An analytical method to select spindle speed variation parameters for chatter suppression in NC machining

3.1. Energy input in constant speed cutting process

The work done by cutting force is the source of self-excited vibration energy. Based on the analysis of phase delay, the process of chatter energy input was found out and discussed. In chatter cutting process with constant spindle speed, the work carried out by dynamic cutting force during the time form $t_1$ to $t_2$ can be derived from Eq. (1) as follows:

where:

Since the phase delay $\beta$ is constant, the mathematical expectation of $E_1$ during integer number of vibration cycles is zero, and so as to $E_2$. Consequently, the work done by cutting force becomes:

During chatter process, the phase delay $\beta$ falls into an unstable zone $\left(2k\pi -\pi ,2k\pi \right)$, $\left(k\in \mathrm{N}\right)$, which leading the cutting force to do positive work. The energy produced by the dynamic cutting force will be continuously transmitted into the cutting system, which may result in continuous chatter. If the phase delay is changed into unstable zones, this continuous energy input process can be interrupted.

3.2. Energy input in vary spindle speed cutting process

The purpose of spindle speed variation after the chatter occurred is to disturb the process of energy input, and then to suppress structural vibration. When the spindle speed is changing, the time delay $\tau$ between two consecutive cuts should be substituted by the time-variable one $\tau \left(t\right)$, and the phase delay $\beta$ by $\beta \left(t\right)$. The Eq. (5) indicates that the variation of phase delay $\beta \left(t\right)$ directly affect the system energy input, and this energy input process can be suspended when phase delay $\beta \left(t\right)$ is changed into stable zones. Thus it can be said that the spindle speed variation parameters affect the chatter suppression efficacy indirectly by influencing the phase delay variation.

In this article, periodic spindle speed variation was considered, i.e. the spindle speed was modulated periodically according to the given range $A$ and acceleration/deceleration time $T$. For convenience in studying the influence of spindle speed acceleration on chatter suppression efficacy, the triangular waveform variation of spindle speed was analyzed in this work, as shown in Fig. 2. The triangular modulation, with the same dynamic characteristics of the spindle, provides larger spindle speed acceleration than the others [21] such as the sinusoidal, random and square-wave modulations.

Fig. 2Relationship between the phase delay βt and the spindle speed Nt

As following, the characteristic of phase delay variation during vari-speed cutting was analyzed. Firstly, the relationship between the mean energy input and the phase delay variation range was discussed. In addition, it was extended to analyze a relationship between the mean energy input and the spindle speed variation parameters (i.e. the spindle speed variation range $A$ and the spindle acceleration $\alpha$). Secondly, the phenomenon of regional energy accumulation enhancement was also seriously considered. Then, influence of the spindle acceleration on chatter suppression efficacy was analytically investigated.

3.2.2. The averaged energy input of the cutting system

Generally, a large number of vibration period may take place during the time $T$ of an acceleration/deceleration process. Substitute Eq. (10) into Eq. (4):

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$E_2=-\frac{a_w{k}_s{a}_0^2\omega \mu }{2}{\int }_{t_0}^{t_0+T}\sin \left[\left(2+60\alpha /N_0^2\right)\omega t-60\omega /N_0\right]dt.$

As mentioned above, the mathematical expectation of $E_1$ is zero. Then the energy input of the chatter cutting system can be drawn out according to Eq. (3) as follows:

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$E\approx \overline{E_3}=-\frac{a_w{k}_s{a}_0^2\omega \mu }{2}{\int }_{t_0}^{t_0+T}\sin \left[\beta \left(t\right)\right]dt.$

Eq. (12) implies that a positive work will be carried out by the cutting force when $\beta \left(t\right)$ falls in the unstable zones $\left(2k\pi -\pi ,2k\pi \right)$, $\left(k\in \mathrm{N}\right)$. Or in other word, vibration energy of the cutting system will be increased when $\beta \left(t\right)$ falls into the unstable zones $\left(2k\pi -\pi ,2k\pi \right)$, $\left(k\in \mathrm{N}\right)$ and will be decreased when $\beta \left(t\right)$ into the stable zones $\left(2k\pi ,2k\pi +\pi \right)$, $\left(k\in \mathrm{N}\right)$.

It should be pointed out that the variation direction of $\beta \left(t\right)$ is reversed at the start and end moment of each acceleration/deceleration process, as shown in Fig. 2. The variation range $\mathrm{\Delta }\beta$and the middle value ${\beta }_m$ of the phase delay were defined as follows:

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$\left\{\begin{array}{l}\mathrm{\Delta }\beta ={\beta }_{max}-{\beta }_{max}=2k_d\pi +{{\rm B}}_d,\\ {\beta }_m=\left({\beta }_{max}+{\beta }_{min}\right)/2=2k_m\pi +{{\rm B}}_m,\end{array}\right.$

where $k_d$, $k_m\in \mathrm{N}$;${\mathrm{}{\rm B}}_d$, ${{\rm B}}_m\in [0,2\pi )$.

During an acceleration/deceleration process as shown in Fig. 2, there are usually more than one stable or/and unstable zones. And the phase delay $\beta \left(t\right)$ goes through stable and unstable zones alternatively. Denote the total stable range of $\beta \left(t\right)$ by ${\beta }_{stable}$ and the total unstable range by ${\beta }_{unstable}$, there is $\mathrm{\Delta }\beta ={\beta }_{stable}+{\beta }_{unstable}$. The ratio of stable range ${\beta }_{stable}$ to the whole variation range $\mathrm{\Delta }\beta$ was defined as the stable range ratio

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$\xi ={\beta }_{stable}/\mathrm{\Delta }\beta .$

The conclusion can be obtained without proof confirming a large value of the ratio $\xi$ (see Eq. (14)) is favorable to the high performance in chatter suppression.

According to Eq. (13), the stable range ratio $\xi$ can also be defined as the function of ${\beta }_m$ and $\mathrm{\Delta }\beta$, i.e. $\xi =\xi \left({\beta }_m,\mathrm{\Delta }\beta \right)$. Usually, the middle phase delay ${\beta }_m$ mainly depends on the mean spindle speed, and the variation range of phase delay $\mathrm{\Delta }\beta$ mainly depends on the variation range of spindle speed. A series of values of $\xi$ with different $\mathrm{\Delta }\beta$ and ${\beta }_m$ were calculated out and shown in Fig. 3.

As shown in Fig. 3, along with the increase of $\mathrm{\Delta }\beta$, the stable range ratio $\xi$ tends to constant value 0.5 in a waving mode. The shapes of the ratio curve are determined by the middle phase delay ${\beta }_m$, or more exactly, by ${{\rm B}}_m$. As shown in Eq. (15), when $\mathrm{\Delta }\beta \in \left(2k\pi ,2k\pi +\pi \right)$, $\left(k\in \mathrm{N}\right)$ with ${{\rm B}}_m=\pi /2$ and when $\mathrm{\Delta }\beta \in \left(2k\pi -\pi ,2k\pi \right)$, $\left(k\in \mathrm{N}\right)$ with ${{\rm B}}_m=3\pi /2$ the maximum $\xi ={\xi }_{max}$ is reached. The maxima points of ${\xi }_{max}$ are obtained when $\mathrm{\Delta }\beta =\pi +2k_d\pi$, while the vibration energy input will reach minimal value.

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${\xi }_{max}=\left\{\begin{array}{cccc}\xi \left({\beta }_m,\mathrm{\Delta }\beta \right),& {\beta }_m=\pi /2+2k_m\pi ,& \mathrm{\Delta }\beta \in \left(2k_d\pi ,2k_d\pi +\pi \right),& k_d\in \mathrm{N},\\ \xi \left({\beta }_m,\mathrm{\Delta }\beta \right),& {\beta }_m=3\pi /2+2k_m\pi ,& \mathrm{\Delta }\beta \in \left(2k_d\pi +\pi ,2k_d\pi +2\pi \right),& k_d\in \mathrm{N}.\end{array}\right.$

Fig. 3The stable range ratio ξ of the phase delay variation

The maxima points of ${\xi }_{max}$ indicate there are two successive stable ranges in the vicinity of each inflection point of $\beta \left(t\right)$, which means a long time for dissipation of vibration energy. More details about this phenomenon are discussed in the third part of this section. Additionally according to Eq. (7) and Eq. (13), it should be mentioned that the middle value ${\beta }_m$ and the variation range $\mathrm{\Delta }\beta$ of the phase delay both are functions of the mean spindle speed $N_m$, the spindle speed variation range $A$ and the spindle acceleration $\alpha$.

The energy input of the cutting system with a constant vibration amplitude was numerically computed under the assumption that the chatter frequency remained unchanged, as shown in Fig. 4 and Fig. 5, where $\alpha =\mathrm{}$1000 rpm/s and chatter frequency is 295.8 Hz. Al-Regib et al. [7] have conducted similar simulations and experiments to analyze the energy input $E$ by analyzing the spindle speed variation ratio. However, they paid no attention to the phase delay variation range $\mathrm{\Delta }\beta$. In this article, Fig. 4 depicts the energy input $E$ (see Eq. (3)) and the stable range ratio $\xi$ during the time of a spindle speed variation period, i.e. during the time $2T$. The aforementioned conclusion is proved by Fig. 4 that the increase of the stable phase variation range ratio causes a reduction of the energy input of the chatter system, and vice versa. In Fig. 4, the minima points of energy input $E$ and the maxima points of stable range ratio $\xi$ have almost the same abscissa values, i.e. the phase variation range $\mathrm{\Delta }\beta$. The subtle difference may attribute to the fact that the phase delay between inner and outer modulations cannot trace exactly the same waveform of spindle speed. Corresponding to Fig. 4, 5 shows the spindle speed variation ranges with the maximum, minimum and mean values, i.e. $N_{max}$, $N_{min}$ and $N_m$. It can be seen that, with a given phase delay variation range $\mathrm{\Delta }\beta$, the required spindle speed is influenced by the middle phase delay ${\beta }_m$ (or say, ${{\rm B}}_m$).

Fig. 4Relationship between the stable range ratio ξ and the energy input E

Fig. 5The spindle speed range required by the phase delay variation

Based on Fig. 3-5, it can be found out that: (1) For an allowed variation range of spindle speed ($\mathrm{\Delta }\beta >3\pi$ during vari-speed cutting process for chatter suppression, the maxima speed can be decreased or/and the minima speed increased to make $\mathrm{\Delta }\beta =2k_d\pi +\pi$ (i.e. ${{\rm B}}_m=3\pi /2\text{,}$${\beta }_m=2k_i\pi$ and $\mathrm{}{\beta }_{\mathrm{m}\mathrm{a}\mathrm{x}}=2k_a\pi +\pi$), by which the stable range ratio $\xi$ reaches a higher value. Therefore, the larger range of spindle speed variation may not mean the higher performance of chatter suppression. (2) Among the maxima points of ${\xi }_{max}$, the one relating to a smaller $\mathrm{\Delta }\beta$ has a larger value of $\xi$, which means higher performance of chatter suppression. (3) In the range of high values of $\mathrm{\Delta }\beta$, the stable range ratio $\xi$ trend to the constant value 0.5, which meaning that an excessively enlargement of spindle speed variation range cannot significantly improve the performance of chatter suppression.

However, in a practical application, the actual mean spindle speed $N_m$ is usually kept constant. Based on this, the energy input of the cutting system was numerically computed, as shown in Fig. 6 where $\alpha =$1000 rpm/s, $N_m=$1000 rpm and chatter frequency is 295.8 Hz. Fig. 6 depicts the relationship between the work $E$ carried out by the cutting force and the spindle speed variation range $A$. The mean cutting force work trends to zero, in a wavelike manner with descending amplitude, as the speed variation range increasing. Correspondingly, the stable range ratio $\xi$ fluctuates with a direction counter to energy input $E$ and trends to the constant value 0.5. Fig. 7 depicts the variation range $\mathrm{\Delta }\beta$ and the middle value ${\beta }_m$ of the phase delay, both of which are influenced by the speed variation range $A$ combined with the constant mean spindle speed $N_m$. That is why the shape of the line representing $\xi$ in Fig. 6 is different from the ones in Fig. 3 and Fig. 4.

Fig. 6Relationship between the energy input E and the corresponding stable range ratio

Fig. 7The variation range and the middle value of the phase delay

3.2.3. Regional enhancement of the energy input

According to Eq. (10) and Eq. (12), the energy input can be approximately calculated by Eq. (16) for a given cutting process suffering chatter:

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$E\approx \frac{\mu a_w{k}_s{N}_0^2{a}_0^2}{120\alpha }{\int }_{{\beta }_{start}}^{{\beta }_{end}}\sin \left(\beta \right)d\beta .$

Usually, during a continuous increase/decrease process of the phase delay $\beta \left(t\right)$, there is $\left|{\beta }_{end}-{\beta }_{start}\right|>\pi$. Near to the maxima and the minima spindle speed, if the phased delay falls in unstable zone for a relative long time, the vibration may be aggravated instantaneously. This phenomenon is shown in Fig. 8, focusing only on the locations of the inflection points of phase delay.

Fig. 8The location of maxima/minima points of the phase delay βt

Therefore, ${\epsilon }_a$ and ${\epsilon }_i$ (see Eq. (7)) were set to be the research object in the following cases:

Case 1: In the vicinity of inflection points, $\beta \left(t\right)$ falls in the stable zone, i.e. ${\epsilon }_{a,i}\in \left[0,\pi \right]$, and for a time relatively shorter than that in the former or the next unstable zone. The vibration energy accumulated in the former unstable zone may not be reduced completely, and then may be enlarged in the following unstable zone.

Case 2: In the vicinity of inflection points, $\beta \left(t\right)$ falls in the unstable zone, i.e. ${\epsilon }_{a,i}\in \left[\pi ,2\pi \right]$, and for a time relatively longer than that in the former or the next stable zone. Therefore, more vibration energy will be accumulated resulting in instantaneous vibration enlargement.

Ignoring the constant coefficients in Eq. (16), the parameters $\eta$, ${\eta }_a$ and ${\eta }_i$ were calculated according to Eq. (17) to discuss the instantaneous energy accumulation:

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$\left\{\begin{array}{l}{\eta }_{a,i}={\int }_{{\beta }_1}^{{\beta }_2}\sin \left(\beta \right)d\beta +{\int }_{{\beta }_2}^{{\beta }_3}\sin \left(\beta \right)d\beta ,\\ \eta =\left({\eta }_a+{\eta }_i\right)/4,\end{array}\right.$

where ${\beta }_1$ indicates the initial $\beta \left(t\right)$, ${\beta }_2$ the minimal or the maximum $\beta \left(t\right)$, ${\beta }_3$ the ending $\beta \left(t\right)$.

In case 1, ${\eta }_{a,i}$ indicates the energy input during a time between the start of the former unstable zone and the end of the current stable zone. In case 2, ${\eta }_{a,i}$ indicates the energy input during a time between the start of the current unstable zone and the end of the following stable zone. When neither case 1 nor case 2 occurs, set $\eta =$0. It shall be obtained that $\eta \in \left[0,1\right]$ and that $\eta$ can be used as the indicator of regional energy accumulation near to the reflection point of phase delay. Higher value of $\eta$ may mean a worse performance of the vari-speed cutting method for chatter suppression.

4.1. Verification with numerical simulation

A well-developed cutting process model [17] established upon Eq. (1) was used in this work. The dynamic model was reshaped as shown in Eq. (18):

where $C_d$ denotes the process damping coefficient and $V\left(t\right)$ the cutting speed.

The influence of the cutting process damping on the cutting force was considered in Eq. (18). Since the work carried out by the damping force is inconspicuously variable because of the limited speed variation, shall not be analyzed in this article. Other influences such as the tool run out and structural nonlinearity were considered in the cutting process simulations. These additional factors would not influence on the aforementioned analysis results.

A series of numerical simulations were conducted with the parameters as shown in Table 1. In order to keep the conditions as close to the real as possible, the test results shown in references [17, 19] were consulted to determine the cutting force coefficients and the structural parameters.

The vibration equation was solved with the fourth order Runge-Kutta method coupled. With low-level random perturbation added in, constant spindle speed was applied at the beginning of the cutting process to excite the chatter. When the chatter occurred with spindle speed 1560 rpm and frequency 206.3 Hz, the spindle speed was modulated in a triangular waveform for the chatter suppression. Fig. 9 depicts some of the simulation results.

Fig. 9 shows that chatter amplitudes were reduced significantly by vari-speed cutting method with the spindle acceleration 1000 rpm/s. The vari-speed cutting process shown in Fig. 9(b) is much stable than that shown in the other two figures.

Successively, the variances of the vibration $x\left(t\right)$ were calculated and compared with the indicating parameters $\eta$ and $\xi$ as shown in Fig. 10(a). The change of $var\left(x\right)$ reflects the variation trend of chatter intensity.

It can be found out form Fig. 10(a) that the change of the vibration variance $var\left(x\right)$ inversely follows a trend of the stable range ratio variation. In addition, the stable range ratio $\xi$ plays a more important role than the regional energy accumulation indicator $\eta$ in chatter suppression. However, it seems that the variation trend of variance $var\left(x\right)$ with the increase of speed variation range $A$ lags behind in comparison with that of the stable range ratio. That may be attributed to the influence of the high level of the regional energy accumulation indicator.

Fig. 9Simulation results of the SSV cutting process: a) A= 400 rpm; b) A= 550 rpm; c) A= 700 rpm

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Fig. 10Influence of the spindle speed variation range on the chatter suppression: a) the vibration variance, stable range ratio and regional energy accumulation indicator; b) the phase delay variation ranges in the simulations of SSV cutting process

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While keeping other parameters constant, the spindle speed acceleration $\alpha$ was reduced to 300 rpm/s and a new group of simulations conducted. According to the aforementioned data processing flow, parts of the simulation results were plotted in Fig. 11. Then the variances $var\left(x\right)$ of the vibration $x\left(t\right)$ were calculated and compared with parameters $\eta$ and $\xi$ in Fig. 12(b).

Fig. 11Simulation results of the SSV cutting process: a) A= 400 rpm; b) A= 550 rpm; c) A= 700 rpm

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Fig. 12The influence of the spindle speed variation range on chatter suppression a) the vibration variance, stable range ratio and regional energy accumulation indicator; b) the phase delay variation ranges in the simulations of SSV cutting process

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Fig. 11 shows that chatter could not been suppressed by varying spindle speed with acceleration 300 rpm/s. By comparing Fig.10(a) and Fig. 12(a), the conclusions can be drawn out that: (1) the chatter suppression efficacy is closely related to the stable range ratio $\xi$ and the regional energy accumulation indicator $\eta$. Therefore, these two coefficients can be calculated according to Eq. (14) and Eq. (17) and be used for parameter selection of the spindle speed variation. (2) Spindle acceleration is of crucial importance for chatter suppression by SSV cutting method. SSV cutting process with low spindle acceleration cannot eliminate chatter regardless of the high stable range ratio and the low regional energy accumulation indicator.

4.2. Verification with experiments

The influence of the stable range ratio $\xi$, the regional energy accumulation indicator $\eta$ and the spindle acceleration $\alpha$ on the chatter suppression efficacy was researched through two groups of turning experiments. Fig. 13 depicts the experimental setup. Although the relative tool-workpiece vibration amplitude is a direct indicator of the chatter intensity, the detection of this item relatively requires more complicated testing equipment and procedures. Moreover, the influence of SSV parameters on chatter suppression efficacy can be expressed by the variation trend of chatter intensity. Therefore, the variation trend of chatter intensity was paid more attention than the absolute vibration amplitude in this paper. In order to monitor the variation trend of chatter intensity, an accelerometer was installed near the tool shank. The acceleration signal was recorded by the LMS Test.lab vibration testing equipment and analyzed with the MATLAB software.

Fig. 13Setup for turning experiments

Although the theory and effectiveness had been affirmed by prior studies, actual implementation of variable speed cutting developed slowly. One of the reasons is that a special integrated module for chatter monitoring and control is required in NC system, which needs high level permission of NC system. The bus based method [23] of the embedded error compensation modular in NC system developed by the authors’ work team was modified, so that it can be used to control the spindle speed in the following experiments.

The cutting conditions are shown in Table 2. The overhang length of the workpiece was set to a large value (i.e. 200 mm), which reduced both the radial dynamic stiffness and the resonance frequency of the workpiece. Considering this fact, it is the workpiece that determines the dynamic characteristics of the whole cutting system. So the chatter occurs with a vibration frequency approximately equal to the workpiece resonance frequency. The relative tool-workpiece displacement fluctuated along an unchanged radial direction of workpiece. Therefore, the SSV parameters selection method proposed based on the analysis of SDOF cutting system, is appropriate to these turning processes.

Each cutting process was finished in a short time, i.e. less than 5 seconds, so that little material was cut off, and the cutting system dynamics were kept almost time-invariance. Constant speed cutting was conducted before each group of experiments to observe the chatter frequency. The actual spindle speed was 999 rpm, and the chatter frequency was 212.5 Hz.

In the first group experiments, the spindle speed variation range $A$ was changed from 40 rpm to 260 rpm with an increment equal to 20 rpm, while the acceleration/deceleration time $T$ keeping constant. In the second group experiments, the acceleration/deceleration time was changed from 0.4 s to 2 s with an increment equal to 0.1 s, while the spindle speed variation range keeping constant. Fig. 14 depicts the parameters of the applied spindle speed waveform.

Fig. 14The spindle speed variation parameters applied in turning experiments

Acceleration signals $y\left(t\right)$ were obtained through the accelerometer fixed near the tool shank. For instance, the tool vibration signals, corresponding to the SSV parameters $P_1$, $P_2$, $P_3$ and $P_4$ in Fig. 14, are shown in Fig. 15.

The photographs of the machined surfaces, corresponding to the SSV parameters $P_1$, $P_2$, $P_3$ and $P_4$, are shown in Fig. 16. For convenient observation, the pictures were magnified five times in width. Fig. 16(a) and Fig. 16(c) show that the SSV method with improper parameters cannot well reduce the chatter intensity. Actually, the machining quality without a SSV cutting will be even worse, while the proper SSV parameters greatly improves the machining quality as shown in Fig. 16(b) and Fig. 16(d).

Fig. 15The tool vibration signals in SSV cutting process: a) SSV parameter P1 (T= 0.5 s, A= 40 rpm/s); b) SSV parameter P2 (T= 0.5 s, A= 260 rpm/s); c) SSV parameter P3 (T= 2.0 s, A= 200 rpm/s); d) SSV parameter P4 (T= 0.4 s, A= 200 rpm/s)

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Fig. 16Machined surfaces in SSV cutting process: a) SSV parameter P1 (T= 0.5 s, A= 40 rpm/s); b) SSV parameter P2 (T= 0.5 s, A= 260 rpm/s); c) SSV parameter P3 (T= 2.0 s, A= 200 rpm/s); d) SSV parameter P4 (T= 0.4 s, A= 200 rpm/s)

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Successively, the variance $var\left(y\right)$ of the vibration acceleration $y\left(t\right)$ was calculated and compared with the spindle acceleration $\alpha$, the stable range ratio $\xi$ and the regional energy accumulation indicator $\eta$ as shown in Fig. 17 and Fig. 18. The variance values are shown in Fig. 17(a) relating to the first group experiments, and shown in Fig. 17(b) relating to the second group experiments. Subsequently, the variance values are drawn against the corresponding spindle acceleration in Fig. 18(a). The corresponding coefficients ${\eta }_{\mathrm{1,2}}$ and ${\xi }_{\mathrm{1,2}}$ are drawn in Fig. 18(b), with the subscripts indicating the group indices.

Fig. 18(a) shows that the cutting chatter was fully suppressed in the range of high spindle acceleration (i.e. the zone II between the lines ${\alpha }_{critical}=\mathrm{}$250 rpm/s and ${\alpha }_{limit}$ in Fig. 14), while the chatter suppression effectiveness was weaker in the range of lower spindle acceleration (i.e. the zone III under the line ${\alpha }_{critical}$). In the lower spindle acceleration zone, the cutting processes of group 2 have larger vibration variances $var\left(y\right)$ than that of group 1, which coupled with the phenomena, as shown in Fig. 18(b), that the cutting processes of group 2 have lower stable range ratios $\xi$ when the spindle acceleration $\alpha <$145 rpm/s and have lower regional energy accumulation indicator $\eta$ when 145 rpm/s$<\alpha <{\alpha }_{critical}$.

According to Eq. (16) and Fig. 18(b), the experimental results can be explained as that: While the spindle acceleration $\alpha$ is kept at a low level, the chatter aroused and kept by the sever energy accumulation, and also would be enhanced by the phenomena of regional energy enlargement and the low stable range ratio. However, the influence of all the factors that would aggravate the energy accumulation were weakened by the increase of spindle acceleration, hence the chatter was suppressed.

Fig. 17Experiment results: a) group 1; b) group 2

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Fig. 18Experiment results comparison between the two groups of experiments: a) vibration variances; b) phase delay variation

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In summary, the spindle acceleration $\alpha$ plays an important role in variable spindle speed cutting process for the chatter suppression. A high level acceleration $\alpha$, coupled with the phase delay $\beta \left(t\right)$ alternating between stable and unstable zones, may be in favor of the chatter suppression. Conversely, when the spindle acceleration $\alpha$ is decreased, influence of the stable range ratio $\xi$ and the regional energy accumulation enhancement coefficient $\eta$ on the chatter suppression efficacy may become more distinct.