Angle and frequency domain force models for a roughing end mill with a sinusoidal edge profile

International Journal of Machine Tools & Manufacture 43 (2003) 1509–1520

Angle and frequency domain force models for a roughing end millwith a sinusoidal edge profile

J.-J. Junz Wang∗, C.S. YangDepartment of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan

Received 5 December 2002; accepted 10 June 2003

Abstract

This paper presents analytical force models for a cylindrical roughing end mill with a sinusoidal edge profile in both the angleand frequency domains. Starting from a general expression for the chip thickness model, it is shown that under normal feedconditions, there exists only one cutting point at any axial position for anN-flute roughing end mill with its chip thicknessN timesthat of a regular end mill, while the effective axial depth of cut is only 1/Nth that of a regular end mill. Based on the chip loadmodel, the analytical force model is subsequently established through convolution integration of the elemental cutting function withthe cutting edge geometry function in the angular domain, followed by Fourier analysis to obtain the frequency domain force model.Distinctive features of the milling forces for a roughing end mill are illustrated and compared with a regular end mill in thefrequency as well as in the angular domain. The effects of the geometric parameters of a roughing end mill on the chip loaddistribution and on the features of milling force are discussed. The force models in both the frequency and angular domains arefinally verified through milling experiments. 2003 Elsevier Ltd. All rights reserved.

Keywords: End milling; Roughing end mill; Sinusoidal cutting edge; Chip load; Milling force

1. Introduction

Numerous researches have been devoted to the mode-ling and analysis of the end milling process; however,most of these studies concentrated on the solid helicalend mills or the finishing end mills[1,2]. Although heli-cal roughing end mills with sinusoidal or square cuttingedge profile are in common use, there exist few pub-lished works on the analysis and force modeling forthese types of cutters. Wang et al.[3] presented anumerical chip load and force model for a cylindricalend mill with sinusoidal cutting edges. It was shown foran N-flute roughing cutter that the chip thickness islarger than, and at mostN times, that of the same endmill with continuous linear cutting edges; the thickerchip thickness results in lower cutting constants andlower cutting energy by as much as 23%. In lack of an

∗ Corresponding author. Tel.:+886-6-2757575; fax: +886-6-2367231.

E-mail address: [email protected] (J.-J. Junz Wang).

0890-6955/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0890-6955(03)00163-9

analytical force model, however, the effects of cuttergeometric parameters on the chip load kinematics andmilling force characteristics were not formulated andsystematically analyzed. Through numerical timedomain simulation of the chip load kinematics in dynam-ics milling, Merdol and Altintas[4] predicted the forceand chatter vibration for a roughing cutter with a cuttingedge represented by a cubic spline. The reduction ofaxial depth of cut and a significant increase of chatterstability was reported. Improvement in machining stab-ility was also observed by Compomanes[5] for a rough-ing cutter with sinusoidal cutting edges.

It is well accepted in the industry as well as by theresearchers that a roughing end mill outperforms a fin-ishing end mill in its higher material removal rate andimproves the machining stability in an end milling oper-ation; however, no analytical work to account for itsmerits has yet been found. Although final dimensionalaccuracy and surface properties of the machined objectsare not the immediate concern of the roughing process,process characteristics of the roughing cutter includingchip form, power consumption, machine/work/fixture

1510 J.-J. Junz Wang, C.S. Yang / International Journal of Machine Tools & Manufacture 43 (2003) 1509–1520

Nomenclature

A amplitude of the sinusoidal cutting edgeAa[k],Ay[k] the kth order Fourier coefficients of the total cutting forcesa helix angle of the cutterb, r, h angular, radial, and axial position variables for cutting points of the cutterba axial immersion angle within axial depth of cutbc immersion angle of the crest cutting regionbp angular spacing between adjacent flutescs cutter sequence functioncwdc chip width density function of the regular cutterCWDc Fourier transform of cwdc

cwdcw chip width density function of the roughing cutterCWDcw Fourier transform of cwdcw

da, dr axial and radial depths of cutfx, fy local cutting forcesf̄x, f̄y total cutting forcesha, hr ba/bp and dr/(2R)hs A/txkt, kr tangential and radial cutting constantsN number of cutting flutesp pitch of the sinusoidal cutting edgepx, py elementary cutting functions in X and Y directionsPx, Py Fourier transforms of px and py

q, q1, q2 cutter position in the work, entry and exit anglesR nominal radius of the roughing cutterRck active cutting region of the kth fluterk, hk radius and axial position of a cutting point on the kth flutetc, t̄c uncut chip thickness, average chip thicknesstck uncut chip thickness of the kth flutets tooth sequence functiontx feed per toothwr, wa radial and axial cutting windowsw, wc normalized frequency and crest passing frequency�h p/N

vibration, and process stability are nevertheless allimportant considerations in planning a roughing processjust as they are in the finish machining process. Theseprocess characteristics all originate from the same basicsof the milling process: the chip load kinematics and theaccompanying force generation. It is therefore the pur-pose of this paper to present the chip load kinematics aswell as an analytic force model for the roughing endmill. In the next section, the chip thickness equation isfirst derived. Built on this chip load model, the angulardomain milling force model is then established in Sec-tion 3 through the convolution integral approach andthen analyzed in the frequency domain in Section 4.Based on the frequency domain force model, a formulais derived in Section 4 for the identification of cuttingconstants from the average forces. The analytical forcemodel is finally verified through milling experiments.

2. Chip load kinematics

The wavy roughing end mill cutter described herein isassumed to be comprised of a cutter body having a helicalthread of sinusoidal profile with a large helix angle as wellas having the conventional N helical gashes with a smallerconstant helix angle, a, separating the cutting flutes. Theexternal profile of the cutter is shown in Fig. 1(a and b)along with the coordinate systems representing the workcoordinate in X–Y–Z and the cutter coordinate in r–b–h.

In the cutter coordinate system, the axial position of thecutting edge on the kth flute for an N-flute end mill, as seenin the unfolded view of Fig. 1(c), can be described as

hk(b) �R

tana(b�(k�1)bp); b�[(k�1)bp, (1)

(k�1)bp � ba]

1511J.-J. Junz Wang, C.S. Yang / International Journal of Machine Tools & Manufacture 43 (2003) 1509–1520

Fig. 1. (a) Top view and (b) front view of the cutter and the coordi-nate systems. (c) The helical cutting edge function, hk(b). (d) Theradius function with sinusoidal cutting edge, rk(b) for a four-fluteroughing end mill.

where ba = (da /R)tan a and bp = 2π/Nba and bp are theaxial immersion angle and the flute spacing angle,respectively. The radius of a cutting point on this kthflute as a function of its axial position, h, can be shownto be

rk(h) � R�Asin�2πhp

�(k�1)bp�; h�[0,da] (2)

where p is the wavelength of the sinusoidal edge profile.Fig. 1(d) illustrates the radius functions of the wavy cut-ting edges for a four-flute cutter.

Substituting Eq. (1), Eq. (2) can be expressed as afunction of b in

rk(b) � R�Asin� 2πRptana

(b�(k�1)bp) (3)

�(k�1)bp�; b�[(k�1)bp, (k�1)bp � ba]

The uncut chip thickness for a cutting point on a reg-ular true end mill is commonly represented by: [6]

tc(q) � txsinq (4)

This equation is applicable when all flutes have the samecutting edge radius and each cutting point always cutsover the surface left by the previous flute. For a cutterwith wave shape edge, the effective radii vary amongcutting points, and the chip thickness of a cutting pointat (bk, rk, h) depend not only on the radius of the cuttingpoint at (bk�l , rk�l , h) but also on the radii of other cuttingpoints at the same axial position, h. Thereby the

expression in Eq. (4) needs to be modified to reflect thechip thickness variation among different cutting pointsas a result of the uneven cutting point radii. The analysisof actual uncut chip thickness under this circumstancecan be treated in a similar way to the analysis of chipload for a cutter with radial offset [7]. The chip thicknessat any cutting point (rk, bk, h) on the kth flute can bemathematically expressed as the minimum positive poss-ible chip thickness that can be formed between this cut-ting point and other cutting points on the previous flutes,i.e. points at (bk�m, rk�m, h) with m = 1 to N. Assumingthe work surface has been machined through severalpasses and its surface is completely engaged in the cut-ting, a general expression for the chip thickness cut bythe kth flute with uneven radii can be written as [7]

tck(q,h) � minm � 1 to N

{tck,m(q,h)|tck,m(q,h) � mtxsinq (5)

� rk(h)�rk�m(h)}

where tck,m(q, h) represents the possible uncut chip thick-ness formed between cutter points at (bk, rk, h) on thekth flute and a cutting point at (bk�m , rk�m , h) on the(k�m)th flute at a cutting position of q; and tck,m(q, h)is set to zero if it is negative. Substituting Eq. (3) intoEq. (5) and through numerical computation, it can beshown for a common sinusoidal wave geometry undernominal feed per tooth that the chip thickness expressionin Eq. (5) can be simplified and approximated by

tck(q,b) � �Ntxsinq for b�Rck

0 otherwise(6)

where Rck is the engaged cutting regions for the kth flutein the b coordinate. The validity of Eq. (6) solelydepends on the dimensionless sinusoidal amplitude tofeed per tooth ratio, hs = A / tx. The value of hs requiredfor Eq. (6) to be applicable is dependent on the numberof the flute and the ratio of the radial depth of cut to thecutter diameter, hr, or the normalized radial depth of cut.The minimum feed ratios which result in less than 2%chip load error by using Eq. (6) are found throughnumerical computation and are plotted in Fig. 2 for N= 2, 3 and 4 in a normal up or down cut configurationfor the radial depth of cut up to the cutter radius. For aradial depth of cut larger than the cutter radius, the con-dition for hr = 0.5 shall apply. Fig. 2 shows that, for alarger radial depth of cut and a larger number of cutterflutes, the feed ratio for which Eq. (6) is applicable needsto be larger, indicating a smaller feed will be required.For the typical edge geometry of a roughing end millwith A = 0.5–1 mm, the recommended feed per toothfrom the tool manufacturers should satisfy the constraintimposed in Fig. 2 for most cutting situations; thereforeEq. (6) can be appropriately used as the chip load modelfor the roughing end mill.

The engaged cutting regions, Rck, appearing in Eq. (6)


Fig. 2. The minimum feed ratio, hs, as function of the normalizedradial depth of cut, hr, required to satisfy the chip thickness Eq. (6)for cutters of different flute number.

are shown in Fig. 3 as the shaded regions for a slot mill-ing condition for a typical four-flute roughing end millwith R = 5 mm, a = 30°, and the sinusoidal geometryof A = 1 mm and p = 1 mm. The undulation amplitudeto feed ratio, hs, has to be larger than 10 as required bythe constraints shown in Fig. 2. Cutting points withinthis cutting region cut over the surface left by themselvesin the previous revolution, thus m = N in Eq. (5), andthe chip thickness is N times that of a regular end mill.The horizontal sections of these cutting regions in Fig.

Fig. 3. Cutting regions for a four-flute roughing cutter. (a), (b), (c) and (d) are for flute number k = 1, 2, 3, and 4, respectively. Cutter geometry:N = 4, R = 5mm, a = 30°, A = 1 mm, p = 1 mm, tx � 0.1 mm.

3 correspond to the crest portions of each kth cuttingedge, which is shown in Fig. 1(d) as the upper non-inter-secting edge portion of each cutter flute. The verticalaxis indicates their angular range of cutter engagementin the work. The horizontal axes of these four figuresshow the b coordinate for each flute of the four-flutecutter. Although their coordinate values have a shift of90° for each consecutive flute, they correspond to thesame axial position in h, starting from the bottom of thecutter at h = 0. It is shown that the axial positions corre-sponding to the b coordinates of the cutting regions foreach flute do not overlap with those of other flutes.Therefore, at any axial position, there exists only onecutting point located on one of the cutting flutes. Theperiod of the sinusoidal edge in b can be shown to bep tan a/R and the width of each crest cutting region, bc,is 1/N of this period with

bc �ptana

NR(7)

The width of crest window is therefore proportional tothe pitch of the sinusoidal edge and inversely pro-portional to the number of the flutes and the correspond-ing chip width in the axial direction is simply p/N. Forthe roughing end mill in Fig. 3, the width of the crestwindow is 1.6° and the chip width is only 0.25 mm.Therefore, for each cutting flute of a roughing end millunder a normal feed rate, only 1/Nth of the axial lengthis engaged in the cutting, while the chip thickness is Ntimes that of a regular end mill, thus having the same


amount of material removed per cutter revolution as aregular end mill. Unlike the regular end mill, each fluteof which will generate a single piece of chip having thewidth of the axial depth of cut, the roughing end mill isshown to generate multiple separated, narrower andthicker chips, resulting in better chip evacuation in theroughing operation.

Although a specific roughing end mill has been chosenin the illustration in this and the following section, thechip load analysis presented above is analytical and canbe readily extended for a general cylindrical roughingend mill with an external profile represented by R, N anda, and with a sinusoidal cutting edge of amplitude, A,and pitch, p. Based on the chip load model presented inEq. (6), the analytical milling force model for the rough-ing end mill will be derived next.

3. Analytical force model for a roughing end millwith sinusoidal edge profile

Since the geometry of a roughing end mill only variesslightly from that of a regular end mill, force models ofboth types of cutter are expected to share some commoncharacteristics in their composed structures of the mill-ing forces. A convolution milling force model for a reg-ular end mill has been presented in Ref. [8], where thetotal milling forces were shown to be composed as theconvolution integral of three process functions: the toothsequence function, ts, the chip width density function,cwd, and the elementary cutting functions, [px, py]T;that is,

�f̄x(f)

f̄y(f)� � kttx·ts∗cwd∗�px

py� (8)

where ∗ indicates the convolution integration and thethree process functions were shown to be:

ts(f) � ��k � ��

d(f�kbp) (9)

cwd(b) �dh1

db� � R

tanab�[0, ba]

0 otherwise

(10)

and

�px(q)

py(q)� � �1 kr

�kr 1��sinq cosq

sinq2 �wr(q) (11)

kt in Eq. (8) is the tangential cutting coefficient andkr in Eq. (11) is the radial cutting coefficient representedby the ratio of the local radial cutting force to the tangen-

tial force. Both cutting coefficients are assumed to beconstant and independent of the chip thickness variationfor a given set of cutting condition. wr(q) in Eq. (11)is the radial cutting window function representing thesweeping range of the cutting point in the work definedby the entry and exit angles, q1 and q2, so that

wr(q) � �1 q�[q1,q2]

0 otherwise(12)

The dimensionless elementary cutting functions in Eq.(11) specify the trajectories of the normalized local cut-ting forces in the X and Y directions as a function of thecutting point position in the work; the chip thicknessvariation in Eq. (4), the effect of radial cutting constantas well as the radial sweeping range are all taken intoaccount in this elementary cutting vector of the millingprocess. cwd in Eq. (10) physically corresponds to theeffective chip width generated per unit radial rotation ofthe cutting edge within the axial immersion angle, ba,and is mathematically equivalent to the helix lead of thecutter, h�(b). The periodic tooth sequence function in Eq.(9) signifies the cutting sequence of the periodic cuttingflutes and their flute spacing. The same convolutionstructure of the force generating process shall still applyfor the roughing end mill since the integrating mech-anism involved in the composition of the total forcesremains the same. These process functions, however,need to be modified to reflect the differences in the chipload distribution owing to the presence of crest cuttingedges discussed in the preceding section.

For a roughing end mill, the force trajectory for apoint within the crest cutting region is similar to that ofa regular end mill except that the feed becomes N timeslarger; hence, the local tangential and radial forces perunit chip width can be written as

�ft(q)

fr(q)� � �kttc

kr ft� � �kt(Ntxsinq)

kr ft�wr(q) (13)

where the chip thickness, tc, is substituted by Ntxsin qwr(q), the actual chip thickness seen by the cuttingpoints. Through coordinate transformation, these twolocal force components can be expressed in the X–Ycoordinates as

�fx(q)

fy(q)� � �cosq �sinq

sinq cosq��ft(q)

fr(q)� � kt(Ntx)�px(q)

py(q)� (14)

where

�px(q)

py(q)� � �1 kr

�kr 1��sinqcosq

sinq2 �wr(q) (15)

are the dimensionless elementary cutting functions andare the same as those of a regular end mill in Eq. (11),while the local forces in Eq. (14) are N times larger sincethe feed per tooth at the crest cutting region hasbecome Ntx.


The chip width density function for a roughing endmill is perceivably more complicated than that of a reg-ular end mill shown in Eq. (10). The axial functions ofboth a regular and a roughing end mill are superimposedand illustrated in Fig. 4(a) for a four-flute cutter. Thesame roughing end mill as in Fig. 3 is used. For theregular end mill, represented as thin lines in the figure,the cwdk function for the kth flute is defined by the rec-tangular periodic function:

cwdk(b) �dhk

db(16)

� � Rtana

b�[(k�1)bp,(k�1)bp � ba]

0 otherwise

The magnitude of the cwd function is geometrically theslope of the cutting edge in Fig. 4(c) and only dependson two geometric parameters, R and a. The cwdk func-tions are only defined within a range of axial immersionangle, ba, and are separated from each other by the flutespacing angle, bp. The complete chip width density func-tion of a regular cutter with N flutes as shown in Fig.4(c) is mathematically the sum of each respective cwdi

function:

Fig. 4. (a) The cutting edge function with crest cutting region in thicklines. (b) The axial cutting window function for the roughing cutter.(c) The chip width density function for the regular end mill. (d) Thechip width density function for the roughing end mill.

cwdc(b) � �Nk � 1

cwdk(b) (17)

For the roughing end mill, the axial position functionsof the crest cutting regions are shown in Fig. 4(a) insegments of thick lines. As discussed earlier, thelocations of these crest regions possess a deterministicperiodic pattern and therefore can be described by ananalytical expression. Fig. 4(b) illustrates the projectionof these crest cutting regions onto the b axis, showingthat the crest regions in the b coordinate can be rep-resented by a sequence of narrow rectangular unit win-dow functions. The basic building block of thesesequence functions is a short unit window functiondefined by

u(b) � �1 b�[0,bc]

0 otherwise(18)

The axial crest window function for the first flute shownin Fig. 4(b) can therefore be represented by a periodicsequence of this unit window function with a periodof Nbc

wa1(b) � � ��m � 0

u(b � mNbc), b�[0,ba]

0 otherwise

(19)

Like the chip width density function in Eq. (16), Eq.(19) is only defined within the range of the axial immer-sion angle. The axial crest window function of the suc-ceeding kth flute, as shown in Fig. 4(d), is similar to Eq.(19), but shifted by a phase lag of (k�1)(bc + bp), that is

wak(b) � wa1(b�(k�1)(bp � bc)) (20)

and the complete crest window function for the N-flutecutter is therefore

wa(b) � �Nk � 1

wak(b) (21)

Combining Eqs. (17) and (21), the chip width densityfunction of a roughing end mill is therefore,

cwdcw(b) � cwdc(b)wa(b) (22)

which is the modulation of the cwdc function of a regularend mill by the crest cutting windows of the roughingend mill.

Based on the definition of the cwdk function in Eq.(16), the actual chip width density of a cutting point onthe roughing end mill will be slightly different from thatof a regular end mill, since its radius function varies withh and b as shown in Fig. 1(d). However, for most rough-ing cutters, the amplitude of the sinusoidal wave is muchsmaller than the nominal radius of the cutter body so that


variation of radius among cutting points can be ignored.Hence, the chip width density values for both types ofcutters can be treated the same.

The third process function, termed as the cuttersequence function, signifies the rotation and the periodicengagement of cutter and is represented by a periodicimpulse function with a period of 2π

cs(f) � ��k � ��

d(f�2kπ) (23)

Following the same convolution structure of millingforces for a regular end mill as expressed in Eq. (10),the total milling force of a roughing end mill is therefore

�f̄x(f)

f̄y(f)� � cs∗cwdcw∗�fx

fy� (24)

� kt(Ntx)·cs∗(cwdc·wa)∗�px

py�

It can be shown that the ts∗cwd term in Eq. (8), whichrepresents the periodic chip width density function of therotating cutter, is equivalent to the cs∗cwdc and Eq. (8)for the milling forces of a regular end mill can also bewritten as

�f̄x(f)

f̄y(f)� � kttx·cs∗cwdc∗�px

py� (25)

Comparing Eq. (24) with Eq. (25) shows that the mill-ing force expressions for the two types of cutters possessthe same structure and similar process functions, but dif-fer only in two aspects: (1) the feed per tooth is N timeslarger for the roughing end mill, and (2) the chip widthdensity function of a regular end mill, cwdc, is modu-lated by the unit crest window function, resulting in afinely dispersed chip width density function, cwdcw, forthe roughing end mill.

Numerical implementation of the convolution inte-gration can be easily carried out with many generalsoftware packages such as Matlab. The formation of thetotal milling force through convolution integral of Eqs.(24) and (25) is illustrated in Fig. 5 for the X and Y forcecomponents of a down milling process. The elementarycutting functions are shown in Fig. 5(a) along with theradial cutting window, wr. Using the X force as theexample, the total X force at any instant f is the sum ofthe product of the two functions px and cwdw within theradial cutting window. As the cutter rotates, the cwdw

function, shown in reversed horizontal axis, moves tothe right and the crest cutting region represented by therectangular pulses enter and leave the radial cutting win-dow in a prescribed sequence, generating a pulsing pat-tern in the total cutting force. As shown in Fig. 5(c andd), the separated crest cutting edges in Fig. 5(b) generatelocal force spikes superimposed on the familiar periodicforce pulsation of a regular end mill. These local spikes

Fig. 5. Illustration of the convolution integration of the elementarycutting function with the chip width density functions for a regularand a roughing end mills. (a) The elementary cutting functions. (b)The chip width density functions for the regular cutter (dashed line)and the roughing cutter (shaded). (c) The total X force as the outputof the convolution process and (d) the total Y force. (Same cutter asin Fig. 3. kt = 2588 N/mm2, kr = 0.46, da=6.5 mm, dr=0.5 mm, tx =0.0516 mm/tooth, down cut.)

result from the frequent entry and exit cuts of the localcrest cutting edges. For comparison purpose, the X andY force components of a regular end mill are also shownin Fig. 5(c and d) assuming the same cutting conditions,cutting constants, and external cutter geometry but with-out the sinusoidal cutting edge. The pulsations due to theperiodic engagement and disengagement of each regularcutting flute follow the same trend as the roughing endmill, however, without the short spikes caused by thecrest cutting edges.

4. Frequency domain analysis of the milling forces

The characteristics of the periodic milling forces aremore suitably analyzed and compared in the frequencydomain. The Fourier transform of the milling force fora roughing end mill in Eq. (24) can be shown to be

�F̄x(w)

F̄y(w)� � (Nkttx)·CS(w)·CWDcw(w)·�Px(w)

Py(w)� (26)

� (Nkttx)· ��

k � �

d(w�k)·CWDcw·�Px

Py�


where CWDcw(w), Px(w) and Py(w) are the Fourier trans-forms of their angular domain functions. The Fouriertransform of the periodic impulse function, cs, is also aperiodic impulse function with a frequency period of 1and has been substituted in the second equality of Eq.(26). The periodic milling forces can also be representedin a Fourier series expansion form with the Fourier coef-ficient of the kth harmonic being the weighting of theimpulse at w = k in Eq. (26) divided by 2π [9], that is

�Ax[k]

Ay[k]� �

kttx2π

N·CWDcw(k)�Px(k)

Py(k)� (27)

Similarly, the Fourier coefficients of the milling forcefor a regular end mill can be shown from Eq. (25) to be

�Ax[k]

Ay[k]� �

kttx2π

CWDc(k)�Px(k)

Py(k)� (28)

Like in the angular domain, a comparison of Eqs. (27)and (28) shows that the main differences between theforce spectra for the two types of end mills lie in the Nfactor and chip width density functions: N·CWDcw in Eq.(27) and CWDc in Eq. (28). Since these two functionshave a multiplicative effect, the differences in the totalforce spectra can be investigated through the frequencyspectra of these two chip width density functions. Thespectra of N·CWDcw and CWDc for the four-flute rough-ing and regular end mills are shown in Fig. 6 for fourdifferent axial depths of cut with ha = ba /bp = 0.25,0.5, 0.75, and 1.0. Although these frequency spectrashould be sampled at each discrete normalized frequencywhere w = k, as required by Eqs. (27) and (28), they areshown for the sake of clarity in the continuous frequencyvariable, w. These plots clearly indicated that the regularend mill has significant force components at the funda-mental tooth-passing frequency, w = N, and at the har-monic frequencies where w = Nk, while the roughingend mill has significant frequency contents at a widerrange of discrete frequencies where w = k, with its nor-malized fundamental frequency being w = 1, the spindlefrequency. There exists a strong presence of forces forthe roughing cutter at the higher frequency regions,reflecting the short spikes of forces in the angulardomain. As a result of the modulation effect of the crestwindow function, these higher frequency forces appearat multiples of the fundamental frequency of the periodiccrest window function, wc = 2π/ (Nbc) = 2πR / (Ptan a), which is 56.3 for the same cutter as in Fig. 3.This frequency can be called the crest passing frequencyin contrast to the tooth passing frequency of a regularend mill. As the pitch of the sinusoid undulation getslarger, the crest window becomes wider and the crestpassing frequency will decrease. Around the multiplesof the crest passing frequency at w = 0, wc, and 2wc areshown to have significant presence of side peaks, whichare the scaled-down replicas of the force spectra of a

Fig. 6. Spectra of N·CWDcw for the roughing cutter and CWDc forthe regular cutter. (a) ha=0.25, (b) ha=0.5, (c) ha=0.75, and (d) ha=1.

regular end mill. The effect of the tooth passing fre-quency on the roughing end mill therefore shows up asthe side bands around the multiples of the crest passingfrequency. The frequency interval between the peaks isequal to the tooth passing frequency, N. At w = 0, theDC components of the force spectra for both types ofcutter are shown to be the same, relating to the fact thatthe same amount of material is removed for both cutters.Under the special case of ba = bp as in Fig. 6(d), thecwdc function in Fig. 4(c) for a regular cutter becomesa constant function so that CWDc is nonzero only at w= 0. Therefore, the dynamic component of the millingforce will vanish and only the average force remains aspresented in [8]. The spectrum of a roughing end mill,however, shows that significant presence of dynamicforces at frequencies w = kwc might still exist.

As indicated by Eqs. (27) and (28), the spectra of thetotal milling forces are the products of the spectra of theelementary cutting functions and the chip width densityfunction. It has been shown in [8] that the spectra of Px


and Py decrease in w�1. Therefore, the high peaks of theCWDcw function in the higher frequency regions due tothe narrow crest cutting window will be attenuated bythe elementary cutting function, and the resulting forcespectra of the total milling forces in the higher frequencyregions will not be as strong as those of the CWDca

shown in Fig. 6. From the angle domain viewpoint, thehigh frequency contents of the periodic short pulses inthe cwdca function are smoothed out by the wider radialcutting window in the convolution integration process asillustrated in Fig. 5.

5. Average forces and cutting constantsidentification

The analytical expression for the average forces canbe found from Eq. (27) by letting k = 0:

�Ax[0]

Ay[0]� �

kttx

2πN·CWDcw(0)�Px(0)

Py(0)� (29)

where CWDcw(0), Px(0) and Py(0) are the DC values ofthe Fourier transforms and are geometrically the signedarea of their respective angular domain functions. Theycan be found to be

CWDcw(0) � da (30)

and

�Px(0)

Py(0)� � �1 kr

�kr 1��D

E� (31)

where D �14

(cos2q1�cos2q2) (32)

and E �12

(q2�q1) �14(sin2q1�sin2q2)

Substituting Eqs. (30)–(32), Eq. (29) becomes

�Ax[0]

Ay[0]� �

Nkttxda

2π �1 kr

�kr 1��D

E� (33)

The expression for the average forces of the regularend mill is similarly found from Eq. (28) with k = 0,

�Ax[0]

Ay[0]� �

kttx

2πCWDc(0)�Px(0)

Py(0)� (34)

�Nkttxda

2π �1 kr

�kr 1��D

E�

where

CWDc(0) � Nda (35)

and Eq. (31) are substituted. Although the averageforces for both types of cutters are the same, Eqs. (30)and (35) show that DC components of their chip width

density functions are N times different. The physical sig-nificance of this DC component lies in the effective axialdepth of cut or the effective chip width of the rotatingcutter. This effective axial depth plays an important rolein determining the stability units of the milling process,which is proportionally decreased with increasing axialdepth of cut [4,10,11]. It is evident that, due to the crestcutting geometry, the roughing cutter takes N times thechip load for the engaged cutting regions while havingonly 1/Nth of the effective axial depth of a regular cutter.Given that all other cutting conditions remain the same,a roughing end mill thus will allow larger axial depth ofcut than a regular end mill and still maintain system stab-ility.

The closed form expression of the milling forcesallows the specific cutting constants of the roughing endmill to be identified from the measured milling forces.This is easily achieved using the averages of the millingforces. By rearranging Eq. (33), the specific cutting con-stants can be found from

�kt

ktkr� �

2πNtxda(D2 � E2)�D E

E �D��Ax[0]

Ay[0]� (36)

This identification formula for the roughing end mill isalso the same as that of a regular end mill presentedin [8].

6. Experimental verification and discussions

Experiments use a four-flute roughing end mill of 10mm diameter and a helix angle of 30°. The sinusoidalroughing thread has a pitch of 1 mm and an amplitudeof 1 mm. The work material is AL2024-T3. Verificationof the force model and analysis is carried out in twostages. Cutting constants are identified in the first stageof experiments for the given pair of roughing cutter andwork, followed by force prediction for milling testsusing different cutting conditions. A series of millingtests are first undertaken using cutting conditions listedin Table 1. The tangential and radial cutting constants foreach cutting condition are identified from the averages oftheir measured X and Y forces through Eq. (36) and areshown in Fig. 7 with respect to the average chip thick-ness. The magnitude of these cutting constants and thetrend of increasing magnitude with decreasing averagechip thickness are similar to those observed of a regularend mill as presented in [8]. These cutting constants arerelated to the average chip thickness and approximatedby the following exponential expressions

kt � 1281(t̄c)�0.237 MPa; kr � 0.089(t̄c)�0.338 (37)

where t̄c is the average chip thickness with

t̄c �2Ntxhr

q2�q1

(38)


Table 1Experimental cutting conditions and average forces for the identifi-cation of cutting constants. Work material: AL2024-T3. Geometry ofthe HSS roughing end mill: N = 4, R = 5 mm, a = 30°, p = 1 mm,A = 1 mm. Cutting depths: da=5 mm, dr=0.5 mm. rpm = 1000. Dry,down cut

No. tx (mm/tooth) t̄c (mm) Ax[0] (N) Ay[0] (N)

1 0.1575 0.1397 �73.78 166.22 0.1050 0.0931 �36.90 117.93 0.0788 0.0699 �27.77 95.104 0.0630 0.0559 �21.59 75.655 0.0525 0.0466 �17.84 65.056 0.0450 0.0399 �15.24 59.657 0.0394 0.0349 �13.34 50.258 0.0350 0.0297 �12.59 45.809 0.0315 0.0279 �11.33 38.06

Fig. 7. Tangential cutting constant, kt, and radial cutting constant, kr,as functions of average chip thickness.

Given the same cutting conditions, the average chipthickness for a roughing cutter is also N times that of aregular end mill owing to the crest cutting effect; this,consequently, leads to a smaller cutting constants andfurther bolsters the capability of a roughing end mill inachieving a more stable milling process with theoreti-cally N times the material removal rate of a regularend mill.

The predictive capability of the analytical force modelis verified by experiments with cutting conditions differ-ent from those of Table 1. The average chip thicknessis first obtained from Eq. (38) and the cutting constantsfrom Eq. (37). The predicted forces in frequency domainare then calculated from Eq. (27) and, from these forcespectra, forces in the angular domain are obtainedthrough the inverse Fourier transform. Both the force

trajectories and spectra are shown in Figs. 8 and 9 toagree well with the measured values. Forces in Fig. 8show more significant spikes than in Fig. 9 due to theshallower radial depth of cut taken for the cutting con-ditions in Fig. 8 with dr = 0.5 mm. The wider radialcutting range dr = 2 mm in Fig. 9 has a stronger smoo-thing effect on the short pulsation of the chip width den-sity function.

Alternatively, the theoretical milling forces can bepredicted starting from the angular domain through Eq.(24) and then converted to the frequency domain by theFourier transform method. Results will be the same start-ing from either the angular or frequency domains sinceboth Eqs. (24) and (27) are simply the transform of theother. Required data processing for the force predictionin both domains including convolution, Fourier trans-form, and inverse Fourier transform are all standard

Fig. 8. Predicted and measured forces. The total forces in angledomain (a and b) and their spectra in normalized frequency domain (cand d). (Same cutter as in Fig. 3. kt=2356 N/mm2, kr=0.43, da=9 mm,dr=0.5 mm, tx=0.0267 mm/tooth, down cut.)


Fig. 9. Predicted and measured forces. The total forces in angledomain (a and b) and their spectra in normalized frequency domain (cand d). (Same cutter as in Fig. 3. kt=2483 N/mm2, kr=0.47, da=6 mm,dr=2 mm, tx=0.0417 mm/tooth, down cut.)

commands available in many general software packages;thus the programming task for predicting the millingforce can be greatly simplified.

7. Conclusions

For a common roughing end mill with sinusoidal cut-ting edges, its special chip load distribution and analyti-cal force models in the angle and frequency domainshave been presented. The two geometric parameters ofthe sinusoidal cutting edge, the wave amplitude and thepitch, play definite and distinct roles in effecting the chipload kinematics of the roughing end mill. It is shown,for an N-flute roughing end mill with feed ratio largerthan a given undulation amplitude to the feed per tooth

ratio, that only the crest portion of the cutting edge isengaged in the cutting and the chip width is 1/Nth ofthe pitch of the sinusoidal wave edge. The total engagedcutting width on each flute is also 1/Nth that of a regularend mill while the chip thickness is increased to N times.This reduction of effective axial depth of cut is of specialsignificance in promoting machining stability andallowing higher apparent axial depth of cut for a rough-ing cutter.

Based on the chip load analysis, analytical force mod-els have been established in the angular domain throughconvolution integration and in the frequency domain byFourier analysis. While the milling forces of a regularend mill are characterized by dynamic forces having har-monic frequencies at w = Nk, the multiples of the toothpassing frequency, roughing end mills have their fre-quency spectra densely distributed around the multiplesof the crest passing frequency at kwc=2πkR/(Ptan a),k=0, 1, 2... with significant presence of side peaks similarto the spectra structure of a regular end mill. These mill-ing forces characteristics of the regular and roughing endmills are illustrated and compared in the frequency aswell as in the angular domain and finally verifiedthrough milling experiments.

Although a specific roughing end mill has been chosenfor analysis and verification throughout this paper, thepresented chip load analysis and force models can bereadily extended for a general cylindrical roughing endmill with an external profile represented by the cutterradius, R, flute number, N, and the helix angle, a, andwith a sinusoidal cutting edge of amplitude, A, andpitch, p.

Acknowledgements

The authors wish to express their gratitude to theNational Science Council of Taiwan for the financialsupport extended to this research through Grant No.NSC-83-0425-E-006-003.

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Documents

Angle and frequency domain force models for a roughing end mill with a sinusoidal edge profile