Programming spindle speed variation for machine tool chatter suppression

International Journal of Machine Tools & Manufacture 43 (2003) 1229 1240 Programming spindle speed variation for machine tool chatter suppression Emad Al-Regib a, Jun Ni a,, Soo-Hun Lee b 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 method is based on varying the spindle speed for minimum energy input by the cutting process. The work done by the cutting force during sinusoidal spindle speed variation S 3 V is solved numerically over a wide range of spindle speeds to study the effect of S 3 Von stable and unstable systems and to generate charts by which the optimum S 3 V amplitude ratio can be selected. For on-line application, a simple criterion for computing the optimal S 3 V amplitude ratio is presented. Also, a heuristic criterion for selecting the frequency of the forcing speed signal is developed so that the resulting signal ensures fast stabilization of the machining process. The proposed criteria are suitable for on-line chatter suppression, since they only require knowledge of the chatter frequency and spindle speed. The effectiveness of the developed S 3 V 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 performance of machine tools is chatter. Chatter not only limits productivity of cutting processes but also causes poor surface finish and reduced dimensional accuracy, increases the rate of tool wear, results in a noisy workplace and reduces the life of a machine tool. Chatter can be avoided by keeping a low depth of cut, however this leads to low productivity. Over the years, various methods have been developed to avoid regenerative chatter without reducing the depth of cut. The basic principle of these techniques is to prevent the dynamic of the machining process from locking on the most favorable phase for chatter. Slavicek [16] and Vanherck [25] proposed the use of milling 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). of these methods in chatter suppression has been verified by simulation and experiments [3,6,23]. These techniques can be applied to the design of a non-uniform pitch cutter for a specific cutting condition, but cannot be applied to single point machining. Weck et al. [26] utilized on-line generated stability lobes to select a spindle speed so that maximizes the depth-of-cut limit. Later, Smith and Tlusty [17], Delio et al. [2] and Tarng et al. [22] avoided the need for the knowledge of the stability lobes and proposed that the best tooth passing frequency be made equal to the chatter frequency. This minimizes the phase between the inner and outer modulations. This approach is adaptive in the sense that the spindle speed is changed based on feedback measurement of the chatter frequency. This method is practical for high spindle speed machining when the stability lobes are well separated. Another technique to suppress regenerative chatter is sinusoidal spindle speed variation (S 3 V) around the mean speed to disturb the regenerative mechanism. Since this technique was introduced by Stoferle and Grab [19], there have been many research efforts to verify its effec- 0890-6955/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/s0890-6955(03)00126-3

1230 E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229 1240 tiveness on machining stability by numerical simulation and experiments in turning [7,8,14,15,21,28] and in milling [1,8,9]. Despite the above research efforts, this technique has not been implemented widely in industry because there is no systematic way to select the proper amplitude and frequency of the sinusoidal forcing signal. The selection of these parameters depends on the dynamics of the machining system and is constrained by the spindle-drive system response and its ability to track the forcing speed signal. In addition, variable speed machining can result in an adverse effect and may even cause chatter in an otherwise stable process [5,10,15,24]. This usually occurs when this method is applied to high speed machining. Recently, Soliman and Ismail [18] proposed using fuzzy logic to select on-line the amplitude and frequency of the forcing speed signal. Yilmaz et al. [27] generalized sinusoidal spindle speed variation technique by introducing multi-level random spindle speed variation, where the spindle speed is varied in random fashion within the maximum amplitude ratio allowed by the spindle-drive. In this paper, a systematic procedure for designing a stabilizing spindle speed, by selecting the effective amplitude and frequency of the forcing speed signal, is developed. The remainder of the paper is summarized as follows. In Section 2, the theoretical background of machining process modeling is reviewed. Based on energy analysis, the effect of S 3 V amplitude ratio on stability is investigated in Section 3. The work done by the cutting force during S 3 V is solved numerically over a wide range of spindle speed to generate charts by which the optimum S 3 V amplitude can be selected. Section 4 develops a simple criterion for computing the optimum S 3 V amplitude ratio, based on the spindle speed and chatter frequency. Also, a criterion to select the minimum effective S 3 V frequency is proposed. Section 5 presents experimental verification results. Conclusions follow in Section 6. 2. Theoretical background Machine tool chatter is a self-excited vibration caused by the interaction of the chip removal process and the structure of the machine tool. The most important type of chatter is regenerative chatter, which occurs mainly when a favorable phase relationship develops between the inner and outer modulations caused by vibration during two consecutive tooth passes. The conventional model of a single degree of freedom machining system is shown in Fig. 1. In this model, the resultant cutting force F(t) is proportional to the instantaneous uncut chip thickness h(t) as expressed by: F x (t) K c bh(t), (1) where b is the axial depth of cut and K c is the static Fig. 1. Model for single degree-of-freedom machining system. cutting stiffness. The instantaneous uncut chip thickness h(t) composed of the mean uncut chip thickness h o, the inner modulated cut surface x(t), due to the current tooth pass, and the outer modulated surface x(t t), due to the previous tooth pass. Hence, the instantaneous uncut chip thickness can be written as h(t) h 0 x(t) m x(t t). (2) Here t is the time delay between two consecutive cuts and represents the regenerative feedback effect. It is related to the spindle speed, S in (rpm), by t 60 zs, (3) where z is the number of teeth on the cutter (z = 1in turning). The quantity zs/60 is the tooth passing frequency in (Hz) and m is the overlapping factor and will be assumed here to be m = 1 for maximum regenerative effect. The structure dynamics is represented by the oriented transfer function G(s). The regenerative chatter can be represented by control block diagram as shown in Fig. 2 [11]. The closed loop system can be represented by the second order system: ẍ(t) 2zw n ẋ(t) w 2 nx(t) K c b [x(t) x(t t) (4) h o ] Fig. 2. Block diagram of regenerative chatter loop [13].

E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229 1240 1231 where z is the damping ratio and w n is the natural frequency of the machining system. This model can be used to study the stability of the machining system under both constant and variable spindle speeds where the spindle speed S = S(t) and the time delay t = t(t) are time-varying functions. The characteristic equation of the constant speed machining system can be derived as 1 K c b (1 e ts )G(s) 0. (5) Let s = jw, then the above equation can be rewritten as 1 2 j sinwt 2(1 coswt) K c b [Real[G] (6) jimag[g]]. The quantity wt is the phase angle f (radians) of the regenerative wave on the machined surface, where w is the vibration frequency (rad/sec). From Eq. (3), the relation between the spindle speed, vibration frequency, and the phase angle can be expressed: wt f 60w zs. (7) It is convenient to express this phase angle as an integer number of waves N plus a fractional portion of a wave e/2p such that if there are N + e/2p vibration waves during one revolution of the workpiece, where N = 0, 1, 2,% and 0 e/2p 1, then the relation between the spindle speed, the vibration frequency, and the phase angle is: 60 w f 2p N e (8) zs Eqs. (6) and (8) can be utilized to generate the stability lobe diagram in terms of the limiting depth of cut vs. the spindle speed as shown in Fig. 3. In this figure, each lobe corresponds to a number N = 0, 1, 2,%, where the smaller 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 stabilization effect of variable speed on machining systems. This relationship is best described by energy analysis. In the next section, this relationship will be investigated quantitatively to study the relation between the variable spindle speed signal and stability. 3. Energy-based stability analysis of variable speed machining It has been shown that varying the spindle speed using sinusoidal function is the most feasible profile to suppress chatter since it is more convenient for the spindledrive system to track and easier for CNC realization [10]. Typical sinusoidal spindle speed signal has the following form: S(t) S m [1 asin(2p f s t)] (9) where S m is the mean spindle speed in (rpm), a is the amplitude ratio, and f s is the signal frequency in Hz. The application of variable spindle speed is constrained by the spindle-drive system, which has a low pass filter response to time-varying signals. The frequency of the time-varying signal should be within the bandwidth of the spindle-drive system, f s w bw /2p, and the amplitude ratio is also a function of the spindle-drive system and usually constrained to values in the range 0 a a a, where a a is the maximum allowable amplitude ratio. The effect of varying the spindle speed on stability can be studied by considering the balance between the work done by the regenerative cutting force and the energy absorbed by the machine s structural damping. The kinetic energy of the mass and the potential energy of the system are conservative over the vibration period and hence, they are not considered. A machining system is stable as long as the total change in the energy of the system due to the structural damping U d and due to the work done by the regenerative force U F being negative over one cycle of vibration: U d U F 0. (10) Since varying the spindle speed affects the chip thickness, which in turn affects the cutting force, the work done by the regenerative force will be computed. Here, for illustration purposes, the traditional one-dimensional regenerative force model [15,24] will be considered. Assuming the machining system is vibrating with a frequency w and has a small sinusoidal displacement: x(t) Xcos(w t) (11) then, ẋ(t) Xwsin(w t) (12) and

1232 E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229 1240 x(t t) Xcos(wt wt). (13) The validity of this assumption for variable speed machining has been verified in Zhang [28]. For variable speed machining, the phase angle wt is time-varying [12,28], which is expressed using Eqs. (7) and (9) as: wt 60 w zs(t) 60 w zs m [1 asin(2pf s t)]. (14) Substituting in Eq. (13): 60w x(t t) Xcos w t (15) zs m [1 asin(2pf s t)]. The work undertaken by the regenerative force over n integer cycles of vibration can be computed from the relation: n/w U F 2p F(t) ẋ(t) dt. (16) 0 Substituting for the regenerative force from Eqs. (1) and (2) into Eq. (16): n/w U F 2p K c b [x(t) x(t t) (17) 0 h 0 ] ẋ(t) dt. After substituting Eqs. (11), (12), and (15) into Eq. (17) U F K c b 2p n/w 0 Xcos(wt) Xcos wt 60w h zs m [1 asin(2pf s t)] 0 [ wxsin(w t)]dt and performing part of the integration, the work done by the regenerative force can be found to be: U F w K c bx 2 2p n/w 0 (18) 60w cos w t sin(w t) dt zs m [1 asin(2pf s t)] where the number of vibration cycles is expressed by [12]: n w. (19) (2p) 2 f s The average work done by the regenerative force over one vibratory cycle is computed by dividing Eq. (18) by the number of cycles (n) from Eq. (19): U F (2p) 2 f s K c bx 2 2p n/w 60w cos w t sin(w t) dt. zs m [1 asin(2pf s t)] 0 (20) The closed form solution of this equation has been approximated using Bessel functions [28] to investigate the effect of S 3 V parameters on stability. Radulescu et al. [19] solved it numerically to explain qualitatively the robustness of variable speed machining on stability augmentation. Here, Eq. (20) will be solved numerically for wide range of mean spindle speed by considering the following relation between the vibration frequency and the mean spindle speed: 60 w 2p N zs m e m (21) m where N m is the lobe number and e m is the fractional phase associated with the mean spindle speed. Substituting in Eq. (20) and rearranging: 2pn/w U F K c bx 2 (2p)2 f s (22) 0 cos w t 2p N m e m sin(w t) dt. [1 asin(2pf s t)] For constant speed machining, where the amplitude ratio is a = 0 and the number of cycles is n = 1, Eq. (22) can be solved analytically [4]: U F K c bx sin e m. (23) 2 When 0 e m p, the variation in the chip thickness leads the tool movement and consequently, the energy introduced to the system in the cycle is smaller than the energy dissipated and the system is stable. When p e m 2p, the variation in the chip thickness lags the tool movement and consequently, the energy introduced to the system in the cycle is larger than the energy dissipated and the system is unstable [9,13,29]. For variable speed machining, Eq. (22) does not have a closed form solution. The right hand side of the above equation can be computed numerically to investigate the effect of the variable spindle speed signal on the work done by the regenerative force. Eq. (22) is only a function of the lobe number N m, the fractional phase e m, and the amplitude ratio a: U F K c bx 2 fun(n m,e m,a) (24) since varying the variable spindle speed signal s frequency f s has no effect on the computed integration. Eq. (22) is solved numerically over wide range of N m, e m, and a to investigate the effect of S 3 V amplitude ratio on the stability when S 3 V is applied to unstable and stable machining systems especially at high speed machining. In addition, the optimum S 3 V amplitude ratio, which minimizes the work undertaken by the regenerative cut-

E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229 1240 1233 ting force in Eq. (22) is computed numerically and tabulated in charts as function of N m and e m. 3.1. S 3 V effect on stable system In order to investigate the effect of S 3 V amplitude ratio on the stability when S 3 V is applied to a stable system, the work done by the regenerative force is computed for the amplitude ratio in the range 0 a 0.4, using a wide range of N m while keeping the fractional phase in the stable range 0 e m p. Here, the maximum allowable amplitude ratio is assumed, a a = 0.4 for illustration purposes. In Fig. 4, the results are plotted for mean spindle speeds with N m = 0, 1,%, 10 and fractional phase of e m = p/2. This fractional phase corresponds to applying S 3 V to the most stable case when the regenerative force under constant speed (a = 0) results in maximum energy dissipation. This figure shows that applying S 3 V to a stable system causes the regenerative force to dissipate less energy compared to the constant speed case and even deliver energy under certain amplitude ratios. For example, in Fig. 4 if S 3 V with amplitude ratio a5 is applied to the case where N m = 5 and e m = p/2, the regenerative work delivers maximum energy compared to applying S 3 V with other amplitude ratios in the allowable range 0 a 0.4. However, when S 3 V with amplitude ratio b5 is applied, the regenerative force dissipates energy. This offers an explanation for the adverse effect of variable speed machining on stability [5,10,15,24]. Similar results are obtained for other N m and other values of the fractional phase in the range 0 e m p. The above analysis shows that it is not advantageous to apply S 3 V to stable system. 3.2. S 3 V effect on unstable system The work done by the regenerative force when S 3 V is applied to an unstable machining system is computed numerically from Eq. (22). In this case, the fractional phase for the unstable machining system under constant spindle speed is in the range p e m 2p. When S 3 V is applied, the plot of the resulting work done vs. the amplitude ratio has a damped harmonic form. In Fig. 5, Fig. 4. Effect of S 3 V amplitude on the work done by the regenerative force when S 3 V is applied to a stable system. Fig. 5. Effect of S 3 V amplitude on the work done by the regenerative force when S 3 V 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 for mean spindle speeds with N m = 0, 1,%, 10 and fractional phase of e m = 3p/2. This fractional phase corresponds to the maximum energy delivered by the regenerative force to the system under constant speed, i.e. for amplitude ratio a = 0. Fig. 5 shows that applying S 3 Vtoan unstable system always reduces the work done by the regenerative force compared to the constant speed case and consequently S 3 V enhances the stability of the system. However, some amplitude ratios are more effective than others. For example, in Fig. 5 if S 3 V with amplitude ratio a5 is applied to the case where N m = 5, the regenerative work dissipates maximum energy compared to applying S 3 V with other amplitude ratios in the allowable range of 0 a 0.4. However, when S 3 V with amplitude ratio b5 is applied, the regenerative force delivers energy. The figure also shows that there is only a single optimal amplitude ratio that results in maximum energy dissipation by the regenerative force. In Fig. 5, the points (a2, a3, a4, a5) correspond to the optimal amplitude ratios for the five cases under consideration. Also, it can be noticed from the figure that the higher the nominal speed (the smaller N m ) the larger the amplitude ratio required for the work to start dissipating energy. For example, for N m = 0,1, there is no optimal amplitude ratio in this allowable range 0 a 0.4. This explains why S 3 V is less effective in stabilizing machining systems with low dominant frequency at high spindle speed than it is at lower speeds [14,17]. This is because at such high spindle speeds, applying S 3 V with amplitude ratio in the range of 0 a 0.4 always causes the regenerative force to deliver energy to the system. However, this energy is less than the one delivered under constant speed machining. 3.3. Optimal S 3 V amplitude ratio In Fig. 6, the optimal S 3 V amplitude ratios are tabulated for a wide range of mean spindle speeds (N m = 2,%, 8 and p e m 2p) by minimizing the work done by the cutting force in Eq. (22) with respect to the amplitude ratio a: U F K c bx 2 a 2 0 and U F K c bx 2 a 2 0. (25) This figure can be used to select the optimal S 3 V amplitude ratio whenever chatter frequency is available by computing the lobe number N m and the fractional phase e m from Eq. (21) and then selecting the corresponding optimal amplitude ratio from Fig. 6. The above analysis shows that selecting the proper S 3 V amplitude is crucial when S 3 V is applied to suppress chatter in machining. This motivates the development of a criterion for selecting the proper S 3 V amplitude ratio, Fig. 6. Chart for selecting optimal S 3 V amplitude ratio for known N m and e m. which can be practical for on-line chatter suppression. In the next section, a simple criterion to compute the optimum S 3 V amplitude ratio will be presented. 4. Programming spindle speed variation 4.1. Criterion for computing the optimal S 3 V amplitude The energy-based analysis in the previous section suggests that the optimum S 3 V amplitude ratio which minimizes the work done by the cutting force is a function of the lobe number N m and the fractional phase e m, whose relation to the mean spindle speed and chatter frequency, w c = 2p f c, is described by e m +2p N m = 60 w c /zs m. The optimum S 3 V amplitude ratio in Fig. 6 can be approximated with the following relation: e m a opt. (26) 2p N m This is a simple relation for computing the optimum S 3 V amplitude ratio on-line since it requires only the knowledge of the spindle speed and chatter frequency. It has been shown in Section 3.2 that the optimum S 3 V amplitude ratio at high spindle speed machining (small N m ) is very high and may be beyond the range allowed by the spindle-drive system response. Consider the maximum spindle speed corresponding to the optimum S 3 V amplitude ratio computed from Eq. (26): S Max opt 60 w c 2p zn m. (27) A closer look at Eq. (27), shows that changing the spindle speed to the maximum speed is equivalent to the spindle speed selection method [17,22], where the stabilizing spindle speed is selected without variation so

E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229 1240 1235 that the ratio between the chatter frequency and the spindle speed in (rad/sec) (tooth passing frequency in milling) is an integer. Hence, when the optimal amplitude ratio computed using Eq. (26) is higher than the allowable range for the spindle-drive system, the spindle speed selection method can be applied instead of the S 3 V. 4.2. Criterion for selecting S 3 V frequency Although, S 3 V frequency is not as critical as the amplitude ratio [8,12], researchers showed by simulation and experiments that the effectiveness of S 3 V cannot be realized unless the S 3 V frequency is increased beyond a minimum value [7,8,14,15]. Also, selecting S 3 V frequency is constrained by the spindle-drive? system. Hence, a criterion for selecting the minimum effective S 3 V frequency is required to determine whether such minimum value exceeds the bandwidth of the spindledrive system. Since the S 3 V frequency determines how fast the energy is dissipated from the machining system when variable spindle speed is applied, the proposed criterion can be stated as follows. The work done by the regenerative force should start dissipating energy from the system within at most one rotation of the spindle (one tooth pass in milling) after applying the spindle speed variation. This means that if the variable spindle speed after one tooth pass (at time t = 60 zs m ) is denoted by S f, then the corresponding fractional phase denoted by e f and defined by the relation e f 2pN m 60 w c (28) zs f should reach the value e f = p (i.e. leaving the unstable region p e m 2p) within one spindle rotation after applying the S 3 V. The spindle speed at time (t = 60/z S m ) is expressed by: S(t) S 60 60 w zs m c S f (p 2pN m ) z. (29) From the sinusoidal spindle speed function S(t), another relation can be obtained for S f : S(t) S 60 zs m S f S m 1 asin 2p f s 60 zs m (30) where f s is the S 3 V frequency to be computed. Eqs. (29) and (30) can be solved simultaneously for the S 3 V frequency: sin 1 f s zs m 60 w c 120 p a zs m (p 2pN m ) 1 (31) a. For on-line application, this equation is computed using the optimal amplitude ratio obtained from Eq. (26). 5. Experimental results 5.1. Application to turning process This section analyzes the effects of S 3 V on chatter suppression in turning process of a cylindrical workpiece. Experiments were carried out on a Novamat N 50-1 horizontal CNC lathe with no tailstock. The cutting conditions are shown in Table 1. A LabView software module and an I/O board are used to generate the S 3 V signal. The signal is then sent through the spindle speed override to the CNC, which controls the AC motor driving the lathe spindle through belt and gear group transmission. The input speed command, actual spindle speed signal from the tachometer, and the acceleration signal from a PCB W353B15 (10 mv/g sensitivity) accelerometer placed on top of the turret were first passed through a PCB signal conditioner and recorded during the experiments. In order to determine the allowable range for the parameters of the sinusoidal signal, the spindle-drive system s response to sinusoidal signal has been determined experimentally. The bandwidth of the spindle-drive system is found to be 1 Hz. This is the maximum allowable frequency for the variable spindle speed signal. It is also found that the maximum allowable amplitude ratio is 0.25. 5.1.1. S 3 V effect on stable system In this section, an experimental case is presented where the S 3 V can have an adverse effect on machining. S 3 V with different amplitude ratios is applied at spindle speed S m = 600 rpm and depth of cut, b = 2mmtoa stable process. Fig. 7 shows the acceleration signal for the stable process under constant speed and the acceleration signals when S 3 V is applied with amplitude ratios a = {0.05, 0.10, 0.20} and S 3 V frequency f s = 0.5 Hz. The experimental results reveal how the vibration amplitude increases with increasing the S 3 V amplitude ratio. In Fig. 8, the maximum power in the spectra corresponding to the signals in Fig. 7 is plotted with respect to the S 3 V amplitude ratios. Fig. 9 depicts the workpiece topography along with the R a values of the surface roughness for the constant speed cutting and S 3 V case with a = 0.20. The figure shows that the surface finish Table 1 Cutting conditions Workpiece material Carbon steel 1018 Workpiece dimensions Length, 180 mm; diameter, 35 mm Cutting insert TP 100 coated Carboloy SNMA 644 Insert dimensions 3/4 3/4 1/4 with 1/16 nose radius Tool holder Carboloy MSDNN85-6, neutral shank with 45 o side cutting edge angle Feed-rate 60 mm/rev

1236 E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229 1240 Fig. 9. Effect of S 3 V amplitude ratio on the surface finish of a stable process. Fig. 7. Effect of S 3 V amplitude ratio on a stable process. Fig. 8. Effect of S 3 V amplitude ratio on the spectrum of a stable process. for the S 3 V case with a = 0.20 is worse than the one for the constant speed case. 5.1.2. S 3 V amplitude ratio effect on unstable system To study the effect of S 3 V on unstable system experimentally, S 3 V with different amplitude ratios is applied at spindle speed S m = 600 rpm and depth of cut, b = 5 mm where the process is unstable under constant speed cutting. Fig. 10a shows the acceleration signal for the Fig. 10. Effect of S 3 V amplitude ratio on an unstable process. unstable process under constant speed. The chatter frequency is f c = 229 Hz, which corresponds to N m = 22 and e m = 2p 0.9. The optimal S 3 V amplitude ratio computed from Eq. (26) is a opt = 0.04. Fig. 10 depicts the acceleration signals when S 3 Vis applied with amplitude ratios a = {0.02, 0.04, 0.10, 0.20, 0.25} and S 3 V frequency, f s = 1.0 Hz. The experimental results illustrate how the vibration amplitude decreases after increasing the S 3 V amplitude ratio to a opt = 0.04 and beyond. Comparing the signal for a opt = 0.04 with the signals for higher S 3 V amplitude ratio,

E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229 1240 1237 the figure shows that the acceleration signal for S 3 V with a opt = 0.04 does not have transients with relatively high vibration amplitude. Fig. 11 shows the workpiece topography along with the R a values of the surface roughness for the constant speed case, S 3 V with a opt = 0.04, and S 3 V with a = 0.20, respectively. The surface finishes are better for the cases with S 3 V cutting than the one for constant speed, and the surface finish corresponding to a opt = 0.04 has better quality than the one for S 3 V with a = 0.20. The maximum power in the spectra corresponding to the signals in Fig. 10 are plotted with respect to the S 3 V amplitude ratios in Fig. 12a. In this figure, the power in the spectrum drops sharply after increasing the S 3 V amplitude ratio to a opt = 0.04 and beyond. Also, the figure shows that the spectrum corresponding for S 3 V with a opt = 0.04 has the minimum power. Experiments are also conducted to investigate the effect of S 3 V amplitude ratio on the power spectrum when cutting with higher spindle speeds where the lobe number, N m, is lower. Fig. 12b depicts the maximum power in the spectra with respect to the S 3 V amplitude ratios for spindle speed S m = 1060 rpm and depth of cut b = 4 mm, where the process is unstable under constant speed cutting. The chatter frequency in this case is f c = 242 Hz, which corresponds to N m = 13 and e m = 2p 0.7. The optimal S 3 V amplitude ratio can be computed from Eq. (26), a opt = 0.054. Fig. 12c shows the experimental results for spindle speed S m = 1300 rpm and depth of cut b = 4. The chatter frequency in this case is f c = 234.5 Hz, which corresponds to N m = 10 and e m = 2p 0.82. The optimal S 3 V amplitude ratio computed from Eq. (26) is a opt = 0.082. In all cases in Fig. 12, the S 3 V cutting with the optimal amplitude ratio has the minimum power in the spectrum compared to the other cuttings. 5.1.3. S 3 V frequency effect To study the effect of S 3 V frequency on unstable system experimentally, S 3 V with different frequency is applied at spindle speed S m = 600 rpm and depth of cut, b = 5 mm, where the process is unstable under constant speed cutting. Fig. 13a shows the acceleration signal for Fig. 11. Effect of S 3 V amplitude ratio on the surface finish of an unstable process. Fig. 12. Effect of S 3 V amplitude ratio on the spectrum when S 3 Vis applied to an unstable process (a)s m = 600 rpm and b = 5 mm; (b) S m = 1060 rpm and b = 4 mm; (c)s m = 1300 rpm and b = 4 mm. the unstable process under constant speed. The chatter frequency is f c = 229 Hz, which corresponds to N m = 22 and e m = 2p 0.9. Fig. 13 depicts the acceleration signals when S 3 Vis applied with frequency f s = {0.25, 0.50, 0.75, 1.00} Hz and S 3 V amplitude ratio, a opt = 0.04, which is the optimal amplitude ratio computed from Eq. (26). When this ratio is used to compute the minimum effective S 3 V frequency from Eq. (31), it results in f s = 0.72. The experimental results illustrate how the vibration amplitude decreases with increasing the S 3 V frequency until f s = 0.75 where after this value, the reduction in the vibration amplitude is insignificant. Fig. 14 shows the workpiece topography along with R a values of the surface roughness for the constant speed case, S 3 V with f s = 0.25 Hz, and S 3 V with f s = 0.75 Hz, respectively. Although, the

1238 E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229 1240 Effect of S 3 V frequency on the spectrum of an unstable pro- Fig. 15. cess. Fig. 14. process. Fig. 13. Effect of S 3 V frequency on an unstable process. Effect of S 3 V frequency on the surface finish of an unstable surface finishes are better for the cases with S 3 V cutting than the one for constant speed, the surface finish corresponding to f s = 0.75 Hz has better quality than the one for S 3 V with f s = 0.25 Hz. The maximum power in the spectra corresponding to the signals in Fig. 13 is plotted with respect to the S 3 V frequency in Fig. 15a. In this figure, the power in the spectrum decreases with increasing the S 3 V frequency until f s = 0.75 Hz where after this value, the power level almost stays the same. 5.1.4. Example: Application of S 3 V programming on chatter suppression In this experiment, the proposed method to program S 3 V, by selecting the optimal effective amplitude ratio and effective frequency, is applied to suppress chatter in turning process of a cylindrical workpiece. The mean spindle speed is S m = 1050 rpm and the depth of cut is b = 5 mm. Under constant speed cutting, chatter develops with frequency,f c = 250 Hz. When chatter is fully developed, S 3 V is applied with optimum amplitude ratio, a opt = 0.05 and frequency f s = 0.9 Hz, which are computed from Eqs. (26) and (31), respectively. Fig. 16a shows the acceleration signal during both constant and variable speed cutting. The actual spindle speed signal from the tachometer is depicted in Fig. 16b. The spectra of the acceleration signals, before and after applying S 3 V, are shown in Fig. 16c. Fig. 16d depicts the surface finish of the workpiece, during constant speed and S 3 V cuttings, along with the corresponding surface roughness, R a values. These figures reveal clear comparisons between the acceleration signal amplitude level together with the surface finish in the case of constant speed cutting and S 3 V cutting regions. 5.2. Application to boring process The criteria developed in this paper to program S 3 V are verified by comparing the results obtained from Eqs. (26) and (31) with experimental results from design of experiment approach conducted by an industrial partner to set the effective S 3 V amplitude ratio and frequency to suppress chatter in a boring process. The cutting conditions are shown in Table 2. In this process, constant speed results in unstable cutting with a high-pitched sound and chatter marks are clearly visible on the machined surface. The chatter frequency of 200 Hz is clearly seen in the spectrum. The spindle-drive system s response to sinusoidal signal is determined experimentally. The bandwidth of the spindle-drive system is found to be 2.3 Hz and the maximum allowable amplitude ratio is 0.30. In the dsign of experiment, S 3 V is applied with a range of S 3 V amplitude ratios, a = {0.05, 0.1, 0.15, 0.20,

E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229 1240 1239 0.25, 0.30}, and frequencies, f s = {0.5, 1.0, 1.5, 2.0}. That is, 24 combinations of amplitude and frequency are applied consuming 24 workpieces. Out of the 24 combinations, the combination of S 3 V amplitude ratio a exp = 0.05 and frequency f s exp = 0.5 Hz was the most successful in suppressing the chatter while all other combinations resulted in minor improvement. In Fig. 17a c, the input spindle speed, the accelerometer signal and its power spectrum are shown for constant cutting and the S 3 V case (with amplitude ratio of a exp = 0.05 and frequency f s exp = 0.5 Hz). The intensity in the power spectrum for S 3 V cutting is smaller than the one for the constant spindle speed. The surface finish of the workpiece is shown in Fig. 17d for both the constant spindle speed and S 3 V cases where the effect of S 3 V can be visualized more clearly. In order to verify the effectiveness of the proposed criteria in predicting the effective amplitude and frequency, Eqs. (26) and (31) are computed for chatter frequency f c = 200 Hz and mean spindle speed S m = 673 rpm to give a opt = 0.049 and f s = 0.69 Hz, respectively. These results are in close agreement with the parameters found by the design of experiment procedure. 6. Conclusions In this paper, a systematic procedure for programming spindle speed variation signal s amplitude and frequency Fig. 16. Application of S 3 V programming on chatter suppression in turning process. Table 2 Cutting conditions Workpiece material Carbon steel 4040 Workpiece dimensions Length, 100 mm; inner diameter, 115 mm; outer diameter, 135 mm Spindle speed 673 rpm Feed-rate 300 mm/min Depth of cut 2 mm Fig. 17. Application of S 3 V programming on chatter suppression in boring process.

1240 E. Al-Regib et al. / International Journal of Machine Tools & Manufacture 43 (2003) 1229 1240 is developed. The criteria for selecting the S 3 V amplitude ratio give results close to the optimum amplitude ratio computed numerically from minimizing the work done by the machining force. At high spindle speed machining, the optimum S 3 V amplitude ratio is very high and beyond the allowable range by available spindle-drive systems. Hence, applying another technique, such as the spindle speed selection method [17], is more feasible. The proposed criterion for selecting the S 3 V frequency is based on how fast the regenerative energy is dissipated from the machining system. The proposed criteria are suitable for on-line S 3 V programming since the only requirement is knowledge of the chatter frequency and the spindle speed. The effectiveness of the developed method is verified experimentally. 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