A New Algorithm for the Numerical Simulation of ...

Wei-Hong Zhang1 Professor e-mail: [email protected]

Gang Tan Min Wan Tong Gao Sino-French Laboratory of Concurrent Engineering, The Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, School of Mechatronic Engineering, Northwestern Polytechnical University, P.O. Box 552 Shaanxi, Xi’an 710072, China

David Hicham Bassir FEMTO-ST, Départment LMARC, UMR—CNRS 6174, 25000 Besancon, France

1

A New Algorithm for the Numerical Simulation of Machined Surface Topography in Multiaxis Ball-End Milling In milling process, surface topography is a significant factor that affects directly the surface integrity and constitutes a supplement to the form error associated with the workpiece deformation. Based on the tool machining paths and the trajectory equation of the cutting edge relative to the workpiece, a new and general iterative algorithm is developed here for the numerical simulation of the machined surface topography in multiaxis ball-end milling. The influences of machining parameters such as the milling modes, cutter runout, cutter inclination direction, and inclination angle upon the topography and surface roughness values are studied in detail. Compared with existing methods, the basic advantages and novelties of the proposed method can be resumed below. First, it is unnecessary to discretize the cutting edge and tool feed motion and rotation motion. Second, influences of cutting modes and cutter inclinations are studied systematically and explicitly for the first time. The generality of the algorithm makes it possible to calculate the pointwise topography value on any sculptured surface of the workpiece. Besides, the proposed method is proved to be more efficient in saving computing time than the time step method that is commonly used. Finally, some examples are presented and simulation results are compared with experimental ones. 关DOI: 10.1115/1.2815337兴 Keywords: ball-end milling, surface topography, simulation algorithm, sculptured surface

Introduction

Sculptured surfaces are widely used for designs of complex structures and functional products in the die/mould, automobile, and aerospace industries. Ball-end milling is one of the most common processes in machining the sculptured surfaces. The surface topography of a workpiece is characterized by the cutting remainders that depend mainly on the tool geometry, cutting modes, machining parameters, machining temperature, rigidity, and dynamic property of the machine system as well. As a measure of machining quality, the surface topography is, among others, of great importance to ensure the workpiece assembly, to prevent the microcrack initiation and fatigue failure. In practice, with a reasonable selection of machining parameters and cutting modes, an improved surface quality can be achieved and the finish milling process work can be reduced considerably or even omitted for the benefit of productivity in cost and time. For this reason, it is worthwhile to develop reliable simulation methods that are able to predict machined surface topography and roughness before setting up the machining process. Today, numerical simulation of machined surface topography and roughness is a challenging subject that attracts many researchers. Relevant works can be summarized below. Imani and Eibestawi 关1兴 and Sadeghi et al. 关2兴 used solid modeling techniques and Boolean operations to deal with the geometric simulation of the ball-end milling process. Kim and Chu 关3,4兴 analyzed theoretically the effect of cutter marks on scallop height and pointed out that the cutter marks are important for surface finish. A texture superposition method was developed to include effects of both the cutter marks and the cutter runout on scallop height. 1 Corresponding author. Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received May 16, 2006; final manuscript received August 29, 2007; published online January 30, 2008. Review conducted by William J. Endres.

Chen et al. 关5,6兴 studied the surface scallop generation mechanism of ball-end milling and proposed a classification of two kinds of scallops, i.e., the pick-interval scallop and the feed-interval scallop. The feedrate optimization and tool profile modification were carried out for high-speed ball-end milling process in which the generated scallop effect was taken into account. Based on the surface generation mechanism of ball-end milling process, Jung et al. 关7,8兴 developed the so-called ridge method in which three types of ridges are defined in vertical milling to predict the characteristic lines of the cut remainder relative to two kinds of cutting modes, i.e., unidirectional mode and bidirectional mode. In addition to these, other representative simulation methods are recognized depending on how the cutting edge is modeled. The first type is to discretize the cutting edge into piecewise segments. Typically, Altintas and Lee 关9兴 proposed a combined static and dynamic model in which the cutter was discretized into axial elements to predict the cutting force, surface finish, and chatter stability lobes of ball-end milling process. Antoniadis et al. 关10兴 presented a Milling Software Needle program that was able to simulate the tool kinematics and reveal the effect of cutting geometry upon the resulting roughness for ball-end milling. Bouzakis et al. 关11兴 described a ball-end milling model where the cutting edge was segmented into elementary linear sections with constant width and the workpiece geometry was represented by parallel reference section levels. In this way, the engagement between the tool and the workpiece on all reference levels gave rise to the milled surface. Yan et al. 关12兴 developed a simulation algorithm in which the cutting edge and workpiece were coherently meshed. The second type of these methods refers to the so-called Z-map modeling method. To use it, the cutting edge is represented by its parametric equation and the tool-cutting rotation motion is discretized. In this manner, the final surface topography is determined by comparing the point on the cutting edge with the node on the workpiece surface in the height direction. For example, Soshi et al. 关13兴 applied such a method to predict surface topog-

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raphy in five-axis ball-end milling without considering the cutter runout and wear. In contrast, only the runout effect was considered in the work of Zhao et al. 关14兴. More importantly, both runout and wear were taken into account by Liu et al. 关15兴 and Sriyotha et al. 关16兴. Besides, the topography was studied by Toh 关17兴 in high-speed milling of inclined hardened steel surface experimentally. However, almost all of above methods are limited to the threeaxis ball-end plane milling simulation. There lacks a general mathematic simulation model, which is suitable to sculptured surface simulation. Although research works 关13–16兴 attempted to predict topography of multiaxis ball-end milling, any simulation result has not yet been found to predict the surface topography of five-axis ball-end milling systematically and explicitly. Also, many factors that affect the surface topography and roughness, e.g., cutter runout, tool inclination, and feed modes, have not been fully studied yet. Besides, it is necessary to remark that the time step method employed by many authors 关12–16兴 is much more straightforward due to the fact that the local topography is evaluated with the advancement of the cutter. However, as discussed later, the accuracy and computing time are seriously affected by the time step and the thickness of the tool slices. Based on the previous work of Gao et al. 关18兴, this paper proposes a new and general iterative method to simulate the arbitrary machined surface topography and the roughness in multiaxis ballend milling. First, simulation coordinate systems are established to describe the trajectory equation of the cutting edge relative to the workpiece. Then, a numerical algorithm is developed to solve the trajectory equation and to gain the topography value. To highlight the work, influences of cutting modes, tool inclination direction, and inclination angle upon the topography and roughness are studied. Furthermore, because there is no need to discretize the cutting edge, the tool-cutting rotation motion, and feed motion, the proposed algorithm is able to determine the exact tool position and cutting edge point. As a result, the topography value of any node on the machined surface can be acquired in a continuous way. The following discussions and numerical results will show that the algorithm proposed in this paper is also faster than the time step method. This is the advantage over the time step method. To validate the algorithm, some examples of plane milling and cylindrical surface milling are considered. Simulation results are compared with experimental ones and those found in the literature.

2 Mathematical Formulation of Machined Surface Topography In this section, different coordinate systems related to multiaxis ball-end milling are established first. Then, the trajectory equation of the cutting edge relative to the workpiece is formulated. Due to the mathematical complexity, a numerical iterative algorithm is developed to solve the trajectory equation. In this work, as we are focused on the finish milling process with the depth of cut being small, effect of machining temperature, the cutter-workpiece flexibility and the dynamic property of the machine system are ignored. 2.1 General Formulation of the Cutting Edge Trajectory Equation. For an arbitrary linear feed portion of the whole tool trajectory, the involved set of coordinate systems shown in Fig. 1 is established in order to obtain the parametric trajectory equation of the cutting edge relative to an arbitrary point, N, on the workpiece. Note that R and ␤ denote the radius and helix angle of the cutter, respectively. 共1兲 OXYZ: global coordinate system in which the workpiece machining surface and the tool path are described. 共2兲 OWXWY WZW: simulation coordinate system attached to the 011003-2 / Vol. 130, FEBRUARY 2008

Fig. 1 Coordinate systems in the ball-end milling process

共3兲

共4兲

共5兲共6兲

workpiece with the axis OWZW being parallel to the tool axis OAZA. OAXAY AZA: local coordinate system attached to the main shaft of the mill machine and moves in pure translation of feed speed f relative to the workpiece. OAZA is along the main shaft. OAXA and OAY A are parallel to OWXW and OWY W, respectively. OCXCY CZC: local coordinate system fixed on the cutter. OCY C and the vector of spindle runout, e, are aligned. The cutter revolves around the spindle at the angle speed ␻. The angle between OCY C and OAY A is ␻t + ␮, with ␮ being the initial eccentricity angle. OCX jY jZ j: local coordinate system attached to the jth cutting edge. OCX j intersects with the jth cutting edge and has an angle ␾ j with OCXC. NXNY NZN: local coordinate system of any given node on the machined surface. NZN is the normal direction of node N. Note that axis NF is parallel to the linear feed velocity vector ˜f. All axes can be derived below. NXN = ␶1共x␶1,y ␶1,z␶1兲 = ˜␶1/储˜␶1储,

˜␶1 = ˜f ⫻ ñ

NY N = ␶2共x␶2,y ␶2,z␶2兲 = ˜␶2/储˜␶2储,

˜␶2 = ñ ⫻ ˜␶1

˜储 NZN = n共xn,y n,zn兲 = ñ/储n

共1兲

Let ␪ and z be the curvilinear parameters denoting the position angle of the cutter and the z coordinate of the given point on the cutting edge, respectively. Then, coordinates of Point P on the jth cutting edge can be expressed as 关x y z 1兴Tj = 关r cos ␪ r sin ␪ z 1兴Tj

共2兲

in which r = R冑tan ␤ − ␪ / tan ␤ and z = −R␪ / tan ␤, 0 艋 ␪ 艋 tan ␤. 2

2

After multiple coordinate system transformations, the overall transformation matrix, denoted by M jN, from coordinate OCX jY jZ j to coordinate NXNY NZN consists of M jN = MWNMAWMCAM jC

共3兲

in which M jC, MCA, MAW, and MWN denote the transformation matrix from OCX jY jZ j to OCXCY CZC, OCXCY CZC to OAXAY AZA, OAXAY AZA to OWXWY WZW, and OWXWY WZW to NXNY NZN for each of them. Hence, in NXNY NZN, the trajectory equation of a given point on the jth cutting edge relative to the workpiece is written as Transactions of the ASME


冤冥冤冥 x

x

冤

= K ␶2

y

= M jN y z N z

K ␶1

j

R 冑tan2 ␤ − ␪2 cos共␻t + ␮ − ␾ j − ␪ + ␥␶ 兲 − z␶ R␪ − K␶ e sin共␻t + ␮ + ␥␶ 兲 + d␶ 1 1 tan ␤ 1 1 1 tan ␤

R 冑tan2 ␤ − ␪2 cos共␻t + ␮ − ␾ j − ␪ + ␥␶ 兲 − z␶ R␪ − K␶ e sin共␻t + ␮ + ␥␶ 兲 + d␶ + 共x␶ x f + y␶ y f + z␶ z f 兲t 2 2 tan ␤ 2 2 2 2 2 2 tan ␤

Kn

R 冑tan2 ␤ − ␪2 cos共␻t + ␮ + ␾ j − ␪ + ␥n兲 − zn R␪ − Kne sin共␻t + ␮ + ␥n兲 + dn + 共xnx f + yny f + znz f 兲t tan ␤ tan ␤ Ki = 冑xi2 + y i2,

cos ␥i = xi/Ki,

sin ␥i = y i/Ki

di = xi共xA − xN兲 + y i共y A − y N兲 + zi共zA − zN兲

where ␾ j = ␾1 − 共j − 1兲2␲ / nz 共j = 1 , 2 , . . . , nz兲. nz and ␾1 denote the number of teeth and the angle between the 1th cutting edge and axis OCXC, respectively. 关xN , y N , zN兴 denotes the coordinates of node N in OWXWY WZW, and 关xA , y A , zA兴 denotes the coordinates of cutter center,OA, in OWXWY WZW. 2.2 Coordinate System Transformation of Multiaxis BallEnd Milling. In fact, Eq. 共4兲 holds under the assumption that the local coordinate system OAXAY AZA is parallel to the global coordinate system OXYZ. However, in most cases, the axes of OXYZ and OWXWY WZW are not parallel in multiaxis ball-end milling. Namely, the tool cutter axis is typically inclined with respect to the normal of workpiece machining surface and moves ahead with varying angles in the feed direction and pick-feed 共cross-feed兲 direction. As the workpiece and tool paths are described under the global coordinate system OXYZ, we have to determine 关xN , y N , zN兴, 关xA , y A , zA兴 and the normal at Point N involved in Eq. 共4兲 in advance by means of coordinate transformation from OXYZ into OWXWY WZW. As shown in Fig. 2, such a transformation constitutes two steps in a five-axis ball-end milling. 共1兲 OXYZ revolves around axis Y with an angle ␥ to OWX⬘Y ⬘Z⬘. 共2兲 OWX⬘Y ⬘Z⬘ revolves around axis X⬘ with an angle ␩ to OWXWY WZW. It is noticed that OWZW is parallel to the cutter axis and the signs of ␥ and ␩ are determined using the right hand helix rule. Denoting M1 and M2 to be the transformation matrices of the circumrotation around axis Y and axis X⬘, respectively, we have M1 =

冤

cos ␥ 0 − sin ␥ 0

1

sin ␥ 0

0 cos ␥

冥

共5兲

Fig. 2 Coordinate system transformation in multiaxis ball-end milling

Journal of Manufacturing Science and Engineering

冥

共i = ␶1, ␶2,n兲

共4兲

冤

1

0

0

M2 = 0 cos ␩ sin ␩ 0 − sin ␩ cos ␩

冥

共6兲

Consequently, the overall transformation matrix M from OXYZ to OWXWY WZW is written as

冤

cos ␥

0

− sin ␥

M = M2M1 = sin ␩ sin ␥ cos ␩ sin ␩ cos ␥ cos ␩ sin ␥ − sin ␩ cos ␩ cos ␥

冥

共7兲

2.3 Numerical Topography Simulation Algorithm and Accuracy. From Eq. 共4兲, it can be seen that when parameters R, ␤, ␻, ␮, e, and ␾ j are given in advance, the trajectory equation is a nonlinear system in function of both parameters ␪ and t. When a considered point on the workpiece is in cut, the general form of the trajectory equation written in NXNY NZN is x共␪,t兲 = 0 y共␪,t兲 = 0

共8兲

␪ and t will be evaluated numerically due to the implicit relations. Their substitution into the z expression of Eq. 共4兲 will thereafter give rise to the remainder height of the considered point being in cut. The resulting topography value of this point then corresponds to the minimum of all remainder heights formed by all cutting edges. Here, the key issue associated with topography simulation is how to solve Eq. 共8兲 in an efficient way. The Newton–Raphson iteration scheme is employed in this work and the appropriate selection of the starting iteration point is investigated below. First, the curve defined by Eq. 共8兲 with e = 0 is plotted in Fig. 3. It can be seen that 兩x共␪ , t兲兩e=0 = 0 is a periodical function of parameter t with the periodicity being 2␲ / ␻.

Fig. 3 Isovalue curves of equation system without cutter runout „e = 0…

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The curve families indicate that the particular point 共␪max , t0兲 is quite close to the analytical solution point of Eq. 共8兲. The former can be hence chosen as the starting iteration point. From the x expression of Eq. 共4兲, we have

␪0 = ␪max = 共z␶1d␶1 + K␶1冑R2 − d2␶1兲tan ␤/R

共9兲

Considering Eq. 共9兲 together with 兩x共␪ , t兲兩e=0 = 0 of Eq. 共8兲, we have t0 = 共cos−1 q − 共␮ − ␾ j − ␪0 + ␥␶1兲 + 2k␲兲/␻

k苸Z

共10兲

with q = 1 for z␶1R␪ / tan ␤ − d␶1 艌 0 and q = −1 for z␶1R␪ / tan ␤d␶1 ⬍ 0. Similarly, when cutter runout e exists, the above method is still effective since the runout value is far smaller than cutter radius R in practice. For detailed explanations, see Ref. 关18兴. With the starting point 共␪0 , t0兲, the solution of Eq. 共4兲 can be quickly obtained. In the Newton–Raphson iteration, the convergence will be controlled when x共␪ , t兲艋 ␧ and y共␪ , t兲艋 ␧ are satisfied. Here, ␧ is the prescribed tolerance set to be 10−5. Simulation examples demonstrate that a suitable value of ␧ is needed to ensure both the precision and the computing efficiency. To have an overall idea, the following procedure is provided to show how the final topography of an arbitrary machined surface is evaluated in multiaxis ball-end milling. Step 1. Input the tool paths, cutting parameters, and the geometrical surface model of the workpiece to be machined. Step 2. Determine the linear feed velocity vector ˜f of each segmented linear feed path and then carry out coordinate transformation by means of Eq. 共7兲. Step 3. Determine the cutting areas swept along this linearized tool path according to the cutting depth and cutter radius. Step 4. Calculate the cutter remainder height of the considered point inside the workpiece cutting areas swept by the current cutting edge following Eqs. 共4兲 and 共8兲. Step 5. Calculate the cutter remainder height for the next cutting edge and update the cutter remainder height by comparing the relative magnitude between the current value and the existing one. Step 6. Repeat Steps 4 and 5 for all cutting edges and calculate cutter remainder height of the next point on the workpiece until all the points in cut are calculated. Step 7. Repeat Steps 2–6 and treat the following linear tool path until the whole cutting process is carried out. Step 8. Output the final topography and roughness of the simulation process. Alternatively, the time step method 关12–16兴 is a straightforward one. In such a method, the concept of discrete cutting tooth is employed to denote tool model. Time steps are used to index both the cutter forward motion and tool rotation. The important thing is how to determine the time step, the axial slice thickness, and the tolerance ␧ that affect considerably the accuracy and the computing time. The first parameter should satisfy the condition that the feeding distance formed by a time step is less than the basic grid length along the feed direction. As the final cutter remainder height of a node is finally determined by the lowest axial slice that sweeps this node, the second parameter should take a value small enough for the trade-off between the accuracy and computing time. Comparatively, the proposed iterative method is an indirect method, whose result accuracy is merely affected by the tolerance ␧. However, due to the discretization of the workpiece surface and the tool-cutting motion, the simulation results of both iterative method and time step method could not accurately represent the surface characteristics in the microscale. To have a clear idea, consider the simulation case where the workpiece is meshed into a 34⫻ 34 grid and the basic grid length is 0.03 mm. Assume that the roughness computation accuracy is 0.001 mm. Then, the axial slice thickness used in the time step method should be less than 0.001 mm. As a result, the computing times are 514 s and 246 s when the time step method and the 011003-4 / Vol. 130, FEBRUARY 2008

iterative method are used, respectively. Therefore, in the same condition, the iterative method proposed in this paper is much faster.

3 Simulation Examples and Experimental Verifications Several simulation examples are illustrated for both plane milling and sculptured surface milling. Influences of the feedrate, cutting modes, inclination direction, and inclination angle of the cutter upon the surface topography and roughness are studied. As done in Refs. 关7,8兴, the maximum peak value of the machined surface topography, RZ, is evaluated to characterize the surface roughness. To validate the simulation accuracy, experimental tests are performed on the machine bed of a MIKRON 1350ucp machine center with coolant. The ball-end mill is a high-speed steel 共HSS兲 cutter with 30 deg helix angle. The workpiece material is Al7075-T6. RZ is measured by a Taylor–Hobson contour device. The micrograph of the machined surface is taken by a Nikon SMZ800 microscope. 3.1 Typical Plane Cutting Modes and Tool Path. As shown in Figs. 4-共1兲–4-共3兲, each represents the unidirectional upmilling, unidirectional downmilling, and bidirectional milling modes, respectively, when the cutter axis has an inclined angle ␥ along the pick-feed direction. Similarly, in Figs. 4-共4兲–4-共6兲, case corresponds to the cutter axis of an inclined angle ␩ along the feed direction. When both angles ␥ and ␩ take no zero values simultaneously, this becomes the case of a five-axis milling, as shown in Fig. 2. In fact, any kinematics of five-axis milling modes can be described by a proper combination of the above single inclined angle cases given in Fig. 4. In the milling process, as cutting edges are in a combined motion of pure translation and rotation, the trajectory of a cutting edge point is cycloid. Therefore, the final machined surface topography is generated by overlapping of several sweeping cutting paths. Meanwhile, the overlapping of sweeping edges’ trajectories will be different for each cutting mode. Figures 5-共1兲–5-共3兲 show different cases of a two-flute ball-end cutter with the same initial phase. Geometrically, it can be observed that there is no big difference between the overlapping of the unidirectional downmilling mode and that of the unidirectional upmilling mode. Here, AB and CD are two continuous tool paths in vicinity and the arrows denote the feed direction. Regions surrounded by the thick black lines are the cutter remainders left on the workpiece, which form the final topography. Practically, due to the dynamic and static tool deflections, some differences do exist between the machined topography of unidirectional upmilling and that of unidirectional downmilling 关17兴. Based on the machining parameters given in Table 1 and the simulation procedure of Sec. 2.3, we obtain the topography results under different feedrates of F 共mm/min兲 in Fig. 6. As only the geometric aspects are considered in our simulation of finish milling with the negligence of cutting static and dynamic effect, the results obtained will be insensitive to the upmilling mode and downmilling mode. Therefore, only the topography of the unidirectional up milling mode is given here. Following conclusions can be drawn. •

•

With the same pick feed f p, the scallop height and the roughness of the machined surface are enlarged rapidly with the increase of feedrate F. In contrast to the conventional roughness theory, the simulation algorithm is able to capture the feed-interval scallop and pick-feed interval scallop. As shown in Fig. 7, the feed-interval scallop increases with the augmentation of the feedrate while the pick-feed interval scallop increases with the augmentation of the pick feed. The roughness is dominated by the feed-interval scallop in the case of high feedrate F = 2000 mm/ min. Different cutting modes lead to different machined surface topographies, and the topography related to unidirectional Transactions of the ASME


Fig. 4 Typical plane cutting modes: „1… unidirectional upmilling with inclined angle ␥, „2… unidirectional downmilling with inclined angle ␥, „3… bidirectional milling with inclined angle ␥, „4… unidirectional upmilling with inclined angle ␩, „5… unidirectional downmilling with inclined angle ␩, and „6… bidirectional milling with inclined angle ␩

milling mode is relatively smooth. This observation is, however, indistinct following the conventional roughness theory. The feed-interval scallop depends on the cutting mode and changes along with the latter in magnitude and in location. The difference caused by both cutting modes is, however, small when the feedrate is relatively slow but it becomes large with the increase of the feedrate. From authors’ knowl-

edge, none of existing researches has yet studied the influences of feed modes systematically and explicitly. For comparison, the measured surface topography given in Ref. 关8兴 is shown in Fig. 8. Obviously, simulation results presented in this paper show good agreement. Meanwhile, roughness curves are compared in Fig. 9 versus feed per tooth f t in both unidirectional cutting mode and bidirectional cutting mode with a constant pick feed of 0.5 mm. Results in this paper are found to agree well with those in Ref. 关8兴. The roughness of both feed modes increases with f t. The difference of the roughness between these two modes is tiny when f t is less than 0.6 mm and it becomes significant when f t is larger than 0.6 mm. However, the existing method 关8兴 is only suitable to vertical milling simulation 共␥ = ␩ = 0 deg兲 and also it ignores the influences of cutter runout and phase difference. Alternatively, as the cutting speed at the tip of the ball-end cutter equals zero in vertical milling process, this unfavorable cutting condition leads to poor machined surface quality and tool wear. This is why the cutter axis is often inclined with an angle of ␥ or ␩ to obtain good surface quality in practical milling. To make one such simulation, parameters are chosen below. S = 2000 rpm, F = 2000 mm/ min, ␥ = 10 deg, and ␩ = 0 deg. Other parameters are the same as those listed in Table 1. Figures 10-共1兲 and 10-共2兲 illustrate the simulated topographies related to the unidirectional upmilling mode and bidirectional milling mode, respectively. Clearly, the roughness decreases sharply and the texture of the machined surface is much more smooth when compared with the simulation results given in Fig. 6. This is the reason why the inclination of the cutter axis is widely applied to

Table 1 Specification of machining parameters

Fig. 5 Overlapping of two sweeping points of the cutting edges: „1… unidirectional downmilling mode, „2… unidirectional upmilling mode, and „3… bidirectional milling mode


␤ dp ␥ 共deg兲 ␩ 共deg兲 R 共mm兲 S 共rpm兲共deg兲 f p 共mm兲 e 共mm兲共mm兲 nz 0

0

5

2000

30

0.5

0.000

0.100

2

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Fig. 6 Simulated topography results under different feedrates of F „mm/min…: „1… unidirectional upmilling mode and „2… bidirectional milling mode

improve the surface quality. Figure 11-共1兲 shows the effect of the inclined angle upon the surface roughness in unidirectional milling mode. An inclined angle of 5 – 10 deg is suitable to ensure a good surface quality. Similarly, Fig. 11-共2兲 shows the effect of the inclined angle in bidirectional milling mode. An inclined angle of about 20 deg leads to the good surface quality. In both cases, the roughness remains nearly unchanged when the inclined angle varies beyond a certain value. The simulation results agree with the results given in Refs. 关11,14兴. Now, more complicated simulations of multiaxis ball-end milling are addressed. In addition to the specific parameters given in

Fig. 7 Simulation topography of unidirectional upmilling „F = 2000 mm/ min…

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Table 1, assume that a ball-end mill of four right-handed flutes is employed. Two cases of cutter-axis inclination are considered with ␥ = 15 deg, ␩ = 10 deg and ␥ = 15 deg, ␩ = 0 deg. Figures 12共1兲–12-共4兲 represent the simulated topographies in both milling modes. Correspondingly, Figs. 12-共5兲–12-共8兲 represent machined surface micrographs obtained by a Nikon SMZ800 microscope. The main difference from 共1兲 to 共4兲 is that the elliptical topographies are orientated differently depending on the milling mode. In Fig. 12-共1兲, the ellipses incline right, whereas in Fig. 12-共2兲, the ellipses incline left and right alternatively. In Fig. 12-共3兲, the ellipses orient upright and in Fig. 12-共4兲, the ellipses orient horizontally and upright alternately. The same orientations can be found in the micrographs of Figs. 12-共5兲–12-共8兲. Note that arrows in these figures denote the cutter feed direction. Relevant roughness is listed in Table 2 for comparison. Although the surface topography shows a good level of accordance in certain cases, some deviations do exist between the simulated roughness and the measured one. From the mechanistic point of view, it is known that the cutter will be pushed away in downmilling and be pulled over in upmilling. Here, the unidirectional milling mode shown in Fig. 12 corresponds to the upmilling while the bidirectional milling mode corresponds to an alternant effect of upmilling and downmilling. Hence, the difference between the measured roughness and the simulated roughness is relatively large in bidirectional milling mode. Besides, further investigations Transactions of the ASME


Fig. 8 Measured machined surface topography †8‡ „F = 2000 mm/ min…: „1… surface topography in unidirectional cutting mode and „2… surface topography in bidirectional cutting mode

should also be made about the effect of the cutter deflection caused by cutting force upon the machined surface topography and roughness.

Fig. 9 Comparison of roughness curves versus feed per tooth ft „mm/tooth…: „1… results given in Ref. †8‡ and „2… our results Rz


Fig. 10 Machined surface topography with inclined angle ␥ = 10 deg and ␩ = 0 deg: „1… unidirectional upmilling and „2… bidirectional milling

3.2 Typical Cutting Modes of Sculptured Surfaces and Tool Path. As described in Refs. 关19,20兴, Fig. 13 represents the representative milling modes referred to as contouring and ramping for ball-end milling of convex and concave cylindrical surfaces. Here, contouring means that the tool paths are parallel to the axis of the cylindrical workpiece. Ramping means that the cutting tool follows the path in the circumferential direction of the

Fig. 11 Roughness curves versus inclined angles ␥ , ␩: „1… unidirectional milling mode and „2… bidirectional milling mode

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Fig. 12 Simulations and experiments of surface topographies: „1… simulated unidirectional upmilling „␥ = 15 deg, ␩ = 10 deg…; „2… simulated bidirectional milling „␥ = 15 deg, ␩ = 10 deg…; „3… simulated unidirectional upmilling „␥ = 15 deg, ␩ = 0 deg…; „4… simulated bidirectional milling „␥ = 15 deg, ␩ = 0 deg…; „5… measured unidirectional upmilling „␥ = 15 deg, ␩ = 10 deg…; „6… measured bidirectional milling „␥ = 15 deg, ␩ = 10 deg…; „7… measured unidirectional upmilling „␥ = 15 deg, ␩ = 0 deg…; and „8… measured bidirectional milling „␥ = 15 deg, ␩ = 0 deg…

workpiece. In these pictures, ⌽ and f denote the milling position angle and feed direction, separately. The curved tool paths of multiaxis ball-end milling can be discretized into small segmented linear feeds. As shown in Fig. 14, the thick circular lines denote the profile of the machined workpiece with radius R0, and the thin circular lines denote the trajectory of the ball-end mill center. d0 is the initial distance of the cutter axis from the origin O in the X direction. f p is the pick feed and d is the increment of the linear feed tool path. The contouring path discretization is illustrated in Figs. 14-共1兲 and 14-共3兲, respectively. The ramping path discretization is illustrated in Figs. 14-共2兲 and 14-共4兲, respectively. In contouring, suppose Points A and B are starting points of two successive linear feed tool paths, respectively. A⬘ and B⬘ are corresponding contact points between the cutter and workpiece. The feed direction vector along Y axis is here vertical to the paper plane and remains unchanged. Therefore, the tool paths of contouring are as follows. Convex contouring,

␣0 = sin−1共d0/共R + R0兲兲 ␣0 + ␣ = sin−1共共d0 + d兲/共R + R0兲兲 xA = 共R + R0兲sin共␣0兲

Fig. 13 Unidirectional cutting mode: „1… convex contouring, „2… convex ramping, „3… concave contouring, and „4… concave ramping

zA = 共R + R0兲cos共␣0兲

xB = 共R + R0兲sin共␣0 + ␣兲

zB = 共R + R0兲cos共␣0 + ␣兲

共11兲 xB = 共R0 − R兲sin共␣0 + ␣兲

Concave contouring,

␣0 = sin 共d0/共R0 − R兲兲 −1

In ramping, suppose that A, B, and C are the starting points of three successive linear feed segments of tool paths and that A⬘, B⬘, and C⬘ are the contact points between the cutter and workpiece. Coordinates of A, B, and C are the following. Convex ramping,

␣0 + ␣ = sin−1共共d0 + d兲/共R0 − R兲兲 xA = 共R0 − R兲sin共␣0兲

zB = − 共R0 − R兲cos共␣0 + ␣兲共12兲

zA = − 共R0 − R兲cos共␣0兲

␣0 = sin−1共d0/共R0 + R兲兲 Table 2 Roughness comparison

␣ = 2 sin−1共共d/2兲/共R0 + R兲兲

␥ 共deg兲

␩

Feed mode

共deg兲

Simulated RZ 共␮m兲

Experimental RZ 共␮m兲

Unidirectional up Bidirectional Unidirectional up Bidirectional

15 15 15 15

10 10 0 0

8.006 7.996 8.649 7.777

7.907 11.620 10.790 9.036

xA = 共R0 + R兲sin共␣0兲 xB = 共R0 + R兲sin共␣0 + ␣兲 xC = 共R0 + R兲sin共␣0 + 2␣兲

zA = 共R0 + R兲cos共␣0兲 zB = 共R0 + R兲cos共␣0 + ␣兲 zC = 共R0 + R兲cos共␣0 + 2␣兲共13兲

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Fig. 15 Simulated topographies „␮m… of the cylindrical surface: „1… convex contouring with ⌽ = †5 deg, 30 deg‡ and „2… concave contouring with ⌽ = †5 deg, 30 deg‡

xC = 共R0 − R兲sin共␣0 + 2␣兲

zC = − 共R0 − R兲cos共␣0 + 2␣兲共14兲

Fig. 14 Tool path discretization: „1… convex contouring, „2… convex ramping, „3… concave contouring, and „4… concave ramping

Simulation parameters of convex contouring and concave contouring are listed in Table 3. The cutter position angle ⌽ ranges within 关5 deg, 30 deg兴. Simulated topographies related to unidirectional cutting mode are illustrated in Figs. 15-共1兲 and 15-共2兲, respectively. The simulated roughness of convex surface is smaller than that of concave surface and each of both is 5.467 ␮m and 10.950 ␮m, respectively. Such results can be schematically illustrated by means of Fig. 16. The upper part denotes the convex contouring while the lower part denotes the concave contouring. r p denotes the cutter remainder height estimated roughly. In such a simplified case, the cutter remainder height can be evaluated by the following equations. For convex milling,

␺ = sin−1共共d0 + f p兲/共R0 + R兲兲 − sin−1共d0/共R0 + R兲兲 ␣0 = sin−1共d0/共R0 − R兲兲

rp = R −

␣ = 2 sin−1共共d/2兲/共R0 − R兲兲 xA = 共R0 − R兲sin共␣0兲 xB = 共R0 − R兲sin共␣0 + ␣兲

冑

R2 − 共R + R0兲2 sin2

冉冊

冉

冉冊冊

␺ ␺ − 共R + R0兲 1 − cos 2 2

共15兲

zA = − 共R0 − R兲cos共␣0兲

For concave milling,

zB = − 共R0 − R兲cos共␣0 + ␣兲

␺ = sin−1共共d0 + f p兲/共R0 − R兲兲 − sin−1共d0/共R0 − R兲兲

Table 3 Specification of machining parameters R0 共mm兲

R 共mm兲

e 共mm兲

F 共mm/min兲

S 共rpm兲

nz

␤ 共deg兲

d p 共mm兲

f p 共mm兲

15

5

0.0

1000

1000

4

30

0.1

0.5


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共2兲共3兲共4兲共5兲共6兲

Fig. 16 Illustration of remainder height in convex and concave ball-end milling

rp = R −

冑

R2 − 共R0 − R兲2 sin2

冉冊

冉

冉冊冊

␺ ␺ − 共R0 − R兲 1 − cos 2 2

共16兲 Therefore, the remainder height, r p, is larger in concave milling than in convex milling. 共1兲 Rz = 12.650

Rz = 15.108 Rz = 5.682 Rz = 7.434 Rz = 9.819 Rz = 11.740

To have a detailed insight into the convex milling, Figs. 17-共1兲– 17-共4兲 give correspondingly the local simulated surface topographies and the micrographs located at different position angles of ⌽. The simulated roughness changes with the position and equals 12.650 ␮m and 5.682 ␮m in each case of 共1兲 and 共3兲, and the measured Rz is 15.108 ␮m and 7.434 ␮m, respectively. The difference may be caused by the effect of static and dynamic deformation. As the sculptured surface can be reasonably approximated as a finite number of small plane patches, the milling process of the portioned surface can be considered as a sequence of plane milling operations. Therefore, the underlying mechanism can be clarified from conclusions drawn in Sec. 3.1. With this idea in mind, Fig. 17-共1兲 can be simplified as vertical plane milling while Fig. 17-共3兲 can be simplified as inclined milling with varying inclined angles relative to the pick-feed direction. Therefore, the surface roughness is bigger in Case 共1兲. Meanwhile, an upward contouring is present at the left side and the front faces of the cutter are engaged in the cutting. Oppositely, a downward contouring is present at the right side and the front and back faces are

Fig. 17 Simulated and measured topographies „␮m… versus cutter position angle ⌽ in convex contouring milling: „1… simulated ⌽ = †−5 deg, 5 deg‡; „2… measured ⌽ = †−5 deg, 5 deg‡; „3… simulated ⌽ = †5 deg, 15 deg‡; „4… measured ⌽ = †5 deg, 15 deg‡; „5… simulated ramping ⌽ = †5 deg, 15 deg‡; and „6… measured ramping ⌽ = †5 deg, 15 deg‡

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Transactions of the ASME


engaged in the cutting. Consequently, the topography is asymmetrically distributed in Fig. 17-共1兲 even the machined surface is axisymmetrical. Also, Figs. 17-共3兲–17-共5兲 give the simulated surface topographies and micrographs under the same condition. Figs. 17-共3兲 and 17-共4兲 correspond to the convex contouring milling, whereas Figs. 17-共5兲 and 17-共6兲 correspond to the convex ramping milling. It can be seen that the roughness varies distinctly with the milling mode. The predicted roughness RZ is 5.682 ␮m in contouring milling and is 9.819 ␮m in ramping milling. In contrast, the measured roughness RZ is 7.434 ␮m and is 11.740 ␮m in each milling mode.

4

Conclusions

A new and general iterative algorithm is developed to predict the machined surface topography and roughness in multiaxis ballend milling of plane and sculptured surfaces. In plane cutting mode, influences of feed modes, inclined direction, and inclined angles upon surface topography and roughness are fully studied and some instructional conclusions are figured out to help selections of machining parameters. In sculptured surface milling, contouring and ramping in unidirectional milling mode are studied. It is shown that appropriate cutting parameters will greatly improve the machined surface quality. Besides, due to the effect of cutter rotation direction, it is shown that the surface topography becomes asymmetrically distributed even for an axisymmetric cylindrical surface. These results are essential in developing a versatile simulation tool of the complete comoputer aided manufacturing computer aided manufacture 共CAM兲 system that is able to predict the surface topography and to optimize machining parameters.

Acknowledgment Thanks are expressed to Jun-Xue Ren for the help. This work is supported by the 111 Project 共Grant No. B07050兲 and the National Natural Science Foundation of China 共Grant No. 50435020兲.

Nomenclature ␤ ␪ ␾j ␥

⫽ ⫽ ⫽ ⫽

␩ ⫽ ␻ ⫽ ␮ ⫽ dp e fp ft F

⫽ ⫽ ⫽ ⫽ ⫽

helix angle 共deg兲 curvilinear parameter of cutting edge 共rad兲 angle between axis OCX j and OCXC 共rad兲 cutter inclined angle along pick-feed direction 共deg兲 cutter inclined angle along feed direction 共deg兲 angle speed of the spindle 共rad/s兲 initial angle between axis OCY C and OAY A 共rad兲 depth of cut 共mm兲 cutter runout 共mm兲 pick feed 共mm兲 feed per tooth 共mm兲 feedrate 共mm/min兲


nz R S t

⫽ ⫽ ⫽ ⫽

number of cutter flutes cutter radius 共mm兲 rotational speed of spindle 共rpm兲 curvilinear parameter of cutting edge 共s兲

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