Detection of Chatter in Machining Processes by the Multiscale Maximum Approximate Entropy and Continuous Wavelet Transform

Abstract

Chatter is a complex dynamic instability in machining processes and presents nonlinear and nonstationary behavior. Detection of this phenomenon before a catastrophic failure occurs has great importance in the industry today. This behavior demands online monitoring signal-processing techniques suitable for facing these kinds of dynamics such as approximate entropy (AE) and wavelet transform. Moreover, AE is useful for dealing with noisy signals and requires a relatively small amount of observations. In this study, we propose an improved AE methodology, the multiscale maximum approximate entropy (MMAE), to detect chatter in milling processes. The maximum AE is achieved by the calculation of the parameter r proposed by Sheng and Chon. In the past, the calculation of this parameter was a drawback of the AE technique. The results show the effectiveness of this proposed technique in detecting clearly different gradual and drastic changes in chatter conditions. Moreover, a more known technique is presented: the time–frequency maps provided by continuous wavelet transform (CWT). The results also show the efficacy of this technique in detecting different levels of chatter. The results are corroborated by the machining piece observation of the chatter phenomenon. MMAE is also compared with sample entropy (SE) and the Hurst exponent obtained by the R/S analysis. At the end, a comparison analysis of the mentioned techniques is carried out, showing that they all have advantages and disadvantages. However, the disadvantages of MMAE and CWT can be solved, as mentioned in the comparison section. Thus, the conclusion is that MMAE and CWT techniques are optimal for the online monitoring of chatter in machining processes.

1. Introduction

Online vibration monitoring plays a very important role in the detection of failures in machines in the industry. Since 1990, vibration monitoring has not only been applicable to special machines such as turbines but all types of machines in the industry. Since then, it has had an important role in predictive maintenance programs, which has generated great economic savings. The boom of online vibration monitoring happened due to two reasons: first, the rise of piezoelectric sensors, which can detect high frequencies; second, the technological development of signal-processing methods, such as the Fast Fourier Transform (FFT). The problem with the FFT is that its basis has linear principles, so it is not optimal for nonlinear and nonstationary systems. However, it is important to mention that for practical situations, FFT has given acceptable results in a great variety of systems [1].

Later, another method called Short-Time Fourier Transform (STFT) arose. This method provides not only frequency information but also time information. STFT offers maps where frequency variations localized on time can be seen. The STFT has two drawbacks: first, it continues using sine and cosine functions as the FFT, and second, it uses a fixed-size window. It causes resolution problems, and it is not an easy task to select the optimal size of the window. Due to this fact, the wavelet transform (WT) appeared. It was introduced by Grossman and Morlet in 1984 [2]. The advantage of this technique is that it uses a variable window size and does not use sine and cosine functions but a function called the mother function, which resembles a function for nonlinear systems. There are several types of mother functions such as Morlet, Paul, Haar, and Daubechies. There are many references for the theory of the WT [3,4]. The WT has been widely used for online failure detection monitoring [5,6,7,8]. In this research, the Morlet function was used. This is due to the fact that Morlet is the function that is the most similar to a transient second-order system response. It allows us to distinguish nonlinear and transient from stable behaviors. That is why the Morlet function is one of the most used to analyze vibration signals.

Frequently, machine vibrations present nonlinear and nonstationary behavior. For instance, in mechanical elements such as gears and bearings, in machining processes, and in cases with large deformations and friction. Nonlinear systems can present complex phenomena such as bifurcations, hysteresis, limit cycles, jump phenomena, chaos, and so on. The causes of nonlinearities in mechanical elements can be varied; the most common are discontinuous stiffness and damping or a combination of both. Generally, discontinuous stiffness is modeled with a cubic polynomial term and is one of the parameters more frequently found in a nonlinear dynamic system.

More specifically, machining is a very common process and plays a very important economic role in the industry today. Chatter is a very undesired and complex phenomenon in the machining process. Chatter presents nonlinear behavior, and phenomena such as limit cycles and jump phenomenon [9]. Chatters are self-excited vibrations that have nonlinear and nonstationary behavior, which makes it difficult to detect them in an online monitoring process. Due to this reason, besides frequency and time–frequency techniques such as FFT and WT, other methods suitable for nonlinear and nonstationary dynamics have been proposed in the literature to detect this phenomenon before a catastrophic event occurs that causes economic and productivity losses. Ref. [10] proposes an R/S scaling analysis approach to detect chatter in milling processes. The R/S technique is used to obtain the Hurst exponent, which is a fractal parameter that indicates the persistent and anti-persistent behavior of the process. Anti-persistent behavior coincides with the presence of chatter in milling processes. The Hurst exponent obtained by the R/S analysis was used in this research for comparison purposes. However, according to the results, more work has to be conducted to improve the algorithm of the R/S analysis to obtain better results. Ref. [11] uses a kurtosis-based method called Variational Mode Decomposition (VMD) and energy entropy to detect chatter in a milling process. VMD has already been employed in feature extraction from nonstationary signals. A constraint of VMD is that the decomposition parameters K and α (the mode number and balancing parameter) are required to be defined in advance [12]. Ref. [13] realizes nonlinear feature extraction using approximate entropy, the Holder exponent, and the Lyapunov exponent in order to construct machine learning models to detect chatter. The nonlinear features extracted from the sensors’ signatures allowed them to identify chatter successfully. Ref. [14] proposes nonlinear feature extraction by means of multiscale permutation entropy and multiscale fuzzy entropy to detect chatter in milling processes. Before, for the signal preparation, they used techniques such as empirical mode decomposition and intrinsic mode functions. They showed that the proposed technique provides good results in detecting chatter based on the best scale. Ref. [15] proposes a novel technique for machinery vibro-acoustic fault diagnosis for a multi-sensor fusion network. The proposed method, called CDTFAFN, can obtain more comprehensive fault information and is able to work in highly noisy environments by means of two stages of fusion: coarse-grained feature fusion and fine-grained feature fusion. Ref. [16] uses the advantages of noise in stochastic resonance to enhance weak signals of interest to detect incipient faults in machinery systems that present nonlinear behavior. They use overdamped and underdamped harmonic-Gaussian double-well potential stochastic resonance to successfully enhance weak fault characteristics in bearings for incipient fault detection.

Refs. [17,18] propose artificial intelligence (AI) techniques to detect chatter in machining processes. Ref. [17] utilizes a convolutional neural network by adopting expert knowledge, and [18] proposes a support vector machine to detect chatter. However, the identification of chatter by AI algorithms is still supported by conventional signal-processing techniques. Some key difficulties for AI techniques are as follows: the learning and classification process requires a considerable amount of CPU power limits, human knowledge, and time for labeling data. Mainly when there are many classes, before training the model [19].

Though some techniques have given good results, techniques for nonlinear and nonstationary dynamics are not completely well established. Thus, more research is being carried out to find techniques suitable for nonlinear and nonstationary dynamics.

In previous work, chatter has been detected by using entropy concepts. In [20], the approximate entropy (AE) method with a multiscale approach is used, and [21] utilizes an entropy randomness index. In this paper, the method is improved by adding a maximum approximate entropy with the calculation of the parameter r, and continues using a multiscale approach. AE is useful for dealing with noisy signals and requires a relatively small amount of observations. Entropy is a term related to disorder. Traditional entropy methods are widely used in chaos theory to determine the existence of this. However, they require infinite time series without noise such as the Kolmogorov–Sinai entropy [22]. Nevertheless, the experimental data are short and noisy time series. Due to this fact, Pincus [23] proposed AE in 1991, which no longer focuses on looking at the process attractor but on what irregular is the time series. This is an advantage for AE, since it requires much less data for its calculation than the entropy method used in chaos theory [24]. AE measures the logarithmic frequency, in which similar blocks of a predefined length remain similar for blocks augmented by one position with a predetermined tolerance [25]. So that, AE is an indicator of how irregular (complexity) related to randomness is presented in the time series. AE has been widely applied in the medicine area; however, in the mechanical area, it has been less utilized.

This paper proposes the use of maximum AE (MAE) given by the selection of the parameter r. This was proposed by Sheng [26] and Chon [27]; they demonstrated that the MAE gives better results and provided formulas to calculate the parameter r to obtain the MAE. In addition to this, the multiscale approach proposed by Costa [28] was utilized. This paper proposes the multiscale maximum approximate entropy (MMAE) to detect different levels of chatter in the milling process. The results show the effectiveness of this proposed technique in clearly detecting gradual and drastic changes in chatter conditions. Moreover, MMAE is compared with sample entropy, introduced by Richman and Moorman [29]. The main difference between AE and SE (sample entropy) is that SE utilizes logarithmic calculus until the end. And for SE, there is no comparison between the same sequences. AE gives better results. Furthermore, a continuous wavelet transform analysis is performed. Time–frequency maps given by this technique are also used to detect different levels of chatter in the milling process. The results show the effectiveness of the CWT time–frequency maps. In order to see the good results of MMAE, it is compared with the Hurst exponent obtained by the R/S technique. Finally, a comparison analysis of the mentioned techniques is performed.

2. Materials and Methods

2.1. Experimental Setup

The experiment was taken from the work by [10]. These experiments were performed based on some instability curves to produce chatter. Such curves show the zones where chatter is present as a function of the depth of the cut and the spindle speed. The experiments were carried out on a CNC 15kWVF-3DHAASVOP-D milling machine (Haas Automation Inc., Kannapolis, NC, USA)with a cutter of 25.4 mm APK-D1.00-1.5-W1-FE with two inserts R390-11T308M-PM mounted on a CAT40ER40X3.359 E tool holder. The machining piece was an aluminum 6061 block with the following dimensions: 50.8 mm × 203.2 mm × 250 mm. The first step to obtain the instability curves was to calculate the modal parameters by impact testing. A Kistler 9724 instrumented hammer (Kistler, Winterthur, Switzerland) was used with a biaxial Analog Device ADXL321+/−18g, 100 mV/g, 0.5 Hz, 2.5 kHz accelerometer. The signals were treated with an anti-alias filter and sampled to 40 kHz with a National Instruments NI-USB 6251 16-bit 1.25MS/S data acquisition card (National Instruments, Austin, AT, USA). The card was connected to a Dell Precision M2300 laptop (Dell, Round Rock, TX, USA) with an Intel Core 2 Duo processor (see Figure 1). The experiments were conducted with the cutting tool at different immersion rates. In some cases, the cutting parameter that varied was the depth of the cut, and in others, the spindle speed. The spindle speed was varied from 3500 to 5900 rpm with changes of 300 rpm. Each spindle speed was kept constant for 30 mm with a feed of 400 mm/min. The depth of the cut was varied from 0 to 10 mm by tilting the block of aluminum by 1.8°. Acceleration signals were sampled at 40 kHz.

2.2. Chatter Generation

The acceleration signals were obtained by means of modal analysis to calculate the dominant frequency and the model parameters by fitting the transfer function of the system. The estimated parameters were ω_nx = 1133 Hz, k = 3.8 × 10⁶ N/m and ζ_x ≈ ζ_y = 3.4%. Moreover, the cutting coefficients were K_t = 6 × 10⁸ N/m² and K_n = 0.35 K_t N/m², which can be used for aluminum according to Altintas [30]. A stability analysis was conducted and stability curves were obtained with full immersion and quarter immersion of the cutting tool, with the depth of the cut versus spindle speed shown in Figure 2 and Figure 3.

In Figure 2, the vectors for test 1 (T1) and test 2 (T2) correspond to a full immersion of the cutting tool. The vector T1, with blue dotted arrow, starts with a stable zone and finishes with a chatter zone. The vector T2, with blue dotted arrow, is in a stable zone from the beginning to the end. Experimental test 1 was selected for this research to see the transition from stable to unstable conditions as shown in the figure. In Figure 3, the vectors for tests 3 and 4 (T3 and T4, respectively) correspond to a quarter immersion of the cutting tool. The experimental T3 and T4 were selected to compare different levels of unhealthy conditions due to the presence of chatter. The vector T3 is in a stable zone according to the stability diagram, however, there is the presence of chatter according to the machined piece. The vector T4 is almost completely in a chatter zone.

2.3. Entropy

The entropy concept is used in different areas such as thermodynamics, information theory, mathematics, and sociology. In different fields, entropy is related to the concepts of randomness and disorder.

2.4. Approximate Entropy

A disadvantage of the traditional methods for calculating entropy is that their processing requires infinite and noiseless data, which is the opposite of real time series. Due to this fact, in 1991, Pincus introduced AE, which is a statistical method that can be used for finite and noisy time series to quantify the complexity and irregularity of such series. The calculation steps of AE are shown in many scientific papers; please see [20].

2.5. Multiscale Approximate Entropy (MSAE)

The MSAE consists of obtaining the entropy at different time scales. Costa [28,31] utilized a process to obtain coarse-grained time series versions of the original signal as a function of the scale parameter τ. The version of coarse-grained time series at a certain scale τ is obtained by means of the average of τ consecutive observations without overlapping. For the time series {x₁, x₂,…,x_N}, the coarse-grained procedure is applied through the scale factor τ with non-overlapping windows in the following manner [32]:

(1)${Y_j}^{(\tau )}=(1/\tau ){\displaystyle \sum _{i=(j-1)\tau +1}^{j\tau }{X}_i}$

where $1\le j\le {\displaystyle \raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$\tau $}\right.}$.

Once the versions of coarse-grained time series are obtained, the AE is calculated for each scale. For τ = 1, the version of the coarse-grained time series is the original one. Another way of seeing this process is a moving average filter.

2.6. Maximum Approximate Entropy (MAE)

For the calculation of AE, it is necessary to provide a value for two input parameters: the parameters m and r. The selection of the m parameter is a less complicated task than the selection of the parameter r. The most common value used in the literature for the parameter m is m = 2, which provides good results. On the other hand, for the r parameter, values from 0.1 to 0.25 times the standard deviation of the series are taken, which have given good results in medicine. Nevertheless, wrong results have been shown in other cases. For instance, Sheng et al. [26] showed that the selection of r = 0.25 times the standard deviation applied to a series with Brownian movement wrongly suggested that the signal had low complexity. Considering this, they showed that the maximum value of AE provided better results, and it was not always obtained with r = 0.25 times the standard deviation of the series. Based on this, they proposed formulas for the calculation of the parameter r to obtain the maximum value of AE. In order to achieve this, they used a heuristic stochastic model and Monte Carlo simulations. In a later work, Chon [27] improved them. The equation that Chon provided for m = 2 is the following:

(2)$m=2:r_{\max }=(-0.036+0.26\sqrt{{\displaystyle \frac{s{d}_1}{s{d}_2}}})/\sqrt[4]{{\displaystyle \frac{N}{1000}}}$

where sd₁ is the standard deviation of the series X(i) and sd₂ is the standard deviation of the series X(i) − X(i−1) for i = 1, …, N.

With these values, the MAE is calculated in the following manner:

(3)$MAE(m,r_{\max },N)={\phi }^m(r_{\max })-{\phi }^{m+1}(r_{\max })$

2.7. Multiscale Maximum Approximate Entropy (MMAE)

This paper proposed applying the process of multiscale used by Costa to the MAE. It results in the MMAE given by the following expression:

(4)$MMAE(m,r_{\max },N,\tau )={{\phi }_{\tau }}^m(r_{\max })-{{\phi }_{\tau }}^{m+1}(r_{\max })$

where τ is the scale parameter.

2.8. Continuous Wavelet Transform

The wavelet transform, introduced by Grossman and Morlet (1984) [2], solved the problem of the STFT, since it deals with a variable window size and uses a mother function that is similar to the response of a nonlinear system instead of sine or cosine functions utilized by Fourier.

For this research, the time–frequency maps provided by the continuous wavelet transform were used. For a description of the CWT methodology, please see [1].

Several wavelet functions have been applied to vibration analysis, one of the most known is the Morlet wavelet function which was used for this research work.

The Morlet wavelet function is given by the following expression:

(5)$\psi \left({\displaystyle \frac{t-\tau }{s}}\right)=e^{\left({\displaystyle \frac{j2\pi f_0(t-\tau )}{s}}\right)}{e}^{\left({\displaystyle \frac{-\alpha {(t-\tau )}^2}{s^2{\beta }^2}}\right)}$

where f₀, α, and β are constants.

The process consists of seeking the correlation between the wavelet function and the parts of the signal at different scales. The correlation is registered in vectors known as wavelet coefficients.

2.9. Time–Frequency Maps of the CWT

The time–frequency maps of the CWT provide information on three variables: the time, frequency, and amplitude of the processed signal. They are very useful because they allow the transient behavior of the processed signal intensity to be visualized at different frequencies. The processed signal intensities are shown with different colors. This transient behavior is typical of signals from unstable systems.

3. Results and Discussion

Four experimental tests were selected to validate the effectiveness of the proposed techniques to detect chatter. Two correspond to the total immersion rate of the cutting tool and two to a quarter immersion. Table 1 depicts the characteristics of the selected experimental tests:

The input parameters selected for MMAE were m = 2, N = 5000, and τ = 5. τ = 1, 2, 3 and 4 were tested but the results were not the most optimal. The objective is to select the minimum scale factor τ for the processing time to be faster. The algorithm was developed in Matlab 2023 software. For the CWT, AutoSignal 1.7 commercial software was used to process the test signals. The same parameters as MMAE were selected for the SE technique, except the parameter r, which was r = 0.25 times the standard deviation of the series taken. The number of observations used for the R/S analysis to obtain the Hurst exponent was N = 1024. SE and R/S algorithms were developed in Matlab 2023 software.

For test 1, the signal and the machining piece are shown in Figure 4:

Chatter is seen at the end of the process as the depth of the cut increases. At the beginning, the process is stable. MMAE can clearly detect this, as shown in Figure 5. Values greater than 1.6 indicate the stable part of the milling process, and values of MMAE less than 0.5 refer to the intense presence of chatter from 13 to 17 s. The MMAE averages are 1.93 for the stable region and 0.28 for the unstable zone, as shown in Table 2. The minimum value for the stable region is 1.65. The maximum value of the chatter zone is 0.28, far from the threshold of 0.5.

The results of sample entropy for test 1 are shown in Figure 6, which are similar to those for MMAE. However, the distance between the stable and unstable regions is reduced, which makes the establishment of thresholds for the different levels of chatter difficult. Figure 7 shows the results of the CWT for experimental test 1. It can be seen that there is an unstable frequency at about 2500 Hz at the end of the process. The transient and distortional behavior of the frequency reflects this unstable dynamic called chatter. The different intensities of the processed signal are shown with different colors. From lower to higher intensities are indicated by yellow, green, light blue, navy blue, and pink respectively. This unstable frequency is very intense since it includes all the colors.

The Hurst exponent shown in Figure 8 depicts that the persistent behavior of the processed signal indicated by values above 0.5 coincides with the stable section. In contrast to this, the anti-persistent behavior coincides with the unstable process. However, in order to have better results, anti-persistent values of about 0.2 would be expected to have a clear difference between the stable and unstable sections related to persistent and anti-persistent behaviors, respectively.

The experimental test 2 signal and machining piece are depicted in Figure 9.

A stable process is shown in the machining piece since it does not have any presence of chatter. This is indicated clearly by MMAE in Figure 10, with values above 1.6 during the whole milling process. The MMAE values shown in Table 3 are very similar to the stable region for test 1.

The results of SE are very similar to MMAE, as shown in Figure 10 and Figure 11.

Figure 12 illustrates the results of CWT for test 2. The time–frequency map shows a stable frequency at about 100 Hz. This frequency is stable since it does not present transient behavior, which is a characteristic of unstable dynamics. The frequency around 2500 Hz appears but with a very low intensity (light yellow marks), so that, it is not taken into account.

The Hurst exponent results shown in Figure 13 depict that the persistent behavior is related to the stable process.

The machining piece in Figure 14 shows light chatter marks and a decrease in stability compared to test 2.

This is indicated by MMAE shown in Figure 15. The values are around 1.6, in contrast to those in test 2, which are clearly above 1.6. The results of MMAE clearly depict how the stability decreases, which is indicated by the appearance of chatter marks on the machining piece. The average MMAE shown in Table 4 decreases considerably compared to test 2, from 1.96 to 1.67.

The SE results for test 3 are shown in Figure 16; it can be seen that the values decrease compared to the previous test, which coincides with the decrement of stability.

The results of the CWT for test 3 are shown in Figure 17. The slight appearance of the unstable frequency of about 2500 Hz can be clearly seen with yellow and green colors, corresponding to the decrease in stability indicated by the presence of chatter marks on the machining piece.

There is a decrement of Hurst exponent values shown in Figure 18, which coincides with the appearance of chatter in the milling process. However, the values are very close to the previous test, making it difficult to set up offsets.

For test 4, chatter marks are stronger than those shown in test 3, as shown in Figure 19. This is clearly indicated by the results of the MMAE shown in Figure 20.

The values of MMAE are about 1. They decrease compared to test 3, indicating a decrease in stability that corresponds to the stronger presence of chatter shown in the machining piece. It is important to notice that the values are about 1, greater than those of test 1 since the chatter in test 1 was more intense. This shows the effectiveness of the MMAE for quantifying the chatter in milling processes. The average MMAE for test 4 is 1.05, as shown in Table 5, indicating a decrease in stability. The SE results for test 4 are shown in Figure 21; the values decrease compared to the results of test 3 in a similar way to MMAE values. Nevertheless, the decrement is very small, which makes it difficult to establish a threshold.

The results of the CWT for test 4 are shown in Figure 22. The time–frequency map indicates an increase in the intensity of the unstable frequency at about 2500 Hz compared to the results of test 3, which correspond to the stronger chatter marks shown on the machine piece of the current test. This increase of instability compared to the test 3 is shown with the appearance of colors light blue and navy blue in the time-frequency map.

The Hurst exponent results depicted in Figure 23 for test 4 show a slight decrement in the Hurst values. Nevertheless, the change in chatter appearance on the machining piece is more notable. As the previous test, it is complicated to establish a threshold.

Table 6 shows the MMAE values that correspond to the chatter severity levels. Below 0.5, a severe presence of chatter is presented, as can be seen in the MMAE result graphs.

Comparative Analysis of the Techniques Used in This Research

Wavelet and MMAE have given very good results in detecting chatter; however, both have advantages and disadvantages. One clear advantage of the proposed MMAE compared to the time–frequency map of the CWT is that with MMAE, it is easier to quantify the levels of chatter because it is a parametric method. It is very important when an alarm is set in an online monitoring system. On the other hand, the time–frequency map of the CWT requires an expert to identify the chatter. However, machine learning or automated pattern recognition can be used in the future to overcome this disadvantage. A drawback of the MMAE is that it requires a previous reference for the failure condition parameters to identify different levels of chatter. However, thresholds can be standardized for the different setups, so that having the standardized threshold for different arrangements is not a drawback anymore. In contrast to MMAE, the maps of the CWT do not need a previous reference. Another general advantage of MMAE is that it can effectively work with noisy signals and with a relatively small amount of observations. Hurst exponents obtained by the R/S analysis do not show suitable results; however, it is a promising technique because the 0.5 value, which divides persistent and anti-persistent behavior, can be used as a threshold in the future. In addition to this, the processing time is fast. However, more work has to be conducted to improve the R/S algorithm. Regarding the processing time of the MMAE and wavelet techniques, they have relatively slow algorithms; however, this is not a problem for an online monitoring system due to the fact that there is powerful electronic technology to make a suitable online monitoring process. Finally, the sample entropy results were similar to those of MMAE; however, MMAE gave better results.

4. Conclusions

In this paper, chatter present in a milling process was successfully detected by two techniques. First, MMAE proposed in this research can gradually detect the presence of chatter, which was corroborated with the machining piece. It is shown that the irregularity—randomness of the signal measured by MMAE, is clearly related to the unstable nonlinear dynamics of the chatter phenomenon. The selection of the parameter r with a maximum AE successfully and clearly indicates the gradual and drastic changes in the process’s stability. In the past, a drawback of AE was the selection of the parameter r. Now, by selecting r with a maximum AE, we can avoid this, providing excellent results as shown in this research. Second, the time–frequency maps of the CWT also provided good results in detecting the gradual presence of the chatter phenomenon. A comparison of the techniques was carried out, showing that they have advantages and disadvantages. However, the disadvantages can be overcome, as mentioned in the previous section. In conclusion, MMAE and wavelet are optimal techniques for the online monitoring of chatter in machining processes, considering that the disadvantage of the relatively slow processing time can be solved with electronic technology such as FPGA systems.

It is important to mention that both techniques are expected to work in real-world manufacturing environments. They were tested in a lab-scale milling experiment; however, it was corroborated that the instability of the process was clearly related to the complexity and randomness of the vibration signal in the case of MMAE. In real-world manufacturing environments, there are variations in tool wear, fixture variability, thermal effects during prolonged operations, and so on. If these changes in machine dynamics affect the stability of the process with the appearance of chatter, this will be indicated by MMAE values. In the case of CWT, the transient behavior of the frequency is related to the instability of the process. In the same way, if these dynamic variations affect the stability of the process, this will be reflected on the transient behavior of the unstable frequency. Then,, the time-frequency maps of CWT will be able to show.

Author Contributions

Conceptualization, D.P.-C. and J.C.J.-C.; methodology, D.P.-C., J.Á.-R., L.V.-M. and J.C.J.-C.; software, D.P.-C., L.V.-M. and J.C.J.-C.; validation, D.P.-C., J.C.J.-C. and J.Á.-R.; formal analysis, D.P.-C. and J.C.J.-C.; investigation, D.P.-C. and J.Á.-R.; resources, J.C.J.-C. and L.V.-M.; data curation, L.V.-M.; writing—original draft preparation, D.P.-C.; writing—review and editing, D.P.-C. and J.C.J.-C.; supervision, J.C.J.-C., J.Á.-R. and L.V.-M.; funding acquisition, J.C.J.-C. and L.V.-M. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not Applicable.

Informed Consent Statement

Not Applicable.

Data Availability Statement

The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author.

Acknowledgments

The authors acknowledge Universidad Autónoma de Querétaro for allowing the use of its installations and equipment.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. Schematic diagram of the experimental setup.

Figure 2. Stability diagram for full immersion of the cutting tool.

Figure 3. Stability diagram for quarter immersion of the cutting tool.

Figure 4. The acceleration signal corresponding to experimental test 1. The machining piece is shown below.

Figure 5. MMAE results for experimental test 1.

Figure 6. SE results for experimental test 1.

Figure 7. Time–frequency map corresponding to the CWT of experimental test 1.

Figure 8. Hurst exponent results for experimental test 1.

Figure 9. Experimental test 2 acceleration signal and the corresponding machining piece.

Figure 10. MMAE for experimental test 2.

Figure 11. SE results for experimental test 2.

Figure 12. Time–frequency map corresponding to the CWT of experimental test 2.

Figure 13. Hurst exponent results for experimental test 2.

Figure 14. Acceleration signal and machining piece corresponding to test 3.

Figure 15. MMAE for experimental test 3.

Figure 16. SE results for experimental test 3.

Figure 17. Time–frequency map corresponding to the CWT of experimental test 3.

Figure 18. Hurst exponent results for experimental test 3.

Figure 19. The acceleration signal corresponding to experimental test 4. The machining piece is shown below.

Figure 20. MMAE for experimental test 4.

Figure 21. SE results for experimental test 4.

Figure 22. Time–frequency map corresponding to the CWT of experimental test 4.

Figure 23. Hurst exponent results for experimental test 4.

Table 1

Characteristics of the selected experimental tests.

Experimental Test	Immersion	Depth of Cut (mm)	Spindle Speed (rpm)
T1	Total	0–10	5800
T2	Total	3	3500–5900
T3	A quarter	3	3500–5900
T4	A quarter	6	3500–5900

Table 2

Values of MMAE for stable and unstable regions of test 1.

	AVERAGE MMAE	MIN MMAE VALUE	MAX MMAE VALUE
STABLE REGION	1.93	1.65	2.12
UNSTABLE REGION	0.28	0.08	0.31

Table 3

MMAE values for stable region of test 2.

	AVERAGE MMAE	MIN MMAE VALUE	MAX MMAE VALUE
STABLE REGION	1.96	1.68	2.13

Table 4

MMAE values for unstable region of test 3.

	AVERAGE MMAE	MIN MMAE VALUE	MAX MMAE VALUE
UNSTABLE REGION	1.67	1.25	1.86

Table 5

MMAE values for unstable region of test 4.

	AVERAGE MMAE	MIN MMAE VALUE	MAX MMAE VALUE
UNSTABLE REGION	1.05	0.74	1.46

Table 6

MMAE values vs. chatter severity levels.

CHATTER STATE	MMAE VALUES
NO CHATTER	>1.6
MILD	0.5–1.6
SEVERE	<0.5

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