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published online 24 September 2012 2013 227: 3 originallyProceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics

Krzysztof Kecik, Rafal Rusinek and Jerzy WarminskiModeling of high-speed milling process with frictional effect

  

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Special Issue Article

Modeling of high-speed milling processwith frictional effect

Krzysztof Kecik, Rafal Rusinek and Jerzy Warminski

Abstract

This article deals with the problem of chatter vibrations in high-speed milling process taking into account both regen-

eration and frictional chatter. In this aim, a nonlinear model of high-speed down milling is developed. The proposed

model includes friction force produced between an edge of a tool and a workpiece, modeled by nonlinear and non-

smooth function and also the time delay effect, which is responsible for a vibration regeneration. The influence of friction

on the process stability is compared with results obtained in a classical way using a commercial software. Finally,

numerical tests are compared with the stability lobe diagrams obtained experimentally during real machining of nickel

superalloys.

Keywords

Chatter, frictional chatter, stability lobe diagram, milling

Date received: 5 March 2012; accepted: 30 March 2012

Introduction

Presently, high-speed machining, especially high-speed milling (HSM) plays an important role in man-ufacturing. Milling is a process of generatingmachined surfaces by progressively removing anamount of material from the workpiece by a millingcutter rotating at a comparatively high speed (HSM).The characteristic feature of the milling process is thateach milling cutter tooth removes a share of the stockin the form of small individual chips. The cutting ofmetals is frequently accompanied with vibrations ofworkpiece and cutting tool which are known asmachine-tool chatter. Chatter denotes self-excitedvibrations, which are induced and maintained bychanging forces generated by the cutting process.Chatter may arise during machining, at specific com-bination of cutting depth and spindle speed. This phe-nomenon is harmful, mainly because of heavyvibration of the tool which causes low surface quality.Usually, chatter is divided into two categories. Theone called regenerative is caused by regeneration ofwaviness of the workpiece surface. The second typeof chatter is caused by factors associated with frictionbetween a tool and a workpiece and therefore is calledfrictional chatter. Prediction of chatter is usually esti-mated through generating stability lobes diagram(SLD), which partitions a rotational speed versus adepth-of-cut plane into stable and unstable regions.An important characteristic feature of SLD is a max-imal depth-of-cut that is dependence of the maximumstable depth-of-cut, which does not generate chatter

vibrations regardless of the cutting speed. This bound-ary determines area of absolute stability of process.Generally, the value of maximal depth-of-cut dependson the material and geometry of the cutting tool aswell as the workpiece material. Additionally, the SLDshape indicates that for some speeds it is possible toobtain higher cutting depth without stability loss. Bydetermining of SLD it is possible to achieve the high-est applicable material removal rate (MRR) for amachining process.

Several physical mechanism causing chatter phe-nomena can be found in papers.1,2 Usually, the regen-erative chatter is one of most important causes ofinstability in the cutting process. However,Wiercigroch and Krivtsov,3 Grabec4 and Foonget al.5 show that frictional effect is also importantand causes self-excited vibrations in the cutting pro-cess. Moreover, the frictional models have shown evi-dence of chaotic vibrations, which are mainly due tothe nonlinear nature of the dry friction and the inter-mittent contact between the cutting tool and theworkpiece.

Proc IMechE Part K:

J Multi-body Dynamics

227(1) 3–11

! IMechE 2012

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DOI: 10.1177/1464419312458636

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Department of Applied Mechanics, Lublin University of Technology,

Poland

Corresponding author:

Krzysztof Kecik, Department of Applied Mechanics, Lublin University

of Technology, Nadbystrzycka 35 St., Lublin 20-618, Poland.

Email: k.kecik@pollub.pl

Faassen et al.6 in his work, milling process is mod-eled based on dedicated experiments on both thematerial behavior of the workpiece material and themachine dynamics are dependent on the spindlespeed. The modeled chatter boundaries are comparedto the experimental results in order to validatethe model and the stability analysis. Stepan7 useda four-dimensional phase-space representationembedded in the infinite-dimensional phase space ofregenerative chatter to investigate the chatter vibra-tions of an industrial machine tool. Elimination ofchatter vibration is important and is still an existingproblem. Interesting method of active vibration sup-pression, based on the delayed resonator is presentedin Olgac and Hosek.8 It is an active vibration absorbertuning of which is achieved utilizing a simple timedelayed feedback. For the milling process, the domin-ant (chatter) frequency of the perturbation vibrationslies in general close to the eigenfrequency of the spin-dle dynamics.9 Sims et al.10 modeled variable helixand variable pitch milling tools using a semi-discreti-zation method (SDM). However, they only predictedthe chatter stability, and did not optimize the tooldesign for minimizing chatter. Stability of thin flexiblepart was studied by Abrari et al.11 by finite elementsmethod (FEM). Flexible tools are modeled as canti-levers; flexible parts are created as solid models, aremeshed and are dynamically solved by FEM.

The aim of this article is to show the SLD takinginto account both regenerative and frictional chatter.In order to predict the stability boundaries, an accur-ate dynamic model for the down-milling process isprepared. The nonlinear tool-part dynamics is ana-lyzed during milling of material, which is difficult tocut (nickel superalloys) in the chatter and chatter freeregimes. The presented model consists of the part,which describes frictional effect between the edge ofa tool and a workpiece and additionally part respon-sible for the regenerative effect. The results obtainednumerically from the mathematical model are

compared with an experimentally determined SLDby means of CutPro software.

Model of milling process with frictionaland regeneration effects

In the milling process, material is removed from aworkpiece by a rotating cutting tool with a certainspeed. A schematic representation of the milling pro-cess is shown in Figure 1. In order to calculate stabil-ity of the milling process it is necessary to haveinformation regarding the structural dynamics of themilling model. A machine tool is modeled as onedegree of freedom oscillator with undamped naturalfrequency !0 ¼

ffiffiffiffiffiffiffiffiffik=m

p, where k, m are stiffness and

mass of the cutter-holder system (Figure 1) and cdenotes a damping coefficient.

Milling forces can be modeled in a different wayfor given cutter geometry, cutting conditionsand work material. In the literature various modelsof cutting forces have been proposed.12–14 Usually,the forces are function of the depth-of-cut andthe feed rate. For a single cutting tooth (j), the tan-gential and radial forces are denoted as Ftj and Fnj,respectively (Figure 1(b)). During the milling pro-cess the cutter – workpiece interaction results in anacting force Fj in x-direction which can beexpressed as

FjX ¼ Ftj sin ’j � Fnj cos ’j� �

gj ð1Þ

where, j¼ 0,1 , . . . , z (z is number of teeth of thecutter), ’j is rotation angle of j-th tooth, gj is equalto 1 if j-th tooth is active and 0 if not active. In otherwords, gj determines whether the tooth cut is in or outof cut. The tooth is in cut if ’s4’j4’e, where ’s and’e are the start and exit angles. Additionally, in thisexample the radial depth-of-cut (ae) is assumed tobe half diameter of cutter, then ’s¼ 0, ’e¼�/2(Figure 1(b)). Moreover, the function gj is applied to

Figure 1. One degree of freedom model of down-milling process: (a) 3D and (b) 2D view.

4 Proc IMechE Part K: J Multi-body Dynamics 227(1)

control of positive value of chip thickness becauseonly then the tool cuts workpiece, not air. Thus, fordown-milling gj is given by function

gj ¼ H sin ’j� �

H cos’j � cos ’e� �

Hðhj Þ ð2Þ

where H denotes Heaviside function and gj isdefined as

gj ¼1, ’s4’j4’e, and hj ðtÞ4 0

0, �=25 ’j 5 2� or hj ðtÞ40

( )ð3Þ

Note, that chip thickness decreases during down-milling. Tangential and radial cutting forces action onthe tooth are proportional to the axial depth-of-cut apand the chip thickness hj

Ftj ¼ Ktaphj ðtÞ�, Fnj ¼ Knaphj ðtÞ

�ð4Þ

where Kt, Kn are constant cutting coefficients whichdepend on the cutting material properties, the coeffi-cient � also depending on the material, is usually esti-mated about 0.9.15 The chip thickness hj(t) is functionof a feed rate. The function of a chip thickness con-sists of the static part (hSTAT) caused by feed of thecutter fz, and the dynamics part (hDYN) caused by thevibrations of the cutter at the present time t and pre-vious flute at time period (t��). It means that cuttingforce (1) depends not only on the current cutter dis-placement, but also on the displacement of the previ-ous tooth

hjSTATðtÞ¼ fz cos’j, hjDYNðtÞ¼ xðtÞ�xðt��Þð Þcos’j

ð5Þ

When the spindle speed n is given in r/min, toothpassing period is defined as �¼ 60/zn.

The frictional chatter is the result of frictionbetween the cutter edge and workpiece. Therefore,a nonlinear function of the cutting force FTj,which depends on relative velocity vr, is proposed asfollows16

FTj ¼ FjX�, � ¼ 1� �vr þ �v3r ð6Þ

where � and � are constants, which shapes the frictioncurve (Figure 2). Similar friction characteristic hasbeen applied by Wiercigorch and Krivtsov.3 The rela-tive velocity, which depends on the vibration velocityand cutting speed (vc), is defined as

vr ¼ _xþ vc sin ’j ð7Þ

Equation (6) is responsible for the energy thatis supplied to the vibrating system, and thus mayrender self-excited vibration in the course of contactfriction.17

The parameters � and � in case of static millingprocess, that is when there are no vibrations in thesystem, are defined by equation16

� ¼ 3�v2c ,

� ¼1� �

2v3cð8Þ

The equation of motion, for a one degree of free-dom milling model, presented in Figure 1, and for asingle tooth (j¼ 1), can be written as

m €xþ c _xþ kx ¼ aphðtÞ� 1� �vr þ �v

3r

� �Kt sin ’� Kn cos’ð Þ gj ð9Þ

where, the gj function determines the instant of con-tact loss between the tool and a workpiece. The pre-sented structure is assumed to be flexible in the x-direction only. The differential equation of motionof milling process (9) is nonlinear non-autonomouswith time delay and discontinuous because ofHeaviside step functions.

Numerical and experimental results

Numerical simulations are carried out in order toevaluate the model capability and to describe down-milling process accurately. The governing equation (9)is solved with the help of the fourth-order Runge–Kutta method with fixed time step 0.0001. SLDshave been generated using parameters obtained fromexperimental modal tests of a real spindle–toolsystem. A numerical SLD is obtained by point-by-point investigation of a gird in the parameter spaceconsisting of the spindle speed resolution 20 r/min anddepth-of-cut resolution 0.1mm. Based on the modal

Figure 2. Characteristic of friction versus relative velocity.

Kecik et al. 5

analysis performed on a milling machine with a fixedcutter (diameter of 12mm), the modal parameters are:m¼ 0.485 kg, k¼ 1.949� 107N/m, c¼ 90,077Ns/m.The radial depth-of-cut and the feed equal ae¼ 6mm,fz¼ 0.02mm/tooth, respectively. The cutting resist-ance coefficients obtained in experiment for Inconel718 alloy take values: Kt¼ 97.66� 108N/m1þk andKn¼ 29.3� 108N/m1þk. Figure 3(a) represents SLDobtained numerically from the model including onlyregenerative chatter, while Figure 3(b) shows numer-ical SLD for regenerative and frictional chattertogether.

Here, a technical stability criterion is appliedduring numerical simulations. Namely, if vibrationsamplitude is greater than xcr¼ 0.5mm, the process isclassified as unstable (the black color in Figure 3(a)and (b)). The critical axial depth-of-cut apcr is about1.2mm. Note, that it is possible to increase the depth-of-cut (or spindle speed) without chatter generation.This leads to significant improvement of materialremoval rates (MMR). Both diagrams (Figure 3(a)and (b)) are very similar, this suggests that for theseparameters, friction phenomenon influences stabilitydiagrams very little. This is primarily due to the factthat the friction between the tool and the workpiece isnot so big. In order to show differences the moreadvanced analysis is applied later in this article. Forthe same parameters, the SLD has been obtained bycommercial software – CutPro9 (Figure 4). In thisexample, the critical depth-of-cut is a little smaller(apcr¼ 1mm) compared to numerical result. This dif-ference is due to the fact that SLD from CutPro isobtained for two directions (a model with twodegree of freedom), and also depends on the stabil-ity criterion engaged in the proposed model.Additionally, in the CutPro software, the processis modeled by using the linear stability theory andnonlinearities such as a tool jump out of cut, multipleregeneration, process damping, run-out and nonlinearcutting coefficients are neglected in linear stabilityanalysis.

As it is mentioned before, the appearance of thestability diagram depends on the index applied to esti-mate the stability boundary of process, therefore inthe literature one can find e.g. the index developedon the basis of the recurrence quantification ana-lysis18,19 or on the vibration amplitude.20 As far asthe method of chatter prediction is concerned, otherresearchers have used fast Fourier transform (FFT),21

time series analysis,22 Kalman filters,23 and neuralnetworks24 or customized models25 to find instabilityparameters.

In the unstable region, the amplitude of vibration ismuch higher compared to stable milling. For examplefor, n¼ 16,000 r/min (Figure 5(a)) and depth-of-cutap¼ 1.5mm, the amplitude of vibrations is muchsmaller than n¼ 14,000 r/min (Figure 5(b)). Theseresults clearly show that the applied higher spindlespeed can cause the smaller vibration and cuttingforces. This means that it is possible to increase thedepth-of-cut, without the occurrence of chatter.

Bifurcation methodology can be applied hereto look for dramatic changes in the topology of

Figure 3. Exemplary numerical stability lobe diagram, for model with regenerative effect for �¼ 0.9, �¼ 0, �¼ 0 (a) and

regenerative and frictional effects, �¼ 3�v2r , �¼ 0.025/v3

r (b).

Figure 4. Stability lobe diagram obtained from commercial

software CutPro9.

6 Proc IMechE Part K: J Multi-body Dynamics 227(1)

dynamic behavior. On observing the influence ofimportant stability parameters as depth-of-cut andspindle speed it is possible to identify critical bifurca-tion points. These studies also allow one to designcontrollers to suppress unwanted dynamics. InFigure 6 on ordinate axis, the value of maximal andminimal peaks of amplitudes is shown, on changingthe rotational speed in steps of 100 r/min. InFigure 6(a), the influence of spindle speed versus amp-litude of vibration is shown for model with regenera-tive effect only. One can easily find the dangerousspindle speeds at which the chatter phenomenonoccurs. This figure agrees with the correspondingSLD (Figure 3(a)). The similar result obtained formodel with frictional effects (Figure 6(b)).Both diagrams are very similar, but the introducedfrictional part causes slightly smaller amplitude ofvibration.

Similar results are obtained in case of depth-of-cutas presented in Figures 7 and 8. These diagramshave been obtained for two different spindle speeds.

Figure 7 shows workpiece vibration versus depth-of-cut for n¼ 16,000 r/min, and Figure 8 forn¼ 14,000 r/min.

Interesting results have been obtained for analysisof stability process on the plane: depth-of-cut versusfeed. In the case of the model without the friction(Figure 9(a)) the boundary between stable andunstable technological parameters is practically a ver-tical line, but for model with added friction, theboundary is very similar, but shifted to the rightside (Figure 9(b)).

These slight differences show that stability ofmilling process in the model with friction dependsmore on the feed of a cutter. The numerical andexperimental studies exhibit that stability processdepends on the choice of feed rate.26 This suggeststhat model with friction is closer to realistic millingprocess than model without effect.

Generally, friction phenomenon in this model, mayreduce the average value of chip thickness, have(t).This situation is presented in Figure 10(a) and (b).

Figure 5. Amplitude of vibration during stable n¼ 16,000 r/min (a) and unstable n¼ 14,000 r/min (b) for down-milling process with

regenerative effect ap¼ 1.5 mm.

Figure 6. Influence of rotational speed on the amplitude vibration for ap¼ 1.5 mm, without (a) and with frictional effect (b).

Kecik et al. 7

Figure 7. Influence of axial depth-of-cut on the amplitude vibration for n¼ 16,000 r/min, without frictional effect (a) and with friction

effect (b).

Figure 8. Influence of axial depth-of-cut on the vibration amplitude for n¼ 14,000 r/min, without friction effect (a) and friction

effect (b).

Figure 9. Influence of axial depth-of-cut versus feed per tooth for n¼ 16,000 r/min, without friction effect (a) and with friction

effect (b).

8 Proc IMechE Part K: J Multi-body Dynamics 227(1)

Additionally, the height peaks of chip thickness inFigure 10(b) are flattened.

The peak values of chip thickness for the chatterspindle speed are higher, this suggests that this par-ameter can be used as an index of stability. Similarconclusions can be drawn from Figure 11. If the thick-ness of a chip increases (near ap¼ 2mm) then theunstable region arises, especially in case of regenera-tive chatter (Figure 11(a)).

Conclusions and final remarks

This study presents the problem of chatter duringmilling process. The nonlinear one degree of freedommodel, which includes the regenerative and frictionalchatter is proposed. For comparison, the stabilitylobes diagram obtained from the proposed model ispresented together with SLD drawn by commercialsoftware CutPro9. On the basis of one-point modalanalysis, the modal parameters for the mathematicalmodel and SLD are obtained. On analyzing the clas-sical stability lobes of regenerative chatter, it is foundthat practically there is no difference when friction isadded (Figure 3). It is noticed that for certain spindle

speeds the friction may slightly reduce the criticaldepth-of-cut and can be the additional source ofinstability. For larger values of speed of cut, in theanalyzed cases the vibration amplitude in stable orunstable regions is smaller compared to classicalmodel with only regeneration component (Figures 6and 7). Numerical SLDs are similar to that of theresults from commercial software. However, thenumerical SLD depends on the assumed stability cri-terion (a technical indicator of stability). In practice,it is essential to find suitable indicator of the processstability to create the adequate numerical or experi-mental stability regions of milling process.

Obtained results suggest using a thickness ofchip or better, the ratio between depth-of-cut andchip thickness as index of stability (Figures 10 and11). This index seems most promising to apply inpractice.

In future work, the efficient method ofinstabilities detection and development of themethod of milling process stabilization will be studied.Additionally, this model will be expanded to twodegrees of freedom with consideration of the gyro-scopic effect.

Figure 10. Average thickness of chip for model without (a) and friction effect (b), for ap¼ 1.5 mm versus spindle speed of cutter.

Figure 11. Average thickness of a chip for the model without (a) and with friction (b), for n¼ 16,000 r/min versus depth-of-cut.

Kecik et al. 9

Funding

This work was supported by the European Regional

Development Fund - Project ‘‘Modern material technolo-gies in aerospace industry’’ (grant no. POIG.0101.02-00-015/08).

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Appendix

Notation

ae radial depth of cutap axial depth of cutapcr critical depth of cutc damping of tool-holder systemfz feed of cutterFnj radial cutting forceFtj tangential cutting forceFTj nonlinear function of the cutting forceFXj acting force in x directiongj function determinates cut or not cuthDYN dynamic chip thicknesshj chip thicknesshSTAT static chip thicknessH Heaviside functionk stiffness of tool-holder systemKn constant cutting coefficientKt constant cutting coefficientm mass of tool-holder systemn spindle speedt timevc cutting speedvr relative velocity between tool and

workpiecex amplitude vibrationz number of teeth of the cutter

10 Proc IMechE Part K: J Multi-body Dynamics 227(1)

� constant coefficient describes frictionshape curve

� constant coefficient describes frictionshape curve

� friction coefficient� material coefficient

� passing period’e exit angles of cutting’j rotation angle of cutter edge’s start angles of cutting!0 undamped natural frequency

Kecik et al. 11