,t兲 = 0

M=

冤

冎

共3兲

0

In the coordinate system OCX jY jZ j, consider an arbitrary point P on the jth cutting edge, whose coordinates are expressed as 关x

y

z

1兴Tj = 关R cos ␪

R␪/tan ␤

1兴Tj

共4兲

where ␪ is the curvilinear parameter of P. The latter can be expressed in the coordinate system NXNY NZN through coordinate transformation 关x

y

z

1兴NT = M关x

y

z

1兴Tj

共5兲

with M being the overall transformation matrix

0

− sin共␻t + ␮ − ␾ j兲 cos共␻t + ␮ − ␾ j兲

−1

− zT + zN

0

e cos共␻t + ␮兲 + f · t + y T − y N

cos共␻t + ␮ − ␾ j兲

sin共␻t + ␮ − ␾ j兲

1

e sin共␻t + ␮兲 + xT − xN

0

0

0

1

Journal of Manufacturing Science and Engineering

R sin ␪

冥

共6兲

FEBRUARY 2006, Vol. 128 / 97

Fig. 3 Trajectory of the point on the jth cutting edge

By inserting Eqs. 共4兲 and 共6兲 into Eq. 共5兲, the equation system reads

冤冥 冤

− R␪/tan ␤ − zT + zN

x

y = − R sin共␻t + ␮ − ␾ j − ␪兲 + e cos共␻t + ␮兲 + f · t + y T − y N R cos共␻t + ␮ − ␾ j − ␪兲 + e sin共␻t + ␮兲 + xT − xN z

冥

Fig. 4 Coordinate systems in the ball-end milling process

共7兲

To evaluate the minimum value of z for the topography, ␪ will be first obtained in a closed form by solving x共␪ , t兲 = 0. From the first equation in 共7兲, we have

␪ = 共zN − zT兲tan ␤/R

共8兲

Therefore, node N is cut by such particular points P of all cutting edges, having the same parameter ␪ 共independent of j兲. As shown in Fig. 3, for a workpiece section with x共␪兲 = 0, the trajectory of the point P in the y-z plane with respect to the given node N on the workpiece is trochoid. Here, the gray and hatched parts represent the workpiece and the remainder cut by one tooth, respectively. Note that nt is the tooth number and f t is the feed per tooth. In this case, the z value of point A with y共␪ , t兲 = 0 is what we should evaluate. Because of the nonlinearity of the second equation in Eq. 共7兲, it is impossible to give rise to the analytic expression of t. In this paper, parameter t is derived numerically. To stabilize the iteration process, the initial value t0 is set to be the parameter of a particular point B 共see Fig. 3兲, satisfying following conditions:

冦

冏冏 冏 冏 ⳵z ⳵t

⳵z ⳵t2

=0

t=t0

2

t=t0

艌 0, 兩y共␪,t0兲兩 艋

nt f t 2

冧

冉

冊

1 R sin共␾ j + ␪兲 + e ␮ − arctan + k␲ ␻ R cos共␾ j + ␪兲

冤

NXN = ␶1共x␶1,y ␶1,z␶1兲 = ˜␶1/储˜␶1储 ˜␶1 = f ⫻ ˜n NY N = ␶2共x␶2,y ␶2,z␶2兲 = ˜␶2/储˜␶2储 ˜␶2 = ˜n ⫻ ˜␶1 ˜储 NZN = n共xn,y n,zn兲 = ˜n/储n

k苸Z

共10兲

关x

y

z

1兴Tj = 关r cos ␪

r sin ␪

z

1兴Tj

2

− Kxe sin共␻t + ␮ + ␥x兲 + dx

Ky cos共␻t + ␮ − ␾ j + ␥y兲 Ky sin共␻t + ␮ − ␾ j + ␥y兲 z␶2 − Kye sin共␻t + ␮ + ␥y兲 + dy + 共x␶2x f + y ␶2y f + z␶2z f 兲t Kz cos共␻t + ␮ − ␾ j + ␥z兲 Kz sin共␻t + ␮ − ␾ j + ␥x兲 zn 0

共12兲

with r = R冑tan ␤ − ␪ / tan ␤共0 艋 ␪ 艋 tan ␤兲 and z = −R␪ / tan ␤. In this case, the overall transformation matrix M4⫻4 corresponds to 2

Kx cos共␻t + ␮ − ␾ j + ␥x兲 Kx sin共␻t + ␮ − ␾ j + ␥x兲 z␶1

0

共11兲

In OCX jY jZ j, let ␪ and z be curvilinear parameters, then the coordinates of point P on the jth cutting edge can be expressed as

Note that k is an integer whose value is determined by the second

M=

2.3 Topography Simulation in Ball-End Milling Process. In the same way, coordinate systems can be established for the ballend milling process with the inclusion of the spindle runout as shown in Fig. 4. By definition, N关xN , y N , zN兴 is a given point on the machined surface and ˜n is the normal vector of the machined surface at N. f关x f , y f , z f 兴 denotes the feed direction vector. In the ball-end milling process, the equation of the cutting edge and the geometrical shape of machined surface are more complicated. In order to simplify the expression of the trajectory equation, we establish a specific local coordinate system NXNY NZN located at node N, whose axes NXN, NY N, and NZN are as follows:

共9兲

From the first equation in Eq. 共9兲, it follows that t0 = −

equation in Eq. 共9兲. Consequently, the value of t satisfying y共␪ , t兲 = 0 can be obtained by means of the Newton-Raphson method, which proves to be highly convergent in our applications. Note that according to Eq. 共9兲, for any cutting edge j, the initial value and solution of t are both unique. Hence, for any node on the machined surface, the number of calculating z ’ s is equal to the number of cutter teeth nt.

− Kze sin共␻t + ␮ + ␥z兲 + dz + 共xnx f + y ny f + znz f 兲t

0

1

冥

共13兲

where K␰ = 冑x␩2 + y ␩2

cos ␥␰ =

x␩

冑x␩ + y␩ 2

2

sin ␥␰ =

y␩

冑x␩ + y␩2 2

d␰ = x␩共xT − xN兲 + y ␩共y T − y N兲 + z␩共zT − zN兲 98 / Vol. 128, FEBRUARY 2006

共14兲 Transactions of the ASME

␰ = x,y,z

␩ = ␶1, ␶2,n

Note that ␰ and ␩ have a correspondence when used jointly. Consequently, the trajectory equation system of point P in NXNY NZN is as follows:

冤冥 x

冤

Kx

R 冑tan2 ␤ − ␪2 cos共␻t + ␮ − ␾ j − ␪ + ␥x兲 − z␶ R␪ − Kxe sin共␻t + ␮ + ␥x兲 + dx 1 tan ␤ tan ␤

R 冑tan2 ␤ − ␪2 cos共␻t + ␮ − ␾ j − ␪ + ␥y兲 − z␶ R␪ − Kye sin共␻t + ␮ + ␥y兲 + dy + 共x␶ x f + y␶ y f + z␶ z f 兲t y = Ky 2 tan ␤ 2 2 2 tan ␤ z R 冑tan2 ␤ − ␪2 cos共␻t + ␮ + ␾ j − ␪ + ␥z兲 − zn R␪ − Kze sin共␻t + ␮ + ␥z兲 + dz + 共xnx f + yny f + znz f 兲t Kz tan ␤ tan ␤

Similarly, since all trajectory equations are nonlinear functions of both ␪ and t, the latter will be evaluated by the NewtonRaphson iterative scheme and then evaluated to obtain the topography values. To figure out the dependance of the equation system x共␪ , t兲 = 0 and y共␪ , t兲 = 0 on ␪ and t, first consider the simplified problem without spindle runout 共e = 0兲. Both equations are plotted in Fig. 5. Note that x共␪ , t兲 = 0 is periodic with respect to t, whereas y共␪ , t兲 = 0 is not. Now, the aim is to first determine ␪ and t at the intersection point between two families of curves and, thereafter, the corresponding z value. Without the loss of generality, consider point A. Suppose p = cos共␻t + ␮ − ␾ j − ␪ + ␥x兲

p 苸 关− 1,1兴

␪=

冑

tan ␤

共17兲

As shown in Fig. 6, ␪ is symmetrical with respect to p and each bifurcation is monotone. Accordingly, when p takes the extreme value with 兩p兩 = 兩cos共␻t + ␮ − ␾ j − ␪ + ␥x兲兩 = 1

共18兲

the maximum value of ␪ occurs with

共20兲

then Eq. 共19兲 can be simplified as

␪max =

z␶1dx + Kx冑R2 − d2x R

tan ␤

共21兲

Such a ␪max is, in fact, the parameter of the extreme point B shown in Fig. 5. To ensure the convergence of the iteration, the initialization is made by setting ␪0 = ␪max. By inserting Eq. 共21兲 into x兩共␪ , t兲兩e=0 = 0 of Eq. 共15兲, the corresponding analytical value of t0 reads t0 =

z␶1dx + Kx兩p兩 共z2␶1 + p2K2x 兲R2 − d2x R共z2␶1 + p2K2x 兲

z2␶1 + K2x = z2␶1 + 共x2␶1 + y 2␶1兲 = 储␶1储2 = 1

共16兲

by solving x兩共␪ , t兲兩e=0 = 0, we have

冥

共15兲

冋

z␶1R␪0 − dx tan ␤ 1 cos−1 − 共␮ − ␾ j − ␪0 + ␥x兲 + 2k␲ ␻ KxR冑tan2 ␤ − ␪20

册

k苸Z

共22兲

According to Eq. 共18兲, Eq. 共22兲 can be further simplified as t0 =

1 兵cos−1关sign共z␶1R␪0 − dx tan ␤兲兴 − 共␮ − ␾ j − ␪0 + ␥x兲 ␻ + 2k␲其

k苸Z

共23兲

Furthermore, in the specific coordinate system NXNY NZN, due to the fact that

A different k gives a different t0. Hence, the latter together with the given ␪0 will result in a set of z values whose minimum is the desired topography value. From such a starting point B 共␪0 , t0兲, usually the iteration process converges quickly to point A following numerical experiences. Likewise, when the runout exists, initial parameter values given above are also reliable to stabilize the iteration process because of eⰆR. In addition, numerical experience confirms that the convergence is not influenced by other parameters, such as feed rate and spindle speed. In Fig. 7, the convergence curves are illustrated for a test case with spindle runout e = 0.5mm, which is more than five times the feed per tooth 共f t = 0.094 mm兲.

Fig. 5 Curves of equation system without spindle runout „e = 0…

Fig. 6 Relationship between ␪ and p

␪max =

冑

z␶1dx + Kx 共z2␶1 + K2x 兲R2 − d2x R共z2␶1 + K2x 兲

tan ␤

Journal of Manufacturing Science and Engineering

共19兲

FEBRUARY 2006, Vol. 128 / 99

Fig. 7 Convergence curves of the iteration process: „a… convergence curves of x and y, and „b… convergence curves of ␪ and t

It is important to remark that since all developments are concerned with an arbitrary feed direction and an arbitrary node on the machined surface, the proposed method is therefore applicable to any machined curved surface.

3

Examples

Several numerical examples are considered, and the surface topography of both end milling and ball-end milling processes is numerically simulated. Ra or RMS is evaluated to characterize the machined surface roughness quantitatively. A Taylor-Hobson contour device is used to measure the roughness of the machined surface for comparison to the simulation results. The surface roughness given in this section is measured along the feed direction for the end milling process. As to the ball-end milling process, the measurement is made vertically to the feed direction. 3.1 End Milling Process. Cutting conditions are described in Table 1. Simulation results given in Fig. 8 show that the up and down milling processes result in different surface textures. From Table 2, it is noted that the simulated Ra associated with up milling is less than that of the down milling and is very similar to the measured outcome. This result agrees with the basic knowledge of machining experiences. A comparison of the profile between the simulation and experimental results is shown in Fig. 9. Obviously, the simulation curve is periodic since the roughness is predicted kinematically without considerations of vibration, tool wear, and other factors. The cor-

relation between these results is evident. Another example is from Xu 关12兴. The cutting conditions are described in Table 3, and the simulation results are presented in Table 4. The root mean square 共rms兲 of simulation result is used as an alternative index to quantitatively characterize the surface roughness. The proposed simulation method demonstrates its computing accuracy over the existing method. The predicted result is close to the measured one. Finally, effects of various manufacturing parameters, such as cutter radius and feed speed, on the machined surface roughness are investigated. For the up milling process, it is shown in Fig. 10 that an increase of the cutter radius will lead to a decrease of surface roughness. Contrarily, an increase of the feed speed will lead to an increase of the roughness as shown in Fig. 11. These numerical results conform to practical experiences. It is noted that with the increase of the feed speed and decrease of the cutter radius, simulation results are closer to experimental ones. The reason is that the effect of the relative movement of the cutting edge and the workpiece to the surface roughness will increase, correspondingly, with those changes. Consequently, effects of other factors, e.g., vibration and tool wear, will decrease. Another important comment has to be made for the above results. As opposed to the curve in Fig. 10, the experimental roughness is usually bigger than the simulation one because only some of all factors influencing the surface roughness are considered in the latter. However, some other unconsidered factors are not always sure to increase the roughness. For example, the influence of flank wear on the roughness was studied in 关8兴. It turned out that both experimental and simulated roughness decreased with the tool wear. The simulation rms’s obtained in Table 1 were also bigger than experimental ones. Similar behaviors can also found in 关3兴. 3.2 Ball-End Milling Process. The cutting conditions are shown in Table 5. Milling kinematics is illustrated in Fig. 12共a兲 with the unidirectional mode being applied. In this test, suppose that the cutter has an inclined angle of ␣ and the feed direction has a deviation angle of ␬ with respect to the reference line. Now, the interests are focused to understand the influences of both ␣ and ␬ on the surface topographies and roughness. Simulation results are presented in Fig. 12 for three pairs of ␣ and ␬. The results show that the undulation along the cross-feed direction is more violent than that along the feed direction in all three cases. With the kinematic simulation, the topographies always hold the periodicity along both directions provided that the phase difference of the cutter between paths is not considered. Additionally, for ␣ = 30 deg and ␬ = 0 deg, Ra is measured experimentally and compared to the predicted value in Table 6. Effects of the cross feed and cutter radius on surface roughness are investigated here. For the milling process with ␣ = 30 deg and ␬ = 90 deg, a decrease of the cross feed and an increase of the cutter radius will both lead to a decrease of surface roughness, as shown in Figs. 13 and 14, respectively. These numerical results conform to practical experimental ones.

4

Conclusions

A new method is developed for the prediction of the machined surface topography in the milling process. The key issue related to the proper parameter initialization and computing scheme is in-

Table 2 Simulation results in the end milling process Table 1 Cutting conditions in the end milling process R

nt

␤

n

ft

a

e

3 mm

3

30 deg

800 rev/ min

0.167 mm

1 mm

0.012 mm

100 / Vol. 128, FEBRUARY 2006

Milling type

Measured Ra 共␮m兲

Simulated Ra 共␮m兲

Relative error 共%兲

Up milling Down milling

2.075 2.225

2.368 2.495

14.1 12.1

Transactions of the ASME

Fig. 8 Simulation results in the end milling process: „a… up milling and „b… down milling Table 3 Cutting conditions in the down milling process †12‡ R

nt

␤

n

a

ft

4 mm

3

30 deg

400 rev/ min

2 mm

0.375 mm

Table 4 Simulation results in the down milling process Measured rms 关12兴

Simulated rms 关12兴

Relative error 关12兴

Simulated rms

Relative error

2.0 ␮m

3.2 ␮m

60%

1.8 ␮m

10%

Table 5 Cutting conditions in the ball-end milling process R

nt

␤

n

ft

fp

a

e

5 mm 4 30 deg 585 rev/ min 0.094 mm 0.6 mm 1 mm 0.01 mm

Fig. 10 Effects of the cutter radius on the surface roughness in the end milling process

Table 6 Simulation results in the ball-end milling process „␣ = 30 deg and ␬ = 0 deg… Measured Ra

Simulated Ra

Relative error

1.950 ␮m

2.395 ␮m

22.8%

Fig. 9 Comparison of the profile between simulation and experimental results

Journal of Manufacturing Science and Engineering

Fig. 11 Effects of the feed speed on the surface roughness in the end milling process

FEBRUARY 2006, Vol. 128 / 101

Fig. 12 Simulation results in ball-end milling process: „a… simulation model; „b… ␣ = 30 deg, ␬ = 0 deg, Ra = 2.395 ␮m; „c… ␣ = 30 deg, ␬ = 45 deg, Ra = 2.328 ␮m; „d… ␣ = 60 deg, ␬ = 0 deg, Ra = 2.421 ␮m

102 / Vol. 128, FEBRUARY 2006

Transactions of the ASME

Fig. 13 Effects of the cross feed on the surface roughness in the ball-end milling process

vestigated to ensure the convergence and efficiency of the iteration process. With the proposed method, the topography value of any point on the machined surface can be calculated directly without discretizing the cutting edge and meshing the workpiece. The effects of the milling parameters on the surface roughness are analyzed. The validity of the method is confirmed experimentally by solving several end milling and ball-end milling problems. Numerical results indicate that the algorithm is general and efficient.

Acknowledgment This work is supported by the National Natural Science Foundation of China 共Grant No. 50435020兲 and Shaanxi Province Natural Science Foundation 共Grant No. 2004E217兲.

Nomenclature a e e f f fp

⫽ ⫽ ⫽ ⫽ ⫽ ⫽

ft g共z兲 M n ˜兲 n共n

⫽ ⫽ ⫽ ⫽ ⫽

nt ⫽ O WX WY WZ W ⫽ O AX AY AZ A ⫽ O CX CY CZ C ⫽ O CX j Y j Z j ⫽ NXNY NZN ⫽ R ⫽ t ⫽ ␤ ⫽

depth of cut 共mm兲 vector of spindle runout spindle runout, mm feed vector feed rate, mm/s the path interval in the ball-end milling process, mm feed per tooth, mm/tooth parametric equation of the generator overall transformation matrix spindle speed, rev/min normal vector of the machined surface at a certain point tooth number coordinate system fixed on the workpiece local coordinate system fixed on the main shaft of the milling machine local coordinate system fixed on the cutter local coordinate system fixed on the jth cutting edge local coordinate system fixed on node N on the machined surface cutter radius, mm time parameter, s spiral angle, deg

Journal of Manufacturing Science and Engineering

Fig. 14 Effects of the cutter radius on the surface roughness in the ball-end milling process

␾j ␮ ␪ ␶1共˜␶1兲 , ␶2共˜␶2兲

⫽ ⫽ ⫽ ⫽

angle between axis OCX j and OCXC, rad initial angle between axis OCY C and OAY A, rad curvilinear parameter of cutting edge, rad tangent vector of the machined surface at a certain point ␻ ⫽ angle speed of the spindle, rad/s

References 关1兴 Kline, W. A., DeVor, R. E., and Shareef, I. A., 1982, “The Prediction of Surface Accuracy in End Milling,” ASME J. Eng. Ind., 104, pp. 272–278. 关2兴 Jung, T. S., Yang, M. Y., and Lee, K. J., 2004, “A New Approach to Analysing Machined Surfaces by Ball-End Milling, Part I: Formulation of Characteristic Lines of Cut Remainder,” Adv. Manuf. Technol., 25, pp. 833–840. 关3兴 Jung, T. S., Yang, M. Y., and Lee, K. J., 2004, “A New Approach to Analysing Machined Surfaces by Ball-End Milling, Part II: Roughness Prediction and Experimental Verification,” Adv. Manuf. Technol., 25, pp. 841–849. 关4兴 Imani, B. M., Sadeghi, M. H., and Elbestawi, M. A., 1998, “An Improved Process Simulation System for Ball-End Milling of Sculptured Surface,” Int. J. Mach. Tools Manuf., 38, pp. 1089–1107. 关5兴 Imani, B. M., and Elbestawi, M. A., 2001, “Geometric Simulation of Ball-End Milling Operations,” ASME J. Manuf. Sci. Eng., 123, pp. 177–184. 关6兴 Sadeghi, M. H., Haghighat, H., and Elbestawi, M. A., 2003, “A Solid Modeler Based Ball-End Milling Process Simulation,” Int. J. Adv. Manuf. Technol., 22, pp. 775–785. 关7兴 Elbestawi, M. A., Ismail, F., and Yuen, K. M., 1994, “Surface Topography Characterization in Finish Milling,” Int. J. Mach. Tools Manuf., 34共2兲, pp. 245–255. 关8兴 Ismail, F., Elbestawi, M. A., Du, R., and Urbasik, K., 1993, “Generation of Milled Surfaces Including Tool Dynamics and Wear,” ASME J. Eng. Ind., 115共2兲, pp. 245–252. 关9兴 Bouzakis, K. D., Aichouh, P., and Efstathiou, K., 2003, “Determination of the Chip Geometry, Cutting Force and Roughness in Free Form Surfaces Finishing Milling, With Ball End Tools,” Int. J. Mach. Tools Manuf., 43, pp. 499–514. 关10兴 Ehmann, K. F., and Hong, M. S., 1994, “A Generalized Model of the Surface Generation Process in Metal Cutting,” CIRP Ann., 43, pp. 483–486. 关11兴 Xu, A. P., Qu, Y. X., and Li, W. M., 2001, “Generalized Simulation Model for Milled Surface Topography—Application to Peripheral Milling,” Chin. J. Mech. Eng., 14, pp. 121–126. 关12兴 Xu, A. P., 1998, “Physical Simulation of NC Milling Process by Considering the Cutter Flexibility,” Ph.D. Thesis, Tianjin University. 关13兴 Yan, B., Zhang, D. W., Xu, A. P., Huang, T., and Zeng, Z. P., 2001, “Modeling and Simulation of Ball-End Milling Surface Topology,” J. Comput.-Aided Des. Comput. Graphics, 13, pp. 135–140. 关14兴 Lazoglu, I., 2003, “Sculpture Surface Machining: A Generalized Model of Ball-End Milling Force System,” Int. J. Mach. Tools Manuf., 43, pp. 453–462. 关15兴 Li, S. J., Liu, R. S., and Zhang, A. J., 2002, “Study on an End Milling Generation Surface Model and Simulation Taking Into Account of the Main Axle’s Tolerance,” J. Mater. Process. Technol., 129, pp. 86–90. 关16兴 Antoniadis, A., Savakis, C., Bilalis, N., and Balouktsis, A., 2003, “Prediction of Surface Topomorphy and Roughness in Ball-End Milling,” Int. J. Adv. Manuf. Technol., 21, pp. 965–971.

FEBRUARY 2006, Vol. 128 / 103