International Journal of Machine Tools & Manufacture 43 (2003) 1229–1240

Programming spindle speed variation for machine tool chattersuppression

Emad Al-Regiba, Jun Nia,∗, Soo-Hun Leeb

a Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, 1023 H.H. Dow Building, 2300 Hayward Street,Ann Arbor, MI 48109, USA

b School of Mechanical and Industrial Engineering, Ajou University, Suwon, South Korea

Received 22 June 2000; received in revised form 15 April 2003; accepted 8 May 2003

Abstract

This paper presents a novel method for programming spindle speed variation for machine tool chatter suppression. This methodis based on varying the spindle speed for minimum energy input by the cutting process. The work done by the cutting force duringsinusoidal spindle speed variation S3V is solved numerically over a wide range of spindle speeds to study the effect of S3V onstable and unstable systems and to generate charts by which the optimum S3V amplitude ratio can be selected. For on-line application,a simple criterion for computing the optimal S3V amplitude ratio is presented. Also, a heuristic criterion for selecting the frequencyof the forcing speed signal is developed so that the resulting signal ensures fast stabilization of the machining process. The proposedcriteria are suitable for on-line chatter suppression, since they only require knowledge of the chatter frequency and spindle speed.The effectiveness of the developed S3V programming method is verified experimentally. 2003 Elsevier Ltd. All rights reserved.

Keywords:Chatter; Precision machining; Spindle speed variation; Stability; Numerical analysis; Vibration control

1. Introduction

One of the most significant factors affecting the per-formance of machine tools is chatter. Chatter not onlylimits productivity of cutting processes but also causespoor surface finish and reduced dimensional accuracy,increases the rate of tool wear, results in a noisy work-place and reduces the life of a machine tool. Chatter canbe avoided by keeping a low depth of cut, however thisleads to low productivity. Over the years, variousmethods have been developed to avoid regenerativechatter without reducing the depth of cut. The basic prin-ciple of these techniques is to prevent the dynamic ofthe machining process from locking on the most favor-able phase for chatter.

Slavicek[16] and Vanherck[25] proposed the use ofmilling cutters with non-uniform tooth pitch and Stone[20] used end mills with alternating helix. Effectiveness

∗ Corresponding author. Fax:+1-734-936-0363.E-mail address:junni@umich.edu (J. Ni).

0890-6955/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0890-6955(03)00126-3

of these methods in chatter suppression has been verifiedby simulation and experiments[3,6,23]. These tech-niques can be applied to the design of a non-uniformpitch cutter for a specific cutting condition, but cannotbe applied to single point machining.

Weck et al.[26] utilized on-line generated stabilitylobes to select a spindle speed so that maximizes thedepth-of-cut limit. Later, Smith and Tlusty[17], Delioet al. [2] and Tarng et al.[22] avoided the need for theknowledge of the stability lobes and proposed that thebest tooth passing frequency be made equal to the chatterfrequency. This minimizes the phase between the innerand outer modulations. This approach is adaptive in thesense that the spindle speed is changed based on feed-back measurement of the chatter frequency. This methodis practical for high spindle speed machining when thestability lobes are well separated.

Another technique to suppress regenerative chatter issinusoidal spindle speed variation (S3V) around themean speed to disturb the regenerative mechanism. Sincethis technique was introduced by Stoferle and Grab[19],there have been many research efforts to verify its effec-

1230 E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229–1240

tiveness on machining stability by numerical simulationand experiments in turning [7,8,14,15,21,28] and in mill-ing [1,8,9]. Despite the above research efforts, this tech-nique has not been implemented widely in industrybecause there is no systematic way to select the properamplitude and frequency of the sinusoidal forcing signal.The selection of these parameters depends on thedynamics of the machining system and is constrained bythe spindle-drive system response and its ability to trackthe forcing speed signal. In addition, variable speedmachining can result in an adverse effect and may evencause chatter in an otherwise stable process [5,10,15,24].This usually occurs when this method is applied to highspeed machining. Recently, Soliman and Ismail [18] pro-posed using fuzzy logic to select on-line the amplitudeand frequency of the forcing speed signal. Yilmaz et al.[27] generalized sinusoidal spindle speed variation tech-nique by introducing multi-level random spindle speedvariation, where the spindle speed is varied in randomfashion within the maximum amplitude ratio allowed bythe spindle-drive.

In this paper, a systematic procedure for designing astabilizing spindle speed, by selecting the effectiveamplitude and frequency of the forcing speed signal, isdeveloped. The remainder of the paper is summarizedas follows. In Section 2, the theoretical background ofmachining process modeling is reviewed. Based onenergy analysis, the effect of S3V amplitude ratio onstability is investigated in Section 3. The work done bythe cutting force during S3V is solved numerically overa wide range of spindle speed to generate charts bywhich the optimum S3V amplitude can be selected. Sec-tion 4 develops a simple criterion for computing the opti-mum S3V amplitude ratio, based on the spindle speedand chatter frequency. Also, a criterion to select theminimum effective S3V frequency is proposed. Section5 presents experimental verification results. Conclusionsfollow in Section 6.

2. Theoretical background

Machine tool chatter is a self-excited vibration causedby the interaction of the chip removal process and thestructure of the machine tool. The most important typeof chatter is regenerative chatter, which occurs mainlywhen a favorable phase relationship develops betweenthe inner and outer modulations caused by vibration dur-ing two consecutive tooth passes.

The conventional model of a single degree of freedommachining system is shown in Fig. 1. In this model, theresultant cutting force F(t) is proportional to the instan-taneous uncut chip thickness h(t) as expressed by:

Fx(t) � Kc b h(t), (1)

where b is the axial depth of cut and Kc is the static

Fig. 1. Model for single degree-of-freedom machining system.

cutting stiffness. The instantaneous uncut chip thicknessh(t) composed of the mean uncut chip thickness ho, theinner modulated cut surface x(t), due to the current toothpass, and the outer modulated surface x(t�t), due to theprevious tooth pass. Hence, the instantaneous uncut chipthickness can be written as

h(t) � h0 � x(t)�m x(t�t). (2)

Here t is the time delay between two consecutive cutsand represents the regenerative feedback effect. It isrelated to the spindle speed, S in (rpm), by

t �60zS

, (3)

where z is the number of teeth on the cutter (z = 1 inturning). The quantity zS/60 is the tooth passing fre-quency in (Hz) and m is the overlapping factor and willbe assumed here to be m = 1 for maximum regenerativeeffect. The structure dynamics is represented by the ori-ented transfer function G(s). The regenerative chatter canbe represented by control block diagram as shown inFig. 2 [11].

The closed–loop system can be represented by thesecond order system:

x(t) � 2zwnx(t) � w2nx(t) � Kc b [x(t)�x(t�t) (4)

� ho]

Fig. 2. Block diagram of regenerative chatter loop [13].

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where z is the damping ratio and wn is the natural fre-quency of the machining system. This model can be usedto study the stability of the machining system under bothconstant and variable spindle speeds where the spindlespeed S = S(t) and the time delay t = t(t) are time-vary-ing functions.

The characteristic equation of the constant speedmachining system can be derived as

1 � Kc b (1�e�ts)G(s) � 0. (5)

Let s = jw, then the above equation can be rewritten as

�12

� jsinwt

2(1�coswt)� Kc b [Real[G] (6)

� jImag[G]].

The quantity wt is the phase angle f (radians) of theregenerative wave on the machined surface, where w isthe vibration frequency (rad/sec). From Eq. (3), therelation between the spindle speed, vibration frequency,and the phase angle can be expressed:

wt � f �60wzS

. (7)

It is convenient to express this phase angle as aninteger number of waves N plus a fractional portion ofa wave e /2p such that if there are N + e /2p vibrationwaves during one revolution of the workpiece, whereN = 0, 1, 2,% and 0�e / 2p � 1, then the relationbetween the spindle speed, the vibration frequency, andthe phase angle is:

60 wzS

� f � 2p N � e (8)

Eqs. (6) and (8) can be utilized to generate the stabilitylobe diagram in terms of the limiting depth of cut vs.the spindle speed as shown in Fig. 3. In this figure, eachlobe corresponds to a number N = 0, 1, 2,%, where thesmaller the number, the higher the spindle speed.

The relationship between the fractional phase and

Fig. 3. Typical stability lobe diagram.

stability plays an important role in explaining the stabil-ization effect of variable speed on machining systems.This relationship is best described by energy analysis.In the next section, this relationship will be investigatedquantitatively to study the relation between the variablespindle speed signal and stability.

3. Energy-based stability analysis of variable speedmachining

It has been shown that varying the spindle speed usingsinusoidal function is the most feasible profile to sup-press chatter since it is more convenient for the spindle-drive system to track and easier for CNC realization[10]. Typical sinusoidal spindle speed signal has the fol-lowing form:

S(t) � Sm[1 � asin(2p fs t)] (9)

where Sm is the mean spindle speed in (rpm), a is theamplitude ratio, and fs is the signal frequency in Hz.

The application of variable spindle speed is con-strained by the spindle-drive system, which has a lowpass filter response to time-varying signals. The fre-quency of the time-varying signal should be within thebandwidth of the spindle-drive system, fs�wbw / 2p, andthe amplitude ratio is also a function of the spindle-drivesystem and usually constrained to values in the range0 � a � aa, where aa is the maximum allowable ampli-tude ratio.

The effect of varying the spindle speed on stabilitycan be studied by considering the balance between thework done by the regenerative cutting force and theenergy absorbed by the machine’s structural damping.The kinetic energy of the mass and the potential energyof the system are conservative over the vibration periodand hence, they are not considered. A machining systemis stable as long as the total change in the energy of thesystem due to the structural damping �Ud and due tothe work done by the regenerative force �UF being nega-tive over one cycle of vibration:

�Ud � �UF�0. (10)

Since varying the spindle speed affects the chip thick-ness, which in turn affects the cutting force, the workdone by the regenerative force will be computed. Here,for illustration purposes, the traditional one-dimensionalregenerative force model [15,24] will be considered.Assuming the machining system is vibrating with a fre-quency w and has a small sinusoidal displacement:

x(t) � Xcos(w t) (11)

then,

x(t) � �Xwsin(w t) (12)

and

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x(t�t) � Xcos(wt�wt). (13)

The validity of this assumption for variable speed mach-ining has been verified in Zhang [28].

For variable speed machining, the phase angle wt istime-varying [12,28], which is expressed using Eqs. (7)and (9) as:

wt �60 wzS(t)

�60 w

zSm[1 � asin(2pfst)]. (14)

Substituting in Eq. (13):

x(t�t) � Xcos�w t�60w

zSm[1 � asin(2pfst)]�. (15)

The work undertaken by the regenerative force overn integer cycles of vibration can be computed fromthe relation:

�UF � �2p n /w

0

�F(t) x(t) dt. (16)

Substituting for the regenerative force from Eqs. (1) and(2) into Eq. (16):

�UF � �2p n /w

0

�Kc b [x(t)�x(t�t) (17)

� h0 ] x(t) dt.

After substituting Eqs. (11), (12), and (15) into Eq. (17)

�UF � �Kcb�2p n /w

0�Xcos(wt) � Xcos�wt

�60w

zSm[1 � asin(2pfst)]� � h0�[�wXsin(w t)]dt

and performing part of the integration, the work doneby the regenerative force can be found to be:

�UF � �w Kc b X2 �2p n /w

0

(18)

cos�w t�60w

zSm[1 � asin(2pfst)]� sin(w t) dt

where the number of vibration cycles is expressed by[12]:

n �w

(2p)2fs. (19)

The average work done by the regenerative force overone vibratory cycle is computed by dividing Eq. (18) bythe number of cycles (n) from Eq. (19):

�UF � � (2p)2 fs Kcb X2 �2p n /w

0

(20)

cos�w t�60w

zSm[1 � asin(2pfst)]� sin(w t) dt.

The closed form solution of this equation has beenapproximated using Bessel functions [28] to investigatethe effect of S3V parameters on stability. Radulescu etal. [19] solved it numerically to explain qualitatively therobustness of variable speed machining on stability aug-mentation.

Here, Eq. (20) will be solved numerically for widerange of mean spindle speed by considering the follow-ing relation between the vibration frequency and themean spindle speed:

60 wzSm

� 2p Nm � em (21)

where Nm is the lobe number and em is the fractionalphase associated with the mean spindle speed. Substitut-ing in Eq. (20) and rearranging:

�UF

KcbX2 � � (2p)2 fs �2pn/w

0

(22)

cos�w t�2p Nm � em

[1 � asin(2pfst)]� sin(w t) dt.

For constant speed machining, where the amplituderatio is a = 0 and the number of cycles is n = 1, Eq.(22) can be solved analytically [4]:

�UF

KcbX2 � �sin em. (23)

When 0 � em � p, the variation in the chip thicknessleads the tool movement and consequently, the energyintroduced to the system in the cycle is smaller than theenergy dissipated and the system is stable. When p �em � 2p, the variation in the chip thickness lags the toolmovement and consequently, the energy introduced tothe system in the cycle is larger than the energy dissi-pated and the system is unstable [9,13,29].

For variable speed machining, Eq. (22) does not havea closed form solution. The right hand side of the aboveequation can be computed numerically to investigate theeffect of the variable spindle speed signal on the workdone by the regenerative force. Eq. (22) is only a func-tion of the lobe number Nm, the fractional phase em, andthe amplitude ratio a:

�UF

KcbX2 � fun(Nm,em,a) (24)

since varying the variable spindle speed signal’s fre-quency fs has no effect on the computed integration. Eq.(22) is solved numerically over wide range of Nm, em,and a to investigate the effect of S3V amplitude ratio onthe stability when S3V is applied to unstable and stablemachining systems especially at high speed machining.In addition, the optimum S3V amplitude ratio, whichminimizes the work undertaken by the regenerative cut-

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ting force in Eq. (22) is computed numerically and tabu-lated in charts as function of Nm and em.

3.1. S3V effect on stable system

In order to investigate the effect of S3V amplituderatio on the stability when S3V is applied to a stablesystem, the work done by the regenerative force is com-puted for the amplitude ratio in the range 0 � a �0.4, using a wide range of Nm while keeping the frac-tional phase in the stable range 0�em�p. Here, themaximum allowable amplitude ratio is assumed, aa =0.4 for illustration purposes. In Fig. 4, the results areplotted for mean spindle speeds with Nm = 0, 1,%, 10and fractional phase of em = p /2. This fractional phasecorresponds to applying S3V to the most stable casewhen the regenerative force under constant speed (a =0) results in maximum energy dissipation. This figureshows that applying S3V to a stable system causes theregenerative force to dissipate less energy compared tothe constant speed case and even deliver energy under

Fig. 4. Effect of S3V amplitude on the work done by the regenerativeforce when S3V is applied to a stable system.

certain amplitude ratios. For example, in Fig. 4 if S3Vwith amplitude ratio a5 is applied to the case whereNm = 5 and em = p /2, the regenerative work deliversmaximum energy compared to applying S3V with otheramplitude ratios in the allowable range 0 � a � 0.4.However, when S3V with amplitude ratio b5 is applied,the regenerative force dissipates energy. This offers anexplanation for the adverse effect of variable speedmachining on stability [5,10,15,24]. Similar results areobtained for other Nm and other values of the fractionalphase in the range 0�em�p. The above analysis showsthat it is not advantageous to apply S3V to stable system.

3.2. S3V effect on unstable system

The work done by the regenerative force when S3Vis applied to an unstable machining system is computednumerically from Eq. (22). In this case, the fractionalphase for the unstable machining system under constantspindle speed is in the range p � em � 2p. When S3Vis applied, the plot of the resulting work done vs. theamplitude ratio has a damped harmonic form. In Fig. 5,

Fig. 5. Effect of S3V amplitude on the work done by the regenerativeforce when S3V is applied to an unstable system.

1234 E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229–1240

the work done by the regenerative force is plotted formean spindle speeds with Nm = 0, 1,%, 10 and fractionalphase of em = 3p /2. This fractional phase correspondsto the maximum energy delivered by the regenerativeforce to the system under constant speed, i.e. for ampli-tude ratio a = 0. Fig. 5 shows that applying S3V to anunstable system always reduces the work done by theregenerative force compared to the constant speed caseand consequently S3V enhances the stability of the sys-tem. However, some amplitude ratios are more effectivethan others. For example, in Fig. 5 if S3V with amplituderatio a5 is applied to the case where Nm = 5, the regener-ative work dissipates maximum energy compared toapplying S3V with other amplitude ratios in the allow-able range of 0 � a � 0.4. However, when S3V withamplitude ratio b5 is applied, the regenerative forcedelivers energy. The figure also shows that there is onlya single optimal amplitude ratio that results in maximumenergy dissipation by the regenerative force. In Fig. 5,the points (a2, a3, a4, a5) correspond to the optimalamplitude ratios for the five cases under consideration.Also, it can be noticed from the figure that the higherthe nominal speed (the smaller Nm) the larger the ampli-tude ratio required for the work to start dissipatingenergy. For example, for Nm = 0,1, there is no optimalamplitude ratio in this allowable range 0 � a � 0.4.This explains why S3V is less effective in stabilizingmachining systems with low dominant frequency at highspindle speed than it is at lower speeds [14,17]. This isbecause at such high spindle speeds, applying S3V withamplitude ratio in the range of 0 � a � 0.4 alwayscauses the regenerative force to deliver energy to thesystem. However, this energy is less than the one deliv-ered under constant speed machining.

3.3. Optimal S3V amplitude ratio

In Fig. 6, the optimal S3V amplitude ratios are tabu-lated for a wide range of mean spindle speeds (Nm=2,%, 8 and p � em � 2p) by minimizing the work doneby the cutting force in Eq. (22) with respect to the ampli-tude ratio a:

∂ � �UF

KcbX2�∂ a

� 0 and∂2 � �UF

KcbX2�∂ a2 � 0. (25)

This figure can be used to select the optimal S3V ampli-tude ratio whenever chatter frequency is available bycomputing the lobe number Nm and the fractional phaseem from Eq. (21) and then selecting the correspondingoptimal amplitude ratio from Fig. 6.

The above analysis shows that selecting the properS3V amplitude is crucial when S3V is applied to suppresschatter in machining. This motivates the development ofa criterion for selecting the proper S3V amplitude ratio,

Fig. 6. Chart for selecting optimal S3V amplitude ratio for knownNm and em.

which can be practical for on-line chatter suppression.In the next section, a simple criterion to compute theoptimum S3V amplitude ratio will be presented.

4. Programming spindle speed variation

4.1. Criterion for computing the optimal S3Vamplitude

The energy-based analysis in the previous section sug-gests that the optimum S3V amplitude ratio which mini-mizes the work done by the cutting force is a functionof the lobe number Nm and the fractional phase em,whose relation to the mean spindle speed and chatterfrequency, wc = 2p fc, is described by em + 2p Nm =60 wc /zSm. The optimum S3V amplitude ratio in Fig. 6can be approximated with the following relation:

aopt �em

2p Nm

. (26)

This is a simple relation for computing the optimum S3Vamplitude ratio on-line since it requires only the knowl-edge of the spindle speed and chatter frequency.

It has been shown in Section 3.2 that the optimum S3Vamplitude ratio at high spindle speed machining (smallNm) is very high and may be beyond the range allowedby the spindle-drive system response. Consider themaximum spindle speed corresponding to the optimumS3V amplitude ratio computed from Eq. (26):

SMax |opt �60 wc

2p z Nm

. (27)

A closer look at Eq. (27), shows that changing thespindle speed to the maximum speed is equivalent tothe spindle speed selection method [17,22], where thestabilizing spindle speed is selected without variation so

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that the ratio between the chatter frequency and thespindle speed in (rad/sec) (tooth passing frequency inmilling) is an integer. Hence, when the optimal ampli-tude ratio computed using Eq. (26) is higher than theallowable range for the spindle-drive system, the spindlespeed selection method can be applied instead of theS3V.

4.2. Criterion for selecting S3V frequency

Although, S3V frequency is not as critical as theamplitude ratio [8,12], researchers showed by simulationand experiments that the effectiveness of S3V cannot berealized unless the S3V frequency is increased beyond aminimum value [7,8,14,15]. Also, selecting S3V fre-quency is constrained by the spindle-drive? system.Hence, a criterion for selecting the minimum effectiveS3V frequency is required to determine whether suchminimum value exceeds the bandwidth of the spindle-drive system.

Since the S3V frequency determines how fast theenergy is dissipated from the machining system whenvariable spindle speed is applied, the proposed criterioncan be stated as follows. The work done by the regener-ative force should start dissipating energy from the sys-tem within at most one rotation of the spindle (one toothpass in milling) after applying the spindle speed vari-ation. This means that if the variable spindle speed after

one tooth pass (at time t =60zSm

) is denoted by Sf, then

the corresponding fractional phase denoted by ef anddefined by the relation

ef � 2pNm �60 wc

z Sf

(28)

should reach the value ef = p (i.e. leaving the unstableregion p � em � 2p) within one spindle rotation afterapplying the S3V. The spindle speed at time (t = 60/zSm) is expressed by:

S(t) � S� 60z Sm

� � Sf �60 wc

(p � 2pNm) z. (29)

From the sinusoidal spindle speed function S(t), anotherrelation can be obtained for Sf:

S(t) � S� 60z Sm

� � Sf � Sm�1 � asin�2p fs60

z Sm�� (30)

where fs is the S3V frequency to be computed. Eqs. (29)and (30) can be solved simultaneously for the S3V fre-quency:

fs �z Sm

120 psin�1� 60 wc

a z Sm(p � 2pNm)�

1a�. (31)

For on-line application, this equation is computed usingthe optimal amplitude ratio obtained from Eq. (26).

5. Experimental results

5.1. Application to turning process

This section analyzes the effects of S3V on chattersuppression in turning process of a cylindrical work-piece. Experiments were carried out on a Novamat N50-1 horizontal CNC lathe with no tailstock. The cuttingconditions are shown in Table 1.

A LabView software module and an I/O board areused to generate the S3V signal. The signal is then sentthrough the spindle speed override to the CNC, whichcontrols the AC motor driving the lathe spindle throughbelt and gear group transmission. The input speed com-mand, actual spindle speed signal from the tachometer,and the acceleration signal from a PCB W353B15 (10mV/g sensitivity) accelerometer placed on top of the tur-ret were first passed through a PCB signal conditionerand recorded during the experiments.

In order to determine the allowable range for the para-meters of the sinusoidal signal, the spindle-drive sys-tem’s response to sinusoidal signal has been determinedexperimentally. The bandwidth of the spindle-drive sys-tem is found to be 1 Hz. This is the maximum allowablefrequency for the variable spindle speed signal. It is alsofound that the maximum allowable amplitude ratio is0.25.

5.1.1. S3V effect on stable systemIn this section, an experimental case is presented

where the S3V can have an adverse effect on machining.S3V with different amplitude ratios is applied at spindlespeed Sm = 600 rpm and depth of cut, b = 2 mm to astable process. Fig. 7 shows the acceleration signal forthe stable process under constant speed and the acceler-ation signals when S3V is applied with amplitude ratiosa = {0.05, 0.10, 0.20} and S3V frequency fs = 0.5 Hz.The experimental results reveal how the vibration ampli-tude increases with increasing the S3V amplitude ratio.In Fig. 8, the maximum power in the spectra correspond-ing to the signals in Fig. 7 is plotted with respect tothe S3V amplitude ratios. Fig. 9 depicts the workpiecetopography along with the Ra values of the surfaceroughness for the constant speed cutting and S3V casewith a = 0.20. The figure shows that the surface finish

Table 1Cutting conditions

Workpiece material Carbon steel 1018Workpiece dimensions Length, 180 mm; diameter, 35 mmCutting insert TP 100 coated Carboloy SNMA 644Insert dimensions 3/4� × 3 /4� × 1 /4� with 1/16�nose radiusTool holder Carboloy MSDNN85-6, neutral shank

with 45o side cutting edge angleFeed-rate 60 mm/rev

1236 E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229–1240

Fig. 7. Effect of S3V amplitude ratio on a stable process.

Fig. 8. Effect of S3V amplitude ratio on the spectrum of a stableprocess.

for the S3V case with a = 0.20 is worse than the onefor the constant speed case.

5.1.2. S3V amplitude ratio effect on unstable systemTo study the effect of S3V on unstable system exper-

imentally, S3V with different amplitude ratios is appliedat spindle speed Sm = 600 rpm and depth of cut, b = 5mm where the process is unstable under constant speedcutting. Fig. 10a shows the acceleration signal for the

Fig. 9. Effect of S3V amplitude ratio on the surface finish of a stableprocess.

Fig. 10. Effect of S3V amplitude ratio on an unstable process.

unstable process under constant speed. The chatter fre-quency is fc = 229 Hz, which corresponds to Nm = 22and em = 2p∗0.9. The optimal S3V amplitude ratio com-puted from Eq. (26) is aopt = 0.04.

Fig. 10 depicts the acceleration signals when S3V isapplied with amplitude ratios a = {0.02, 0.04, 0.10,0.20, 0.25} and S3V frequency, fs = 1.0 Hz. The experi-mental results illustrate how the vibration amplitudedecreases after increasing the S3V amplitude ratio toaopt = 0.04 and beyond. Comparing the signal for aopt

= 0.04 with the signals for higher S3V amplitude ratio,

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the figure shows that the acceleration signal for S3Vwith aopt = 0.04 does not have transients with relativelyhigh vibration amplitude. Fig. 11 shows the workpiecetopography along with the Ra values of the surfaceroughness for the constant speed case, S3V with aopt =0.04, and S3V with a = 0.20, respectively. The surfacefinishes are better for the cases with S3V cutting thanthe one for constant speed, and the surface finish corre-sponding to aopt = 0.04 has better quality than the onefor S3V with a = 0.20.

The maximum power in the spectra corresponding tothe signals in Fig. 10 are plotted with respect to the S3Vamplitude ratios in Fig. 12a. In this figure, the powerin the spectrum drops sharply after increasing the S3Vamplitude ratio to aopt = 0.04 and beyond. Also, the fig-ure shows that the spectrum corresponding for S3V withaopt = 0.04 has the minimum power.

Experiments are also conducted to investigate theeffect of S3V amplitude ratio on the power spectrumwhen cutting with higher spindle speeds where the lobenumber, Nm, is lower. Fig. 12b depicts the maximumpower in the spectra with respect to the S3V amplituderatios for spindle speed Sm = 1060 rpm and depth of cutb = 4 mm, where the process is unstable under constantspeed cutting. The chatter frequency in this case is fc= 242 Hz, which corresponds to Nm = 13 and em =2p∗0.7. The optimal S3V amplitude ratio can be com-puted from Eq. (26), aopt = 0.054. Fig. 12c shows theexperimental results for spindle speed Sm = 1300 rpmand depth of cut b = 4. The chatter frequency in thiscase is fc = 234.5 Hz, which corresponds to Nm = 10and em = 2p∗0.82. The optimal S3V amplitude ratiocomputed from Eq. (26) is aopt = 0.082. In all cases inFig. 12, the S3V cutting with the optimal amplitude ratiohas the minimum power in the spectrum compared tothe other cuttings.

5.1.3. S3V frequency effectTo study the effect of S3V frequency on unstable sys-

tem experimentally, S3V with different frequency isapplied at spindle speed Sm = 600 rpm and depth of cut,b = 5 mm, where the process is unstable under constantspeed cutting. Fig. 13a shows the acceleration signal for

Fig. 11. Effect of S3V amplitude ratio on the surface finish of anunstable process.

Fig. 12. Effect of S3V amplitude ratio on the spectrum when S3V isapplied to an unstable process (a)Sm = 600 rpm and b = 5 mm; (b)Sm = 1060 rpm and b = 4 mm; (c)Sm = 1300 rpm and b = 4 mm.

the unstable process under constant speed. The chatterfrequency is fc = 229 Hz, which corresponds to Nm =22 and em = 2p∗0.9.

Fig. 13 depicts the acceleration signals when S3V isapplied with frequency fs = {0.25, 0.50, 0.75, 1.00} Hzand S3V amplitude ratio, aopt = 0.04, which is the opti-mal amplitude ratio computed from Eq. (26). When thisratio is used to compute the minimum effective S3V fre-quency from Eq. (31), it results in fs = 0.72. The experi-mental results illustrate how the vibration amplitudedecreases with increasing the S3V frequency until fs =0.75 where after this value, the reduction in the vibrationamplitude is insignificant. Fig. 14 shows the workpiecetopography along with Ra values of the surface rough-ness for the constant speed case, S3V with fs = 0.25 Hz,and S3V with fs = 0.75 Hz, respectively. Although, the

1238 E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229–1240

Fig. 13. Effect of S3V frequency on an unstable process.

Fig. 14. Effect of S3V frequency on the surface finish of an unstableprocess.

surface finishes are better for the cases with S3V cuttingthan the one for constant speed, the surface finish corre-sponding to fs = 0.75 Hz has better quality than the onefor S3V with fs = 0.25 Hz. The maximum power in thespectra corresponding to the signals in Fig. 13 is plottedwith respect to the S3V frequency in Fig. 15a. In thisfigure, the power in the spectrum decreases with increas-ing the S3V frequency until fs = 0.75 Hz where after thisvalue, the power level almost stays the same.

5.1.4. Example: Application of S3V programming onchatter suppression

In this experiment, the proposed method to programS3V, by selecting the optimal effective amplitude ratioand effective frequency, is applied to suppress chatter inturning process of a cylindrical workpiece. The meanspindle speed is Sm = 1050 rpm and the depth of cut is

Fig. 15. Effect of S3V frequency on the spectrum of an unstable pro-cess.

b = 5 mm. Under constant speed cutting, chatterdevelops with frequency,fc = 250 Hz. When chatter isfully developed, S3V is applied with optimum amplituderatio, aopt = 0.05 and frequency fs = 0.9 Hz, which arecomputed from Eqs. (26) and (31), respectively.

Fig. 16a shows the acceleration signal during bothconstant and variable speed cutting. The actual spindlespeed signal from the tachometer is depicted in Fig. 16b.The spectra of the acceleration signals, before and afterapplying S3V, are shown in Fig. 16c. Fig. 16d depictsthe surface finish of the workpiece, during constantspeed and S3V cuttings, along with the correspondingsurface roughness, Ra values. These figures reveal clearcomparisons between the acceleration signal amplitudelevel together with the surface finish in the case of con-stant speed cutting and S3V cutting regions.

5.2. Application to boring process

The criteria developed in this paper to program S3Vare verified by comparing the results obtained from Eqs.(26) and (31) with experimental results from design ofexperiment approach conducted by an industrial partnerto set the effective S3V amplitude ratio and frequencyto suppress chatter in a boring process. The cutting con-ditions are shown in Table 2.

In this process, constant speed results in unstable cut-ting with a high-pitched sound and chatter marks areclearly visible on the machined surface. The chatter fre-quency of 200 Hz is clearly seen in the spectrum. Thespindle-drive system’s response to sinusoidal signal isdetermined experimentally. The bandwidth of thespindle-drive system is found to be 2.3 Hz and themaximum allowable amplitude ratio is 0.30.

In the dsign of experiment, S3V is applied with a rangeof S3V amplitude ratios, a = {0.05, 0.1, 0.15, 0.20,

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Fig. 16. Application of S3V programming on chatter suppression inturning process.

Table 2Cutting conditions

Workpiece material Carbon steel 4040Workpiece dimensions Length, 100 mm; inner diameter,

115 mm; outer diameter, 135 mmSpindle speed 673 rpmFeed-rate 300 mm/minDepth of cut 2 mm

0.25, 0.30}, and frequencies, fs = {0.5, 1.0, 1.5, 2.0}.That is, 24 combinations of amplitude and frequency areapplied consuming 24 workpieces. Out of the 24 combi-nations, the combination of S3V amplitude ratio a�exp

= 0.05 and frequency fs�exp = 0.5 Hz was the most suc-

cessful in suppressing the chatter while all other combi-nations resulted in minor improvement. In Fig. 17a–c,the input spindle speed, the accelerometer signal and itspower spectrum are shown for constant cutting and theS3V case (with amplitude ratio of a�exp = 0.05 and fre-quency fs�exp = 0.5 Hz). The intensity in the power spec-trum for S3V cutting is smaller than the one for the con-stant spindle speed. The surface finish of the workpieceis shown in Fig. 17d for both the constant spindle speedand S3V cases where the effect of S3V can be visualizedmore clearly.

In order to verify the effectiveness of the proposedcriteria in predicting the effective amplitude and fre-quency, Eqs. (26) and (31) are computed for chatter fre-quency fc = 200 Hz and mean spindle speed Sm = 673rpm to give aopt = 0.049 and fs = 0.69 Hz, respectively.These results are in close agreement with the parametersfound by the design of experiment procedure.

6. Conclusions

In this paper, a systematic procedure for programmingspindle speed variation signal’s amplitude and frequency

Fig. 17. Application of S3V programming on chatter suppression inboring process.

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is developed. The criteria for selecting the S3V ampli-tude ratio give results close to the optimum amplituderatio computed numerically from minimizing the workdone by the machining force. At high spindle speedmachining, the optimum S3V amplitude ratio is veryhigh and beyond the allowable range by availablespindle-drive systems. Hence, applying another tech-nique, such as the spindle speed selection method [17],is more feasible. The proposed criterion for selecting theS3V frequency is based on how fast the regenerativeenergy is dissipated from the machining system. Theproposed criteria are suitable for on-line S3V program-ming since the only requirement is knowledge of thechatter frequency and the spindle speed. The effective-ness of the developed method is verified experimentally.

Acknowledgements

The authors acknowledge the partial financial supportfrom NSF grant EEC-9526519 and the NSF-I/UCRCenter for Dimensional Measurement and Control inManufacturing at the University of Michigan. E. Al-Regib acknowledges the financial support by King Fai-sal Foundation.

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