[2014] a Mathematical Model for Grinding Ball End Mill

Computer-Aided Design 50 (2014) 16–26

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Computer-Aided Design

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Amathematical model for simulating and manufacturing ball end millHien Nguyen, Sung-Lim Ko ∗

Department of Mechanical Design & Production Engineering, Konkuk University, 120 Neungdong-ro, Gwangjin-gu, Seoul, 143-701, South Korea

h i g h l i g h t s

• A new mathematical model for grinding the ball end mill is proposed.• The rake face of ball end mill is modeled as a developable surface.• The conditions of engagement between wheel and the rake face are established.• The configuration of the flute surface was directly computed.

a r t i c l e i n f o

Article history:Received 7 January 2013Accepted 12 January 2014

Keywords:ModelingBall end millCNC grindingRake faceCutting simulation

a b s t r a c t

The performance of ball end mill cutters in cutting operations is influenced by the configuration of therake and clearance faces in the ball component. From the mathematical design of a cutting edge curve,the rake face can be defined by the rake angle and the width of the rake face at each cross section alongthe cutting edge. We propose the fundamental conditions that must govern the engagement between thegrinding wheel and the designed rake face in order to avoid interference while machining a ball end mill.As a result, a new mathematical model for determining the wheel location and a software program forsimulating the generation of the rake face of a ball endmill are proposed. In addition,methods for grindingthe clearance face in both concave and flat-shapes are introduced. The flute surface generated by a diskwheel during the grinding process is determined on the basis of a tangency condition. The results of theexperiment and the simulation are compared to validate the proposed model.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The helical flute, which located in both the cylindrical part andthe ball part of a ball endmill, plays an important role inmachininga sculptured surface. Manymathematical models have been devel-oped to machine different kinds of end mills.

For a cylindrical end mill, two basic approaches for determin-ing the wheel location and profile were developed to achieve a de-sired endmill. In the first approach, a CADmodel and an algorithmwere proposed to determine the relative positions of thewheel theworkpiece to grind an end mill according to certain design factorssuch as the rake angle and the core radius [1–3]. The second ap-proach aims to generate a wheel profile for a prescribedworkpiececross-section on the basis of the fundamental condition of engage-ment between the wheel and the helical groove [4,5].

The basic geometrical components of the ball part of a ballend mill are depicted in workpiece coordinates, OXYZ, as shownin Fig. 1(a). The ball part is machined using a grinding wheel, asshown in Fig. 1(b) and (c). The rake face and the flute surface are

∗ Corresponding author. Tel.: +82 2 450 3465.E-mail address: [email protected] (S.-L. Ko).

0010-4485/$ – see front matter© 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.cad.2014.01.002

generated by the side face and the revolution surface of the grind-ing wheel, respectively (Fig. 1(b)). The cutting edge is the inter-section curve between the rake face and the clearance face; thebottom curve is the intersection curve between the rake faceand the flute surface. The spiral rake face with a constant leadand rake angle is usually required to improve the cutting perfor-mance. This requirement complicates the ball end mill machiningprocess. To machine a ball end mill, the equation of the cuttingedge is derived first, and then the wheel location is determinedalong the cutting edge. Several optimal cutting edges were pre-sented [6,7]. The second approach mentioned above [4,5]was usedin combinationwith a cutting edge equation to generate a grindingwheel profile for a given cross-section of the ball end mill [8–10].Namely, the grinding wheel was predicted to grind a ball end millwith a constant helix angle cutting edge [8], circular-arc ball endmill [9], and concave-arc ball end mill [10]. However, researchersdemonstrated that the resulting manufactured grinding wheelsstill generate residual surfaces on theworkpiece andproduce a sidecutting edge strip [8,9]. The rake face of a taper ball end millwas produced using special grinding wheels such as those with atoroidal or spherical shape [11,12]; thismethod produces a smoothcutting edge and maintains a normal rake angle along the cuttingedge. However, the wheel location was chosen to grind a maxi-mum cutting depthwithout using the tangency condition between

H. Nguyen, S.-L. Ko / Computer-Aided Design 50 (2014) 16–26 17

Fig. 1. Ball end mill and grinding wheel geometry: (a) Geometry of the ball part of ball end mill, (b) Wheel for grinding rake face, (c) Wheel for grinding clearance face.

the surface of the wheel and the defined rake face. Therefore, forthis case, the maximum depth of the cut differs from the designeddepth. In addition, the unconventional shape of the wheels (torusor sphere) prevents widespread application in industry.

The limitations of previous works were due to their exclusivefocus on the cutting edge design and dynamic behavior analysis[6,7]. Additionally, to obtain a precise rake angle, the rake face wasmachinedwith specific wheel shapes (torus or sphere) [11,12]. Therake face and flutewere producedwith a predictedwheel using thesecond approach [8–10], but still there is a deviation between thedesigned and the produced ball end mill.

In this paper, a newmathematical model for manufacturing theball part of a ball end mill, including the rake and clearance faces,will be presented in continuation of the studies on the cylindri-cal part [1,2]. The rake face is modeled as a developable surfaceusing a cutting edge equation and a proposed bottom curve, asshown in Fig. 1(a). The rake face can thus be generated by usingthe side face (flat face) of the grinding wheel shown in Fig. 1(b).A cutting edge with a constant lead and normal rake angle is em-ployed in the proposed model. First, the equation for the cuttingedge curve is derived in Section 2. In Section 3.1.1, we propose thebottom curve of the rake face, which is obtained by determiningthe width of the rake face for each cross-section where the normalrake angle is measured. On the basis of the suggested fundamentalconditions for avoiding interference between the grinding wheelwith the designed rake face, the location of the wheel center andthe orientation of the wheel axis are determined in Section 3.1.2.Section 3.2 presents the two designs for grinding the clearanceface: concave and flat. The rake face is ground by the side face of thegrindingwheel, whereas the flute surface is generated by the revo-lution surface of the grinding wheel (Fig. 1(b)). This flute surface isdetermined in Section 4 by using the tangency condition betweenthe wheel and flute surfaces. Finally, in Section 5, the results of themodel will be verified on the basis of both simulation and experi-mental results.

2. Mathematical model of cutting edge curve

The geometrical accuracy of the cutting edge determines theprecision of the product that is machined by a ball end millingcutter. For this reason, the cutting edgemust be precisely designedand machined. The aim of this section is to establish the equationof a constant lead helix cutting edge for a ball end mill.

The cutting edge of a ball end mill is located in both the balland cylindrical parts. A coordinate system, OXYZ, is applied to aworkpiece in which the origin O is placed at ball center and theOX-axis coincides with the longitudinal axis of the workpiece. Inthis paper all derivations are carried out in this coordinate system(OXYZ). The cutting edge in the ball part is modeled in Fig. 2. A

Fig. 2. Cutting edge curve of the ball part of a ball end mill.

general position vector of a point Ci, along the cutting edge ispresented in the workpiece coordinate system OXYZ as follows:

r =

x,R2 − x2 sinϕ,

R2 − x2 cosϕ

(1)

where R is the radius of the ball head andϕ is the lag angle betweenthe tool tip (x = R) and the current point Ci on the cutting edge.The definition of a helix angle is the angle between the tangentvector of the helix, Ti, and the tangent vector of the longitudinalaxis, T ′

i . The angles β and βb are defined as the helix angles in thecylindrical and in the ball parts, respectively.

A mathematical model of the cutting edge with a constantlead or a constant helix was previously published [7]. Our modelconsiders the case of the cutting edge with a constant lead, whichis preferred by the cutter grinders to save the material during theregrinding operation. Eq. (2) can be applied to both the ball andcylindrical parts to obtain a constant lead cutting edge curve:

dϕ = −tanβR

dx. (2)

In addition, the lag angle is zero at the tool tip (ϕ = 0, at x = R);therefore, the lag angle can be determined along the cutting edgeas follows:

ϕ = −tanβR

(x − R). (3)

18 H. Nguyen, S.-L. Ko / Computer-Aided Design 50 (2014) 16–26

Fig. 3. Modeling of the rake face.

The variable helix angle along the cutting edge curve in the ballpart can be represented as follows:

βb = tan−1R2

− x2

R2tanβ

. (4)

The equation of the cutting edge in the ball part is obtained by sub-stituting Eq. (3) into Eq. (1):

x = R1 −

ϕ

tanβ

y = R

1 −

1 −

ϕ

tanβ

2

sinϕ

z = R

1 −

1 −

ϕ

tanβ

2

cosϕ.

(5)

3. Mathematical model for machining ball end mill

3.1. Mathematical model for grinding rake face in ball part

In this section, the tangent and normal vectors to the sphericalsurface along the cutting edge are introduced as the basiccomponents of the cutting edge. On the basis of these componentsand the designed depth of the cut in the normal plane of the rakeface, a bottom curve of the rake face is proposed, which is the keypoint for determining the wheel location along the cutting edge inSection 3.1.2.

The tangent vector of the cutting edge Ti, the normal vector ofthe ball surface Ni, and vector Bi that is the cross product of the vec-tors Ni and Ti at a point Ci in the cutting edge curve are described inFig. 3. Considering the position vector r = [x, y, z] of a point alongthe cutting edge, the vectors Ti, Ni, and Bi are expressed by Eq. (6):

Ti =r

∥r∥= [Txi , Tyi, Tzi]

Ni =r

∥r∥= [Nxi, Nyi, Nzi]

Bi = Ni × Ti = [Bxi, Byi, Bzi].

(6)

The normal rake angle, γ , is measured in the so-called normalplane Pn, which is normal to the elementary cutting edge. The rakeface is modeled as a developable surface, with a cross-section atthe normal plane that consists of the line segment CiKi, as shownin Fig. 3. Corresponding to a point on the cutting edge, the radialdepth ∥CiKi∥ = hi is given; this depth is zero at the tip and in-creases in the negative direction of the X-axis. The dotted curve inFig. 3 represents the bottom curve of the rake face. In this paper,the following equation of radial depth was utilized for the resultsof the simulation and the experiment:

∥CiKi∥ = hi = (c1 + c2x/R)R2 − x2 (7)

where 0 ≤ x ≤ R. The two coefficients c1 and c2 can be adjustedto yield a suitable bottom curve. For instance, the rake faces areillustrated in Fig. 4 using the cutting edge described in Eq. (5) forball radius of R = 6 mm, a helix angle of β = 25°, a rake angle ofγ = 0° and coefficients of radial depth of c1 = 0.25, c2 = 0.15 andc1 = 0.4, c2 = 0 for Fig. 4(a) and (b), respectively.

3.1.1. Construction of equation for bottom curve of rake faceOur goal is to precisely machine the rake face of a ball end mill

with a constant normal rake angle and a designed cut depth. There-fore, the equation of the bottom curve, which is formed by thedesigned cut depth in normal plane, is required to determine thewheel location. Moreover, both the principal normal vector of thebottom curve and normal vector of the rake face along the line seg-ment in normal plane are derived in this section. These derivationsare used in the next section to ensure that the condition of engage-ment sufficiently avoids interference.

The position vector of point Ki on the bottom curve to the pointCi on the cutting edge is described in Fig. 3. It is calculated in theOXYZ coordinate system as follows:

rOKi = rOCi + CiKi = [xki , yki , zki ] (8)

where,

CiKi = −hi cos γ Ni − hi sin γ Bi. (9)

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7

7

6

5

4

3

2

1

00 1 2 3 4 5 6 7

a b

Fig. 4. Rake face configuration: (a) bottom curve with c1 = 0.25, c2 = 0.15, (b) bottom curve with c1 = 0.4, c2 = 0.

H. Nguyen, S.-L. Ko / Computer-Aided Design 50 (2014) 16–26 19

Fig. 5. Relative wheel location and workpiece to avoid interference with bottomcurve.

The bottom curve of the rake face is defined uniquely in Eq. (8), andit can be parameterized as follows:

rOKi = rOK (ϕ) = [xk(ϕ), yk(ϕ), zk(ϕ)]. (10)

Eq. (11) represents the principal normal vector of the bottom curveof the rake face, NKi at point Ki, which will be used in the nextsection:

NKi =dr2OKi/dϕ

2dr2OKi/dϕ2 = [NKxi ,NKyi ,NKzi ]. (11)

The parameterized equation of the rake face can be determinedusing the cutting edge and bottom curves as follows:

SR(ϕ, v) = (1 − v)rOCi + vrOKi = rOCi − vhi(cos γ Ni + sin γ Bi)

with 0 ≤ v ≤ 1. (12)

The modeled rake face has an important property: the normalvector to the rake face is constant (free from the variable v) alongthe line segment CiKi. Later, this property will be implementedin the tangency condition between the side face of the grindingwheel and the rake face. To prove this property exists, a point Liis assumed to lie on the segment CiKi such that ∥CiLi∥ = v ∥CiKi∥

with 0 ≤ v ≤ 1. The normal of the rake face at point Li is given inBox I.

3.1.2. Wheel location and conditions of engagement to avoidinterference with rake face

In this section, the conditions of engagement between thegrinding wheel and the rake face are proposed, which allows pre-cise machining of the rake face using the side face of the grindingwheel. As a consequence, the wheel center location and the ori-entation of wheel axis are determined along the cutting edge tomachine the rake face.

The rake face is machined by grinding each line, CiKi, using theside face of the grinding wheel with a radius of Rw , as shown inFigs. 1(b) and 5. The side face of the grinding wheel with centerposition of Gi is the outer face of the wheel and perpendicularto wheel’s axis. To grind the segment CiKi on the rake facewithout interfering with the defined cutting edge, the side faceof the grinding wheel must contain the tangent vector, Ti, of thecutting edge curve at the current grinding point Ci. Many potentiallocations of the wheel exist that allow the line CiKi to be groundwithout interference with the cutting edge, as shown in Fig. 5;however, the grindingwheel at the location-2 exhibits interference

Fig. 6. Model for determination of wheel location in grinding rake face.

with the defined bottom curve of the rake face. Therefore, onlylocation-1 of the grinding wheel can grind the defined rake facewithout interference. Thus, to avoid interference between thegrinding wheel and the defined bottom curve, the direction of thevector, N ′

Ki, must pass through the center of the wheel’s side face,

Gi. The vector N ′

Kiis the projection of the principal normal vector

of the bottom curve, NKi , at point Ki into the plane of the wheel’sside-face. The fundamental conditions to avoid interference whilegrinding the rake face are summarized as follows:

Condition 1: The side face of the grinding wheel must passthrough the line CiKi and contain the tangent vector of the cuttingedge, Ti, at the current grinding point Ci.

Condition 2: The center of the wheel’s side face, Gi, must becontained in the projection of the normal vector of the bottomcurve to the side-face of the grinding wheel, N ′

Ki.

Note that the principle normal vector of the bottom curve, NKi ,intersects the wheel axis, Ii, as shown in Fig. 6 when conditions1 and 2 are satisfied. This characteristic will be utilized for thecalculation of the location of the wheel center.

On the basis of the condition 1, the wheel axis, Ii, can berepresented by Eq. (14):

Ii =Ti × KiCiTi × KiCi

= cos γ Bi − sin γ Ni. (14)

Using the condition 1 and condition 2 regarding the proposed en-gagement, the position of the wheel center in workpiece coordi-nates is determined by Eq. (15):

rOGi = rOKi + N ′

KiRw. (15)

Eq. (15) guarantees having only one commonpoint,Ki, between thegrindingwheel and the bottom curve of rake face. In addition, fromthe condition 2 vectors Ii, NKi are intersected, therefore the three

20 H. Nguyen, S.-L. Ko / Computer-Aided Design 50 (2014) 16–26

3)

NLi =

∂SR(ϕ,v)∂ϕ

×∂SR(ϕ,v)∂v ∂SR(ϕ,v)∂ϕ

×∂SR(ϕ,v)∂v

=

∥r∥ Ti − vhi

∥r∥R cos γ Ti + kn sin γ (Ni × NFi)

×

−hi(cos γ Ni + sin γ Bi)

∥r∥ − vhi

∥r∥R cos γ + kn sin γ sinψ

hi

= cos γ Bi − sin γ Ni (1

in which NFi, kn and ψ are the principal normal vector, normal curvature and the angle between the principal normal vector NFi andthe normal of the sphere Ni at point Ci on the cutting edge, respectively.

Box I.

Fig. 7. Model for grinding concave-shaped clearance face: (a) wheel location in the normal plane, (b) wheel location in OXYZ coordinate system.

vectors Ii, N ′

Ki, NKi are coplanar. Thus, the vector N ′

Kiin Eq. (15) can

be determined as:

N ′

Ki =(Ii × NKi)× Ii

∥(Ii × NKi)× Ii∥. (16)

The normal of the rake face is proven to be constant along the linesegment CiKi, as indicated by Eq. (13). Thereforewhen the side faceof the grinding wheel contains the line segment CiKi, the wheelaxis orientation is assigned to be equivalent to the normal of therake face at point Ci. Then the contact line between the designedrake face and the side face of the grinding wheel is this line seg-ment (CiKi). The position of the wheel center and the orientation ofwheel axis that are required for precise machining of the rake faceare determined by Eqs. (14) and (15), respectively.

3.2. Mathematical model for grinding clearance face in ball part

In the case of machining, the clearance angle of a ball end milldirectly affects the cutting performance and the tool life. The shapeof the clearance face is also an important factor for a cutter, and itvaries depending on the grindingmethod. In this paper, models forgrinding the clearance face of a ball end mill into flat and concaveshapes will be introduced.

The clearance angle, α0, is defined as the angle between thetangent vector at point Ci and the extension of the vector Bi in thenormal plane. The clearance face is formed by the outer circle ofthe cup wheel in the concave method and by the side face of thecup wheel in the flat method; the outer circle of the cup wheel hasa radius of Rc , as shown in Fig. 1(c).

3.2.1. Grinding of concave clearance faceIn this section, the cutting edge and its basic components are

used to determine the wheel center location and the wheel axis

orientation in order to generate a concave-shaped clearance facewith a constant clearance angle along the cutting edge.

The model for grinding a concave-shaped clearance face isshown in Fig. 7. Here, the clearance face formed in grinding processis represented by the segment CiDi, which is a segment on theouter circle of the grinding wheel along the cutting edge. Togrind a clearance face with clearance angle α0, the vector CiGi,must make the angle α0 with the normal vector Ni, as shown inFig. 7(a). Therefore, the relative position of the wheel center tothe workpiece and the orientation of wheel axis are determinedas follows:

rOGi = rOCi + CiGi (17)

where,

CiGi = Rc cosα0Ni − Rc sinα0Bi. (18)

The orientation of the wheel axis can be obtained as follows:

Ii = Bi × Ni = Ti. (19)

3.2.2. Grinding of flat clearance faceIn this section, the cutting edge and its basic components are

used to determine the wheel center location and the wheel axisorientation in order to generate a flat-shaped clearance face witha constant clearance angle along the cutting edge.

The model for grinding a flat-shaped clearance face is shown inFig. 8. The clearance face is formedby the side-face of the cupwheelwhen the wheel moves along the cutting edge. In this model, theformed clearance face is a developable surface and its cross-sectionin the normal plane is a straight line CiDi, as shown in Fig. 8(a). Theclearance angle is defined as the angle between the vector CiDi andthe extension of the vector Bi. The outer circle of the side face is

H. Nguyen, S.-L. Ko / Computer-Aided Design 50 (2014) 16–26 21

Fig. 8. Model for grinding a flat-shaped clearance face: (a) wheel location in the normal plane, (b) wheel location in top view.

positioned to pass through points Ci and Di. The side face of thecup wheel is perpendicular to the normal plane; therefore, it willcontain the vector CiDi and the tangent vector of the cutting edgeTi as shown in the model in Fig. 8. Thus, the position of the wheelcenter is calculated as follows:

rOGi = rOCi + CiGi

CiGi = −R sinα0(sinα0Ni + cosα0Bi)

−

R2c − (R sinα0)2Ti.

(20)

The orientation of the wheel axis is calculated as follows:

Ii = Ti × (sinα0Ni + cosα0Bi). (21)

The NC-code is generated from the position of thewheel center,rGi, and its axis orientation, Ii. This NC code is used for a 5-axisCNC grinder for fabricating the rake and clearance face of ball endmill [11].

4. Determining flute surface of ball part

When the cutting edge equation, normal rake angle and de-signed cut depth are given, the rake face can be modeled. Addi-tionally, the wheel center location and the wheel axis orientationto machine this rake face can be determined, as explained in theprevious section. However, the generated surface also consists ofa flute surface, which is ground by the revolution surface of thegrinding wheel when the wheel moves along the cutting edge. Theaim of this section is to calculate the generated surface for the pur-pose of visualization and simulation. The generated flute surface ofa ball end mill will be calculated by using the tangency conditionat any point of the wheel surface.

It is important to determine the flute surface (Fig. 1(a)) of a ballend mill because the flute of the ball component determines thestiffness and the chip evacuation capability of the cutter. It is help-ful to calculate and optimize the grinding wheel profile to obtaina suitable flute. Although the flute surface can be obtained by car-rying out a simulation with commercial software, this paper willinstead present a direct calculation for the points of the flute sur-face. For a designed rake face and a given grindingwheel geometry,the formation of the flute surface during the grinding process forthe rake face of a ball end mill will be presented.

The flute surface of the ball part is determined by envelopingthe surface generated by the grindingwheel’s movement along thecutting edge, and it is limited by the ball surface of the workpiece.This flute surface consists of three components (1) rear surface of

Fig. 9. Generated surface formed by moving wheel: (1) rear surface, (2) sweptsurface and (3) front surface.

the grinding wheel at the initial position, (2) swept surface gen-erated during the motion of the grinding wheel and (3) the frontsurface of the grinding wheel at the final position as shown inFig. 9.

The points on the swept surface (2) are determined on the ba-sis of the key idea that, the grinding wheel is tangent at all timesto the swept envelope along a sweeping profile, as shown in Fig. 9.Therefore, at any point P along the swept surface (2), the unit nor-mal vector, N(P), is parallel to the vector normal to the swept sur-face at that instant. In addition, the velocity vector, V (P), of point Pmust be tangent to the swept surface. Therefore, the swept surface(2) can be determined based on the tangency condition [13,14]:

N(P) · V (P) = 0. (22)

However, the rear surface (1) is a part of the wheel surface,which is limited by the ball surface of the workpiece at the initialposition, and satisfies N(P) · V (P) < 0. The front surface (3) isa part of the wheel surface at the final position, which satisfiesN(P) · V (P) > 0.

The location of the wheel center and the unit vector (∥I∥ = 1)of the wheel axis orientation in Eqs. (14) and (15)in workpiece

22 H. Nguyen, S.-L. Ko / Computer-Aided Design 50 (2014) 16–26

Fig. 10. Notation for calculating flute surface: (a) wheel profile, (b) local coordinate system setting.

coordinates can be parameterized as follows:

rG(t) = (rOGix , rOGiy , rOGiz ) (23)

I(t) = (Iix, Iiy, Iiz). (24)

The movement of the grinding wheel can be defined using thetrajectory of the wheel center, rG(t), and the tool axis orientation,I(t), as shown in Fig. 10(b). A local coordinate system XLYLZL islocated at the wheel center, and it is established as follows:

ZL = IXL = I/∥I∥YL = ZL × XL.

(25)

The axial cross-section of the grinding wheel with the width,L, in the XLZL plane of the local coordinate system is modeled, asin Fig. 10(a). The parametric equations of the revolution surface ofthe wheel can be obtained by rotating a curve along the ZL-axis, asdescribed in Eq. (26):

SW (u, θ, t) = rG(t)+ r(u) cos θ XL + r(u) sin θ YL + uZL. (26)

Eq. (22) was written for the general case [12]. Finding theexpression of V (P) and N(P) is difficult, as previously stated [13].However, the expressions of V (P) and N(P) in Eq. (22) can bedetermined by following the procedures given below.

In the cross section of the wheel surface that is normal to thewheel axis I , offset from the wheel center a distance u in the ZLaxis, taking a point P in that cross-section at an angle, θ , measuredfrom the axis XL as shown in Fig. 10(a). The velocity of point P andthe normal ofwheel surface at point P are determined inworkpiececoordinates as follows:

V (P) = V (G)+ΩG × rGP

= rG + ∥I∥YL × (r(u) cos θ XL + r(u) sin θ YL + uZL) (27)

where, ΩG =I×I∥I∥2

= ∥I∥YL, is the instantaneous angular velocityof the wheel about point G. Thus:

V (P) = rG + u∥I∥XL − ∥I∥r(u) cos θ ZL. (28)

The normal of the wheel surface at point P can be calculated asfollows:

N(P) =

∂SW (u,θ,t)∂θ

×∂SW (u,θ,t)

∂u ∂SW (u,θ,t)∂θ×

∂SW (u,θ,t)∂u

=

cos θ XL + sin θ YL − r(u)ZL1 + r2(u)

. (29)

By applying V (P) · N(P) = 0 to Eqs. (28) and (29) we obtain thefollowing:

(rG · XL + ∥I∥r(u)r(u)+ ∥I∥u) cos θ + rG · YL sin θ

− r(u)rG · ZL = 0. (30)

By solving Eq. (30), the contact line between the wheel surfaceand the swept surface is obtained. The points on the generatedsurface (2) in Fig. 9 are determined by θ(u, t), which is obtainedfrom Eq. (30). The generated surface types (1) and (3) made by themoving wheel when it moves from the initial position (t = 0) tothe final position (t = tf ) can be easily calculated. The generatedsurface (1) is a part of the wheel surface that satisfies

V (P) · N(P) < 0. (31)

For type (3), it satisfies

V (P) · N(P) > 0. (32)

The flute surface generated by moving wheel in Eq. (31) can bedetermined through the results of Eqs. (30)–(32). Moreover, thissurface is limited by the ball surface in the positive direction of theX-axis in workpiece coordinates; thus, Eqs. (33)(b) and (c) must beadded to determine the points in the generated surface of a ballend mill.r(θ, u, t) = rG(t)+ uZL + r(u)(XL cos θ + YL sin θ) (a)

∥r(θ, u, t)∥ ≤ R (b)rx(θ, u, t) ≥ 0. (c)

(33)

In addition, the rake face is designed to be a developable surfacewith a constant normal vector along the line segment CiKi asestablished in Eq. (13) and a wheel location determined by Eqs.(14) and (15). Then, the contact line of the side face of the wheelis always the line segment CiKi as shown in Appendix A. There isalso one portion of the contact line that lies in the edge part of thegrindingwheel. In this paper, wewill consider only the contact linein the edge of the side face that directly generates the rake facebecause the width of the wheel (L) is chosen to be large enough so

H. Nguyen, S.-L. Ko / Computer-Aided Design 50 (2014) 16–26 23

543210

-1-2-3-4-5

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

a b c d

6543210

-10 1 2 3 4 5 6 7 8

-4-2

02

4

Fig. 11. Generation of flute surface: (a) cutting edge, (b) wheel profile, (c) flute surface generated by VERICUT, (d) flute surface calculated from the proposed model.

Fig. 12. Results of simulation of the grinding of ball end mill (R = 6 mm, γn = 0°, β = 25° and αo = 11°).

that only the edge part of the side face is involved in the materialremoval process. This portion of the contact line is the transitioncontact line between the side face and the revolution surface. Theends of this portion of the contact line are Ki at the bottom line andKi1 which can be obtained from Eq. (30) at u = 0.

5. Verification of model via simulation and experiment

To verify themodel of ball endmill manufacturing, a simulationof the grinding process to fabricate the ball part of a ball end millwas performed. Additionally, a ball end mill was manufacturedunder the same conditions as those used in the simulation, thusenabling a comparison between the two. A target ball endmill wasmade with a radius of R = 6 mm, a normal rake angle of γn = 0°,a helix angle of β = 25°, clearance angle, α0 = 11°. The cuttingedge was determined by Eq. (5) and it is shown in Fig. 11(a). Theequation of the bottom curve is described by Eq. (34) (c1 = 0.25,c2 = 0.15):

CiKi = (0.25 + 0.15x/R)R2 − x2 where, 0 ≤ x ≤ R. (34)

Fig. 11(c) and (d) show simulation results of the generatedhelical flute of the ball part with a given wheel profile, which isdescribed in the XLZL plane via wheel coordinates, as shown inFig. 11(b). Fig. 11(c) was generated using the commercial softwareVERICUT for rake face generation and Fig. 11(d) is the result of thecalculation of the flute surface using Eqs. (30)–(33).

Fig. 11(d) was created using the simulation program thatwas implemented using MATLAB version 7.9; the program wasexecuted on a PC with the following components: processor:Intel(R) Core(TM) 2 Duo; CPU: 2.2 GHz; and 2 GB of RAM equippedwith an Intel graphics card of 256 MB. First, the cutting edge andgrinding wheel are divided into 180 points each (Ci in Fig. 11(d))and 100 points along the wheel profile (∆u = L/100). Then, the

contact lines are calculated and depicted as a black line for therevolution part and a pink line for the side face. The transitioncontact line along the edge of the side face of the grinding wheel isalso calculated. The contact lines are shown at points Ci (i = 30, 60,90, 120, 150) in Fig. 11(d), in which the two end points Ki and Ki1 ofthe transition contact line are too close to be observed distinctly inthe figure. The generated flute surface (Fig. 11(d)) is rendered bytiling the contact lines with the Patch-Function in MATLAB to becompared with Fig. 11(c). The computation time of our programis approximately 4 s, and the computation time of VERICUT isapproximately 4 s for the lowest resolution and 12 s for the highestresolution; however, the computation time depends heavily on thenumber of divided points in the cutting edge and wheel profile.The simulation results in Fig. 11(c) and (d) are well matchedwhichmeans that carrying out analytical calculation is faster and moreeconomical than using commercial software (VERICUT).

To validate the mathematical analysis developed in the previ-ous sections, simulations were performed with the designed ballend mill using VERICUT as shown in Fig. 12. The ball end mill wasmanufactured with a 5-axis CNC grinding machine on the basis ofthe model developed in this paper, as shown in Fig. 13.

To simulate the grinding process, the machine parameters ofthe 5-axis CNC grinding machine are measured. The parameters ofthe wheel geometry and the value of the machine setup are givenas inputs. The wheel locations are calculated and transformed intoNC code to be used in VERICUT and the 5-axis CNC grinding ma-chine. The geometry of the ball end mill generated by the math-ematical model shown in Fig. 13 agrees well with the simulationresult shown in Fig. 12. Detailed comparisons of the results fromexperiments and simulations show that the rake and clearancefaces of the designed ball end mill were reasonably fabricated onthe basis of the results of the experiment, simulation, anddesign, asshown in Figs. 14 and 15. In this section, all the figures are attachedwith grids to show the matching between Figs. 11–14. With this

24 H. Nguyen, S.-L. Ko / Computer-Aided Design 50 (2014) 16–26

Fig. 13. Ball end mill produced by the NC code from the manufacturing model (R = 6 mm, γn = 0°, β = 25° and αo = 11°).

Fig. 14. Comparison of the performance for the rake face: (a) by experiment, (b) by simulation (VERICUT), (c) design.

Fig. 15. Measurement of designed parameters at the normal plane.

grid, the results of experiment and simulation can be comparedquantitatively.

The rake face is designed according to the geometry of thecutting edge and bottom curves, which are given by Eqs. (5) and(33)(b), respectively, as shown in Fig. 14(c). Themachined rake face(Fig. 14(a)) agreeswellwith the simulated (Fig. 14(b)) anddesigned(Fig. 14(c)) rake face. The direct measurement of the generatedsample that is shown in Fig. 13 is complicated; thus, a cross sectionof the simulation result that is perpendicular to the cutting edgeat the connection point between the ball and the cylinder parts isanalyzed. The result shows that the measured design factors suchas the rake angle, clearance angle, radial depth of the rake face, and

the width of the clearance match well with the designed one, asshown in Fig. 15.

In the experiment, CNC grinder (Maker: TIC) that consists of5-axeswith two rotationalmotions (A,W) and three linearmotions(X, Y, Z) is used, as shown in Fig. 16. The wheel center locationand the wheel axis orientation required for grinding the ball partof a ball end mill are calculated in the workpiece coordinatesystem using Eqs. (14) and (15). Then, the inverse kinematicproblem for the CNC grinder is solved to convert this data intoan NC program on the basis of the method that was presented inprevious study [11]. The machine measurements, wheel positionsin workpiece coordinates and the corresponding NC program in

H. Nguyen, S.-L. Ko / Computer-Aided Design 50 (2014) 16–26 25

Fig. 16. 5-axis CNC grinder: (a) External view, (b) Machine structure.

absolute mode (G90) for grinding the rake face are provided in asupplemental file.

6. Conclusion

A new mathematical model for grinding the ball part of a ballend mill is proposed. The complete process for manufacturing aball end mill is formulated, including processes such as modelingthe rake and clearance faces. The helical flute surface of the ball endmill is analytically calculated and verified by means of simulationand experiment. The study is summarized as follows:

1. The rake face is modeled by using cutting edge and bottomcurve equations as a developable surface, which allows the rakeface to be generated by using the side face of the grinding wheel.Models for grinding the clearance face with both flat and concaveshapes are suggested.

2. On the basis of the cutting edge and the suggested bottomcurve, we propose the conditions for engagement between thewheel surface and the designed rake face, such that interferencebetween the two is avoided. Consequently, the wheel locationalong the cutting edge is determined to machine the rake face andthe clearance with reasonable precision.

3. The configuration of the flute surface is computed directlyfrom the designed rake face and the given wheel profile withoutusing commercial software for simulation, which not only offersmore convenience, but also lowers the cost of the process.

Acknowledgments

This research was supported by the Leading Foreign ResearchInstitute Recruitment Program through the National ResearchFoundation of Korea (NRF) funded by the Ministry of Education,Science and Technology (MEST, 2011-00260), the IndustrialCore Technology Development Project through the Ministryof Knowledge Economy (Grant number:10035641) and KonkukUniversity. Finally, we thank the anonymous reviewers for theirsuggestions for improving this paper.

Appendix A

In this appendix the required derivations to show that thecontact line in the side face of the grinding wheel is the CiKi aregiven.

As grinding wheel moves along the cutting edge, the side faceof grindingwheel generates a one-parameter family of planes. Thisone-parameter family of planes is represented as:

U(ϕ) : I(ϕ) · (r − rC(ϕ)) = 0 (A.1)

where r is position vector of the points that lie in the side face, andI(ϕ) is wheel axis orientation and is the normal vector of the sideface.

The envelope S of the family U(ϕ) is found by intersecting theplane U(ϕ)with the first derivative planes U(ϕ) [15,16]:

S :

U(ϕ) : I(ϕ) · (r − rC(ϕ)) = 0

U(ϕ) :d[I(ϕ) · (r − rC(ϕ))]

dϕ= 0.

(A.2)

The equation of plane U(ϕ) is simplified as follows:

U(ϕ) :d[I(ϕ) · (r − rC(ϕ))]

dϕ

= I(ϕ) · (r − rC(ϕ))−rC(ϕ) I(ϕ) · T (ϕ)

= I(ϕ) · (r − rC(ϕ)) = 0. (A.3)At certain ϕ, this yields a straight line r(ϕ). Hence, the envelope

surface S is a ruled surface which is tangent along each ruling r(ϕ)to a single plane (namely U(ϕ)). It is well known in differential ge-ometry that this characterizes the surface as developable surface.

The next step is to show the intersecting line, r(ϕ), is the lineCiKi. Note that Eq. (A.3) is equation of the plane with normal, I(ϕ),which has the same direction with tangent vector of the cuttingedge as shown as follows:

I(ϕ) =dI(ϕ)dϕ

=(cos γ d(N × T )− sin γ dN)

dϕ

= cos γ

∥r∥R

T × T + N ×dTdϕ

− sin γ

∥r∥R

T

= kn cos γ N × NF − sin γ∥r∥R

T

=

kn cos γ sinψ − sin γ

∥r∥R

T (A.4)

where NF , kn andψ are the principal normal vector, normal curva-ture and the angle between the principal normal vector NF and thenormal of the sphere N at point C on the cutting edge, respectively.

It is obvious that from Eq. (A.4) the plane U(ϕ) passes throughpoint C, and its normal vector, I(ϕ), is the same direction with tan-gent vector of cutting edge as shown in Eq. (A.4). Thus, it can beconcluded that the contact line is the intersecting line betweenU(ϕ) and U(ϕ), it is also the line CiKi as shown in Fig. A.1.

Appendix B. Supplementary data

Supplementary material related to this article can be foundonline at http://dx.doi.org/10.1016/j.cad.2014.01.002.

26 H. Nguyen, S.-L. Ko / Computer-Aided Design 50 (2014) 16–26

Fig. A.1. The contact line the side face of the grinding wheel.

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