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ORIGINAL ARTICLE

Modeling and analysis for clearance machining process of endmills

Guochao Li & Jie Sun & Jianfeng Li

Received: 2 March 2014 /Accepted: 7 July 2014 /Published online: 30 July 2014# Springer-Verlag London 2014

Abstract Clearance of end mills has great impact on theperformance of milling, and therefore a high demand for itsmachining theory and process is put forward. Based on theanalysis of practical machining process, enveloping theory,principles of spatial geometry are introduced to establish theclearances processing model, as well as considering the ge-ometries, orientations, and locations of wheels used to ma-chine the clearances with convex, eccentric, or elliptic shapes.Accordingly, limitations of wheel geometry and location tomachine a desired clearance are discussed. A commercialcomputer aided design system with API function program-ming is used to visualize the machining process. The solidmodel is finally obtained.

Keywords Clearance . Endmill . Process

1 Introduction

Clearance is one of the most important structures of end mills[1]. It influences the intensity and sharpness of cutter edges,

the tool life as well as the quality of machined surfaces.However, the research on the end mill manufactureprocess has mainly focused on the helical groove grindingprocesses [2–5]. There is little analysis on clearance grindingprocess.

In many cases, clearance was designed and calculated inthe groove design process, based on the assumption that it wasa straight line at the cross section of end mills [6–12]. Obvi-ously, clearance models built with this approach could notreflect the actual processing precisely. With different researchideas, Chong described the re-sharpen processes for clear-ances of end mills [13]. The principle of eccentric clearancere-shaping process was emphasized, including the re-shapingorder, the position of end mill, wheel, and tooth rest. Based ona five-axis CNC tool grinder, Chen et al. planed the grindingprocesses of clearance by using a tool grinding CAM system[14]. Translation and revolution sequences of every grinderaxis in flat, concave, and eccentric machining process wereintroduced. Uhlmann et al. machined several clearance faceswith cup-shaped grinding wheels and discussed the influenceof the diamond grain size of the used grinding wheels [15].Besides, the clearance processing technology has been men-tioned by some other researchers [16–18]. However, thesestudies have not provided details of how to establish themathematical model of clearance machining process.

In this paper, according to the practical machining process,enveloping theory and principles of spatial geometry model-ing technology, clearance machining processes are detailed.Mathematical models of wheel geometry, orientation, andlocation are calculated. Principle and problems in eccentricclearance machining process are emphasized. Accordingly,the clearance process simulation system is developed by usingUG second development technology. Furthermore, examplesof convex, eccentric, and elliptic clearance machining pro-cesses are carried out.

G. Li : J. Sun (*) : J. LiSchool of Mechanical Engineering, Shandong University,Jinan 250061, Chinae-mail: [email protected]

G. Lie-mail: [email protected]

J. Lie-mail: [email protected]

G. Li : J. Sun : J. LiKey Laboratory of High Efficiency and Clean MechanicalManufacture, Ministry of Education, Shandong University,Jinan 250061, China

Int J Adv Manuf Technol (2014) 75:667–675DOI 10.1007/s00170-014-6154-3

Nomenclature

α0 Clearance angle φ1 Inclination angle of wheelmachining the flat clearance

β Helical angle φ2 Inclination angle ofwheel machining theeccentric clearance

Φ Adjacentcutter angle

n1 Orientation of wheel machiningthe flat clearance

P Lead of thecutting edge

n2 Orientation of wheelmachining the eccentricclearance

mR Radius ofend mill

n3 Orientation of wheelmachining the convexclearance

gR Radius ofgrindingwheel

O1(xO1,yO1,zO1) Location of wheel machiningthe flat clearance

gb Thickness ofgrindingwheel

O2(xO2,yO2,zO3) Location of wheel machiningthe eccentric clearance

O3(xO3,yO3,zO3) Location of wheel machiningthe convex clearance

2 Clearance machining processes

According to the difference of wheel types and grinding pro-cesses, the clearance geometry of end mill is generally dividedinto three kinds: flat, eccentric, and convex (Fig. 1). The flatclearance is the most common shape of end mills. As shown inFig. 1a, cup or cone-shaped wheel is used and the relativeposition to the end mill is shown. At the beginning, the wheelaxis (n1) is perpendicular to the end mill axis. To avoid thephenomenon of grinding burn caused by the overlarge contactarea, the wheel is then rotated. Therefore, the flat clearance ismainly ground by the wheel circumference. The eccentricclearance has a higher intensity and a longer tool life than theother two shapes. It is generally machined with a cylinderwheel, whose axis (n2) is coplanar with the axis of end mill(Fig. 1b). To grind a clearance angle, there must be a certainangle (φ2) between the two axes. The convex clearance has theminimum intensity. On the other hand, it has the easiest ma-chining process. Fig. 1c gives the wheel position relative to theendmill. The wheel axis (n3) is parallel to the endmill axis andits location could be deduced by the clearance angle α0.

In practice, the grinding process of the clearance face isusually done from shank side to end teeth. In order to calculateconveniently, all of the process is beginning at the end teeth inthis paper, which will not influence the machining result.

3 Flat clearance machining process

As indicated in Fig. 2, coordinate system XgYgZg representsthe wheel frame while XYZ represents the stationary end mill

frame. In this paper, the clearance is machined only after thehelical groove has been ground, the cutter tip is located atcoordinates (mR, 0, 0) and equations are expressed in the endmill frame.

Figure 3 shows the transformation procedures of wheelcoordinate system for flat clearance machining. Assume thatthe wheel coordinate system is coincident with the end millcoordinate XYZ at the beginning, coordinate system XgY gZ g

can be deduced by two translation parameters mR and -gRfrom O along X-axis and Z-axis, respectively. Rotating theframe XgY gZ g by an angle α0 about its Zg-axis (get thecoordinate system Xg1Y g1Z g1), flat clearance angle with avalue of α0 could be machined. In order to avoid the phenom-enon of grinding burn, which is caused by too much contactarea between wheel and clearance, the wheel is then rotated tothe coordinate system Xg2Y g2Z g2 by an angle φ1 (Fig. 3).Now, the initial wheel position relative to the end mill isobtained. The wheel orientation can be expressed by thefollowing vector:

n1 ¼ cos φ1ð Þ⋅ cos α0ð Þ; cos φ1ð Þ⋅ sin α0ð Þ; sin φ1ð Þ½ �T ð1Þ

and the wheel location can be expressed by

xO1 ¼ mRþ gR ⋅ sin φ1ð Þ ⋅ cos α0ð ÞyO1 ¼ gR ⋅ sin φ1ð Þ ⋅ sin α0ð ÞzO1 ¼ −gR ⋅ cos φ1ð Þ

8<: ð2Þ

4 Eccentric clearance machining process

4.1 Machining principle

According to the wheel position provided in Fig. 1b, coordi-nates are built to express the initial positions of wheels to startthe machining process (Fig. 4). The parameter φ2 is therotational angle of the wheel frame about its Xg-axis. Thewheel orientation can be given by the vector

n2 ¼ sin φ2ð Þ; 0; cos φ2ð Þ½ �T ð3Þ

and the wheel location can be expressed as

xO2 ¼ gR ⋅ cos φ2ð Þ þ mR−gb ⋅ sin φ2ð ÞyO2 ¼ 0zO2 ¼ −gR ⋅ sin φ2ð Þ−gb ⋅ cos φ2ð Þ

8<: ð4Þ

Figure 5a gives the wheel position relative to the end mill ineccentric machining process. The machining result shows thattwo faces are ground after the grinding process: one is theclearance face enveloped by the cylindrical face of wheel and

668 Int J Adv Manuf Technol (2014) 75:667–675

the other is the unwanted face produced by the wheel circum-ference. Figure 5b shows the machining result based onFig. 1b (i.e., the point p4 is coincident with point Q and gb=10 mm). P1 is the intersection point of contact line and end

mill contour. P2 is the intersection point of contact line and thewheel circumference. P3 is the intersection point of end millcontour and the wheel circumference. Obviously, two prob-lems can be observed from the figure: the envelope surface

Fig. 1 Machining process of clearance with flat, eccentric, and convex shapes

(a) Wheel coordinate (b) End mill coordinate

Fig. 2 Coordinates for wheel and end mill calculation. a Wheel coordinate. b End mill coordinate

Int J Adv Manuf Technol (2014) 75:667–675 669

boundary (i.e., the clearance face boundary) is not coincidedwith the cutting edge and the unwanted machining face willaffect other teeth when Φ’>Φ.

Besides, the inclination angle φ2 and clearance angle α0

could not be calculated so far. To solve these three problems,further study is detailed in the following sections.

4.2 Analysis of clearance face boundary

From Fig. 5b, it is learned that the envelope surface boundarywill overlap with the cutting edge, as long as the point p1 islocated on the cutting edge. Based on Eqs. (3) and (4), thecylinder face of the wheel as shown in Fig. 4 can be repre-sented in this form by

rg ¼xgygzg

24

35 ¼

xO2 þ m1 ⋅ sin φ2ð Þ þ gR ⋅ cos φ2ð Þ ⋅ cos m2ð ÞgR ⋅ sin m2ð Þ

zO2 þ m1 ⋅ cos φ2ð Þ‐gR ⋅ cos m2ð Þ⋅sin φ2ð Þ

24

35ð5Þ

Then, a family of wheel surfaces can be obtained bymoving the wheel along the cutting edge with a constant leadP and expressed as

rgt m1;m2; tð Þ ¼xgtygtzgt

24

35 ¼

xg ⋅ cos tð Þ‐gR ⋅sin m2ð Þ ⋅ sin tð Þxg ⋅sin tð Þ þ gR ⋅sin m2ð Þ ⋅ cos tð Þ

zg þ P⋅tð Þ= 2 ⋅ πð Þ

24

35 ð6Þ

where t represents the rotation angle that the wheel movesaround the end mill axis. According to the principle of enve-lope, the fundamental relationship for the clearance machiningis formed as

∂rgt=∂m1; ∂rgt=∂m2; ∂rgt=∂t� � ¼ 0

where

∂rgt=∂m1 ¼ sin φ2ð Þ⋅cos tð Þ; sin φ2ð Þ⋅sin tð Þ; cos φ2ð Þ½ �T

∂rgt=∂m2¼‐gR⋅cos m2ð Þ⋅sin tð Þ‐gR⋅cos φ2ð Þ⋅cos tð Þ⋅sin m2ð ÞgR⋅cos m2ð Þ⋅cos tð Þ‐gR⋅cos φ2ð Þ⋅sin m2ð Þ⋅sin tð Þ

gR⋅sin φ2ð Þ⋅sin m2ð Þ

24

35

∂rgt=∂t ¼‐sin tð Þ⋅ xO2 þ m1⋅sin φ2ð Þ þ gR⋅cos φ2ð Þ⋅cos m2ð Þð Þ‐gR⋅cos tð Þ⋅sin m2ð Þcos tð Þ⋅ xO2 þ m1⋅sin φ2ð Þ þ gR⋅cos φ2ð Þ⋅cos m2ð Þð Þ‐gR⋅sin m2ð Þ⋅sin tð Þ

P= 2⋅πð Þ

24

35

It can be simply denoted as

P⋅cos m2ð Þ⋅sin φ2ð Þ= 2⋅πð Þ‐xO2⋅sin m2ð Þ‐m1⋅sin φ2ð Þ⋅sin m2ð Þ ¼ 0 ð7Þ

or

m1 ¼ P⋅cot m2ð Þ= 2πð Þ‐xO2=sin φ2ð Þ ð8Þ

Now, the envelope equation derived from the grindingprocess is expressed as Eq. (7), where m1∈[0, gb] andm2∈[0, 2π]. Substituting Eq. (8) and t=0 into Eq. (6), the

Fig. 3 Transformationprocedures of wheel coordinatesystems for flat clearancemachining

Fig. 4 Transformation procedures of wheel coordinate system for eccen-tric clearance machining


contact line (i.e., line p1p2) of the wheel and the clearance atthe initial processing position is obtained in the form

rgt t¼0ð Þ m2ð Þ ¼xgt t¼0ð Þygt t¼0ð Þzgt t¼0ð Þ

24

35

¼P⋅sin φ2ð Þ ⋅ cot m2ð Þð Þ=2π þ gR ⋅ cos m2ð Þ⋅cos φ2ð Þ

gR ⋅ sin m2ð Þzo2 þ cos φ2ð Þ⋅ P ⋅ cot m2ð Þ=2π‐xo2=sin φ2ð Þð Þ‐gR⋅cos m2ð Þ⋅sin φ2ð Þ

24

35

ð9Þ

Finally, the coordinates of p1 (xp1,yp1,zp1) could be calcu-lated by solving the equation x2gt(t=0)+y

2gt(t=0)=mR

2. In orderto make the envelope surface boundary coincide with thecutting edge, the point p1 must be moved to point p4 at thebeginning (i.e., t=0 and point p4 coincides with point Q).Therefore, before the machining process, the wheel should

rotate and moves along the Z-axis from the frame XgYgZg (asshown in Fig. 4) with an angle λ and a displacement |zp1|.Where, λ is the angle between X-axis and the projection line ofOp1 on the cross section of end mill, which can be shown as

λ ¼ cos‐1 xp1⋅mR� �

= mR⋅ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2p1þ y2

p1

� �r� �� ð10Þ

(a) Eccentric clearance machining process

(b) Contact lines of eccentric clearance machining

Fig. 5 Eccentric clearance machining mechanism. a Eccentric clearance machining process. b Contact lines of eccentric clearance machining

Table 1 Process parameters of eccentric clearance manufacturing

mr P gR gb φ2 β

10 mm 60 mm 75 mm 5 mm 10.47° 46.3°

Table 2 Points on the contact line

Points Point coordinates Variable

x y z m2(°)

pt1 9.8765 −1.555 −0.7578 181.188

pt2 9.75 −1.5573 −1.4431 181.1898

pt3 9.6239 −1.5597 −2.1263 181.1926

pt4 9.4982 −1.5621 −2.8074 181.1934

pt5 9.3728 −1.5644 −3.4865 181.1952

pt6 9.2478 −1.5668 −4.1636 181.1970

pt7 9.1233 −1.5691 −4.8386 181.1988


4.3 Definition of the inclination angle

Substituting parameters listed in Table 1 into Eq. (9), thepoints on the contact line could be calculated. The results areshown in Table 2. Figure 6 shows the positions of points in athree-dimensional coordinates. Connecting the first and lastpoint (pt1 and pt7) with a straight line, it is found that all of theother points lie in the line, which means that the contact line isalso a straight line.

According to Fig. 5b, the relationship between φ2 andα0 is expressed in Figs. 7, and the following equations areconcluded:

CD ¼ BD ⋅ tanφAD ¼ BD ⋅ tanβtanα ¼ CD=AD

9=;⇒tanφ ¼ tanα⋅tanβ

from which it can be easily shown that

φ2 ¼ tan‐1 tan α0ð Þ⋅tan βð Þð Þ ð11Þ

This equation shows that the inclination angle φ2 onlydepends on the helical angle β and the clearance angle α0.

4.4 Noninterference conditions

It is obvious that only when the angle Φ’ is smaller thanΦwillclearance be machined without interference. From Fig. 5b, welearn that the angle Φ’ is produced by line p1p2 and line p2p3on the cross section of end mill. So, the calculation of theangle Φ’ is divided into two parts in the next studies.

① Analysis of surface produced by line p1p2Substituting zgt=0 into Eq. (6), a relationship can be

deduced as

t ¼ f 1 m1;m2ð Þ ð12Þ

Now, by substituting Eqs. (8) and (12) into Eq. (6),equation of the machined contour (p2p5 in Fig. 5b) pro-duced by line p1p2 on the cross section of end mill isobtained. Then, the coordinates (xp5,yp5,0) of point p5could be calculated through solving equations of ma-chined and end mill contours.

② Analysis of surface produced by line p2p3Substituting m1=0 into Eq. (6), the helical surface

generated by the wheel circumference moving along thecutting edge is obtained in the form

Fig. 6 Points on the contact line pt1pt2

Fig. 7 Relationship between theclearance angle (α0) and theinstall angle (φ2)


rgt m1¼0ð Þ m2; tð Þ ¼xgt m1¼0ð Þygt m1¼0ð Þzgt m1¼0ð Þ

24

35

¼xO2 þ gR⋅cos φ2ð Þ⋅cos m2ð Þð Þ⋅cos tð Þ‐gR⋅sin m2ð Þ⋅sin tð Þ

gR⋅sin m2ð Þ⋅sin tð Þ þ gR⋅sin m2ð Þ⋅cos tð ÞzO2‐gR⋅cos m2ð Þ⋅sin φ2ð Þ þ P⋅tð Þ= 2⋅πð Þ

24

35

ð13ÞBy solving the equation zgt(m1=0)=0, a function t=

f2(m2) can be derived. Substituting it into Eq. (13), equa-tion of the machined contour produced by line p2p3 on thecross section of end mill is obtained. Now, the coordinates(xp3,yp3,0) of point p3 are deduced.

Based on the coordinates of point p5 and p3, the angleΦ’ can be expressed as

ϕ0 ¼ cos‐1 xp3 ⋅ xp5 þ yp3 ⋅ yp5

� �. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2p3 þ y2p3

q⋅

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2p5 þ y2p5

q� ��

③ Simplification algorithmThe approach mentioned above could obtain a precise

value of Φ’, but the calculate process is very complicated.As the angleΦ’ do not influence the clearance geometry, a

rapid evaluation method whose solution may have a per-missible error will be more utility in practice.

Based on Fig. 7, the machined contour produced byline p1p2 on the cross section of end mill can be expressedas a segment of circular arc SAD, which is defined as

sAD ¼ BD⋅tanβ ¼ gb⋅cos φ2ð Þ⋅tanβ

According to Fig. 5b, the machined contour producedby line p2p3 can be approximately represented in the form

sp2p3≈

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigb ⋅ sin φ2ð Þð Þ2 þ yp1

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimR2 − mR − gb ⋅ sin φ2ð Þð Þ2

q� �2s

Now, the angle Φ’ can be given by

ϕ0≈180⋅ sAD þ sp2p3

� �= mR⋅πð Þ ð14Þ

Considering that mR, β, and φ2 are parameters of endmill which could not be changed, the wheel thickness gbis the only variable to reach the noninterference condition:Φ’<Φ.

Fig. 8 Simulation of clearance machining process

Fig. 9 Initial position of wheel for flat clearance machining

Fig. 10 Simulation result of flat clearance machining

Fig. 11 Initial position of wheel for eccentric clearance machining


5 Convex clearance machining process

As shown in Fig. 1c, the wheel orientation used tomachine theconvex clearance is easily obtained in the form

n3 ¼ 0; 0; 1½ �T

and origin of the wheel frame is located at coordinates(xo3,yo3,zo3)

xO3 ¼ gR ⋅ cos α0ð Þ þ mRyO3 ¼ gR ⋅ sin α0ð ÞzO3 ¼ 0

8<: ð15Þ

6 Examples

In order to verify themathematical models established above, aclearance grinding simulation system is established by the UGsecondary development technology. In the simulation, just likethe actual grinding process, the wheel moves along a helicalline (i.e., the helical cutting edge) step by step. Each step takesa Boolean subtraction operation: wheel as the “cutter” andblank as the “objective”. Then, the clearance model could beestablished by these Boolean operations (Fig. 8).

End mills with parameters of mR=10 m, p=60 mm, andα0=10° are taken as examples to verify the mathematicalmodels. The wheels used in the examples have a radius of75 mm.

(a) gb=15mm, n2=[0.1817,0,0.9834]T, coordinates of O2 are (81.4675,0,-28.3762)

(b) gb=5mm, n2=[0.1817,0,0.9834]T, coordinates of O2 are (82.8470,0,-18.5288)

Fig. 12 Simulation result ofeccentric clearance machining. agb=15 mm,n2=[0.1817,0,0.9834]T,coordinates of O2 are(81.4675,0,–28.3762). b gb=5 mm, n2=[0.1817,0,0.9834]

T,coordinates of O2 are(82.8470,0,–18.5288)

Fig. 13 Initial position of wheel for convex clearance machining Fig. 14 Simulation result of convex clearance machining


6.1 Example of flat clearance machining

Substituting φ1=2° into Eqs. (1) and (2), the wheel orientationcould be calculated as n1=[0.98,0.17,0.03]

T, and coordinatesof O1 are (12.58,0.45,–74.95). The simulation process isshown in Fig. 9 and the result is presented in Fig. 10.

6.2 Example of eccentric clearance machining

Based on Eq. (11), the inclination angle (φ2) of wheel used tomachine the eccentric clearance can be deduced as 10.4675°.Fig. 11 provides the simulation process.

By substituting gb=15 mm into Eqs. (3) and (4), the wheelp o s i t i o n p a r am e t e r s c o u l d b e c a l c u l a t e d a sn2= [0.1817,0,0.9834]T and coordinates of O2 are(81.4675,0,–28.3762). According to Eq. (14), the angle Φ’(Fig. 5b) can be calculated as 142.53°. The simulation resultis shown in Fig. 12a. Obviously, interference will be occurredwhen Φ<Φ’ (i.e., end mills with more than four teeth could notbe machined correctly). Besides, the cutting edge is over-cutted, for it is inconsistent with the envelope surface boundary.

Based on Eq. (9) and (10), coordinates of p1 are deduced as(9.8784,–1.5550,–0.7476) and λ is equal to 8.9458°. Rotatingand moving the wheel along the Z-axis from frame XgYgZg(as shown in Fig. 4) with 8.9458° and 0.7476 mm, andreducing gb from 15 to 5 mm, a new simulation result canbe obtained (Fig. 12b). Now, a clearance could be groundproperly as long as the adjacent cutter angle Φ is larger than62.39° (because Φ’=62.39°).

6.3 Examples of convex clearance machining

Based on Eq. (15), coordinates of O3 can be calculated as(81.4675,0,–28.3762). The simulation process and result areshown in Fig. 13 and Fig. 14.

7 Conclusions

Based on the practical processes analysis, mathematicalmodels for machining flat, eccentric, and convex shapes ofclearance are established.Wheel orientations and locations arecalculated and conditions to machine a desired clearance arediscussed. Furthermore, a simulation system is developed toverify the mathematical models and visualize the machiningprocess. The main conclusions are summarized as follows:

1. Examples show that mathematical models of clearancemachining have been derived correctly.

2. For eccentric clearance machining, undercutting problemson the adjacent cutting edge can be solved by reducing thethickness of wheel. Furthermore, to machine a desiredeccentric clearance, the envelope surface boundary should

be coincidedwith the cutting edge,which leads to additionaltransformations for the wheel before starting the process.

3. Clearances obtained from the simulation program aresolid models, which could be used as an input model forFEM analysis.

Acknowledgment It is a project supported by the Important NationalScience & Technology Specific Projects (No.2012ZX04003-021) and“Taishan Scholar Program Foundation of Shandong”.

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