The research of surface waviness control method for 5-axis ... · waves, which has about 1 –5 mm...

ORIGINAL ARTICLE

The research of surface waviness control method for 5-axisflank milling

Lei Jiang & Elssawi Yahya & Guofu Ding & Minghua Hu &Shengfeng Qin

Received: 14 January 2013 /Accepted: 29 April 2013# Springer-Verlag London 2013

Abstract This paper presents a comprehensive control meth-od of surface waviness for 5-axis flank milling, reducingcutting force variation, and improving unstable rotation stateof rotary axis to suppress low-frequency milling chatter. Wepropose a cutting force-sweep area model and define an aver-age cutting force term, so that the control of cutting force turnsinto sweep area rate (SAR) control, realized by feed rate. Thefeed rate might be dynamic, adjusted to make the correspond-ing SAR constant. In addition, the method can improve theunstable rotation state, such as discontinuous or reciprocatingrotation in micro angle, by means of decomposing the unstablerotation to more micro continuous rotation or tool axis orien-tation adjustments based on the geometric relationship of toolaxis and machining tolerance. Finally, a series of comparisonexperiments were conducted on a 5-axis machine tool. Fromthe comparison of cutting force and surface scan, the resultsshowed that the control method could suppress low-frequencymilling chatter and reduce surface waviness effectively.

Keywords 5-axis flank milling . Surface waviness . Millingchatter . Cutting force . Unstable rotation

1 Introduction

With the development of NC machining technology, 5-axisflank milling process is widely applied in machining parts

with complex sculptured surfaces such as aircraft structuralparts, turbines, and blades with cylindrical or conical tool.Since 5-axis flank milling of developable ruled surfacesdoes not contain geometrical errors, the main research offlank milling focuses on the generation of optimal tooltrajectory for non-developable ruled surfaces, even genericfree-form surfaces [1].

Compared with end milling, the axial depth of flank millingis quite deep. It has higher cutting force, which is prone tomilling chatter. One of the resulting problems is that themachined surface consists of a range of components withwaves, which has about 1–5 mm interval and parallels to toolaxis. From the definition of a typical surface profile, thesurface feature is defined as surface waviness [2], whichaffects surface accuracy and assembly precision. Recently,with the development of precision machining technology, thewaviness problem has become more prominent than before.

The source of surface waviness is the unstable low-frequency milling chatter between tool and workpiece [3].From the relationship analysis of the waviness interval andfeed rate, the frequency range is mainly limited to 10–50 Hz.Therefore, besides determining the tool path and cutting depth,how to select proper milling parameters to reduce millingchatter or improve milling stability is an important issue.Manyresearchers have investigated the machining dynamics for it,involving tool vibration model and cutting force model, etc.

Based on the tool vibration model, some researchers fo-cused on controlling spindle speed to suppress the millingchatter excited by transient cutting force. With Floquet theoryand Fourier series expansion, Minis and Yanushevsky appliedthe Nyquist criterion to obtain the solution of stable millingchatter range [4]. Al-Regib et al. proposed a spindle speedplanning method to suppress milling chatter [5]. Bravo et al.studied a method for identifying chatter stability region formilling systems with similar dynamics parameters [6]. Gagnolet al. produced a chatter–range prediction model and

L. Jiang : E. Yahya :G. Ding (*) :M. HuInstitute of Advanced Design and Manufacturing,Southwest Jiaotong University, Chengdu,People’s Republic of China 610031e-mail: [email protected]

S. QinSchool of Engineering and Design, Brunel University,Uxbridge, UK

Int J Adv Manuf TechnolDOI 10.1007/s00170-013-5041-7

introduced spindle speed frequency response function of thetip point [7]. Budak had some research about the chattersuppression theory of irregular tooth pitch cutter [8]. Ahmadiand Ismail proposed a dynamic model to predict chatter in theflank milling of curved surfaces on a 5-axis machine [9]. Thechatter simulation was also conducted in the time domain andthe entry and exit angles of the tool into workpiece wereresearched analytically to reduce milling chatter. The aboveresearches established the dynamics relationship of millingchatter in high-frequency range (more than 50 Hz) with spin-dle speed, while the high-frequency chatter has major effecton roughness of which interval is shorter than waviness [10].

Some other researchers focus on suppressing millingchatter by reducing cutting force variation, which is realizedby feed rate adjustment. Ip et al. proposed a fuzzy modelbased on the material removal rate optimization approach tocalculate an optimum feed rate for each cutting point in themachining of sculptured surfaces, and the cutting force waskept constant for the ball-nosed cutters [11]. Li et al. studieda solid model based on milling process simulation andoptimization system integrated with CAD/CAM to predictmilling force, and according to simulation results, feed ratecan be optimized to satisfy machining quality for end mill-ing [12]. Bae et al. proposed a simplified-cutting forcemodel based on the 2D chip-load analysis for the concaveline-line segment of a NC tool path to keep feed rate andcutting load of end milling constant [13]. Guzel and Lazogluintroduced a force model based on the physics of cuttingprocess to integrate the physics of ball-end milling processinto the optimization of feeds during the sculpture surfacemachining [14]. Jung and Oh proposed a discrete time first-order model between the feed rate and tool deflection withconsideration for the 3-axis machine tool compliance toreduce cutting force variation. The experiments showed thatthe feed rate optimization could suppress low-frequencymilling chatter and reduce waviness of 3-axis machiningand end milling drastically [15].

Although reducing cutting force variation by feed rateadjustment is effective for waviness, the control of cuttingforce for 5-axis flank milling is more complex. Firstly, evenif the cutting parameters were initially optimized, the mill-ing state might be deteriorated as the orientation variation ofthe tool axis goes bigger. There are few researches about thecutting force of 5-axis flank milling. Larue and Altintasproposed a prediction method of cutting force about flankmilling ruled surfaces with tapered, helical, ball-end cuttingtools by a sweep volume model [16]. Sun and Guo studied asweep volume model to predict effectively the instantaneouscutting force in 5-axis milling process with radial cutter runout based on tool motion analysis [17]. They established therelationship among cutting force, tool path, and feed rate,but not control. Secondly, for most 5-axis machine tools, therotary axis is driven by gear or worm, its strength and

stiffness might be lower than the linear axis due to theconstraints of mechanical structure, especially the rotaryaxis of spindle. When machining a complex shape at highspeed, most feed steps of G01 instructions are about 0.5–2 mm in the constraint of machining tolerance, and thecorresponding machining time is below 0.2 s or frequencyis about 5–20 Hz. If the cutter location point (CL point) isnear singularity position, the rotary axis might be recipro-cating micro rotation abruptly [18]. The unstable rotationcould also cause the rotary axis to vibrate or chatter in lowfrequencies. So the surface waviness control for 5-axis ismore difficult than 3-axis.

Upon further analyses about the cutting force model, thispaper proposes a cutting force-sweep area model of flankmilling, and the feed rate was adjusted accordingly to controlthe cutting force more stable in low-frequency range. Inaddition, the method can improve the unstable rotation stateby means of decomposing the original rotation to more microcontinuous rotation or tool axis orientation adjustment.

This paper takes a 5-axis machine tool of XFYZBA kineticstructure as example to explain our method. The remainder ofthe paper is organized as follows. Section 2 provides thetheory analysis of the proposed control method. Section 3expresses the preprocessing about constructing dual non-uniform rational B-spline (NURBS) to describe 5-axis flankmilling tool path. Section 4 and 5 discuss the adjustment aboutfeed rate and the unstable rotation in detail. The experimentsand results are described in section 6, and finally, conclusionsare drawn in section 7.

2 The theory analysis of control method

In this paper, the source of 5-axis flank machining surfacewaviness focuses on low-frequency milling chatter betweentool and workpiece, which might be caused by the cuttingforce variation and unstable rotation.

2.1 The definition of cutting force-sweep area model

During flank milling, the machined surface is obtained by agenerating point R, which moves along the cutting genera-tor. When the vertical height of milling layer ap and radialmilling depth ae are constant, the milling region could beapproximated as a ruled surface with uniformity thickness.The integrated cutting force f(t) which is instantaneousacting on the cutter changes continuously along the cuttingedge at time t as tool rotating. Suppose the tool isdecomposed into elementary discs along the cutting axis ofwhich thickness is dz. Moreover, the cutting force acting atthe micro-cutting edge dz can be decomposed into a tangen-tial force dft, a radial force dfr, and an axial force dfa. Set thecoordinate origin of the cutting force coordinate system

Int J Adv Manuf Technol

(FCS) at the tool tip, where the Z-axis is tool axis, and X-axis is tangent to the tool path. The transient cutting forcemodel is shown in Fig. 1.

Suppose the cutter rotation is determined by the bladeangle ε, which is the projection of R and Y-axis on XOYplane. Set the instantaneous cutting thickness as ep:

ep ¼ Fz sin ε; ð1Þwhere Fz is the feed distance of segment dz perrevolution[19]. So the tangential cutting force of segmentdz is obtained as follows:

d f t εð Þ ¼ epKtg εð Þdz; ð2Þwhere Kt is the tangential cutting force coefficient[20, 21].Although the Kt might vary with ep, it could be approximatedas constant when ep is very small[13]. The g(ε) is the windowfunction considering the intermittent cutting process:

g εð Þ ¼ 1 0≤ε≤δ0 else

;

�ð3Þ

where δ is the exit cutting angle of the cutter.

The dfa and dfr have the following relationships with dft:

d f r εð Þ ¼ Krd f t εð Þd f a εð Þ ¼ Kad f t εð Þ ;

�ð4Þ

where the Kr and Ka are the radial and axial cutting forcescoefficient respectively[22]. So, the cutting force of segmentdz based on FCS can be expressed as

d f x εð Þd f y εð Þd f z εð Þ

24

35 ¼ cos εð Þ sin εð Þ 0−sin εð Þ cos εð Þ 0

0 0 1

24

35� d f t εð Þd f r εð Þ

d f a εð Þ

24

35

¼epKtg εð Þ cosεþ Krsinεð ÞdzepKt g εð Þ Krcosε−sinεð Þdzep Kt g εð Þ Ka dz

24

35 ð5Þ

In high-speed milling, most of the spindle speed is above10,000 rpm, and the high-frequency transient force is dis-crete which could not describe the integrated cutting stateeffectively. In this paper, the average cutting force of eachtool revolution is defined to improve the description. It isexpressed as

d f x−ave ¼1

2π

Z 2π0d f x εð Þdε ¼

NKt Fz2π

1

4−δcos2δ

4þ Krδ

2−sin2δ4

� �dz

d f y−ave ¼1

2π

Z 2π0d f y εð Þdε ¼

NKt Fz2π

sin2δ4

−δ2þ Kr 1−cos2δð Þ

4

� �dz

d f z−ave ¼1

2π

Z 2π0d f z εð Þdε ¼

NKtKaFz2π

1−cosδð Þdz

;

8>>>>>>><>>>>>>>:

ð6Þ

where N is the number of tool teeth.Setting S as the sweep area of the cutting tool axis per

revolution, the cutting force has the following relationshipwith sweep area:

f x−ave ¼Z ap

0d f x−ave ¼ S

NKt2π

1

4−δcos2δ

4þ Krδ

2−sin2δ

4

� �

f y−ave ¼Z ap

0d f y−ave ¼ S

NKt2π

sin2δ4

−δ2þ Kr 1−cos2δð Þ

4

� �

f z−ave ¼Z ap

0d f z−ave ¼ S

NKtKa2π

1−cosδð Þ

:

8>>>>>><>>>>>>:

ð7ÞThe above analyses show that the average cutting force

has the proportional relationship with the sweep area pertool revolution, which is adopted for 3- or 5-axis flankmilling. From the cutting force-sweep area model, the cut-ting force can be controlled by sweep area rate (SAR).

2.2 The definition of unstable rotation of rotary axis

Suppose a tool path is expressed by G01 NC instruc-tions. Set the θi-2, θi-1, θi, θi+1 as the rotation angles of

A/B axis in the (i-2)-th, (i-1)-th, i-th, (i+1)-th G01 in-struction respectively. In high-speed feeding, if the CLpoint is near the singularity position, the rotation state ofrotary axis is unstable or the rotary axis might be recip-rocating abrupt in micro angle. It could cause the rotaryaxis to vibrate or chatter in low frequency. Define thelimit of micro angle as σ, which can be obtained frommachining experiments. Due to the different combinationsof the above angles, the unstable rotation can be definedas two types:

(1) Discontinuous rotation in micro angleIf the rotation angles satisfy the following condition:

θi−1 ¼ θiθi−1 − θi−2j j þ θiþ1−θij j≤2σ ;

�ð8Þ

it could be defined as discontinuous rotation in microangle. In this definition, two cases such as discontinu-ous monotonic ((θi-1-θi-2)·(θi+1-θi|)≥0), and discontinu-ous non-monotonic ((θi-1-θi-2)·(θi+1-θi|)

(2) Reciprocating rotation in micro angleIf the rotation angles satisfy the following condition:

θi−1 ≠ θiθi−1−θi−2j j þ θiþ1−θij j≤ 2σθi−1−θi−2ð Þ � θi−θi−1ð Þ < 0θi−θi−1ð Þ � θiþ1−θið Þ < 0

;

8>><>>:

ð9Þ

it could be defined as reciprocating rotation in microangle, as shown in Fig. 3.

2.3 The control process

The control process of surface waviness has two aspects:reducing variation of cutting force and improving unstablerotation state of rotary axis. The control flowchart is shownin Fig. 4.

The tool path expressed by G01 NC instructions, ofwhich the coordinates are discrete, should be converted intoparametric spline expression for the sweep area calculationin preprocessing.

According to the cutting force-sweep area model, thecutting force could be controlled by SAR. When the ap isconstant, it is realized by feed rate. However, the definitionof 5-axis machine tool feed rate is dimensionless and is notestablished in the programming coordinate system (PCS).According to the definition of SIEMENS840D NC system,when G90 and G94 instructions are effective, the feed ratecan be defined as

Fi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

Δγið Þ2q

ti

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔxið Þ2 þ Δyið Þ2 þ Δzið Þ2 þ ΔAið Þ2 þ ΔBið Þ2

qti

;

ð10Þwhere Δγi (γ=x, y, z, A, B) is the axis movement of i-th NCfeed instruction in the machine coordinate system(MCS), andti is the corresponding process time. Based on the inversekinematical transformation and RTCP/RPCP function, thefeed rate adjustment might be under constraint condition ofSAR in MCS.

By means of decomposing the unstable rotation to moremicro continuous rotation or adjusting tool axis orientation basedon the geometric relationship of tool axis and machining toler-ance, the unstable rotation state can be improved. For the dis-continuous non-monotonic or reciprocating rotation,interpolating the CL points in an unstable region is adopted. Inaddition, for the discontinuous monotonic rotation, adjusting thetool axis under the constraint of machining tolerance is adopted.

Fig. 1 The cutting force model of flank milling

a. Discontinuous monotonic rotation b. Discontinuous nonmonotonic rotation

Fig. 2 The definition ofdiscontinuous rotation in microangle


3 The preprocessing

A 5-axis tool path could be defined as two NURBS curveswhich are based on PCS [23, 24]. Where P(u)=(x(u), y(u),z(u)) defines the locus of CL points, and Q(uQ)=(x(uQ),y(uQ), z(uQ)) defines the locus of a up cutter location points(UCL points) corresponding to the vertical height of millinglayer ap, as shown in Fig. 5.

Set the position vectors of CL points as {P0, P1, …,Pn}. The P(u) is calculated with the position vectorsresulting in Pi∈P(u) where (i=0,1,…, n). At first, theknot vector is calculated by using the method of stan-dard chord length parameterization. The resultant knotvector is U=[u0, u1,…, un+6], in which [u3, un+3] be-longs to [0, 1], u0=u1=u2=u3=0 and un+3=un+4=un+5=un+6=1. The control points di (i=0, 1, …, n+2) couldbe achieved as follows[25]

a1 b1 c1a2 b2 c2⋱ ⋱ ⋱

an bn cnanþ1 bnþ1 cnþ1

266664

377775

d1d2⋮dndnþ1

266664

377775 ¼

e1e2⋮enenþ1

266664

377775; ð11Þ

where ai¼ Δiþ2ð Þ2

ΔiþΔiþ1þΔiþ2 ; bi¼Δiþ2 ΔiþΔiþ1ð ÞΔiþΔiþ1þΔiþ2 þ

Δiþ1 Δiþ2þΔiþ3ð ÞΔiþ1þΔiþ2þΔiþ3 ;

ci¼ Δiþ1ð Þ2

Δiþ1þΔiþ2þΔiþ3 ; ei¼ Δiþ1 þΔiþ2ð ÞPi−1; Δi¼uiþ1−ui.With the knot vector U and control points di, the NURBS

curve P(u) could be obtained as follows

P uð Þ ¼

Xni¼0

Ni;3 uð ÞdiXni¼0

Ni;3 uð Þ; ð12Þ

where Ni,3(u) is the basis function for the three-orders B-spline. And the Q(uQ) could be defined in the same way.

4 The reducing of average cutting force variation

4.1 Calculation of flank sweep area

Set Pi-1, Pi as the start and end CL points of i-th G01 NCinstruction,Qi-1 andQi as the corresponding UCL points. Pi-1,Pi, Qi-1 and Qi are the four corners of a sweep-ruled surface(PiQi is the rectilinear generator), as shown in Fig. 6.

Set point P on the trajectory of Pi-1Pi, and define that

v ¼ u−uiþ2uiþ3−uiþ2

¼ uQ−uQ iþ2ð ÞuQ iþ3ð Þ−uQ iþ2ð Þ

; w ¼ Pi−1Rj jPi−1Qi−1j j

: ð13Þ

From the above parameters, the parameter equation ofruled surface can be defined as:

r v;wð Þ ¼ P uð Þ þ w Q uQ� �

−P uð Þ� �: ð14ÞSo the sweep area Si from Pi-1 to Pi can be obtained:

Si ¼ ∬0≤v≤10≤w≤1

r0v v;wð Þ � r

0w v;wð Þ

�� dvdw

¼ ∬0≤v≤10≤w≤1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir02wr

02w− r

0v � r0w

� �2qdvdw; ð15Þ

where

r02v ¼ x

02v þ y

02v þ z

02v

r02w ¼ x

02w þ y

02w þ z

02w

r0w � r

0w ¼ x

0vx

0w þ z

0vy

0w þ z

0vz

0w

:

8<: ð16Þ

Fig. 3 The definition of reciprocating rotation in micro angle

Calculate sweep area

The surface waviness controlmethod for 5-axis flank milling

Feedrate adjustment

Discontinuousmonotonic

Discontinuousnonmonotonic

Analyze the rotary state

Adjust toolaxis

Reducing cuttingforce variation

SAR constraint Machining dynamicconstraints

Improving unstablerotary state

Construct dual NURBS todescribe tool pathPreprocessing

Reciprocating

InterpolateCL/UCL points

Suppress the lowfrequency milling chatter

Fig. 4 The surface waviness control flowchart of 5-axis flankmachining


4.2 The feed rate adjustment

The geometric integrated model of Pi in MCS could bedescribed as Eq. (17), which is based on inverse kinematicaltransformation [26].

Pi ¼ xi yi zi 1½ �T ¼ −TTð Þ � Tx � Ty � Tz � TB � TA � TT � 0 0 0 1½ �T

¼1 0 0 sxi0 1 0 syi0 0 1 szi þ L0 0 0 1

2664

3775�

cosBi 0 sinBi 00 1 0 0

−sinBi 0 cosBi 00 0 0 1

2664

3775�

1 0 0 00 cosAi −sinAi 00 sinAi cosAi 00 0 0 1

2664

3775�

1 0 0 00 1 0 00 0 1 −L0 0 0 1

2664

3775�

0001

2664

3775: ð17Þ

Where TA, TB, Tx, Ty, Tz are kinematical matrixes about A,B, X, Y, Z axis respectively, TT is a position matrix of the tooltip about A axis, L is the distance between tool tip and Aaxis. sxi, syi and szi are the coordinates of X, Y, Z axis relativeto initial position in MCS.

Define an equivalent displacement of i-th NC instructionas Δsi, so

Δ si ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

Δγið Þ2q

γ ¼ x ; y ; z ; A ; BΔ xi ¼ sxi − sxi ¼ xi−xi þ L sin Bi cos Ai−L sin Bi−1 cos Ai−1Δ yi ¼ s yi − s yi ¼ yi − yi − L sin Ai þ L sin Ai−1Δzi ¼ szi−szi ¼ zi−zi þ Lcos Bi cos Ai−Lcos Bi−1 cos Ai−1ΔAi ¼ Ai−Ai−1ΔBi ¼ Bi−Bi−1

:

8>>>>>>><>>>>>>>:

ð18Þ

Set the SAR as S′, the feed rate adjustment of i-th G01NC instruction should be

Fi ¼ Δsiti ¼ Δsi �S0

Si: ð19Þ

In machining process, each axis servomotor should with-stand forces including cutting force, inertia force, and fric-tion. In G01 instructions, the machining velocity andacceleration at the transition point between two segmentsmight be discontinuous; this would cause impact to degrademachining stability and precision. In addition, the impactload might exceed the machine’s motor rated power evenburn it. Therefore, in order to improve machining quality,each axis possesses limits of its kinematical characteristics:

Fig. 5 The dual NURBScurves of flank milling

QQ(u )

P(u)

R

P

u

iQQi-1

Pi

i-1P

X

Z

Y

O

Fig. 6 The cutting area of flank millingFig. 7 The adjustment of discontinuous non-monotonic rotation inmicro angle


maximal velocity vmax and maximal acceleration amax. The-se act as a limiting factor for axis coordination in MCS [27].

Suppose the Acc/Dec rule is linear, the velocities of fiveaxes should not exceed their saturation limits vmax:

Fi ¼ Δsiti ≤Δsi

maxΔγivγ−max

� � γ ¼ x; y; z; A; B : ð20Þ

The accelerations of five axes should not exceed theirsaturation limits amax:

aγi ¼2 vγi−vγ i−1ð Þ� �ti þ ti−1 ¼

2FiΔγiΔsi

−2Fi−1Δγi−1

Δsi−1ΔsiFi

þ Δsi−1Fi−1

≤aγ−max

γ ¼ X ; Y ; Z;A;B

:

ð21Þ

Therefore, the adjusted feed rate of i-th G01 NC instruc-tion should satisfy the above three constraint conditions asfollows:

Fi ¼ min Δsi � S0

Si;

Δsi

maxΔγivγ−max

� � ; Δsimax

vγ ið Þ−vγ i−1ð Þaγ−max

� �0BB@

1CCA

γ ¼ x; y; z; A; B

:

ð22Þ

5 The adjustment for improving unstable rotation state

5.1 The adjustment for discontinuous non-monotonicor reciprocating rotation

By means of decomposing the unstable rotation to moremicro continuous rotation, an unstable rotation state couldbe more stable than before. For the discontinuous non-monotonic or reciprocating rotation in micro angle,

Fig. 8 The adjustment of reciprocating rotation in micro angle

Interpolate new CL pointsparameter ui-1(new), ui(new), unew

Calculate A and B angle

Calculate the group ofinterpolation points

The end

Rotating stateimproved?

Y

Satisfy thetolerance?

Entire regionis traversed?

N

Y

N

Y

Start

N

Fig. 9 The adjustment algorithm of discontinuous monotonic rotationin micro angles

Fig. 10 The geometric diagram of CL/UCL points and machiningtolerance

Fig. 11 The adjustment of discontinuous monotonic rotation in microangles


interpolating the CL/UCL points in an unstable region isadopted. The interpolation parameter of a new point is

unew ¼ uiþ2 þ uiþ32: ð23Þ

From Eq. (12), the interpolated coordinates of CL/UCLpoints can be obtained, and the tool axis orientation V andA/B axis angle can be obtained from Eq. (24) by geometricintegrated model. The adjustment results are shown inFigs. 7 and 8.

V ¼ TB � TA � 0 0 1 0½ �T

¼cosB 0 sinB 00 1 0 0

−sinB 0 cosB 00 0 0 1

2664

3775�

1 0 0 00 cosA −sinA 00 sinA cosA 00 0 0 1

2664

3775

0010

2664

3775

¼ Q uQ� �

−P uð Þap

ð24Þ

5.2 The adjustment for discontinuous monotonic rotation

For the discontinuous monotonic rotation, the tool axis orien-tation could be adjusted under the constraint condition ofmachining tolerance, and the unstable rotation state might beconverted into continuous monotonic. In addition, the adjust-ment should be iterative, and the algorithm is shown as Fig. 9.

There are three CL/UCL points should be interpolated toadjust tool axis orientation at least. Firstly, two points can be

interpolated to substitute the original Pi-1/Qi-1 and Pi/Qi. Theunew(i+2) is interpolated from ui+2 to minus adjacent region,and unew(i+3) is interpolated from ui+3 to plus adjacent re-gion. In addition, another new interpolated point locatesbetween them.

unew iþ2ð Þ ∈ uiþ1; uiþ2½ Þunew ¼

unew iþ2ð Þ þ unew iþ3ð Þ2

unew iþ3ð Þ ∈ uiþ3; uiþ4ð �:

8><>: ð25Þ

So the interpolation point Pnew(i-1), Pnew, Pnew(i) and cor-responding UCL points could be calculated from Eq. (13).They compose a group of interpolation points.

Secondly, every CL/UCL points are on the NURBS toolpath, there is no radial error. However, due to chord substi-tute curve, the new CL/UCL points should satisfy the fol-lowing machining tolerance constraint condition [28]:

lCL

.UCL

� ρCL

.UCL

−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρCL

.UCL

2−ΔL2

� �2vuut ≤l: ð26Þ

Where 1CL/UCL is a chord error, 1 is a machining toler-ance, ρCL/UCL is the curve radius and ΔL is the distancebetween the adjacent CL/UCL points, as shown in Fig. 10.Only the group satisfying the above conditions will beretained, or else another group should be interpolated andverified.

The last step is to determine whether the interpolationpoint group retained could improve the rotation state. TheA/B rotation angles of the new CL/UCL group can beobtained by Eq. (24). If the discontinuous rotation state isimproved, the points group can be used to substitute the oldones, as shown in Fig. 11. Otherwise, another group pointshould be interpolated and verified iteratively until the entireunstable region is traversed.

6 Performance evaluations

To verify the effectiveness of the proposed control method,a series of milling experiments were conducted on aPARPAS BF-100-TTm 5-axis machine tool of which kinetic

Fig. 12 The contrast specimen

a. Origin NC program b. Optimized NC program

Fig. 13 The segmentscomparison of NC programs(segment)


structure was XFYZBA. The velocity limits of X/Y/Z axiswere 10,000 mm/min, and 1.39 rpm for A/B axis, the accel-erations limits of X/Y/Z axis were 500 mm/s2, and 0.05 r/s2

for A/B axis. The machining tolerance 1 was set as 0.2 mm.In the finish flank milling process, the height of each

milling layer ap was 4.75 mm, the radial milling depth aewas 3.5 mm. The tool used was carbide 20 mm in diameter,and it had four teeth with a helix angle of 30 °. The spindlespeed was set at 12,000 rpm, and the origin program feedrate was 900 mm/min. After mounting the tool, the L was373.64 mm. Refer 3-axis milling, the constraint condition ofSAR was set at 4,900 mm2/min, and the limit of micro anglewas set at 0.5 °, which appeared rather stable and moderate.Two same-type aluminum specimens were flank milled, asshown in Fig. 12. One was milled with the original NCprogram and the other was milled with the optimized NCprogram by the proposed method. A segment comparison ofNC program was shown in Fig. 13. The feed rate wasconverted from original constant into dynamic adjustment.Such as feed rate was changed from constant 900 to 803.6 in

N3438 or 797.2 in N3439 instruction. More feedrate com-parison was shown in Fig. 14. As a result, the correspondingSAR was converted from original dynamic variation (4,300to 7,400) into constant 4,900 mm2/min. More SAR compar-ison was shown in Fig. 15. In addition, the unstable rotationof some specific NC instruction was improved, such as anew N3445 NC instruction was interpolated between orig-inal N3444/N3445 instructions about discontinuous non-monotonic rotation.

During flank milling, cutting forces in X and Y directionswere measured by a dynamometer. As the average SAR ofthe original NC program equaled the optimized one, theamplitude-time character of original cutting force was notmuch different from the optimized one. Moreover, throughthe frequency spectrum analysis based on the functionalFourier transform, the dynamic component of cutting forcewas calculated. The cutting force spectra showed some veryclear peaks at 8, 30, and 38 Hz in low-frequency range, asshown in Fig. 16 and 17. From the comparisons of thevibration modes at the above frequencies, it can be found

100

200

300

400

500

600

700

800

900

1000

1100

0.0 6.6 11.7 14.9 20.7 24.5 27.0

Time (S)F

eedr

ate

(mm

/min

)

The feedrate of optimized NC program The feedrate of original NC programFig. 14 The feed ratescomparison of NC programs(segment)

4000

4500

5000

5500

6000

6500

7000

7500

8000

0.0 6.6 11.7 14.9 20.7 24.5 27.0

Time (S)

SA

R (

mm

2 /m

in)

The SAR of optimized NC program The SAR of original NC programFig. 15 The SAR comparisonof NC programs (segment)


that the amplitude–frequency character of cutting force waschanged obviously, such as the amplitude at the abovefrequency reduced effectively. More comparisons are de-tailed in Table. 1. This indicated that the cutting state be-came more stable and the low-frequency milling chatter wassuppressed effectively.

The specimen surfaces were measured by a Nanomap500dual mode 3D profilometer, the results were shown in Figs. 18and 19. The machined surface profile of the original NCprogram had obvious waviness of which interval was about3 mm and depth was about 0.01 mm. The machined surface

a. The force of original NC program b. The force of optimized NC program

Fig. 16 The comparison of cutting force in X direction (segment)

a. The force of original NC program b. The force of optimized NC program

Fig. 17 The comparison of cutting force in Y direction (segment)

Table 1 The amplitude–frequency character comparison of cutting force

Frequency point(0–50 Hz)

Originalamplitude

Optimizedamplitude

8 Hz(X-direction) 20 N 12 N

8 Hz(Y-direction) 31 N 28 N






profile of the optimized NC program had not obviouswaviness except some cutting marks of tool rotation, ofwhich interval was about 0.5 mm and depth was about0.002 mm. From the comparisons of the two specimens’surface scans, it can be found that the surface wassmoother than before.

7 Conclusions

In this paper, a comprehensive control method for 5-axisflank milling was proposed based on low-frequency millingchatter. The control method includes reducing of cuttingforce variation and improving unstable rotation state ofrotary axis. The highlights of the this paper are

(1) Average cutting force of each tool revolution is definedto improve the description of the integrated cuttingstate.

(2) Upon the proposed cutting force-sweep area model, thecontrol of cutting force can be converted into SAR,which can be realized by feed rate.

(3) The derivation of how to calculate the flank milling areaand adjust the feed rate is explained by constructing dualNURBS curves to describe tool paths.

(4) The unstable rotation state is defined including the dis-continuous and reciprocating rotation in micro angle.

(5) By means of decomposing the unstable rotation to moremicro continuous rotation or adjusting tool axis orienta-tion based on the geometric relationship of tool axis andmachining tolerance, the unstable rotation state can beimproved.

Finally, a series of comparison experiments was conductedin a 5-axis machine tool. The feed rate was converted from theoriginal constant into dynamic adjustments, while the corre-sponding SAR was converted from the original dynamicvariations into a constant. From the comparison of cuttingforce and surface waviness scans, it can be found that theamplitude of force vibration components in low-frequencyrange was reduced effectively and surface was smoother thanbefore. The results showed that the proposed control methodwas effective to suppress low-frequency milling chatter andimprove surface waviness for 5-axis flank milling.

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a. The machined surface profile of original NC program b. The machined surface profile of optimized NC program

Fig. 19 The comparison of the machined surface profile (segment)

a. Machined surface of original NC program b. Machined surface of optimized NC program

Fig. 18 The comparison of themachined surface scan(segment)


Acknowledgments This work is supported by the Special Fund ofHigh-end CNC Machine Tools and Basic Manufacturing Equipment(2010ZX04015-011), China. It is also supported by the FundamentalResearch Funds for the Central Universities under Grant(SWJTU09ZT06, SWJTU11CX144 and SWJTU11CX026), New Cen-tury Excellence Plan Grand (NCET-09-0665), and the National Scienceand Technology Pillar Program during the Eleventh Five-Year PlanPeriod (no.2009BAG12A01-A01).

Appendix

Notation

γi (γ=x, y, z, A, B, C) The CL point coordinates in PCSabout i-th NC instruction

ap The vertical height of millinglayer

ae The radial milling deptht The cutting timedz The micro-thickness of

elementary discdft/dfr/dfa The tangential/radial/axial cutting

force of segment dzdfx/dfy/dfz The cutting force of segment

dz based on FCSdfx-ave/dfy-ave/dfz-ave The average cutting force

of segment dz in each tool rotationε The angle projection of R

and Y-axis on XOY plane in FCSep The instantaneous cutting thicknessKt/Kr/Ka The tangential/radial/axial cutting

force coefficientΔ The exit cutting angle of the

cutterFz The feed distance of segment

dz per revolutionN The number of teethS The sweep area of tool axis per

revolutionfx-ave/fy-ave/fz-ave The average cutting force of tool

in each tool revolutioni The sequence number of G01

NC instructionθi The rotation angle of i-th G01

NC instructionσ The limit of micro angleΔγi (γ=x, y, z, A, B, C) The axis moving distance

in MCS about i-th NC instructionFi The feed rate of i-th NC

instructionPi The CL point of i-th G01 NC

instruction

Qi The UCL point of i-th G01 NCinstruction

u/uQ The parameter of NURBSU The knot vector of NURBSD The control point of NURBS

curveSi The ruled surface area from

Pi-1 to PTγ (γ=x, y, z, A, B, C) The axis kinematical matrixTT The position matrix of tool tip

about A axissx, sy, sz The coordinates of X, Y, Z-axis

relative to initial position in MCS.Δsi The equivalent displacement

of i-th NC instructionS' The SARL The distance between tool tip

and A axis.vγ-max (γ=x, y, z, A, B,C) The velocity limits of five axesaγ-max (γ=x, y, z, A, B,C) The acceleration limits of five axesVi The tool axis orientation of i-th

NC instruction1CL/UCL The chord error of CL/UCL pointsρCL/UCL The curve radius of CL/UCL points1 The machining toleranceΔL The distance between adjacent

CL/UCL points

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The research of surface waviness control method for 5-axis flank millingAbstractIntroductionThe theory analysis of control methodThe definition of cutting force-sweep area modelThe definition of unstable rotation of rotary axisThe control process

The preprocessingThe reducing of average cutting force variationCalculation of flank sweep areaThe feed rate adjustment

The adjustment for improving unstable rotation stateThe adjustment for discontinuous non-monotonic or reciprocating rotationThe adjustment for discontinuous monotonic rotation

Performance evaluationsConclusionsAppendixReferences

The research of surface waviness control method for 5-axis ... · waves, which has about 1 –5 mm...

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